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IntroductionChain closure
Tameness
Compactness and powerful images once
and for all
Michael Lieberman(Joint with Will Boney)
Masaryk University Department of Mathematics and Statisticshttpwwwmathmunicz~lieberman
Brno-Prague Algebra WorkshopMay 31 2019
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in the infinitarylanguage Lκκ if a theory T is lt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
Recall that finitary first-order logic is compact if a theory T inLωω is finitely satisfiable it is satisfiable
Strongly compact κ uncountable and in Lκκ if a theory T islt κ-satisfiable T is satisfiable
Weakly compact κ inaccessible and in Lκκ if a theory T with|T | le κ is lt κ-satisfiable T is satisfiable
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
We work in a familiar interval in the large cardinal hierarchy
[ weakly compact strongly compact ]
DefinitionWe say that a cardinal κ is (δ θ)-compact δ le κ le θ if whenevera theory T of size θ in Lδδ is lt κ-satisfiable it is satisfiable
This gets pretty much everything
κ (κκ)-compact weakly compact
κ (δinfin)-compact δ-strongly compact
κ (lt κinfin)-compact almost strongly compact
κ (κinfin)-compact strongly compact
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
Here κ-accessibility is basically free κ-accessible embeddability ieclosure under κ-directed colimits follows from the large cardinalassumption
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and κ-accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
Here lt κ-tameness is reformulated as κ-accessibility of thepowerful image of a particular forgetful functor
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
Theorem (BoneyUnger)
Let κ satisfy microω lt κ for all micro lt κ The following are equivalent
(1) κ is almost strongly compact
(2) The powerful image of any accessible functor F K rarr Lbelow κ is κ-accessible and accessibly embedded in L
(3) Every AEC below κ is lt κ-tame
Proof(1) rArr (2) Makkai and Pare refined by Brooke-Taylor andRosicky
(2) rArr (3) Lieberman and Rosicky
(3) rArr (1) Boney and Unger
They make use of a combinatorial construction which producesresults for eg (δ θ)-compact cardinals
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Large cardinal notionsand their equivalents
A careful reworking of these arguments gives
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder θ+-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
We will not focus too much on the detailsmdashκ-closed microLetcmdashand instead try to get the flavor of the correspondence
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 1 Realize the powerful image of F as the full image of
H P rarr L
where P is the category of monos L rarr FK K isin K Everything isstill nicely accessible
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We will prove the following
TheoremIf κ is (δ θ)-compact the powerful image of any (suitablysize-preserving) accessible functor F K rarr L below δ has powerfulimage closed under θ+-small κ-directed colimits of κ-presentableobjects
ProofThree steps almost exactly as in Brooke-TaylorRosickyStep 2 Embed P and L into categories of structures and realizeP as the category of models of an infinitary theory T in someLδδ(Σ)mdashin this way one realizes the full image of F ultimately as
RedL(T )
the category of reducts of models of T to the signature of L
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
Crucially careful reading reveals that RedL(T ) is κ-preaccessible
Step 3 We wish to show that RedL(T ) is closed under θ+-smallκ-directed colimits of κ-presentable objectsmdashtake such a diagram〈Mi i isin I 〉 in RedL(T ) and consider its colimit M in Str(L) Itsuffices to show there is a mono M rarr N N isin RedL(T ) M isθ+-presentable so |U(M)| = θ Take
TM atomicnegated atomic diagram of M names cm for m isin M
It suffices to exhibit a model N |= T cup TM Any Γ sube T of size lt κ is included in T plus a piece TMΓ of TM
involving lt κ of the cm We may take Mi with U(Mi ) containingtheir realizations cMm by κ-directedness so Mi |= T cup TMΓ By(δ θ)-compactness we are done
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We work in an AEC K although an arbitrary concrete accessiblecategory will do
DefinitionA Galois type over M isin K is an equivalence class of pairs (f a)f M rarr N and a isin UN We say (f0 a0) sim (f1 a1) if there is anobject N and morphisms gi Ni rarr N such that the followingdiagram commutes
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with U(g0)(a0) = U(g1)(a1)
Assuming the amalgamation property which we do this is in factan equivalence relation
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
DefinitionWe say that Galois types in K are (lt κ lt θ)-tame if for everyM isin Kltθ and types p ∕= q over M there is M0≺KM of size lessthan κ such that p M0 ∕= q M0
We turn this into a question about powerful images in precisely thesame way as in LRosicky
L1 Category of diagrams witnessing equivalence of pairs
N0g0 983587983587 N
M
f0983619983619
f1983587983587 N1
g1983619983619
with selected elements ai isin UNi U(g0)(a0) = U(g1)(a1)
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
L2 Category of pairs
N0
M
f0983619983619
f1983587983587 N1
with selected elements ai isin UNi
Let F L1 rarr L2 be the obvious forgetful functor
Facts
1 The functor F is accessible If K is below κ so is F
2 The powerful image of F consists of precisely the equivalentpairs of (representatives of) types
With these facts there is essentially nothing to do
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
In LRosicky it is shown that κ-accessibility of the powerful imageof F implies lt κ-tameness We proceed similarly but assumingonly closure under θ-small κ-directed colimits of κ-presentableobjects
ProofSuppose M is the κ-directed colimit of θ-presentable objects
M = colimiisinIMi
with colimit maps φi Mi rarr M Suppose types (fj aj) j = 0 1are equivalent over any Mi ie
N0 Mif0φi
983651983651f1φi
983587983587 N1
is always in the powerful image of F Since the original pair is thecolimit of this systemmdashof the permitted formmdashand the powerfulimage is closed under such colimits they are equivalent as well
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images
IntroductionChain closure
Tameness
We have nearly proven
Theorem (BoneyL)
Let δ be an inaccessible cardinal and θ a κ-closed strong limitcardinal The following are equivalent
1 κ is logically (δ lt θ)-strong compact
2 If F K rarr L is below δ and preserves microL-presentable objectsthen the powerful image of F is κ-accessible and closed in Lunder lt θ-small κ-directed colimits of κ-presentables
3 Any AEC K with LS(K) lt δ is (lt κ lt θ)-tame
The final link (3) rArr (1) is by BoneyUnger whence cometh allthe technical conditions in the preamble
Lieberman Compactness and powerful images