An Elementary Approach to Elementary ToposTheory

Todd Trimble

Western Connecticut State UniversityDepartment of Mathematics

October 26, 2019

Back Story

I Tierney’s approach: private communication.

I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.

I Tierney’s approach: constructions are more natively”set-theoretical”.

Back Story

I Tierney’s approach: private communication.

I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.

I Tierney’s approach: constructions are more natively”set-theoretical”.

Back Story

I Tierney’s approach: private communication.

I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.

I Tierney’s approach: constructions are more natively”set-theoretical”.

Back Story

I Standard approach to deduce existence of colimits:P : E op → E is monadic.

I Construction of coproducts: X + Y is an equalizer:

X+Y P(PX×PY )

uP(PX×PY )−→−→P〈P(PπPX uX ),P(PπPY uY )〉

PPP(PX×PY )

u : 1E → PP is unit uX (x) = A : PX |x ∈ A

Back Story

I Standard approach to deduce existence of colimits:P : E op → E is monadic.

I Construction of coproducts: X + Y is an equalizer:

X+Y P(PX×PY )

uP(PX×PY )−→−→P〈P(PπPX uX ),P(PπPY uY )〉

PPP(PX×PY )

u : 1E → PP is unit uX (x) = A : PX |x ∈ A

Notation and Preliminaries

I Power-object definition of topos: finite limits, universalrelations 3X → PX × X .

R → X × Y

X →χR

PY

R 3Y

X × Y PY × Y

i

y

χi×1Y

I singX : X → PX classifies δX : X → X × X .

Notation and Preliminaries

I Power-object definition of topos: finite limits, universalrelations 3X → PX × X .

R → X × Y

X →χR

PY

R 3Y

X × Y PY × Y

i

y

χi×1Y

I singX : X → PX classifies δX : X → X × X .

Notation and Preliminaries

I 31= 1→ P1× 1, aka t : 1→ Ω.

I All monos are regular:

A X Ω

1

i χi

!t

I Epi-mono factorizations are unique when they exist.

I Toposes are balanced.

Notation and Preliminaries

I 31= 1→ P1× 1, aka t : 1→ Ω.

I All monos are regular:

A X Ω

1

i χi

!t

I Epi-mono factorizations are unique when they exist.

I Toposes are balanced.

Cartesian closure

I Exponentials PZY exist, namely P(Y × Z ) ∼= (PZ )Y :

X → P(Y × Z )

R → X × Y × Z

X × Y → PZ

X → PZY

I

X 1 XY 1Y

PX P1 PXY P1Y

singXy

ty

tY

τ τY

Slice theorem

I If E is a topos, then for any object X , the category E/X isalso a topos. The change of base X ∗ : E → E/X is logicaland has left and right adjoints.

I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical.

I Colimits in E/Y , when they exist, are stable under pullbackf ∗ : E/Y → E/X .

Slice theorem

I If E is a topos, then for any object X , the category E/X isalso a topos. The change of base X ∗ : E → E/X is logicaland has left and right adjoints.

I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical.

I Colimits in E/Y , when they exist, are stable under pullbackf ∗ : E/Y → E/X .

Internal logic

1× 1t×t→ Ω× Ω

∧ = χt×t : Ω× Ω→ Ω

[≤] → Ω× Ω

⇒ = χ[≤] : Ω× Ω→ Ω

Internal logic

1× 1t×t→ Ω× Ω

∧ = χt×t : Ω× Ω→ Ω

[≤] → Ω× Ω

⇒ = χ[≤] : Ω× Ω→ Ω

Internal logic

X!→ 1

t→ Ω

tX : 1→ ΩX = PX

∀X = χtX : PX → Ω

Define⋂

X : PPX → PX by⋂F = x : X | ∀A:PX A ∈PX F ⇒ x ∈X A

Internal logic

X!→ 1

t→ Ω

tX : 1→ ΩX = PX

∀X = χtX : PX → Ω

Define⋂

X : PPX → PX by⋂F = x : X | ∀A:PX A ∈PX F ⇒ x ∈X A

Construction of coproducts

I Initial object: define 0 → 1 to be “intersection all subobjectsof 1”, classified by

1tP1→ PP1

⋂→ P1

I Lemma: 0 is initial.

I Uniqueness: if f , g : 0⇒ X , then Eq(f , g) 0 is an equality,by minimality of 0 in Sub(1).

I Existence: consider

P X

0 1 PX

ysingX

tX

Coproducts

I 0 is strict by cartesian closure, so 0→ X is monic.

I Given X ,Y , disjointly embed them into PX × PY :

X × 1 PX × PY 1× Y PX × PYχδ×χ0 χ0×χδ

X t Y is the “disjoint union”: the intersection of thedefinable family of subobjects of PX × PY containing theseembeddings.

Coproducts

I 0 is strict by cartesian closure, so 0→ X is monic.

I Given X ,Y , disjointly embed them into PX × PY :

X × 1 PX × PY 1× Y PX × PYχδ×χ0 χ0×χδ

X t Y is the “disjoint union”: the intersection of thedefinable family of subobjects of PX × PY containing theseembeddings.

Coproducts

I Lemma: Any two disjoint unions of X ,Y are isomorphic.

I Proof: If Z = X ∪Y via i : X → Z and j : Y → Z , then mapZ into PX × PY via

X〈1X ,i〉→ X × Z Y

〈1Y ,j〉→ Y × Z

Z → PX Z → PY

Then Z → PX × PY is monic. Both Z and X t Y are leastupper bounds of X and Y in Sub(PX × PY ).

Coproducts

I Theorem: X t Y is the coproduct.

I Proof: Given f : X → B and g : Y → B, form

X〈1X ,f 〉→ X × B, Y

〈1Y ,g〉→ Y × B.

Then (X t Y )× B ∼= (X × B) t (Y × B). So both X ,Yembed disjointly in (X t Y )× B. Obtain

X t Y → (X t Y )× B.

Image factorization

I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.

IB X B

X Y X Y

1Xy

f f

I im(f ) =⋂

Y B : PY | f ∗B = X

Image factorization

I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.

IB X B

X Y X Y

1Xy

f f

I im(f ) =⋂

Y B : PY | f ∗B = X

Image factorization

I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.

IB X B

X Y X Y

1Xy

f f

I im(f ) =⋂

Y B : PY | f ∗B = X

Image factorization

I Lemma: f : X → Y indeed factors through im(f ) : I → Y .

I Proof: We must show f ∗(im(f )) = X . But

f ∗

( ⋂B | f ∗B=X

B

)=

⋂B | f ∗B=X

f ∗B [E/Yf ∗→ E/X is logical]

=⋂

B | f ∗B=X

X

= X

Image factorization

I Lemma: X → im(f ) → Y is the epi-mono factorization off : X → Y .

Proof: Put I = im(f ); suppose X → I equalizes g , h : I ⇒ Z .Then

X → Eq(g , h) I → Y

makes Eq(g , h) a subobject through which f factors. HenceEq(g , h) = I and g = h.

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .

CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:

I Form the image factorization of 〈f , g〉 : X → Y × Y :

X → R Y × Y

I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.

I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).

I Form the image factorization of χE :

Y Q PY

I Theorem: Y → Q is the coequalizer of f , g .