Pretorsion theories in general categories

Alberto FacchiniUniversita di Padova

Parnu, 17 July 2019

Based on three joint papers,

A. Facchini and C. Finocchiaro, Pretorsion theories, stable categoryand preordered sets, submitted for publication, arXiv:1902.06694,2019.

A. Facchini, C. Finocchiaro and M. Gran, Pretorsion theories ingeneral categories, we will submit it and put it in arXiv next week.

A. Facchini and L. Heidari Zadeh, An extension of properties ofsymmetric group to monoids and a pretorsion theory in thecategory of mappings, submitted for publication, arXiv:1902.05507,2019.

From the symmetric group Sn to the monoid Mn

Several properties we teach every year to our first year studentsabout the symmetric group Sn can be easily extended or adaptedto the monoid Mn.

Here n ≥ 1 denotes a fixed integer,X will be the set {1, 2, 3, . . . , n},Sn is the group of all bijections (permutations) f : X → X , andMn is the monoid of all mappings f : X → X .

The operation in both cases is composition of mappings.

Standard properties of Sn

(1) Every permutation can be written as a product of disjointcycles, in a unique way up to the order of factors.

(2) Disjoint cycles permute.

(3) Every permutation can be written as a product oftranspositions.

(4) There is a group morphism sgn: Sn → {1,−1}. (The numbersgn(f ) is called the sign of the permutation f . )

(5) For n ≥ 2, Sn is the semidirect product of An and anysubgroup of Sn generated by a transposition.

Every permutation is a product of disjoint cycles

Given any mapping f : X → X , it is possible to associate to f adirected graph Gd

f = (X ,Edf ) (the graph associated to the function

f ), having X as a set of vertices and Edf := { (i , f (i)) | i ∈ X } as a

set of arrows. Hence Gdf has n vertices and n arrows, one arrow

from i to f (i) for every i ∈ X . In the directed graph Gdf , every

vertex has outdegree 1.

Every permutation is a product of disjoint cycles

If f : X → X is a permutation, every vertex in Gdf has outdegree 1

and indegree 1. Any finite directed connected graph in which everyvertex has outdegree 1 and indegree 1 is a cycle. Therefore thegraph Gd

f , disjoint union of its connected components, is a disjointunion of cycles in a unique way. Hence any permutation f is aproduct of disjoint cycles.

For an arbitrary mapping f : X → X . . .

For any mapping f : X → X , we can argue in the same way, butinstead of a disjoint union of cycles, we get as Gd

f a disjoint unionof forests on cycles:

A forest on a cycle

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An arbitrary mapping f : X → X . . .

Any mapping f : X → X consists a lower part (a disjoint union ofcycles, i.e., a bijection) and an upper part (a forest).

For a mapping f : X = {1, 2, 3, . . . , n} → X = {1, 2, 3, . . . , n}:(1) f is a bijection if and only if f n! = 1X .(2) The graph Gd

f is a forest (i.e., the only cycles on Gdf are the

loops) if and only if f n = f n+1.

The category of mappings M

Let M be the category whose objects are all pairs (X , f ), whereX = {1, 2, 3, . . . , n} for some n ≥ 0 and f : X → X is a mapping.Hence M will be a small category with countably many objects.

A morphism g : (X , f )→ (X ′, f ′) in M is any mappingg : X → X ′ for which the diagram

Xg//

f��

X ′

f ′

��

X g// X ′

commutes.

The category of mappings M

The category M can also be seen from the point of view ofUniversal Algebra. It is a subcategory of the category (variety) ofall algebras (X , f ) with one unary operation f and no axioms. Themorphisms in the category M are exactly the homomorphisms inthe sense of Universal Algebra.

The product decomposition of f as a product of disjoint forests oncycles corresponds to the coproduct decomposition in this categoryM as a coproduct of indecomposable algebras.

A congruence on (X , f ), in the sense of Universal Algebra, is anequivalence relation ∼ on the set X such that, for all x , y ∈ X ,x ∼ y implies f (x) ∼ f (y).

The pretorsion theory (C,F) on M

Now let C be the full subcategory of M whose objects are thepairs (X , f ) with f : X → X a bijection. Let F be the fullsubcategory of M whose objects are the pairs (X , f ) where f is amapping whose graph is a forest. (Here, C stands for cycles and Fstands for forests, or torsion-free objects, as we will see.)

Clearly, an object of M is an object both in C and in F if and onlyif it is of the form (X , 1X ), where 1X : X → X is the identitymapping. We will call these objects (X , 1X ) the trivial objectsof M. Let Triv be the full subcategory of M whose objects are alltrivial objects (X , 1X ).

Call a morphism g : (X , f )→ (X ′, f ′) in M trival if it factorsthrough a trivial object. That is, if there exists a trivial object(Y , 1Y ) and morphisms h : (X , f )→ (Y , 1Y ) and` : (Y , 1Y )→ (X ′, f ′) in M such that g = `h.

The pretorsion theory (C,F) on M

Proposition

If (X , f ) and (X ′, f ′) are objects of M, where f is a bijection andthe graph of f ′ is a forest, then every morphismg : (X , f )→ (X ′, f ′) is trivial.

For every object (X , f ) in M there is a “short exact sequence”

(A0, f |A0A0

)ε↪→ (X , f )

π� (X/∼, f )

with (A0, f |A0A0

) ∈ C and (X/∼, f ) ∈ F .

An example: an object (X , f ) in M

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The partition of X modulo ∼.

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The torsion-free quotient (X/∼, f ) of (X .f ) modulo ∼.

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Figure: The quotient set X/∼.

Preorders, partial orders and equivalence relations

A preorder on a set A is a relation ρ on A that is reflexive andtransitive.

Main examples of preorders on A:

(1) partial orders (i.e., ρ is also antisymmetric).

(2) equivalence relations (i.e., ρ is also symmetric).

Proposition

Let A be a set. There is a one-to-one correspondence between theset of all preorders ρ on A and the set of all pairs (∼,≤), where ∼is an equivalence relation on A and ≤ is a partial order on thequotient set A/∼.

The correspondence associates to every preorder ρ on A the pair('ρ,≤ρ), where 'ρ is the equivalence relation defined, for everya, b ∈ A, by a 'ρ b if aρb and bρa, and ≤ρ is the partial order onA/'ρ defined, for every a, b ∈ A, by [a]'ρ ≤ [b]'ρ if aρb.

Conversely, for any pair (∼,≤) with ∼ an equivalence relation onA and ≤ a partial order on A/∼, the corresponding preorder ρ(∼,≤)on A is defined, for every a, b ∈ A, by aρ(∼,≤)b if [a]∼ ≤ [b]∼.

A a set.

{ ρ | ρ is a preorder on A }l 1−1

{ (∼,≤) | ∼ is an equivalence relation on A

and ≤ is a partial order on A/∼}

ρ, preorder on A

↓('ρ,≤ρ), where 'ρ is the equivalence relation on A

defined, for every a, b ∈ A, by a 'ρ b if aρb and bρa,

and ≤ρ is the partial order on A/∼ defined,

for every a, b ∈ A, by [a]'ρ ≤ [b]'ρ if aρb.

The category of preordered sets

Let Preord be the category of all preordered sets.Objects: all pairs (A, ρ), where A is a set and ρ is a preorder on A.Morphisms f : (A, ρ)→ (A′, ρ′): all mappings f of A into A′ suchthat aρb implies f (a)ρ′f (b) for all a, b ∈ A.

ParOrd: full subcategory of Preord whose objects are all partiallyordered sets (A, ρ), ρ a partial order.

Equiv: full subcategory of Preord whose objects are all preorderedsets (A,∼) with ∼ an equivalence relation on A.

Trivial objects, trivial morphisms

Triv:=Equiv ∩ ParOrd, full subcategory of Preord whose objectsare all the objects of the form (A,=), where = denotes the equalityrelation on A. We will call them the trivial objects of Preord.Hence Triv is a category isomorphic to the category of all sets.

A morphism f : (A, ρ)→ (A′, ρ′) in Preord is trivial if it factorsthrough a trivial object, that is, if there exist a trivial object (B,=)and morphisms g : (A, ρ)→ (B,=) and h : (B,=)→ (A′, ρ′) inPreord with f = hg .

Prekernels

Let f : A→ A′ be a morphism in Preord. We say that a morphismk : X → A in Preord is a prekernel of f if the following propertiesare satisfied:

1. fk is a trivial morphism.

2. Whenever λ : Y → A is a morphism in Preord and f λ istrivial, then there exists a unique morphism λ′ : Y → X inPreord such that λ = kλ′.

Prekernel of a morphism f : A→ A′ in Preord

For every mapping f : A→ A′, the equivalence relation ∼f on A,associated to f , is defined, for every a, b ∈ A, by a ∼f b iff (a) = f (b).

Proposition

Let f : (A, ρ)→ (A′, ρ′) be a morphism in Preord. Then aprekernel of f is the morphism k : (A, ρ ∩∼f )→ (A, ρ), where kthe identity mapping and ∼f is the equivalence relation on Aassociated to f .

Precokernels

Let f : A→ A′ be a morphism in Preord. A precokernel of f is amorphism p : A′ → X such that:

1. pf is a trivial map.

2. Whenever λ : A′ → Y is a morphism such that λf is trivial,then there exists a unique morphism λ1 : X → Y withλ = λ1p.

Let f : X → Y and g : Y → Z be morphisms in Preord. We say

that Xf // Y

g// Z is a short preexact sequence in Preord if

f is a prekernel of g and g is a precokernel of f .

A canonical short preexact sequence for every (A, ρ) inPreord.

Let A be any set, let ρ be a preorder on A and let 'ρ be theequivalence relation on A defined by a 'ρ b if aρb and bρa and ≤ρis the partial order on A/'ρ induced by ρ, then

(A,'ρ)k // (A, ρ)

π // (A/'ρ,≤ρ)

is a short preexact sequence in Preord with (A,'ρ) ∈ Equiv and(A/'ρ,≤ρ) ∈ ParOrd.

Stable category

In Preord we have two special properties:(1) Z := Equiv ∩ ParOrd (the class of trivial objects) coincideswith the class of all projective objects in Preord.(2) It is possible to factor out the morphisms that factor though anobject of Z, constructing a quotient category of the categoryPreord, getting the stable category Preord.

Stable category

As far as quotient categories are concerned, we follow [MacLane,“Categories for the Working Mathematician”, 2nd edn., pp. 51–52].

For every pair of objects (A, ρ), (A′, ρ′) in Preord, let RA,A′ be therelation on the set Hom(A,A′) defined, for everyf , g : (A, ρ)→ (A′, ρ′), by fRA,A′g if there is a coproductdecomposition A = B

∐C of A (=disjoint union) such that f |B

and g |B are two trivial morphisms and f |C = g |C .

For instance, for any pair f , g : A→ B trivial morphisms, we getthat fRA,Bg .

LemmaThe assignment (A,A′) 7→ RA,A′ is a congruence on the categoryPreord.

Stable category

It is therefore possible to construct the quotient categoryPreord := Preord/R. We will call it the stable category. Itsobjects are all preordered sets (A, ρ), like in Preord. Themorphisms (A, ρ)→ (A′, ρ′) are the equivalence classes ofHom(A,A′) modulo RA,A′ , that is,

HomPreord(A,A′) := HomPreord(A,A′)/RA,A′ .

Notice, that ideals in a category are C are exactly subfunctors ofthe functor Hom : Cop × C → Sets.The quotients categories C/R in Mac Lane, with R a congruencein C, are exactly the quotient functors of the functorHom : Cop × C → Sets.

Stable category

There is a canonical functor F : Preord→ Preord, which iscoproduct preserving.For every preordered set (A, ρ), let A∗ be the subobject of Adefined by

A∗ :=⋃

[a]≡ρ∈T[a]≡ρ ,

where T := { [a]≡ρ ∈ A/≡ρ | |[a]≡ρ | > 1 }. We will say that (A, ρ)is reduced if A = A∗. Therefore every preordered set A is thecoproduct (=disjoint union), in a unique way, of two subobjects:the reduced object A∗ and the trivial object A \ A∗.

Stable category

Proposition

The following conditions are equivalent for two non-emptypreordered sets (A, ρ), (A′, ρ′):

1. (A, ρ), (A′, ρ′) are isomorphic objects in Preord;

2. (A∗, ρ), (A′∗, ρ′) are isomorphic objects in Preord.

3. There exist trivial objects X ,X ′ in Preord such thatA∐

X ∼= A′∐

X ′ in Preord.

The pointed category Preord∗

Pointed category = category with a zero object.

Preord is not a pointed category, but if we remove from it theinitial object ∅, we get a pointed category Preord∗, so that inPreord∗ we have a notion of kernel, cokernel and exact sequences.

Proposition

If f : (A, ρ)→ (A′, ρ′) is a morphism in Preord with A,A′ 6= ∅ andk : A, ρ ∩∼f )→ (A, ρ) is a prekernel of f , then k is a kernel of fin the pointed category Preord∗.

Similarly for precokernels and short preexact sequences.

Pretorsion theories

Fix an arbitrary category C and a non-empty class Z ofobjects of C. For every pair A,A′ of objects of C, we indicate byTrivZ(A,B) the set of all morphisms in C that factor through anobject of Z. We will call these morphisms Z-trivial.

Let f : A→ A′ be a morphism in C. We say that a morphismε : X → A in C is a Z-prekernel of f if the following properties aresatisfied:

1. f ε is a Z-trivial morphism.

2. Whenever λ : Y → A is a morphism in C and f λ is Z-trivial,then there exists a unique morphism λ′ : Y → X in C suchthat λ = ελ′.

Pretorsion theories

Proposition

Let f : A→ A′ be a morphism in C and let ε : X → A be aZ-prekernel for f . Then the following properties hold.

1. ε is a monomorphism.

2. If λ : Y → A is any other Z-prekernel of f , then there exists aunique isomorphism λ′ : Y → X such that λ = ελ′.

Dually, a Z-precokernel of f is a morphism η : A′ → X such that:

1. ηf is a Z-trivial morphism.

2. Whenever µ : A′ → Y is a morphism and µf is Z-trivial, thenthere exists a unique morphism µ′ : X → Y with µ = µ′η.

Pretorsion theories

If Cop is the opposite category of C, the Z-precokernel of amorphism f : A→ A′ in C is the Z-prekernel of the morphismf : A′ → A in Cop.

Let f : A→ B and g : B → C be morphisms in C. We say that

Af // B

g// C

is a short Z-preexact sequence in C if f is a Z-prekernel of g andg is a Z-precokernel of f .

Clearly, if Af // B

g// C is a short Z-preexact sequence in C,

then

Cg// B

f // A is a short Z-preexact sequence in Cop.

Pretorsion theories: definition

Let C be an arbitrary category. A pretorsion theory (T ,F) for Cconsists of two replete (= closed under isomorphism) fullsubcategories T ,F of C, satisfying the following two conditions.

Set Z := T ∩ F .

(1) HomC(T ,F ) = TrivZ(T ,F ) for every object T ∈ T , F ∈ F .

(2) For every object B of C there is a short Z-preexact sequence

Af // B

g// C

with A ∈ T and C ∈ F .

Like torsion theories in the abelian case

In the rest of the talk, whenever we will deal with a pretorsiontheory (T ,F) for a category C, the symbol Z will always indicatethe intersection T ∩ F .Notice that if (T ,F) is a pretorsion theory for a category C, then(F , T ) turns out to be a pretorsion theory in Cop.

Proposition

Let (T ,F) be a pretorsion theory in a category C, and let X beany object in C.

1. If HomC(X ,F ) = TrivZ(X ,F ) for every F ∈ F , then X ∈ T .

2. If HomC(T ,X ) = TrivZ(T ,X ) for every T ∈ T , then X ∈ F .

Main properties

As a corollary, from Proposition 1.6 we have that given apretorsion theory (T ,F) in a category C, any two of the threeclasses T ,F ,Z determine the third.

First of all, we have that the short Z-preexact sequence given inAxiom (2) of the definition of pretorsion theory is uniquelydetermined, up to isomorphism.

Uniqueness of the short Z-preexact sequence

Proposition

Let C be a category and let (T ,F) be a pretorsion theory for C. If

Tε // A

η// F and T ′

ε′ // Aη′// F ′

are Z-preexact sequences, where T ,T ′ ∈ T and F ,F ′ ∈ F , thenthere exist a unique isomorphism α : T → T ′ and a uniqueisomorphism σ : F → F ′ making the diagram

Tε //

�

Aη//

=��

F

σ��

T ′ε′ // A

η′// F ′

commute.

Torsion subobject and torsion-free quotient object arefunctors

Proposition

Let (T ,F) be a pretorsion theory for a category C. Choose, forevery X ∈ C, a short Z-preexact sequence

t(X )εX // X

ηX // f (X ) ,

where t(X ) ∈ T and f (X ) ∈ F . Then the assignments A 7→ t(A),(resp., A 7→ f (A)) extends to a functor t : C → T (resp.,f : C → F).

Torsion subobject and torsion-free quotient object arefunctors

If, for every X ∈ C, we chose another short Z-preexact sequence

t ′(X )λX // X

πX // f ′(X )

with t ′(X ) ∈ T , f ′(X ) ∈ F , and t ′ : C → T , f ′ : C → F are thefunctors corresponding to the new choice, then there is a uniquenatural isomorphism of functors t → t ′ (resp., f → f ′).

T is a coreflective subcategory of C

TheoremLet (T ,F) be a pretorsion theory for a category C. Then thefunctor t is a right adjoint of the category embedding eT : T ↪→ C,so that T is a coreflective subcategory of C.Dually, f is a left adjoint of the embedding eF : F ↪→ C and F is areflective subcategory of C.

When the objects in Z are projective

By projective object we mean projective with respect toepimorphisms.

A regular epimorphism is a morphism that is the coequalizer ofsome parallel pair of morphisms.

Proposition

Let C be a category and let Z be a non-empty subclass of Cconsisting of projective objects. If f : A→ B is any morphism,then any Z-precokernel of f (if it exists) is a regular epimorphism.

When the objects in Z are projective

A strong epimorphism is a morphism f : X → Y with the followingproperty: whenever there is a commutative square

Xf //

u��

Y

v��

U m// V

with m a monomorphism, there is a unique arrow α : Y → U suchthat αf = u and mα = v .

Any regular epimorphism is a strong epimorphism.

When the objects in Z are projective

Corollary

Let (T ,F) be a pretorsion theory on a category C. Suppose thatZ := T ∩ F consists of projective objects. Then:

(a) The class T is closed under strong quotients.

(b) The class F is closed under subobjects.

Some examples

(1) We have already seen the category M whose objects are allpairs (X , f ), where X = {1, 2, 3, . . . , n} for some positive integer nand f : X → X is a mapping. The morphisms in M from (X , f ) to(X ′, f ′) are the mappings ϕ : X → X ′ such that f ′ϕ = ϕf . In thisexample, the pretorsion theory in M is the pair (C,F), where Cconsists of all objects (X , f ) of M with f : X → X a bijection, andF consists of all objects (X , f ) of M with f n = f n+1, wheren = |X |. The trivial objects, that is, the objects in Z := C ∩ F , arethe pairs (X , 1X ), where 1X : X → X is the identity mapping.

Some examples

(2) The previous example can be extended to infinite sets X . LetM′ be the category whose objects are all pairs (X , f ), where X isany set and f : X → X is any mapping. Now C′ consists of allobjects (X , f ) of M′ for which the connected components of Gu

f

are either isolated points or finite circuits. Equivalently, C′ consistsof all objects (X , f ) of M′ for which f is a bijection and for everyx ∈ X there exists t > 0 such that x = f t(x).The class F ′ consists of all objects (X , f ) of M′ for which Gu

f areforests, i.e., Gu

f does not contains circuits (of length > 0).Equivalently, C′ consists of all objects (X , f ) of M′ such that, forevery x ∈ X and and every integer t > 0, if x = f t(x), thenx = f (x).The intersection Z ′ := C′ ∩ F ′ consists of all pairs (X , f ) withf : X → X the identity mapping of X .

Some examples

(3) The pretorsion theory (Equiv,ParOrd) in the category Preordof preordered sets.

(4) Let M be a commutative monoid. For every commutativemonoid M, let U(M) denote its group of units (=invertibleelements). A monoid M is reduced if U(M) = 1. In the pointedcategory of commutative monoids, there is a pretorsion theory(Ab,R), where Ab is the class of abelian groups and R is theclass of all reduced commutative monoids. The trivial objects arethe monoids with one element. For any two monoids M,N, thereis a unique trivial morphism z : M → N. It sends every element ofM to 1N . For every commutative monoid M, the canonical shortpreexact sequence is the sequence U(M)→ M → M/U(M).

Some examples

(5) Let C be the category Grp(Top) of topological groups, T thecategory Grp(Ind) of topological groups endowed with the trivialtopology, and F the category of Grp(T0) of T0 topological groups.Then (Grp(Ind),Grp(T0)) is a pretorsion theory in Grp(Top).

(6) The Kolmogorov quotient of a topological space.This is a generalization of the previous example. Let C = Top bethe category of topological spaces and T0 be the subcategory of allT0 topological spaces. A topology on a set X is a partitiontopology if there is an equivalence relation on X for which theequivalence classes are a base for the topology. Let P be the fullsubcategory of Top whose objects are all topological spaces whosetopology is a partition topology. The trivial objects in Z := P ∩ T0

are exactly the discrete topological spaces.

Some examples

(7) Let C be the category of all finite categories, and let T be thecategory of all totally disconnected finite categories, that is, thefinite categories for which all morphisms are endomorphisms, andFinitePreord the category of finite preordered sets (=finitetopological spaces). Then (T ,FinitePreord) is a pretorsion theoryin C.

Further references

[1] A. V. Arhangel’skiı and R. Wiegandt, Connectednesses anddisconnectednesses in topology, General Topology and Appl. 5(1975), 9–33.[2] M. Barr, Non-abelian torsion theories, Canad. J. Math. 25(1973) 1224–1237[3] B. A. Rattray, Torsion theories in non-additive categories,Manuscripta Math. 12 (1974), 285–305.[4] D. Bourn and M. Gran, Torsion theories in homologicalcategories, J. Algebra 305, 18–47 (2006).[5] A. Buys, N. J. Groenewald and S. Veldsman, Radical andsemisimple classes in categories. Quaestiones Math. 4 (1980/81),205–220.

Further references

[6] M. M. Clementino, D. Dikranjan and W. Tholen, Torsiontheories and radicals in normal categories, J. Algebra 305 (2006),92–129.[7] M. Grandis and G. Janelidze, From torsion theories to closureoperators and factorization systems, to appear, 2019.[8] M. Grandis, G. Janelidze and L. Marki, Non-pointed exactness,radicals, closure operators, J. Aust. Math. Soc. 94 (2013),348–361.[9] G. Janelidze and W. Tholen, Characterization of torsiontheories in general categories, in “Categories in algebra, geometryand mathematical physics”, A. Davydov, M. Batanin, M. Johnson,S. Lack and A. Neeman Eds., Contemp. Math. 431, Amer. Math.Soc., Providence, RI, 2007, pp. 249–256.

Further references

[10] J. Rosicky and W. Tholen, Factorization, fibration and torsion,arxiv/0801.0063, to appear in Journal of Homotopy and RelatedStructures.[11] S. Veldsman, On the characterization of radical andsemisimple classes in categories. Comm. Algebra 10 (1982),913–938.[12] S. Veldsman, Radical classes, connectednesses and torsiontheories, Suid-Afrikaanse Tydskr. Natuurwetenskap Tegnol. 3(1984), 42–45.[13] S. Veldsman and R. Wiegandt, On the existence andnonexistence of complementary radical and semisimple classes,Quaestiones Math. 7 (1984), 213–224.