Pretorsion theories in general categories Alberto Facchini - - PowerPoint PPT Presentation

Pretorsion theories in general categories

Alberto Facchini Universit` a di Padova P¨ arnu, 17 July 2019

Based on three joint papers,

• A. Facchini and C. Finocchiaro, Pretorsion theories, stable category

and preordered sets, submitted for publication, arXiv:1902.06694, 2019.

• A. Facchini, C. Finocchiaro and M. Gran, Pretorsion theories in

general categories, we will submit it and put it in arXiv next week.

• A. Facchini and L. Heidari Zadeh, An extension of properties of

symmetric group to monoids and a pretorsion theory in the category 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 students about the symmetric group Sn can be easily extended or adapted to 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, and Mn 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 disjoint cycles, in a unique way up to the order of factors. (2) Disjoint cycles permute. (3) Every permutation can be written as a product of transpositions. (4) There is a group morphism sgn: Sn → {1, −1}. (The number sgn(f ) is called the sign of the permutation f . ) (5) For n ≥ 2, Sn is the semidirect product of An and any subgroup 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 a directed graph G d

f = (X, E d f ) (the graph associated to the function

f ), having X as a set of vertices and E d

f := { (i, f (i)) | i ∈ X } as a

set of arrows. Hence G d

f has n vertices and n arrows, one arrow

from i to f (i) for every i ∈ X. In the directed graph G d

f , every

vertex has outdegree 1.

Every permutation is a product of disjoint cycles

If f : X → X is a permutation, every vertex in G d

f has outdegree 1

and indegree 1. Any finite directed connected graph in which every vertex has outdegree 1 and indegree 1 is a cycle. Therefore the graph G d

f , disjoint union of its connected components, is a disjoint

union of cycles in a unique way. Hence any permutation f is a product of disjoint cycles.

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

For any mapping f : X → X, we can argue in the same way, but instead of a disjoint union of cycles, we get as G d

f a disjoint union

• f forests on cycles:

A forest on a cycle

An arbitrary mapping f : X → X . . .

Any mapping f : X → X consists a lower part (a disjoint union of cycles, 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 G d

f is a forest (i.e., the only cycles on G d f 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 ), where X = {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 mapping g : X → X ′ for which the diagram X

g

• f
• X ′

f ′

• X

g

X ′

commutes.

The category of mappings M

The category M can also be seen from the point of view of Universal Algebra. It is a subcategory of the category (variety) of all algebras (X, f ) with one unary operation f and no axioms. The morphisms in the category M are exactly the homomorphisms in the sense of Universal Algebra. The product decomposition of f as a product of disjoint forests on cycles corresponds to the coproduct decomposition in this category M as a coproduct of indecomposable algebras. A congruence on (X, f ), in the sense of Universal Algebra, is an equivalence 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 the pairs (X, f ) with f : X → X a bijection. Let F be the full subcategory of M whose objects are the pairs (X, f ) where f is a mapping whose graph is a forest. (Here, C stands for cycles and F stands 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 only if it is of the form (X, 1X), where 1X : X → X is the identity

• mapping. We will call these objects (X, 1X) the trivial objects
• f M. Let Triv be the full subcategory of M whose objects are all

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

The pretorsion theory (C, F) on M

Proposition

If (X, f ) and (X ′, f ′) are objects of M, where f is a bijection and the graph of f ′ is a forest, then every morphism g : (X, f ) → (X ′, f ′) is trivial. For every object (X, f ) in M there is a “short exact sequence” (A0, f |A0

A0) ε

֒ → (X, f )

π

։ (X/∼, f ) with (A0, f |A0

A0) ∈ C and (X/∼, f ) ∈ F.

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

• •

The partition of X modulo ∼.

• •
• 900000000km
• 90000000000000000nm

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

• •
• 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 and transitive. 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 the set 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 the quotient set A/∼. The correspondence associates to every preorder ρ on A the pair (≃ρ, ≤ρ), where ≃ρ is the equivalence relation defined, for every a, b ∈ A, by a ≃ρ b if aρb and bρa, and ≤ρ is the partial order on A/≃ρ defined, for every a, b ∈ A, by [a]≃ρ ≤ [b]≃ρ if aρb. Conversely, for any pair (∼, ≤) with ∼ an equivalence relation on A and ≤ a partial order on A/∼, the corresponding preorder ρ(∼,≤)

• n A is defined, for every a, b ∈ A, by aρ(∼,≤)b if [a]∼ ≤ [b]∼.

A a set. { ρ | ρ is a preorder on A }

• 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′ such that aρb implies f (a)ρ′f (b) for all a, b ∈ A. ParOrd: full subcategory of Preord whose objects are all partially

• rdered sets (A, ρ), ρ a partial order.

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

Trivial objects, trivial morphisms

Triv:=Equiv ∩ ParOrd, full subcategory of Preord whose objects are all the objects of the form (A, =), where = denotes the equality relation 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 factors through a trivial object, that is, if there exist a trivial object (B, =) and morphisms g : (A, ρ) → (B, =) and h: (B, =) → (A′, ρ′) in Preord with f = hg.

Prekernels

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

• 1. fk is a trivial morphism.
• 2. Whenever λ: Y → A is a morphism in Preord and f λ is

trivial, then there exists a unique morphism λ′ : Y → X in Preord 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 if f (a) = f (b).

Proposition

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

Precokernels

Let f : A → A′ be a morphism in Preord. A precokernel of f is a morphism 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 with λ = λ1p. Let f : X → Y and g : Y → Z be morphisms in Preord. We say that X

f

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) in Preord.

Let A be any set, let ρ be a preorder on A and let ≃ρ be the equivalence relation on A defined by a ≃ρ b if aρb and bρa and ≤ρ is 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) coincides with the class of all projective objects in Preord. (2) It is possible to factor out the morphisms that factor though an

• bject of Z, constructing a quotient category of the category

Preord, 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 the relation on the set Hom(A, A′) defined, for every f , g : (A, ρ) → (A′, ρ′), by fRA,A′g if there is a coproduct decomposition 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 get that fRA,Bg.

Lemma

The assignment (A, A′) → RA,A′ is a congruence on the category Preord.

Stable category

It is therefore possible to construct the quotient category Preord := Preord/R. We will call it the stable category. Its

• bjects are all preordered sets (A, ρ), like in Preord. The

morphisms (A, ρ) → (A′, ρ′) are the equivalence classes of Hom(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 of the functor Hom: Cop × C → Sets. The quotients categories C/R in Mac Lane, with R a congruence in C, are exactly the quotient functors of the functor Hom: Cop × C → Sets.

Stable category

There is a canonical functor F : Preord → Preord, which is coproduct preserving. For every preordered set (A, ρ), let A∗ be the subobject of A defined 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 the coproduct (=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-empty preordered 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 that

A 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 the initial object ∅, we get a pointed category Preord∗, so that in Preord∗ 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′ = ∅ and k : A, ρ ∩∼f ) → (A, ρ) is a prekernel of f , then k is a kernel of f in the pointed category Preord∗. Similarly for precokernels and short preexact sequences.

Pretorsion theories

Fix an arbitrary category C and a non-empty class Z of

• bjects of C. For every pair A, A′ of objects of C, we indicate by

TrivZ(A, B) the set of all morphisms in C that factor through an

• bject of Z. We will call these morphisms Z-trivial.

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

• 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 such that λ = ελ′.

Pretorsion theories

Proposition

Let f : A → A′ be a morphism in C and let ε: X → A be a Z-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 a

unique 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, then

there 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 a morphism f : A → A′ in C is the Z-prekernel of the morphism f : A′ → A in Cop. Let f : A → B and g : B → C be morphisms in C. We say that A

f

B

g

C

is a short Z-preexact sequence in C if f is a Z-prekernel of g and g is a Z-precokernel of f . Clearly, if A

f

B

g

C is a short Z-preexact sequence in C,

then C

g

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 C consists of two replete (= closed under isomorphism) full subcategories 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 A

f

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 pretorsion theory (T , F) for a category C, the symbol Z will always indicate the 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 be any 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 a pretorsion theory (T , F) in a category C, any two of the three classes T , F, Z determine the third. First of all, we have that the short Z-preexact sequence given in Axiom (2) of the definition of pretorsion theory is uniquely determined, 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 T

ε

A

η

F

and T ′

ε′

A

η′

F ′

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

ε

• α
• A

η

• =
• F

σ

• T ′

ε′

A

η′

F ′

commute.

Torsion subobject and torsion-free quotient object are functors

Proposition

Let (T , F) be a pretorsion theory for a category C. Choose, for every 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 → t(A), (resp., A → f (A)) extends to a functor t : C → T (resp., f : C → F).

Torsion subobject and torsion-free quotient object are functors

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 the functors corresponding to the new choice, then there is a unique natural isomorphism of functors t → t′ (resp., f → f ′).

T is a coreflective subcategory of C

Theorem

Let (T , F) be a pretorsion theory for a category C. Then the functor 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 a reflective subcategory of C.

When the objects in Z are projective

By projective object we mean projective with respect to epimorphisms. A regular epimorphism is a morphism that is the coequalizer of some parallel pair of morphisms.

Proposition

Let C be a category and let Z be a non-empty subclass of C consisting 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 following property: whenever there is a commutative square X

f

• u
• Y

v

• U

m

V

with m a monomorphism, there is a unique arrow α: Y → U such that 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 that Z := 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 all pairs (X, f ), where X = {1, 2, 3, . . . , n} for some positive integer n and 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 this example, the pretorsion theory in M is the pair (C, F), where C consists of all objects (X, f ) of M with f : X → X a bijection, and F consists of all objects (X, f ) of M with f n = f n+1, where n = |X|. The trivial objects, that is, the objects in Z := C ∩ F, are the pairs (X, 1X), where 1X : X → X is the identity mapping.

Some examples

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

• bjects (X, f ) of M′ for which the connected components of G u

f

are either isolated points or finite circuits. Equivalently, C′ consists

• f all objects (X, f ) of M′ for which f is a bijection and for every

x ∈ X there exists t > 0 such that x = f t(x). The class F′ consists of all objects (X, f ) of M′ for which G u

f are

forests, i.e., G u

f does not contains circuits (of length > 0).

Equivalently, C′ consists of all objects (X, f ) of M′ such that, for every x ∈ X and and every integer t > 0, if x = f t(x), then x = f (x). The intersection Z′ := C′ ∩ F′ consists of all pairs (X, f ) with f : X → X the identity mapping of X.

Some examples

(3) The pretorsion theory (Equiv, ParOrd) in the category Preord

• f preordered sets.

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

Some examples

(5) Let C be the category Grp(Top) of topological groups, T the category Grp(Ind) of topological groups endowed with the trivial topology, 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 be the category of topological spaces and T0 be the subcategory of all T0 topological spaces. A topology on a set X is a partition topology if there is an equivalence relation on X for which the equivalence classes are a base for the topology. Let P be the full subcategory of Top whose objects are all topological spaces whose topology 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 the category of all totally disconnected finite categories, that is, the finite categories for which all morphisms are endomorphisms, and FinitePreord the category of finite preordered sets (=finite topological spaces). Then (T , FinitePreord) is a pretorsion theory in C.

Further references

[1] A. V. Arhangel’ski˘ ı and R. Wiegandt, Connectednesses and disconnectednesses 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 homological categories, J. Algebra 305, 18–47 (2006). [5] A. Buys, N. J. Groenewald and S. Veldsman, Radical and semisimple classes in categories. Quaestiones Math. 4 (1980/81), 205–220.

Further references

[6] M. M. Clementino, D. Dikranjan and W. Tholen, Torsion theories and radicals in normal categories, J. Algebra 305 (2006), 92–129. [7] M. Grandis and G. Janelidze, From torsion theories to closure

• perators and factorization systems, to appear, 2019.

[8] M. Grandis, G. Janelidze and L. M´ arki, Non-pointed exactness, radicals, closure operators, J. Aust. Math. Soc. 94 (2013), 348–361. [9] G. Janelidze and W. Tholen, Characterization of torsion theories in general categories, in “Categories in algebra, geometry and 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. Rosick´ y and W. Tholen, Factorization, fibration and torsion, arxiv/0801.0063, to appear in Journal of Homotopy and Related Structures. [11] S. Veldsman, On the characterization of radical and semisimple classes in categories. Comm. Algebra 10 (1982), 913–938. [12] S. Veldsman, Radical classes, connectednesses and torsion theories, Suid-Afrikaanse Tydskr. Natuurwetenskap Tegnol. 3 (1984), 42–45. [13] S. Veldsman and R. Wiegandt, On the existence and nonexistence of complementary radical and semisimple classes, Quaestiones Math. 7 (1984), 213–224.