From torsion theories to closure operators and factorization systems

Torsion theories are here extended to categories equipped with an ideal of ‘null morphisms’, or equivalently a full subcategory of ‘null objects’. Instances of this extension include closure operators viewed as generalised torsion theories in a ‘category of pairs’, and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15]. Introduction Classically, a torsion theory (T,F) in an abelian category A (see Borceux [3], Section 1.2) is a pair (T,F) of full, replete subcategories of A such that: (i) every morphism T → F from an object of T to an object of F is null, (ii) for every object A in A there exists a short exact sequence T A F, with T in T and F in F. The objects of T and F are called, respectively, the torsion objects and the torsion-free objects of the theory. One easily proves that:


Introduction
Classically, a torsion theory (T, F) in an abelian category A (see Borceux [3], Section 1.2) is a pair (T, F) of full, replete subcategories of A such that: (i) every morphism T → F from an object of T to an object of F is null, (ii) for every object A in A there exists a short exact sequence T A F, with T in T and F in F. All this can be extended, with the same words, to any pointed category E with kernels and cokernels, a short exact sequence (m, p) being any pair of consecutive maps where m is a kernel of p and the latter a cokernel of m (see e.g. [16]).
In fact, we are interested in wider extensions, to non-pointed categories where we still have some form of exactness with respect to an ideal of 'null morphisms', as in [11]- [13]; these extensions, which we call multi-pointed categories and pre-pointed categories, are briefly analysed in Sections 1 and 4.
Section 2 studies torsion theories in a multi-pointed category E, essentially defined as above. More generally, a torsion operator (τ, ϕ) consists of two endofunctors T, F : E → E and two natural transformations τ : T → 1 and ϕ : 1 → F giving a short exact sequence (0.1) for every object A. This operator is a torsion theory if and only if τ and ϕ give invertible transformations T τ = τ T and F ϕ = ϕF , which means that T and F can be (uniquely) extended to an idempotent comonad and monad, respectively.
In Section 3 we show that a closure operator for topological spaces amounts to a torsion operator on the category Top 2 of pairs of topological spaces (in the sense of Algebraic Topology, see 1.2), which is multi-pointed with respect to a natural ideal of null morphisms. More generally, this works for any category C 2 'of pairs', constructed over a category C equipped with a suitable choice of 'distinguished monomorphisms'. The stronger case of a torsion theory corresponds to a closure operator which is weakly hereditary and idempotent. All this is outlined in the first table below In Sections 4 and 5 we develop the second table. Torsion operators and torsion theories are generalised to pre-pointed categories. Any category of morphisms C 2 comes equipped with a subcategory of null objects, the isomorphisms of C; in this sense C 2 is pre-pointed and has a notion of short pre-exact sequence, which amounts to a factorization x = x ·x of a morphism in C. Now a torsion operator in C 2 amounts to a functorial factorization in C; the former is a torsion theory if and only if the latter is an (orthogonal) factorization system. This correspondence can be refined: we already remarked that a torsion theory is a torsion operator (τ, ϕ) where (T, τ ) 'is' an idempotent comonad and (F, ϕ) an idempotent monad. This suggests to consider an intermediate case: an algebraic torsion operator, where the endofunctors T and F are respectively enriched to a comonad and a monad. In the previous case of a multi-pointed category, the additional structure is automatically idempotent and we just get a torsion theory. For a pre-pointed category E this is not the case: in fact, for a category of morphisms E = C 2 , we get an algebraic factorization on C, introduced in [14] under the name of a 'natural weak factorization system'; the more explicit term 'algebraic', instead of 'natural', was used in [10].

Short exact sequences in categories with an ideal of null morphisms
Short exact sequences are a basic ingredient of torsion theories. We explore some fairly general situations where they can be defined. Categories with an assigned ideal of null morphisms and (co)kernels with respect to this ideal have been considered by various authors, including C. Ehresmann [7] and R. Lavendhomme [17]. The present results have been developed in [11]- [13]; we generally refer to the recent book [13], following its terminology and notation. The symbol ⊂ denotes weak inclusion.
After recalling the notion of semiexact category, we briefly explore two extensions: a multi-pointed category, and -more generally -a pre-pointed category.
1.1 Semiexact categories An ex0-category, as defined in [13], is a category E equipped with a set of null morphisms N , where: The condition that N be an ideal means that every composite with a null morphism is null, while the 'closedness' of N means that every null morphism factorizes through a null object, that is, an object whose identity belongs to N . Equivalently, one can assign a full subcategory E 0 closed under retracts, called the subcategory of null objects. (The equivalence comes out of an obvious covariant Galois connection between subsets of morphisms and subsets of objects, in any category.) A semiexact category is an ex0-category E where: (ex.1) every morphism of E has a kernel and a cokernel, with respect to N , written as: The kernel is characterised up to isomorphism by the 'usual' universal property, written with respect to the ideal of null morphisms (see the left diagram below): -f· kerf is null, The cokernel is defined by the dual property, represented in the right diagram above. It follows easily that kerf is mono and cokf is epi. A normal mono is, by definition, a kernel (of some morphism), and a normal epi is a cokernel; the arrows , are reserved here for such morphisms. One proves that the morphism f has a unique normal factorization f = mgp ( [13], Section 1.1.5) through its normal coimage p (the cokernel of the kernel morphism) and its normal image m (the kernel of the cokernel morphism) This factorization is natural; f is said to be an exact morphism if this g is an isomorphism.
A sequence in E will be said to be short exact if m is a kernel of p, and p a cokernel of m.
The semiexact category E is said to be pointed (or p-semiexact) if it has a zero object 0 and its null morphisms are precisely the zero morphisms (those which factorize through 0). In this case kernels and cokernels acquire the usual meaning. Incidentally, we recall that a p-semiexact category where each morphism is exact is the same as an exact category in the sense of Puppe and Mitchell [19,20], called a p-exact category in [13]; such a category is abelian if and only if it has all finite products, if and only if it has all finite sums.
The ideal N cannot be empty unless E is. In the trivial case N = E all kernels and cokernels are identities. The opposite category E op is equipped with the ideal N op , and is semiexact.
A functor between semiexact categories is said to be exact if it preserves kernels and cokernels.

A basic example
The general behaviour of semiexact categories can be very different from that of the pointed ones, as one can see analysing the classical category Top 2 of pairs of topological spaces, in the sense of Algebraic Topology (cf. [8]). An the composition is plain. Top is fully embedded in Top 2 , identifying the space X with the pair (X, ∅).
The object (X, A) is usually read as X modulo A, and viewed as a sort of 'formal quotient'. It is, therefore, quite natural to define the morphism f : (X, A) → (Y, B) to be null whenever f (X) ⊂ B. The null objects are thus the 'diagonal' pairs (T, T ) and f is null if and only if it factorizes through one of them, for instance (X, X) (or (B, B)).
Kernels and cokernels exist and the normal factorization of f can be presented as The morphism f is exact if and only if it is injective and f (X) ⊃ B. Every short exact sequence in Top 2 is (up to isomorphism) of the form where both maps carry all elements to themselves. This also determines the normal subobjects and the normal quotients of the object (X, A).
Now the pair (X, A) is indeed the normal quotient in Top 2 . Every homology theory for pairs of spaces, in the sense of Eilenberg and Steenrod [8], carries the sequence (1.6) to a long exact sequence of abelian groups, called the homology sequence of the triple (X, B, A). Note the following facts, in contrast with the behaviour of abelian (or just pointed) categories. A null morphism (X, A) → (Y, B) between two given objects need neither exist (take X = ∅ = B) nor be unique. A monomorphism (given by an injective map) need not have a null kernel. A null morphism need not be exact, a null monomorphism need not be normal. An exact monomorphism need not be a normal mono: for example, the normal quotient (X, A) (X, B) is mono but it is not a normal mono (unless A = B). The initial object (∅, ∅) and the terminal object ({ * }, { * }) are distinct and both null, but do not determine the null morphisms in any useful way.
Every object (X, A) has a least normal subobject, the kernel of its identity, and a least normal quotient, which are null objects, generally nonisomorphic Further properties show that Top 2 is a 'homological category', in the sense of [11]- [13]. Similarly, the homological category Gp 2 of 'pairs of groups' is the domain of relative (co)homology of groups, and a source of modified categories where exact sequences and spectral sequences coming out of homotopy theory can be studied.
General 'categories of pairs' C 2 on a category C 'with distinguished subobjects' have been studied in [13], Section 2.5. This topic will be reviewed in Section 3 under more general assumptions on C, still sufficient to characterise the short exact sequences of C 2 as in diagram (1.6): here we need not have a semiexact (or homological) category, but just a 'multi-pointed' one, as defined below.
1.3 Multi-pointed categories A first extension of semiexact categories will allow us to view closure operators as torsion operators, in full generality.
By a multi-pointed category we mean a category E equipped with a closed ideal N (or equivalently with a full subcategory E 0 closed under retracts in E) where we assume the existence of the kernel and cokernel of any identity (and a choice of them). The latter are written as This can be viewed as a 'multi-object extension' of the notion of pointed category: in fact the latter amounts to a multi-pointed category where the category E 0 is equivalent to the singleton category.
Kernels and cokernels are defined as in 1.1, but we do not assume their existence, in general. We only note that the kernel of a morphism f : A → B amounts to the pullback of 0 B : B 0 B along f while the cokernel of f is the same as the pushout of 0 A : A A 0 along f . Moreover, A short exact sequence (m, p) is defined as in 1.1 (see (1.4)); the morphisms m : M A and p : A P appearing there will be said to be a normal monomorphism and a normal epimorphism, respectively. Note that a kernel morphism that does not have a cokernel is not considered as a normal mono.
Incidentally we note that, in a category equipped with a closed ideal, (co)kernels can only be defined as (co)limits under some additional assumption, as in (1.8). To wit, the category Set of sets, equipped with the closed ideal of constant mappings (that factorize through a singleton), has all cokernels, which are pushouts, but lacks kernels -even though all limits exist.

Normal subobjects and quotients Extending topics studied
in [13] for semiexact categories, in the multi-pointed category E every object A has a (possibly large) ordered set Nsb(A) of normal subobjects and an ordered set Nqt(A) of normal quotients. They are linked by an antiisomorphism produced by cokernels and kernels which will be called kernel-cokernel duality, or kernel duality for short.
Let us remark that kernel duality only operates on normal subobjects and normal quotients of E and reverses their order relation. On the other hand, categorical duality, which operates on all items, takes a normal mono of E to a normal epi of the opposite multi-pointed category E op and preserves their order. When speaking of 'duality' without specification we shall mean the categorical one.
These ordered sets are bounded: the object A has a least normal subobject 0 A : A 0 A and a greatest one Similarly, it has a least normal quotient 0 A : A A 0 and a greatest one (If E is semiexact these ordered sets are lattices; if it is p-exact they are modular lattices [13].) A is a null object if and only if 0 A = 1 A , if and only if 0 A = 1 A , and then A 0 = A = A 0 . An object is null if and only if it has precisely one normal subobject (the null one), if and only if it has precisely one normal quotient.

The subcategory of null objects A multi-pointed category E
can be simply described by properties of its full subcategory E 0 of null objects, in the same line as in [15] for semiexact categories.
In fact 0 A : A 0 A is a universal arrow from the embedding E 0 → E to the object A, while 0 A : A A 0 is a universal arrow the other way round. Therefore the subcategory E 0 satisfies the following condition: (ex.0a) E 0 is a full, replete, reflective and coreflective subcategory, with a componentwise-mono counit and a componentwise-epi unit (for the coreflector D and the reflector C, respectively) (1.10) This is indeed an equivalent definition of multi-pointed categories, as we shall see in Proposition 1.3. The pointed case is characterised by E 0 being equivalent to the singleton category, that is, being an indiscrete category.
We now introduce a further extension by dropping the cancellation conditions inside axiom (ex.0a). This will allow us to include factorization structures in the reach of torsion structures.
Definition 1.1 (Pre-pointed categories). By a pre-pointed category we mean a pair (E, E 0 ) satisfying the following axiom: (ex.0b) E is a category and E 0 is a full replete reflective and coreflective subcategory.
The adjunctions of the inclusion U : E 0 → E will be written as E 0 is closed in E under retracts (as any reflective subcategory), and therefore determined by the ideal N of the morphisms which factorize through its objects.
The pair (E, E 0 ) will often be denoted as E, provided no ambiguity arises.
We shall see in Section 4 that every category of morphisms C 2 is prepointed, in a natural way.
(a) For every object A, the objects A 0 and A 0 are null. The counit component κA : A 0 → A is cancellable with respect to pairs of morphisms Z −→ −→ A 0 defined on a null object. Dually for γA. Proof. (a) is obvious. This implies the non-trivial part of (b): if A is null then 1 A factorizes as κA.v, whence κA is a split epi in E 0 , and therefore invertible.
where Z is null, and the latter factorizes through κB. Proof. Follows easily from the previous lemma.

Torsion theories and torsion operators in multi-pointed categories
Torsion theories in abelian categories are dealt with in the book by Borceux [3], Section 1.2. We show here that the basic results can be easily extended to multi-pointed categories. Throughout this section E is a multi-pointed category. For a category C, writing X ∈ C we mean X ∈ ObC.
Definition 2.1 (Torsion theories). By a torsion theory in the multi-pointed category E we mean a pair (T, F) of full, replete subcategories of E such that (tor.1) every morphism T → F , with T ∈ T and F ∈ F, is null, The objects of T are called the torsion objects of the theory, those of F the torsion-free objects. Categorical duality gives a torsion theory (F op , T op ) in the opposite category E op .
Condition (tor.1) means that T⊥F with respect to the orthogonality relation in ObE, defined by: A stronger result, in the presence of (tor.2), will be given in Corollary 2.3(b).
We shall see that the torsion objects are always closed under normal quotients. A torsion theory is said to be hereditary if the torsion objects are also closed under normal subobjects.
Proposition 2.2 (The torsion exact sequence). Let (T, F) be a torsion theory in the multi-pointed category E.
(a) Every object A has a short exact sequence, determined up to isomorphism and called the torsion exact sequence of A. (Implicitly, we are making a choice of a distinguished sequence for every A, in its isomorphism class.) The normal subobject in this sequence is the greatest normal subobject of A with domain in T. This defines the torsion function τ : ObE → MorE of the theory.
(b*) The normal quotient is the greatest normal quotient of A with codomain in F. This defines the torsion-free function ϕ : ObE → MorE of the theory.
(c) All this determines a subfunctor T and a quotient-functor F of the identity of E.
Proof. The points (a), (b) and (b*) can be proved by a single argument. By hypothesis there exists a short exact sequence (m, p) as in the diagram below, with T ∈ T and F ∈ F Given any normal monomorphism m : T A with domain in T, the morphism pm is null, whence m ≺ m. Therefore, in our sequence, m is determined as the greatest normal subobject of A with domain in T. By duality p is determined as in (b*), and the sequence itself is determined (up to isomorphism).
is null. This gives a (unique) commutative diagram with distinguished short exact rows and defines the functors T and F on the morphisms of E.
In particular, T is closed under normal quotients.
(c*) If B ∈ F and there is a morphism h : B → B with a null kernel, then B ∈ F. In particular, F is closed under normal subobjects.
(d) An object of E belongs to T and F if and only if it is null.
Proof. (a) and (a*) follow trivially from the previous proposition.
(b) By (tor.1) every morphism T → F , with T ∈ T and F ∈ F, is null. Let an object A be given, such that every morphism A → F with F ∈ F is null. Then ϕA : A F (A) is the null normal quotient of A, and the latter belongs to T by (a).
(c) In the given hypotheses, take a morphism f : A → B with B ∈ F. Then f h is null, and f is also; therefore A ∈ T, by (b).
(d) If A belongs to T and F then 1 A is null. The converse follows from (b).
Theorem 2.4 (The torsion function). In the multi-pointed category E it is equivalent to assign: (i) a torsion theory (T, F), (ii) a torsion function τ : ObE → MorE that satisfies the following three axioms: A is a normal subobject producing a subfunctor T of the identity (that is, every map f : Proof. (ii) ⇒ (i). Define T and F as in (2.7). For (tor.1) we take a morphism f : A → B with A ∈ T and B ∈ F; then we have a commutative diagram with τ A invertible and τ B null, so that f is null. For (tor.2) we take the short exact sequence where T A is in T by (tf.2) and A/T (A) is in F, by (tf.3).

2.1
The torsion-free function Dually, one can give axioms for a torsion-free function on the multi-pointed category E. It is thus equivalent to assign: (i) a torsion theory (T, F), (ii) a torsion-free function ϕ : ObE → MorE that satisfies the following three axioms: (ff.1) for every A in E, ϕA : A F (A) is a normal quotient producing a quotient functor F of the identity (that is, every map f : is invertible (and the functor F can be made idempotent), Given a torsion theory (T, F), the corresponding torsion-free function is defined as in (2.3). Given a torsion-free function ϕ, the corresponding torsion theory (T, F) is defined as: (2.10) Definition 2.5 (Torsion operators). More generally a torsion operator (τ, ϕ) on the multi-pointed category E will consist of assigning: (top.1) for every object A, a natural short exact sequence (τ A, ϕA) with central object A. Naturality, of course, means that for every morphism f : A → B there is a (unique) commutative diagram with distinguished short exact rows This defines the endofunctors T and F of E, as, respectively, a subfunctor and a quotient-functor of the identity.

Closure operators as torsion operators in categories of pairs
Given a category C with assigned 'distinguished subobjects', we construct a 'category of pairs' C 2 which is multi-pointed with respect to a natural ideal of null morphisms. Torsion operators in C 2 amount to closure operators on the distinguished subobjects of C, while a torsion theory corresponds to the idempotence and weak hereditariness of the associated closure operator. For closure operators we follow the terminology of Dikranjan and Tholen [6].

Categories with distinguished subobjects
We have recalled the classical category Top 2 of pairs of topological spaces, based on a choice of distinguished subobjects in Top, namely the subspaces. This construction has a natural extension to a semiexact category of pairs C 2 over a 'ds-category' C ( [13], Section 2.5). Forsaking the existence of all kernels and cokernels, this procedure can be further generalised as follows.
A ds1-category is a category C equipped with a set d of distinguished subobjects, called d-subobjects for short. We assume that: (ds.0) for each object X, 1 X represents a d-subobject, (ds.1) if a : A → X and b : B → X represent d-subobjects and a ≤ b, the monomorphism u : A → B such that a = bu is equivalent to a d-subobject.
We speak of a ds2-category if, moreover: (ds.2) every morphism factorizes through a smallest d-subobject of its codomain.
We shall write d X for the ordered set of d-subobjects of X, and d X for the preordered set of the d-monomorphisms with values in X, equivalent to the previous ones. We show below that conditions (ds.0) and (ds.1) are sufficient to construct a multi-pointed category of pairs C 2 and to characterise its short exact sequences, while the additional condition (ds.2) will be used to introduce closure operators in C.
Remark. If in (ds.1) we assume that the monomorphism u : A → B is indeed a d-subobject, our construction below can be simplified (up to equivalence of categories) taking as objects of C 2 the d-subobjects instead of all the dmonomorphisms. This stronger assumption is 'sound' in the usual concrete categories, and gives a category of pairs Top 2 as classically constructed.

General categories of pairs
Given a ds1-category C, we construct the multi-pointed category of pairs C 2 = Pair(C, d) as a full subcategory of the category of morphisms C 2 .
Its objects are the distinguished monomorphisms in d (the union of all d X ), its morphisms f = (f, f ) : a → b are the commutative squares as in the left diagram below, with obvious composition Such a morphism f : a → b is assumed to be null if one can insert a map h (necessarily unique), forming a commutative diagram. The null objects are the isomorphisms of C and f : a → b is null if and only if it factorizes through 1 : B → B as in the right diagram above (or equivalently through 1 : X → X).
The kernel 0 a : a 0 → a of 1 a and its cokernel 0 a : a → a 0 are shown in the left diagram below, proving that C 2 is multi-pointed The short exact sequences of C 2 , the normal subobjects and the normal quotients of the object a are -up to isomorphism -of the form shown in the right diagram above, determined by an inequality a ≤ b in d X (since the monomorphism u is in d , by (ds.1).) In fact, it is easy to check that (b, 1 A ) : c → a is a kernel of (1 X , c) : a → b, and the latter a cokernel of the former. Conversely, the diagonal embedding C → C 2 (that sends X to 1 X ) has for reflector the codomain-projection (with unit 0 a : a → 1 X ); therefore the functor C preserves the existing colimits. Now, if a morphism (f, f ) : t → a has a cokernel, the latter is the

Closure operators on distinguished subobjects
Now we take C to be a ds2-category. We shall simplify notation, denoting distinguished subobjects a : A → X, b : B → X, ... by their domain A, B, ... and writing A ⊂ B for a ≤ b; which is appropriate for the usual concrete categories (see the last remark in 3.1). Following (and extending) the definition of [6] in Chapter 2, by a closure operator on the ds2-category C we mean a family c = (c X ) of operators on the ordered sets d X (for X ∈ ObC) that satisfies the following axioms, for every A, B in d X and every morphism Here f * A is the smallest d-subobject of Y through which the composite A ⊂ X → Y factorizes. As usual, a subobject A is said to be closed in X if A = A, and dense in X if A = X.
One says that the closure operator c is idempotent if it satisfies the following condition (cl.4), and weakly hereditary if it satisfies (cl.5): Theorem 3.1 (Closure operators as torsion operators). Let C be a ds2category and C 2 = Pair(C, d) its multi-pointed category of pairs. To assign a closure operator c on (the distinguished subobjects of) C is equivalent to assigning a torsion operator (τ, ϕ) in C 2 , as defined in Section 2. Moreover, c is idempotent and weakly hereditary if and only if (τ, ϕ) is a torsion theory.
Proof. (a) Given a closure operator, we define the associated torsion operator (τ, ϕ) in C 2 by letting Then T is indeed a subfunctor of the identity.
(b) Conversely, let (τ, ϕ) be a torsion operator in C 2 . We define the associated closure operator on distinguished subobjects of C by noting that T (X, A), as a normal subobject of (X, A), determines a d-subobject c X (A) of X This is indeed a closure operator on (C, d), in the sense of 3.3. - -Continuity: every morphism f : show that the weak hereditariness and idempotence of c amount to (τ, ϕ) being a torsion theory (by (top.2 ) in 2.5).

Pre-pointed categories and categories of morphisms
After a brief analysis of pre-pointed categories, defined in 1.1, we show that any category of morphisms E = C 2 is pre-pointed, and that a 'short preexact sequence' in C 2 amounts to a factorization x = x .x of a morphism in C.
4.1 Pre-kernels and pre-cokernels Let (E, E 0 ) be a pre-pointed category.
For every morphism f : A → B, (a choice of) the pullback of κB : B 0 → B along f , as in the left diagram below will be written as kf : Kf → A and called a pre-kernel of f . We only fix the choice k(1 A ) = κA. We show below that pre-kernels are weak kernels, in a natural way. But this property puts a weak constraint on the object Kf , while pullbacks are determined up to isomorphism, of course. Dually, the pushout of γA : A → A 0 along f (see the right diagram above) will be written as cf : B → Cf and called the pre-cokernel of f . Again we fix c(1 A ) = γA.
A sequence will be said to be short pre-exact if m is a pre-kernel of p and p is a precokernel of m. This means that f.kf is null, and for every map u in E such that f u is null, there is some map v such that u = kf.v, as in the left diagram below Dually cf : B → Cf , if it exists, is a weak cokernel of f , with a weak universal property as in the right diagram above.
In particular, κA = k(1 A ) : A 0 → A is a weak kernel of 1 A and γA = c(1 A ) : A → A 0 a weak cokernel of the latter.
Proof. If f u is null, then it factorizes through κB : B 0 → B as f u = κB.g, whence the pair (u, g) factorizes uniquely through the pullback Kf and u just factorizes through kf . Proposition 4.2. Let E = (E, E 0 ) be a pre-pointed category with all prekernels and all pre-cokernels.
(a) The universal property of pullbacks determines a functor K : E 2 → E defined on the category of morphisms of E, and a natural transformation k : K → Dom : E 2 → E given by the family (kf ) f .
Equivalently, we can present the pair (K, k) as a functor k : (a*) Dually, the universal property of pushouts determines a functor C : E 2 → E and a natural transformation c : Cod → C : E 2 → E given by the family (cf ) f . Equivalently, we have a functor c : : cf → cg, as in the right square above. (c) E is p-semiexact if and only if E is pointed.
Remark 4.3. Therefore, in a pre-pointed category E with all pre-kernels, the family (kf ) f can be extended to a natural assignment of weak kernels, that is, a natural transformation k : K → Dom : E 2 → E such that, for every map f in E 2 , kf : Kf → Domf is a weak kernel of f . Again, let us note that the weak universal property in (a) ties down the object Kf in a weak form, and would not give any naturality, by itself. The root of the question is the fact that κA : A 0 → A is determined by the universal property of the adjunction U D in (1.11); this means that κA, besides being a weak kernel of 1 A , also satisfies the restricted cancellation property of 1.2(a).

Categories of morphisms
We begin to study the category of morphisms E = C 2 of a category C, along the lines above. An object of E is a morphism x : X → X of C. A morphism f = (f , f ) : x → y is a commutative square of C. E will always be equipped with the full subcategory E 0 of null objects: the isomorphisms of C. We now prove that E 0 is indeed a full replete reflective and coreflective subcategory of E: the embedding U : E 0 → E has adjoints C U D with x → y is null if and only if it factorizes through κy : 1 Y → y, which amounts to the existence of a morphism h making the left diagram below commutative Equivalently, f factorizes through γx : x → 1 X . Note that C 2 is not multi-pointed, in general: the counit κ need not be pointwise mono, nor the unit γ pointwise epi.
Any category of pairs over C is a full subcategory of C 2 , with a consistent choice of null objects and null morphisms.

Pre-kernels and pre-cokernels
If the category C has pullbacks and pushouts, the same holds for its category of morphisms E = C 2 , which is thus a pre-pointed category with all pre-kernels and pre-cokernels (with respect to E 0 ).
More precisely, without any hypothesis on C, it is easy to see that a morphism f : x → y in E has a pre-kernel if and only if, in the left diagram below, the pair (y, f ) has a pullback (P, u, x ): then we define Kf as the induced map Kf : X → P , and kf = (1, x ) : Kf → x. Note that f·kf is proved to be null by u Dually, the morphism f : x → y has a pre-cokernel cf if and only if, in the right diagram above, the pair (f , x) has a pushout (Q, y , v): then we define Cf as the induced map Cf : Q → Y and cf = (y , 1) : y → Cf . Again cf·f is proved to be null by v.
Therefore a pre-kernel of any morphism f : This remark clearly shows that the 'special form' (1, x ) : x → x does not give, by itself, a choice of pre-kernels: it only fixes i = 1 but leaves free the other isomorphism j. (To obtain a 'distinguished' form of pre-kernels one should also choose a distinguished form for any factorization x = x ·x in C, which cannot be done in a natural way in the usual categories of structured sets.) 4.4 Short pre-exact sequences For every object x : X → X of the pre-pointed category E = C 2 , a factorization x = x .x in C gives a short pre-exact sequence of C 2 , that is, a sequence (f, g) : • → x → • where f is the pre-kernel of g and the latter is the pre-cokernel of f Conversely, we have seen that any short pre-exact sequence of C 2 is of this type, up to isomorphism in E.
Without any hypothesis on the category C, a sequence • → x → • of the form (4.9) will be called a special short pre-exact sequence of C 2 (with respect to E 0 , the full subcategory of isomorphisms of C).

Functorial factorizations as torsion operators in categories of morphisms
We show now that a functorial factorization in a category C can be viewed as a torsion operator in the pre-pointed category E = C 2 . Such a torsion operator is algebraic, or is a torsion theory, if and only if the functorial factorization is algebraic, or is a factorization system. For the sake of uniformity, as expressed in the tables of the Introduction, we are slightly adapting the existing terminology on weak factorization systems, from [2,4,10,14,21]. 'Algebraic factorizations', in the present sense, were introduced in [14] under the name of 'natural weak factorization systems', and studied in [4,5,10] as 'algebraic weak factorization systems'. We also note that a 'functorial realisation of a weak factorization system', in the sense of [21], is intermediate between functorial and algebraic factorizations, in the present sense.
The domain and codomain functors of a category of morphisms are written as Dom and Cod.

Torsion operators and torsion theories (a) Extending
Definition 2.5, a torsion operator (ε, η) in the pre-pointed category E = (E, E 0 ) will consist of: (to.1) two endofunctors L, R : E → E and two natural transformations ε : L → 1 and η : 1 → R such that, for every object A, the pair (εA, ηA) is a short pre-exact sequence of E (with central object A) Equivalently, we can assign (to.1 ) two functors ε, η : E → E 2 with Cod.ε = idE = Dom.η, that satisfy the 'pre-exactness' condition above.
One recovers L = Dom.ε and R = Cod.η. Then, for every morphism f : A → B in E, εf and ηf are the 'naturality squares' of the transformations ε and η where (L, ε) and (R, η) are, respectively, an idempotent comonad and an idempotent monad. In other words, we are requiring that: If E is multi-pointed this agrees with the characterisation of torsion theories given at the end of 2.5.
Again this is a torsion theory when the comonad L and the monad R are idempotent (that is, when δ and µ are invertible).

Functorial factorizations
Let C be any category and E = C 2 its pre-pointed category of morphisms (see 4.2).
A functorial factorization (L, R) in the category C consists of two endofunctors L, R : C 2 → C 2 such that: Equivalently, we have a functor E : C 2 → C and two natural transformations L : Dom → E and R : E → Cod satisfying (ii).
These data can be reorganised as a pair (ε, η) of natural transformations ε : L → 1 : E → E and η : 1 → R : E → E such that, for every object x : X → X in C 2 , the pair (εx, ηx) is a short pre-exact sequence of E, and actually a special one (in the sense of 4.4) But every short exact sequence in E = C 2 is of this type, up to isomorphism. We have thus proved the following characterisation. The action of the functors L, R : Proof. The first part was proved above; the second plainly follows from 5.2(i).
(b) Then the four natural transformations ε, δ, η, µ have components as in the diagram below (c) An algebraic factorization in C amounts to -three functors L, R : C 2 → C 2 and E : C 2 → C, -two natural transformations δ : E → EL and µ : ER → E, such that, for every morphism x : X → X of C, (i) Lx : X → Ex, Rx : Ex → X , (iv) δ Lx · δ x = E(1 X , δ x) · δ x, µ x · µ Rx = µ x · E(µ x, 1 X ), Proof. Point (a) is obvious. As to (b), we already know from 5.2 that the components of ε and δ are as shown in diagram (5.8). Since the first component of ε is an identity, the counit axiom of L (see (5.5)) implies that the same holds for δ, as in the diagram. Dually for µ. For (c), conditions (i), (ii) allow us to reconstruct the natural transformations ε, δ, η, µ, as in diagram (5.8). The left part of condition (iii) is the counit axiom of L on the non-trivial components ε = R and δ , taking into account that, by (5.3): (L(εx)) = E(εx) = E(1 X , Rx).
Dually, the right-hand part of condition (iii) is the unit axiom of R on the non-trivial components η = L and µ .

The distributive law
Continuing the analysis of the algebraic case, these components δ x and µ x give raise to a natural transformation λ : LR → RL : C 2 → C 2 , with λ x = δ x and λ x = µ x Ex δ x G G The latter is automatically a distributive law of the comonad L over the monad R [1], that is, it makes the following diagrams commutative In fact, the upper part of the first diagram comes out of: (λx) · (Lηx) = δ x · (ηx) = δ x · Lx = δ x · E(1, x) = EL(1, x) · δ (idX ) = L 2 x = (ηLx) , because of the naturality of δ : E → EL over the map (1, x) : idX → x, and the unit axiom of µ, as expressed in 5.4(iii). The lower part of the same diagram is dealt in a dual way. The commutativity of the second diagram comes from the (co)associativity axioms of δ and µ.