Theory and Applications of Categories, Vol. 13, No. 15, 2004, pp. 235–251.

INTERNAL MONOTONE-LIGHT FACTORIZATION FOR CATEGORIES VIA PREORDERS

Dedicated to Aurelio Carboni on the occasion of his sixtieth birthday

JOÃO J. XAREZ

Abstract. It is shown that, for a finitely-complete category C with coequalizers of kernel pairs, if every product-regular epi is also stably-regular then there exist the reflections (R)Grphs(C) → (R)Rel(C), from (reflexive) graphs into (reflexive) relations in C, and Cat(C) → Preord(C), from categories into preorders in C. Furthermore, such a sufficient condition ensures as well that these reflections do have stable units. This last property is equivalent to the existence of a monotone-light factorization system, provided there are sufficiently many effective descent morphisms with domain in the respective full subcategory. In this way, we have internalized the monotone-light factorization for small categories via preordered sets, associated with the reflection Cat → Preord, which is now just the special case C = Set.

1. Introduction

Monotone-light factorization of morphisms in an abstract category C, with respect to a full reflective subcategory X, was studied by A. Carboni, G. Janelidze, G.M. Kelly, and R. Paré in [2].

According to [2], the existence of such factorization requires strong additional conditions on the reflection C → X, which hold in the (Galois theory of the) adjunction between compact Hausdorff and Stone spaces, needed to make the classical monotonelight factorization of S. Eilenberg (cf. [4]) and G.T. Whyburn (cf. [9]) a special case of the categorical one.

In fact, A. Carboni and R. Paré studied in categorical terms the classical monotonelight factorization, relating this factorization system with the reflection CompHaus → Stone of compact Hausdorff spaces into Stone spaces. But the connection between adjunctions and factorization systems was already known by M. Kelly (from [3]), and surprisingly the (reflective) factorization system associated to CompHaus → Stone did not coincide with the one of Carboni and Paré. The connection between the two distinct factorization systems was to be made by the categorical Galois theory of G. Janelidze: the

The author acknowledges partial financial assistance by Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT

Received by the editors 2004-05-18 and, in revised form, 2004-10-07. Published on 2004-12-05. 2000 Mathematics Subject Classification: 18A32, 12F10. Key words and phrases: (reflexive) graph, (reflexive) relation, category, preorder, factorization sys-

tem, localization, stabilization, descent theory, Galois theory, monotone-light factorization. \begin{matrix}c \\ ◯ \end{matrix} João J. Xarez, 2004. Permission to copy for private use granted.

right-hand class of the former factorization system (E′, M^{∗}) is the class of all coverings, while the right-hand class of the latter (E, M) only includes the trivial coverings.

In our Ph.D. thesis we showed that monotone-light factorization also does exist for C = Cat, the category of all categories, X being the category Preord of preordered sets (cf. [10]). A crucial observation here is that the reflection Cat → Preord has stable units in the sense of [2].

What we do in our present work is to internalize this reflection and associated monotone-light factorization system. For a given finitely-complete category C with coequalizers of kernel pairs, we start with the categories of internal categories Cat(C) and internal preorders Preord(C). Then, we give sufficient conditions for the existence of a full reflection Cat(C) → Preord(C) with stable units, and for the existence of a monotone-light factorization associated.

Hence, the reflection Cat → Preord turns out to be a special case of our construction, when C = Set. In the process, we also found that having stable units is necessary for the existence of a monotone-light factorization, in case a certain very weak condition holds, as it happens with both Cat and CompHaus (see Corollary 6.2).

Notice that there are few known similar situations where there is a (categorical) monotone-light factorization. This work provides an helpful frame for the search of new ones.

In fact, our results are also valid for internal (reflexive) graphs, which may be reflected onto the internal (reflexive) relations. This corresponds to the reflection (R)Grphs → (R)Rel for the case C = Set, presented below in Example 6.5.

2. The reflection Cat(C) → CEqRel(C) and related ones

For any finitely-complete category C, there is the category Cat(C) of categories in C. That is, the category whose objects C are the diagrams in C of the form

[figure: C = C_{1} ×_{C_{0}} C_{1}; π_{1}; γ; π_{2}; C_{1}; d_{1}; i; d_{0}; C_{0}]  (2.1)

satisfying the conditions

d_{0}i = 1_{C_{0}} = d_{1}i, d_{0}π_{1} = d_{1}π_{2}, d_{0}γ = d_{0}π_{2}, and d_{1}γ = d_{1}π_{1},

where the square represented by the second equation is a pullback and the composition operation γ satisfies the associative and unit laws. The obvious morphisms f = (f_{1}, f_{0}) :

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C → D are called internal functors and are displayed in the commutative diagram

[figure: B_{1} ×_{B_{0}} B_{1}; π_{1}; γ; π_{2}; B_{1}; d_{1}; i; d_{0}; B_{0} .; f_{1} × f_{1}; f_{1}; f_{0}; C_{1} ×_{C_{0}} C_{1}; π_{1}; γ; π_{2}; C_{1}; d_{1}; i; d_{0}; C_{0}]

There is also the adjunction

D ⊣ H : C → Cat(C),  (1)

whose left adjoint takes C in Cat(C) to D(C) = C_{0} in C, forgetting all the other structure of C (see diagram 2.1). The right adjoint takes C_{0} in C to the object

[figure: H(C_{0}) = C_{0} × C_{0} × C_{0}; π_{1}; γ; π_{2}; C_{0} × C_{0}; d_{1}; i; d_{0}; C_{0}]

in Cat(C), wherein:

• the morphisms d_{0}, d_{1} and i =< 1_{C_{0}}, 1_{C_{0}} > are respectively the product projections and the diagonal map in the product diagram

[figure: C_{0}; C_{0} × C_{0}; C_{0} ;; C_{0}; 1_{C_{0}}; 1_{C_{0}}; i; d_{0}; d_{1}]

• the morphisms π_{1} and π_{2} are the projections in the pullback diagram

[figure: C_{0} × C_{0}; C_{0} × C_{0} × C_{0}; C_{0} ;; C_{0} × C_{0}; π_{1}; π_{2}; d_{1}; d_{0}]

• the morphism γ = d_{0} × d_{1} is the one in the product diagram

[figure: C_{0} × C_{0}; C_{0}; d_{0}; π_{2}; d_{0}; C_{0} × C_{0}; C_{0} × C_{0} × C_{0}; C_{0} .; C_{0} × C_{0}; γ; π_{1}; d_{1}; d_{1}]

Hence, the adjunction 1 can be thought as the full reflection

Cat(C) → CEqRel(C),  (2)

from categories in C into connected equivalence relations in C, such that its unit is displayed by

δ_{C} = (d_{C}, 1_{C_{0}}) : C → D(C),  (3)

for each category C in C, where d_{C} =< d_{0}, d_{1} > is the morphism pictured in the product diagram

[figure: C_{0}; C_{0} × C_{0}; C_{0} .; C_{1}; d_{1}; d_{0}; d_{C}]  (2.2)

Moreover, it is easy to check that its reflector D : Cat(C) → CEqRel(C) is a left exact functor.

Therefore, one knows from [3] and [2] that the factorization system (E, M) associated with the full reflection 2 is stable. Meaning that, the class E of vertical morphisms coincides with the class E′ of stably-vertical morphisms in Cat(C); and so also the class M of trivial coverings coincides with the class M^{∗} of coverings in Cat(C). In other words, the reflective factorization system coincides with the monotone-light factorization system (E′, M^{∗}), where both classes are stable under pullbacks in Cat(C).

The class E = E′ consists of those morphisms f : C → B in Cat(C) such that D(f) = (f_{0}×f_{0}, f_{0}) is an isomorphism in CEqRel(C), i.e., of those morphisms f = (f_{1}, f_{0}) in Cat(C) such that f_{0} : C_{0} → B_{0} is an isomorphism in C.

And M = M^{∗} consists of those morphisms f = (f_{1}, f_{0}) : C → B in Cat(C) such that the commutative diagram

[figure: B; C; D(B); D(C); f; δ_{C}; D(f); δ_{B}]

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is a pullback in Cat(C). Hence, since pullbacks are calculated componentwise in Cat(C), a morphism f = (f_{1}, f_{0}) in Cat(C) belongs to M = M^{∗} if and only if the commutative diagram

[figure: B_{1}; C_{1}; B_{0} × B_{0}; C_{0} × C_{0}; f_{1}; d_{C}; f_{0} × f_{0}; d_{B}]

is a pullback in C.

The functor D : Cat(C) → CEqRel(C) and the adjunction unit δ : 1_{Cat(C)} → HD can be thought respectively as an endofunctor on Cat(C) and a natural transformation from 1_{Cat(C)} to that endofunctor D.

In this way, the fact that CEqRel(C) is a reflective full subcategory of Cat(C) is expressed equivalently by stating that the pair (D, δ) is an idempotent pointed endofunctor of Cat(C), in the sense of [7], i.e., (D, δ) is well-pointed (Dδ = δD, see [5]) and δD is an iso. The same reasoning is also valid for the simpler cases of graphs in C and reflexive graphs in C, when diagram 2.1 is reduced respectively to

[figure: C = C_{1}; d_{1}; d_{0}; C_{0}]

and

[figure: C = C_{1}; d_{1}; i; d_{0}; C_{0} ,]

with d_{0}i = 1_{C_{0}} = d_{1}i.

For these two simpler cases we have therefore the full reflections

D ⊣ H : C → Grphs(C)  (4)

and

D ⊣ H : C → RGrphs(C),  (5)

from the functor categories of graphs and reflexive graphs in C, respectively.

In the next, we shall show that, for the last two reflections 4 and 5, the natural transformation δ : 1 → D can be factorized as

δ = μη : 1 → I → D,

with (I, η) a well-pointed endofunctor, i.e., Iη = ηI.

The same happens to be true for the reflection 1 under a certain sufficient condition, which implies (for all the three reflections) that (I, η) is idempotent and gives rise to monotone-light factorization systems.

3. The well-pointed endofunctors (I, η) of (R)Grphs(C) and Cat(C)

Throughout the rest of the paper C denotes a finitely-complete category with coequalizers of kernel pairs.

We shall also use the notation (R)Grphs(C) when stating results that are valid both for the category Grphs(C) of graphs in C and for the category RGraphs(C) of reflexive graphs in C. The translation from one realm to another shall always be trivial.

Let

ker(d_{C}) = (p_{1}, p_{2})

be the kernel pair of the morphism d_{C} : C_{1} → C_{0} × C_{0} in diagram 2.2.

And let

e_{C} = coker(p_{1}, p_{2}) : C_{1} → I(C_{1})

be the coequalizer of p_{1} and p_{2}, as in the diagram

[figure: I(C_{1}); C_{0} × C_{0} ,; d_{C}; m_{C}; C_{1}; e_{C}; C_{1} ×_{C_{0}×C_{0}} C_{1}; p_{2}; C_{1}; e_{C}; p_{1}; d_{C}]  (3.1)

where m_{C} : I(C_{1}) → C_{0} × C_{0} is the unique morphism such that d_{C} = m_{C}e_{C}.

We shall call product-regular epi of the graph C in C such a morphism e_{C} : C_{1} → I(C_{1}) in C.

We will denote by I(d_{0}) and I(d_{1}) the components of m_{C} =< I(d_{0}), I(d_{1}) > in the product diagram

[figure: C_{0}; C_{0} × C_{0}; C_{0} .; I(C_{1}); I(d_{1}); I(d_{0}); m_{C}]

Next Lemma 3.1 is just an easy consequence of the fact that any kernel pair (p_{1}, p_{2}) is always the kernel pair of its coequalizer e = coker(p_{1}, p_{2}). It helps to prove the following Proposition 3.2, which introduces a certain endofunctor I on the category of (reflexive) graphs in C (and such that it may induce a reflection into the full subcategory of (reflexive) relations in C, as we shall see in next section 4).

3.1. Lemma. If m_{C}e_{C}f = m_{C}e_{C}g, with m_{C} and e_{C} the morphisms in diagram 3.1, then e_{C}f = e_{C}g, for any morphisms f and g in C.

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3.2. Proposition. There is a well-pointed endofunctor (I, η) of (R)Grphs(C), in the sense of [7] (i.e., a functor I : (R)Grphs(C) → (R)Grphs(C) and a natural transformation η : 1_{(R)Grphs(C)} → I with ηI = Iη), such that:

(a) the functor I : (R)Grphs(C) → (R)Grphs(C) takes a general morphism of (reflexive) graphs f = (f_{1}, f_{0}) : C → B to I(f) = (I(f_{1}), f_{0}) : I(C) → I(B);

(b)

[figure: I(C) = I(C_{1}); I(d_{1}); e_{C}i; I(d_{0}); C_{0} and; I(B) = I(B_{1}); I(d_{1}); e_{B}i; I(d_{0}); B_{0}]

are obtained through the process described above;

(c) the morphism I(f_{1}) in C is the one uniquely determined in the diagram

[figure: C_{1} ×_{C_{0}×C_{0}} C_{1}; p_{1}; p_{2}; C_{1}; B_{1}; f_{1}; e_{C}; e_{B}; I(B_{1}); I(C_{1}); I(f_{1}); m_{C}; m_{B}; B_{0} × B_{0} ;; C_{0} × C_{0}; f_{0} × f_{0}] (3.2)

(d) the natural transformation η : 1_{(R)Grphs(C)} → I is displayed by

η_{C} = (e_{C}, 1_{C_{0}}) : C → I(C),

for each (reflexive) graph

[figure: C = C_{1}; d_{1}; i; d_{0}; C_{0}]

in C. Proof. We only have to show that:

(i) I(f_{1}) is indeed uniquely determined in the diagram 3.2; (ii) η_{I(C)} = I(η_{C}), for every (reflexive) graph C in C.

(i) As m_{B}e_{B}f_{1} = (f_{0} × f_{0})m_{C}e_{C}, we have that (m_{B}e_{B}f_{1})p_{1} = (m_{B}e_{B}f_{1})p_{2}. Which implies, by the previous Lemma 3.1, that (e_{B}f_{1})p_{1} = (e_{B}f_{1})p_{2}.

(ii) First remark that the diagram

[figure: C_{1}; I(C_{1}); e_{C}; e_{C}; e_{I(C)}; I(I(C_{1})); I(C_{1}); I(e_{C})]

is commutative, since it is an instance of the left square in diagram 3.2.

Hence, I(e_{C}) = e_{I(C)} because e_{C} is an epimorphism.

Therefore,

I(η_{C}) = I(e_{C}, 1_{C_{0}}) = (I(e_{C}), 1_{C_{0}}) = (e_{I(C)}, 1_{C_{0}}) = η_{I(C)},

for every (reflexive) graph C in C.

3.3. Proposition. If, for every product-regular epi e_{C} in C, the obvious morphisms e_{C} ×_{C_{0}} e_{C} : C_{1} ×_{C_{0}} C_{1} → I(C_{1}) ×_{C_{0}} I(C_{1}) and e_{C} ×_{C_{0}} e_{C} ×_{C_{0}} e_{C} : C_{1} ×_{C_{0}} C_{1} ×_{C_{0}} C_{1} → I(C_{1})×_{C_{0}} I(C_{1})×_{C_{0}} I(C_{1}) are all regular epis in C, then there is a well-pointed endofunctor (I, η) of Cat(C) such that conditions (a), (b), (c) and (d), at the statement of the previous Proposition 3.2, are valid (when one considers of course only the reflexive graph part of internal categories in C).

In particular, if we just demand that every product-regular epi e_{C} : C_{1} → I(C_{1}) is a stably-regular epi in C (i.e., that any pullback p^{∗}(e_{C}) of it along any morphism p : E → I(C_{1}) in C is a regular epi), there is also such a well-pointed endofunctor (I, η) of Cat(C).

Proof. We still need to show that:

(1) for every category C in C (see diagram 2.1), there exists a unique morphism I(γ) : I(C_{1}) ×_{C_{0}} I(C_{1}) → I(C_{1}) making the square in the following diagram

[figure: p′_{1}; p′_{2}; C_{1} ×_{C_{0}} C_{1}; C_{1}; γ; e_{C} ×_{C_{0}} e_{C}; e_{C}; I(C_{1}); I(C_{1}) ×_{C_{0}} I(C_{1}); I(γ); C_{0} × C_{0}; d_{C}; m_{C}]

commute, where (p′_{1}, p′_{2}) = ker(e_{C} ×_{C_{0}} e_{C}) is the kernel pair of e_{C} ×_{C_{0}} e_{C};

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(2) the diagram

[figure: I(C) = I(C_{1}) ×_{C_{0}} I(C_{1}); I(π_{1}); I(γ); I(π_{2}); I(C_{1}); I(d_{1}); e_{C}i; I(d_{0}); C_{0}]

is a category in Cat(C), wherein I(π_{1}) and I(π_{2}) denote the obvious pullback projections.

(1) We observe then: d_{C}γ =< d_{0}, d_{1} > γ =< d_{0}γ, d_{1}γ > =< d_{0}π_{2}, d_{1}π_{1} > (since C is a category in C) =< I(d_{0})e_{C}π_{2}, I(d_{1})e_{C}π_{1} > =< I(d_{0})I(π_{2})e_{C} ×_{C_{0}} e_{C}, I(d_{1})I(π_{1})e_{C} ×_{C_{0}} e_{C} > =< I(d_{0})I(π_{2}), I(d_{1})I(π_{1}) > e_{C} ×_{C_{0}} e_{C} ⇒ d_{C}γp′_{1} = d_{C}γp′_{2} ⇒ e_{C}γp′_{1} = e_{C}γp′_{2} (by Lemma 3.1) ⇒ there exists one and only one morphism I(γ) such that

I(γ)e_{C} ×_{C_{0}} e_{C} = e_{C}γ,

since e_{C} ×_{C_{0}} e_{C} is a regular epi by hypothesis.

(2) We have by definition that

I(d_{0})(e_{C}i) = 1_{C_{0}} = I(d_{1})(e_{C}i) and I(d_{0})I(π_{1}) = I(d_{1})I(π_{2}).

Then, we observe: m_{C}I(γ)e_{C} ×_{C_{0}} e_{C} = d_{C}γ =< d_{0}π_{2}, d_{1}π_{1} > (since C is a category in C) =< I(d_{0})I(π_{2}), I(d_{1})I(π_{1}) > e_{C} ×_{C_{0}} e_{C} ⇒ m_{C}I(γ) =< I(d_{0})I(π_{2}), I(d_{1})I(π_{1}) >, since by hypothesis e_{C} ×_{C_{0}} e_{C} is an epi. So, there is only left to prove that I(γ) satisfies the associative and unit laws.

The unit law holds for I(γ) if in the following obvious commutative diagrams

[figure: C_{0} ×_{C_{0}} C_{1}; C_{1} ×_{C_{0}} C_{1}; i × 1; 1 ×_{C_{0}} e_{C}; e_{C} ×_{C_{0}} e_{C}; e_{C}; I(C_{1}) ×_{C_{0}} I(C_{1}); C_{0} ×_{C_{0}} I(C_{1}); I(i) × 1; C_{1}; γ; I(γ); I(C_{1}); C_{1} ×_{C_{0}} C_{0}; C_{1} ×_{C_{0}} C_{1}; 1 × i; e_{C} ×_{C_{0}} 1; e_{C} ×_{C_{0}} e_{C}; e_{C}; I(C_{1}) ×_{C_{0}} I(C_{1}); I(C_{1}) ×_{C_{0}} C_{0}; 1 × I(i); C_{1}; γ; I(γ); I(C_{1})]

where I(i) stands for e_{C}i, the arrows 1 ×_{C_{0}} e_{C} and e_{C} ×_{C_{0}} 1 are regular epis.

But this is indeed so because 1 ×_{C_{0}} e_{C} and e_{C} ×_{C_{0}} 1 are isomorphic to e_{C}, as it is easy to check by drawing the respective pullback diagrams.

The associative law holds for I(γ) if, in the obvious commutative diagram

[figure: C_{1} ×_{C_{0}} C_{1} ×_{C_{0}} C_{1}; C_{1} ×_{C_{0}} C_{1}; γ × 1; 1 × γ; C_{1}; γ; e_{C} ×_{C_{0}} e_{C} ×_{C_{0}} e_{C}; e_{C} ×_{C_{0}} e_{C}; e_{C}; I(C_{1}) ×_{C_{0}} I(C_{1}); I(C_{1}) ×_{C_{0}} I(C_{1}) ×_{C_{0}} I(C_{1}); 1 × I(γ); I(γ) × 1; I(γ); I(C_{1}) ,]

where γ(1×γ) = γ(γ×1) because C is a category in C, the morphism e_{C}×_{C_{0}}e_{C}×_{C_{0}}e_{C} is a regular epi.

At last, it is easy to check that if e_{C} is a stably-regular epi then e_{C} ×_{C_{0}} e_{C} and e_{C} ×_{C_{0}} e_{C} ×_{C_{0}} e_{C} are also stably-regular epis (one should use the well-known fact that the class of stably-regular epis is closed under composition).

4. The reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C)

In this section we are going to investigate under which conditions the well-pointed endofunctors of the last section become reflections into the relations and preorders in C, respectively.

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4.1. Proposition. Consider the pullback diagram

[figure: I(C_{1}) ×_{C_{0}×C_{0}} I(C_{1}); I(C_{1}); I(C_{1}); C_{0} × C_{0} ,; m_{C}; m_{C}; p′_{1}; p′_{2}; C_{1}; e_{C}; C_{1} ×_{C_{0}×C_{0}} C_{1}; e_{C} ×_{C_{0}×C_{0}} e_{C}; p_{2}; C_{1}; e_{C}; p_{1}]  (4.1)

where the morphisms p_{1}, p_{2}, e_{C} and m_{C} are those in diagram 3.1.

Then, the following three conditions are equivalent:

(a) the morphism m_{C} : I(C_{1}) → C_{0} × C_{0} is monic in C;

(b) the morphism e_{C} ×_{C_{0}×C_{0}} e_{C} : C_{1} ×_{C_{0}×C_{0}} C_{1} → I(C_{1}) ×_{C_{0}×C_{0}} I(C_{1}) is epi in C;

(c) the morphism e_{C} ×_{C_{0}×C_{0}} e_{C} : C_{1} ×_{C_{0}×C_{0}} C_{1} → I(C_{1}) ×_{C_{0}×C_{0}} I(C_{1}) is a regular epi in C.

In particular, if the product-regular epi e_{C} : C_{1} → I(C_{1}) is a stably-regular epi in C then the morphism m_{C} : I(C_{1}) → C_{0} × C_{0} is monic in C.

Proof. If m_{C} is monic in C then the pullback projections p′_{1} and p′_{2} in diagram 4.1 are isos, since they constitute its kernel pair. Therefore, we have that e_{C} ×_{C_{0}×C_{0}} e_{C} is for instance isomorphic to e_{C}p_{2}. It follows that e_{C} ×_{C_{0}×C_{0}} e_{C} is a regular epi in C, since e_{C} is a regular epi and p_{2} is a split epi.

On the other way, let us assume that e_{C} ×_{C_{0}×C_{0}} e_{C} is an epimorphism. By Lemma 3.1 we know that e_{C}p_{2} = e_{C}p_{1}, so that the pullback projections p′_{1} and p′_{2} are identical. It follows that m_{C} is monic in C.

Finally, we are going to suppose that e_{C} : C_{1} → I(C_{1}) is a stably-regular epi. Observe then that e_{C} ×_{C_{0}×C_{0}} e_{C} is the composite of the morphisms 1×_{C_{0}×C_{0}} e_{C} : I(C_{1})×_{C_{0}×C_{0}} C_{1} → I(C_{1}) ×_{C_{0}×C_{0}} I(C_{1}) and e_{C} ×_{C_{0}×C_{0}} 1 : C_{1} ×_{C_{0}×C_{0}} C_{1} → I(C_{1}) ×_{C_{0}×C_{0}} C_{1}. Each of these morphisms is a pullback of e_{C}, and therefore stably-regular in case e_{C} is so.

The next theorem follows trivially from the above considerations.

4.2. Theorem. The well-pointed endofunctors given at Proposition 3.2 and Proposition 3.3 induce respectively the reflections (R)Grphs(C) → (R)Rel(C), from the category of (reflexive) graphs in C to its full subcategory of (reflexive) relations in C, and Cat(C) → Preord(C), from the category of categories in C to its full subcategory of preorders in C, if and only if for every product-regular epi e_{C} : C_{1} → I(C_{1}) in C the associated morphism e_{C} ×_{C_{0}×C_{0}} e_{C} in diagram 4.1 is epi in C.

In particular, for the existence of these reflections it is sufficient that all product-regular epis are stably-regular epis in C.

5. The stabilization of (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C)

The following Proposition 5.1 states that if every product-regular is stably-regular then the reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C) do have stable units in the sense of [3] and [2].

5.1. Proposition. Every unit morphism η_{C} = (e_{C}, 1_{C_{0}}) : C → I(C) is stablyvertical in (R)Grphs(C) or Cat(C) (in the sense of [7]; see section 2), provided that all product-regular epis e_{B} : B_{1} → I(B_{1}) are stably-regular in C.

Proof. We have to show that any pullback g^{∗}(η_{C}) = (g_{1}^{∗}(e_{C}), g_{0}^{∗}(1_{C_{0}})) : P → B of η_{C} = (e_{C}, 1_{C_{0}}) : C → I(C), along any morphism g = (g_{1}, g_{0}) : B → I(C), is vertical.

As pullbacks in (R)Grphs(C) and Cat(C) are calculated pointwise, the question amounts to ask if the morphism I(g_{1}^{∗}(e_{C})) in the following diagram

[figure: P_{1}; B_{1}; g_{1}^{∗}(e_{C}); e_{P}; e_{B}; I(B_{1}); I(P_{1}); I(g_{1}^{∗}(e_{C})); m_{P}; m_{B}; B_{0} × B_{0}; P_{0} × P_{0}; g_{0}^{∗}(1_{C_{0}}) × g_{0}^{∗}(1_{C_{0}})]  (5.1)

is invertible in C.

Remark that diagram 5.1 is analogous to diagram 3.2 and that within it:

• g_{1}^{∗}(e_{C}) stands for the pullback of e_{C} along g_{1} : B_{1} → I(C_{1}) in C;

• g_{0}^{∗}(1_{C_{0}}) is the pullback of the identity morphism 1_{C_{0}} along g_{0} in C, and so is an iso in C as well as g_{0}^{∗}(1_{C_{0}}) × g_{0}^{∗}(1_{C_{0}});

• P_{1} = B_{1} ×_{I(C_{1})} C_{1} and P_{0} ≅ B_{0}.

We then observe:

the morphisms g_{1}^{∗}(e_{C}), e_{B} and e_{P} are stably-regular epis in C ⇒

(since the class of stably-regular epis is known to be closed under composition) the morphisms e_{B}g_{1}^{∗}(e_{C}) and e_{P} are stably-regular epis in C ⇒

(since e_{B}g_{1}^{∗}(e_{C}) = I(g_{1}^{∗}(e_{C}))e_{P}) the morphisms I(g_{1}^{∗}(e_{C}))e_{P} and e_{P} are stably-regular epis in C ⇒

(since the stably-regular epis are known to have the strong right cancellation property) the morphism I(g_{1}^{∗}(e_{C})) is a stably-regular epi.

Furthermore, I(g_{1}^{∗}(e_{C})) is also monic in C, since m_{B}I(g_{1}^{∗}(e_{C})) ≅ m_{P} and m_{P} : I(P_{1}) → P_{0} × P_{0} is monic in C by Proposition 4.1. Being simultaneously monic and regular epi, the morphism I(g_{1}^{∗}(e_{C})) must be invertible in C.

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The fact that both reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C) have stable units, is known to induce respectively a reflective factorization system on (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C) (cf. [2]).

Corollary 5.2 singles out the already stated consequences of product-regular epis being as well stably-regular.

5.2. Corollary. If all product-regular epis are as well stably-regular epis in C, then there are reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C), from the category of (reflexive) graphs to the category of relations in C and from the category of categories to the category of preorders in C, which do have stable units (in the sense of [3] and [2]). Therefore these reflections induce reflective factorization systems (E, M) on (R)Grphs(C) and Cat(C) such that (cf. [2]):

• E is the class of vertical morphisms, in the sense of [7];

• M is the class of trivial coverings, in the sense of categorical Galois theory.

5.3. Examples. If C is a topos (for instance, C = Set), an abelian category (C = Ab), Barr-exact like C = Grps, or only regular (with finite limits!), then C is in the conditions of last Corollary 5.2. Therefore, in all these examples, there are reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C) with stable units: indeed, in all these cases, regular epimorphisms are also stably-regular.

6. The monotone-light factorization systems for the reflections (R)Grphs(C) → (R)Rel(C) and Cat(C) → Preord(C)

In this section we shall be interested in sufficient conditions for the existence of monotonelight factorization systems (E′, M^{∗}) associated with the reflections of last section.

E′ denotes the class of stably-vertical morphisms, and M^{∗} the class of coverings, either in (R)Grphs(C) or in Cat(C).

A morphism f = (f_{1}, f_{0}) : C → B belongs to M^{∗} if and only if there exists an effective descent morphism (abbr. e.d.m.) p = (p_{1}, p_{0}) : E → B such that the pullback p^{∗}(f) of f along p is in the class of trivial coverings M.

We start by recalling the main result of [2], and then derive from it a corollary giving some conditions under which the existence of a monotone-light factorization system can be identified with the stable units property.

6.1. Theorem. [2] Let A be a finitely-complete category and (E, M) a factorization system on A. Then, the pair (E′, M^{∗}), obtained by simultaneously stabilizing E and localizing M is a factorization system if and only if every morphism α : A → B in A is locally stable (i.e., there is an e.d.m. p : E → B in A such that, in the (E, M)-factorization p^{∗}(α) = me, of the pullback p^{∗}(α) : E ×_{B} A → E of α along p, the morphism e belongs to E′).

6.2. Corollary. Let A be a finitely-complete category, X a full reflective subcategory of A, and (E, M) the prefactorization system on A associated to X (cf. [2]).

Consider now the following two hypothesis (1), (2) and the two statements (a), (b):

(1) the unique morphism 1 : 1 → 1 between terminal objects in A is projective with respect to effective descent morphisms (i.e., for every e.d.m. p : E → 1 there is a morphism q : 1 → E in A such that 1 = pq);

(2) for each object B in A, there is an object E in X and an e.d.m. p : E → B from E to B in A;

(a) (E, M) and (E′, M^{∗}) are factorization systems;

(b) the reflection A → X has stable units (in the sense of [3] and [2]).

If (1) holds then (a) implies (b).

If (2) holds then (b) implies (a).

Remark also that:

• under hypotheses (1) and (2) the statements (a) and (b) are equivalent;

• the monotone-light factorization system (E′, M^{∗}) may coincide with the reflective one (E, M), being therefore crucial to establish whether the class of morphisms E is stable under pullbacks or not.

Proof. We shall first prove that under hypothesis (1) the statement (a) implies (b). The reader should refer to the following pullback diagram in A:

[figure: A; HI(A); η_{A}; f; E × A; π_{1}; 1; !_{HI(A)}; E; A; 1 .; p; π_{2}; !_{A}]

The unique morphism !_{A}, from any object A to the terminal object 1 in A, can be factorized through the unit morphism η_{A} as

!_{A} =!_{HI(A)}η_{A} : A → HI(A) → 1,

where H and I denote respectively the inclusion and the reflector functor.

We know that η_{A} is vertical, i.e., it is in E; and that !_{HI(A)} is a trivial covering, i.e., it is in M, since

!_{HI(A)} = H(!_{I(A)}) : HI(A) → H(1) = 1.

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By the preceding Theorem 6.1, there is an e.d.m. p : E → 1 such that the (E, M)- factorization of the product projection π_{1} : E × A → A is stable, i.e., π_{1} = me with e a stably-vertical morphism (e ∈ E′). But then, the pullback f^{∗}(π_{1}) of π_{1} along the morphism f : 1 → E such that 1 = pf (f exists by hypothesis (1)), also has a stable (E, M)-factorization (M and E′ are both pullback stable classes).

Finally, remark that the morphism f^{∗}(π_{1}) is isomorphic to !_{A} : A → 1. Therefore, as the (E, M)-factorization is unique up to isomorphism, we conclude that η_{A} is a stablyvertical morphism.

In order to prove that under hypothesis (2) the statement (b) implies (a), we consider for any morphism α : A → B in A the pullback diagram

[figure: E; E ×_{B} A; B ,; A; π_{1}; α; p; π_{2}]

where p is an e.d.m. such that E is in X (it exists by hypothesis (2)).

The projection π_{1} in the previous diagram can be factorized through the unit morphism η_{E×_{B}A} as

π_{1} = π′_{1}η_{E×_{B}A} : E ×_{B} A → HI(E ×_{B} A) → E.

Hence, as π′_{1} belongs to M and η_{E×_{B}A} is stably-vertical by statement (b), we just proved that every morphism α : A → B in A is locally stable (cf. Theorem 6.1). Since the reflection has stable units, it follows that (E, M) is a factorization system (cf. [2]).

Now, we can state, in the next Corollary 6.3, sufficient conditions for the existence of monotone-light factorization systems on (R)Grphs(C) and Cat(C).

Parentheses will be used to state simultaneously the three cases.

6.3. Corollary. There is a full reflection (R)Grphs(C) → (R)Rel(C) (Cat(C) → Preord(C)) with stable units, and giving rise to a monotone-light factorization system on (R)Grphs(C) (Cat(C)), provided the following two conditions hold:

(a) all product-regular epis are stably-regular epis in C;

(b) for each (reflexive) graph (category) B in C there is respectively a (reflexive) relation (category) E in C and an e.d.m. p = (p_{1}, p_{0}) : E → C in (R)Grphs(C) (Cat(C)) from E to C.

6.4. Remark. The morphism p = (p_{1}, p_{0}) is an e.d.m. in (R)Graphs(C) in case p_{0} and p_{1} are effective descent morphisms in C.

For the case of reflexive graphs demanding that p_{0} and p_{1} are effective descent morphisms in C amounts to ask the first component p_{1} of p to be an e.d.m. in C.

In effect, being p_{1}, d′_{0} and d_{0} effective descent morphisms, p_{0}d_{0} = d′_{0}p_{1} is an e.d.m.. Hence, p_{0} must also be an e.d.m., by the strong right cancellation property for effective descent morphisms.

6.5. Example. For C = Set we have that:

• Cat(Set) = Cat, the category of all small categories;

• Preord(Set) = Preord, the category of preordered sets;

• (R)Grphs(Set) = (R)Grphs, the category of (reflexive) graphs;

• (R)Rel(Set) = (R)Rel, the category of (reflexive) relations.

The reflection Cat → Preord was studied in [10]. There, we concluded that it verifies hypotheses (1) and (2) of Corollary 6.2, and that it has stable units. The monotone-light factorization system associated was also shown to be non-trivial, i.e., (E′, M^{∗}) ≠ (E, M).

Being (R)Grphs a topos, its effective descent morphisms coincide with the epimorphisms, i.e., with surjections on arrows. Then, it is easy to verify that hypotheses (1) and (2) of Corollary 6.2 hold for the reflection (R)Grphs → (R)Rel. Using the same techniques of [10] for Cat → Preord, one concludes that the reflection (R)Graphs → (R)Rel also has stable units, and one easily characterizes the classes E′ and M^{∗} of morphisms in (R)Graphs with respect to this reflection:

• a morphism α : A → B in (R)Graphs is stably-vertical (α ∈ E′) if and only if it is bijective on objects (the vertices!) and surjective on arrows;

• a morphism α : A → B in (R)Graphs is a covering (α ∈ M^{∗}) if and only if it is faithful (on arrows!).

In fact, the reflection (R)Grphs → (R)Rel is in every aspect an extension of the studied reflection Cat → Preord, and so we shall give no further details.

We want also to mention the counter-example corresponding to the reflection RGrphs → Preord. Which although admissible (also called semi-left-exact), and satisfying hypotheses (1) and (2) of Corollary 6.2, has not stable units, and therefore it does not have a monotone-light factorization system.

References

[1] Borceux, F. Handbook of categorical algebra 2, Cambridge University Press, 1994.

[2] Carboni, A., Janelidze, G., Kelly, G. M., Paré, R. On localization and stabilization for factorization systems. App. Cat. Struct. 5, (1997) 1–58.

[3] Cassidy, C., Hébert, M., Kelly, G. M. Reflective subcategories, localizations and factorization systems. J. Austral. Math. Soc. 38A (1985) 287–329.

[4] Eilenberg, S. Sur les transformations continues d’espaces métriques compacts. Fundam. Math. 22 (1934) 292–296.

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[5] Kelly, G.M. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on Bull. Austral. Math. Soc. 22 (1980) 1–83.

[6] Janelidze, G., Sobral, M., Tholen, W. Beyond Barr Exactness: Effective Descent Morphisms in Categorical Foundations. Special Topics in Order, Topology, Algebra and Sheaf Theory, Cambridge University Press, 2004.

[7] Janelidze, G., Tholen, W. Functorial factorization, well-pointedness and separability J. Pure Appl. Algebra 142 (1999) 99–130.

[8] Mac Lane, S. Categories for the Working Mathematician, 2nd ed., Springer, 1998.

[9] Whyburn, G. T. Non-alternating transformations. Amer. J. Math. 56 (1934) 294– 302.

[10] Xarez, J. J. The monotone-light factorization for categories via preorders. Galois theory, Hopf algebras and semiabelian Categories, 533–541, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004.

Departamento de Matemática, Universidade de Aveiro. Campus Universitário de Santiago. 3810-193 Aveiro. Portugal.

Email: jxarez@mat.ua.pt

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