Theory and Applications of Categories, Vol. 15, No. 7, 2006, pp. 178–193.

A GALOIS THEORY WITH STABLE UNITS FOR SIMPLICIAL SETS

JOÃO J. XAREZ

Abstract. We recall and reformulate certain known constructions, in order to make a convenient setting for obtaining generalized monotone-light factorizations in the sense of A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. This setting is used to study the existence of monotone-light factorizations both in categories of simplicial objects and in categories of internal categories. It is shown that there is a non-trivial monotone-light factorization for simplicial sets, such that the monotone-light factorization for reflexive graphs via reflexive relations is a special case of it, obtained by truncation. More generally, we will show that there exists a monotone-light factorization associated with every full subcategory Mono(F_{n}), n ≥ 0, consisting of all simplicial sets whose unit morphisms are monic for the localization F_{n} : Set^{Δ^{op}} → Set^{Δ_{n}^{op}}, which truncates each simplicial set after the object of n-simplices. The monotone-light factorization for categories via preorders is as well derived from the proposed setting. We also show that, for regular Mal’cev categories, the reflection of internal groupoids into internal equivalence relations necessarily produces monotone-light factorizations. It turns out that all these reflections do have stable units, in the sense of C. Cassidy, M. Hébert and G. M. Kelly, giving rise to Galois theories.

1. Introduction

Essentially every reflection C → X, from a category C into its full subcategory X, gives rise to a factorization system (E, M). Then, by respectively stabilizing and localizing the classes E and M of morphisms in C, in the sense of [1], one may obtain another one (E′, M^{∗}), to be called a monotone-light factorization system. The main result of [1] gives a necessary and sufficient condition for what seems to be a quite rare occurrence. Recall also from [1], that M^{∗} is exactly the class of covering morphisms in the sense of Galois theory of G. Janelidze.

In my PhD thesis [10] (see also [11]), I studied in particular the reflection Cat → Preord from categories into preordered sets. It proved to be another example giving a Galois theory for the category Cat of all categories and a non-trivial monotone-light factorization (i.e., (E′, M^{∗}) ≠ (E, M)). A unit morphism η_{A} : A → I(A) of this reflection is the coequalizer of the kernel pair of another unit morphism φ_{A} : A → F (A), associated to

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

Received by the editors 2004-11-05 and, in revised form, 2006-06-22. Published on 2006-07-07 in the volume of articles from CT2004. 2000 Mathematics Subject Classification: 18A32, 18A40, 18G30, 12F10, 55U10, 08B05, 18B25. Key words and phrases: simplicial object, simplicial set, internal category, internal preorder, regular

category, Mal’cev category, descent theory, Galois theory, reflection with stable units, monotone-light factorization, Kan extension, elementary topos, geometric morphism.

\begin{matrix}c \\ ◯ \end{matrix} João J. Xarez, 2006. Permission to copy for private use granted.

the reflection of categories into indiscrete categories, as displayed in the pullback diagram 2.1.

In the present paper we study this coequalizer of the kernel pair process in a more general fashion, by beginning not with a reflection but simply with a pointed endofunctor (F, φ), i.e., a natural transformation φ : 1_{C} → F, from the identity functor of a category C to an endofunctor F : C → C. We impose then additional conditions, under which the regular epimorphisms η_{A} : A → I(A) define a reflection (I, η). After that we give sufficient conditions for (I, η) to have stable units, in the sense of [2] and [1].

In another instance of this process, the pointed endofunctor (F, φ), associated with the reflection RGrphs → LEqRel of reflexive graphs into connected equivalence relations, gives rise to the reflection of reflexive graphs into reflexive relations. This reflection RGrphs → RRel has stable units, as follows from Corollary 4.4 (see Section 4).

It is known that if a reflection (I, η) has stable units it is necessarily admissible in the sense of categorical Galois theory [5], also called semi-left exact in [2]. There is therefore a Galois theory for reflexive graphs via reflexive relations.

Furthermore, if a reflection C → X from a finitely-complete category C has stable units then there exists an associated monotone-light factorization in C, provided that for each object B there is an effective descent morphism p : E → B in C such that its domain E belongs to the full subcategory X (see Corollary 6.2 in [9], which follows from the main result of [1]). This is really the case for both Cat → Preord and RGrphs → RRel.

The two reflections just considered are of course respectively a special case of Cat(S) → Preord(S), categories in S into preorders in S, and RGrphs(S) → RRel(S), reflexive graphs in S into reflexive relations in S, when S = Set. Sufficient conditions on S for successful internalizations of the former reflections were given in [9]. G. Janelidze suggested me to extend the results obtained in that paper [9] to simplicial objects, looking at them as higher dimensional graphs, with possible future applications in homotopy theory. And so I did, applying the coequalizer of the kernel pair process to the reflection of the category of simplicial sets Smp into its full subcategory of the nerves of indiscrete categories; and obtaining as well a new reflection with stable units and a non-trivial monotone-light factorization, into the category OSmC of ordered simplicial complexes. Note that there is a non-trivial monotone-light factorization system associated to the reflection Smp_{n} → OSmC_{n}, for each integer n ≥ 1. This reflection is just a special case of Smp → OSmC after truncating the simplicial sets right after the object of nsimplices. The previous reflection RGrphs → RRel is therefore simply the special case Smp_{1} → OSmC_{1} having n = 1.

More generally, we could begin with the localizations F_{n} : Smp → Smp_{n}, n ≥ 0 (n = 0 corresponding to the reflection of simplicial sets into the nerves of indiscrete categories analyzed in the last paragraph), apply the coequalizer of the kernel pair process and obtain reflections with stable units and monotone-light factorizations for Smp. In fact, these localizations F_{n} are examples of geometric morphisms F^{∗} ⊣ F_{∗} : E → F between elementary topoi which are embeddings having Mono(F^{∗}) dense in F (i.e., every object of F is a colimit of objects of Mono(F^{∗})). This turns out to be enough to conclude that there is a reflection I : F → Mono(F^{∗}) with stable units and a monotone-light factorization (see Proposition 7.3).

If S is a Mal’cev variety of universal algebras then every internal category in S is a groupoid. Hence, the above-mentioned process produces in such a case a reflection with stable units Grpd(S) → EqRel(S), from internal groupoids into equivalence relations. As M. Gran pointed out to me, for each internal groupoid G in S, there is an internal functor (σ, d_{1}) : Eq(d_{0}) → G from an equivalence relation (see Example 5.2). This guarantees the existence of monotone-light factorizations for internal groupoids via equivalence relations, exactly as in the other reflections studied. The category of groupoids in groups is known to be equivalent to the category of crossed modules, Grpd(Grp) ≃ CrossMod. So, in particular, there is a monotone-light factorization for crossed modules via normal subgroup inclusions.

2. The coequalizer of a pointed endofunctor’s kernel pair is well-pointed

Throughout all this paper, (F, φ) will denote a pointed endofunctor on a finitely-complete category C, such that for every object A in C the kernel pair of φ_{A} : A → F (A) has a coequalizer. That is:

• φ : 1_{C} → F is a natural transformation from the identity functor of a finitelycomplete category C to the endofunctor F : C → C;

• for every object A in C, the kernel pair of the morphism φ_{A} does have a coequalizer η_{A} = coeq(ker(φ_{A})), and the morphisms involved will be displayed as

[figure: I(A); F (A) .; φ_{A}; μ_{A}; A; η_{A}; A ×_{F (A)} A; π_{2,A}; A; η_{A}; π_{1,A}; φ_{A}]  (2.1)

Our next lemma follows immediately from the fact that, in the category C^{C} of endofunctors on C, both the kernel pair (π_{1}, π_{2}) of φ and its coequalizer η = coeq(ker(φ)) : 1_{C} → I are computed pointwise.

2.1. Lemma. Diagram 2.1 defines a pointed endofunctor (I, η) on C.

2.2. Proposition. The pair (I, η) obtained as above is a well-pointed endofunctor in the sense of [7], i.e., Iη = ηI.

Proof. By naturality of η, I(η_{A})η_{A} = η_{I(A)}η_{A}. Hence, being η_{A} an epimorphism, one obtains I(η_{A}) = η_{I(A)} for each A in C.

2.3. Example. Consider the category Smp(S) = S^{Δ^{op}} of simplicial objects in S, where Δ is the category of positive ordinal numbers [n] (n ≥ 0).

If S is a finitely-complete category with coequalizers of kernel pairs, so is Smp(S), since it is well known that in any functor category the limits and colimits can be calculated pointwise. Hence, by the results above, one can state that, for any pointed endofunctor (F, φ) on Smp(S), the pair (I, η) obtained by the coequalizer of the kernel pair process is well-pointed, i.e., Iη = ηI.

2.4. Example. We are now beginning the analysis of an example already studied in [9]. In what follows, we will just adapt the results known for this case to the convenient setting introduced in the present paper. More details are given in [9].

For any finitely-complete category S there is the category Cat(S) of categories in S. That is, the category whose objects are the diagrams in S 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}]  (2.4)

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.

Consider now, for a finitely-complete category S with coequalizers of kernel pairs, the pointed endofunctor (F, φ) on Cat(S) for which:

[figure: F (C) = C_{0} × C_{0} × C_{0}; C_{0} × C_{0}; C_{0}]  (2.5)

where C_{0} is the object of objects of C, and the morphisms are the obvious ones between the powers of C_{0}; for every C in Cat(S), φ_{C} = (d_{C}, 1_{C_{0}}) : C → F (C), where d_{C} = ⟨d_{0}, d_{1}⟩ is the morphism determined by the commutative diagram

[figure: C_{0}; C_{0} × C_{0}; C_{0} .; C_{1}; d_{1}; d_{0}; d_{C}]  (2.6)

By the results above, in order to conclude that there is a well-pointed endofunctor (I, η) obtained by the coequalizer of the kernel pair process, one needs only to show that the coequalizer η_{C} of the kernel pair of every internal functor φ_{C} does exist in Cat(S).

Sufficient conditions for the existence of such coequalizers η_{C} were given in [9, Proposition 3.3]. In particular, if the coequalizer e_{C} : C_{1} → I(C)_{1} of the kernel pair (p_{C}, q_{C}) of d_{C} : C_{1} → C_{0} × C_{0} is a pullback stable regular epimorphism in S, then the following diagram displays the coequalizer η_{C} = (e_{C}, 1_{C_{0}}) of the kernel pair of φ_{C} = (d_{C}, 1_{C_{0}}) in Cat(S):

[figure: (C_{1} ×_{C_{0}×C_{0}} C_{1}) ×_{C_{0}} (C_{1} ×_{C_{0}×C_{0}} C_{1}); C_{1} ×_{C_{0}×C_{0}} C_{1}; C_{0}; p_{C} × p_{C}; q_{C} × q_{C}; p_{C}; q_{C}; 1_{C_{0}}; C_{1} ×_{C_{0}} C_{1}; π_{1}; γ; π_{2}; C_{1}; d_{1}; i; d_{0}; C_{0}; 1_{C_{0}}; e_{C} × e_{C}; e_{C}; I(C)_{1} ×_{C_{0}} I(C)_{1}; π_{1}^{I(C)}; γ^{I(C)}; π_{2}^{I(C)}; I(C)_{1}; d_{1}^{I(C)}; e_{C}i; d_{0}^{I(C)}; C_{0} .] (2.7)

3. Idempotency of (I, η)

It is known that a pointed endofunctor (I, η) on C is idempotent (i.e., Iη = ηI and ηI is an isomorphism) if and only if the full replete subcategory F ix(I, η) of (I, η)-fixed objects in C is reflective in C with reflection η (an object A is (I, η)-fixed if η_{A} is an isomorphism; see [6]).

Proposition 3.1 below states that, if (I, η) is the well-pointed endofunctor obtained from the pointed endofunctor (F, φ) by the coequalizer of the kernel pair process, then F ix(I, η) is equal to Mono(F, φ), the full subcategory of C formed by the objects A with φ_{A} monic. It follows trivially that ηI is an isomorphism if and only if φI is a monomorphism in C^{C}. This new characterization of the idempotency of the well-pointed endofunctor (I, η) will be given in Corollary 3.2.

3.1. Proposition. Consider the well-pointed endofunctor (I, η) on C obtained from a pointed endofunctor (F, φ), through the coequalizer of the kernel pair process displayed in diagram 2.1.

Then, the two full subcategories F ix(I, η) and Mono(F, φ) of C are identical, F ix(I, η) = Mono(F, φ).

Proof. Consider the pullback diagram 2.1.

If φ_{A} is a monomorphism then η_{A} must also be a monomorphism. This implies that η_{A} is an isomorphism, since it is in addition a regular epimorphism.

Conversely, if η_{A} is an isomorphism then π_{1,A} = π_{2,A} is an isomorphism, since η_{A} is the coequalizer of the kernel pair (π_{1,A}, π_{2,A}). Therefore, φ_{A} is a monomorphism provided η_{A} is an isomorphism.

3.2. Corollary. The endofunctor (I, η) on C, obtained by the process displayed in diagram 2.1, is idempotent, in the sense of [7] and [6], if and only if φ_{I(A)} is a monomorphism for each object A in C.

Corollaries 3.3 and 3.4 below will be useful in the examples.

3.3. Corollary. If μ is a monomorphism then (I, η) is idempotent provided either F φ or F η is a monomorphism.

Proof. Let us suppose first that both μ and F η are monomorphisms. By naturality of φ one has F (η_{A})φ_{A} = φ_{I(A)}η_{A} for each object A in C. Therefore F (η_{A})μ_{A} = φ_{I(A)}, since φ_{A} = μ_{A}η_{A} and η_{A} is a regular epimorphism. Hence, φ_{I(A)} is a monomorphism because it is the composite of two monomorphisms, for each object A in C. It follows then from Corollary 3.2 that (I, η) is idempotent.

To complete this proof, one needs only to note that if the composite F φ = F μ · F η is a monomorphism then F η is necessarily also a monomorphism.

3.4. Corollary. If C is a regular category or, more generally, admits a (regular epi, mono)-factorization, and (F, φ) is idempotent then (I, η) is idempotent.

Proof. Since C is regular, μ is a monomorphism because φ_{A} = μ_{A}η_{A} is a regular epimono factorization. F φ is an isomorphism since (F, φ) is idempotent. Hence, (I, η) is idempotent by Corollary 3.3.

3.5. Example. For any finitely-complete category S with coequalizers of kernel pairs, consider the pointed endofunctor (F, φ) on Smp(S) = S^{Δ^{op}} for which

[figure: F (A) = ⋯ A_{0} × A_{0} × A_{0}; A_{0} × A_{0}; A_{0} .]  (3.1)

The 0-component of φ(A) : A → F (A) is the identity morphism 1_{A_{0}} : A_{0} → A_{0}, as shown in the diagram

[figure: ⋯ A_{2}; A_{1}; A_{0}; 1_{A_{0}}; d_{A}; ⋯; d_{A} × d_{A}; ⋯ A_{0} × A_{0} × A_{0}; A_{0} × A_{0}; A_{0} .]  (3.2)

Then, one knows that the pair (I, η) is a well-pointed endofunctor, by the reasons given in Section 2.

Remark now that:

• F (A) and F (F (A)) are the same simplicial objects, since their objects of points are the same, (F (A))_{0} = A_{0} = (F (F (A)))_{0}, for every simplicial object A;

• F (φ_{A}) is the identity morphism of F (A) for every simplicial object A, since (F (φ_{A}))_{0} = (φ_{A})_{0} = 1_{A_{0}} (any morphism of simplicial objects with codomain F (A) is completely determined by its 0-component!).

So, according to Corollary 3.3, one has only to show that every morphism μ_{A} in Smp(S) is a monomorphism in order to prove that (I, η) is idempotent. But that is the case when S admits a (regular epi, mono)-factorization, for then the factorization φ_{A} = μ_{A}η_{A} is calculated pointwise using the regular epi-mono factorizations in S.

One could as well arrive to the same conclusion, for a category S which is regular or, more generally, admits a (regular epi, mono)-factorization, by using only Corollary 3.4: a functor category like Smp(S) admits regular epi-mono factorizations (respectively, Smp(S) is regular) whenever S admits regular epi-mono factorizations (respectively, S is regular); it is also easy to show that (F, φ) is idempotent in this example.

3.6. Example. Let us consider again, for a finitely-complete category S with coequalizers of kernel pairs, the pointed endofunctor (F, φ) on Cat(S) of Example 2.4. It is also required for each internal category C in S that the coequalizer η_{C} of the kernel pair of φ_{C} does exist in Cat(S), which is known to be the case when in particular S is a regular category (see [9, Proposition 3.3]).

Note that for this example F (φ_{C}) is the identity morphism of F (C). Indeed, one has that F (C) = F (F (C)) since they have the same object of objects (F (C))_{0} = C_{0} = (F (F (C)))_{0}, and (F (φ_{C}))_{0} = (φ_{C})_{0} = 1_{C_{0}} the identity morphism of C_{0} in S.

Hence, by Corollary 3.3, if μ_{C} = (m_{C}, 1_{C_{0}}) : I(C) → F (C) is a monomorphism for each internal category C in Cat(S), then (I, η) is idempotent. Equivalently, if m_{C} is a monomorphism for each C in C, then F ix(I, η) is a full reflexive subcategory of Cat(S). Remark that this is obviously the case when S is regular, since then d_{C} = m_{C}e_{C} is a regular epi-mono factorization in S (the morphisms d_{C} : C_{1} → F (C)_{1}, e_{C} : C_{1} → I(C)_{1} and m_{C} : I(C)_{1} → F (C)_{1} in S, are respectively the first components of the internal functors φ_{C} = (d_{C}, 1_{C_{0}}), η_{C} = (e_{C}, 1_{C_{0}}) and μ_{C} = (m_{C}, 1_{C_{0}}); see diagram 2.7).

4. Stabilization of the idempotent (I, η)

Our next proposition gives sufficient conditions for the coequalizer of the kernel pair process to produce a reflection (I, η) such that, for every pullback g^{∗}(η_{A}) of a unit morphism η_{A} along any morphism g in C, I(g^{∗}(η_{A})) is an isomorphism.

4.1. Proposition. Consider the well-pointed endofunctor (I, η), obtained from the pointed endofunctor (F, φ) by the coequalizer of the kernel pair process displayed in diagram 2.1.

Then, (I, η) is idempotent with stable units, in the sense of [2] and [1], provided the following four conditions hold:

(i) μ is a monomorphism;

(ii) F η is an isomorphism;

(iii) the functor F preserves the pullback diagrams of the form

[figure: C; C ×_{I(A)} A; I(A) ;; A; η_{A}; g]

(iv) all morphisms η_{A} are pullback stable regular epimorphisms.

Proof. Consider the commutative diagram

[figure: C ×_{I(A)} A; C; g^{∗}(η_{A}); η_{C×_{I(A)}A}; η_{C}; I(C); I(C ×_{I(A)} A); I(g^{∗}(η_{A})); φ_{I(C×_{I(A)}A)}; φ_{I(C)}; F I(C) ,; F I(C ×_{I(A)} A); F I(g^{∗}(η_{A}))]

where g^{∗}(η_{A}) is the pullback of the unit morphism η_{A} : A → I(A) along g : C → I(A).

As, by Corollary 3.3, conditions (i) and (ii) in the statement imply that (I, η) is idempotent, it only remains to verify that I(g^{∗}(η_{A})) is an isomorphism.

Conditions (ii) and (iii) in the statement imply that both F (η_{C}) and F (g^{∗}(η_{A})) are isomorphisms in C. F I(g^{∗}(η_{A})) is therefore also an isomorphism, since F I(g^{∗}(η_{A}))F (η_{C×_{I(A)}A}) = F (η_{C})F (g^{∗}(η_{A})). Hence, we conclude that I(g^{∗}(η_{A})) is a monomorphism, since φ_{I(C)}I(g^{∗}(η_{A})) = F I(g^{∗}(η_{A}))φ_{I(C×_{I(A)}A)} and φ_{I(C×_{I(A)}A)} is a monomorphism by Corollary 3.2.

Furthermore, by condition (iv) in the statement, g^{∗}(η_{A}), η_{C×_{I(A)}A} and η_{C} are all pullback stable regular epimorphisms, which are known to be closed under composition and to have the strong right cancellation property. I(g^{∗}(η_{A})) is therefore a regular epimorphism since I(g^{∗}(η_{A}))η_{C×_{I(A)}A} = η_{C}g^{∗}(η_{A}).

Hence, being I(g^{∗}(η_{A})) a regular epimorphism which is simultaneously a monomorphism, it is necessarily an isomorphism.

Corollary 4.3 below, which follows trivially from our next lemma and Proposition 4.1, restates the latter in terms of factorization systems.

We will now suppose that both pointed endofunctors (F, φ) and (I, η), respectively the “input” and “output” of the coequalizer of the kernel pair process, are idempotent. It is well known that in this case the full reflective subcategories F ix(F, φ) and F ix(I, η) give rise respectively to prefactorization systems (E_{F} , M_{F}) and (E_{I}, M_{I}), where E_{F}, respectively E_{I}, is the class of morphisms f in C such that F f, respectively If, is an isomorphism (see [1]). We have already showed in Proposition 3.1 that F ix(I, η) = Mono(F, φ), which implies F ix(F, φ) ⊆ F ix(I, η). Hence, by the properties of prefactorization systems associated to reflective subcategories, M_{F} ⊆ M_{I} and E_{I} ⊆ E_{F} (see [1, §3]).

4.2. Lemma. If the pointed endofunctors (F, φ) and (I, η) are idempotent then μ is a monomorphism and F η is an isomorphism.

Proof. It is known that every unit morphism η_{A} belongs to E_{I} (see [1]). Therefore, η_{A} also belongs to E_{F} ⊇ E_{I}, i.e., F (η_{A}) is an isomorphism.

Then, as F φ is an isomorphism by hypothesis, it is easy to conclude that F μ must also be an isomorphism, since φ = μ · η. This implies that μ is a monomorphism, since F (μ_{A})φ_{I(A)} = φ_{F (A)}μ_{A} by naturality of φ, for every object A in C, and φI is a monomorphism by hypothesis (see Corollary 3.2).

4.3. Corollary. Suppose that the pointed endofunctors (F, φ) and (I, η) of Section 2 are both idempotent, i.e., they are in fact reflections with the respective associated prefactorization systems (E_{F} , M_{F}) and (E_{I}, M_{I}) (see [1]).

Then, the reflection (I, η) does have stable units, in the sense of [2], provided the following two conditions hold:

(i) the functor F preserves the pullback diagrams of the form

[figure: C; C ×_{I(A)} A; I(A) ,; A; η_{A}; g]

where η_{A} is any unit morphism of the reflection (I, η);

(ii) all unit morphisms η_{A} are pullback stable regular epimorphisms.

Condition (i) is equivalent to the following condition (i’) provided (F, φ) and (I, η) are both idempotent:

(i’) for every object A in C, the unit morphism η_{A} of I belongs to E′_{F}, the largest class of morphisms contained in E_{F} which is stable under pullbacks.

Our next corollary is a direct consequence of Corollaries 3.3 and 4.3.

4.4. Corollary. If C is a regular category, (F, φ) is idempotent and F is a left exact functor, then (I, η) is idempotent with stable units.

Proof. If C is a regular category and (F, φ) is idempotent, then Corollary 3.4 states that (I, η) is also idempotent.

The left exactness of F is equivalent to the condition that every pullback of a morphism in E_{F} is in E_{F} (see [2, Th. 4.7]), i.e., E_{F} = E′_{F}, which implies that every unit morphism η_{A} is in E′_{F}, since it is known to be in E_{I} ⊆ E_{F}. We have just checked that condition (i’) in Corollary 4.3 holds.

Finally, the fact that C is a regular category validates condition (ii) in Corollary 4.3.

4.5. Example. In Examples 3.5 and 3.6, of pointed endofunctors (F, φ) on Smp(S) and Cat(S) respectively, for a finitely-complete category S with coequalizers of kernel pairs, it is easy to check that (F, φ) is idempotent and F is a left exact functor.

Hence, if S is a regular category then all the conditions in Proposition 4.1 hold for those examples. We can conclude that the two general examples do give rise to (I, η) idempotent with stable units, provided S is a regular category.

5. Monotone-light factorization for the idempotent (I, η)

Proposition 5.1 below is a weaker version of [9, Corollary 6.2]. The latter is a consequence of the main result of [1], which gives necessary and sufficient conditions for the existence of a monotone-light factorization system (E′, M^{∗}).

5.1. Proposition. [9] Suppose that the following two conditions hold:

• the pointed endofunctor (I, η) on C, obtained as in Section 2, is idempotent and does have stable units;

• for each object B in C, there is an effective descent morphism p : E → B in C such that its domain E is an object of F ix(I, η)(=Mono(F, φ)).

Then, (E, M) and (E′, M^{∗}) are factorization systems. The pair (E, M) stands for the prefactorization system associated to the reflection (I, η). The latter pair (E′, M^{∗}) is obtained from the former by simultaneous stabilization of E and localization of M, in the sense of [1], and it is called a monotone-light factorization system.

5.2. Example. For S = Set the category of sets, the idempotent (I, η) with stable units on Cat(Set) is simply the reflection Cat → Preord of small categories into preordered sets, which was studied in [11]. For each object B in Cat, there is the effective descent morphism ε_{B} : Cat([3], B) · [3] → B in Cat, the obvious projection from the coproduct of sufficiently many copies of the ordinal number [3]. Note that the morphism ε_{B} is the counit of the adjunction (−) · [3] ⊣ Cat([3], −) : Cat → Set.

Hence, the conditions of Proposition 5.1 hold and one can conclude that there is a non-trivial monotone-light factorization on Cat which arises from the process described in this paper. The idempotent (F, φ) is in this case the reflection of the category Cat of all categories into the category of indiscrete categories.

As M. Gran pointed out to me, Proposition 5.1 applies to the reflection Grpd(S) → EqRel(S) between internal groupoids and equivalence relations, for any regular category S. This is exactly the reflection Cat(S) → Preord(S) in the case S is a (regular) Mal’cev category (i.e., a category with finite limits in which the internal equivalence relations coincide with the reflexive relations, EqRel(S) = RRel(S)), simply because Cat(S) = Grpd(S) and Preord(S) = EqRel(S) in that case. In fact, for each internal groupoid G in S, there is an internal functor (σ, d_{1}) : Eq(d_{0}) → G, with σ = γ(1_{G_{1}} × s), as shown in the following diagram

[figure: G_{1} ×_{G_{0}} G_{1}; γ; G_{1}; s; d_{1}; i; d_{0}; G_{0} .; σ × σ; σ; d_{1}; G_{1} ×_{G_{0}} G_{1} ×_{G_{0}} G_{1}; p_{1} × p_{2}; G_{1} ×_{G_{0}} G_{1}; p_{1}; < 1, 1 >; p_{2}; G_{1}]

The internal functor (σ, d_{1}) is an effective descent morphism in the category Grpd(S) of internal groupoids in S. This is so because σ < 1_{G_{1}}, id_{0} >= 1_{G_{1}} and d_{1}i = 1_{G_{0}}.

Hence, being S a regular Mal’cev category, we conclude from Proposition 5.1 that there is a monotone-light factorization associated to the reflection (I, η). This is of course the case when S is a Mal’cev variety of universal algebras. In particular, if S = Grp the category of groups, the coequalizer of the kernel pair process produces a reflection with stable units and monotone-light factorization for crossed modules. Since Cat(Grp) = Grpd(Grp) is equivalent to the category CrossMod of crossed modules.

6. The monotone-light factorization for simplicial sets via ordered simplicial complexes

The category of ordered simplicial complexes in S will be denoted by OSmC(S); obviously we can simply define it as Mono(F, φ), where F and φ are as in Example 3.5. The following proposition gives an equivalent reformulation; although the readers familiar with simplicial objects will find it straightforward, we will recall the proof.

6.1. Proposition. For a regular category S, a simplicial object A in Smp(S) is in its full reflective subcategory OSmC(S) if and only if its face maps d_{i}^{A} : A_{j+1} → A_{j}, 0 ≤ i ≤ j + 1, are jointly monic, for each j ≥ 0.

Proof. Consider the commutative diagram

[figure: A_{j+1}; d_{i}^{A}; d_{i}^{F (A)}; A_{j}; A_{0}^{j+2}; φ_{j+1}; φ_{j}; A_{0}^{j+1} ,]  (6.1)

associated to the pointed endofunctor (F, φ) of Example 3.5. We know from Proposition 3.1 that the simplicial object A belongs to OSmC(S) = F ix(I, η) if and only if φ_{j} : A_{j} → A_{0}^{j+1} is a monomorphism for every j ≥ 0, i.e., it belongs to Mono(F, φ).

We are going to suppose first that φ_{j} : A_{j} → A_{0}^{j+1} is a monomorphism for every j ≥ 0. Let d_{i}^{A}f = d_{i}^{A}g, for every 0 ≤ i ≤ j + 1. Then, φ_{j}d_{i}^{A}f = φ_{j}d_{i}^{A}g, which implies by the commutativity of the diagram just above that d_{i}^{F (A)}φ_{j+1}f = d_{i}^{F (A)}φ_{j+1}g, having 0 ≤ i ≤ j + 1. This last equality and the fact that the face maps (of any fixed dimension) of F (A) are jointly monic imply that φ_{j+1}f = φ_{j+1}g. Hence, we have just proved that every object in Smp(S) of the form I(A) does have jointly monic face maps.

We have to prove the converse now. Suppose that the face maps d_{i}^{A} : A_{j+1} → A_{j}, 0 ≤ i ≤ j + 1, are jointly monic, for each j ≥ 0. Under such an assumption, one has to show that if φ_{j} is a monomorphism (induction hypothesis) then so is φ_{j+1}, since φ_{0} is the identity morphism of A_{0}. In this way, if φ_{j+1}f = φ_{j+1}g then d_{i}^{F (A)}φ_{j+1}f = d_{i}^{F (A)}φ_{j+1}g, having 0 ≤ i ≤ j + 1, and φ_{j}d_{i}^{A}f = φ_{j}d_{i}^{A}g by the commutativity of diagram 6.1. Hence, d_{i}^{A}f = d_{i}^{A}g, having 0 ≤ i ≤ j+1, since φ_{j} is a monomorphism by the induction hypothesis. It follows that φ is a monomorphism provided the morphisms d_{i}^{A} are jointly monic, 0 ≤ i ≤ j + 1.

According to Proposition 5.1, there is a monotone-light factorization associated with the reflection Smp → OSmC of simplicial sets into ordered simplicial complexes, provided there is an effective descent morphism ε_{B} : E → B such that E is in OSmC, for each simplicial set B in Smp.

Indeed, we may choose ε_{B} to be the B-component of the counit of the adjunction (−) · ω ⊣ Smp(ω, −) : Smp → Set, where ω is the first infinite ordinal considered as a simplicial set via the usual nerve functor. That is, ε_{B} : Smp(ω, B) · ω → B is the canonical morphism from the coproduct of “sufficiently many” copies of ω to B.

In fact, as Smp = Set^{Δ^{op}} is a presheaf category, a morphism of simplicial sets is an effective descent morphism if it is an epimorphism. Every counit morphism ε_{B} : Smp(ω, B) · ω → B, of the adjunction Smp → Set, is an effective descent morphism if and only if the right adjoint Smp(ω, −) is a faithful functor. As this latter statement is easy to check, we have guaranteed the existence of a monotone-light factorization for simplicial sets via ordered simplicial complexes.

Note also that, as will see below, such a monotone-light factorization (E′, M^{∗}) is a non-trivial one, i.e., it does not coincide with the reflective factorization (E, M).

For each n ≥ 0, the reflection Smp_{n}(S) → OSmC_{n}(S), of the presheaf category of ntruncated simplicial objects to n-truncated ordered simplicial complexes in S, is induced by the reflection Smp(S) → OSmC(S). For the case S = Set, there is a monotonelight factorization associated to each reflection Smp_{n} → OSmC_{n}, n ≥ 0, which is a straightforward restriction of the one for Smp. In these cases one can replace ω by the ordinal number [n], when displaying the suitable effective descent morphisms for each object B in Smp_{n}.

In particular, having n = 1, the reflection Smp_{1} → OSmC_{1} is just the reflection RGrphs → RRel of reflexive graphs into reflexive relations, which has a non-trivial monotone-light factorization

(E′, M^{∗}) = (Bijections on V ertices and Surjections on Arrows, “F aithful”),

similar to the one for Cat → Preord. This gives the desired conclusion of non-triviality for all monotone-light factorizations above except the case n = 0, where the reflection itself is trivial, i.e., it is the identity functor of the category of sets.

7. Geometric morphisms and monotone-light factorizations

In last Section 6, we have started with the reflection of the presheaf category Smp = Set^{Δ^{op}} into its full subcategory in which the objects are the nerves of indiscrete categories, F_{0} : Smp → Smp_{0} = Set^{Δ_{0}^{op}}(≃ Set). This can be generalized to the reflections F_{n} : Smp → Smp_{n} = Set^{Δ_{n}^{op}}, n ≥ 0, which truncate each simplicial set right after the object of n-simplices:

[figure: A ⟼; A_{n}; ⋮; ⋯; A_{1}; A_{0}]  (7.1)

(Note that for the obvious localizations

F_{n}^{k} : Smp_{k} → Smp_{n}, k ≥ n ≥ 0,

we could copy all the results obtained below for the adjunctions 7.1).

Given a functor K : B → A, let S be a regular and complete category. Then, the induced functor

S^{K} : S^{A} → S^{B}

is continuous, i.e., preserves all limits, and has a right adjoint, since every object T : B → S of the presheaves category S^{B} does have a right Kan extension along K : B → A. Furthermore, it is known that if K is fully faithful then the right adjoint of S^{K} is also so. Hence, one can apply Corollary 4.4 factorizing the localization S^{K} : S^{A} → S^{B} through a reflection with stable units. Remark that we could just ask for finitely completeness of S, provided B were a finite category and A had finite hom-sets, as is the case for the full inclusion Δ_{n}^{op} ⊂ Δ^{op}, n ≥ 0 (see Theorem 1 in [8, X.3]: right Kan extension as a point-wise limit).

The localizations F_{n} : Smp → Smp_{n}(n ≥ 0) are in fact examples of essential geometric morphisms, between elementary topoi, which are also embeddings. We shall call geometric morphism to any adjunction between finitely-complete categories such that the left adjoint preserves finite limits; it is called an embedding when the right adjoint is fully faithful; it is called essential if the left adjoint is also a right adjoint of some other functor.

We will now prove that all these localizations F_{n} : Smp → Smp_{n}, besides giving rise to reflections I_{n} : Smp → Mono(F_{n}) with stable units, by applying the coequalizer of the kernel pair process, produce in addition monotone-light factorizations, as for the case n = 0 already worked out in last Section 6. According to Proposition 5.1, we still need to exhibit an effective descent morphism p : E → B (i.e. an epimorphism, since Smp is a topos) for each simplicial set B, and such that φ_{E}^{n} : E → F_{n}(E) is monic in Smp, i.e., E ∈ Mono(F_{n}) = F ix(I_{n}) (φ^{n} is of course the unit of the adjunction associated to F_{n}). We have already shown in Section 6 that there exists an effective descent morphism p : E → B with E in Smp_{0} for every simplicial set B; therefore one can state Conclusion 7.2.

Alternatively, we are going to give a more general argument, which also provides the needed effective descent morphisms p : E → B with E in Smp_{n}, for each simplicial set B. We care to do so since the proof of the following Proposition 7.3 is a simple generalisation of the considerations below.

It is known that every B ∈ Smp = Set^{Δ^{op}} is a colimit of representable functors. Hence, there is a canonical presentation of B,

[figure: E′; q; r; E; p; B ,]  (7.2)

where p = ker(q, r) is the coequalizer in Smp of q and r, and E is the coproduct of a family of representable functors.

7.1. Lemma. Consider the reflection F_{n} : Smp → Smp_{n} given in diagram 7.1, whose unit morphism is φ^{n}.

Every unit morphism of any representable functor

φ_{Δ(−,[p])}^{n} : Δ(−, [p]) → F_{n}(Δ(−, [p])), p ≥ 0,

is a monomorphism in Smp = Set^{Δ^{op}}.

Proof. First, note that the face maps

d_{i} : Δ([j + 1], [p]) → Δ([j], [p]), 0 ≤ i ≤ j + 1,

of the simplicial set Δ(−, [p]) are jointly monic for each j ≥ 0. This is so because Δ(−, [p]) is the nerve of the ordinal number [p] considered as a category.

The fact that the face maps are jointly monic implies that any two simplicial morphisms f and g from any simplicial set A into Δ(−, [p]) are equal if and only if their two 0-components f_{0}, g_{0} : A_{0} → Δ([0], [p]) are equal.

Hence, φ_{Δ(−,[p])}^{n} is a monomorphism if and only if its 0-component is an injection: φ_{Δ(−,[p])}^{n} is a monomorphism if and only if each of its components is an injection; the converse is also easy to prove,

φ_{Δ(−,[p])}^{n}f = φ_{Δ(−,[p])}^{n}g ⇒ (φ_{Δ(−,[p])}^{n})_{0}f_{0} = (φ_{Δ(−,[p])}^{n})_{0}g_{0}

⇒ f_{0} = g_{0}, because (φ_{Δ(−,[p])}^{n})_{0} is an injection by hypothesis

⇒ f = g, because two morphisms into a representable functor are completely determined by their 0-components.

Indeed, the 0-component (φ_{Δ(−,[p])}^{n})_{0} is always the identity function, since the reflection φ_{A}^{n} : A → F_{n}(A) does not change the first n objects of simplices A_{i}, 0 ≤ i ≤ n.

If F ⊣ G : X → C is a generic adjunction with unit φ, then ∐_{i∈I} φ_{C_{i}} = φ_{∐_{i∈I} C_{i}} for any family {C_{i} ∈ C | i ∈ I} such that its coproduct does exist. Therefore, E = ∐_{i∈I} Δ(−, [n_{i}]) belongs to Mono(F_{n}) if and only if φ_{∐_{i∈I} Δ(−,[n_{i}])} = ∐_{i∈I} φ_{Δ(−,[n_{i}])} is a monomorphism. That is so because Lemma 7.1 tells us that each φ_{Δ(−,[n_{i}])} is injective componentwise, which implies that ∐_{i∈I} φ_{Δ(−,[n_{i}])} is injective componentwise, i.e., a monomorphism.

7.2. Conclusion. There is a monotone-light factorization for each of the reflections obtained through the coequalizer of the kernel pair process from (F_{n}, φ^{n}), n ≥ 0.

7.3. Proposition. Let F : E → F be a geometric morphism between regular categories, F^{∗} ⊣ F_{∗} : E → F, which is an embedding.

Then, the reflection I : F → Mono(F^{∗}), obtained from the localization F^{∗} : F → E through the coequalizer of the kernel pair process, does have stable units. Moreover, there is a monotone-light factorization associated to the reflection I : F → Mono(F^{∗}) provided the following four conditions also hold:

1. the category F is cocomplete;

2. the full subcategory Mono(F^{∗}) is dense in F, i.e., every object of F is a colimit of objects of Mono(F^{∗}).

3. in F the coproduct of monomorphisms is a monomorphism;

4. regular epis are effective descent morphisms in F.

References

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

[3] Gabriel, P., Zisman, M. Calculus of fractions and homotopy theory. Ergebnisse der mathematik, Vol. 35. Berlin-Heidelberg-New York: Springer 1967.

[4] Janelidze, G. Internal categories in Mal’cev varieties. Preprint, York University, Toronto (1990).

[5] Janelidze, G. Pure Galois theory in categories. J. Algebra 132 (1990) 270–286.

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

[7] 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.

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

[9] Xarez, J. J. Internal monotone-light factorization for categories via preorders. Theory Appl. Categories 13 (2004) 235–251.

[10] Xarez, J. J. The monotone-light factorization for categories via preordered and ordered sets. PhD thesis, University of Aveiro (Portugal), 2003.

[11] Xarez, J. J. The monotone-light factorization for categories via preorders. in 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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