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Fundamental groupoids for simplicial objects in Mal'tsev categories

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and in fact a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).


Introduction
Categorical Galois theory, as developed by G. Janelidze ([25, 29, 3, 27]), is a general framework that allows the study of central extensions or coverings of the objects of a category. A large collection of examples has been given, ranging from the Galois theory of commutative rings of Magid ([32,10]) and the theory of coverings of locally connected spaces to the central extensions of groups, Lie algebras, or more generally exact Mal'tsev categories [28].
The main ingredient of this theory is the notion of Galois structure, which is defined as an adjunction, with the right adjoint often taken to be fully faithful, and a class of morphisms in the codomain of the right adjoint, satisfying suitable conditions, in particular admissibility, which amounts to the preservation by the left adjoint of certain pullbacks. For example, the inclusion of any Birkhoff subcategory of an exact Mal'tsev together with the class of regular epimorphisms always forms an admissible Galois structure ( [28]).
In [8], Brown and Janelidze used this theory to describe what they called second order coverings for simplicial sets, using the adjunction given by the nerve functor and the fundamental groupoid, and the class of Kan fibrations. In fact, they restriced their analysis to Kan complexes, as this condition implies the admissibility of these objects for the corresponding Galois structure. Later Chikhladze introduced relative factorization systems, and showed that the induced relative factorization system for Kan fibrations is locally stable, so that the Galois structures induces a relative monotone-light factorization ( [15]).
On the other hand regular Mal'tsev categories were characterized in [11] as the categories in which the Kan condition holds for every simplicial object, thus extending a theorem of Moore stating that the underlying simplicial set of a simplicial group is always a Kan complex. Moreover, regular epimorphisms in the category of simplicial objects then coincide with Kan fibrations. This suggests that the inclusion of groupoids into simplicial objects in any exact Mal'tsev category might induce an admissible Galois structure.
The main objective of this paper is to show that this is indeed the case, and more precisely that the category of groupoids in an exact Mal'tsev category is always a Birkhoff subcategory of the category of simplicial objects. The paper is organised as follows : we begin with some preliminaries, to fix notations and provide the background notions. We then construct the reflection of the category of simplicial objects into the subcategory of internal groupoids. Next, we give a characterization of the central extensions for the Galois structure. In the next section we compare our construction with the homotopy relations for the simplices in a Kan complex, which are used to define its homotopy groupoid. Then we prove that the Galois structure admits a relative monotone-light factorization system. We end the paper with a discussion of reflexive graphs, seen as truncated reflexive graphs.
1. Preliminaries 1.1. Simplicial objects. Let ∆ denote the category of finite nonzero ordinals, with monotone functions as morphisms. For a given category C, the category Simp(C) of simplicial objects in C is the category of functors ∆ op → C. Equivalently, an object X of Simp(C) is a collection of objects (X n ) n∈N together with face maps d i : X n → X n−1 for all n > 0 and 0 ≤ i ≤ n, and degeneracy maps s i : X n → X n+1 for n ≥ 0 and 0 ≤ i ≤ n, satisfying the following simplicial identities, whenever they make sense: When necessary, we will write d X i or s X i to distinguish the face or degeneracy maps of different simplicial objects. A morphism f : X → Y in Simp(C) is then a collection of morphisms f n : X n → Y n that commute with face and degeneracy maps, in the sense that d Y i f n+1 = f n d X i and s Y i f n = f n+1 s X i for all i, n. If X is a simplicial object, we will denote Dec(X) the simplicial object (X n+1 ) n≥0 , whose face and degeneracies are the same as those of X, without the last faces d n+1 : X n+1 → X n and last degeneracies s n : X n → X n+1 for all n ≥ 1. The simplicial identities imply that the maps d n+1 : X n+1 → X n form a morphism of simplicial objects ǫ X : Dec(X) → X. Since all these maps are split (and thus regular) epimorphisms, ǫ is a regular epimorphism in Simp(C), although it need not be a split epimorphism. Notice that Dec defines an endofunctor of Simp(C), and ǫ is a natural transformation from Dec to the identity endofunctor.
∆ is a skeleton of, and thus equivalent to, the category of non-empty finite totally ordered sets. In particular, since this category contains the poset P f,n.e. (N) of non-empty finite subsets of N (ordered by inclusion) as a subcategory, there is a canonical functor Φ : P f,n.e. (N) → ∆ that maps any set with n + 1 elements to {0, . . . , n} and any inclusion map to an injective morphism in ∆.
For a given simplicial object X, and for every n ≥ 2, one can consider the restriction of Φ to the poset of proper subsets of {0, 1, . . . , n}; taking the opposite functor and composing with X : ∆ op → C gives a diagram in C. The limit of this diagram is the n-th simplicial kernel of X, and denoted K n (X). In particular, we have maps µ i : K n (X) → X n−1 for i = 0, . . . , n, satisfying d i µ j = d j−1 µ i for all 0 ≤ i < j ≤ n, and the maps µ i are universal with this property. Thus the face maps d 0 , . . . , d n : X n → X n−1 induce a canonical map κ n : X n → K n (X). Following [18], we say that X is exact at X n−1 if κ n is a regular epimorphism, and exact if it is exact at X n for all n ≥ 1.
Moreover, for every n ≥ 2 and 0 ≤ k ≤ n, we can also restrict Φ to the poset of proper subsets of {0, . . . , n} that contain k, and then compose the opposite functor with X. The limit of this diagram is the object of (n, k)-horns Λ n k (X), and it is equipped with maps ν i : Λ n k (X) → X n−1 for 0 ≤ i ≤ n and i = k that satisfy the identities d i ν j = d j−1 ν i for all 0 ≤ i < j ≤ n and i = k = j, and are universal with this property. There is then also a canonical arrow λ n k : X n → Λ n k (X) induced by the face maps d i : X n → X n−1 for i = k, and X is said to satisfy the Kan property if all these maps are regular epimorphisms. Moreover, a map f : X → Y between simplicial objects is called a Kan fibration if for all n and k the canonical arrow θ n k in the diagram (where the inner square is a pullback) is a regular epimorphism. For every n ≥ 1, we denote ∆ n the full subcategory of ∆ consisting of the ordinals with n + 1 elements or less, and Simp n (C) the category of functors ∆ op n → C, whose objects we called n-truncated simplicial objects. The inclusion ∆ n ֒→ ∆ then induces by precomposition the truncation functor Simp(C) → Simp n (C).
An internal reflexive graph in C is simply a 1-truncated simplicial object. A multiplicative graph is then a reflexive graph endowed with a partial multiplication m : X 1 × X 0 X 1 → X 1 that is unital and compatible with the domain and codomain maps ( [13]), and an internal category is a multiplicative graph whose multiplication is associative. All these conditions can be summarized by saying that an internal multiplicative graph is an object of Simp 2 (C), such that the square is a pullback. Internal functors are also the same thing as (restricted) simplicial morphisms. Moreover, any internal category can be extended to a simplicial object by simply taking its nerve. From now on we will thus consider Cat(C) and Grpd(C) as full subcategories of Simp(C); more precisely, a simplicial object X is an internal category if and only if the commutative square is a pullback for all n ≥ 2.

1.2.
Mal'tsev categories and higher extensions. A finitely complete category C is called a Mal'tsev category if every internal reflexive relation is an equivalence relation [11,12,13,2]; in a regular category, this condition is equivalent to the fact that the composition R•S of two equivalence relations R, S on the same object X is an equivalence relation. When this is the case, R • S is in fact the join of R and S in the poset of equivalence relations of X. Accordingly this poset is a lattice. In fact this is a modular lattice ( [12]), i.e. we have the identity for all equivalence relations R, S, T on X such that R ≤ T . An important property of Mal'tsev categories is that the inclusion of the category Grpd(C) of internal groupoids into the category MRG(C) of multiplicative reflexive graphs is an isomorphism, and that the truncation functor MRG(C) → RG(C) is fully faithful ( [13]).
For a variety, this is also equivalent to the existence of a ternary operation p satisfying the equations p(x, y, y) = x and p(x, x, y) = y. In particular, the categories of groups, R-modules, rings, Lie algebras and C * -algebras are all examples of Mal'tsev categories; other examples include the category of Heyting algebras, the dual of any topos [5] or any additive category.
In any regular category, a commutative square We can also define a triple extension as a commutative cube for which all faces, as well as the induced commutative square are double extensions. Triple extensions satisfy the same properties as in the previous proposition : in particular, a split cube between double extensions is always a triple extension.
1.3. Categorical Galois theory and monotone-light factorization systems. We recall some definitions from [28,29]. A Galois structure Γ = (C, X , I, U, F ) consists of a category C, a full reflective subcategory X of C, with reflector I : C → X and inclusion U : X → C and a class F of morphisms of C containing all isomorphisms, stable under pullbacks and composition, and preserved by I. We will call the morphisms in F extensions. Let us write, for any object B of C (resp. of X ), C ↓ B (resp. X ↓ B) for the full subcategory of the slice category C/B (resp. X /B) consisting of extensions f : X → B. Then any arrow p : E → B induces a functor p * : C ↓ B → C ↓ E defined by pulling back. If p is an extension, this functor has a left adjoint p ! defined by composition with p; the extension p is said to be of effective F -descent, or simply a monadic extension, if the functor p * is monadic.
Moreover, the reflector I induces for every B a functor I B : C ↓ B → X ↓ I(B) which maps f : X → B to I(f ) : I(X) → I(B); and every such functor has a right adjoint U B : X ↓ I(B), defined for any g : Y → I(B) by the pullback The object B is then said to be admissible if U B is fully faithful, which is equivalent to the reflector I preserving all pullback squares of the form above. A Galois structure Γ is said to be admissible if every object is admissible. Given an admissible Galois structure, an extension f : X → B in C ↓ B is said to be • trivial if it lies in the replete image of U B , or equivalently if the square is a pullback; • central, or alternatively a covering, if there exists a monadic extension p : E → B such that p * (f ) is trivial; • normal, if it is a monadic extension and if f * (f ) is a trivial extension (that is, if the projections of the kernel pair of f are trivial).

Example 1.
If C is an exact Mal'tsev category and X is any Birkhoff (i.e. full reflective and closed under quotients and subobjects) subcategory of C, and F is the class of regular epimorphisms, then the Galois structure Γ is admissible, and moreover every extension is monadic and every central extension is also normal ( [28]). When C is the category of groups and X the subcategory of abelian groups, the central extensions in this sense are exactly the surjective group homomorphisms whose kernel is included in the center of the domain ( [28]). More generally, in any exact Mal'tsev category with coequalizers, the central extensions of the Galois structure defined by the subcategory of abelian objects are the extensions such that the Smith-Pedicchio commutator [Eq[f ], ∇ X ] is trivial ( [22]).
If Γ is a Galois structure where F is the class of all morphisms in C, admissibility is equivalent to the reflector I being semi-left-exact in the sense of [14]. In that case any morphism f : X → B in C induces a commutative diagram .
Since the reflector I preserves the pullback in this diagram I(f ′ ) is an isomorphism, and f ′′ is a trivial extension by definition. Moreover in that case the classes E of morphisms inverted by I and the class M of trivial extensions are orthogonal to one another, and thus the two classes form a factorisation system (E, M) in C ( [14]). The trivial extensions are then stable under pullbacks, but the class E does not have this property in general.
In order to obtain a stable factorization system, one can localize M and stabilize E, as in [9]; this means that we replace E by the class E ′ of maps for which every pullback along a monadic extension is in E, and M by the class M * of maps f that are locally in M, in the sense that there exists a monadic extension p such that p * (f ) ∈ M. In the context of Galois Theory these are precisely the central extensions. The two classes E ′ and M * are orthogonal, but in general they do not form a factorization system. When this is the case, the resulting factorization system is called the monotone-light factorization system In the case where F is no longer the class of all morphisms in C, it need not be true that every morphism admits a (E, M)-factorization. Nevertheless, this is still true for extensions; it is then natural to extend the notion of factorization system to the case where only some morphisms have a factorization. This was done by Chikhladze in [15] : If C is a category and F a class of morphisms of C containing the identities and closed under composition and pullbacks, a relative factorization system for F consists of two classes E and M of maps such that Then any admissible Galois structure Γ = (C, X , I, F ) yields a relative factorization system for F with E and M consisting of the maps inverted by I and the trivial extensions, respectively. When moreover this factorization system can be stabilized, then the stable factorization system (E ′ , M * ) is called a relative monotone-light factorization system for F .

Example 2.
If C is the category of simplicial sets, X the category of groupoids, I the fundamental groupoid functor, and F the class of Kan fibrations, then every Kan complex is an admissible object, and the central extensions were called second order coverings in [8].
This Galois structure admits a relative monotone-light factorization system, as shown in [15].
Example 3. In a finitely complete category, any object X has a corresponding discrete internal groupoid. This defines a fully faithful functor D : C → Grpd(C). If C is exact, then this functor admits a semi-left-exact left adjoint π 0 : Grpd(C) → C ( [4]). When C is moreover Mal'tsev, C is in fact a Birkhoff subcategory of Grpd(C), and the central extensions of the Galois structure (Grpd(C), C, π 0 , F ) (where F is the class of regular epimorphisms) are precisely the regular epimorphic discrete fibrations ( [21]). This Galois structure admits a relative monotone-light factorization system ( [16]).

The reflection of simplicial objects into groupoids
Convention. From now on, C will denote a regular Mal'tsev category. For a given simplicial object (X n ) n≥0 with face maps d i : X n → X n−1 for n ≥ 1 and 0 ≤ i ≤ n, we will denote D i the kernel pair of d i .
Note that Simp(C), being a functor category, is also regular Mal'tsev. Lemma 1. If X is a simplicial object in C, all the commutative squares given by Proof. If i < j − 1, the two squares in the diagram commute. On the other hand, if j = i+1, then at least one of the inequalities 1 ≤ j ≤ n+2 is strict, hence at least one of the squares will commute; in any case, the commutative square is a double extension.
Moreover, any morphism f : X → Y of simplicial objects has to commute with the face and degeneracies; hence, when f is a regular epimorphism, every square is a double extension. The resulting cube will then always be a split epimorphism between double extensions, hence a triple extension.
Remark. The pullback X 1 × X 0 X 1 of d 0 along d 1 coincides with the object of (2, 1)-horns Λ 2 1 (X), and similarly the other two pullbacks X 1 × X 0 X 1 , which define the kernel pairs of d 0 and d 1 , coincide with the objects of (2, 2) and (2, 0)-horns, respectively. In particular, the previous lemma shows that every simplicial object satisfies the Kan property and that every regular epimorphism is a Kan fibration for 2-horns. The proof for the higher order horns can be done in the same way, using n-fold extensions for n ≥ 3, as in [18].
As a consequence we have Proof. By the previous lemma f : X → Y induces a triple extension. In particular the square Moreover, d k can be identified with a component of the regular epimorphism ǫ X : Dec(X) → X, and thus the cube are all double extensions, which implies the desired equalities.
Lemma 2. For any simplicial object X, the following equivalence relations in X 1 are all equal : Proof. We prove the first identity; the other one is obtained in a similar way. Since Definition 2. We will call H 1 (X) this equivalence relation.
Proposition 2. Let X be a simplicial object in C. Then for all n ≥ 2 the following conditions are equivalent : Moreover, X is an internal groupoid if and only if it satisfies these conditions for all n ≥ 2.
We first consider the case where n = 2; for this case it is enough to prove that Assuming now that the condition holds for n, we prove that it holds for n + 1. Assume that D i ∧ D j = ∆ X n+1 ; then taking images by d k (for k / ∈ {i, j}) on both sides shows that D i ′ ∧ D j ′ = ∆ Xn for some i ′ , j ′ , and thus, by the induction hypothesis, for all i ′ , j ′ . In particular, for any 0 ≤ r < s ≤ n + 1, we have for some r ′ , s ′ Now X is an internal groupoid if and only if the squares are all pullbacks. Since we know already that they are all double extensions, this is equivalent to the fact that the pair d 0 , d n is jointly monic, and this is equivalent to (2).
Thus any internal category always satisfies the second condition, and conversely any simplicial object satisfying the first one is an internal category where the square is a pullback. This condition is equivalent to the internal category being a groupoid.
Note that in the above proof we only needed to know that X was an internal category to prove that it satisfied the conditions; so this gives us a new proof of the fact that any internal category in a regular Mal'tsev category is an internal groupoid. Proof. All the intersections that characterize internal groupoids in the previous proposition are preserved by regular epimorphisms of simplicial objects, which shows that groupoids are closed under quotients. Moreover they are also closed under subobjects where the horizontal sides are monomorphisms and the right-hand vertical side is an isomorphism, and thus the left-hand vertical side is a monomorphism. Since it is also a regular epimorphism (by Lemma 1), this means d i , d j is an isomorphism, hence X is an internal groupoid.
Remark. In fact the previous corollary also characterizes Mal'tsev categories among the regular (or even finitely complete) ones : indeed a reflexive relation R ֒→ X × X is just a subobject of the reflexive graph (X × X, X, π 1 , π 2 , δ X ), and by taking iterated simplicial kernels, one can extend this to a monomorphism in Simp(C), whose codomain is just the nerve of the indiscrete equivalence relation/groupoid on X. Thus every reflexive relation is a subobject of a groupoid, and a relation is a groupoid if and only if it is an equivalence relation. Accordingly : Convention. We now assume that the category C is also exact.
In this setting, we have Proof. Note first that since by definition H 1 (X) ≤ D 0 ∧ D 1 , d 0 and d 1 : X 1 → X 0 both factor through the coequalizer η 1 of H 1 (X), and their factorizations have a common section η 1 s 0 , which we will also denote s 0 , so that we get a morphism of reflexive graphs Let us then form the pullback H 1 is a regular epimorphism, and as a consequence so is which we will denote η 2 . We also define H 2 (X) = Eq[η 2 ]. Now we need to show that η 1 d 1 : X 2 → X 1 H 1 (X) factorizes through η 2 ; for this it is enough to show that η 1 d 1 (H 2 (X)) = ∆ X 2 , which is equivalent to d 1 (H 2 (X)) ≤ H 1 (X). Since d 0 and d 2 are jointly monic by construction, we find that graph, which is then automatically a groupoid, which we denote I(X). We also denote η X : X → I(X) the morphism of simplicial objects induced by 1 X 0 , η 1 and η 2 . We can show that η n is is a regular epimorphism for all n, by iterating the argument showing that η 2 is a regular epimorphism.
The only thing that remains to be checked is that this construction is universal. For this we must prove that for every morphism f : X → Y to a groupoid Y, there exists a factorization of f n through η n : X n → I(X) n for all n (note that such a factorization is unique, as every η n is a regular epimorphism). The case n = 0 is trivial as η 0 is the identity. For n = 1, it is enough to prove that Eq[f 1 ] ≥ H 1 (X), or equivalently f 1 (H 1 (X) This shows that the truncation of f to a morphism (f 1 , f 0 ) of reflexive graph factors through the groupoid X 1 /H 1 (X), with a factorization (g 1 , g 0 = f 0 ); applying the nerve functor allows us to extend this factorization to higher levels, resulting in morphisms g n : I(X) n → Y n . Then the factorizations f n = g n η n for n ≥ 2 can be obtained from the universal property of the pullbacks defining each X n and Y n . Then since each η n is a regular epimorphism, the morphisms g n define a morphism of simplicial objects.
Let us denote H n (X) the kernel pair of η n . We have proved already that . For the next section, it will be useful to prove a similar formula for H n (X) for n ≥ 3 : Proof. We prove the result by induction on n. The case n = 2 was done in the proof of Theorem 1. Now let us assume that it holds for n; since by construction the square is a pullback, so that the two maps d 0 , d n+1 are jointly monic, we have for n + 1

Moreover, by the induction hypothesis we have the identities
Combining all these, we get the identity From there we already see that For the converse inequality, first note that and thus, since the lattice of equivalence relations of X n+1 is modular, we have

Now to conclude the proof it is enough to prove that
for all m ≥ 1, which we will do by induction. The case where m = 1 is trivial, so let us now assume that (3) holds for some m. Then we have and as a consequence we have It follows that the left-hand side must be equal to Now since 0<j≤m+1 (D 0 ∧ D j ) ≤ D 0 , using again the modularity law, we find that and this is smaller than D m+1 ∨ 0≤i<j<m+1 (D i ∧ D j ) , which concludes the proof.
Remark. If the category C is not only exact Mal'tsev but also arithmetical ( [36]), then the category Grpd(C) coincides with the category of equivalence relations, which is thus a Birkhoff subcategory of Simp(C). Note that in that case, H 1 (X) = d 0 (D 1 ∧D 2 ) = D 0 ∧D 1 , since direct images preserve intersections of equivalence relations (by Theorem 5.2 of [7]). Accordingly our reflection becomes a reflection of Simp(C) into Eq(C).

Corollary 4. An exact Mal'tsev category is arithmetical if and only if Eq(C) is a Birkhoff subcategory of Simp(C).
Remark. Note that, by contrast with the Smith-Pedicchio commutator, whose quotient gives a left adjoint of the forgetful/inclusion functor Grpd(C) → RG(C), we don't need to assume the existence of any colimits to define H 1 (X).

Characterization of central extensions
Being a Birkhoff subcategory of the exact Mal'tsev category Simp(C), Grpd(C) is admissible in the sense of categorical Galois theory, when F is the class of all regular epimorphisms. In this section we will characterize the central extensions with respect to this reflection.
Convention. If f : X → Y is a map in Simp(C), we will denote F n the kernel pair of the corresponding map f n : X n → Y n , for all n ≥ 0. Similarly, for maps g : Z → W and f ′ : X ′ → Y ′ in Simp(C), we will denote the corresponding kernel pairs G n and F ′ n (for n ≥ 0), respectively.
First, we note that Proposition 4.2 of [28] implies, in our case, that trivial extensions f : X → Y are characterized by the property that F n ∧ H n (X) = ∆ Xn for all n ≥ 0, that is : Our characterization of central extensions is then obtained simply by "distributing" the intersection with F n appearing in these equations with the join or image. In other words we have and for all n ≥ 2.
To prove this we will need a couple of lemmas.
be a pullback square of regular epimorphisms in Simp(C), and let n ≥ 2 and 0 ≤ i < j ≤ n. Let us denote d ′ i the face maps of the simplicial object P, and D ′ i their kernel pairs. Then Proof. Since pullbacks in Simp(C) are computed "levelwise" in C, for all n the square is a pullback. Since moreover limits commute with limits, in the cube the top and bottom faces are pullbacks; one can then show that the square In particular, For the converse, the equation (6) shows already that if F n ∧ D i ∧ D j = ∆ Xn , then Since it is also smaller than F ′ n , and since f ′ n and g ′ n are jointly monic by construction, we have F ′ n ∧ D ′ i ∧ D ′ j = ∆ Pn . Lemma 4. Let f : X → Y be a split epimorphism, with section s : Y → X, and let A, B be two equivalence relations on X, with respective coequalizers q A , q B . Assume that we have a diagram where the vertical downwward arrows are split epimorphisms, and the upward and downward squares commute. Then the following conditions are equivalent : Proof. First of all, we have the inequality which immediately proves that the first condition implies the second.
For the converse, we can complete the diagram (7) by taking the pushouts of the top and bottom spans. This yields a cube which is a split epimorphism between double extensions, hence a triple extension. In particular, the square The first equality implies that q A , f is a mono, hence an iso; then so is γ in the diagram above, and thus the left and right faces of the cube are pullbacks. Similarly, the second equality B ∧ Eq[f ] implies that the top face is a pullback as well, and then so is the square Proof of Theorem 2. Let us consider the diagram Now assume first that f is a central extension, so that the left-hand square is a pullback. Since by construction I(X × Y X) is an internal groupoid, (5) holds for I(π 1 ), and then by Lemma 3 it also holds for π 1 and thus for f .
Assuming now that (5) holds, then again by Lemma 3 it also holds with π 1 : X × Y X → X, so that Eq[(π 1 ) n ] ∧ D ′ i ∧ D ′ j = ∆ Xn× Yn Xn ∀0 ≤ i < j ≤ n. But π 1 is a split epimorphism in the category of simplicial objects of C. Thus in particular, for all 0 ≤ i < j ≤ n, (π 1 ) n and D ′ i ∧ D ′ j satisfy the assumptions of Lemma 4, and thus we have This implies that the left-hand square is a pullback; thus π 1 is a trivial extension, and f is a central extension.
The equivalence relation F 2 ∧D 0 ∧D 1 is the kernel pair of the arrow θ 2 2 : is a pullback. The triviality of F 2 ∧ D 0 ∧ D 2 and F 2 ∧ D 1 ∧ D 2 can be interpreted in the same way with the horn objects Λ 2 1 and Λ 2 0 . Moreover, the higher order conditions F n ∧D i ∧D j = ∆ Xn imply that all the morphisms θ n k for n ≥ 2 are isomorphisms, and thus that all squares are isomorphisms. One can prove that the converse is true as well.

Comparison with simplicial sets
As noted before, the left adjoint to the nerve functor between groupoids and simplicial sets is the fundamental groupoid functor [19]. For a simplicial set X which satisfies the Kan condition, also called a quasigroupoid, this left adjoint can alternatively be described as the homotopy groupoid (see [1,31]). One defines the homotopy relation on X 1 by saying that two elements (or 1-simplices) f, g ∈ X 1 are homotopic if and only if there exists α ∈ X 2 such that d 0 (α) = f , d 1 (α) = g and d 2 (α) = s 0 d 1 f = s 0 d 1 g. This is always a reflexive relation (since for a given f one can take α = s 0 f ), and using the Kan condition one can then prove that this is actually an equivalence relation. The homotopy groupoid is then the groupoid whose objects are just the elements of X 0 , arrows are homotopy classes of 1-simplices, identities defined by the classes of degenerate 1-simplices, and composition defined by the existence of fillers for (2, 1)-horns (with two sided inverses defined by the existence of fillers for the outer horns).
This relation can be interpreted in any regular category as follows : first take the pullback and then factorize the map (d 0 , d 1 )π 1 : X 0 × X 1 X 2 → X 1 × X 1 as a regular epimorphism q : P → R followed by a monomorphism r = (ρ 1 , ρ 2 ) : R → X 1 × X 1 , so that R is a relation on X 1 . As in the case of sets, this is a reflexive relation; indeed, the simplicial identities imply that This relation is in fact equal to d 0 (D 1 ∧ D 2 ) whenever X satisfies the Kan condition, as we shall now see. In fact it will be helpful to prove a slightly more general result: Given any regular epimorphism f : X → Y between two simplicial objects, if we take the pullback The case where f is the morphism ǫ X : Dec(X) → X gives the desired identity.
Proof. Consider the diagram where the top and bottom faces of the cube are pullbacks. Since all the vertical arrows are split by a degeneracy map s 0 , and the horizontal maps commute with these sections, the dotted arrow is a split epimorphism as well. In particular, the image factorization of (d X 0 , d X 1 )π 1 is the same as that of (d X , which would concludes the proof.
Since we have a decomposition of f 2 given by the diagram we can rewrite the top pullback in (9) as the upper rectangle in the following diagram : , the composition λ 2 0 s 1 is a monomorphism, and thus so is ϕ 1 m. Since λ 2 0 is the regular epimorphism in the factorization of (d X 1 , d X 2 ) : On the other hand, the right-hand rectangle above coincides with the left-hand square in the rectangle Since the two squares are pullbacks, the whole rectangle is one as well. But this is the same as the outer rectangle in where the two squares are again pullbacks. Thus P coincides with the intersection D 1 ∧F 1 , which concludes the proof.
Remark. If one sees a Kan complex as a quasigroupoid or ∞-groupoid, then the left adjoint to the nerve or inclusion functor Grpd → Kan is in a sense a "strictification", which turns quasigroupoids into actual groupoids.
The equivalence relation d 0 (D 1 ∧ D 2 ∧ F 2 ) which appears in our characterization of central extensions admits an alternative construction, similar to that of H 1 (X). More precisely, if we take now L to be the limit of the lower part of the diagram (with the dotted arrows forming the limit cone) then Indeed, the limit in diagram (10) can also be obtained as the pullback Moreover, we have Thus the pullback square above factorises as a rectangle and one can easily show that the right-hand square is a pullback, and as a consequence so is the left-hand side square. But this square is exactly the pullback that appears if we apply Lemma 5 to the induced map ǫ X , Dec(f ) : Dec(X) → X × Y Dec(Y), which is a regular epimorphism between simplicial objects because the square is a double extension in C for all n. Thus Lemma 5 implies that the two constructions are equal.

The relative monotone-light factorization system
In order to prove that our Galois structure admits a relative monotone-light factorization system, we use the following criterion, due to Carboni, Janelidze, Kelly and Paré in the absolute case and to Chikhladze in the relative case : Proposition 4 ( [9,15]). Let (C, X , I, F ) be an admissible Galois structure. The class F admits monotone-light factorization if for each object B of C there is an effective Fdescent morphism p : C → B where C is a stabilizing object, i.e. an object such that if h = me is the (E, M)-factorization of any morphism h : X → C, then any pullback of e along a map in F is still in E.
We will prove that, in our case, the shifting Dec(X) of a simplicial object X is always stabilizing. For this it suffices to prove that exact objects are stabilizing since we have : Proposition 3.9). Any simplicial object that is contractible and also satisfies the Kan condition is exact.
As a consequence, if X satisfies the Kan condition, then its shifting Dec(X) is exact. We will need the following characterization of images in regular categories: Proposition 6 ( [11]). Let f : X → Z and g : Y → Z be two morphisms in a regular category C. Then g factors through the regular image of f if and only if there exist an object W of C with a morphism h : W → X and a regular epimorphism q : W → Y such that f h = gq.
Note that this equality means that an extension whose codomain is exact is trivial if and only if it is central.
Proof. The inequality always hold. To prove the converse, we consider the inclusion ϕ = (ϕ 1 , ϕ 2 ) of the equivalence relation d 0 (D 1 ∧ D 2 ) ∧ F 1 into X 1 × X 1 . Since this relation is smaller than d 0 (D 1 ∧ D 2 ), by the characterization given in Proposition 6 and the alternative construction of d 0 (D 1 ∧ D 2 ) given in Section 4, there must exist a regular epimorphism p : Z → d 0 (D 1 ∧ D 2 ) ∧ F 1 and a morphism α = α 1 , α 2 : Z → X 2 × X 1 X 0 such that d 0 α 1 = ϕ 1 p and d 1 α 1 = ϕ 2 p; and since moreover it is smaller than F 1 we have Now consider the maps One can check that the identity d i y j = d j−1 y i holds for all 0 ≤ i < j ≤ 3, so that these maps determine a map y from Z to the third simplicial kernel K 3 (Y), and we can consider the pullback Y being exact at Y 2 means that κ 3 is a regular epimorphism, and as a consequence so is p ′ . Consider now the maps One can check that the identity d i x j = d j−1 x i holds for all i < j and i = 2 = j, thus they determine a map x : Z ′ → Λ 3 2 (X) ; and moreover we have Since θ 3 2 is a regular epimorphism, so is p ′′ , and by construction we have d i α ′′ = x i p ′′ for i = 0, 1, 3 and f 3 α ′′ = α ′ p ′′ . Now the map d 2 α ′′ : Z ′′ → X 2 is such that (10)) such that ρ 1 β = α 2 p ′ p ′′ , ρ 2 β = d 2 α ′′ and ρ 3 β = f 1 d 1 α 1 p ′ p ′′ . Now we can check that Lemma 7. If Y is exact, then Y is stabilizing : given any morphism f : X → Y, the induced map f, η X : X → Y × I(Y) I(X) is stably in E.
Proof. To simplify the diagrams, we denote P = Y × I(Y) I(X). Let us consider a pullback square with g a regular epimorphism in Simp(C). We need to prove that I(h) : I(Q) → I(Z) is invertible. Since it is a map between internal groupoids, it is enough to prove that I(h) 0 and I(h) 1 are invertible. Note that the functor I leaves the objects X 0 unchanged, and thus f 0 , η 0 is an isomorphism, and thus so are h 0 and I(h) 0 . So we only need to prove is that I(h) 1 is an isomorphism.
Since Grpd(C) is a Birkhoff subcategory of Simp(C) and h is a regular epimorphism, the square is a double extension in Simp(C), and thus the square is a (regular) pushout in C. This already proves that h 1 = I(h) 1 is a regular epi. Now if there exists a map t : Z 1 → Q 1 /H 1 (Q) such that th 1 = (η Q ) 1 , then using the universal property of the pushout (12) we can construct a retraction for h 1 , which proves that it is an isomorphism. So we are left to prove that such a map t exists ; since h 1 is a regular epimorphism, it is enough to prove that Eq[h 1 ] ≤ H 1 (Q).
To prove this, we denote ψ 1 , ψ 2 : Eq[h 1 ] → Q 1 the two projections of the kernel pair. Then the commutativity of (11) (or rather, the corresponding commutative square involving h 1 in C) implies that where the last equality is the preceding lemma. As a consequence, we know that there must exist an arrow α : A → L and a regular epimorphism p : We now prove that f 2 , η 2 ρ 2 α factors through a degeneracy of P. More precisely, we prove that Since the degeneracy map s P 0 is induced by those of I(X) and Y, it is enough to prove that f 2 ρ 2 α and η 2 ρ 2 α factorize in the same manner through s Y 0 and s I(X) 0 respectively. By construction we must have By construction the two maps d are jointly monic, and thus these equalities implies that and this in turn implies that (13) hold. From this we find that Since Q 2 is the pullback of g 2 along f 2 , η 2 , there exists a unique map α ′ : A → Q 2 such that h 2 α ′ = s Z 0 h 1 ψ 1 p and g ′ 2 α ′ = ρ 2 α. From this, we find that 1 and h 1 are jointly monic, we have d Q 0 α ′ = ψ 1 p, and similarly d Q from the definition of Q 1 it suffices to check that the identity holds after composition with h 1 and g ′ 1 . We have α ′ Thus α ′ factorizes through the pullback of s Q 0 along d Q 2 , and thus (ψ 1 , ψ 2 )p = (d Q 0 , d Q 1 )α ′ factorizes through the inclusion of H 1 (Q) in Q 1 × Q 1 , which concludes the proof.
As a consequence, we then have Theorem 3. The Galois structure (Simp(C), Grpd(C), I, U, F ) admits a relative monotonelight factorization system (E ′ , M * ), where E ′ is the class of maps stably inverted by I and M * is the class of central extensions of this Galois structure.

Truncated simplicial objects and weighted commutators
For all n ≥ 2, the inclusion Grpd(C) ֒→ Simp n (C) factorizes through Simp(C), and the characterization of groupoids in truncated simplicial objects is identical. Moreover the construction of the equivalence relations H n (X) does not depend on the objects X m for m > n. Thus Grpd(C) can also be seen as a Birkhoff subcategory of Simp n (C), with the reflection defined in the same way, in the sense that the reflectors commute with the truncation functor. The characterization of central extensions also extends in the same way.
The inclusion Grpd(C) ֒→ Simp 1 (C) = RG(C) also factors through Simp(C), as every reflexive graphs admits at most one groupoid structure ( [13]). On the other hand, this time the reflection does not commute with the truncation, as the construction of H 1 (X) is dependent on X 2 and the face maps X 2 → X 1 . In fact, the reflection RG(C) → Grpd(C) is obtained by taking the quotient of X 1 by the Smith-Pedicchio commutator [D 0 , D 1 ] SP ( [35]). The central extensions of reflexive graphs in exact Mal'tsev categories (with coequalizers) with respect to this adjunction have been characterized in [17]. Note that this commutator is preserved by regular images, and is always smaller than the intersection; as a consequence, we always have the inequalities It turns out that this reflection can also be obtained by applying our results, at least when the category C is finitely cocomplete. Indeed, in that case the truncation functor Simp(C) → RG(C) has a left adjoint G, defined by taking left Kan extensions along the inclusion ∆ op 2 → ∆ op . Now since the inclusion Grpd(C) → RG(C) is the composition of the nerve functor and the truncation T , the functor IG must be a left adjoint to this inclusion. Thus our work can be used to give an alternative description of the Smith-Pedicchio commutator as the equivalence relation H 1 (G(X 1 , X 0 , d 0 , d 1 , s 0 )).
Let us make this construction explicit. The object X 2 = (G(X 1 , X 0 , d 0 , d 1 , s 0 )) 2 is the pushout X 1 + X 0 X 1 of s 0 : X 0 → X 1 along itself, with s 0 and s 1 the two canonical maps X 1 → X 1 + X 0 X 1 . In order to satisfy the simplicial identities we must then define d 0 to be the unique map for which d 0 s 0 = 1 and d 0 s 1 = s 0 d 0 , which we denote [1, s 0 d 0 ] : X 1 + X 0 X 1 → X 1 ; similarly, we must have d 1 = [1, 1] and d 2 = [s 0 d 1 , 1].
In the case where C is not only exact Mal'tsev but also semi-abelian ( [30,2]), there is for every object an order-preserving bijection between equivalence relations and normal subobjects, which is also compatible with regular images. Accordingly, our results can be easily translated in terms of normal subobjects, by replacing every kernel pair by the kernel of the corresponding morphism.
Proof. Let us denote K i ≤ X 1 the kernel of d i : X 1 → X 0 (for i = 0, 1). We recall from [23] the construction of the weighted commutator [K 0 , K 1 ] X 0 : we first define ψ as the map making the diagram