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Monotone-light factorization for Kan fibrations of simplicial sets with respect to groupoids
Abstract
We show that the class of second order covering maps of simplicial sets in the sense of R. Brown and G. Janelidze is a part of a factorization system for the class of Kan fibrations of simplicial sets.
... Our counterexample 2. 6 shows that the answer to this question is negative, in general. However, as we explain in Section 3, when C is an exact Mal'tsev category, one can see the comprehensive factorization for regular epimorphisms in Gpd(C) as a relative monotone-light factorization system (in the sense of [6]). ...
... Our counterexample 2. 6 shows that the answer to this question is negative, in general. However, as we explain in Section 3, when C is an exact Mal'tsev category, one can see the comprehensive factorization for regular epimorphisms in Gpd(C) as a relative monotone-light factorization system (in the sense of [6]). This result relies in particular on the fact that any internal groupoid X admits a (relative) stabilising object [4] that is the image Dec(X) of X under the décalage functor recalled in Section 2. ...
... In [6], the author proposes a notion of factorization system relative to a given subclass of arrows. Namely, given a category C and a class F of arrows in C containing identities, closed under composition, and pullback stable, a factorization system for F is a pair of classes of maps (E, M) such that: ...
Given an exact category , it is well known that the connected component reflector from the category of internal groupoids in to the base category is semi-left-exact. In this article we investigate the existence of a monotone-light factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation system in , where is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where is an exact Mal'tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in is the relative monotone-light factorisation system (in the sense of Chikhladze) in the category corresponding to the connected component reflector, where is the class of final functors and the class of regular epimorphic discrete 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]). ...
... 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] : ...
... This Galois structure admits a relative monotone-light factorization system, as shown in [15]. ...
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).
... Our counterexample 2. 6 shows that the answer to this question is negative, in general. However, as we explain in Sect. ...
... However, as we explain in Sect. 3, when C is an exact Mal'tsev category, one can see the comprehensive factorization for regular epimorphisms in Gpd(C) as a relative monotonelight factorization system (in the sense of [6]). This result relies in particular on the fact that any internal groupoid X admits a (relative) stabilising object [4] that is the image Dec(X) of X under the décalage functor recalled in Sect. 2. ...
... This allows one to view the trivial coverings of our Galois structure as part of a relative factorization system for the class F , as explained below. In [6], the author proposes a notion of factorization system relative to a given subclass of arrows. Namely, given a category C and a class F of arrows in C containing identities, closed under composition, and pullback stable, a factorization system for F is a pair of classes of maps (E, M) such that: ...
Given an exact category , it is well known that the connected component reflector from the category of internal groupoids in to the base category is semi-left-exact. In this article we investigate the existence of a monotone-light factorisation system associated with this reflector. We show that, in general, there is no monotone-light factorisation system in , where is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where is an exact Mal'tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in is the relative monotone-light factorisation system (in the sense of Chikhladze) in the category corresponding to the connected component reflector, where is the class of final functors and the class of regular epimorphic discrete fibrations.
... In this section we show that there is a factorisation system of surjective homomorphisms induced by the reflective subcategory Qnd * of Qnd, and we describe it explicitly. We refer the reader to the reference [5] for a discussion of general factorisation systems (see also [8] for the notion of factorisation system for a given class of morphism). ...
We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and, moreover, 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).
We study and compare two factorization systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorization system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorization system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorization systems.
If (, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:AB is in if each of its pullbacks lies in (that is, if it is stably in ), and is in M
* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (, M
*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M
*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.
A classical theory gives an equivalence between the category of covering maps of a space and the category of actions on sets of the fundamental groupoid of the space. We give a corresponding theory in dimension 2 for simplicial sets as a consequence of a Generalised Galois Theory. This yields an equivalence between a category of 2-covering maps of a simplicial set B and a category of actions on groupoids of a certain double groupoid constructed from B.