A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules

Abstract

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids. Indeed, for extensions with abelian kernel, the push forward construction makes the following classical assignment functorial: Opext C (Y, −) : [Abelian objects in C ↓ Y ] → [Abelian groups] (see [1] for groups and other algebraic categories, and [7] for pointed protomodular categories). Here, the supplementary conditions ensure that a gives rise to a morphism of abelian group objects in C ↓ Y. Indeed, the same situation can be treated also with the alternative approach developed by Bourn in [4], where a direction functor assigns to each abelian extension of the object Y a Y-module. Under this perspective, the push forward construction recovers the fact that such direction functor is a pseudo cofibration (see [4]). A generalization of the push forward construction arises when we consider the normal monomorphism appearing in any short exact sequence as an instance of a pre-crossed module. When the base category is the category of groups, under suitable hypothesis, it is possible to push forward along a map not only a normal monomorphism, but any pre-crossed module. This way we obtain a crossed module with the same cokernel (to the best of our knowledge, the push forward of a pre-crossed module was introduced by Noohi in [20]). The purpose of the present work is to develop a push forward construction in the intrinsic setting of a semi-abelian category C, where the notion of internal crossed module was introduced by Janelidze in [12]. In fact, in Theorem 3.6, we present necessary and sufficient conditions, expressed in terms of internal actions, for the push forward of a given pre-crossed module to exist in C (for the case of a crossed module, a push forward construction with different conditions was independently developed by Hartl [11]). These conditions simplify when the category C is action accessible [8], as presented in Theorem 3.10. The last result is very useful for the construction of push forward in many algebraic examples, like those of rings, Lie algebras, associative algebras and, more in general, any category of interest in the sense of Orzech [21]. The push forward construction, together with its main property (see Theorem 3.6, (PF)), turns out to be strongly related to the comprehensive factorization of internal functors (see [3]). Our investigation shows that push forwards can be used in order to factorize morphisms of crossed modules, so that final functors between internal groupoids can be characterized as push forward squares.

Fibered aspects of Yoneda's regular span

“…As a consequence of Theorem 2.13 in [17], j ′ is a normal monomorphism and the morphism (β, q · 0, 1 ) of crossed modules induces an isomorphism on cokernels.…”

“…With reference to the general case of strongly semi-abelian categories, one observes that the pushout computed in k-Vect in the above diagram, turns out to be a push forward when the diagram is considered in AssAlg. This can be proved by means of the characterization of push forwards given in [17,Theorem 2.13]: the square α · j = j ′ · µ presents j ′ as the push forward of j along µ since (α, µ) induces an isomorphism between the cokernels of j and j ′ (both are kernels of p) and a regular epimorphism between the kernels (in fact, the last is an isomorphism too, since j and j ′ are monomorphisms).…”

“…However, the square of vertexes B, G 2 , B ′ and B ′ × B G 2 does commute, since in fact ρ is the coequalizer of the pair (ι 1 · β, ι 2 · j). The homomorphism ρ · ι 1 is the push forward of j along β [17]. It is possible to prove that the ρ · ι 1 is in fact a normal monomorphism and that it has the same (isomorphic) cokernel as j so that we can obtain a homomorphism δ such that δ · ρ · ι 2 = ∂.…”

“…If we also regard B ′ as a trivial G 2 -module, then β is clearly equivariant with respect to the G 2 -actions and the semidirect product B ′ ⋊ G 2 is in fact a direct product. It is immediate then to see that the conditions for the push forward are fulfilled and we can describe explicitly the construction following the lines of Section 2.3 in [17]. The precrossed module of Lemma 2.11 in [17] reduces, in our case, to the morphism β, −j : B → B ′ × G 2 , which is monomorphic (since its second projection −j is), hence it is a kernel (notice that the existence of −j is ensured by the centrality of B in G 2 ).…”

“…the opcartesian liftings in the fibers giving rise to a fiberwise opfibration) we have to replace pushouts by push forwards. The internal version of the latter, developed in [17], is the main tool to treat crossed n-fold extensions in any strongly semi-abelian category, as detailed in Section 4.3, where we explicitly describe opcartesian liftings as push forwards of crossed extensions. The proof of Lemma 4.10 gives a recipe for performing the push forward of crossed extensions in several categories of algebraic varieties, as (non-unital) associative algebras, Lie algebras and Leibniz algebras among others.…”