ACTIONS IN MODIFIED CATEGORIES OF INTEREST WITH APPLICATION TO CROSSED MODULES Dedicated to Teimuraz Pirashvili on his 60th birthday
نویسندگان
چکیده
The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss ' Cat-Ass we consider this problem in Cat-Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of category, which will be called modified category of interest. Applying the results obtained in this direction and the equivalence of categories we find a condition under which there exists the split extension classifier of a crossed module and give the corresponding construction.
منابع مشابه
Limits in modified categories of interest
We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows. These constructions unify many corresponding results for the cases of groups, commutative algebras and can also be adapted to ...
متن کاملThe category of generalized crossed modules
In the definition of a crossed module $(T,G,rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate som...
متن کاملINTERNAL CROSSED MODULES AND PEIFFER CONDITION Dedicated to Dominique Bourn on the occasion of his 60th birthday
In this paper we show that in a homological category in the sense of F. Borceux and D. Bourn, the notion of an internal precrossed module corresponding to a star-multiplicative graph, in the sense of G. Janelidze, can be obtained by directly internalizing the usual axioms of a crossed module, via equivariance. We then exhibit some sufficient conditions on a homological category under which this...
متن کاملMonoidal Categories of Comodules for Coquasi Hopf Algebras and Radford’s Formula
We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford’s S formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a ...
متن کاملACTION ACCESSIBILITY FOR CATEGORIES OF INTEREST Dedicated to Dominique Bourn on the occasion of his 60th birthday
We prove that every category of interest (in the sense of [17]) is action accessible in the sense of [8]. This fact allows us to give an intrinsic description of centers and centralizers in this class of categories. We give also some new examples of categories of interest, mainly arising from Loday’s paper [11], [12] and [14].
متن کامل