Central extensions of precrossed and crossed modules
نویسنده
چکیده
The notion of centrality for crossed modules was introduced by Norrie in her thesis [7], in which she studied the category of crossed modules CM from an algebraic point of view, showing suitable generalizations of group theoretic concepts and results. Subsequently, Norrie’s approach was followed by Carrasco, Cegarra and R.-Grandjeán. In [5] they proved that CM is an algebraic category (i.e. there is a tripleable underlying functor from CM to the category of sets). This fact leads to the construction of cotriple homology and cohomology theories for crossed modules. In [3], Arias, Ladra and R.-Grandjeán began a similar study for the category of precrossed modules PCM. In this work we extended to precrossed modules Norrie’s definition of center, and we proved that PCM is also an algebraic category. In the aforementioned works some results on central extensions of crossed modules were developed. For example, Norrie studied the existence of universal central extensions in CM. Moreover, Carrasco, Cegarra and R.Grandjeán also classified the central extensions in CM with their second cohomology group. The analogue results for precrossed modules can be found in [4] and [1]. This talk will cover some of these and other results developed in the last years concerning central extensions in PCM and CM, with special attention to the connection between universal central extensions in both categories [2]. In general, universal central extensions in PCM and CM don’t coincide for a perfect crossed module. For example, for a two-sided ideal I of a ring R there is a perfect crossed module (E(I), E(R), i) of elementary matrices,
منابع مشابه
Relative Commutator Theory in Varieties of Ω-groups
We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ω-groups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules.
متن کاملMore about Homological Properties of Precrossed Modules
Homology groups modulo q of a precrossed P-module in any dimensions are deened in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second ho-mology group modulo q of a precrossed P-module which shows that for q = 0 our deenition is a natural extension of Conduch e and El-lis' deenition CE]. Some other properties of homologies of precross...
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 کامل