Central extensions of precrossed and crossed modules

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,