theory, and change of base for groupoids and multiple

This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.

and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A

Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Čech derived functors of the lower central series functors Zk. The paper ends with an application to algebraic K-theory. Introduction and Summary The well known Hopf formula for the second integral homology of a group says that for a given group G there is an isomorphism H2(G) ∼ = R ∩ [F, F]

Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed P-module which shows that for q = 0 our definition is a natural extension of Conduch'e and Ellis ' definition [CE]. Some other properties of homologies of precrossed P-modules are investigated. Introduction The homology of precrossed modules was introduced by Conduch'e and Ellis in [CE]. The aim of this paper is to pursue their line of investigation homological properties of precrossed modules. Let P be a group. A precrossed P-module (M; ) is a group homomorphism : M ! P together with an action of P on M denoted by p m for p 2 P and m 2 M , which satisfies the following condition: ( p m) = p(m)p \Gamma1 : If in addition the following Peiffer identity holds (m) m 0 = mm 0 m \Gamma1 ; (M; ) is a crossed P-module (see e.x. [BH]). A mor...

Higher extensions and higher central extensions, which are of importance to non-abelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. 0