Well-known techniques from homological algebra and algebraic topology allow one to construct a cohomology theory for groups on which the action of a fixed group is given. After a brief discussion on the modules to be considered as coefficients, the first section of this paper is devoted to providing some definitions for this cohomology theory and then to proving that they are all equivalent. The second section is mainly dedicated to summarizing certain properties of this equivariant group cohomology and to showing several relationships with the ordinary group cohomology theory.

In this paper certain central extensions of crossed modules are classified. For these extensions, we obtain several results which extend the classical ones for central extensions of groups. In particular, central stem extensions of a perfect crossed module are classified in terms of a second integral homology crossed module.

In this paper we examine higher order Peiffer elements in simplicial Lie algebras and apply them to the Lie 2-crossed module and Lie crossed squares introduced by Ellis.

We associate to a continuous map between pointed spaces a long 2-exact sequence of homotopy pointed groupoids. The usual homotopy sequence of a map follows from this 2-exact sequence taking, for each groupoid, the set of connected components. We also study a condition of strong 2-exactness for a sequence of cat-groups and pointed groupoids.

In this paper we construct a functor $\Phi : \rm{pro}\mathcal{T}op \to \rm{pro}\mathcal{ANR}$ which extends Mardešic correspondence that assigns to every metrizable space its canonical $\mathcal{ANR}$-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that $\mathcal{ANR}$-spaces play for ordinary shape equivalences. In the last section we characterize SSDR-promaps, as defined by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.

We give a sufficient condition for the evaluation fibration, whose total space is the free iterated loop space, not to be totally non cohomologous to zero with respect to a given field.

This paper is an attempt to study extensions of symmetric categorical groups from a structural point of view. Using in a systematic way bilimits in the 2-category of symmetric categorical groups, we develop a theory which closely follows the classical theory of abelian group extensions. The basic results are established for any proper class of extensions, and a cohomological classification is obtained for those extensions whose epi part has a categorical section.

We apply recent work of A. Lazarev which develops an obstruction theory for the existence of $R$-algebra structures on $R$-modules, where $R$ is a commutative $S$-algebra. We show that certain $MU$-modules have such $A_\infty$ structures. Our results are often simpler to state for the related $BP$-modules under the currently unproved assumption that $BP$ is a commutative $S$-algebra. Part of our motivation is to clarify the algebra involved in Lazarev's work and to generalize it to other important cases. We also make explicit the fact that $BP$ admits an $MU$-algebra structure as do $E(n)$ and $\widehat{E(n)}$, in the latter case rederiving and strengthening older results of U. Würgler and the first author.

We characterize the categories which are projective covers of regular protomodular categories. Our result gives in particular a characterization of the categories with weak finite limits with the property that their exact completions are semi-abelian categories. As an application, we obtain a categorical proof of the recent characterization of semi-abelian varieties.

We compute the Taylor tower for Hochschild homology as a functor from augmented commutative simplicial $\dbQ$-algebras, to chain complexes over $\dbQ$. We use this computation to obtain the layers for the Taylor tower of rational algebraic $K$-theory. We also show that the Hodge decomposition for rational Hochschild homology is the decomposition of the Taylor tower of the augmentation ideal functor into its homogeneous layers when evaluated at a suspension.

This paper defines monoidal control, which is a specialization of control by entourage as presented in [5]. It is shown that on metric spaces monoidal control generalizes bounded control, and describes continuous control. A systematic way of obtaining results from bounded control, in the sense of [1], as results of monoidal-, and hence continouos-, control, is developed. Especially this provides versions of the Hurewicz and Whitehead theorems with monoidal control, thus simultaneously establishing them for continouos control over metric spaces.