Antonio M. Cegarra

Group-theoretic algebraic models for homotopy types

In this paper a nonabelian version of the Dold-Kan-Puppe theorem is provided, showing how the Moore-complex functor defines a full equivalence between the category of simplicial groups and the category of what is called ‘hypercrossed complexes of groups’, i.e. chain complexes of nonabelian groups (Gn,δn) with an additional structure in the form of binary operations G1 × G1 → Gk. We associate to a pointed topological space X a hypercrossed complex (X); and the functor induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationship between (X) and Whiteheads crossed complex II(X) is established by a canonical surjection p:(X) → II(X), which is a quasi-isomorphism if and only if X is a J-complex. Algebraic models consisting of truncated chain-complexes with binary operations are deduced for n-types, and as an application we deduce a group-theoretic interpretation of the cohomology groups Hn(G, A).

Co)Homology of crossed modules

Abstract This paper begins with the observation that the category of crossed modules is tripleable over the category of sets, so that it is an algebraic category. This leads to a cotriple cohomology theory for crossed modules, whose basic study this work is mainly dedicated to.

On the Geometry of 2-Categories and their Classifying Spaces

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillens Theorem A.

Nerves and classifying spaces for bicategories

This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory C . It proves the fact that the geometric realizations of all of these possible candidate “nerves of C ” are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason’s “Homotopy Colimit Theorem” to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the “Grothendieck construction on the diagram”. Our results provide coherence for all reasonable extensions to bicategories of Quillen’s definition of the “classifying space” of a category as the geometric realization of the category’s Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental “delooping” construction. 18D05; 55U40 1 Introduction and summary Higher-dimensional categories provide a suitable setting for the treatment of an extensive list of subjects with recognized mathematical interest. The construction of nerves and classifying spaces of higher categorical structures, and bicategories in particular, discovers ways to transport categorical coherence to homotopical coherence and it has shown its relevance as a tool in algebraic topology, algebraic geometry, algebraic K ‐theory, string theory, conformal field theory and in the study of geometric structures on low-dimensional manifolds. This paper explores the relationship amongst the various simplicial and pseudosimplicial objects that have been (or might reasonably be) functorially and characteristically associated to any bicategory C . It outlines and proves in detail the far from obvious fact that the geometric realizations of all of these possible candidate “nerves of C ” are homotopy equivalent. Any one of these realizations could therefore be taken as the

On Graded Categorical Groups and Equivariant Group Extensions

In this article we state and prove precise theorems on the homotopy classication of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

On catn-groups and homotopy types☆

Abstract We give an algebraic proof of Lodays ‘Classification theorem’ for truncated homotopy types. In particular we give a precise construction of the homotopy cat n -group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusies ‘total Dec’.

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,⊗) over a small category ℬ. We show a natural bijection between the set of equivalence classes of such torsors and [B({ℬ}),B(G,⊗)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with πi(Y)=0 for all i≠1,2.

Cohomology of cofibred categorical groups

Abstract In this paper we study and interpret a certain non-abelian cohomology H i ( B , G ) , 0≤ i ≤2, of a small category B with coefficients in a B -cofibred categorical group G . The work is developed in a purely abstract setting, but several examples that indicate a very close connection with algebraic and topological problems are discussed explicitly. For instance, we obtain two new interpretations for the Brauer group of a Galois extension of commutative rings: an algebraic one in terms of equivalence classes of torsors over the Galois group and a topological one in terms of homotopy classes of cross-sections for a fibration over an Eilenberg–MacLane space of type K ( G ,1).

Classifying Spaces for Monoidal Categories Through Geometric Nerves

The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.

Higher dimensional obstruction theory in algebraic categories

In this paper we generalize Duskins low dimensional obstruction theory, established for the Barr-Becks cotriple cohomogy HG2, to higher dimensions by giving a new interpretation of HGn+1 in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to HGn+1(S,A) in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over HGn(S,A).