Antonio R. Garzón

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

Closed model structures for algebraic models of n-types"'

In this paper we give a general method to obtain a closed model structure, in the sense of Quillen, on a category related to the category of simplicial groups by a suitable adjoint situation. Applying this method, categories of algebraic models of connected types such as those of crossed modules of groups (2-types), 2-crossed modules of groups (3-types) or, in general. n-hypercrossed complexes of groups ((n + I)-types), as well as that of n-simplicial groups (all types), inherit such a closed model structure.

Schreier theory for singular extensions of categorical groups and homotopy classification

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ϕ: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ϕ.

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.

Obstruction Theory for Extensions of Categorical Groups

For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1→Z(H)→H→Aut(H)→Out(H)→1 is raised to a categorical group level by using a suitable notion of exactness. Breens Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg–Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.

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).

Graded extensions of categories

The problem of extending categories by groups, including theory of obstructions, is studied by means of factor systems and various homological invariants, generalized from Schreier{ Eilenberg{Mac Lane group extension theory. Explicit applications are then given to the classication of several algebraic constructions long known as crossed products, appearing in many dierent contexts such as monoids, Cliord systems or twisted group rings. c 2000 Elsevier Science B.V. All rights reserved. MSC: 18D30; 18G50; 16S35

Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n≥1.

Homotopy theory for truncated weak equivalences of simplicial groups

In this paper we give for any r, n ,0 % r % n, a Quillen’s model structure to the category of simplicial groups where the weak equivalences are those morphisms f E such that p q (f E ) is an isomorphism for r % q % n. This is carried out by studying the cases r fl 0 and n U¢ previously and, in each one of them, we make explicit some constructions for the associated homotopy theories, such as the cylinder and path objects and the loop and suspension functors, and we also relate the simplicial homotopy relation to the homotopy relation obtained from these structures.

Equivariant Extensions of Categorical Groups

If Γ is a group, then the category of Γ-graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from Γ. In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreier’s theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).