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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 Quillen’s Theorem A. Mathematics Subject Classifications (2000): 18D05, 18D10, 55U40, 55P15, 55U10.
In this article we state and prove precise theorems on the homotopy classification 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.
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… (More)
We consider monads over varying categories, and by defining the morphisms of Kleisli and of Eilenberg-Moore from a monad to another and the appropriate transformations (2-cells) between morphisms of Kleisli and between morphisms of Eilenberg-Moore, we obtain two 2-categories MndKl and MndEM. Then we prove that MndKl and MndEM are, respectively, 2-isomorphic… (More)
By considering the notion of action of a categorical group G on another categorical group H we define the semidirect product HnG and classify the set of all split extensions of G by H. Then, in an analogous way to the group case, we develop an obstruction theory that allows the classification of all split extensions of categorical groups inducing a given… (More)
This paper explores the relationship amongst the various simplicial and pseudo-simplicial 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… (More)
This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors. Mathematical Subject Classification: 18D05, 55P15, 18F25.
Graded monoidal categories were introduced by Frohlich and C.T.C. ̈ Wall in 8 , where they presented a suitable abstract setting to study the Brauer group in equivariant situations. This paper is concerned with the analysis and classification of these graded monoidal categories, following a parallel treatment to that made in 2 for the non-monoidal case. In… (More)
The homotopy classification of graded categorical groups and their homomorphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts. Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or… (More)
The main result in this paper states that every strongly graded bialgebra whose component of grade 1 is a finite-dimensional Hopf algebra is itself a Hopf algebra. This fact is used to obtain a group cohomology classification of strongly graded Hopf algebras, with 1-component of finite dimension, from known results on strongly graded bialgebras. 2002… (More)