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.
For a commutative cancellative semigroup S, we define the rank of S intrinsically. This definition implies that the rank of S equals the usual rank of its group of quotients. We also characterize the rank in terms of embeddability into a rational vector space of the greatest power cancellative image of S.
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.
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)
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 classi-ÿcation of several algebraic constructions long known as crossed products,… (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.
This paper explores the relationship amongst the various simpli-cial and pseudo-simplicial objects characteristically associated to any bicate-gory 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… (More)
The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over… (More)
The homotopy classification of graded categorical groups and their homo-morphisms 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)
This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors… (More)