On Categorification (original) (raw)

Categorification and group extensions

Applied Categorical Structures, 2002

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a "coboundary" condition (Proposition 4.5).

Group-theoretic algebraic models for homotopy types

Journal of Pure and Applied Algebra, 1991

In this paper a nonabelian version of the Dold-Kan-Puppe theorem is provided. showing hou the Moore-complex functor defines a full equivalence between the category of simplicial groups and the category of what is called 'hypcrcrcssed complexes of groups'. i.e. chain complexes of nonabelian groups (G,,. S,,) with an additional structure in the form of binary operations G, x G,* G,. We associate to a pointed topological space X a hypercrossed complex A(X): and the functor 2 induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationshjp between s(X) and Whitehead's crossed complex II(X) is established by a cano:!ical surjection p : ii(X)-* II(X). which is a quasi-isomorphism if and only if X is a J-complex. Algebraic rnodels consisting of truncated chain-complexes with binary operations I;re deduced for rl-types. and as an application we deduce a group-theoretic interpretation of the cohomology groups H"(G. A).

Some properties of group-theoretical categories

Journal of Algebra, 2009

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following [O2]) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category.

On catn-groups and homotopy types

Journal of Pure and Applied Algebra, 1993

Bullejos, M., A.M. Cegarra and J. Duskin, On cat"-groups and homotopy types, Journal of Pure and Applied Algebra 86 (1993) 135-154. We give an algebraic proof of Loday's 'Classification theorem' for truncated homotopy types. In particular we give a precise construction of the homotopy cat"-group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusie's 'total Dee'. modules.

Cohomology of cofibred categorical groups

Journal of Pure and Applied Algebra, 1999

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 coe cients in a B-coÿbred 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 ÿbration over an Eilenberg-MacLane space of type K(G; 1).

Categories: History, basic and developments

vii Acknowledgements viii Preface ix Bibliography 58 vi Abstract This project was an opportunity for the author to be introduced to the basic notions of Category Theory and their applications to Algebraic Topology during the early stages of the development of the theory. In terms of the basics, the notions of Category, Functor and Natural Transformation are introduced. Within this context, the process of formulation of various Homology and Cohomology groups in terms of Functors are examined. The notion of Adjoint Functors are also developed to some extent. Finally,

Categorical, Homological, and Homotopical Properties of Algebraic Objects

Journal of Mathematical Sciences, 2017

This monograph is based on the doctoral dissertation of the author defended in the Iv. Javakhishvili Tbilisi State University in 2006. It begins by developing internal category and internal category cohomology theories (equivalently, for crossed modules) in categories of groups with operations. Further, the author presents properties of actions in categories of interest, in particular, the existence of an actor in specific algebraic categories. Moreover, the reader will be introduced to a new type of algebras called noncommutative Leibniz-Poisson algebras, with their properties and cohomology theory and the relationship of new cohomologies with well-known cohomologies of underlying associative and Leibniz algebras. The author defines and studies the category of groups with an action on itself and solves two problems of J.-L. Loday. Homotopical and categorical properties of chain functors category are also examined.

Fusion categories and homotopy theory

Quantum Topology, 2010

We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically.

On Graded Categorical Groups and Equivariant Group Extensions

Canadian Journal of Mathematics, 2002

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.