[Show abstract][Hide abstract] ABSTRACT: Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion
allows, paralleling the group case, to introduce the notion of K-categorical groups of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then,
and are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated
to the fibre homotopy sequence As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.
[Show abstract][Hide abstract] ABSTRACT: Thomason's Homotopy Colimit Theorem has been extended to bicategories and
this extension can be adapted, through the delooping principle, to a
corresponding theorem for diagrams of monoidal categories. In this version, we
show that the homotopy type of the diagram can be also represented by a genuine
simplicial set nerve associated with it. This suggests the study of a homotopy
colimit theorem, for diagrams $\b$ of braided monoidal categories, by means of
a simplicial set {\em nerve of the diagram}. We prove that it is weak homotopy
equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained
from composing $\b$ with the geometric nerve functor of braided monoidal
categories.
[Show abstract][Hide abstract] ABSTRACT: 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 to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves. Comment: This a revised version (with 59 pages now) of our paper on realizations of braided categories, where we have taken into account the referee's report. Indeed, we are much indebted to the referee, whose useful observations greatly improved our manuscript
Advances in Mathematics 07/2009; 226(1). DOI:10.1016/j.aim.2010.06.027 · 1.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 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.
[Show abstract][Hide abstract] ABSTRACT: Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.
Topology and its Applications 02/2007; 154(4-154):834-847. DOI:10.1016/j.topol.2006.09.005 · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we introduce and study the first cohomology categorical group H 1 (G, A) of a categorical group G with coefficients in a braided categorical group A provided of a coherent G-action. The fundamental exact sequence connecting H 0 and H 1 in this context is then established.
Communications in Algebra 10/2006; 10(10). DOI:10.1080/00927870600860841 · 0.39 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we present a cohomological description of the equivariant Brauer group relative to a Galois finite extension of fields endowed with the action of a group of operators. This description is a natural generalization of the classic Brauer–Hasse–Noether's theorem, and it is established by means of a three-term exact sequence linking the relative equivariant Brauer group, the 2nd cohomology group of the semidirect product of the Galois group of the extension by the group of operators and the 2nd cohomology group of the group of operators.
Journal of Algebra 02/2006; 296(1):56-74. DOI:10.1016/j.jalgebra.2005.11.032 · 0.60 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: If G is a group, then the category of G-graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from G. 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).
[Show abstract][Hide abstract] ABSTRACT: 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. Breen's 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.
[Show abstract][Hide abstract] ABSTRACT: In this paper we prove that, for any Galois finite field extension $F/K$ on which a separated group of operators $\Gamma$ is acting, there is an isomorphism between the group of equivariant isomorphism classes of finite dimensional central simple $K$-algebras endowed with a $\Gamma$-action and containing $F$ as an equivariant strictly maximal subfield and the second equivariant cohomology group of the Galois group of the extension.
Bulletin of the Belgian Mathematical Society, Simon Stevin 07/2003; 10(3). · 0.44 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 groups, or rings, or rings-groups or algebras as well as for graded Clifford systems with operators, equivariant Azumaya algebras over Galois extensions of commutative rings and for strongly graded bialgebras and Hopf algebras with operators. These specialized classifications follow from the theory of graded categorical groups after identifying, in each case, adequate systems of factor sets with graded monoidal functors to suitable graded categorical groups associated to the structure dealt with.
Theory and Applications of Categories 06/2003; 11(1). · 0.49 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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 pi(Y)=0 for all i?1,2.
[Show abstract][Hide abstract] ABSTRACT: 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 Γ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings.
Journal of Algebra 07/2001; 241(2):620-657. DOI:10.1006/jabr.2001.8769 · 0.60 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we do phrase the obstruction for realization of a generalized group character, and then we give a classification
of Clifford systems in terms of suitable low-dimensional cohomology groups.
[Show abstract][Hide abstract] ABSTRACT: 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 classification of several algebraic constructions long known as crossed products, appearing in many different contexts such as monoids, Clifford systems or twisted group rings.
Journal of Pure and Applied Algebra 12/2000; 154(1):117-141. DOI:10.1016/S0022-4049(99)00059-6 · 0.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: . In this paper we use Quillen's model structure given by Dwyer-Kan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case. 1. Introduction 1.1. Summary. A well-known and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the t...
Theory and Applications of Categories 11/2000; · 0.49 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: If S⊆Z is a multiplicative system and C is the class of the S-torsion abelian groups, we study Serre modC homotopy theory in the subcategories of simplicial groups whose objects have trivial Moore complex in dimensions less than r and greater than n for 0≤r≤n. This is carried by giving a closed model structure in these categories and then studying the associated homotopy theory. When n→∞ we obtain the Serre homotopy theory for r-reduced simplicial groups studied by Quillen in Ann. Math. 90 (1969) 205–295. If S=Z−{0} and r=1 we have the corresponding rational homotopy theory. The case n=r+1 allows to consider Serre homotopy theory in categories of cat-groups or crossed modules of groups.
Journal of Pure and Applied Algebra 03/2000; 147(2):107-123. DOI:10.1016/S0022-4049(98)00143-1 · 0.47 Impact Factor