Abstract: Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.
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, â),
a categorical crossed module that algebraically represents the homotopy 3-type of X.
Abstract: For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory
of G-spaces and it is called the âabelian cohomology of G-modulesâ. Then, as the main results of this paper, natural one-toone
correspondences between elements of the 3rd cohomology groups of G-modules, G-equivariant pointed simply-connected
homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all
these objects with equivariant quadratic functions between G-modules is also discussed.
Abstract: The codiagonal functor W transfers a Quillen closed model structure on the
bisimplicial set category from the ordinary model category of simplicial sets. This bisimplicial
model structure is different from the so called Moerdijk model structure, which is
similarly transferred from simplicial sets but through the diagonal functor. We show the
mutual relationship of these two closed model structures on the category of bisimplicial sets.
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 BrauerâHasseâNoetherâs 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.
Abstract: The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by
the simplicial set WX, the bar construction on X. We stress the interest of this result by showing two
relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem
of Thomason, for diagrams of small categories, and the generalized EilenbergâZilber theorem of
DoldâPuppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model
for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish
in degree 4 and higher.
Abstract: The usual constructions of classifying spaces for monoidal categories produce
CW-complexes with many cells that, moreover, do not have any proper geometric
meaning. However, geometric nerves of monoidal categories are very handy simplicial
sets whose simplices have a pleasing geometric description: they are diagrams with
the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper
is to prove that geometric realizations of geometric nerves are classifying spaces for
monoidal categories.
Abstract: 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 Quillenâs Theorem A.
Abstract: In this paper we prove that, for any Galois finite field extension
F/K on which a separated group of operators G is acting, there is an isomorphism
between the group of equivariant isomorphism classes of finite dimensional
central simple K-algebras endowed with a G-action and containing F as
an equivariant strictly maximal subfield and the second equivariant cohomology
group of the Galois group of the extension.
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.
Abstract: 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.
Abstract: Categorical B-groups, or (co)fibred categorical groups over a small category B, are
relevant in algebraic topology, since they are algebraic homotopy 2-types of homotopy
coherent diagrams of path-connected spaces, and also in algebra, since they
include the graded categorical groups. The main objective of this paper is to state
and prove a classification theorem for these categorical B-groups. In this classification,
two categorical B-groups which are connected by a monoidal B-equivalence
are considered the same. Hence, we solve the problem of giving a complete invariant
of this equivalence relation by means of triples (G, A, c), consisting of a diagram of
groups, G : B ! Gp, with the shape of the category B, a G-module A and a cohomology
class c 2 H3B(G,A). Here, Hn
B (G,A), n 0, are the cohomology groups of
the diagram of groups G, whose study the article is also mainly dedicated to. They
are defined by using simplicial methods: from a diagram of groups one obtains a
diagram of simplicial sets, by pointwise applying the classifying space construction
Abstract: 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.
Abstract: This paper begins with the observation that the category of crossed modules is tripleable
over the category of sets, so that it is an algebraic category. This leads to a cotriple cohomology
theory for crossed modules, whose basic study this work is mainly dedicated to.
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,â) over a small
category B. We show a natural bijection between the set of equivalence classes of such torsors
and [B(B),B(G,â)], 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 Ïi(Y ) = 0 for all i = 1, 2.
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.
Abstract: The long-known results of SchreierEilenbergMac 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 commutative rings.
