Papers by Antonio Cegarra
This work contributes to clarifying several relationships between certain higher categorical stru... 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 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 ...
Like categories, small 2-categories have well-understood classifying spaces. In this paper, we de... more Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend to homotopy colimits of 2-functors lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the Homotopy Invariance Theorem (by Bousfield and Kan), the Homotopy Colimit Theorem (Thomason), Theorems A and B (Quillen), or the Homotopy Cofinality Theorem (Hirschhorn).
Journal of Pure and Applied Algebra, 2007
For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises fr... more 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 "abelian cohomology of G-modules". Then, as the main results of this paper, natural one-toone correspondences between elements of the 3 rd 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.
A.: On graded categorical groups and equivariant group extensions
Abstract. In this article we state and prove precise theorems on the homotopy classification of g... more 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.
Nerves and Classifying Spaces for Bicategories
This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects... 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 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 ...
Classifying Spaces for Braided Monoidal Categories and Lax Diagrams of Bicategories
This work contributes to clarifying several relationships between certain higher categorical stru... 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 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
On the category of bisimplicial sets there are different Quillen closed model structures associat... more On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. 1. Introduction and
On the category of bisimplicial sets there are different Quillen closed model structures associat... more On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. 1. Introduction and
STRUCTURE AND CLASSIFICATION OF MONOIDAL Groupoids
This work contributes to clarifying several relationships between certain higher categorical stru... more This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural 'filling condition'. Any such double groupoid characteristically has associated to it 'homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of 'weak equivalence' between such double groupoids, and a corresponding 'homotopy category' is defined. Our main result in the paper states that the geometric realization functor induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy 2-types (that is, the homotopy category of all topological spaces with the property that the n th homotopy group at any base point vanishes for n ≥ 3). A quasi-inverse functor is explicitly given by means of a new 'homotopy double groupoid' construction for topological spaces.
Bicategorical Homotopy Fiber Sequences
Homology, Homotopy and Applications, vol.4(1), 2002, pp.1–23 COHOMOLOGY OF GROUPS WITH OPERATORS
Well-known techniques from homological algebra and alge-braic topology allow one to construct a c... more Well-known techniques from homological algebra and alge-braic topology allow one to construct a cohomology theory for groups on which the action of a fixed group is given. After a brief discussion on the modules to be considered as coefficients, the first section of this paper is devoted to providing some def-initions for this cohomology theory and then to proving that they are all equivalent. The second section is mainly dedicated to summarizing certain properties of this equivariant group cohomology and to showing several relationships with the or-dinary group cohomology theory. 1.
Graded extensions of monoidal categories
The long-known results of Schreier-Eilenberg-Mac Lane on group extensions are raised to a categor... 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 commutative rings.
www.elsevier.com/locate/jalgebra Equivariant Brauer groups and cohomology ✩
In this paper we present a cohomological description of the equivariant Brauer group relative to ... more 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.
THE RANK OF A COMMUTATIVE Cancellative Semigroup
For a commutative cancellative semigroup S, we define the rank of S intrinsically. This definitio... more 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.
Homotopy Fibre Sequences Induced by 2-FUNCTORS
This paper contains some contributions to the study of the relationship between 2-categories and ... 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.
On the geometry of 2-categories and their classifying spaces
In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categor... more 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.
arXiv: Category Theory, 2014
The homotopy theory of higher categorical structures has become a relevant part of the machinery ... more The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between B\'enabou's bicategories and the homotopy types of their classifying spaces. Mainly, we state and prove an extension of Quillen's Theorem B by showing, under reasonable necessary conditions, a bicategory-theoretical interpretation of the homotopy-fibre product of the continuous maps induced on classifying spaces by a diagram of bicategories $\mathcal{A}\to\mathcal{B}\leftarrow \mathcal{A}'$. Applications are given for the study of homotopy pullbacks of monoidal categories and of crossed modules.
We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded cha... more We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded chain complexes of Lie algebras L = (Ln, ∂n) endowed with an additional structure by means of a suitable set of bilinear maps Lr × Ls → Ln. The Moore complex of any simplicial Lie algebra acquires such a structure and, in this way, we prove a Dold-Kan type equivalence between the category of simplicial Lie algebras and the category of hypercrossed complexes of Lie algebras. Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented.
Some Algebraic Applications of Graded Categorical Group Theory
ABSTRACT. The homotopy classification of graded categorical groups and their homomorphisms is app... more 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. 1.
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Papers by Antonio Cegarra