A. R. Garzón

Publications (7)0.6 Total impact

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  Available from: ArXiv

  Article: A homotopy colimit theorem for diagrams of braided monoidal categories

  A. R. Garzón, R. Pérez
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  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.
  03/2011;
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  Available from: ArXiv

  Article: Classifying spaces for braided monoidal categories and lax diagrams of bicategories

  P. Carrasco, A. M. Cegarra, A. R. Garzón
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  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
  07/2009;
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  Available from: ArXiv

  Article: Nerves and classifying spaces for bicategories

  P. Carrasco, A. M. Cegarra, A. R. Garzón
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  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.
  04/2009;
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  Available from: hera.ugr.es

  Article: Obstruction Theory for Extensions of Categorical Groups

  P. Carrasco, A. R. Garzón
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  ABSTRACT: For any categorical group H, we introduce the categorical group O ut(H) and then the well-known group exact sequence 1Z(H)HAut(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 O ut(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.
  Applied Categorical Structures 01/2004; 12(1):35-61. · 0.60 Impact Factor
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  Available from: ugr.es

  Article: Higher dimensional obstruction theory in algebraic categories

  A.M. Cegarra, M. Bullejos, A.R. Garzón
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  ABSTRACT: In this paper we generalize Duskin's low dimensional obstruction theory, established for the Barr-Beck's cotriple cohomogy , to higher dimensions by giving a new interpretation of in terms of obstructions to the existence of non-singular n-extensions or realizations to n-dimensional abstract kernels. We find a surjective map Obs from the set of all n-dimensional abstract kernels with center a fixed S-module A to in such a way that an abstract kernel has a realization if and only if its obstruction vanishes, the set of equivalence classes of such realizations being in this case a principal homogeneous space over .
  Journal of Pure and Applied Algebra.
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  Available from: ugr.es

  Article: Equivariant group cohomology and Brauer group

  A. M. Cegarra, A. R. Garzón
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  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.
• Article: The homotopy categorical crossed module of a CW-complex

  P. Carrasco, A.M. Cegarra, A.R. Garzón
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  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.