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Classifying spaces for braided monoidal categories and lax diagrams of bicategories
- Source :
- Advances in Mathematics. 226(1):419-483
- Publication Year :
- 2011
- Publisher :
- Elsevier BV, 2011.
-
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.<br />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
- Subjects :
- Classifying space
Pure mathematics
Mathematics(all)
Monoidal category
Homotopy colimit
General Mathematics
Loop space
18D05, 18D10, 55P15, 55P48
Mathematics::Algebraic Topology
Braided monoidal category
Bicategory
Grothendieck construction
Mathematics::Category Theory
FOS: Mathematics
Algebraic Topology (math.AT)
Category Theory (math.CT)
Mathematics - Algebraic Topology
Mathematics
Functor
Tricategory
Nerve
Mathematics - Category Theory
Homotopy type
Subjects
Details
- ISSN :
- 00018708
- Volume :
- 226
- Issue :
- 1
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics
- Accession number :
- edsair.doi.dedup.....5595becfd499aae53736dfcdb3fa07e7
- Full Text :
- https://doi.org/10.1016/j.aim.2010.06.027