Nerves and classifying spaces for bicategories

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.Comment: 42 page

Spectra associated to symmetric monoidal bicategories

“…By [4,Theorem 6.4], the geometric realization of the 2-nerve of a strict 2-category C is equivalent to the classifying space of the category T C (see Appendix A for details). Applying this to -(2-categories) levelwise, we get the middle equivalence in the following corollary.…”

“…More precisely, if M is a monoidal category, then the simplicial category B M of [2] is equal to N †M. Furthermore (see [4,Remark 3.3]), since N p †M ' M p , the geometric realization jN †Mj gives a model for the delooping of the A 1 -space BM, that is, BBM ' jN †Mj.…”

“…An alternative construction for the classifying space of a bicategory C is described in [4]. It is the classifying space of the pseudosimplicial category whose n simplices are given by a C 0 ;:::;C n C.C 0 ; C 1 / C.C n 1 ; C n /:…”

“…As pointed out in [4,Remark 3.2], if C is a 2-category, then BC can be constructed by first taking the classifying space of the categories of morphisms, to obtain T C , which is a category enriched over topological spaces, and then taking BT C .…”

“…In fact, there are many distinct constructions of the classifying space of a bicategory [4]. All these constructions give equivalent spaces.…”