Volume 19, issue 4 (2019)

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Semisimplicial spaces

Johannes Ebert and Oscar Randal-Williams

Algebraic & Geometric Topology 19 (2019) 2099–2150

This is an exposition of homotopical results on the geometric realisation of semisimplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen’s Theorems A and B for nonunital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the group-completion theorem.

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semisimplicial space, geometric realisation, Quillen Theorem A, Quillen Theorem B, group-completion
Mathematical Subject Classification 2010
Primary: 18G30, 55R35, 55U10
Received: 29 August 2018
Revised: 20 December 2018
Accepted: 3 January 2019
Published: 16 August 2019
Johannes Ebert
Mathematisches Institut
Universität Münster
Oscar Randal-Williams
Centre for Mathematical Sciences
University of Cambridge
United Kingdom