Semisimplicial spaces. (English) Zbl 1461.55014

This paper is intended as a technical reference for statements about semisimplicial spaces and selected applications of the techniques (to topological categories and the group completion theorem). Practically every statements comes with a proof, in a few instances references to existing literature are provided.
A semisimplicial space is a contravariant functor from the category of finite totally ordered non-empty sets and their order-preserving injections to the category of (compactly generated) topological spaces. (Thus a semisimplicial space is a “simplicial space without degeneracies”.) There is a notion of geometric realisation, obtained from a semisimplicial space by gluing (topological) simplices. These are the topic of Sections 1 and 2.
In Section 3, the authors discuss “nonunital topological categories” (which may not have identity morphisms); such a structure gives rise to a semisimplicial space, its nerve, and hence via geometric realisation to a topological space, the classifying space.
Section 4 contains generalisations of Quillen’s theorems A and B to nonunital topological categories.
Next, Section 5 contains a generalisation of the result that making the topology of a (suitable) topological category discrete does not change the classifying space up to homotopy equivalence.
In Section 6 the authors discuss, as an application of the semi-simplicial techniques developed so far, the group completion theorem for topological monoids.
The exposition closes with a discussion of the product of simplicial spaces in Section 7: the (semisimplicial!) geometric realisation functor, restricted to simplicial spaces, is shown to preserve products up to homotopy equivalence


55U10 Simplicial sets and complexes in algebraic topology
57R90 Other types of cobordism
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55P47 Infinite loop spaces
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