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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

MSC:

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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