Homotopy Theory of Monoids and Group Completion
type: CreativeWork
description: This thesis presents several complete and partial models for the homotopy theory of monoids and the derived functor of group completion. We show that there is a simplicial model structure on the category of reduced simplicial sets that is Quillen equivalent to the Quillen model structure of simplicial monoids. Using this Quillen equivalence we recover the fact that the derived functor of group completion is isomorphic to the homotopy type of loops on the classifying space of a monoid. We use the Street nerve to show that the derived functor of group completion of monoids in the category of ω-groupoids for the Gray tensor product is isomorphic to group completion for simplicial monoids in low degrees. Finally we exploit the connection of ω-groupoids with the theory of rewriting for presentations of monoids to calculate the second homotopy group of the classifying space BM of a monoid M in terms of a chosen presentation by generators and relations.
abstract: This thesis presents several complete and partial models for the homotopy theory of monoids and the derived functor of group completion. We show that there is a simplicial model structure on the category of reduced simplicial sets that is Quillen equivalent to the Quillen model structure of simplicial monoids. Using this Quillen equivalence we recover the fact that the derived functor of group completion is isomorphic to the homotopy type of loops on the classifying space of a monoid. We use the Street nerve to show that the derived functor of group completion of monoids in the category of ω-groupoids for the Gray tensor product is isomorphic to group completion for simplicial monoids in low degrees. Finally we exploit the connection of ω-groupoids with the theory of rewriting for presentations of monoids to calculate the second homotopy group of the classifying space BM of a monoid M in terms of a chosen presentation by generators and relations.
author: Burke, Nigel
language: eng
keywords: Homotopy theory, Model categories, Monoids, Term rewriting
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/2d309ba6-0bb0-41fd-83a8-33bbbc80dc4a/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2021-02
date modified: 2026-07-22T22:24:14Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.72815