A Simplicial Approach to Stratified Homotopy Theory



Nand-Lal, SJ
(2019) A Simplicial Approach to Stratified Homotopy Theory. Doctor of Philosophy thesis, University of Liverpool.

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Abstract

This thesis provides a framework to study the homotopy theory of stratified spaces, in a way that is compatible with previous approaches. In particular our approach will be closely related to the work of Frank Quinn on homotopically stratified sets. We introduce a stratified analogue of the geometric realisation-singular simplicial set adjunction, allowing us to relate simplicial sets to stratified spaces. This allows us to cofibrantly transfer the Joyal model structure from simplicial sets to the category of fibrant stratified spaces. We have chosen to use the Joyal model structure on simplicial sets over the Quillen model structure. This choice allows a partial (ordered) composition of simplices, which under the stratified adjunction corresponds to concatenation of stratified paths. One of the biggest advantages of working in a simplicially enriched model structure is the ability to exploit the combinatorial nature of simplicial sets, which helps us to prove results about stratified spaces. By studying the cofibrations and fibrations that we transfer to the category of stratified spaces, we see that the cofibrant stratified spaces satisfy one condition that Quinn imposed for homotopically stratified sets, and that the fibrant stratified spaces satisfy the other condition. Consequently, the cofibrant-fibrant stratified spaces in our model structure are closely related to homotopically stratified sets. To use our framework to study homotopy theory, we need a notion of basepoint for a stratified space. We define the basing of a stratified space to be a factoring of the counit on the underlying poset through a choice of continuous map to the underlying topological space. The requirement of a stratified space to be based provides a restriction on the stratified spaces, and as such there are examples of cofibrant-fibrant stratified spaces which cannot be based. To justify this approach we are able construct an adjunction between stratified suspension and loop space functors. In addition, we are able to construct an indexed family of categories for a based fibrant stratified space, which we call the homotopy categories of a stratified space. Importantly, in the case of a trivially stratified connected space, the homotopy categories coincide with the homotopy groups of the underlying topological space. The homotopy categories of a based fibrant stratified space behave analogously to homotopy groups. For example, we are able to extract a long exact sequence of homotopy categories from a stratified fibration. Furthermore, we are able to provide partial results towards construction of a Postnikov Tower of a based fibrant stratified space. Further research is required to complete this construction, which would hopefully lead to a stratified analogue of Eilenberg-Mac Lane spaces.

Item Type: Thesis (Doctor of Philosophy)
Additional Information: stephennandlal@gmail.com
Uncontrolled Keywords: Homotopy Theory, Stratified Spaces, Simplicial Homotopy Theory, A Simplicial Approach to Stratified Homotopy Theory, Stratified Homotopy Theory, Abstract Homotopy Theory, Quasi-Categories, Based Stratified Spaces, Stratified Fibration, Homotopy Categories of a Fibrant Stratified Space
Divisions: Fac of Science & Engineering > School of Mathematics
Depositing User: Symplectic Admin
Date Deposited: 29 Apr 2019 15:47
Last Modified: 09 Jan 2021 04:06
DOI: 10.17638/03036209
Supervisors:
  • Woolf, Jon
URI: https://livrepository.liverpool.ac.uk/id/eprint/3036209