A convenient category of locally stratified spaces



Nicotra, Stefano
(2020) A convenient category of locally stratified spaces. PhD thesis, University of Liverpool.

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Abstract

In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is an ∞-category has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and show that the canonical functor θ_A from the fundamental category of a simplicial set A to the fundamental category of its realisation is essentially surjective. We show that the functor θ_A sends split monomorphisms to isomorphisms, in particular we show that θ_A is not necessarily an equivalence of categories. On the other hand, we show that the fundamental category of the realisation of the simplicial circle is equivalent to the monoid of the natural numbers. To conclude, we define left covers of locally stratified spaces and we show that, under suitable assumptions, the category of representations of the fundamental category of a simplicial set is equivalent to the category of left covers over its realisation.

Item Type: Thesis (PhD)
Divisions: Fac of Science & Engineering > School of Mathematics
Depositing User: Symplectic Admin
Date Deposited: 09 Apr 2020 13:51
Last Modified: 09 Jan 2021 02:20
DOI: 10.17638/03078614
Supervisors:
  • Woolf, Jon
URI: https://livrepository.liverpool.ac.uk/id/eprint/3078614