Homotopy coherent category theory

Homotopy Coherent Category Theory

A Coherent Homotopy Category of 2-track Commutative Cubes

We consider a category H ⊗ (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of H ⊗ to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct ...

Coherent Functors in Stable Homotopy Theory

such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [1], but it is also implicit in the work of Freyd on stable homotopy [9]. In this paper we characterize coherent functors in a number of ways and use the...

Homotopy Theory and TDA with a View Towards Category Theory

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view. In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories wit...

A Model Category for the Homotopy Theory of Concurrency

We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories. 1. Geometric models of concurrency Algebra...