Cofibration and Model Category Structures for Discrete and Continuous Homotopy

Abstract

We show that the categories PsTop and Lim of pseudotopological spaces and limit spaces, respectively, admit cofibration category structures, and that PsTop admits a model category structure, giving several ways to simultaneously study the homotopy theory of classical topological spaces, combinatorial spaces such as graphs and matroids, and metric spaces endowed with a privileged scale, in addition to spaces of maps between them. In the process, we give a sufficient condition for a topological construct which contains compactly generated Hausdorff spaces as a subcategory to admit an $I$-category structure. We further show that, for a topological space $X\in C$, the homotopy groups of $X$ constructed in the cofibration category on PsTop are isomorphic to those constructed classically in Top$^*$.