The two out of three property in ind-categories and a convenient model category of spaces

TL;DR

This paper establishes intrinsic conditions for weak cofibration categories to induce model structures on their indcategories, with applications to spaces and C*-algebras, facilitating homotopical analysis in these contexts.

Contribution

It provides sufficient intrinsic conditions for the two out of three property in ind-categories of weak cofibration categories, enabling new model structures on categories of spaces and C*-algebras.

Findings

Model structure on ind-category of compact metrizable spaces

Conditions ensuring the two out of three property in ind-categories

Application to categories related to topological spaces and C*-algebras

Abstract

In [BaSc2], the author and Tomer Schlank introduced a much weaker homotopical structure than a model category, which we called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its indcategory, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a model structure on its ind-category. This model structure is defined on a category that is closely related to a category of topological spaces and has many convenient formal properties. A more general application of these results, to the (opposite) category of separable $C^*$-algebras, appears in a paper by the author, Michael Joachim and Snigdhayan Mahanta [BJM].