From weak cofibration categories to model categories

TL;DR

This paper explores how certain properties and structures in weak cofibration categories are preserved when inducing model category structures on their ind-categories, focusing on properness and monoidal structures.

Contribution

It proves that properties like left properness and monoidal structures in weak cofibration categories are retained in the induced model structures on ind-categories.

Findings

Left properness is preserved in the induced model category.

Monoidal and tensored structures are maintained in the ind-category.

Results apply to almost model structures, broadening applicability.

Abstract

In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, 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 ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists in the weak cofibration category, then the same property or structure also holds in the induced model structure on the ind-category. Namely, we consider the property of being left proper and the structures of a monoidal category and a category tensored over a monoidal category (in a way that is compatible with the weak cofibration structure). For the purpose of future reference, we consider the more general situation where we only have an "almost model structure" on the ind-category.