Homotopical Algebra in Categories with Enough Projectives

Abstract

For a complete and cocomplete category $\mathcal{C}$ with a well-behaved class of `projectives' $\bar{\mathcal{P}}$, we construct a model structure on the category $s\mathcal{C}$ of simplicial objects in $\mathcal{C}$ where the weak equivalences, fibrations and cofibrations are defined in terms of $\bar{\mathcal{P}}$. This holds in particular when $\mathcal{C}$ is $\mathcal{U}$, the category of compactly generated, weakly Hausdorff spaces, and $\bar{\mathcal{P}}$ is the class of compact Hausdorff spaces. We also construct a new model structure on $\mathcal{U}$ itself, where the cofibrant spaces are generalisations of CW-complexes allowing spaces, rather than sets, of $n$-cells to be attached. The singular simplicial complex and geometric realisation functors give a Quillen adjunction between these model structures. For a space in $\mathcal{U}$, these structures allow the definition of homotopy group objects in the exact completion of $\mathcal{U}$, which are invariant under weak equivalence and have a lot of the nice properties usually expected of homotopy groups. There is a long exact sequence of homotopy group objects arising from a fibre sequence in $\mathcal{U}$. Working along similar lines, we study homological algebra in categories of internal modules in $\mathcal{U}$, getting in particular a Lyndon--Hochschild--Serre spectral sequence for extensions of topological groups in $\mathcal{U}$.