Homotopical Algebra in Categories with Enough Projectives

TL;DR

This paper develops new homotopical and homological algebra structures in categories with enough projectives, especially in spaces, enabling advanced analysis of homotopy and homology with applications to topological groups.

Contribution

It constructs novel model structures on simplicial objects and spaces, defining homotopy groups and spectral sequences in categories of topological spaces with projective properties.

Findings

Established a model structure on simplicial objects in categories with enough projectives.

Defined homotopy group objects invariant under weak equivalences.

Derived a Lyndon–Hochschild–Serre spectral sequence for topological group extensions.

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 $\mathcal{U}$.