On Colimits and Model Structures in Various Categories of Manifolds

TL;DR

This paper investigates the limitations of manifold categories as model categories due to missing colimits, and explores enlargements like presheaf categories to establish model structures, solving an open problem related to Poincaré spaces.

Contribution

It demonstrates that standard manifold categories lack finite colimits, and develops methods to create model structures on enlarged categories, including presheaves, addressing an open problem.

Findings

Manifold categories do not have all finite colimits.

Enlarged categories like presheaves can be equipped with model structures.

Resolved an open problem involving Poincaré spaces.

Abstract

After explaining the importance of model categories in abstract homotopy theory, we provide concrete examples demonstrating that various categories of manifolds do not have all finite colimits, and hence cannot be model categories. We then consider various enlargements of our categories of manifolds, culminating in categories of presheaves. We explain how to produce model structures on these enlarged categories, culminating with answering an open problem involving Poincar\'{e} spaces.