[Paper Review] A model structure on the category of small categories for coverings

Kohei Tanaka|arXiv (Cornell University)|Jul 30, 2009

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper establishes a model structure on the category of small categories—called the 1-type model structure—where fibrant objects are groupoids and fibrant replacement corresponds to groupoidification. It shows that coverings in small categories correspond precisely to fibrations with discrete fibers, and universal covers and groupoidification arise from functorial factorizations in this model structure, linking categorical coverings to homotopy theory via Quillen equivalences with simplicial sets and topological spaces.

ABSTRACT

We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification.

Motivation & Objective

• To define a model structure on the category of small categories that captures the homotopy-theoretic behavior of coverings.
• To show that fibrant objects in this model structure are precisely groupoids, and fibrant replacement corresponds to groupoidification.
• To characterize coverings in small categories as fibrations with discrete fibers, establishing a categorical analogue of covering space theory.
• To demonstrate that universal covers and the groupoidification functor emerge naturally from functorial factorizations in the model structure.
• To establish Quillen equivalences between the 1-type model structure on small categories and corresponding model structures on simplicial sets and topological spaces.

Proposed method

• Construct the 1-type model structure on Cat as the left Bousfield localization of the Joyal-Tierney model structure.
• Define weak equivalences as weak 1-equivalences, cofibrations as injective-on-objects functors, and fibrations as functors that are both fibered and cofibered in groupoids.
• Use the nerve functor N and categorization functor c to relate the model structure on Cat to those on simplicial sets and topological spaces.
• Establish that the adjoint pair (c, N) forms a Quillen equivalence between the 1-type model structure on Cat and the 1-type model structure on simplicial sets.
• Prove that the counit map X → NcX is a weak 1-equivalence for any Kan complex X, using the fact that cX is a groupoid when X is Kan.
• Use functorial factorization to construct universal covers and describe groupoidification as a fibrant replacement in the model category.

Experimental results

Research questions

• RQ1 Can a model structure on small categories be defined such that fibrant objects are groupoids and fibrant replacement corresponds to groupoidification?
• RQ2 How are coverings in small categories related to fibrations in the 1-type model structure?
• RQ3 What is the relationship between the 1-type model structure on small categories and the 1-type model structures on simplicial sets and topological spaces?
• RQ4 Can universal covers in small categories be constructed via functorial factorization in this model structure?
• RQ5 Is the adjunction between small categories and simplicial sets a Quillen equivalence when restricted to the 1-type model structures?

Key findings

• The 1-type model structure on small categories is a left Bousfield localization of the Joyal-Tierney model structure, with weak equivalences defined as weak 1-equivalences.
• An object in the 1-type model structure is fibrant if and only if it is a groupoid, confirming that fibrant replacement is groupoidification.
• A functor is a covering in Cat if and only if it is a fibration with discrete fibers, providing a homotopical characterization of coverings.
• Universal covers in Cat and the groupoidification functor are both described via functorial factorization in the model structure, linking them to homotopy-theoretic constructions.
• The adjunction (c, N) between simplicial sets and small categories is a Quillen equivalence when restricted to the 1-type model structures, establishing a Quillen equivalence between Cat₁ and SSet₁.
• The counit map X → NcX is a weak 1-equivalence for any Kan complex X, and this fact underpins the Quillen equivalence between Cat₁ and SSet₁.