model category

for ∞-groupoids

# Contents

## Idea

In general, canonical model structures are model category structures on the categories of some flavor of n-categories for $1\le n\le \infty$ (note that $n=\infty$ or $\omega$ is allowed), which are intended to capture the correct “categorical” theory of these categories.

Canonical model structures are sometimes called “folk” model structures, but the appropriateness of this term is very questionable, especially in the cases $n>1$. Other alternatives are ‘endogenous’, ‘standard’, ‘natural’, and ‘categorical’.

While ultimately the collection of all n-categories should form an $\left(n+1\right)$-category, restricting that to just invertible higher morphisms will yield an (n+1,1)-category, and thus in particular an (∞,1)-category. It is this (∞,1)-category which is intended to be presented by a canonical model structure. In particular, the weak equivalences in a canonical model structure should be the category-theoretic equivalences.

This is to be contrasted with Thomason model structures in which the weak equivalences are the morphisms that induce a weak homotopy equivalence of nerves. This amounts to regarding each category, or rather its nerve, as a placeholder for its groupoidification (Kan fibrant replacement) and then considering the standard notion of equivalence.

In a canonical model structure for some flavor of $n$-categories, usually

• a fibration is a functor that lifts equivalences in all dimensions,
• an acyclic fibration is a functor which is k-surjective for all $0\le k\le n$,
• a weak equivalence is a functor which is essentially k-surjective for all $0\le k\le n$, and
• a cofibration is a functor which is injective on objects and “relatively free” on $k$-morphisms for $1\le k. These can also be described as the morphisms generated? by the inclusions $\partial {G}_{k}↪{G}_{k}$ of the boundary of the $k$-globe into the $k$-globe for $0\le k<\infty$.

## References on particular cases

• The canonical model structure for 1-categories was known to experts for some time before being written down formally (this is the origin of the adjective “folk”).

• It was apparently first published (for categories internal to a Grothendieck topos) by Joyal and Tierney, Strong Stacks and Classifying Spaces, Category theory (Como, 1990) Springer LNM 1488, 213-236.

• A more elementary writeup by Charles Rezk can be found here.

• A general internal version relative to a Grothendieck coverage can be found in

T. Everaert, R.W. Kieboom, T. Van der Linden, Model structures for homotopy of internal categories Theory and Applications of Categories, Vol. 15, CT2004, No. 3, pp 66-94. (tac).

though it seems that the assumptions in this article apply to ambient categories that are semiabelian categories, but do not apply to ambient categories like Top.

• A brief summary, together with a generalization when one assumes only weaker versions of the axiom of choice, can be found at folk model structure on Cat.

• The canonical model structures for 2-categories and bicategories are due to Steve Lack.

• A Quillen Model Structure for 2-Categories, K-Theory 26: 171–205, 2002. (website)
• A Quillen Model Structure for Bicategories, K-Theory 33: 185-197, 2004. (website)
• For $n=3$, Gray-categories:

• for $n=\omega$:

• for $n=\omega$ and all morphisms invertible, there is the model structure on strict omega-groupoids:

• R. Brown and M. Golasinski, A model structure for the homotopy theory of crossed complexes, Cah. Top. Géom. Diff. Cat. 30 (1989) 61-82 (pdf)

## Internalization

A common problem is to transport the (a) model structure on plain $\omega$-categories, i.e. $\omega$-categories internal to $\mathrm{Sets}$ to another internal context, notably for the case that $\mathrm{Sets}$ is replaced with some kind of category of $\mathrm{Spaces}$. This is relevant for the discussion of the homotopy theory of topological and smooth $\omega$-categories.

Usually, such internalization of model structures has the consequence that some properties invoked in the description of the original model structure, notably some of the lifting properties, will only continue to hold “locally”. One way to deal with this is to pass to a notion slightly weaker than that of a model category called a category of fibrant objects as used in homotopical cohomology theory.

But there are also full model structures for such situations. Notice that under a suitable nerve operation all n-categories usually embed into simplicial sets. The models for infinity-stack (infinity,1)-toposes given by the model structure on simplicial presheaves then serves to present the corresponding $\left(\infty ,1\right)$-category of parameterized or internal $n$-categories. See for instance also smooth infinity-stack.

## Cofibrant resolutions

In

it is shown that cofibrant $\omega$-categories with respect to the canonical model structure are precisely the “free” ones, where “free” here means “generated from a polygraph” as described in