nLab canonical model structure

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.

The archetypical examples are

• the canonical model structure on categories

• the canonical model structure on groupoids.

In categorification of these, one has

• canonical model structures on 2-categories.

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

While ultimately the collection of all $n$-categories should form an $(n+1)$-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 $1\le k \lt n$. These can also be described as the morphisms generated by the inclusions $\partial G_k \hookrightarrow G_k$ of the boundary of the $k$-globe into the $k$-globe for $0\le k \lt \infty$.

Internalization

A common problem is to transport the (a) model structure on plain $\omega$-categories, i.e. $\omega$-categories internal to $Sets$ to another internal context, notably for the case that $Sets$ is replaced with some kind of category of $Spaces$. 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 $(\infty,1)$-category of parameterized or internal $n$-categories. See for instance also smooth infinity-stack.

Cofibrant resolutions

In

• Francois Métayer, Cofibrant complexes are free (arXiv)

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

• Francois Métayer, Resolutions by polygraphs (tac)

We had some blog discussion about this at Freely generated omega-categories.

Related concepts

• basic localizer

• canonical model structure on Cat

• canonical model structure on Grpd

• canonical model structure on 2Cat

• canonical model structure on Operad

• model structure on strict omega-categories

• 2-trivial model structure

References

The canonical model structure on categories was known to experts for some time before being written down formally (this is the origin of the adjective “folk”, as in “folklore”).

The canonical model structure on groupoids is claimed (but not proven)

• Donald W. Anderson, p. 783 (12 of 37) in: Fibrations and geometric realization, Bull. Amer. Math. Soc. 84 5 (1978) 765-788 [euclid:1183541139]

A published proof of this and the canonical model structure on categories, in the further generality of categories internal to a Grothendieck topos (hence of 2-sheaves) is due to

• André Joyal, Myles Tierney, Thm. 2 (p. 221) Strong stacks and classifying spaces, in: Category Theory Lecture Notes in Mathematics 1488, Springer (1991) 213-236 [doi:10.1007/BFb0084222]

and more elementary proofs for the plain case (internal to Set) are given in

• Charles Rezk, A model category for categories (1996) [dvi, pdf]

• André Joyal, Model structures on Cat (2010)

and (for the case of groupoids) in

• Neil Strickland, §6.1 of: $K(n)$-local duality for finite groups and groupoids , Topology 39 4 (2000) [arXiv:math/0011109, doi:10.1016/S0040-9383(99)00031-2]

Further generalization to categories internal to finitely complete categories:

• Tomas Everaert, Rudger W. Kieboom, Tim Van der Linden, Model structures for homotopy of internal categories, Theory and Applications of Categories, Vol. 15, CT2004, No. 3, pp 66-94. (tac:15-03).

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)

and for $n=3$ (Gray-categories) in

• Steve Lack, A Quillen model structure for Gray-categories (arxiv:1001.2366)

For $n = \omega$ (strict omega-categories)

• Yves Lafont, Francois Métayer, Krzysztof Worytkiewicz, A folk model structure on omega-cat (arXiv)

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

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

• Dimitri Ara, Francois Metayer, The Brown-Golasinski model structure on ∞-groupoids revisited (pdf) Homology, Homotopy Appl. 13 (2011), no. 1, 121–142.