nLab
model 2-category

Idea

A strict model 2-category is a strict 2-category $C$ – hence a category enriched over Cat – whose underlying category $C_1$ is equipped with the structure of a model category which is compatible, in some sense, with the folk model structure on the enrichment category Cat.

The compatibility is such that in particular the notion of homotopy limit and of pseudo-limit coincide in such model 2-category.

To state the definition of a strict model 2-category, we need the following notation:

Notation

For $C$ a category enriched over Cat and for

$$u:X\to Y$$u : X \to Y

$$v:V\to W$$v : V \to W

two morphisms in $C$, let

$$[u,v]:C(Y,V)\to C(Y,W){\times }_{C(X,W)}C(X,V)$$[u,v] : C(Y,V) \to C(Y,W) \times_{C(X,W)} C(X,V)

be the unique morphism induced from the commutativity of the diagram

$$\begin{array}{ccc}C(Y,V)& \stackrel{C(u,V)}{\to }& C(X,V)\\ {\downarrow }^{C(Y,v)}& & {\downarrow }^{C(X,v)}\\ C(Y,W)& \stackrel{C(u,W)}{\to }& C(X,W)\end{array}.$$\array{ C(Y,V) &\stackrel{C(u,V)}{\to}& C(X,V) \\ \downarrow^{C(Y,v)} && \downarrow^{C(X,v)} \\ C(Y,W) &\stackrel{C(u,W)}{\to}& C(X,W) } \,.

Definition

The structure of a model 2-category on a strict 2-category $C$ with finite limits and colimits is

• a model category structure on the underlying 1-category $C_1$;

• such that

  • if $u:X\to Y$ is a cofibration and $v:V\to W$ is a fibration in $C_1$ then the functor $[u,v]$ defined above is an isofibration in Cat;

  • and $[u,v]$ is a weak equivalence (in the folk model structure on Cat, i.e. an equivalence of categories) if either of $u$ or $v$ is.

This is just the usual notion of an enriched model category specialized to enrichment over the monoidal model category $\mathrm{Cat}$.

Examples

• Every suitably complete and cocomplete 2-category admits a “trivial” or “natural” model 2-category structure in which the weak equivalences are the categorical equivalences and the fibrations are the internal isofibrations. By duality, there also is a second such model structure; in Cat both coincide with the folk model structure.

• The 2-category $T{\mathrm{Alg}}_s$ of algebras and strict morphisms for a suitably well-behaved (strict) 2-monad $T$ inherits a model structure where the weak equivalences are those that become weak equivalences in the underlying category. These are precisely the morphisms that are equivalences in the 2-category $T\mathrm{Alg}$ of $T$-algebras and pseudo morphisms, so $T\mathrm{Alg}$ is the “homotopy 2-category” of $T{\mathrm{Alg}}_s$. Cofibrant replacements in this model structure can also be identified with flexible? replacements in the theory of 2-monads.

• The 2-category $[C,K]$ of diagrams in any 2-category $K$ also inherits two different, but Quillen equivalent, model structures, called the “projective” and “injective” model structures.

Remarks

• Every model 2-category has a homotopy (weak) 2-category? which can be constructed by formally making the weak equivalences into categorical equivalences. It can alternately be described using fibrant and cofibrant replacements, just like the ordinary homotopy category of a model category.

• In a strict model 2-category with a natural model structure, the notions of 2-categorical pseudo-limit and (one canonical construction of) model theoretic homotopy limits coincide. For a general model 2-category, homotopy limits can be construed as representatives of 2-limits in its homotopy 2-category.

References

The original reference, which constructs the natural model structure and its lifting to 2-monads, is:

• Steve Lack, Homotopy theoretic aspects of 2-monads at math.CT/0607646, published as Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 229-260.

The projective and injective model structures on diagrams, and the relation between pseudo-limits and homotopy limits, are discussed in the following (especially section 6).

• Nicola Gambino, Homotopy limits for 2-categories (pdf), published as: Mathematical Proceedings of the Cambridge Philosophical Society 145 (2008) 43-63.)