nLab
model 2-category

Idea

A strict model 2-category is a strict 2-category CC – hence a category enriched over Cat – whose underlying category C 1C_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 CC a category enriched over Cat and for

u:XY u : X \to Y
v:VW v : V \to W

two morphisms in CC, let

[u,v]:C(Y,V)C(Y,W)× 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

C(Y,V) C(u,V) C(X,V) C(Y,v) C(X,v) C(Y,W) C(u,W) C(X,W). \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 CC with finite limits and colimits is

This is just the usual notion of an enriched model category specialized to enrichment over the monoidal model category CatCat.

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 TAlg sT Alg_s of algebras and strict morphisms for a suitably well-behaved (strict) 2-monad TT 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 TAlgT Alg of TT-algebras and pseudo morphisms, so TAlgT Alg is the “homotopy 2-category” of TAlg sT 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][C,K] of diagrams in any 2-category KK 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.)
Revised on March 12, 2009 01:53:44 by Toby Bartels (71.104.234.95)