A strict model 2-category is a strict 2-category – hence a category enriched over Cat – whose underlying category 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.
To state the definition of a strict model 2-category, we need the following notation:
two morphisms in , let
be the unique morphism induced from the commutativity of the diagram
The structure of a model 2-category on a strict 2-category with finite limits and colimits is
a model category structure on the underlying 1-category ;
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 of algebras and strict morphisms for a suitably well-behaved (strict) 2-monad 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 of -algebras and pseudo morphisms, so is the “homotopy 2-category” of . Cofibrant replacements in this model structure can also be identified with flexible replacements in the theory of 2-monads.
The 2-category of diagrams in any 2-category also inherits two different, but Quillen equivalent, model structures, called the “projective” and “injective” model structures.
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.
The original reference, which constructs the natural model structure and its lifting to 2-monads, is:
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).