Homotopy Limits for 2-categories

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits. 1. Quillen model structures in 2-category theory The 2-category of small groupoids, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalences of categories and the fibrations are the Grothendieck fibrations [1, 15]. Similarly, the 2-category of small categories, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalences of categories and the fibrations are the isofibrations, which are functors satisfying a restricted version of the lifting condition for Grothendieck fibrations [15, 24]. Lack has vastly generalised these results by showing that every 2-category K with finite limits and colimits admits a model structure, called here the natural model structure on K, in which the weak equivalences are the equivalences in K and the fibrations are the isofibrations in K [21]. Here, the notions of equivalence and isofibration for a map in a 2-category are obtained by suitably generalising the notions of equivalence and of isofibration for a functor. We take Lack’s theorem as a starting point to study homotopy limits for 2-categories. Our first step is to show that for every small 2-category A and every 2categoryK with finite limits and small colimits, the functor 2-category [A,K] admits a model structure in which the weak equivalences are the pointwise equivalences and the fibrations are the pointwise isofibrations. We refer to this model structure as the projective model structure. When K is assumed to be locally presentable, the existence of the projective model structure follows by a result on the lifting of the natural model structure on a 2category K to 2-categories of algebras for a 2-monad with rank on K [21, Theorem 4.5]. However, the special form of the 2-category [A,K] allows us