HOMOTOPY LIMITS FOR 2-CATEGORIES

Abstract. 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 2-categories. 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 fibra-tions 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 ad-mits a model structure, called here the natural model structure on K, in which th