Limits in n-categories

One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim” for short. The purpose of the this paper is to begin to develop the notion of limit for n-categories, which should be a bridge between the categorical notion of limit and the homotopical notion of holim. We treat Tamsamani’s notion of n-category [36], but similar arguments and results should hold for the Baez-Dolan approach [3], [5], or the Batanin approach [6], [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [32]) ncategory C. Suppose A is an n-category, and suppose φ : A → C is a morphism, which we think of as a family of objects of C indexed by A. For any object U ∈ C we can define the (n − 1)-category Hom(φ, U) of morphisms from φ to U . We say that a morphism ǫ : φ → U (i.e. an object of this (n − 1)-category) is a direct limit of φ (cf 3.2.1 below) if, for every other object V ∈ C the (weakly defined) composition with ǫ induces an equivalence of n− 1-categories from HomC(U, V ) to Hom(φ, V ). An analogous definition holds for saying that a morphism U → φ is an inverse limit of φ (cf 3.1.1 below). The main theorems concern the case where C is the n + 1-category nCAT ′ (fibrant replacement of that of) of n-categories. Theorem (4.0.1 5.0.1) The n + 1-category nCAT ′ admits arbitrary inverse and direct limits.