Abstract

An (∞,1)-category has been shown to support most theorems and constructions of category theory and, in particular, limits in an (∞,1)-category have been constructed as terminal objects in the corresponding (∞,1)-category of cones. In this talk, I will present a generalization of this construction to the (∞,n)-categorical setting for higher n, focusing on the case where n=2. This is joint work with Nima Rasekh and Martina Rovelli.  A good notion of limit in a (strict) 2-category is that of a 2-limit, which is defined as a categorically enriched limit. Unlike its 1-categorical analogue, a 2-limit cannot be characterized as a 2-terminal object in the corresponding 2-category of cones. Instead, a passage to double categories allows such a characterization and a 2-limit is equivalently a double terminal object in the corresponding double category of cones. This issue extends to the ∞-setting and we define limits in an (∞,2)-category as terminal objects in a double (∞,1)-category of cones. In particular, we show that this definition is equivalent to the established definition of (∞,2)-limits as (∞,1)-categorically enriched limits. The case of (∞,n)-categories is analogous, with limits defined in the setting of internal categories to (∞,n-1)-categories.