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.
Unlike their 1-categorical analogues, limits in a (strict) 2-category cannot be characterized as terminal objects in the corresponding 2-category of cones. Instead, a passage to double categories allows for such a characterization. Inspired by the strict case, we define (∞,2)-limits as terminal objects in a double (∞,1)-category of cones and show that this definition is equivalent to the established definition of (∞,2)-limits as (∞,1)-categorically enriched limits.