Quillen cohomology of (∞,2)-categories

Abstract

In this paper we study the homotopy theory of parameterized spectrum objects in the ∞-category of (∞, 2)-categories, as well as the Quillen cohomology of an (∞, 2)-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an (∞,2)-category C, which we call its twisted 2-cell ∞-category. We then establish an equivalence between parameterized spectrum objects over C, and diagrams of spectra indexed by the twisted 2-cell ∞-category of C. Under this equivalence, the Quillen cohomology of C with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between (∞,1)-categories are uniquely determined at the level of the homotopy (3, 2)-category of Cat_∞.