Quillen cohomology of enriched categories

The notion of Quillen cohomology, first defined by Quillen in his seminal book and later extended to include coefficients in spectra, is a universal form of cohomology which enjoys a wide range of applications: it features in obstruction theories and spectral sequences abutting to mapping spaces, controls deformations of objects in the relevant contexts and tends to appear in moduli problems classifying objects realizing a given cohomological invariant. In this talk we will describe an approach to the computation of Quillen cohomology of categories enriched in a well behaved model category S. When S is the category of simplicial sets one recovers a model for the theory of oo-categories, in which case our computation identifies the Quillen cohomology of an oo-category C as the cohomology of spectrum valued functors on the twisted arrow category of C. When S is the category of marked simplicial sets one recovers a model for the theory of (oo,2)-categories, in which case our computation identifies the Quillen cohomology of an (oo,2)-category C with the cohomology of spectrum valued functors indexed by what we call the twisted 2-cell category of C. These results can be used to give explicit and computable obstruction theories for problems such as lifting homotopy commutative diagrams (in either a 1-categorical or a 2-categorical setting). This is joint work with Joost Nuiten and Matan Prasma.