Cohomology of higher categories

Classical obstruction theory studies the extensions of a continuous map along a relative CW-complex in terms of cohomology with local coefficients. In this talk, I will describe a similar obstruction theory for $(\infty, 1)$- and $(\infty, 2)$-categories, using cohomology with coefficients in local systems over the twisted arrow category and the `twisted 2-cell category'.  As an application, I will give an obstruction-theoretic argument that shows that adjunctions can be made homotopy coherent (as proven by Riehl–Verity). This is joint work with Yonatan Harpaz and Matan Prasma.