I am trying to understand how gauge theory interacts with the categorical formulation of TQFT. I will formulate my doubts in two different questions.
I have understood gauging a TQFT in different ways: as cohomology elements $H^1(M,G)$, as charged defects $H_{n-1}(M,G)$ and with the Dijkgraaf-Witten approach through homotopy classes of classifying maps $[M,BG]$ with action in $H^d(BG,U(1))$. This is clear in my mind when $M$ is a closed manifold. How are these cohomology groups affected from the introduction of a boundary? These formulations should no longer holds. (I know the third approach is studied in the original article for manifold with boundaries but I was searching another reference and understand the other two)
As far as I know it is known what happens to 2d and 3d (maybe also general $d$ dimensional) categorical TQFT structure when we introduce one of the previous gauging process. In particular, how does the Hilbert space associated to the functorial partition function on the boundary change?
