  Using the quantum construction of the BV-BFV method for perturbative gauge
theories, we show that the obstruction for quantizing a codimension 1 theory is
given by the second cohomology group with respect to the boundary BRST charge.
Moreover, we give an idea for the algebraic construction of codimension $k$
quantizations in terms of $\mathbb{E}_k$-algebras and higher shifted Poisson
structures by formulating a higher version of the quantum master equation.
