Equivalence of field theories in the BV formalism: the example of (three dimensional) General Relativity

The standard notion of equivalence of field theories roughly requires the Euler-Lagrange loci of the two associated variational problems to be diffeomorphic, possibly modulo the action of the respective symmetry distributions, in some appropriate framework. This can be made sense of more precisely in the Batalin-Vilkovisky setting, where some cohomological presentation of said locus is constructed, but I will try to argue that this is probably just the beginning. I will discuss a series of examples, all related to General Relativity in different spacetime dimensions, that suggest that higher codimension data should play a role in refining equivalence between classical theories, and raise the question of whether (and how) this picture carries over to quantisation.