Equivalence of field theories in the BV-BFV formalism: the case of gravity

The BV-BFV formalism is a powerful framework to deal with gauge symmetries in the presence of boundaries. Developed by Cattaneo, Mnev and Reshetikhin, it succeeds in unifying the Lagrangian Batalin-Vilkovisky with the Hamiltonian Batalin-Fradkin-Vilkovisky formalisms, aiming at a functorial realisation of quantisation of gauge theories over manifolds with boundaries, and possibly corners. While doing so, it provides a refinement of the usual notion of equivalence of field theories, showing that what was previously considered equivalent on manifolds without boundaries, might no longer be considered as such. In this talk I will review the basics of the formalism that are necessary to understand the examples of one dimensional gravity with matter and of the four dimensional case of general relativity.