A note on gluing via fiber products in the (classical) BV-BFV formalism

Abstract: In classical field theory, gluing spacetime manifolds along boundary corresponds to taking a fiber product of the corresponding spaces of fields (as differential graded Fr\'echet manifolds) up to homotopy. We construct this homotopy explicitly in several examples in the setting of BV-BFV formalism (Batalin--Vilkovisky formalism with cutting--gluing).