Introduction

In the study of field theories, one works with vector spaces of an analytical nature, like the space of smooth functions or distributions on a manifold. To use the Batalin–Vilkovisky formalism, we need to perform homological algebra in this setting. The standard approach to working with objects of this nature is to treat them as topological vector spaces, but it is not obvious how to set up a well-behaved version of homological algebra with topological vector spaces.

Our approach here breaks the problem into two steps. First, our cochain complexes are constructed out of very nice topological vector spaces that are already convenient vector spaces. Hence, we view them as cochain complexes of convenient vector spaces, since CVS is a better-behaved category (for our purposes) than LCTVS. Second, we apply the functor difc to view them as cochain complexes of differentiable vector spaces; as CVS is a full subcategory of DVS, nothing drastic has happened. The benefit, however, is that DVS is a Grothendieck Abelian category, so that standard homological algebra applies immediately.

Motivation

There are a few important observations to make about this approach.

Recall that difc does not preserve cokernels, so that difc need not preserve cohomology. Given a complex C* in CVS, the cohomology group HkC* is a cokernel computed in CVS. Hence Hk(difcC*) could be different from difc(HkC*). In consequence, difc need not preserve quasi-isomorphisms. We will view quasi-isomorphisms as differentiable cochain complexes as the correct notion and avoid discussing quasi-isomorphisms as convenient cochain complexes.

The functor difc does preserve cochain homotopy equivalences, however. Thus, certain classical results – such as the Atiyah–Bott lemma (see Appendix D) or the use of partitions of unity (see Section A.5.4 in Appendix A) – play a crucial role for us. They establish explicit cochain homotopy equivalences for convenient cochain complexes, which go to cochain homotopy equivalences of differentiable cochain complexes. Later constructions, such as the observables of BV theories, involve deforming the differentials on these differentiable cochain complexes. In almost every situation, these deformed cochain complexes are filtered in such a way that the first page of the spectral sequence is the original, undeformed differential. Thus we can leverage the classical result in the new deformed situation.