Appendix A - Background

Summary

We use techniques from disparate areas of mathematics throughout this book and not all of these techniques are standard knowledge, so here we provide a terse introduction to

1. • Simplicial sets and simplicial techniques

2. • Operads, colored operads (or multicategories), and algebras over colored operads

3. • Differential graded (dg) Lie algebras and their (co)homology

4. • Sheaves, cosheaves, and their homotopical generalizations

5. • Elliptic complexes, formal Hodge theory, and parametrices

along with pointers to more thorough treatments. By no means do readers need to be expert in all these areas to use our results or follow our arguments. They just need a working knowledge of this background machinery, and this appendix aims to provide the basic definitions, to state the results relevant for us, and to explain the essential intuition.

We do assume that readers are familiar with basic homological algebra and basic category theory. For homological algebra, there are numerous excellent sources, in books and online, among which we recommend the complementary texts by Weibel (1994) and Gelfand and Manin (2003). For category theory, the standard reference Mac Lane (1998) is adequate for our needs; we also recommend the series Borceux (1994a).

Remark: Our references are not meant to be complete, and we apologize in advance for the omission of many important works.We simply point out sources that we found pedagogically oriented or particularly accessible.

Reminders and Notation

We overview some terminology and notations before embarking on our expositions.

For C a category, we often use to indicate that x is an object of C. We typically write C(x, y) to the denote the set of morphisms between the objects x and y, although occasionally we use Hom_C(x, y). The opposite category C^op has the same objects but C^op(x, y) = C(y, x).

Given a collection of morphisms S in C, a localization of C with respect to S is a category C[S^-1] and a functor q : C → C[S^-1] satisfying the following conditions.