Envelopes and the bar complex

TL;DR

This paper provides a detailed reference on the fundamental operations and sign conventions involved in the bar complexes of dg categories, including envelope operations and categorical idempotents.

Contribution

It systematically documents envelope operations, sign rules, and idempotent theory relevant to dg categories and their bar complexes, serving as a foundational reference.

Findings

Clarifies envelope operations on dg categories

Details sign conventions in complex constructions

Includes theory of categorical idempotents

Abstract

This paper is intended as a reference for some basic theory for dg categories and their bar complexes. Our modest goal is to carefully record the most important envelope operations can one perform on dg categories (in which one adjoins shifts, finite direct sums, or twists) and the inescapable sign rules that appear when combining these with opposite categories, tensor products, and the bar resolution. An appendix collects some theory of categorical idempotents that is useful when discussing bar complexes.