Envelopes and the bar complex
Matthew Hogancamp
TL;DR
The paper provides a systematic, concrete treatment of envelope constructions for dg categories, detailing how suspended, additive, twisted, and pretriangulated envelopes ($\mathbf{S},\mathbf{A},\mathbf{Tw},\mathbf{Pretr}$) interact with opposites, tensor products, and bar complexes. It gives explicit descriptions of how these envelopes preserve or reflect dg- and Morita-theoretic properties, notably proving derived Morita equivalences between $\mathcal{C}$ and its envelopes via explicit bar-complex maps. A central technical achievement is the construction of precise chain maps relating the bar complexes of $\mathcal{C}$ and its envelopes, with careful attention to sign conventions and Maurer–Cartan data, thereby providing a hands-on toolkit for computations in dg Morita theory. The results culminate in a Morita-theoretic equivalence between a dg category and its envelopes, while also highlighting limitations (e.g., the Tw envelope need not preserve quasi-equivalences) and situating the theory within a broader framework of counital idempotents and related constructions.
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.