The suspended, additive, twisted, and pretriangulated envelope operations that freely adjoin shifts, finite direct sums, twists, and mapping cones to a differential graded (dg) category, together with the sign rules governing how each interacts with opposite categories, tensor products, and the bar resolution. Establishes the precise relationship between a dg category's bar complex and its envelopes: the bar complex is an idempotent resolution of the identity bimodule, and a dg category is derived Morita equivalent to its suspended, additive, and pretriangulated envelopes.
A reference on the basic theory of differential graded (dg) categories and their bar complexes. It carefully records the main envelope operations on a dg category — adjoining shifts (suspended envelo…