nLab pretriangulated envelope of a dg-category


Any dg-category admits a pretriangulated envelope, which is a fully faithful embedding into a strongly pretriangulated dg-category. When the dg-category is itself pretriangulated, this is in fact an equivalence of dg-categories.


Let TT be a dg-category.


The pretriangulated envelope (or pretriangulated hull, pretriangulated completion) of TT, denoted tri(T)tri(T), is the full sub-dg-category of the derived dg-category D(T)D(T) generated by the representable dg-modules under homotopy fibres and homotopy cofibres.

Here, the notions of homotopy fibre and homotopy cofibre can be taken in a dg-model category presenting D(T)D(T). Explicitly tri(T)tri(T) can be described as follows.


The pretriangulated envelope tri(T)tri(T) coincides with the full sub-dg-category of D(T)D(T) spanned by the finitely generated semi-free dg-modules.


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