nLab pretriangulated envelope of a dg-category

Idea

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.

Definition

Let $T$ be a dg-category.

Definition

The pretriangulated envelope (or pretriangulated hull, pretriangulated completion) of $T$, denoted $tri(T)$, is the full sub-dg-category of the derived dg-category $D(T)$ 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 $D(T)$. Explicitly $tri(T)$ can be described as follows.

Lemma

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

Related concepts

• pretriangulated dg-category
• quasi-equiconic model structure on dg-categories

References

Paragraph 2.3 of

• Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, arXiv:1402.7364.