# 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.

## References

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