# nLab pretriangulated dg-category

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

Pretriangulated dg-categories are models for stable (∞,1)-categories in terms of dg-categories, much like simplicial categories are models for (∞,1)-categories.

The zeroth cohomology category of a pretriangulated dg-category is an ordinary triangulated category, hence a morphism from ${H}^{0}\left(C\right)\to D$ where $C$ is a pretriangulated dg-category and $D$ a triangulated category is called an enhanced triangulated categories.

## Definition

For $E$ a dg-category let $\mathrm{PreTr}\left(E\right)$ be its dg-category of twisted complexes.

$E$ is pretriangulated if for every twisted complex $K\in \mathrm{PreTr}\left(E\right)$ the corresponding dg-functor

$\mathrm{PreTr}\left(-,K\right):{E}^{\mathrm{op}}\to C\left(\mathrm{Ab}\right)$PreTr(-,K) : E^{op} \to C(Ab)

In other words, twisted complexes in $\mathrm{PreTr}\left(E\right)$ have representatives in $E$.

Proposition

For $E$ a pretriangulated dg-category, the homotopy category ${H}^{0}\left(E\right)$ is naturally a triangulated category.

The morphism

${H}^{0}\left(\mathrm{PreTr}\left(E\right)\right)\to {H}^{0}\left(E\right)$H^0(PreTr(E)) \to H^0(E)

## References

See enhanced triangulated category for more links to references.

Revised on September 24, 2012 14:16:47 by Urs Schreiber (82.169.65.155)