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 (see (Cohn 13)).

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

Definition

Let $E$ be a DG category.

Definition

The $n$-translation of an object $X \in E$ is an object $X[n] \in E$ representing the functor

$\Hom(\cdot, X)[n].$

The cone of a closed morphism $f : X \to Y$ of degree zero is an object $\Cone(f) \in E$ representing the functor

$\Cone(\Hom(\cdot, X) \stackrel{f_*}{\to} \Hom(\cdot, Y)).$
Definition

$E$ is called strongly pretriangulated if it admits a zero object, all translations of all objects, and all cones of all morphisms.

Proposition

For every DG category $E$, there exists a strongly pretriangulated DG category $\PreTr(E)$ and a fully faithful DG functor $E \hookrightarrow \PreTr(E)$ such that for any DG functor $F: E \to E'$ to a strongly pretriangulated DG category $E'$, there exists a unique lift $\hat F : \PreTr(E) \to E'$.

See below for a construction of $\PreTr(E)$.

Definition

The DG category $E$ is called pretriangulated if the induced functor $\H^0(E) \to \H^0(\PreTr(E))$ is an equivalence.

Proposition

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

Proposition

The morphism

$H^0(PreTr(E)) \to H^0(E)$
Proposition

If $F : E \to E'$ is a DG functor between two pretriangulated DG categories, then

• $F$ commutes with translation and preserves cones;
• the induced functor $H^0(F)$ is a triangulated functor on the homotopy categories;
• $F$ is a quasi-equivalence? if and only if $H^0(F)$ is a triangulated equivalence.

Definition using twisted complexes

For $E$ a dg-category let $PreTr(E)$ be its dg-category of twisted complexes.

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

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

In other words, twisted complexes in $PreTr(E)$ have representatives in $E$.

References

See enhanced triangulated category for more links to references.

The relation to stable (infinity,1)-categories is discussed in

Revised on March 20, 2014 03:25:21 by Urs Schreiber (89.204.138.150)