nLab
pretriangulated dg-category

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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)DH^0(C)\to D where CC is a pretriangulated dg-category and DD a triangulated category is called an enhanced triangulated categories.

Definition

Let EE be a DG category.

Definition

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

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

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

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

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

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

Definition

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

Proposition

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

Proposition

The morphism

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

is an equivalence of triangulated categories.

Proposition

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

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

Definition using twisted complexes

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

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

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

is representable.

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

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)