pretriangulated dg-category


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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.


Let EE be a DG category.


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)),

which is a mapping cone in chain complexes.


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


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).


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.


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


The morphism

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

is an equivalence of triangulated categories.


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.


See enhanced triangulated category for more links to references.

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

Revised on June 9, 2014 06:46:37 by Tobias Fritz (