nLab full triangulated subcategory

Definition

Recall that a triangulated category is given by a 1-category $C$ together with a suspension or translation endofunctor $X\mapsto X[1]$ and a distinguished family of diagrams of the form

$$X \to Y \to Z\to X[1],$$

called exact triangles, satisfying some axioms.

A triangulated category $B$ is a full triangulated subcategory of a triangulated category $C$ if it is

• a full subcategory $B\subset C$ in 1-categorical sense

• closed under translation functor

• triangulated with respect to triangles in $C$

Localization by a full triangulated subcategory

Given a full triangulated subcategory $B\subset C$, the class $\Sigma$ of morphisms $s$ which fit into a triangle in $C$ of the form

$$X \overset{s}\to Y \to N\to X[1]$$

where $N\in Ob(B)$ forms a (left and right) calculus of fractions. Hence we can form a localization (quotient) category $C[\Sigma^{-1}]$. This category equipped with induced translation functor and the exact triangles which are the images of the exact triangles in $C$ under the localization functor, is triangulated and usually denoted $C/B$. The localization functor is triangulated, sends all objects in $B$ into the $0$ object, and is universal among all triangulated functors with this property.

Literature

For example

• Dmitri Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sbornik: Mathematics 197:12 (2006) 1827 doi