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

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

Created on June 26, 2023 at 11:51:00.
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