nLab full triangulated subcategory


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

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

called exact triangles, satisfying some axioms.

A triangulated category BB is a full triangulated subcategory of a triangulated category CC if it is

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

  • closed under translation functor

  • triangulated with respect to triangles in CC

Localization by a full triangulated subcategory

Given a full triangulated subcategory BCB\subset C, the class Σ\Sigma of morphisms ss which fit into a triangle in CC of the form

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

where NOb(B)N\in Ob(B) forms a (left and right) calculus of fractions. Hence we can form a localization (quotient) category C[Σ 1]C[\Sigma^{-1}]. This category equipped with induced translation functor and the exact triangles which are the images of the exact triangles in CC under the localization functor, is triangulated and usually denoted C/BC/B. The localization functor is triangulated, sends all objects in BB into the 00 object, and is universal among all triangulated functors with this property.


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. See the history of this page for a list of all contributions to it.