Structures · Category theory

CategoryTheory.Pretriangulated

A preadditive category C with an additive shift, and a class of "distinguished triangles" relative to that shift is called pretriangulated if the following hold: * Any triangle that is isomorphic to a distinguished triangle is also distinguished. * Any triangle of the form (X,X,0,id,0,0) is distinguished. * For any morphism f : X ⟶ Y there exists a distinguished triangle of the form (X,Y,Z,f,g,h). * The triangle (X,Y,Z,f,g,h) is distinguished if and only if (Y,Z,X⟦1⟧,g,h,-f⟦1⟧) is. * Given a diagram: `` f g h X ───> Y ───> Z ───> X⟦1⟧ │ │ │ │a │b │a⟦1⟧' V V V X' ───> Y' ───> Z' ───> X'⟦1⟧ f' g' h' ` where the left square commutes, and whose rows are distinguished triangles, there exists a morphism c : Z ⟶ Z' such that (a,b,c)` is a triangle morphism.

Defined in

Mathlib.CategoryTheory.Triangulated.Pretriangulated

Shape

One type argument · adds distinguishedTriangles, isomorphic_distinguished, contractible_distinguished, distinguished_cocone_triangle, rotate_distinguished_triangle, complete_distinguished_triangle_morphism