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An autoequivalence representing an integral degree shift in a category.
A shift functor on a category is an autoequivalence
Its inverse is denoted , and for , the notation denotes the corresponding iterate, with .
For cochain complexes, shifts the grading and changes the sign of the differential according to the chosen convention. In a preadditive category, one usually requires the shift to be additive. Shifted objects supply the fourth vertex of a triangle.
Let be categories.
A functor is an equivalence of categories if there exists a functor and natural isomorphisms
In this case is called a quasi-inverse (or weak inverse) of , and we say and are equivalent.
A category is preadditive if every morphism set is an abelian group and composition is bilinear:
In particular, there is a zero morphism between every ordered pair of objects.
An additive category is a preadditive category with finite biproducts. Preadditivity supplies the sums and signs used in complexes and distinguished triangles.
In a category with a shift functor , a triangle is a diagram
A morphism of triangles is a triple of morphisms between corresponding objects that makes all three resulting squares commute, including the square involving the shifted first object.
Rotating the triangle moves to the first position and produces , with a convention-dependent minus sign on the final map. A chosen subclass may be declared distinguished.