Autoequivalences

An autoequivalence of a category C is an endofunctor F: C → C that is an equivalence. Equivalences are precisely the functors that are full, faithful, and essentially surjective, and they admit a quasi-inverse G: C → C with natural isomorphisms FG ≅ Id_C and GF ≅ Id_C. In particular, F is invertible up to natural isomorphism, with the inverse defined up to isomorphism. The composition of autoequivalences is again an autoequivalence, and, up to natural isomorphism, the autoequivalences form a group under composition. Many categorical properties are preserved by autoequivalences, such as isomorphism classes of objects and morphisms, and, in categories with limits or colimits, the existence of limits or colimits is preserved up to isomorphism.

Standard examples appear in derived categories. The shift functor [1] on a triangulated category is an autoequivalence. In the bounded derived category of coherent sheaves on a scheme X, the operations of pulling back along an automorphism f of X, tensoring by a line bundle L, and shifting generate a large class of autoequivalences; more generally, any composition of these standard operations is an autoequivalence. In categories with a Serre functor, the Serre functor is an autoequivalence.

The study of autoequivalences is central in homological algebra and algebraic geometry, where one often seeks to classify them or understand their action on invariants such as K-theory or cohomology. Autoequivalences reveal symmetries of the categorical or geometric structures under consideration.