Subequivalence

Subequivalence is a concept in mathematics, particularly in the field of category theory, that generalizes the notion of equivalence of categories. While equivalence of categories requires a bijection between objects and isomorphisms between morphisms, subequivalence relaxes these conditions to allow for a surjection on objects and isomorphisms on morphisms.

In a subequivalence, there is a functor F from category C to category D such that:

1. F is surjective on objects, meaning every object in D is the image of some object in C under F.

2. F is full, meaning for every pair of objects X and Y in C, the map F_XY: Hom_C(X, Y) -> Hom_D(F(X), F(Y)) is surjective.

3. F is faithful, meaning for every pair of objects X and Y in C, the map F_XY is injective.

Subequivalences are useful in various areas of mathematics, including homotopy theory and algebraic geometry. They provide a way to compare categories that are not equivalent but share many of the same structural properties.