Subequivalences

Subequivalences are a concept in category theory, a branch of mathematics that studies the relationships between mathematical structures. A subequivalence is a type of morphism, which is a structure-preserving map between objects in a category. Specifically, a subequivalence is a morphism that is both a monomorphism (a one-to-one function) and an epimorphism (a function that is onto or surjective). In other words, a subequivalence is a morphism that is both injective and surjective, making it a bijection in the context of sets.

The concept of subequivalences is particularly useful in the study of toposes, which are categories that generalize the notion of a set. In a topos, subequivalences play a role similar to that of isomorphisms in more traditional categories. They are used to describe the structure of the topos and to understand the relationships between its objects.

Subequivalences are also closely related to the concept of equivalences in category theory. An equivalence is a morphism that has both a left and a right inverse, meaning that it is invertible up to isomorphism. A subequivalence can be thought of as a special case of an equivalence, where the inverse is also a morphism in the category.

In summary, subequivalences are a fundamental concept in category theory, particularly in the study of toposes. They are morphisms that are both injective and surjective, and they play a crucial role in understanding the structure of categories and the relationships between their objects.