biequivalent

Biequivalence is a concept in category theory, a branch of mathematics that studies the relationships between different mathematical structures. It is a generalization of the notion of equivalence in a category, which is a way of saying that two objects are essentially the same.

In a category, two objects are equivalent if there is a zigzag of isomorphisms between them. This means that there is a sequence of morphisms (arrows) that go back and forth between the two objects, with each morphism being an isomorphism (a morphism with an inverse). Biequivalence is a similar concept, but instead of isomorphisms, it uses biequivalences.

A biequivalence is a morphism that is both an equivalence and a coequivalence. An equivalence is a morphism that has a left inverse, meaning there is another morphism that, when composed with the original morphism, results in the identity morphism. A coequivalence is a morphism that has a right inverse, meaning there is another morphism that, when composed with the original morphism, results in the identity morphism.

Therefore, a biequivalence is a morphism that has both a left and a right inverse. This means that there is a zigzag of biequivalences between two objects if and only if the objects are biequivalent.

Biequivalence is a useful concept in category theory because it allows for a more flexible notion of equivalence than the traditional one. It is used in various areas of mathematics, including homotopy theory, where it is used to study the relationships between different spaces.