2isomorphism

In mathematics, particularly in the field of category theory, an isomorphism is a structure-preserving map between two objects that has an inverse. The term "2-isomorphism" extends this concept to the context of 2-categories, where objects, morphisms, and 2-morphisms (also called morphisms of morphisms) are all considered. A 2-isomorphism is a special kind of 2-morphism that is invertible in a higher-dimensional sense.

A 2-isomorphism between two 1-morphisms *f* and *g* in a 2-category consists of a 2-morphism *α*: *f* ⇒ *g* such that there exist 2-morphisms *β*: *g* ⇒ *f* and *γ*: *f* ⇒ *f* satisfying certain coherence conditions. These conditions ensure that *α* and *β* are inverses of each other in a way that respects the composition of 2-morphisms. Specifically, the compositions *γ* ∘ *α* and *β* ∘ *α* must be equal to the identity 2-morphisms on *f* and *g*, respectively, and similarly for *α* ∘ *β* and *γ* ∘ *β*.

2-isomorphisms generalize the notion of isomorphism in ordinary categories by accounting for the additional structure of 2-morphisms. They play a crucial role in higher-dimensional category theory, where higher-dimensional analogies of traditional categorical concepts are explored. The study of 2-isomorphisms is relevant in various areas of mathematics, including algebraic topology, homotopy theory, and higher algebra, where higher-dimensional structures and their relationships are of interest.

The concept of a 2-isomorphism is closely related to the idea of weak equivalences in homotopy theory, where invertibility is understood up to higher-dimensional coherence conditions. This connection highlights the importance of 2-isomorphisms in understanding the homotopical and higher-dimensional aspects of mathematical structures.