2isomorphisms

A 2-isomorphism is a concept in category theory that generalizes the notion of an isomorphism between objects to a relationship between 2-categories. In a 2-category, the hom-sets are themselves categories. A 2-isomorphism between two 2-categories, say $C$ and $D$, is a pair of functors $F: C \to D$ and $G: D \to C$ such that there are natural isomorphisms $\eta: 1_C \Rightarrow G \circ F$ and $\epsilon: F \circ G \Rightarrow 1_D$. These natural isomorphisms are often called unit and counit. This definition is analogous to the definition of an isomorphism between categories, where the unit and counit are required to be identity natural transformations.

The existence of a 2-isomorphism between two 2-categories implies that they are essentially the same from the perspective of 2-categorical structure. They share the same 2-categorical properties and can be interchanged in many contexts. The relationship of being 2-isomorphic is an equivalence relation on the class of 2-categories. This means that any 2-category is 2-isomorphic to itself, and if $C$ is 2-isomorphic to $D$ and $D$ is 2-isomorphic to $E$, then $C$ is 2-isomorphic to $E$.

In simpler terms, while an isomorphism between ordinary categories means they are structurally identical, a 2-isomorphism between 2-categories means their higher-dimensional structure is also essentially the same. This concept is crucial for understanding the relationships between different 2-categorical structures and for developing advanced topics in category theory and its applications, such as in algebraic topology and theoretical computer science.