nLab 2-isomorphism

Idea

In a higher category, invertibility of n-morphisms in its highest dimension is always considered strictly. Thus in an ordinary category, we have 1-morphisms which can be isomorphisms. In a 2-category, it is the 2-morphisms for which, when it comes to invertibility, we always ask for a strict inverse, whereas for 1-morphisms we typically ask only for an equivalence. These 2-morphisms which admit an inverse are known as 2-isomorphisms.

Definition

Definition

Let $\mathcal{A}$ be a 2-category. A 2-isomorphism in $\mathcal{A}$ is a 2-arrow $\phi : f \rightarrow f'$ of $\mathcal{A}$ which admits a (strict) inverse, that is to say, there is a 2-arrow $\phi^{-1}: f' \rightarrow f$ of $\mathcal{A}$ such that $\phi^{-1} \circ \phi = id(f)$ and $\phi \circ \phi^{-1} = id\left(f' \right)$.

Related concepts

A 2-category in which every 2-arrow is a 2-isomorphism is known as a (2,1)-category.