nLab 2-isomorphism

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

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

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

Last revised on July 4, 2020 at 17:50:55. See the history of this page for a list of all contributions to it.