Subsection 8.3.1 (00Z2): Foundations

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Table of Contents
Part 3: Category Theory
Chapter 8: Categories
Section 8.3: Groupoids
Subsection 8.3.1: Foundations (cite)

8.3.1 Foundations

Let $\mathcal{C}$ be a category.

Definition 8.3.1.1.1 ▶ Isomorphisms

A morphism $f\colon A\to B$ of $\mathcal{C}$ is an isomorphism if there exists a morphism $\smash {f^{-1}\colon B\to A}$ of $\mathcal{C}$ such that

\begin{align*} f\circ f^{-1} & = \text{id}_{B},\\ f^{-1}\circ f & = \text{id}_{A}. \end{align*}

Notation 8.3.1.1.2 ▶ The Set of Isomorphisms Between Two Objects in a Category

We write $\textup{Iso}_{\mathcal{C}}\webleft (A,B\webright )$ for the set of all isomorphisms in $\mathcal{C}$ from $A$ to $B$.

Definition 8.3.1.1.3 ▶ Groupoids

A groupoid is a category in which every morphism is an isomorphism.

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