---
date: 2021-10-21 18:42
modification date: Thursday 21st October 2021 18:42:50
title: groupoid
aliases: [groupoid]

---

Tags: 
#higher-algebra/category-theory #higher-algebra/K-theory #higher-algebra/category-theory 
Refs:
[category theory](modern%20category%20theory.md)
<https://www.ams.org/notices/199607/weinstein.pdf>
[Lie algebroid](Unsorted/Lie%20algebroid.md)
[BG](Unsorted/classifying%20space.md)


# groupoid

Idea: simultaneously generalizes groups and equivalence relations.

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

- Every set is a groupoid: just take identity morphisms.
- A groupoid $\cat G$ with one object $X$ is determined by $\Aut_{\cat G}(X)$ which has a group structure, 
- A groupoid with multiple objects and $\Aut_{\cat G}(X) = \pt$ for all $X \in \Ob(\cat G)$ is the same as an equivalence relation on $\Ob(\cat G)$.

# Topological groupoids
![](attachments/Pasted%20image%2020220212182444.png)


# Examples
## Lie groupoids

![](attachments/Pasted%20image%2020220212182656.png)

Definitions of **isotropy groups**:
$$
G_{x}=\left\{g \in G_{1} \mid(s, t)(g)=(x, x)\right\}=(s, t)^{-1}(x, x)=s^{-1}(x) \cap t^{-1}(x) \subset G_{1}
$$

The unit groupoid and translation groupoid:
![](attachments/Pasted%20image%2020220212182923.png)

Manifold groupoid and the fundamental groupoid:
![](attachments/Pasted%20image%2020220212183209.png)

Weighted projective spaces