Tags: #higher-algebra/category-theory #higher-algebra/K-theory #higher-algebra/category-theory Refs: category theory https://www.ams.org/notices/199607/weinstein.pdf Lie algebroid BG

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 \(\mathsf{G}\) with one object \(X\) is determined by \(\mathop{\mathrm{Aut}}_{\mathsf{G}}(X)\) which has a group structure,
• A groupoid with multiple objects and \(\mathop{\mathrm{Aut}}_{\mathsf{G}}(X) = {\operatorname{pt}}\) for all \(X \in {\operatorname{Ob}}(\mathsf{G})\) is the same as an equivalence relation on \({\operatorname{Ob}}(\mathsf{G})\).

Topological groupoids

Examples

Lie groupoids

Definitions of isotropy groups: \begin{align*} G_{x}=\left\{g \in G_{1} \mathrel{\Big|}(s, t)(g)=(x, x)\right\}=(s, t)^{-1}(x, x)=s^{-1}(x) \cap t^{-1}(x) \subset G_{1} \end{align*}

The unit groupoid and translation groupoid:

Manifold groupoid and the fundamental groupoid:

Weighted projective spaces