The Stacks project

79.13 Finite collections of arrows

Let $\mathcal{C}$ be a groupoid, see Categories, Definition 4.2.5. As discussed in Groupoids, Section 39.13 this corresponds to a septuple $(\text{Ob}, \text{Arrows}, s, t, c, e, i)$.

Using this data we can make another groupoid $\mathcal{C}_{fin}$ as follows:

1. An object of $\mathcal{C}_{fin}$ consists of a finite subset $Z \subset \text{Arrows}$ with the following properties:

   1. $s(Z) = \{ u\} $ is a singleton, and

   2. $e(u) \in Z$.

2. A morphism of $\mathcal{C}_{fin}$ consists of a pair $(Z, z)$, where $Z$ is an object of $\mathcal{C}_{fin}$ and $z \in Z$.

3. The source of $(Z, z)$ is $Z$.

4. The target of $(Z, z)$ is $t(Z, z) = \{ z' \circ z^{-1}; z' \in Z\} $.

5. Given $(Z_1, z_1)$, $(Z_2, z_2)$ such that $s(Z_1, z_1) = t(Z_2, z_2)$ the composition $(Z_1, z_1) \circ (Z_2, z_2)$ is $(Z_2, z_1 \circ z_2)$.

We omit the verification that this defines a groupoid. Pictorially an object of $\mathcal{C}_{fin}$ can be viewed as a diagram

To make a morphism of $\mathcal{C}_{fin}$ you pick one of the arrows and you precompose the other arrows by its inverse. For example if we pick the middle horizontal arrow then the target is the picture

Note that the cardinalities of $s(Z, z)$ and $t(Z, z)$ are equal. So $\mathcal{C}_{fin}$ is really a countable disjoint union of groupoids.