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

\[ \xymatrix{ & \bullet \\ \bullet \ar@(ul, dl)[]_ e \ar[ru] \ar[r] \ar[rd] & \bullet \\ & \bullet } \]

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

\[ \xymatrix{ & \bullet \\ \bullet & \bullet \ar[l] \ar[u] \ar@(dr, ur)[]_ e \ar[d] \\ & \bullet } \]

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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04RS. Beware of the difference between the letter 'O' and the digit '0'.