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78.13 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.


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