The Stacks project

Lemma 77.23.1. Assumptions and notation as in Lemmas 77.20.2 and 77.20.3. The vertical composition of

\[ \xymatrix@C=15pc{ \mathcal{S}_{R \times _{s, U, t} R} \ruppertwocell ^{\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_1}} \ar[r]_(.3){\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0} \rlowertwocell _{\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_0}} & [U/R] } \]

is the $2$-morphism $\alpha \star \text{id}_ c$. In a formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1}) $.

Proof. We make two remarks:

  1. The formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1})$ only makes sense if you realize the equalities $\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c$, $\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0$, and $\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c$. Namely, the second one implies the vertical composition $\circ $ makes sense, and the other two guarantee the two sides of the formula are $2$-morphisms with the same source and target.

  2. The reason the lemma holds is that composition in the category fibred in groupoids $[U/_{\! p}R]$ associated to the presheaf in groupoids ( comes from the composition law $c : R \times _{s, U, t} R \to R$.

We omit the proof of the lemma. $\square$

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Your email address will not be published.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 044T. Beware of the difference between the letter 'O' and the digit '0'.