The Stacks project

Lemma 8.2.3. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism of fibred categories over the category $\mathcal{C}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x, y\in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{S}_1)_ U)$. Then $F$ defines a canonical morphism of presheaves

\[ \mathit{Mor}_{\mathcal{S}_1}(x, y) \longrightarrow \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y)) \]

on $\mathcal{C}/U$.

Proof. By Categories, Definition 4.33.9 the functor $F$ maps strongly cartesian morphisms to strongly cartesian morphisms. Hence if $f : V \to U$ is a morphism in $\mathcal{C}$, then there are canonical isomorphisms $\alpha _ V : f^*F(x) \to F(f^*x)$, $\beta _ V : f^*F(y) \to F(f^*y)$ such that $f^*F(x) \to F(f^*x) \to F(x)$ is the canonical morphism $f^*F(x) \to F(x)$, and similarly for $\beta _ V$. Thus we may define

\[ \xymatrix{ \mathit{Mor}_{\mathcal{S}_1}(x, y)(f : V \to U) \ar@{=}[r] & \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{1, V}}(f^\ast x, f^\ast y) \ar[d] \\ \mathit{Mor}_{\mathcal{S}_2}(F(x), F(y))(f : V \to U) \ar@{=}[r] & \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_{2, V}}(f^\ast F(x), f^\ast F(y)) } \]

by $\phi \mapsto \beta _ V^{-1} \circ F(\phi ) \circ \alpha _ V$. We omit the verification that this is compatible with the restriction mappings. $\square$

Comments (2)

Comment #3381 by Job Rock on

I suspsect we want in the statement of the Lemma.

There are also:

  • 2 comment(s) on Section 8.2: Presheaves of morphisms associated to fibred categories

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Your email address will not be published.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 042V. Beware of the difference between the letter 'O' and the digit '0'.