The Stacks project

Remark 85.5.1. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 85.3.3. A morphism $h$ between simplicial sites will mean

  1. Morphisms of sites $h_ n : \mathcal{C}_ n \to \mathcal{C}'_ n$ such that $f'_\varphi \circ h_ n = h_ m \circ f_\varphi $ as morphisms of sites for all $\varphi : [m] \to [n]$.

  2. Cocontinuous functors $v_ n : \mathcal{C}_ n \to \mathcal{C}'_ n$ inducing morphisms of topoi $h_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_ n)$ such that $u'_\varphi \circ v_ n = v_ m \circ u_\varphi $ as functors for all $\varphi : [m] \to [n]$.

In both cases we have $f'_\varphi \circ h_ n = h_ m \circ f_\varphi $ as morphisms of topoi, see Sites, Lemma 7.21.2 for case B and Sites, Definition 7.14.5 for case A.

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Your email address will not be published.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 0D95. Beware of the difference between the letter 'O' and the digit '0'.