The Stacks project

Lemma 85.2.3. Let $f : Y \to X$ be a morphism of simplicial spaces. Then the functor $u : X_{Zar} \to Y_{Zar}$ which associates to the open $U \subset X_ n$ the open $f_ n^{-1}(U) \subset Y_ n$ defines a morphism of sites $f_{Zar} : Y_{Zar} \to X_{Zar}$.

Proof. It is clear that $u$ is a continuous functor. Hence we obtain functors $f_{Zar, *} = u^ s$ and $f_{Zar}^{-1} = u_ s$, see Sites, Section 7.14. To see that we obtain a morphism of sites we have to show that $u_ s$ is exact. We will use Sites, Lemma 7.14.6 to see this. Let $V \subset Y_ n$ be an open subset. The category $\mathcal{I}_ V^ u$ (see Sites, Section 7.5) consists of pairs $(U, \varphi )$ where $\varphi : [m] \to [n]$ and $U \subset X_ m$ open such that $Y(\varphi )(V) \subset f_ m^{-1}(U)$. Moreover, a morphism $(U, \varphi ) \to (U', \varphi ')$ is given by a $\psi : [m'] \to [m]$ such that $X(\psi )(U) \subset U'$ and $\varphi \circ \psi = \varphi '$. It is our task to show that $\mathcal{I}_ V^ u$ is cofiltered.

We verify the conditions of Categories, Definition 4.20.1. Condition (1) holds because $(X_ n, \text{id}_{[n]})$ is an object. Let $(U, \varphi )$ be an object. The condition $Y(\varphi )(V) \subset f_ m^{-1}(U)$ is equivalent to $V \subset f_ n^{-1}(X(\varphi )^{-1}(U))$. Hence we obtain a morphism $(X(\varphi )^{-1}(U), \text{id}_{[n]}) \to (U, \varphi )$ given by setting $\psi = \varphi $. Moreover, given a pair of objects of the form $(U, \text{id}_{[n]})$ and $(U', \text{id}_{[n]})$ we see there exists an object, namely $(U \cap U', \text{id}_{[n]})$, which maps to both of them. Thus condition (2) holds. To verify condition (3) suppose given two morphisms $a, a': (U, \varphi ) \to (U', \varphi ')$ given by $\psi , \psi ' : [m'] \to [m]$. Then precomposing with the morphism $(X(\varphi )^{-1}(U), \text{id}_{[n]}) \to (U, \varphi )$ given by $\varphi $ equalizes $a, a'$ because $\varphi \circ \psi = \varphi ' = \varphi \circ \psi '$. This finishes the proof. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 85.2: Simplicial topological spaces

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Your email address will not be published.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 09VN. Beware of the difference between the letter 'O' and the digit '0'.