The Stacks project

Lemma 4.39.5. Let $\mathcal{C}$ be a category. Let $\mathcal{S}$ be a category over $\mathcal{C}$.

  1. If $\mathcal{S} \to \mathcal{S}'$ is an equivalence over $\mathcal{C}$ with $\mathcal{S}'$ fibred in sets over $\mathcal{C}$, then

    1. $\mathcal{S}$ is fibred in setoids over $\mathcal{C}$, and

    2. for each $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the map $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ identifies the target as the set of isomorphism classes of the source.

  2. If $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in setoids, then there exists a category fibred in sets $p' : \mathcal{S}' \to \mathcal{C}$ and an equivalence $\text{can} : \mathcal{S} \to \mathcal{S}'$ over $\mathcal{C}$.

Proof. Let us prove (2). An object of the category $\mathcal{S}'$ will be a pair $(U, \xi )$, where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $\xi $ is an isomorphism class of objects of $\mathcal{S}_ U$. A morphism $(U, \xi ) \to (V , \psi )$ is given by a morphism $x \to y$, where $x \in \xi $ and $y \in \psi $. Here we identify two morphisms $x \to y$ and $x' \to y'$ if they induce the same morphism $U \to V$, and if for some choices of isomorphisms $x \to x'$ in $\mathcal{S}_ U$ and $y \to y'$ in $\mathcal{S}_ V$ the compositions $x \to x' \to y'$ and $x \to y \to y'$ agree. By construction there are surjective maps on objects and morphisms from $\mathcal{S} \to \mathcal{S}'$. We define composition of morphisms in $\mathcal{S}'$ to be the unique law that turns $\mathcal{S} \to \mathcal{S}'$ into a functor. Some details omitted. $\square$

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Your email address will not be published.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 0045. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 4.39.5 as pdf