Lemma 105.6.3. The category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ constructed in Remark 105.6.1 is fibred in groupoids.

**Proof.**
We claim the fibre categories of $p$ are groupoids. If $(f, f', \gamma ')$ as in (105.6.1.2) is a morphism such that $f : U \to V$ is an isomorphism, then $f'$ is an isomorphism by Lemma 105.5.2 and hence $(f, f', \gamma ')$ is an isomorphism.

Consider a morphism $f : V \to U$ in $W_{spaces, {\acute{e}tale}}$ and an object $\xi = (U, U', a, i, x', \alpha )$ of $\mathcal{C}$ over $U$. We are going to construct the “pullback” $f^*\xi $ over $V$. Namely, set $b = a \circ f$. Let $f' : V' \to U'$ be the étale morphism whose restriction to $V$ is $f$ (More on Morphisms of Spaces, Lemma 75.8.2). Denote $j : V \to V'$ the corresponding thickening. Let $y' = x' \circ f'$ and $\gamma = \text{id} : x' \circ f' \to y'$. Set

It is clear that $(f, f', \gamma ) : (V, V', b, j, y', \beta ) \to (U, U', a, i, x', \alpha )$ is a morphism as in (105.6.1.2). The morphisms $(f, f', \gamma )$ so constructed are strongly cartesian (Categories, Definition 4.33.1). We omit the detailed proof, but essentially the reason is that given a morphism $(g, g', \epsilon ) : (Y, Y', c, k, z', \delta ) \to (U, U', a, i, x', \alpha )$ in $\mathcal{C}$ such that $g$ factors as $g = f \circ h$ for some $h : Y \to V$, then we get a unique factorization $g' = f' \circ h'$ from More on Morphisms of Spaces, Lemma 75.8.2 and after that one can produce the necessary $\zeta $ such that $(h, h', \zeta ) : (Y, Y', c, k, z', \delta ) \to (V, V', b, j, y', \beta )$ is a morphism of $\mathcal{C}$ with $(g, g', \epsilon ) = (f, f', \gamma ) \circ (h, h', \zeta )$.

Therefore $p : \mathcal{C} \to W_{\acute{e}tale}$ is a fibred category (Categories, Definition 4.33.5). Combined with the fact that the fibre categories are groupoids seen above we conclude that $p : \mathcal{C} \to W_{\acute{e}tale}$ is fibred in groupoids by Categories, Lemma 4.35.2. $\square$

## 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Your email address will not be published.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.

## Comments (0)