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

$\beta = \alpha \star \text{id}_ f : x \circ b = x \circ a \circ f \to x' \circ i \circ f = x' \circ f' \circ j = y' \circ j$

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. $\square$

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