Lemma 96.18.1 (06X5)—The Stacks project

Lemma 96.18.1. Generalities on Čech complexes.

If

\[ \xymatrix{ \mathcal{V} \ar[d]_ g \ar[r]_ h & \mathcal{U} \ar[d]^ f \\ \mathcal{Y} \ar[r]^ e & \mathcal{X} } \]

is $2$-commutative diagram of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, then there is a morphism of Čech complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{U} \to \mathcal{X}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{V} \to \mathcal{Y}, e^{-1}\mathcal{F}) \]
if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,
if $f, f' : \mathcal{U} \to \mathcal{X}$ are $2$-isomorphic, then the associated Čech complexes are isomorphic.

Proof. In the situation of (1) let $t : f \circ h \to e \circ g$ be a $2$-morphism. The map on complexes is given in degree $n$ by pullback along the $1$-morphisms $\mathcal{V}_ n \to \mathcal{U}_ n$ given by the rule

\[ (v_0, \ldots , v_ n, y, \beta _0, \ldots , \beta _ n) \longmapsto (h(v_0), \ldots , h(v_ n), e(y), e(\beta _0) \circ t_{v_0}, \ldots , e(\beta _ n) \circ t_{v_ n}). \]

For (2), note that pullback on global sections is an isomorphism for any presheaf of sets when the pullback is along an equivalence of categories. Part (3) follows on combining (1) and (2). $\square$

Comments (2)

Comment #7748 by Mingchen on September 10, 2022 at 10:48

Minor thing: the last comma before the proof should be a period.

Comment #7993 by Stacks Project on January 14, 2023 at 12:31

Thanks and fixed here.

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