Lemma 7.25.4 (00Y1)—The Stacks project

Lemma 7.25.4. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories

\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \]

Proof. Let us denote objects of $\mathcal{C}/U$ as pairs $(X, a)$ where $X$ is an object of $\mathcal{C}$ and $a : X \to U$ is a morphism of $\mathcal{C}$. Similarly, objects of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ are pairs $(\mathcal{F}, \varphi )$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ sends $\mathcal{G}$ to the pair $(j_{U!}\mathcal{G}, \gamma )$ where $\gamma $ is the composition of $j_{U!}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}* = h_ U^\# $.

Let us construct a functor from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$. Suppose that $(\mathcal{F}, \varphi )$ is given. For an object $(X, a)$ of $\mathcal{C}/U$ we consider the set $\mathcal{F}_\varphi (X, a)$ of elements $s \in \mathcal{F}(X)$ which under $\varphi $ map to the image of $a \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) = h_ U(X)$ in $h_ U^\# (X)$. It is easy to see that $(X, a) \mapsto \mathcal{F}_\varphi (X, a)$ is a sheaf on $\mathcal{C}/U$. Clearly, the rule $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ defines a functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$.

Consider also the functor $\textit{PSh}(\mathcal{C})/h_ U \to \textit{PSh}(\mathcal{C}/U)$, $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ where $\mathcal{F}_\varphi (X, a)$ is defined as the set of elements of $\mathcal{F}(X)$ mapping to $a \in h_ U(X)$. We claim that the diagram

\[ \xymatrix{ \textit{PSh}(\mathcal{C})/h_ U \ar[r] \ar[d] & \textit{PSh}(\mathcal{C}/U) \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) } \]

commutes, where the vertical arrows are given by sheafification. To see this ¹, it suffices to prove that the construction commutes with the functor $\mathcal{F} \mapsto \mathcal{F}^+$ of Lemmas 7.10.3 and 7.10.4 and Theorem 7.10.10. Commutation with $\mathcal{F} \mapsto \mathcal{F}^+$ follows from the fact that given $(X, a)$ the categories of coverings of $(X, a)$ in $\mathcal{C}/U$ and coverings of $X$ in $\mathcal{C}$ are canonically identified.

Next, let $\textit{PSh}(\mathcal{C}/U) \to \textit{PSh}(\mathcal{C})/h_ U$ send $\mathcal{G}$ to the pair $(j_{U!}^{PSh}\mathcal{G}, \gamma )$ where $j_{U!}^{PSh}\mathcal{G}$ the presheaf defined by the formula in Lemma 7.25.2 and $\gamma $ is the composition of $j_{U!}^{PSh}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}^{PSh}* = h_ U$ (obvious from the formula). Then it is immediately clear that the diagram

\[ \xymatrix{ \textit{PSh}(\mathcal{C}/U) \ar[r] \ar[d] & \textit{PSh}(\mathcal{C})/h_ U \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# } \]

commutes, where the vertical arrows are sheafification. Putting everything together it suffices to show there are functorial isomorphisms $(j_{U!}^{PSh}\mathcal{G})_\gamma = \mathcal{G}$ for $\mathcal{G}$ in $\textit{PSh}(\mathcal{C}/U)$ and $j_{U!}^{PSh}\mathcal{F}_\varphi = \mathcal{F}$ for $(\mathcal{F}, \varphi )$ in $\textit{PSh}(\mathcal{C})/h_ U$. The value of the presheaf $(j_{U!}^{PSh}\mathcal{G})_\gamma $ on $(X, a)$ is the fibre of the map

\[ \coprod \nolimits _{a' : X \to U} \mathcal{G}(X, a') \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, U) \]

over $a$ which is $\mathcal{G}(X, a)$. This proves the first equality. The value of the presheaf $j_{U!}^{PSh}\mathcal{F}_\varphi $ is on $X$ is

\[ \coprod \nolimits _{a : X \to U} \mathcal{F}_\varphi (X, a) = \mathcal{F}(X) \]

because given a set map $S \to S'$ the set $S$ is the disjoint union of its fibres. $\square$

[1] An alternative is to describe $\mathcal{F}_\varphi $ by the cartesian diagram

\[ \vcenter { \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_ U|_{\mathcal{C}/U} } } \quad \text{for presheaves and}\quad \vcenter { \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_ U^\# |_{\mathcal{C}/U} } } \]

for sheaves and use that restriction to $\mathcal{C}/U$ commutes with sheafification.

Comments (2)

Comment #6029 by Owen on April 02, 2021 at 13:39

To conclude that the functors on the bottom rows are equivalences from the fact that the functors on the top rows are equivalences, I think you need to isolate subcategories of $PSh(C/U)$ and $PSh(C)/h_U$ which are mapped by sheafification fully and essentially surjectively to $Sh(C/U)$ and $Sh(C)/\underline ah_U$ , respectively (I use $\underline a$ for sheafification). As $Sh(C/U)$ is a full subcategory of $PSh(C/U)$ that is obvious, but for $PSh(C)/h_U$ the relevent subcategory is the essential image of the (fully faithful) functor $Sh(C)/\underline ah_U\to PSh(C)/h_U$ that sends $F\to\underline ah_U$ to $iF\times_{i\underline ah_U}h_U\xrightarrow{\operatorname{pr}_2}h_U$ , where $i:Sh(C)\hookrightarrow PSh(C)$ .

Comment #6176 by Johan on May 01, 2021 at 23:00

OK, I think the argument I had in mind is to show that the functors $\textit{Sh}(\mathcal{C}/U) \to \textit{Sh}(\mathcal{C})/h_U^\sharp$ and $\textit{Sh}(\mathcal{C})/h_U^\sharp \to \textit{Sh}(\mathcal{C}/U)$ constructed in the text of the proof are quasi-inverse. The point of having the commutative diagrams involving presheaves, is that it shows that these functors can be computed by computing the result for presheaves and then sheafifying. Thus it suffices to show that the corresponding functors between presheaf categories are quasi-inverse which is what the proof does. OK?

I would be in the market for a clearer proof of this lemma. Anybody?

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