3 Pro-categories
Let \(\mathcal{C}\) be a category. The category \(\mathrm{Pro}(\mathcal{C})\) is the full subcategory of \(\mathrm{Fun}(\mathcal{C}, \mathrm{Set})^{\mathrm{op}}\) on the functors that are cofiltered limits of representable functors. This category satisfies the following universal property:
Let \(\mathcal{D}\) be a category with cofiltered limits. The restriction functor
admits a left-adjoint that induces an equivalence on the full subcategory of \(\mathrm{Fun}(\mathrm{Pro}(\mathcal{C}), \mathcal{D})\) on functors preserving cofiltered limits.
The left-adjoint is the left Kan extension along the inclusion \(\mathcal{C} \to \mathrm{Pro}(\mathcal{C})\).
https://ncatlab.org/nlab/show/pro-object#PropositionCategoriesEquivalentToProC
Let \(\mathcal{A}\) be a category with cofiltered limits and suppose there exists a fully faithful functor \(i\colon \mathcal{C} \to \mathcal{A}\) such that \(i(X)\) is cocompact for all \(X \in \mathcal{C}\). Then
is fully faithful.
Let \(P\) be a property of objects on a category \(\mathcal{A}\) that implies finitely presented. Denote by \(\mathcal{A}_P\) (resp. \(\mathcal{A}_{\mathrm{Pro}(P)}\)) the full subcategory defined by \(P\) (resp. \(\mathrm{Pro}(P)\)). Suppose \(\mathcal{A}\) has cofiltered limits, then \(\lim _{\mathcal{A}}\) induces an equivalence
Follows from 3.2, because by definition \(\mathcal{C}_{\mathrm{Pro}(P)}\) is the essential image of \(\lim _{\mathcal{A}}(\mathrm{Pro}(\mathcal{C}_{P}))\).
Note that 3.3 does not apply to \(\mathcal{A} = \mathrm{Scheme} / X\) and \(P = \mathrm{étale}\), because \(\mathrm{Scheme} / X\) does not have all cofiltered limits. It does apply in the case of \(\mathcal{A} = \mathrm{AffScheme} / X\) though or in the case \(\mathrm{CRing}^{\mathrm{op}} / R\).
Recall the following definition:
A category \(\mathcal{C}\) is precoherent if given a finite effective epimorphism family \(f\colon \colon X_i \to X\) and a morphism \(p\colon Y \to X\), there exists a finite effective epimorphism family \(g\colon Y_j \to Y\) that factors through \(f\).
A morphism \(f\colon X \to Y\) in \(S_{\mathrm{et}}\) is an effective epimorphism if and only if it is surjective.
One direction follows from the fact that the fpqc topology is subcanonical. The converse holds, because étale morphisms are open: If \(f\colon X \to Y\) is an étale morphism we may glue two copies of \(Y\) along the inclusion of the image of \(f\). If \(f\) is not surjective, the two inclusions of \(Y\) into the amalgamated sum are different, but by construction they agree on the image of \(f\). Hence \(f\) is not an epimorphism.
The category of schemes is finitary extensive.
Since the Zariski topology is subcanonical, the category of schemes embeds into the big Zariski topos, which is finitary extensive. Since the Yoneda embedding preserves finite coproducts, the claim follows.
The category \(X_{\mathrm{et}}\) is precoherent.
The category of schemes is finitary extensive, so the same holds for \(X_{\mathrm{et}}\). By 3.5 the effective epimorphisms in \(X_{\mathrm{et}}\) are the surjective morphisms, hence being an effective epimorphism is stable under base change. Thus \(X_{\mathrm{et}}\) is preregular and therefore precoherent.
If \(\mathcal{C}\) is precoherent, also \(\mathrm{Pro}(\mathcal{C})\) is precoherent.
3.1 Coherent topology
Any precoherent category can be equipped with the coherent coverage, where covering families are given by finite effective epimorphic families. The coherent topology is the topology generated by the coherent coverage.
One would hope that the topology on \(X_{\mathrm{et}}\) given by jointly surjective families is the precoherent topology. Since effective epimorphisms are simply surjective maps in \(X_{\mathrm{et}}\), this would mean that every cover in \(X_{\mathrm{et}}\) can be refined by a finite cover. This is not true though, because \(X_{\mathrm{et}}\) also contains non-quasi-compact schemes.
The situation is slightly better if we instead consider the affine étale site \(X^{\mathrm{aff}}_{\mathrm{ét}}\) of \(X\), the category of affine schemes étale over \(X\). Unfortunately, it is not clear to the authors if effective epimorphisms in \(X^{\mathrm{aff}}_{\mathrm{ét}}\) are still surjective.
Let from now on \(\mathcal{C}\) be precoherent equipped with the coherent topology.
Let \(F \colon \mathrm{Pro}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Set}\) be a sheaf. Then \(F|_{\mathcal{C}^{\mathrm{op}}}\) is a sheaf.
By 3.10, the inclusion functor \(\mathcal{C} \to \mathrm{Pro}(\mathcal{C})\) induces a morphism on sites \(\nu \colon \mathrm{Pro}(\mathcal{C}) \to \mathcal{C}\). The direct image functor \(\nu _{*}\colon \mathrm{Shv}(\mathrm{Pro}(\mathcal{C})) \to \mathrm{Shv}(\mathcal{C})\) agrees with the restriction \(F \mapsto F|_{\mathcal{C}^{\mathrm{op}}}\).
Let \(F \colon \mathrm{Pro}(\mathcal{C})^{\mathrm{op}} \to \mathrm{Set}\) be a presheaf. Suppose \(F|_{\mathcal{C}^{\mathrm{op}}}\) is a sheaf and \(F\) preserves filtered colimits. Then \(F\) is a sheaf.
Combining 3.1 and 3.11 yields:
Let \(F\) be a sheaf on \(\mathcal{C}\). Then the unit \(F \to \nu _{*} \nu ^{-1} F\) is an isomorphism.
Denote by \(\mathrm{res}\) the restriction functor \(\mathrm{PShv}(\mathrm{Pro}(\mathcal{C})) \to \mathrm{PShv}(\mathcal{C})\) and by \(\mathrm{ext}\) the left-adjoint of 3.1. By 3.11, \(\mathrm{ext}\) restricts to a functor \(\mathrm{Shv}(\mathrm{Pro}(\mathcal{C})) \to \mathrm{Shv}(\mathcal{C})\) that is the left-adjoint of \(\nu _*\). In particular it agrees with \(\nu ^{-1}\) and the result follows.
3.2 Generated topologies
The previous section established a relationship between the coherent topologies on \(\mathcal{C}\) and \(\mathrm{Pro}(\mathcal{C})\). In the situations we are interested in, the topologies don’t necessarily agree with the coherent topologies though (see Remark 3.9).
To remedy the situation, we make the following definitions:
Let \(\mathcal{U}\) be a one-hypercover in a site \(\mathcal{C}\). We say \(\mathcal{U}\) is exact if taking filtered colimits preserves the multiequalizer diagram of \(\mathcal{U}\).
Any finite one-hypercover is exact, because filtered colimits commute with finite limits.
Let \(F\colon \mathcal{C} \to \mathcal{D}\) be a functor between sites. We say a one-hypercover \(\mathcal{U}\) of an object \(X\) in \(\mathcal{D}\) is relatively pro-representable if it is exact and \(\mathcal{U}\) can be written as the componentwise cofiltered limit of one-hypercovers in \(\mathcal{C}\).
Concretely, a one-hypercover \(\mathcal{U} = (U_{\alpha }, U_{\alpha \beta \gamma })\) is relatively pro-representable if there exists
a cofiltered diagram \(\mathcal{I}\),
functors \(V_{\alpha }\colon \mathcal{I} \to \mathcal{C}\) and \(V_{\alpha \beta \gamma }\colon \mathcal{I} \to \mathcal{C}\),
natural transformations \(p_1 \colon V_{\alpha \beta \gamma } \to V_{\alpha }\) and \(p_2\colon V_{\alpha \beta \gamma } \to V_{\beta }\),
such that
\(U_{\alpha } = \lim _i V_{\alpha }(i)\) and \(U_{\alpha \beta \gamma } = \lim _i V_{\alpha \beta \gamma }(i)\), and
for every \(i \in \mathcal{I}\), the induced pre-one-hypercover \((V_{\alpha }(i), V_{\alpha \beta \gamma }(i))\) is a one-hypercover.
Suppose \(\mathcal{C}\) is coherent, \(\mathcal{D} = \mathrm{Pro}(\mathcal{C})\) and \(F \colon \mathcal{C} \to \mathrm{Pro}(\mathcal{C})\) the inclusion. If both \(\mathcal{C}\) and \(\mathrm{Pro}(\mathcal{C})\) are equipped with the coherent topologies, the topology on \(\mathrm{Pro}(\mathcal{C})\) is generated by relatively pro-representable one-hypercovers.
The analogue of 3.11 is:
Suppose the topology on \(\mathcal{D}\) is generated by relatively pro-representable one-hypercovers and let \(P\) be a presheaf on \(\mathcal{D}\). If \(P \circ F^{\mathrm{op}}\) is a sheaf on \(\mathcal{C}\) and \(P\) preserves filtered colimits, then \(P\) is a sheaf on \(\mathcal{D}\).
By the assumption, it suffices to check the sheaf condition for every relatively pro-representable one-hypercover \(\mathcal{U} = \lim F(\mathcal{V}_i)\). Because \(P\) preserves filtered colimits, one computes that the sheaf condition for \(\mathcal{U}\) is the filtered colimit of the sheaf conditions for \(\mathcal{V}_i\). But \(P \circ F^{\mathrm{op}}\) is a sheaf and filtered colimits preserve the equalizers because \(\mathcal{U}\) is exact.