(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
In the context of the geometry of schemes there is a traditional notion of étale morphism of schemes and an étale topos is a category of sheaves on the étale site of a scheme, consisting of covers by such étale morphisms. This traditional notion we discuss in
More abstractly, given that étale morphisms of schemes may be characterized as modal morphisms with respect to an infinitesimal shape modality, one can consider étale toposes in every context of differential cohesion. This we discuss in
An étale topos is the sheaf topos over an étale site, hence over a site whose “open subsets” are étale morphisms into the base space. The intrinsic cohomology of an étale (∞,1)-topos is étale cohomology.
More generally there is the pro-étale topos over a pro-étale site, which is a bit better behaved. In particular the intrinsic cohomology of a pro-étale (∞,1)-topos includes the Weil cohomology theory ℓ-adic cohomology.
Generally, given that an étale morphism of schemes is a formally étale morphism subject to a size constraint on its fibers – for an actual étale morphism of schemes the fibers are finite sets in the suitable sense (formal duals to étale algebras) while for a pro-étale morphism of schemes they are pro-objects of such fibers – in a suitable ambient context (“differential cohesion”) one can drop all finiteness conditions and consider just opens given by formally étale morphisms as encoded by an infinitesimal shape modality. This we discuss below.
We discuss how in differential cohesion $\mathbf{H}_{th}$ every object $X$ canonically induces its étale topos $Sh_{\mathbf{H}_{th}}(X)$.
For $X \in \mathbf{H}_{th}$ any object in a differential cohesive $\infty$-topos, we formulate
the (∞,1)-topos denoted $\mathcal{X}$ or $Sh_\infty(X)$ of (∞,1)-sheaves over $X$, or rather of formally étale maps into $X$;
the structure (∞,1)-sheaf $\mathcal{O}_{X}$ of $X$.
The resulting structure is essentially that discussed (Lurie, Structured Spaces) if we regard $\mathbf{H}_{th}$ equipped with its formally étale morphisms, def. \ref{FormallyEtaleInHTh}, as a (large) geometry for structured (∞,1)-toposes.
One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive ∞-group with de Rham coefficient object $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection
regarded as an object of the slice (∞,1)-topos $(\mathbf{H}_{th})_{/X}$ almost qualifies as a “bundle of flat $\mathfrak{g}$-valued differential forms” over $X$: for $U \to X$ an cover (a 1-epimorphism) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$.
This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like – but only if $U \to X$ is sufficiently spread out over $X$, hence sufficiently étale. Because, on the extreme, if $X$ is the point (the terminal object), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$.
In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be restricted to plots out of suficiently étale maps into $X$. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion
of the formally étale maps of def. \ref{FormallyEtaleInHTh} into $X$. Since on formally étale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding “cotangent bundle” must be the coreflection along this inclusion. The following proposition establishes that this coreflection indeed exists.
For $X \in \mathbf{H}_{th}$ any object, write
for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on those maps into $X$ which are formally étale, def. \ref{FormallyEtaleInHTh}.
We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$.
The inclusion $\iota$ of def. 1 is both reflective as well as coreflective, hence it fits into an adjoint triple of the form
By the general discussion at reflective factorization system, the reflection is given by sending a morphism $f \colon Y \to X$ to $X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in
Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, being a reflective subcategory of a locally presentable (∞,1)-category, is (as discussed there) itself locally presentable. Hence by the adjoint (∞,1)-functor theorem it is now sufficient to show that the inclusion preserves all small (∞,1)-colimits in order to conclude that it also has a right adjoint (∞,1)-functor.
So consider any diagram (∞,1)-functor $I \to (\mathbf{H}_{th})_{/X}^{fet}$ out of a small (∞,1)-category. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{fet}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at slice category - Colimits).
Therefore we are reduced to showing that the square
is an (∞,1)-pullback square. But since $\mathbf{\Pi}_{inf}$ is a left adjoint it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to
This diagram is now indeed an (∞,1)-pullback by the fact that we have universal colimits in the (∞,1)-topos $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the (∞,1)-pullback of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.
The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an (∞,1)-topos and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a geometric embedding.
By prop. 1 the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is reflective with reflector given by the $(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed)$ factorization system. Since $\mathbf{\Pi}_{inf}$ is a right adjoint and hence in particular preserves (∞,1)-pullbacks, the $\mathbf{\Pi}_{inf}$-equivalences are stable under pullbacks. By the discussion at stable factorization system this is the case precisely if the corresponding reflector preserves finite (∞,1)-limits. Hence the embedding is a geometric embedding which exhibits a sub-(∞,1)-topos inclusion.
For $X \in \mathbf{H}_{th}$ we speak of
also as the (petit) (∞,1)-topos of $X$, or the étale topos of $X$.
Write
for the composite (∞,1)-functor that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. 1, of the projection $A \times X \to X$.
We call $\mathcal{O}_X$ the structure sheaf of $X$.
For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a formally étale morphism in $\mathbf{H}_{th}$ (hence like an open subset of $X$), we have that
where we used the ∞-adjunction $(\iota \dashv Et)$ of prop. 1 and the (∞,1)-Yoneda lemma.
This means that $\mathcal{O}_{X}(A)$ behaves as the sheaf of $A$-valued functions over $X$.
Since $\mathcal{O}_{X}$ is right adjoint to the forgetful functor
it preserves (∞,1)-limits. Therefore this is a structure sheaf which exhibits $Sh_{\mathbf{H}_{th}}(X)$ as a structured (∞,1)-topos over $\mathbf{H}_{th}$ regarded as a (large) geometry (for structured (∞,1)-toposes), with the formally étale morphisms being the “admissible morphisms”.
Let $G \in Grp(\mathbf{H}_{th})$ be an ∞-group and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object.
Then
we may call the $G$-valued flat cotangent sheaf of $X$.
For $U \in \mathbf{H}_{th}$ a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formally étale morphism $U \to X$ is like an open map/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams
By the fact that $Et(-)$ is right adjoint, such diagrams are in bijection to diagrams
where we are now simply including on the left the formally étale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$.
In other words, the sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the domain of the section is constrained to be a formally é patch of $X$.
But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$.
For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive $\infty$-topos, then its petit structured $\infty$-topos $Sh_{\mathbf{H}_{th}}(X)$, according to def. 2, is locally ∞-connected.
We need to check that the composite
preserves (∞,1)-limits, so that it has a further left adjoint. Here $L$ is the reflector from prop. 1. Inspection shows that this composite sends an object $A \in \infty Grpd$ to $\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X$:
By the discussion at slice (∞,1)-category – Limits and colimits an (∞,1)-limit in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an (∞,1)-limit in $\mathbf{H}$ of the diagram with the slice cocone adjoined. By right adjointness of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$.
Now for $A \colon J \to \infty Grpd$ a diagram, it is taken to the diagram $j \mapsto \mathbf{\Pi}_{inf}(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form
Since $\infty$-limits commute with each other this limit is the product of
$\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))$
$\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$).
For the first of these, since the infinitesimal shape modality $\mathbf{\Pi}_{inf}$ is in particular a right adjoint (with left adjoint the reduction modality), and since $Disc$ is also right adjoint by cohesion, we have a natural equivalence
For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is
where the last line follows since ${J \star \Delta^0}$ has a terminal object and hence contractible geometric realization.
In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$.
For $X$ a scheme, and $A \in PSh(X_{et})$ a presheaf, for checking the sheaf condition it is sufficient to check descent on the following two kinds of covers in the étale site
jointly surjective collections of open immersions of schemes;
single surjective/étale morphisms between affine schemes
(all over $X$).
(Tamme, II Lemma (3.1.1), Milne, prop. 6.6)
Since covers by standard open immersions of schemes in the Zariski topology are also étale morphisms of schemes and étale covers, we may take any étale cover $\{Y_i \to Y\}$ over $X$, find an Zariski cover $\{U_i \to X\}$ of $X$, pull back the original cover to that and in turn cover the pullbacks themselves by Zariski covers. The result is still a cover and is so by a collection of open immersions of schemes. Now using compactness assumptions we find finite subcovers of all these covers. This makes their disjoint union be a single morphisms of affines.
For $Z \to X$ any scheme over a scheme $X$, the induced presheaf on the étale site
is a sheaf.
This is due to (Grothendieck, SGA1 exp. XIII 5.3) A review is in (Tamme, II theorem (3.1.2), Milne, 6.2).
By prop. 4 we are reduced to showing that the represented presheaf satisfies descent along collections of open immersions and along surjective maps of affines. For the first this is clear (it is Zariski topology-descent). For the second case of a faithfully flat cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the correspomnding Amitsur complex. See there for details.
This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$. Unique representatives are given by étale schemess over $X$.
(e.g. Tamme, II theorem 3.1)
We consider some examples of sheaves of abelian groups induced by prop. 5 from group schemes over $X$.
The additive group over $X$ is the group scheme
By the universal property of the pullback, the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment
In other words, the sheaf represented by the additive group is the abelian sheaf underlying the structure sheaf of $X$.
Similarly one finds
The multiplicative group over $X$
represents the sheaf $(\mathbb{G}_m)_X$ given by
(e.g. Tamme, II, 3)
For $f \colon X \longrightarrow Y$ a homomorphism of schemes, there is induced a functor on the categories underlying the étale site
given by sending an object $U_Y \to Y$ to the fiber product/pullback along $f$
The morphism in def. 3 is a morphism of sites and hence induces a geometric morphism between the étale toposes
Here the direct image is given on a sheaf $\mathcal{F} \in Sh(X_{et})$ by
while the inverse image is given on a sheaf $\mathcal{F} \in Sh_(Y_{et})$ by
By the discussion at morphisms of sites – Relation to geometric morphisms. See also for instance (Tamme I 1.4).
For $X_{et}$ an étale site, write $\mathcal{D}(X_{et})$ for the derived category of the abelian category $Ab(Sh(X_{et}))$ of abelian sheaves on $X$.
The $q$th derived functor $R^q f_\ast$ of the direct image functor of def. 6 sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the sheafification of the presheaf
where on the right we have the degree $q$ abelian sheaf cohomology group with coefficients in the given $\mathcal{F}$ (étale cohomology).
By the discussion at direct image and at abelian sheaf cohomology. See e.g. (Tamme, II (1.3.4), Milne prop. 12.1).
For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$ two composable morphisms of sites, the Leray spectral sequence for the corresponding direct images exists and is of the form
For the special case that $S_Z = \ast$ and $g^{-1}$ includes an étale morphism $U_Y \to Y$ this yields
For $X$ a scheme and $N$ a quasicoherent module over its structure sheaf $\mathcal{O}_X$, then this induces an abelian sheaf on the étale site by
(e.g. Tamme, II 3.2.1)
A cover in the Zariski topology on schemes is an open immersion of schemes and hence is in particular an étale morphism of schemes. Hence the étale site is finer than the Zariski site and so every étale sheaf is a Zariski sheaf, but not necessarily conversely.
For more see at étale cohomology – Properties – Relation to Zariski cohomology.
The étale topos over the big étale site of commutative rings is the classifying topos for strict local rings.