Is every sheaf a sheaf of sections?

In my first post mentioning sheaves, I introduce sheaves on a topological space $X$ as the fixed points of the nerve-realisation adjunction induced by the canonical inclusion $\mathbf{Op}(X)\to(\mathbf{Top}\downarrow X)$ .

Briefly, the category of presheaves on $X$ is the free cocompletion of $\mathbf{Op}(X)$ , so the canonical inclusion above (since it maps into a cocomplete category) admits an essentially unique cocontinuous extension $|-|:\mathbf{PSh}(\mathbf{Op}(X))\to(\mathbf{Top}\downarrow X)$ , and it turns out that this map admits a right adjoint $\Gamma$ , which sends a space $Y/X$ to its sheaf $\Gamma_{Y/X}$ of local sections.

In this setting, the sheaves on $X$ are those presheaves $F$ for which the adjunction unit $F\Rightarrow \Gamma_{|F|/X}$ is a natural isomorphism. On the flip side, the spaces $Y/X$ for which the adjunction counit $|\Gamma_{Y/X}|\to Y$ is a homeomorphism are precisely the étale spaces over $X$ . Therefore, this adjunction restricts to an equivalence between sheaves on $X$ , and étale spaces over $X$ . In particular, every sheaf is a sheaf of sections for some (étale) space (granted, if sheaves are defined this way, this is tautological).

Sheaves then get a fun generalisation to arbitrary sites, so a question that bugged me when I moved to this generality was if the above formalism somehow carried over. I’m not alone, as this was also asked on MathOverflow (10 years ago, wow), and it seems the answer is—to some extent—yes! In this case, we categorify our spaces and conflate spaces with their Grothendieck topoi.

Fix a site $\mathcal C$ . As described by the top answer therein, the “étale space” associated to any sheaf $F$ on $\mathcal C$ is the slice topos $\mathbf{Sh}(\mathcal C)\downarrow F$ , which canonically lies over $\mathbf{Sh}(\mathcal C)$ . In fact, this canonical projection is part of an essential geometric morphism $\pi_!\dashv\pi^*\dashv\pi_*$ with $\pi_!:\mathbf{Sh}(\mathcal C)\downarrow F\to\mathbf{Sh}(\mathcal C)$ being the canonical projection. This is a particular instance of base change in a topos, where the base change is computed along the unique morphism $F\Rightarrow*$ of sheaves.

Proposition 1. If $\mathcal H$ is a Grothendieck topos, then any morphism $f:X\to Y$ in $\mathcal H$ induces an essential geometric morphism

where $f_!$ is given by post-composition.

Proof. The right adjoint to $f_!$ exists for any category $\mathcal H$ with pullbacks (and is given by pulling back along $f$ , explaining why this is called “base change”). To prove that $f^*$ has a right adjoint as well, it suffices by the adjoint functor theorem (since Grothendieck topoi are locally presentable) to show that it is cocontinuous. For this, it suffices to see that colimits are pullback-stable in $\mathcal H$ . For this: pullback stability is apparent in the category of sets, and thus in any presheaf category. The result thus follows because Grothendieck topoi are exact reflective subcategories of presheaf categories. $\blacksquare$

Essential geometric morphisms (or more generally, any geometric morphism—that is, an adjoint pair $f^*\dashv f_*$ , where the direction of the morphism is given by the right adjoint, and the left adjoint preserves finite limits) are the sensible maps between topoi, hence the sensible generalisation of a continuous map.

Indeed, if $f:X\to Y$ is a continuous map between topological spaces, then we get a geometric morphism between the corresponding categories of sheaves, where the right adjoint $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ is the direct image defined by the formula $(f_*F)(U) := F(f^{-1}(U))$ for a sheaf $F$ on $X$ and an open set $U\subseteq Y$ . The left adjoint makes more sense if you look at the corresponding étale spaces, in which case it truly acts by pullback: the functor $f^*:\mathbf{Sh}(Y)\to\mathbf{Sh}(X)$ sends the sheaf of sections on a space $E/Y$ to the sheaf of sections on $E\times_YX/X$ .

Moreover, if $f:X\to Y$ is étale, then the inverse image functor $f^*$ admits a further left adjoint $f_!$ . Again, it’s easier to see how this behaves by looking at its action on étale spaces, in which case the left adjoint acts by post-composition just as in Proposition 1. In fact, this resulting essential geometric morphism $(f_!\dashv f^*\dashv f_*):\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ is just another instance of base change: identify $X$ with its corresponding sheaf of sections over $Y$ , then $(\mathbf{Sh}(Y)\downarrow X)\simeq\mathbf{Sh}(X)$ via the forgetful functor on the corresponding étale spaces. In fact, the inverse of this map is given by $f_!$ .

Therefore, it makes sense to extend this phenomenon to general Grothendieck topoi and define étale geometric morphisms to be those geometric morphisms $\mathcal K\to\mathcal H$ that factor via an equivalence to a base change $(\pi_!\dashv \pi^*\dashv\pi_*):(\mathcal H\downarrow X)\to\mathcal H$ . By the above discussion, this serves as a conservative extension of the classical setting.

Sections of étale geometric morphisms

How can we use this to make sense of local sections? As local sections are classically defined over open subsets of $X$ , local sections here have to be defined over objects of our site, so we need to upgrade the objects to “spaces” (that is, topoi). Since any object has an associated sheaf (namely by sheafifying its representable presheaf), the natural categorification is to take for any object $U\in\mathcal C$ the topos $(\mathbf{Sh}(\mathcal C)\downarrow h_U^\#)$ (moving forward I will just identify $U$ with its associated sheaf $h_U^\#$ ). Note that this is exactly what you get if you take an open subspace: open subspaces are particularly étale spaces.

Now suppose $\mathcal E\to\mathbf{Sh}(\mathcal C)$ is the étale geometric morphism associated to a sheaf $F$ (i.e., $\mathcal E\simeq(\mathbf{Sh}(\mathcal C)\downarrow F)$ ), then a local section over $U\in\mathcal C$ should be any essential geometric morphism $(\mathbf{Sh}(\mathcal C)\downarrow U)\to\mathcal E$ lying over $\mathbf{Sh}(\mathcal C)$ , entirely analogous to the picture for ordinary local sections:

local sections of $Y/X$ supported in $U\subseteq X$

local sections of $Y/X$ supported in $U\subseteq X$

Indeed, this works! By the Yoneda lemma, $F(U)\cong\mathrm{Nat}(h_U,F)=\mathrm{Nat}(h_U^\#,F)$ , so elements $\sigma\in F(U)$ induce an essential geometric morphism $(\mathbf{Sh}(\mathcal C)\downarrow U)\to\mathcal E$ via base change along $\sigma:h_U^\#\Rightarrow F$ , and this lies over $\mathbf{Sh}(\mathcal C)$ . Conversely, given an essential geometric morphism $\varsigma:(\mathbf{Sh}(\mathcal C)\downarrow U)\to\mathcal E$ that lies over $\mathbf{Sh}(\mathcal C)$ (in the sense that $\varsigma_!$ lies over $\mathbf{Sh}(\mathcal C)$ ), we can extract a canonical element $\varsigma_!(\mathrm{id}_U) : h_U^\#\Rightarrow F$ in $F(U)$ .

Thus, we see that it is entirely reasonable to interpret elements of a sheaf as local sections. However, this presentation does not sit well with me: the whole time, my mind was screaming “Grothendieck construction” into my head, and indeed, this whole phenomenon between sheaves and étale spaces sounds a lot like it.

Grothendieck, again

Recall (from a previous post) how to actually construct slice topoi: if I have a sheaf $F\in\mathbf{Sh}(\mathcal C)$ , then the slice topos is just $(\mathbf{Sh}(\mathcal C)\downarrow F)\simeq\mathbf{Sh}(\int F)$ , where the category of elements of $F$ carries an induced topology. Note that in the special case that $F=h_U^\#$ , we may also take the category $(\mathcal C\downarrow U)=\int h_U$ with its induced topology. Now, we have canonical morphisms of sites $(\mathcal C\downarrow U)\to\mathcal C$ and $\int F\to\mathcal C$ (which are also fibrations), and in fact any essential geometric morphism $(\mathbf{Sh}(\mathcal C)\downarrow U)\to\mathcal E$ lying over $\mathbf{Sh}(\mathcal C)$ corresponds to an essentially unique morphism of sites $(\mathcal C\downarrow U)\to\int F$ ^[cf] over $\mathcal C$ that moreover preserve the fibrational structure.

By applying the inverse of the Grothendieck construction (the Grothendieck deconstruction?), any such morphism of sites over $\mathcal C$ is the Grothendieck construction of an ordinary natural transformation $h_U\Rightarrow F$ , which as we know corresponds to a unique element of $F(U)$ . This is the content of the discussion about local sections done above. This justifies the alternative point of view that the answer to my original question can be answered simply with the Grothendieck construction: the “étale space” of a sheaf $F\in\mathbf{Sh}(\mathcal C)$ can be taken to be $\int F$ . Then, local sections $\sigma\in F(U)$ correspond simply by the Yoneda lemma to morphisms $(\mathcal C\downarrow U)\to\int F$ of fibrations over $\mathcal C$ .

Note that this is not quite consistent with the original story, however. Let $X$ be a topological space, and $F$ is a sheaf over $X$ . Although it is certainly the case that $\mathbf{Op}(|F|)\to\mathbf{Op}(X)$ is a fibration (local homeomorphisms are open, and pullbacks are given by taking preimage), this does not coincide with the category of elements $\int F$ . I believe, however, that we have a canonical inclusion $\int F\hookrightarrow\mathbf{Op}(|F|)$ given by sending a local section $(U,\sigma)\mapsto\sigma(U)$ (the image is open because it is a section of a local homeomorphism). The failure of surjectivity can be loosely chalked up to the fact that general fibrations are far more flexible than categories of opens: in the latter situation, open sets need to be closed under arbitrary unions. If we cocompleted $\int F$ in an appropriate (non-free) way, we would likely recover $|F|$ .

Example 2. I’m drawing my intuition from probably one of the simplest examples of a non-injective local homeomorphism: the canonical projection $\mathbb{R}\twoheadrightarrow\mathbb{R}/\mathbb{Z}\cong\mathbb{S}^1$ . If $F$ is the sheaf of local sections, then $(0,2)\subset\mathbb{R}$ does not lie in the essential image of $\int F\hookrightarrow\mathbf{Op}(\mathbb{R})$ since there are no local sections that can map onto this interval.

(My disappointment is immeasurable, and my day is ruined.) Basically, $\int F$ only remembers the open subsets of $|F|$ capable of admitting a surjective local section.

Circling back with circular reasoning

Let’s return to the more consistent generalisation of taking slice topoi. This almost gives the whole story; all that’s left is an analogue of the adjunction $\mathbf{PSh}(\mathbf{Op}(X))\leftrightarrows(\mathbf{Top}\downarrow X)$ that restricts to the equivalence between sheaves and étale spaces. Here’s my attempt at making this work.

Let $\mathcal C$ be a site, and let $(\mathbf{ShTopos}\downarrow\mathbf{Sh}(\mathcal C))$ be the 2-category of Grothendieck toposes lying over $\mathbf{Sh}(\mathcal C)$ via some essential geometric morphism, where the morphisms are essential geometric morphisms. Then, we have a (2-)functor $\mathcal C\to(\mathbf{ShTopos}\downarrow\mathbf{Sh}(\mathcal C))$ given by sending $U\mapsto(\mathbf{Sh}(\mathcal C)\downarrow U)$ .

Remarkably, the right hand 2-category is 2-cocomplete, so we still get a cocontinuous extension $\mathbf{PSh}(\mathcal C)\to(\mathbf{ShTopos}\downarrow\mathbf{Sh}(\mathcal C))$ (since $\mathcal C$ is a category, its free 2-cocompletion is the same as its free cocompletion). Note that since sheafification is a left adjoint (and is thus cocontinuous), this extension is defined in the expected way: it sends a presheaf $P$ to the topos $(\mathbf{Sh}(\mathcal C)\downarrow P^\#)$ .

This extension is also a left 2-adjoint, just as with ordinary nerve-realisation adjunctions, with right adjoint given by the functor $\Gamma:(\mathbf{ShTopos}\downarrow\mathbf{Sh}(\mathcal C))\to\mathbf{PSh}(\mathcal C)$ sending a topos $\mathcal E$ to the “sheaf of sections” defined by $\Gamma(\mathcal E)(U) = \mathrm{Hom}_{/\mathbf{Sh}(\mathcal C)}((\mathbf{Sh}(\mathcal C)\downarrow U),\mathcal E)$ . Indeed, adjointness follows by the Yoneda lemma for representable presheaves—since $\mathrm{Hom}((\mathbf{Sh}(\mathcal C)\downarrow U,\mathcal E) \stackrel{\mathrm{def}}= \Gamma(\mathcal E)(U) \simeq \mathrm{Hom}(h_U,\Gamma(\mathcal E))$ —then extend cocontinuously.

The adjunction unit on a presheaf $P\in\mathbf{PSh}(\mathcal C)$ is given by

$\displaystyle P(U)\to\Gamma(\mathbf{Sh}(\mathcal C)\downarrow P^\#)(U) = \mathrm{Hom}((\mathbf{Sh}(\mathcal C)\downarrow U),(\mathbf{Sh}(\mathcal C)\downarrow P^\#)) \cong P^\#(U)$

(where the final equivalence has been discussed already). Since we have already established that sheaves correspond to étale geometric morphisms over $\mathbf{Sh}(\mathcal C)$ , this is enough to show that the 2-adjunction

$\mathbf{PSh}(\mathcal C)\leftrightarrows(\mathbf{ShTopos}\downarrow\mathbf{Sh}(\mathcal C))$

restricts to the equivalence of 2-categories between sheaves on $\mathcal C$ and étale topoi over $\mathbf{Sh}(\mathcal C)$ . Moreover, we have just computed for an arbitrary presheaf that taking the sheaf of local sections of the associated étale topos coincides precisely with sheafification, as in the classical picture.

Therefore, we’ve finally returned to the adjunction that begun it all. The only catch is that you already need to understand sheaves before you can even formulate this adjunction. While this would have probably disappointed me when I just learned about sheaves on spaces and their étale spaces, this is an unsurprising conclusion: we needed some way of describing the topology on the site in the adjunction.