The n-Category Café

Beautiful!

Small grammar note: ‘étale map’, but ‘étalé space’.

It is probably a sign of something that I never thought that sheaves inherently belonged to algebraic geometry. (Unlike schemes.)

Hi Tom,

To find these constructions together in print, one possibility
is Mac Lane / Moerdijk, Sheaves in geometry and logic, Ch. 2
(p.90–91), with reference to Chapter 1 for the first general
categorical construction.

But you put it much more forcefully! It is beautiful.

I am not sure I agree with the conclusion regarding algebraic
geometry, though :-) If a conclusion is to be drawn, I would
rather put something like “Algebraic Geometry is more than
what you think!”

Cheers,
Joachim.

Sheaves were developed initially by Leray (although they seem to be traced by Wikipedia back further). He was interested in PDEs rather than anything algebraico-geometric. That was pre-1945. Cartan and Weil used them for a proof of de Rham’s theorem, 1947, then Serre introduced them into Algebraic Geometry only in 1954. The sheaf idea is an extension of analytic continuation. Perhaps therefore sheaf theory is a daughter of analysis, which was mistakenly disowned by its parents and taken under the wing of alg. geom.

Hopefully we can give sheaves back to PDEs and diff. geom. with all current work on approaches to smooth spaces etc. Convincing the analysts that they wrongly disinherited their daughter will be hard though!

On another point, no subject `owns’ sheaf theory, and Joachim’s subversive work for the expansionist Empire of Algebraic Geometry may be a bit like the claims of one world power, A, to dominate a small country, B, essentially because one of A’s historic emperors came from a third small country, C, that conquered B. (I am reading a book on Kublai Khan at the moment if you want to unravel those slightly cryptic comments.)

Hold on a sec. Have you pulled a fast one on us again? Your definition of sheaf is not the usual one. Are they obviously equivalent? If so, can you do this for any Grothendieck topology? I had always thought that abstract sheaf theory had exactly one theorem—that the sheafification functor exists. I’d be impressed if you could get around that!

It’s more general abstract than that. You don’t need to restrict to sheaves on topological spaces.

In full generality:

For $C$ any small category, Grothendieck topologies and categories of sheaves $Sh(C)$ on $C$ are equivalent to fully faithful functors

$$Sh(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh(C)$$

into the category of presheaves $PSh(C) = Func(C^{op},Set)$, that have a left exact left adjoint (a left adjoint that preserves finite limits).

Such a morphism is also called

• a geometric embedding

• a left exact localization

• a reflective subcategory with left exact reflector.

This definition of sheaves is the general abstract one, that allows to define the notion of sheaves in higher category theory, or whenever the notions

• presheaf, adjoint functor, fully faithful functor, left exact functor

are available.

Notably $(\infty,1)$-sheaves (aka $\infty$-stacks) are defined this way, as left exact localizations of presheaf $(\infty,1)$-toposes

$$Sh_{(\infty,1)}(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.$$

An exposition of this central fact of higher topos theory is at models for $\infty$-stack $(\infty,1)$-toposes.

The most clear-sighted account of what geometry is in full generality and how the notion interacts with the notion of sheaves in full generality is described at notions of space.

As you choose different sites and pregeometries (as described there) you get different notions of geometry.

Sheaf theory (higher sheaf theory [higher stack theory]) is perhaps one of my favorite subjects. It is a beautiful theory with connections to every part of mathematics.

I agree with Mike that the statement quoted by Urs is not a generalization of the statement Tom phrased in the introduction. Tom looks at the adjunction between presheaves on X and all spaces over X. Now this adjunction RESTRICTS to an adjunction between a subcategory which can be called sheaves at either side, but Urs’s statement chooses topologically less intuitive side of the “sheaves as presheaves”. By this stage both the info on bigger category Top/X and the geometric look of its subcategory espaces étalé over X is lost. Now Urs explains the generalization of this void-of-usual-spaces-setup corollary (this does not matter at some different levels of questions, but in elementary intuitive foundations it does matter). Now there is a warm up question for Urs: what is REAL higher categorical generalization of étalé spaces (no equivalent categories, description which is analoguous to them, I mean a sort of categorified generalized covering spaces). This is not that simple task. Tim Porter will remember recent thesis at Bangor treating n=2 as one of the issues.

Now, on history. I am so much surprised with the attitude that “sheaves are algebraic geometry”. I was quite knowledgeable about sheaves as undergraduate and studied related algebaric topology a lot then and did not have ANY idea at the time what an algebraic variety is. In our topology seminar which I attended then we used old definitions from 1956 Godement’s excellent book for whom the sheaf is what is here called étalé space and in the Russian edition its total space is called nakryvajushchee prostranstvo, that is covering space; the presheaf point of view is compared of course there in detail (this equivalence of categories proved but not more general adjunction above). Leray, Cartan etc. at the time were interested in analytic continuation like Oka’s problem, Cousin’s problem in cohomology etc. This resulted in EARLY 1950s, that is before Spanier and similar resources irrelevant for early history to lots of results in analytic geometry, Grauert and Remmert taking place on German side and Henri Cartan on French, and of course many others; Cartan’s theorems A and B being some of the famous results. Serre was in the same milleu and transferred in his famous FAC article some of the Cartan seminaire methods to algebraic geometry. Grothendieck was doing tensor products of topological vector spaces at the time. Then he went to Kansas state to do some topology, and there were the breakthrough in sheaf theory, NOT in algebraic geometry. I mean his Kansas report (there are two, the booklet on sheaf theory, and also the official report to the funding agency, where he claims his failure to achieve the goals and then goes on with big discoveries), and his seminar in Spring 1955 where he obtained most of the Tohoku results which were published only 2 years later. Tohoku is still written with main view toward the cohomology of sheaves in topological and equivariant setup. Later AG claimed that the intuition on abelian categories was to isolate the properties he thought of sheaf categories, and that topos theory later was more or less just a later variant of the same initial idea. I mean the characterization of Grothendieck topoi like in Giraud’s theorem is according to him, one of the principal reasons to find the axiomatics of the abelian categories as he phrased it in Tohoku.

In complex analytic geometry according to an intro to a book of Walter Rudin (the one on the complex functions in the unit ball) the sheaves went into second plane only with the discovery of new, quite different, methods of 1970s, which were more effective to then actual questions.

John Baez has recently anounced at the cafe some very recent discovery of Freyd (?) on comparison between abelian and topos case finding a new formulation with closer similarity. I would like to hear a more precise statement and update on this exciting story.

Here’s another interesting aspect of this. It’s certainly true that any adjunction restricts to an equivalence between the full subcategories consisting of the objects for which the unit or counit (as appropriate) is an isomorphism. However, for many adjunctions, these full subcategories can be very small. For instance, for the free-forgetful adjunction of any algebraic theory (monoids, groups, rings, etc.), these full subcategories are empty.

But the adjunction in question here has a special property: it’s idempotent. That means that the unit $\eta_C \colon C\to G F (C)$ is an isomorphism whenever $C$ is of the form $G(D)$, and (equivalently!) that the counit $\varepsilon_D\colon F D(D) \to D$ is an isomorphism whenever $D$ is of the form $F(C)$.

In general, you could think of starting with an object $C$ and trying to get it into one of the full subcategories that are equivalent by successively applying the functors: $C, F(C), G F(C), F G F(C), \dots$. In general, you’ll never get there, while the adjunction being idempotent means that you get there after only one step. (Off the top of my head, I don’t know any examples where you get there in some finite number of steps $\gt 1$.)

Hi I know its an awfully long time since this was posted but i just read it and couldn’t help thinking about this. The question being how does one show “slickly” that the adjunction derived in the first place is idempotent? For this i propose the following Lemma:

Proposed Lemma: Let $\mathbb{C}$ be small, $\mathbb{E}$ co-complete, $z \;:\; \mathbb{C} \rightarrow \mathbb{E}$, and $F \;:\; \hat{\mathbb{C}} \rightharpoonup \mathbb{E} \;:\; G$ the standard adjunction arising from this situation; Then it is idempotent provided that $z$ is full.

If this lemma is correct (i think the proof is below), then the situation with sheaves is a corollary since in that case $z : \mathcal{O}(X) \rightarrow \text{Top}/X$ is clearly full since the only arrows of the form $z(U) \rightarrow z(V)$ in $\text{Top}/X$ must be inclusions.

Here is a proposed proof: Let $\eta$ be the unit of the adjunction. $\eta G$ is trivially a split monic from the triangle identities, so we are only concerned with showing it is epi, for which it suffices to show that each of the embedded components, of the following form, is a surjection of sets:

$\qquad \qquad (\eta_{G D})_C \;:\; \text{Hom}_{\mathbb{C}}(z C,D) \rightarrow \text{Hom}_{\mathbb{C}}(G z C,G D)$

… which function acts by mapping the arrows through $G$, so an arrow $h \;:\; z C \rightarrow D$ on the left becomes $h_{*}$ on the right.

Conversely, consider any arrow $\varphi \;:\; G z C \rightarrow G D$ on the right. Written out in full, this is a natural transformation $\varphi \;:\; \text{Hom}_{\mathbb{E}}(z -,z C) \rightarrow \text{Hom}_{\mathbb{E}}(z -,D)$. Draw a naturality square for this with respect to an arbitrary arrow $g \;:\; C' \rightarrow C$ and chase the unit $\text{id}_{z C}$ around the diagram to learn that $\varphi_{C'} \;:\; (z g) \mapsto \varphi_{C}(\text{id}_C) \cdot (z g)$. Provided that $z$ is full, this allows us to conclude that $\varphi$ is of the form $h_*$ where $h = \varphi_C(\text{id}_C)$ in all cases, i.e. that $(\eta_{G D})_C$ is surjective as required, and the adjunction is idempotent.

This is a really cool post! I was recently reading about nerves and realization, and I realized that in fact this construction described in your blog is a special case this: the right adjoint here is the nerve, and the left adjoint is the realization. So taking the sheaf of local sections of an etale space is just taking the nerve, and constructing the etale space for a sheaf is geometric realization!

https://ncatlab.org/nlab/show/nerve+and+realization

Let $F$ be the functor from presheaves to spaces over $X$, and let $G$ be the functor from spaces over $X$ to presheaves. It is interesting to me how $G$ is right adjoint to $F$, but also $G \circ F$ makes a functor from presheaves to sheaves (sheafification) which is itself left adjoint. I wonder if this can be derived from your formalism here.

So is there an analogue of this story for general sheaves on a site? In the comments above Mike Shulman says there is no analogue of Top/S. In the typeset 2007 note by Tom Leinster, he leaves that as a challenge at the bottom.

I think maybe the adjunction will be instead between presheaves on site C and the category of toposes over Sh(C).

But if that’s the case this adjunction will be of no use to actually define sheaves on C.