## February 5, 2010

### Sheaves Do Not Belong to Algebraic Geometry

#### Posted by Tom Leinster

…and here’s a proof.

They are, of course, very useful in algebraic geometry (as is the equals sign). Also, human beings discovered them while developing algebraic geometry, which is why many of them still make the association.

But as we’ll see, sheaves are an inevitable consequence of general ideas that have nothing to do with algebraic geometry. In fact, sheaves (and various related notions) arise automatically from two completely general categorical constructions, together with one almost imperceptibly small topological observation.

Before I give you the proof, let me make clear that it isn’t due to me. I don’t know who it is due to — I’ve never seen it in print — but I suspect it was known before I was even born. (Update: see Joachim Kock’s comment for a reference.) People who I’ve told this argument to seem to like it, so I wrote it up in a little note a few years ago; then a recent conversation reminded me of it, so I thought I’d air it here.

First categorical construction  Let $\mathbf{A}$ be a small category, $\mathbf{E}$ a category with small colimits, and $J: \mathbf{A} \to \mathbf{E}$ any functor. Then there is an induced adjunction \mathbf{Set}^{\mathbf{A}^{op}} \begin{aligned} \stackrel{\displaystyle\stackrel{\displaystyle - \otimes J}{\longrightarrow}}{\stackrel{\longleftarrow}{Hom(J, -)}} \end{aligned} \mathbf{E}. The right adjoint $Hom(J, -)$ is defined by $(Hom(J, E))(A) = Hom (J(A), E)$ ($E \in \mathbf{E}$, $A \in \mathbf{A}$). The left adjoint $- \otimes J$ is defined by the adjointness, and can be described as a certain coend or colimit.

Example: if $J: \Delta \to \mathbf{Top}$ is the standard simplex functor then $Hom(J, -)$ is the singular simplicial set functor and $- \otimes J$ is geometric realization.

Second categorical construction  Any adjunction restricts canonically to an equivalence between full subcategories.

Precisely, let \mathbf{C} \begin{aligned} \stackrel{\displaystyle \stackrel{F}{\displaystyle\longrightarrow}}{ \stackrel{\longleftarrow}{G}} \end{aligned} \mathbf{D} be an adjunction ($F$ left adjoint to $G$), with unit $\eta: 1 \to G F$ and counit $\varepsilon: F G \to 1$. Let $\bar{\mathbf{C}}$ be the full subcategory of $\mathbf{C}$ consisting of those objects $C$ for which $\eta_C: C \to G F(C)$ is an isomorphism, and dually $\bar{\mathbf{D}}$. Then the adjunction $(F, G, \eta, \varepsilon)$ restricts to an equivalence between $\bar{\mathbf{C}}$ and $\bar{\mathbf{D}}$.

Almost imperceptibly small topological observation  Any open subset of a topological space can be regarded as a space in its own right, and when one open set is contained in another, there is an induced inclusion of spaces.

Precisely, let $S$ be a topological space. Write $\mathbf{O}(S)$ for the poset of open subsets of $S$, regarded as a category (in which each hom-set has at most one element). Write $\mathbf{Top}/S$ for the category of spaces over $S$: objects are continuous maps into $S$, and maps are commutative triangles. Then there is a canonical functor $J: \mathbf{O}(S) \to \mathbf{Top}/S,$ sending an open set $U$ to the inclusion $U \hookrightarrow S$.

Punchline  Fix a topological space $S$. The category $\mathbf{Top}/S$ has small colimits, since $\mathbf{Top}$ does.

Applying the first categorical construction to the functor $J$ just defined produces an adjunction (presheaves on S) = \mathbf{Set}^{\mathbf{O}(S)^{op}} \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Top}/S = (spaces over S). The two functors here are the ones you’d guess.

Applying the second construction now gives an equivalence of categories (sheaves on S) = \mathbf{Sh}(S) \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Et}(S) = (étale spaces over S). This can be interpreted as the definition of sheaf, étale space, etc., or as a theorem, according to taste.

Going right and then left in the adjunction gives the associated sheaf, or sheafification, of a presheaf. Going left and then right gives the ‘étalification’ of a space over $S$.

Posted at February 5, 2010 9:50 PM UTC

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### Re: Sheaves Do Not Belong to Algebraic Geometry

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.)

Posted by: Toby Bartels on February 6, 2010 12:26 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

I wrote earlier:

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

In contrast, I did think that stacks originally belonged to algebraic geometry. Of course, these are categorified sheaves, hence really more general. But I did think that the algebraic geometers developed them first and furthest.

I write ‘did’ only because now I expect somebody to tell me otherwise. But perhaps I was right.

Posted by: Toby Bartels on February 7, 2010 8:53 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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.

Posted by: Joachim Kock on February 6, 2010 12:59 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Joachim. You’re right: M&M’s account of this is pretty close to what I wrote, much closer than I remembered. Pages 90-91 of my dog-eared copy have my pencilled annotations on them, so I clearly did read that part.

As for what algebraic geometry is, well, that’s a subjective matter. (But your comment does remind me a little of something that Jim Stasheff just said.) I guess I was taught that algebraic geometry was, at its heart, the study of solving polynomials. On the other hand, I never quite believed my teachers — I couldn’t believe that all that cool Grothendieck stuff (which I knew zero about at the time) could possibly be about something so unglamorous.

Posted by: Tom Leinster on February 6, 2010 1:20 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Much of “that cool Grothendieck stuff” was specifically about counting solutions of finitely many polynomial equations over a finite field when they define a smooth projective variety (glamour is clearly in the eyes of the beholder). It always seems to me to be a dreadful misunderstanding to imagine that Grothendieck was interested in building towers of abstractions just for the sake of it. My own vague impression from what I’ve read of “Récoltes et semailles” and other writings of his (e.g., the Grothendieck-Serre letters) is that he would agree with my remark, taking into account that a genius like him doesn’t have the same definition of abstract and concrete as common mortals… (For instance, it is striking how so many of the fundamental properties of étale cohomology turn out to hinge, in final analysis, on seemingly boring old facts of algebraic number theory.)

As for the history of sheaves, it might also be interesting to point out how different Leray’s original definition was: it was based on closed subsets instead of open sets (see Chern’s review of Leray’s main paper). This was re-interpreted and redefined in the late 40’s, in particular during the Cartan seminars. Cartan himself used sheaves to prove “Theorems A and B” in complex analytic geometry, but also used them (before) to re-interpret and extend a result of Oka from the same period (the coherence of the relation sheaf between finitely many holomorphic functions). Interestingly, Oka had used a rough definition of a special case of sheaves (corresponding more or less to ideal subsheaves of the structure sheaf of a complex analytic variety), which he called “idéal holomorphe de domaines indéterminés”. Before these works, very little was known concerning complex analysis (or geometry) in several variables – although there are similarities with algebraic geometry, many facts are much harder (e.g., the coherence of the ideal sheaf of a subvariety is a tautology in algebraic geometry, but is quite hard in the complex setting – this was one of Cartan’s first results beyond Oka’s).

The earliest reference I found in Numdam are here (from a lecture of April 9, 1951, where Cartan states that this is a new take on the 1948/49 definition of sheaf, which I can’t find; Cartan works in the context of a general regular topological space in this work) and here (which contains the results about coherence mentioned above).

As already mentioned by others, sheaves and (Cech, if I remember right) sheaf cohomology were present in Serre’s “Faisceaux algébriques cohérents”, which is a few years pre-schemes; so they were absolutely already at the heart of algebraic and analytic geometry when Grothendieck started working on these topics.

Posted by: Emmanuel Kowalski on February 7, 2010 8:28 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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.)

Posted by: Tim Porter on February 6, 2010 7:46 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks, Tim. My history is lousy. I remembered that it was Leray who came up with sheaves, but I didn’t know or remember that it was in the context of PDEs.

Nevertheless, I’m convinced that most mathematicians (or at least, those who’ve heard of sheaves) do associate them with algebraic geometry, and that a significant number think that, in some sense, they “belong” to alg geom.

Would you say that it was algebraic geometry that made sheaves famous?

Posted by: Tom Leinster on February 7, 2010 3:37 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

The context is confused. I am only using a bit of knowledge other than web sources, such as the Wikipedia article and the St. Andrews’ History one.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Leray.html

He seems to have avoided PDEs as he was in a POW camp and feared that he might be forced to do engineering or similar work for the German authorities. He used his expertise turning towards topology. I note the important quote

… algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps.

(This is interesting as another precursor of a categorical viewpoint.)

Was his own motivation for introducing sheaves and spectral sequences topological or analytic? It is not clear.

Sheaves were used in Spanier’s Algebraic Topology book and Hirzebruck’s work, in the 1950s and 1960s, but it is almost certainly EGA and the SGAs and hence Grothendieck, who pushed them to the forefront in Algebraic Geometry, and his prestige has been transfered to them there.

I am not a great one for subdividing mathematics into somewhat arbitrary areas, and tend to emphasise links rather than differences. That is why I think category theory is at the same time very central and often politically marginalised, which route I will turn from as it goes away from the thread! The subdivisions are largely arbitrary.

I have a somewhat related point. Quillen was working in deformation theory in geometry when he started the work on Model Categories, so what area do they fit into? Nowadays people would probably say it was Algebraic Topology, or homotopy theory. The question may be a non-question in some sense.

Posted by: Tim Porter on February 7, 2010 7:36 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Tim Porter makes some nice observations and some remarks I find myself nodding along to, but unable to add much coherent to. I thought I’d just point out, for those who are interested in some of the history behind Leray’s work during and just after the war, that Haynes Miller has written an interesting account of this:

Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences

There’s a copy which can be found on his website.

By the way: I was under the impression, albeit based on little except some browsing in libraries while looking for other things, that there was a period when analysts working in several complex variables took sheaf cohomology (Theorems A, B and all that jazz) pretty seriously and used it.

While this reminds me: has anyone here had cause to look at Finnur Larusson’s recent work advocating a model-categoric/simplicial viewpoint on various parts of holomorphic geometry in higher dimensions? See e.g. this arXiv preprint and various of his sequels.

Posted by: Yemon Choi on February 7, 2010 9:12 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks for that reference to Haynes Miller’s article. I will have a look at it as that period has always interested me for its influence on the maths I grew up with as a postgrad.

I should have mentioned
The Theory of Sheaves. By RICHARD G. SWAN. Chicago lectures in mathematics. (University of Chicago Press), mid 1960s.

They were notes of some of his lectures. (Someone well known to me took the notes so I should not have forgotten them.)

I should also mention Grothenideick’s notes from Kansas. Can someone provide a full title (and link to a scanned copy???)

Posted by: Tim Porter on February 7, 2010 9:34 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Do you mean these lectures?

Posted by: Thomas on February 7, 2010 10:07 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Yes, exactly, that is useful. I have an original hard copy but do love clogging up my hard disc with electronic copies of things!

Posted by: Tim Porter on February 7, 2010 10:15 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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!

Posted by: James on February 6, 2010 9:31 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Ha! You haven’t forgiven me for this. I don’t think I’m pulling a fast one. Like I say, I believe this is something old. And like Joachim and Urs say, it’s in Mac Lane and Moerdijk.

Here you say:

I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

In a sense I agree. If you want to develop the notion of sheaf, starting from the beginning and arriving at the set of basic facts about sheaves that you’re used to, then this won’t save you any work. But I also like the GPS analogy that Urs gives in reply. (Really GPS stands for Gordon, Power and Street, but I also understand this new-fangled usage.) And whenever a definition has several different formulations, people will have subjective feelings about which is the best one.

Posted by: Tom Leinster on February 6, 2010 11:09 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

OK, good. I just wanted to make sure I wasn’t missing a royal road to sheaf theory!

Posted by: James on February 7, 2010 12:09 AM | Permalink | Reply to this

### The abstract definition of sheaves

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

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.

Posted by: Urs Schreiber on February 6, 2010 11:56 AM | Permalink | Reply to this

### Re: The abstract definition of sheaves

OK, but what I intended to ask was whether this formal approach allows you to bypass the hard work of constructing the sheafification functor in the usual approach. I would expect that you’ve just pushed the hard work to proving that this definition of sheaf agrees with the usual one, i.e. a presheaf which behaves in the usual way on covers.

Posted by: James on February 6, 2010 12:50 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

whether this formal approach allows you to bypass the hard work

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level. This helps you to understand what it is that you need to do in any given situation. Then actually doing it may be hard work, but at least you are doing it efficiently then.

Abstract theory is like a GPS sensor: there is a jungle that you have to cross by foot. It will be hard work to make yourself a way through there. But the general abstract GPS sensor that you carry with you will at least ensure that you go on the straight and direct way through. So as to prevent that on top of all the perils that lurk anyway, you’ll not also lose your way and get lost in the thicket.

constructing the sheafification functor in the usual approach

There is no usual and unusual approach here. All there is is the insight that not only does the sheafification functor turn out to be left exact and left adjoint to the inclusion functor, but that conversely it is entirely characterized by this property.

This is not a new insight, mind you. This is the way the standard textbook on abstract sheaf theory discusses it: Sheaves in geometry and logic.

Posted by: Urs Schreiber on February 6, 2010 1:02 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

This formal definition allows you to bypass any confusion as to what the notion of sheaf is on an abstract level.

I think that’s an overly broad statement. I feel that “the notion of sheaf” does not have a unique meaning, and we shouldn’t assert that any one way of thinking about it is the One True Way. This formal definition is one very nice way of thinking about it, which is quick and convenient and gives some good intuition. But the traditional definitions are valid and useful as well, and there are also other valid and useful intuitions.

Posted by: Mike Shulman on February 6, 2010 11:26 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

This formal definition is one very nice way of thinking about it, which is quick and convenient and gives some good intuition. But the traditional definitions are valid and useful as well

Sure, because these traditional definitions are a consequence of this formal definition.

Posted by: Urs Schreiber on February 7, 2010 11:01 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

As I said above, I just wanted to be sure that this point of view doesn’t really cut down on the work you have to do.

That said, I do like this point of view a lot. Somehow I had never appreciated it before. To me, though, it’s particularly worth pointing out that the list of things you need (presheaf, adjoint functor, fully faithful functor, left exact functor) does not include anything like a Grothendieck topology.

Is it true that the point is not so much “sheaves do not belong to algebraic geometry” as much as “sheaves can be defined without sites”?

Posted by: James on February 7, 2010 12:18 AM | Permalink | Reply to this

### Re: The abstract definition of sheaves

You don’t need to restrict to sheaves on topological spaces.

What you wrote is of course correct and interesting, but it doesn’t look to me like a generalization of what Tom wrote. I’m not sure if you were saying that it is, but for sheaves on a general site, there is no analogue of the category $\mathbf{Top}/S$, as far as I know.

Posted by: Mike Shulman on February 6, 2010 11:28 PM | Permalink | Reply to this

### Re: The abstract definition of sheaves

but it doesn’t look to me like a generalization of what Tom wrote. I’m not sure if you were saying that it is,

I read Tom’s post as saying: even though sheaves appeared in algebraic geometry, they can be understood as just being structures in plain category theory.

To which I meant to remark: sure, and that’s called topos theory. Where categories of sheaves are precisely characterized as the geometric subtoposes of presheaf toposes.

Posted by: Urs Schreiber on February 7, 2010 11:05 PM | Permalink | Reply to this

### Where sheaves belong to

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.

Posted by: Urs Schreiber on February 6, 2010 12:09 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic 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.

Posted by: Harry Gindi on February 7, 2010 6:04 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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.

Posted by: Zoran Skoda on February 7, 2010 7:57 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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.

I would posit that for n=2 the answer is: 2-covering spaces, as defined in my thesis (see chapter 4). The thesis you refer to, by Richard Lewis, takes a more algebro-combinatorial approach, with crossed modules acting on simplicial sets.

Posted by: David Roberts on February 8, 2010 12:04 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Thanks Zoran and David.

Yes. I felt from way back when I started working on 2-stack type things (actually before we at Bangor got the start of the Grothendieck correspondence, (in the sense of letters not a bijection!!!!), but after, by several years, AG’s letter to Breen (note not the later letter to Quillen)), that the 2-type of a space classified its 2-covering spaces, that that was very important, and that if I was to attempt to convince analysts, geometers and others that this was important, there had to be fairly concrete models of the covering spaces. The solution using abstract sheaf theoretic models should be doable (and probably is essentially there in places that I should have read!!!) such as David’s thesis, which I have only browsed, but what about Galois theory, Riemann surface theory, ramifications, complex analytic continuation, etc. How were generalisations of those to be found and then sold’ to the non-categorical mathematical audience? That needed something that I could not visualise. Something more concrete than just an abstract construction in a Quillen model category, however pretty, is needed. I had seen algebraic topologists in seminars ‘turn off’ if you mentioned anything non-stable, yet what I wanted was very basic, some sort of elementary approach to those ideas.

I felt those questions were important then and still do now. Some of John’s ideas and a lot of the stuff here in the café and in the Lab go a long way towards answering them.

Grothendieck’s letters to Ronnie, and their transformation into Pursuing Stacks made me very pleased!!! But then I had to start learning all the bits of theory, that would be needed to do put his ideas into practice. I am still learning!

Posted by: Tim Porter on February 8, 2010 7:02 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Hi Zoran,

you write:

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).

I would think this is the topic of section 7.1 in HTT, leading up to theorem 7.1.4.4 on page 555:

For $X$ a topological space, sufficiently nice, there is an adjoint equivalence of $(\infty,1)$-categories

$Top/X \stackrel{\leftarrow}{\to} Sh_{(\infty,1)}(X)$

where left-to-right is taking sections and right-to-left is the analog of forming étale spaces.

Indeed, the +-construction for $\infty$-stacks on $X$ is the immediate generalization of the 1-categorical familiar “form étale space and then take sections”. This is theorem 7.1.4.3 on p. 554.

Posted by: Urs Schreiber on February 8, 2010 7:10 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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 $\gt 1$.)

Posted by: Mike Shulman on February 8, 2010 7:30 PM | Permalink | PGP Sig | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Yes — and it’s important that it’s idempotent. The question is, is there a really slick way to make idempotency pop out of this derivation of the adjunction?

Posted by: Tom Leinster on February 8, 2010 11:59 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

Good question. I don’t immediately see any such slick argument.

Posted by: Mike Shulman on February 9, 2010 5:25 PM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

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.

Posted by: hypnocat on February 19, 2015 2:23 AM | Permalink | Reply to this

### Re: Sheaves Do Not Belong to Algebraic Geometry

I’m afraid I’ve only just sat down to work through this.

I’d love to see a simple criterion for when the adjunction $\widehat{\mathbb{C}} \stackrel{\leftarrow}{\rightarrow} \mathbb{E}$ induced by a functor $\mathbb{C} \to \mathbb{E}$ is idempotent, but I don’t think this lemma can be right.

Here’s what appears to be a counterexample. First recall that an adjunction is idempotent if and only if the comonad it induces is idempotent (that is, the multiplication part of the comonad is an isomorphism).

Now consider the inclusion $FinSet \to Set$ of the full subcategory of finite sets, and form its opposite $FinSet^{op} \to Set^{op}$, which is certainly full (and faithful). The adjunction induced by this latter functor is an adjunction between $Set^{op}$ and $Set^{FinSet}$. It induces a comonad on $Set^{op}$, which of course can be thought of as a monad on $Set$.

As originally shown by Kennison and Gildenhuys, and recounted more recently in Section 2 here, that monad is the ultrafilter monad.

However, the ultrafilter monad $U$ is not idempotent (at least, assuming that $Set$ satisfies the axiom of choice). For if $U$ is idempotent then for each set $X$, the multiplication map $U U X \to U X$ is a bijection, or equivalently the unit map $\eta_{U X} : U X \to U U X$ is a bijection, and therefore every ultrafilter on $U X$ is principal. But if we take $X$ to be infinite, then $U X$ is also infinite, so (by choice) $U X$ admits a non-principal ultrafilter — a contradiction.

What’s wrong with hypnocat’s proof? Well, it seems to me that the codomain of $(\eta_{G D})_C$ (shown in the displayed equation) is wrong. (Here $C \in \mathbb{C}$ and $D \in \mathbb{E}$.) I think that display should read

$(\eta_{G D})_C \colon Hom_{\mathbb{E}}(z C, D) \to Hom_{\mathbb{E}}(z C, F G D).$

If $F$ were right adjoint to $G$ then the codomain could be rewritten as $Hom_{\widehat{\mathbb{C}}}(G z C, G D)$, which is (almost) what hypnocat wrote. But it’s left adjoint.

Posted by: Tom Leinster on August 31, 2015 7:48 PM | Permalink | Reply to this

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