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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 30th 2016
• (edited Nov 30th 2016)

The scan of the writeup of Grothendieck’s 73 Buffalo lecture that we point to at functorial geometry is really badly done. Is there a better scan or any other re-typing available?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2016

This was brought up before, and I think one of the suggestions was to ask at the Categories mailing list. I agree it’s horrendous (don’t people care enough not to make such a mess of things?).

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeNov 30th 2016

This one here seems better.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeNov 30th 2016

In fact, it’s obviously better, so I’ve replaced the link.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 30th 2016

Thanks!

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2016

Yes, thanks! That’s a relief.

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeDec 1st 2016

Zen Lin Low’s thesis is largely about the abstract general notions that are in Grothendieck’s lectures, with the benefit of a couple of decades hindsight.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeDec 1st 2016

Not sure what you mean to imply, but Zhen Lin’s thesis is to linked from the entry.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeDec 1st 2016

Nothing deep. I should have checked the entry before commenting :-/

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 30th 2017
• (edited Oct 3rd 2017)

Going back to the entry functorial geometry it struck me that it didn’t really indicate that there is plenty of functorial geometry on the $n$Lab. So I went and expanded the line that used to read

Of course, the above discussion generalizes to other types of geometry and even higher geometry.

to

Of course, the above discussion generalizes to other types of geometry and even higher geometry, the general perspective being known as synthetic differential geometry or similar. For discussion of functorial (higher) differential geometry see for instance at smooth set (smooth ∞-groupoid), for discussion of functorial supergeometry see at super formal smooth set.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeOct 3rd 2017

10: wrong link, functorial geometry, not geomertry.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 3rd 2017

Thanks. Fixed now.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJun 11th 2018
• (edited Jun 11th 2018)

In his famous 1973 Buffalo Colloquium talk, Alexander Grothendieck urged that his earlier definition of scheme via locally ringed spaces should be abandoned in favour of the functorial point of view.

I would like to augment this remark by a concrete pointer to page and verse where this is said.

I have scanned the pdf scan, but haven’t found it yet. If anyone has the page number, or the energy to find it, I’d be really grateful.

• CommentRowNumber14.
• CommentAuthorThomas Holder
• CommentTimeJun 11th 2018
• (edited Jun 11th 2018)

This remark is apparently based on the message by Lawvere that is quoted in the reference list:

The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme (which had required as a prerequisite the baggage of prime ideals and the spectral space, sheaves of local rings, coverings and patchings, etc.), should be abandoned AS the FUNDAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the “baggage” could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition.

Thus his definition is essentially well-known, and indeed is mentioned in such texts as Demazure-Gabriel, Waterhouse, and Eisenbud; but it is not carried through to the end, resulting in more complication, rather than less. I myself had learned the functorial point of view from Gabriel in 1966 at the Strasbourg-Heidelberg-Oberwolfach seminar and therefore I was particularly gratified when I heard Grothendieck so emphatically urging that it should replace the one previously expounded by Dieudonne’ and himself.

There is hardly a smoking gun page in the lecture notes which are anyway rather a text by Gaeta who actually intends to bridge the gap between traditional algebraic geometry and the scheme language and at various points emphasizes the (“natural”) geometric interpretation as locally ringed spaces. Lawvere’s remark might then be based on another talk of Grothendieck in Buffalo, a private conversation or simply be a distorsion in memory of the effect Grothendieck’s lectures had on Lawvere himself. You might say though that Grothendieck’s presentation as reflected in Gaeta’s text implicitly favors the functorial view but there seems to be hardly an “emphatic urge” to abandon the geometric view which would have conflicted with Gaeta’s own perspective on the material.

I myself had the same problem of pinning down in Gaeta’s text a remark of Lawvere concerning the role of extensive sites there that is reported at Gaeta topos. Note that Lawvere published in 1976 an article in the Eilenberg-Festschrift that contains a passage on the Zariski topos that clearly exhibits the importance of the functorial view and the product preserving property for Lawvere. This is also the first time where he mentions the petit/gros topos division.

• CommentRowNumber15.
• CommentTimeJun 11th 2018
• (edited Jun 11th 2018)

Here are a couple excerpts from e-mails I received from Professor Lawvere in March 2016:

In fact, he gave three extensive courses for students and postdocs here in Buffalo in 1973 and in May 1973 he made the qualitative advance described in his Colloqium talk. [I suggest that you change the reference to Gaeta’s notes; they represent one of the three Courses, NOT the Colloquium talk. Note that the Grothendieck Circle made the same mis-citation!]

Peter Gabriel (who unfortunately died this past November 24) had explained some of the same ideas at Oberwolfach in 1965-66, providing a context in which Grothendieck’s proposal seemed natural. For example, he emphasized the traditional view that the points of an algebraic space form a covariant functor on the category of field extensions of the base. The colimit of that functor gives the abstract set of points (= the prime ideals in the case of Spec); however, that colimit is not exact nor even product preserving, so that the category of abstract sets is not a good base topos unless the base field is algebraically closed.

[…]

There still exists considerable misunderstanding of these matters, so that, for example, fragments of the 1973 idea are pasted on top of the older 1960 view, leading in practice to complication, rather than simplification.

Grothendieck’s advice in his Colloquium talk was that 1960 ingredients (like Zariski opens etc.) are easily extracted from the category of functors, when needed.

[…]

The three courses of Grothendieck were recorded on tapes by Jack Duskin, and recently we had them transferred to a more modern medium. The historian of mathematics David Rowe in Mainz facilitated this transformation, with Michael Wright’s and of course our help. There are no other notes from the time, except Gaeta’s for ’algebraic geometry’, called ’functorial’ (except that the functorial aspect is not emphasized, as it was to be in the Colloquium Talk); the other two courses were ’Toposes’ and ’Algebraic Groups’ of which the audio version seems to be the only remnant.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJun 11th 2018
• (edited Jun 11th 2018)

The functor of points approach has the advantage of making certain constructions much simpler (e.g. the fibered product in the category of schemes), and eliminating the need for certain constructions like the Zariski spectrum. In his famous 1973 Buffalo Colloquium talk, Alexander Grothendieck urged that his earlier definition of scheme via locally ringed spaces should be abandoned in favour of the functorial point of view. This is recalled in Lawvere 03:

The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme (which had required as a prerequisite the baggage of prime ideals and the spectral space, sheaves of local rings, coverings and patchings, etc.), should be abandoned AS the FUNDAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the “baggage” could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition.

and in Lawvere 16:

Peter Gabriel $[$$]$ had explained some of the same ideas at Oberwolfach in 1965-66, providing a context in which Grothendieck’s proposal seemed natural. For example, he emphasized the traditional view that the points of an algebraic space form a covariant functor on the category of field extensions of the base.

Grothendieck’s advice in his Colloquium talk was that 1960 ingredients (like Zariski opens etc.) are easily extracted from the category of functors, when needed.

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeJun 22nd 2018

Added reference to McLarty’s Séminaire Lectures grothendieckiennes talk on Grothendieck’s 1973 lectures (here), with link to the video on YouTube

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJun 22nd 2018
• (edited Jun 22nd 2018)

Thanks, David.

I just looked through it. McLarty’s talk discusses Grothendieck’s development of the concept of topos. He does not seem to mention any functorial geometry.(?)

There is some discussion about objects in a topos as étale spaces, a comment that this only works for what ought to be called petit toposes, and a pointer to Lawvere’s work towards characterizing these. I suppose what is meant here is Lawvere’s work on characterizing the complementary gros toposes. This would be the point that functorial geometry were to come in, but McLarty’s talk does not follow up on this thread. It seems.

• CommentRowNumber19.
• CommentAuthorDavidRoberts
• CommentTimeJun 22nd 2018

Ah, I guess you’re right. I should move it to somewhere else, but I will have to wait for a bit until I am free.

• CommentRowNumber20.
• CommentAuthorDavidRoberts
• CommentTimeJun 25th 2018

Removed reference to McLarty’s talk (now at sheaf and topos theory)

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeJul 8th 2018

I added clarification that after view 1 and 2, the discussion is elaborating on 2.

I also added a link to that Secret blogging seminar discussion on teaching the functor of points approach.

1. Added a remark on the importance and difficulty of $R$-valued points where $R$ is not a field and how the internal language of toposes is helpful.

• CommentRowNumber23.
• CommentAuthorDavidRoberts
• CommentTimeJul 10th 2018
• (edited Jul 10th 2018)

PS Ingo: I like your recent paper on the generic freeness lemma! The proof should go in the Stacks Project, fwiw.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJul 6th 2020
• (edited Jul 6th 2020)

to the gmane archive in this reference item listed in the entry:

produces a timeout error. Maybe (hopefully) that’s just a temporary problem.

But if anyone knows a good way to save that message by Lawvere directly to a pdf file (or similar) which we could then stably host on the nLab server, that would be useful.

(In the same vein, I have just saved copies of those Grothendieck lecture notes to the nLab server, see Grothendieck 65, Grothendieck 73)

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeJul 7th 2020
• (edited Jul 7th 2020)

An alternative way to the archive is through the month, so here:

From: F W Lawvere wlawvere@buffalo.edu X-Sender: wlawvere@hercules.acsu.buffalo.edu Reply-To: wlawvere@acsu.buffalo.edu To: categories@mta.ca Subject: categories: Grothendieck’s 1973 Buffalo Colloquium Message-ID: Pine.GSO.4.05.10303291654120.1044-100000@hercules.acsu.buffalo.edu MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords:
X-UID: 18

In your “Comments on the Development of Topos Theory” you refer to a simpler alternative definition of “scheme” due to Grothendieck. Is this definition available at some place?? Otherwise, it it possible to describe shortly the main idea of this alternative definition??

Since several people have asked the same question over the years, I prepared the following summary which, I hope, will be of general interest:

The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme (which had required as a prerequisite the baggage of prime ideals and the spectral space, sheaves of local rings, coverings and patchings, etc.), should be abandoned AS the FUNDAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the “baggage” could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition.

Thus his definition is essentially well-known, and indeed is mentioned in such texts as Demazure-Gabriel, Waterhouse, and Eisenbud; but it is not carried through to the end, resulting in more complication, rather than less. I myself had learned the functorial point of view from Gabriel in 1966 at the Strasbourg-Heidelberg-Oberwolfach seminar and therefore I was particularly gratified when I heard Grothendieck so emphatically urging that it should replace the one previously expounded by Dieudonne’ and himself.

He repeated several times that points are not mere points, but carry Galois group actions. I regard this as a part of the content of his opinion (expressed to me in 1989) that the notion of topos was among his most important contributions. A more general expression of that content, I believe, is that a generalized “gros” topos can be a better approximation to geometric intuition than a category of topological spaces, so that the latter should be relegated to an auxiliary position rather than being routinely considered as “the” default notion of cohesive space. (This is independent of the use of localic toposes, a special kind of petit which represents only a minor modification of the traditional view and not even any modification in the algebraic geometry context due to coherence). It is perhaps a reluctance to accept this overthrow that explains the situation 30 years later, when Grothendieck’s simplification is still not widely considered to be elementary and “basic”.

To recall some well-known ingredients, let A be the category of finitely-presented commutative algebras over k (or a larger category closed under the symmetric algebra functor, for some purposes). Then the underlying set functor R on A serves as the “line”, and any system of polynomial equations with coefficients in k determines also a functor (sub space of Rn) in the well-known way; in fact, the idea of spec is simply identified with the Yoneda embedding of A^op. For example, R has a subfunctor U of invertible elements and another U’ such that U’(A) = {f|f in A, 1/1-f in A}. The Grothendieck topology for which U and U’ together cover R yields a subtopos Z known as the gros Zariski topos, which turns out to be the classifying topos for local k-algebras in any topos. This Z contains all algebraic schemes over k, but also function spaces Y^X and distribution spaces Hom(R^X,R) for all X,Y in Z. A basic open subspace of any space X is determined as the pullback U sub f of U under any map f: X–>R. One has obviously U sub f intersection U sub g = U sub fg and the intrinsic notion of epimorphism in Z gives a notion of covering. Thus for a space (functor) to have a finite open covering, each piece of which is representable, is a restrictive notion available when needed.

A point of X is a map spec(L) –> X where L is a field extension of k.Thus the “points functor” on spaces goes not to the category of abstract sets but rather is just the restriction operation to the category of functors on fields only. This is part of what Grothendieck seems to have had in mind. A serious discontinuity is introduced by passing to the single underlying set traditionally considered, which is the inductive limit of the functor of fields. The fact that the latter process does not preserve products, and hence for example that an algebraic group “is not a group”, was already for Cartier an indication that the traditional foundation had an unnatural ingredient, but before topos theory one tried to live with it (for example, I recall great geometers from the 1950s struggling to accept the new wisdom that +i and -i is one “point”). The acceptance of the view that, for non-algebraically-closed k, the appropriate base topos consists not of pure sets but rather of sheaves on just the simple objects in A, has in fact many simplifying conceptual and technical advantages; for example this base (in some sense due to Galois!) is at least qd in the sense of Johnstone, and even atomic Boolean in the sense of Barr.

(Technically, to verify that the above limitation to “algebraic” A gives the usual results requires the use of a Birkhoff Nullstellensatz which guarantees that there are “enough” algebras which are finitely-generated as k-modules. The use of a larger A, insuring for example that spaces of distributions are often themselves representable, is quite possible, but the precise description of the kind of double structure which is then topos-theoretically classified needs to be worked out. Gaeta’s notes of Grothendieck’s lecture series at Buffalo point out that A is more closely suited than most categories to serve as a site for a geometric category, because it is what is now called “extensive” )

I believe that Grothendieck’s point of view could be applied to real algebraic geometry as well, in several ways, including the following: Noting that within any topos the adjoint is available which assigns the ring R[-1] to any rig R, let us concentrate on the needed nature of positive quantities R. To include the advantages of differential calculus based on nilpotent elements, let us allow that the ideal of all elements having negatives can be non-trivial, and indeed include many infinitesimals, without disqualifying R from being “nonnegative”. The ring generated by R might appear in a more geometric way as the fiber of R^T, where T is the representor for the tangent-bundle functor. The classifying topos for the theory of “real rigs”, i.e., those for which 1/1+x is a given global operation, contains the classifying topos for “really local rigs” in the following sense (where “really” has the double meaning of (1) a strengthening of localness, but also (2) a concept appropriate to a real (as opposed to complex) environment): The subspace U of invertible elements in the generic algebra R has a classifying map R –> omega which of course as above preserves products; but the distributive lattice omega is in particular also a rig like R, so we can require that the classifying map be a rig homomorphism (i.e., also take + to “union”). (Of course, this elementary condition can be phrased in terms of subspaces of R and of R^2 without involving omega if desired.) The preservation of addition is a strengthening, possible for positive quantities, of the usual notion of localness (which on truth values was only an inequality).

Does the right adjoint to ( )^T restrict to this really local rig classifier?

F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeJul 7th 2020

Thanks! So I have fixed the link in the entry.

• CommentRowNumber27.
• CommentAuthorDavidRoberts
• CommentTimeJul 8th 2020
• (edited Jul 8th 2020)

Gmane broke some years ago. It has moved.

The older postings to the categories mail list are archived in flat text files, one per month up until 2009. Nothing after 2009 or so survives outside the nntp protocol only access to Gmane at its new home (though I haven’t been able to access this to verify). This was brought up on MathOverflow meta, given links to categories postings in answers/questions/comments there, and also raised in the category theory Zulip chat. It’s possible that the info could be extracted and hosted somewhere more stable. Even Bob R doesn’t have an archive, because of how the mail server operates, and he’d like a copy, too!