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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?
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?).
This one here seems better.
In fact, it’s obviously better, so I’ve replaced the link.
Thanks!
Yes, thanks! That’s a relief.
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.
Not sure what you mean to imply, but Zhen Lin’s thesis is to linked from the entry.
Nothing deep. I should have checked the entry before commenting :-/
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.
10: wrong link, functorial geometry, not geomertry.
Thanks. Fixed now.
In rev 6 Adeel added the remarkable remark that:
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.
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.
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.
Thanks, Thomas! Thanks, Adeel!
I have added this to the entry as follows, please check (in particular: Adeel, please check about the citation of the email):
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.
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.
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.
Removed reference to McLarty’s talk (now at sheaf and topos theory)
PS Ingo: I like your recent paper on the generic freeness lemma! The proof should go in the Stacks Project, fwiw.
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