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In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces are proposed for this purpose. I find the most natural and preferable approach: categories of sheaves on suitable sites (among other things, such categories are automatically Grothendieck topoi with all the properties that follow). Ideally, I'm looking for a textbook on differential geometry from scratch that actively uses, wherever appropriate, generalized smooth spaces, which are defined as the category of sheaves on some site (this is my only requirement). If such textbooks do not exist, then any literature on generalized smooth spaces of this kind, where some definitions are given and some theorems are proved, would be useful to me.

Arshak Aivazian
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    I'd go as far to say you're not really doing differential geometry anymore... – Chris Gerig Sep 30 '22 at 14:45
  • @ChrisGerig I don't understand what you mean. – Arshak Aivazian Sep 30 '22 at 15:12
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    I agree with @ChrisGerig here in the sense that this is not how mainstream people working in differential geometry conceive of their subject. This is not to say that it might not be useful for whatever it is you're trying to do, just that any such book won't help you read the differential geometry literature. – Andy Putman Sep 30 '22 at 17:00
  • I wonder the relation of the first sentence (about infinite-dimensional manifolds) and the later. If I understand correctly, smooth spaces / stacks are finitary in nature. – Z. M Sep 30 '22 at 17:14
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    Is synthetic differential geometry appealing to you? The general setting is a topos (sometimes Grothendieck, sometimes just well-pointed or even less than a topos works) — I have a few good texts in mind if that sounds interesting. – Alec Rhea Sep 30 '22 at 18:22
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    @Z.M: The category of sheaves of sets (or simplicial sets) on the site of smooth manifolds is a cartesian closed category. In particular, the internal hom Hom(M,N) between two smooth manifolds M and N exists and has the expected properties. For example, the tangent space at any point can be computed as relative vector fields along a smooth map, etc. Likewise, the Lie algebra of the (infinite-dimensional) group of diffeomorphisms M→M can be computed as the Lie algebra of vector fields on M. The book by Iglesias-Zemmour explains all this. – Dmitri Pavlov Sep 30 '22 at 18:48
  • @DmitriPavlov Thanks. But I am not sure whether this recovers the "infinite-dimensional manifold" structure, or even the topological structure. For example, if we take some form of "geometric realization" of a sheaf in the category of topological spaces, it is a co-end, thus a colimit of finite dimensional manifolds, so there is no Banach structure (on the tangent spaces) or something like this. – Z. M Sep 30 '22 at 19:23
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    @Z.M: The Banach or Fréchet structure can be canonically recovered from the sheaf structure (on finite-dimensional manifolds), see the paper of Losik “Fréchet manifolds as diffeological spaces”. It proves that the category of Fréchet manifolds embeds fully faithfully in the category of diffeological spaces. – Dmitri Pavlov Sep 30 '22 at 19:42
  • @AlecRhea Thank you, I like synthetic differential geometry and I will definitely study it! I have several texts, but please send yours - perhaps I will find new and interesting ones among them. However, it doesn't work for me right now. – Arshak Aivazian Sep 30 '22 at 20:30
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    @AlecRhea The fact is that I give my students a course on differential geometry and, in parallel, a course on the theory of sheaves (more precisely, sheaves on sites and Grothendieck topoi). I would like to use the powerful mechanism of sheaves to construct all the missing spaces, but I would not want the only place where the differential geometry lives for them is only non-classical topoi. I consider constructive mathematics fundamentally more natural and important, but nevertheless it seems inappropriate (today) to force students in the 10th grade to abandon the law of the excluded middle. – Arshak Aivazian Sep 30 '22 at 20:30
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    @DmitriPavlov Do you have any English material which summarizes that proof? There seems no English translation of that paper. I only find an article by the same author which summarizes the theorems but not the proofs. – Z. M Sep 30 '22 at 21:06
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    @Z.M: A translation does exist, but it appears that Springer's online archive only covers years from 2007 on. I ordered it through my library. – Dmitri Pavlov Sep 30 '22 at 21:55
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    I agree that abandoning excluded middle would be ambitious for a 10th grade class, to say the least haha -- the texts I had in mind were Synthetic Geometry of Manifolds and Synthetic Differential Geometry by Anders Kock. – Alec Rhea Oct 01 '22 at 05:03
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    @Z.M: Losik's paper is available here: https://dmitripavlov.org/scans/losik-frechet-manifolds-as-diffeologic-spaces.pdf – Dmitri Pavlov Oct 04 '22 at 00:31
  • @DmitriPavlov Thanks. If it is lawful, maybe also add a link in https://ncatlab.org/nlab/show/Fr%C3%A9chet+manifold#Losik94 and wikipedia page (there is a footnote for Losik's paper) https://en.wikipedia.org/wiki/Diffeology – Z. M Oct 04 '22 at 04:18

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“Diffeology” by Patrick Iglesias-Zemmour is probably the closest match.

He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of diffeological spaces, i.e., concrete sheaves of sets on the site of smooth manifolds. This book is closest in style to a conventional differential geometry textbook.

Another book is “Synthetic geometry of manifolds” by Anders Kock, which treats differential forms, Lie groups and algebras, principal bundles with connections, jets and differential operators. It has a somewhat different focus (e.g., infinitesimals and the internal language of toposes) than the previous book.

In relation to this one can also mention “Models for smooth infinitesimal analysis” by Ieke Moerdijk and Gonzalo Reyes, which covers some foundational topics in differential geometry, like differential forms.

Dmitri Pavlov
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  • Thank you! I asked a related question earlier, and from this comment it seemed to me that diffiological spaces have unpleasant problems at the level of defining a tangent space, so I stopped being interested in them. – Arshak Aivazian Sep 30 '22 at 21:31
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    Now I looked through the book and see that the tangent space is defined just as well as for stacks, and judging by the things that you mention and the content of the book, there are no special obstacles for the development of differential geometry in this context (and in the article mentioned in the comment, it was about problems with other versions of definitions). It will be interesting to see what stops working or becomes more complicate when the condition of concreteness of the sheaves is abandoned. – Arshak Aivazian Sep 30 '22 at 21:31
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    @AivazianArshak: I would not characterize the cited work of Christensen and Wu as “finicky”, in fact, it gives a simple and conceptual treatment of tangent bundles of diffeological spaces. – Dmitri Pavlov Sep 30 '22 at 21:58
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    Is there any analytic version of this? An analytic version might also have a p-adic analogue (aka. locally analytic geometry). – Z. M Oct 01 '22 at 10:46
  • @Z.M: Yes, the functor of points approach can be used to define complex analytic spaces as certain sheaves of sets on the site of Stein spaces. This is completely analogous to how schemes can be defined as certain sheaves of sets on the site of affine schemes, i.e., the opposite category of commutative rings. – Dmitri Pavlov Oct 01 '22 at 15:16
  • No, I am asking an analytic analogue of diffeologies. Complex analytic spaces are of finite type, in this sense it is not an analogue of diffeologies (there are other differences: in some sense the site might be somehow polydiscs instead of Stein's, I guess). – Z. M Oct 01 '22 at 16:34
  • @Z.M: Yes, so (concrete) sheaves of sets on the site of complex manifolds are equivalent to (concrete) sheaves of sets on the site of Stein manifolds, which are equivalent to (concrete) sheaves of sets on the site of polydisks. These are the precise analogues of diffeological spaces in the complex analytic setting. See https://ncatlab.org/nlab/show/complex+analytic+infinity-groupoid on the nLab. – Dmitri Pavlov Oct 01 '22 at 16:52