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Stream: learning: questions

Topic: A topos containing manifolds

Morgan Rogers (he/him) (Sep 23 2024 at 20:43):

I came across this in another topic.
John Onstead said:

John Baez said:

So there's nothing circular about it and we're done while the category theorists are still revving up their engines.

This is humorous, because it certainly seems true. Based on previous questions I asked on here and in general looking up resources on the web, it seems that the subjects of category theory and analysis just don't get along. Ironically this means category theory can explain abstract concepts from the highest levels of mathematics with ease, but finds itself in a bind when describing basic high school calculus or undergrad multivariable calc as per your example!

I don't know whether there is a categorical distinguishing property of smooth maps amongst continuous ones; diffeological spaces and SDG were mentioned earlier; I wonder if they produce the category that I'm about to describe...
I figure it might be interesting (for @David Egolf , at least) to think about how you could present smooth manifolds as a special case of sheaves on the following site: objects are open subsets of $\R^n$ (for any finite $n$), morphisms are smooth maps between these and the covering families are (generated by) the open covers.

Kevin Carlson (Sep 23 2024 at 21:06):

Yes, diffeological spaces are certain sheaves on this site. Objects of SDG usually are sheaves on richer sites that include infinitesimal data.

John Baez (Sep 23 2024 at 21:09):

Diffeological spaces are precisely the 'concrete' sheaves $F$ on the site Morgan mentioned. That means that there's a set $X$ such that for any $U \subseteq \mathbb{R}^n$, $F(U)$ is some subset of maps from $U$ to $X$... and similarly on morphisms.

John Baez (Sep 23 2024 at 21:10):

I blathered about this at length in Convenient categories of smooth spaces.

David Egolf (Sep 23 2024 at 23:21):

Morgan Rogers (he/him) said:

I figure it might be interesting (for David Egolf , at least) to think about how you could present smooth manifolds as a special case of sheaves on the following site: objects are open subsets of $\R^n$ (for any finite $n$), morphisms are smooth maps between these and the covering families are (generated by) the open covers.

That does sound interesting! I haven't yet learned what a "site" is, though. However, "Sheaves in Geometry and Logic" does talk about sheaves on a site! So I might take a little excursion to learn about the concept of "site", before I press on to Part 4 of John Baez's topos theory blog posts...

John Baez (Sep 23 2024 at 23:25):

One can talk about presheaves on any category - do you know about that? A "site" is a category with enough structure on it that you can also talk about sheaves on it.

John Baez (Sep 23 2024 at 23:26):

So far we've only been doing the case where our category is the poset of open sets of some topological space. In this case treating it as a "site" amounts to saying we've decided when a bunch of open sets cover another open set. To make a general category into a site, you need to specify when a bunch of morphisms $f_i : X_i \to X$ 'cover' an object $X$.

John Baez (Sep 23 2024 at 23:27):

To specify this, we can give our category a structure called a 'Grothendieck topology'.

John Baez (Sep 23 2024 at 23:28):

So, while I'm not actually providing any details, you now know that a site is a category equipped with a Grothendieck topology, and this extra structure lets us talk about sheaves on our category.

David Michael Roberts (Sep 24 2024 at 00:03):

[[Hadamard lemma]] seems appropriate to mention here, I think, but not sure how much it helps

David Egolf (Sep 24 2024 at 00:06):

John Baez said:

One can talk about presheaves on any category - do you know about that? A "site" is a category with enough structure on it that you can also talk about sheaves on it.

I see on the nLab that any functor $F:C^{\mathrm{op}} \to \mathsf{Set}$ is a presheaf on $C$. This generalizes the case we've been considering, where we have a presheaf $F:\mathcal{O}(X)^{\mathrm{op}} \to \mathsf{Set}$. The open sets of a topological space form a poset, so the presheafs we've been thinking about so far have been defined on thin categories.

When $C$ is not thin, I suppose that means that it doesn't necessarily convey the right idea to talk about "restricting" a presheaf element. That's because there can be multiple morphisms between objects now! In particular a "smaller" object can now be contained inside a "bigger" one in multiple different ways.

John Baez (Sep 24 2024 at 00:09):

Right. We can say "pulling back along a morphism" instead "restricting", but sometimes people even say "restricting along a morphism".

When you say "presheafs" you remind me of my friend Jim Dolan, who is so fond of forcing plurals to be regular that he even says "serieses".

Kevin Carlson (Sep 24 2024 at 00:09):

That's true, but it's often still useful to think about "restricting". For instance, a presheaf on the category $V\rightrightarrows E$ is a (directed, multi)graph. "Restricting" along each of the two parallel arrows sends an edge to its source, respectively, its target, which does feel like restricting. Geometrically, you even have that the graph corresponding to $V$ via the Yoneda embedding really embeds in the graph corresponding to $E$ in two different ways, so at least up to Yoneda the "restrictions" are really restrictions. We can think quite geometrically of this situation, which is often the case. This story doesn't always work without stretching the concept of "restriction" to the breaking point, but it's pretty good at least for restricting along monomorphisms in the base category of our presheaf.

Morgan Rogers (he/him) (Sep 24 2024 at 08:10):

@David Egolf sorry if I asked this question too soon, feel free to come back to it once you've encountered sites "officially" in John's notes as an extra example :innocent:

John Baez (Sep 24 2024 at 14:50):

Alas, the notes on my blog don't get up to sites! I gave up after I covered sheaves on topological spaces, the definition of elementary topos, and an explanation of how sheaves on a topological space form an elementary topos. Or something like that - I forget the details. I was teaching a class, and writing up notes on my blog, and after a while I gave up writing notes on my blog because nobody was responding to them, which was depressing. So my class went further than these notes.

David Egolf (Sep 24 2024 at 22:18):

Kevin Carlson said:

For instance, a presheaf on the category $V\rightrightarrows E$ is a (directed, multi)graph. "Restricting" along each of the two parallel arrows sends an edge to its source, respectively, its target, which does feel like restricting. Geometrically, you even have that the graph corresponding to $V$ via the Yoneda embedding really embeds in the graph corresponding to $E$ in two different ways, so at least up to Yoneda the "restrictions" are really restrictions.

If I understand you correctly:

• Let $C$ be the category $V\rightrightarrows E$
• Then a presheaf $F:C^{\mathrm{op}} \to \mathsf{Set}$ on $C$ is a directed multigraph
• We can note that $C(-,V):C^{\mathrm{op}} \to \mathsf{Set}$ and $C(-,E):C^{\mathrm{op}} \to \mathsf{Set}$ are also graphs
• In particular, $C(-,V)$ has a singleton vertex set and an empty edge set, while $C(-,E)$ has two vertices and one edge, going from one vertex to another
• So, geometrically $C(-,V)$ is a "single vertex", which can be embedded in the "single arrow" graph described by $C(-,E)$. There are two ways to do this.
• Further, under the Yoneda embedding, $s:V \to E$ and $t:V \to E$ get mapped to natural transformations $:C(-,V) \to C(-,E)$, and in fact describe these embeddings

David Egolf (Sep 24 2024 at 22:23):

Now, if we have a presheaf $F:C^{\mathrm{op}} \to \mathsf{Set}$, we have two sets, $F(V)$ and $F(E)$. Given an edge, we can apply $F(s):F(E) \to F(V)$ or $F(t):F(E) \to F(V)$ to get its source and target vertices.

I guess the idea is to try and think about the source and target vertices of a directed edge as "smaller parts" of that directed edge. (Indeed, we just saw how to embed a single-vertex graph in a single-edge graph, in two ways).

David Egolf (Sep 24 2024 at 22:25):

From this angle, we might think of a restriction (in the context of a presheaf on a topological space) as taking data attached to a larger region and getting just some part of the data, now attached to a smaller region. So, we start with something bigger and "restrict" it to part of itself, to get something smaller.

David Egolf (Sep 24 2024 at 22:27):

I think that $s,t:V \to E$ are both monomorphisms in $C$. Thinking of $V$ as corresponding to a graph with a single vertex, and $E$ as corresponding to a graph with a single directed edge (under the Yoneda embedding), these being monomorphisms makes intuitive sense!

David Egolf (Sep 24 2024 at 22:33):

Morgan Rogers (he/him) said:

David Egolf sorry if I asked this question too soon, feel free to come back to it once you've encountered sites "officially" in John's notes as an extra example :innocent:

As @John Baez noted above, sites aren't covered in that blog series! However, trying to understand how to present smooth manifolds as sheaves on some site might be an enjoyable way to learn a bit about sites! So, I might try to work towards that, in this thread (or in a spinoff thread? whichever would be best!).

Morgan Rogers (he/him) (Sep 25 2024 at 06:16):

Here is fine :wink: I've noticed that interest in various toposes of spaces has been picking up in the last year or two, so I expect other people would also enjoy thinking about this.

Chris Grossack (they/them) (Sep 25 2024 at 15:05):

I've also noticed this, and it's been making me really happy! When I was younger I thought SDG was really cool (despite it never catching on), and wondered why there wasn't a great reference on Synthetic Algebraic Geometry. But in the last few years it seems like SDG and SAG are both starting to pick up some steam. Especially $\infty$-analogues whose internal logics look like HoTT with extra axioms

Chris Grossack (they/them) (Sep 25 2024 at 15:06):

Someday soon I want to start doing research in this kind of stuff, but it's too big a project for me to start on top of a thesis in geometric representation theory, so it'll have to wait for a few years

David Corfield (Oct 01 2024 at 06:32):

Urs Schreiber's article, Higher Topos Theory in Physics, lays out in a reasonable introductory way extended notions of smooth spaces in topos-theoretic terms.