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Stream: learning: questions

Topic: Are manifolds sheaves?

Paolo Perrone (Nov 05 2025 at 19:27):

Possibly basic question, I feel like I should have asked this ages ago, but somehow I never did.
Is a manifold (or at least an atlas) a sheaf or a stack of something? It's all about gluing local data. But unlike, say sections of a bundle on a manifold, I can't quite see how to make the sheaf structure precise.

(Of course it's a representable presheaf in the category of manifolds, but is there something less trivial that one can say? Or does the atlas at least play a special role in making the descent condition explicit?)

John Onstead (Nov 05 2025 at 19:43):

In short, yes, a manifold is a sheaf on the site of (open subsets of) Cartesian spaces and smooth maps between them, with the Grothendieck topology described here. More generally, any smooth manifold, Frechet manifold, and diffeological space is an example of a smooth set, and smooth sets are precisely the sheaves on this site of Cartesian spaces and smooth maps. You can think of a smooth set as "gluing together" copies of Euclidean space with smooth structure.

Even more particularly, the category of smooth manifolds is the Cauchy completion of the category of open subsets of Cartesian space and smooth maps between them. That defines precisely the subcategory of smooth sets on the manifolds.

John Baez (Nov 05 2025 at 20:21):

In short, yes, a manifold is a sheaf on the site of (open subsets of) Cartesian spaces and smooth maps between them, with the Grothendieck topology described here.

Every smooth manifold gives a sheaf on this site, but of course not every sheaf on that site is a manifold. ("Of course" because the category of a manifolds does not have all pullbacks, pushouts, etc.)

Alex Kreitzberg (Nov 05 2025 at 20:42):

Isn't that the best you could hope for here because manifolds aren't well behaved, but sheaves form toposes?

Peva Blanchard (Nov 05 2025 at 20:52):

Alex Kreitzberg said:

Isn't that the best you could hope for here because manifolds aren't well behaved, but sheaves form toposes?

Mmh, I think that is witnessed by John's argument. The category of sheaves has, e.g., pushouts, but manifolds do not. For instance, gluing together two distinct manifolds of different dimensions does not yield a manifold (e.g. a lollipop).

Peva Blanchard (Nov 05 2025 at 20:53):

John Baez said:

Every smooth manifold gives a sheaf on this site, but of course not every sheaf on that site is a manifold.

But is it possible to characterize the manifolds as a "special" subcategory of the category of sheaves on the site of cartesian spaces and smooth maps?

Peva Blanchard (Nov 05 2025 at 20:57):

There might be some condition on the dimension I guess.

John Baez (Nov 05 2025 at 21:33):

Peva Blanchard said:

But is it possible to characterize the manifolds as a "special" subcategory of the category of sheaves on the site of cartesian spaces and smooth maps?

The category of smooth manifolds is fully and faithfully embedded in that category of sheaves, so a smooth manifold can be seen one of these sheaves with an extra property. So, the question is just how nicely one can state that extra property. I don't have anything intelligent to say about that.

Paolo Perrone (Nov 05 2025 at 21:41):

I guess the Cauchy completion condition could be one (e.g. they are the retracts of representables).

David Michael Roberts (Nov 05 2025 at 23:00):

One has to start with saying that the category of manifolds is a full subcategory of that of concrete sheaves on Cart, aka diffeological spaces. Because lots of sheaves have too few (generalised) points, for instance the sheaf of n-forms, which has one point, one curve, one .... and then an infinite-dimensional space of n-dimensional plots.

Fernando Yamauti (Nov 05 2025 at 23:02):

You can say a manifold is a sheaf on cartesian spaces together with an atlas: a jointly epi out of representables that is also representable (i.e., base changing to any representable gives an arrow in the image of Yoneda) and a submersion (after base changing to a representable it's a submersion).

The same thing holds for schemes, analytic spaces etc.

David Michael Roberts (Nov 05 2025 at 23:40):

Yes, I was thinking about how to phrase something like that, and I'm so used to having coproducts in the site, that I got stuck on thinking that you can't get a single map from a representable to the sheaf!

Marco Vianello (Nov 06 2025 at 02:19):

We could probably add that the category of smooth manifolds is equivalent to a subcategory of the category of (locally) ringed spaces. A manifold is then a colimit of "affine" manifolds of the form $(\mathbb R^n, \mathscr C_{\mathbb R^n}^\infty)$.

Jonas Frey (Nov 07 2025 at 10:25):

Fernando Yamauti said:

You can say a manifold is a sheaf on cartesian spaces together with an atlas: a jointly epi out of representables that is also representable (i.e., base changing to any representable gives an arrow in the image of Yoneda) and a submersion (after base changing to a representable it's a submersion).

The same thing holds for schemes, analytic spaces etc.

Can't we even require the cover to be representably étale (local diffeomorphism) rather than a submersion? In my intuition, that's what captures the idea of an open cover of affines.

Fernando Yamauti (Nov 07 2025 at 18:36):

Jonas Frey said:

Fernando Yamauti said:

You can say a manifold is a sheaf on cartesian spaces together with an atlas: a jointly epi out of representables that is also representable (i.e., base changing to any representable gives an arrow in the image of Yoneda) and a submersion (after base changing to a representable it's a submersion).

The same thing holds for schemes, analytic spaces etc.

Can't we even require the cover to be representably étale (local diffeomorphism) rather than a submersion? In my intuition, that's what captures the idea of an open cover of affines.

Yes. Good point. That’s the correct definition. When I wrote that, I was thinking about a smooth atlas, which should again yield a manifold (because the quotient by an eq relation such that one leg, equivalently both, is a submersion is again a manifold).

For the other instances I'd mentioned, you usually will want affine open immersions. In general, you will want the your atlas to output an equivalence relation such that each leg is a covering (over its image) with respect to your chosen topology.

Btw, that’s something curious I'd never thought about before. It seems, for 0-truncated guys, smooth and étale atlases are the same (in all the instances I can remember), but, when we go higher and consider non 0-truncated guys, we usually have two non-equivalent choices of atlases. Is that perhaps a general fact for any geometry (in the sense of structured topos theory)? (What's even a smooth map in the Zariski topology?)

Jonas Frey (Nov 07 2025 at 19:59):

That sounds interesting and I don't have anything intelligent to add. Let me just mention that your point about the coincidence of smooth and étale quotients reminded me of this sentence on page 364 of Moerdijk/Reyes's "Models for smooth infinitesmimal analysis":
image.png

Jonas Frey (Nov 07 2025 at 20:01):

The claim is that adding projections to a site of open covers gives an equivalent sites. Maybe adding projections to the site of open covers can be viewed as giving a site of smooth covers?

Jonas Frey (Nov 07 2025 at 20:03):

What's your recommended reference on the fact that quotients of manifolds by equivalence relations with submersive legs are again manifolds, if you have one?

Jonas Frey (Nov 07 2025 at 20:05):

(I think at some point it has to be pointed out that to get the classical notion of manifold we also have to add a separation condition, since manifolds are supposed to be Hausdorff.)

Fernando Yamauti (Nov 07 2025 at 20:39):

Fernando Yamauti said:

Jonas Frey said:

Fernando Yamauti said:

You can say a manifold is a sheaf on cartesian spaces together with an atlas: a jointly epi out of representables that is also representable (i.e., base changing to any representable gives an arrow in the image of Yoneda) and a submersion (after base changing to a representable it's a submersion).

The same thing holds for schemes, analytic spaces etc.

Can't we even require the cover to be representably étale (local diffeomorphism) rather than a submersion? In my intuition, that's what captures the idea of an open cover of affines.

Yes. Good point. That’s the correct definition. When I wrote that, I was thinking about a smooth atlas, which should again yield a manifold[...].

Hmm...Now I see that the nlab seems to define the covering sieves on cartesian spaces to be jointly surjective open immersions with intersections diffeomorphic to cartesian spaces. That’s even more than étale. So the actually correct definition of an atlas is even more strict.

However, I think both sites (étale and those open immersions) should yield the same topos, but I'm not sure: can every jointly surjective family of euclidean opens be refined to another one where each n-ary intersection is diffeomorphic to an euclidean space?

Fernando Yamauti (Nov 07 2025 at 20:41):

Jonas Frey said:

The claim is that adding projections to a site of open covers gives an equivalent sites. Maybe adding projections to the site of open covers can be viewed as giving a site of smooth covers?

What's a projection? Perhaps I should just check the book...

Fernando Yamauti (Nov 07 2025 at 20:44):

Jonas Frey said:

What's your recommended reference on the fact that quotients of manifolds by equivalence relations with submersive legs are again manifolds, if you have one?

That's called Godement's criterion. Not sure why it's attributed to Godement though...

Peter Arndt (Nov 07 2025 at 20:59):

Here are some old course notes by Bertrand Toen where he gives the characterization of manifolds mentioned above: see Prop 1.2.4

Peter Arndt (Nov 07 2025 at 21:01):

They are a translation, possibly not by Toen himself, of some original French notes, which I can't seem to find right now.

Adrian Clough (Nov 07 2025 at 21:04):

Here are the notes in French in case you are looking for that je ne sais quoi https://ncatlab.org/nlab/show/Master+course+on+algebraic+stacks

Fernando Yamauti (Nov 07 2025 at 21:23):

Fernando Yamauti said:

Jonas Frey said:

The claim is that adding projections to a site of open covers gives an equivalent sites. Maybe adding projections to the site of open covers can be viewed as giving a site of smooth covers?

What's a projection? Perhaps I should just check the book...

Nevermind. I've just checked it. That seems to give the correct definition only when the local models are euclidean spaces if I'm not missing anything. [Lurie's definition of smoothness](https://ncatlab.org/nlab/show/geometry+(for+structured+(infinity%2C1%29-toposes%29#smooth_morphisms) gives the usual def of smoothness in the étale topology, but I don’t know what happens for anything else.

Fernando Yamauti (Nov 07 2025 at 22:49):

Fernando Yamauti said:

Fernando Yamauti said:

Jonas Frey said:

The claim is that adding projections to a site of open covers gives an equivalent sites. Maybe adding projections to the site of open covers can be viewed as giving a site of smooth covers?

What's a projection? Perhaps I should just check the book...

Nevermind. I've just checked it. That seems to give the correct definition only when the local models are euclidean spaces if I'm not missing anything. [Lurie's definition of smoothness](https://ncatlab.org/nlab/show/geometry+(for+structured+(infinity%2C1%29-toposes%29#smooth_morphisms) gives the usual def of smoothness in the étale topology, but I don’t know what happens for anything else.

His smoothness definition depends on a pre-geometry and neither a geometry nor a pre-geom determines the other uniquely. So my question was perhaps not that interesting.

If one chooses the correct pre-geometry (say, compact free algebras if we are talking about an essentially algebraic theory), smoothness is kind of obvious: smooth with respect to some topology means locally isomorphic to the dual of a compact free algebra. So, in particular, a Zariski-smooth atlas of a 0-truncated object, will just give a groupoid scheme with smooth legs, which seems very different from a scheme...

Also, that's even worse: I'm starting to think my initial statement was wrong. In smooth sets, from a smooth (aka submersive) atlas, I can only get a groupoid with smooth legs. I can't remember why I thought the underlying relation was also representable... perhaps it's not?...

Anyways, I'm too sleepy to figure that out now.

Fernando Yamauti (Nov 08 2025 at 17:37):

Fernando Yamauti said:

Also, that's even worse: I'm starting to think my initial statement was wrong. In smooth sets, from a smooth (aka submersive) atlas, I can only get a groupoid with smooth legs. I can't remember why I thought the underlying relation was also representable... perhaps it's not?...

Anyways, I'm too sleepy to figure that out now.

Just clarifying my own confusion from yesterday.

I somehow forgot that the site is a 1-cat, so every representable is 0-truncated. In particular, the groupoid induced from an atlas of some 0-truncated object will always be a relation with smooth legs. Still, the map $R \to X \times X$ is only a mono, so there's no reason for that to define a submanifold. Then, Godement's criterion cannot be used.

However, for schemes in the étale topology, the analogous statement is true: every 0-truncated algebraic stack is an algebraic space.

In the general setting of an essentially algebraic theories with a topology and the respective pre-geometries of compact algebras that are locally (in the chosen topology) compact free algebras, I still don't know when 0-truncated objects with a smooth atlas can be presented by an ordinary atlas (a representable epi $\coprod h_{U_i} \to X$ such that $\{h_U \times_X h_{U_i}\to h_U\}_i$ are admissibles generating a covering sieve).

I don't even know what happens in the Zariski topology. Here smooth maps are Zariski-locally affine spaces, but I don’t know if the argument for the étale topology can be copied here.

Fernando Yamauti (Nov 08 2025 at 19:06):

Fernando Yamauti said:

However, for schemes in the étale topology, the analogous statement is true: every 0-truncated algebraic stack is an algebraic space.

Nah. The proof on Stacks Project relies really on fppf-locality. It doesn't imply the result for the étale topology. It just says that fppf-locally (not necessarily étale-locally) one can construct an étale relation. So I don't even know whether a 0-truncated algebraic stack is an algebraic space in the étale topology.

Dmitri Pavlov (Nov 15 2025 at 17:51):

There is another way to look at manifolds as sheaves: the category of smooth manifolds (not necessarily paracompact or Hausdorff) and etale smooth maps (i.e., local diffeomorphisms) is equivalent to the category of sheaves of sets on the site of cartesian spaces, etale smooth maps, and jointly surjective families as covering families. In particular, the category of manifolds and etale maps is a Grothendieck topos, so admits all small limits, small colimits, and is cartesian closed (and locally cartesian closed). Colimits coincide with colimits in the other categories (e.g., sheaves of sets on manifolds and smooth maps), whereas limits and internal homs are quite different. The inclusion functor from cartesian spaces and etale maps to cartesian spaces and smooth maps induces a left adjoint functor between the corresponding categories of sheaves of sets. The image of this functor can be characterized as the category of manifolds and etale maps, sitting inside the much larger category of smooth sets and smooth maps. Abstractly, this describes what is now called a fractured ∞-topos, a theory developed by Joyal–Moerdijk, Dubuc, Carchedi, Lurie.

Morgan Rogers (he/him) (Nov 17 2025 at 08:22):

Dmitri Pavlov said:

There is another way to look at manifolds as sheaves: the category of smooth manifolds (not necessarily paracompact or Hausdorff) and etale smooth maps (i.e., local diffeomorphisms) is equivalent to the category of sheaves of sets on the site of cartesian spaces, etale smooth maps, and jointly surjective families as covering families. In particular, the category of manifolds and etale maps is a Grothendieck topos, so admits all small limits, small colimits, and is cartesian closed (and locally cartesian closed).

This can't be true: product projections are not étale. The category of manifolds and étale maps doesn't have a terminal object.

Kevin Carlson (Nov 17 2025 at 08:36):

Yeah, the terminal sheaf on the given site would have to correspond to a manifold that admits one local diffeomorphism from each $\mathbb{R}^n.$ Even ignoring the fact that this manifold has every dimension at once, it would have to be rigid, whereas positive-dimensional real manifolds have enormous groups of smooth automorphisms. I can’t guess what you meant here, Dmitri.

Adrian Clough (Nov 17 2025 at 09:19):

Denote by $j: \mathbf{Man}^\mathrm{ét} \to \mathbf{Man}$ the inclusion of the category of smooth manifolds and local diffeomorphisms into the category of smooth manifolds and all smooth maps. The functor $\mathbf{Sh}_{\mathbf{Man}^\mathrm{ét}} \leftarrow \mathbf{Sh}_{\mathbf{Man}} : j^*$ between $\infty$-categories of sheaves, admits a left adjoint $j_!$ which is faithful but not full (but it is full on slices). The restriction of $j_!$ to set-valued sheaves takes values in $1$-groupoid valued sheaves (a.k.a. stacks).

Adrian Clough (Nov 17 2025 at 09:19):

The final object of $\mathbf{Sh}_{\mathbf{Man}^\mathrm{ét}}$ is 0-truncated and is thus given by a set-valued sheaf. Its image under $j_!$ is the disjoint union of all $n$-Haefliger stacks for all dimensions $n$.

Adrian Clough (Nov 17 2025 at 09:26):

The functor $j_!: \mathbf{Sh}_{\mathbf{Man}^\mathrm{ét}} \to \mathbf{Sh}_{\mathbf{Man}}$ preserves pullbacks but not products. For two manifolds $M,N$ the product of $M$ and $N$ in $\mathbf{Sh}_{\mathbf{Man}^\mathrm{ét}}$ is sent to the étalé space of the sheaf on $M$ given by sending any open subset $U \subseteq M$ to the set of local diffeomorphisms $U \to N$. We thus have a canonical map $M \times N \to M$. The map $M \times N \to N$ is given as follows: An element in $M \times N$ is the germ of a local diffeomorphism at $m \in M$; evaluating this germ at $m$ yields a point in $N$.

Adrian Clough (Nov 17 2025 at 09:32):

If we don't truncate, then the image of $j_!$ is equivalent the $\infty$-category of ($\infty$-categorical) Deligne-Mumford stacks (without any sort of separatedness or smallness conditions such as paracompactness) and local diffeomorphisms, which can reasonably be thought of as non-truncated versions of manifolds.

Adrian Clough (Nov 17 2025 at 09:32):

Similar statements are true in algebraic geometry.

Adrian Clough (Nov 17 2025 at 09:33):

These results are due to Carchedi.