Comparative Smootheology

January 3, 2008

Posted by Urs Schreiber

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Here in the $n$ -Café we happen to talk about the various notions of generalized smooth spaces every now and then (last time starting here).

I was dreaming of having, at one point, a survey of the various definitions and their relations in our non-existent wiki. Luckily, while I was just dreaming, Andrew Stacey did it.

Andrew is an expert on the index theorem for Dirac operators on loop spaces (see his list of research articles), and for that work he needs to deal with generalized smooth structures that render loop space a smooth space.

Last time I visited Nils Baas in Trondheim I had the pleasure of talking quite a bit with Andrew. Ever since then I had planned to post something about the intriguing things about loop space Dirac operators he taught me, but never found the time (but see this comment).

Now recently he sent me a link to his new article, which gives a detailed survey of the various definitions of generalized smooth spaces, and a careful and detailed comparison between them:

A. Stacey
Comparative Smootheology
(pdf, arXiv)

Abstract. We compare the different definitions of “the category of smooth objects”.

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Of the four or five different definitions he considers, Andrew favors Frölicher spaces, and he explains why.

I am reading this in particular with an eye towards our recent discussion in Transgression of $n$ -Transport and $n$ -Connections, where I am falling in love with a general definition that Andrew does not discuss explicitly: smooth spaces as general presheaves over the site of manifolds or open subsets of $ℝ \cup ℝ^{2} \cup ℝ^{3} \cup \dots$ .

The set such a presheaf assigns to any object $U$ of the domain category (manifold or open subset of sorts) is to be thought of as the collection of smooth maps from $U$ into the smooth space thus defined.

The slogan here is

A generalized smooth space (in the sense of presheaves on manifolds) is a space which need not locally look like a manifold, but which may be probed by manifolds.

Chen smooth spaces and/or diffeological spaces are a special case of such presheaves, namely presheaves which are quasi-representable: while not in general representable, these are presheaves $X$ for which there exists a set $X_{s}$ such that for $U$ any test domain, we have $X (U) \subset {Hom}_{Set} (U, X_{s})$ . Moreover, morphisms of diffeological or Chen-smooth spaces are morphisms of presheaves $X \to Y$ induced by maps $X_{s} \to X_{y}$ of these sets.

On the other hand, Frölicher spaces and “differentiable spaces”, as in Mostow’s article, are defined not just by specified smooth maps into them, but also by specified smooth maps out of them.

This is extremely useful for instance for having such a standard concept as the chain rule available for generalized smooth spaces. The chain rule for mere presheaves, as above, is, in contrast, a headache.

Moreover, the slick thing about Frölicher’s definition is that he realized that with maps in and out, it is already sufficient to consider all maps from just $ℝ^{1}$ into the generalized smooth space. Hence a Frölicher smooth space is a set together with a specified collection of smooth curves in it, and a specified collection of smooth functions on it, satisfying some compatibility conditions.

I like that. But personally I haven’t quite made up my mind yet.

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

Notice that for presheaves on manifolds it is natural to define all contravariant differential geometric constructions, notably differential forms.

While for co-presheaves on manifolds, it is natural to define all covariant differential geometric constructions, notably vector fields.

Then recall one of the points emphasized in Transgression of $n$ -transport and $n$ -connections: every non-negatively differential graded commutative algebra sits inside the dg-algebra of differential forms on some presheaf-like generalized smooth space.

And then recall from On BV-quantization, Part VIII that we are really looking for a way to generalize this statement to dg-algebras with no restriction on the grading.

So possibly one way to realize this is to consider things that are both presheaves and co-presheaves on manifolds. “Frölicher presheaves”, in a way.

Hm…

Posted at January 3, 2008 10:21 PM UTC

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Re: Comparative Smootheology

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I didn’t put in the “presheaves” version because I didn’t know of it. If someone sends me a precise definition then I’ll add it in (providing I understand it, of course!). Now that, thanks to Urs, I have a copy of Mostow’s paper I’ll add a section on that (again, subject to the proviso of comprehension).

Urs wrote (and I emphasised):

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

It’s the compatibility relationship that leads one to Frölicher spaces. I suspect that you would end up with a non-set-based version of Frölicher spaces which I don’t think anyone would have any qualms about.

As I see it, Frölicher’s main insight was this compatibility relation. The fact that he only used curves and functionals (rather than maps into or out of more general spaces) is more of a technicality. Due to useful results such as Boman’s, one can always reduce a more complicated definition to one using only curves and functionals without losing any information (just “bloat” as I refer to it).

The compatibility relationship can be thought of as “If it looks like a duck and quacks like a duck, then it is a duck.”. Without wishing to start a flame war, the other approaches are more in the line of “It is only a duck if it appears in my list of ‘approved ducks’.”.

Andrew

PS And without wishing to start a subthread on a completely irrelevant issue, a chemist I know (rather well) told me of the following version: “If it looks like a duck and quacks like a duck but doesn’t have the NMR of a duck, it ain’t a duck.”

In categorical terms, that would translate to: “If it looks like a duck and quacks like a duck but doesn’t transform like a duck, it ain’t a duck.” though, of course, it may be a 2-duck.

Posted by: Andrew Stacey on January 4, 2008 9:52 AM | Permalink | Reply to this

Re: Comparative Smootheology

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Andrew, the presheaf version Urs is referring to is but one of many of a class of sheaf models for doing smooth analysis. You may have seen some of this already, so I’ll try to keep this brief.

The specific version Urs is looking at is the category of functors

S^{op} \to Set

where $S$ is the category of finite-dimensional smooth manifolds and smooth maps between them. He denotes this category of functors by $S^{∞}$ .

Besides simplicity, there are a number of pleasant features of this category. For one thing, this category is a topos, which has many wonderful properties: (local) cartesian closure, completeness and cocompleteness, and nice global exactness conditions. Also, the category of ordinary smooth manifolds $S$ embeds fully and faithfully in $S^{∞}$ via the Yoneda embedding, viz. the functor sending $M$ to $hom (-, M)$ . So in some respects it is a paradisal extension of ordinary manifolds.

But not all respects. Notably, the Yoneda embedding fails to preserve some colimits one would very much like to consider in practice, in particular colimits which arise from gluing patches of an open covering:

\sum_{i, j} U_{i} \cap U_{j} \vec{\to} \sum_{i} U_{i} \to M .

More sophisticated sheaf models would repair such defects by considering not all presheaves on $S$ , but sheaves with respect to a Grothendieck topology on $S$ which takes into account such coverings.

The full development of this kind of approach was initiated by Lawvere in the late 60’s. Actually, Lawvere had in mind a development which would incorporate objects of nilpotent infinitesimals (the kind used by Grothendieck in algebraic geometry). The way this works is that one first expands $S$ to a larger category $C$ = $C^{∞}$ -Alg^{op} in which $S$ fully embeds. That is, there is a full and faithful functor

S \to C^{∞} - {Alg}^{op}

which sends $M$ to the $C^{∞}$ -algebra of smooth functions on $M$ . But $C^{∞}$ -Alg, the category of finitely presented (commutative) $C^{∞}$ -algebras, contains other more exotic objects, for example $C^{∞} (ℝ) / I$ where $I$ is the $C^{∞}$ -ideal generated by $x^{2}$ . The corresponding “smooth affine scheme” in $C^{∞} - {Alg}^{op}$ may be regarded as an infinitesimal object consisting of formal elements of $ℝ$ of square zero. It is the “generic tangent vector” $T$ , and one of the beauties is that the tangent bundle of an object $X$ may be constructed as as internal hom $X^{T}$ , which we have available by cartesian closure.

In a nutshell, a typical sheaf model will consider the presheaf topos consisting of functors

C^{∞} - Alg \to Set

and then cut back to sheaves with respect to some topology on $C^{∞} - {Alg}^{op}$ to force various other nice things to occur. Some of the fancier versions also faithfully reflect the use of invertible infinitesimals, à la Abraham Robinson in his nonstandard analysis. (I interpret Urs’s choice as just sticking to one of the simplest types of sheaf models, possibly with an option of switching to something fancier once things are a little more settled.)

There are a number of texts which treat this development. The one I am most familiar with is the one by Moerdijk and Reyes, reviewed here.

Posted by: Todd Trimble on January 4, 2008 12:24 PM | Permalink | Reply to this

Re: Comparative Smootheology

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Hi Andrew,

thanks for your comment!

You write:

I didn’t put in the “presheaves” version because I didn’t know of it. If someone sends me a precise definition then I’ll add it in

I had written the following long response, when I saw that Todd had pipped me. I’ll post it nevertherless. Here goes.

A presheaf is not much more than a contravariant functor. We are usually compelled to think of a contravariant functor as a presheaf if

- its domain is a category with the structure of a site: it’s objects have the essential properties which make them behave roughly like subsets of some topological space

- its codomain is the category of sets, or a category of sets with extra structure.

Sometimes people call any contravariant functor a presheaf. And since contravariant functors are in bijection with covariant functors, that pretty much amounts to calling every functor a presheaf.

But the point of presheaves is, as the name suggests, that some of them are in fact sheaves. And that is a condition which only makes sense when the domain is a site.

And another important point of presheaves is that some of them are representable, and this only makes sense when the codomain is a category that the domain is enriched over.

Let me see, I am starting explaining this from the wrong end now, it seems. But the point I am making so far is, to get started, that:

Generally speaking a presheaf is just any functor $f : S^{op} \to T$ , but usually we want the objects of $S$ to behave like open subsets and, unless we feel a little more sophisticated, want $T$ to be the category of sets.

Then the category of all presheaves on $S$ is simply the category of all contravariant functors from $S$ to $Set$ . We write this category of presheaves on $S$ as ${Set}^{S^{op}} .$

One important point of this is that the category $S$ itself sits inside the category of all presheaves on $S$ :

$Yon : S ↪ {Set}^{S^{op}}$

This is called the Yoneda embedding.

This is one of those almost completely tautological and yet immensely deep things that make abstract nonsense such a delight.

For understanding what presheaves mean, let’s quickly look at how Yoneda’s embedding works:

for every object $U \in Obj (S)$ we obtain a presheaf on $S$ by sending any other object $V$ of $S$ to the set of morphisms from $V$ into $U$ :

$V \mapsto {Hom}_{S} (V, U) .$

Notice that this is indeed a contravariant functor, since the Hom-functor is contravariant in its first argument.

The presheaves isomorphic to one of this form are called the representable presheaves. They are represented by the given object $U$ of $S$ .

What does this tell us? That’s important. It tells us that we should think of the set that any given presheaf $X : S^{op} \to Set$ assigns to any given object $U \in Obj (S)$ as the set of morphisms from $V$ into $X$ . Only that, for presheaves that don’t come from the Yoneda embedding, there is not really anything like a morphism from $V$ into $X$ .

So you should think of the presheaf $X : S^{op} \to Set$ as a rule that assigns to any open set $U$ the set of plots from $U$ to $X$ .

The fact that $X$ is a contravariant functor on $S$ then says that these plots behave sensibly under pullback along morphisms of objects of $S$ .

Moreover, if the presheaf is actually a sheaf, it has the special property that if a bunch of objects $U_{1}, U_{2}, \dots$ of $S$ “cover” an object $V$ in the way open subsets may cover one another, then the plots that $X$ assigns to $V$ are already fixed by the plots it assigns to the $U_{1}, U_{2}, \dots$ .

So you can see now how Chen-smooth spaces and diffeological spaces are examples of certain presheaves:

As I have said, presheaves on any category $S$ are “generalized objects of $S$ ”. More precisely: they are “things that may be probed by objects of $S$ ”.

So for Smootheology, we take one of the following choices:

- $S$ the category whose objects are ordinary (smooth) manifolds, and whose morphisms are ordinary morphisms of (smooth) manifolds.

- $S$ the category of open subsets of $ℝ \cup ℝ^{2} \cup \dots$ and smooth maps between these

- $S$ the category of open convex subsets of $ℝ \cup ℝ^{2} \cup \dots$ and smooth maps between these

Or some variant of that.

Chen-smooth spaces and diffeological spaces are what, it seems, should be called “quasi representable” presheaves on one of these sites, as I indicated in the above entry.

One nice aspect of realizing that Chen-smooth spaces and diffeological spaces are special kinds of presheaves is that we can make use of the immense amount of knowledge about presheaf categories.

Presheaf categories have a bunch of nice properties. They are cartesian closed, for instance. They are even topoi.

Useful for our purposes is the cartesian closedness. It says that the thing of morphisms between any two presheaves on $S$ is itself again a presheaf on $S$ .

For $X$ and $Y$ any two presheaves, the presheaf $hom (X, Y)$ is the one whose assignment of sets works as $hom (X, Y) : U \mapsto {Hom}_{{Set}^{S^{op}}} (U \times X, Y),$

where on the right $U$ denotes the presheaf represented by $U$ , and where the cartesian product $\times$ of presheaves is the componentwise one

$X \times Y : U \mapsto X (U) \times_{Set} Y (U) .$

You can easily check that this general – powerfully elegant – notion of internal homs in presheaves restricts to the ordinary internal hom for Chen-smooth spaces and diffeological spaces. I discuss that in a little more detail in my notes.

Posted by: Urs Schreiber on January 4, 2008 1:06 PM | Permalink | Reply to this

Re: Comparative Smootheology

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Thanks, Todd and Urs, for that; someone has finally given me a reason to be interested in (pre)sheaves!

There’s a lot to absorb there and the semester’s starting next week so it’ll take me a while to ponder all of it, and a bit longer to incorporate it into what I wrote.

One thing that strikes me immediately is that in saying why presheaves are so wonderful, you both start listing fantastic properties of the category, starting with completeness, cocompleteness, and closed. If I have an arbitrary category, in this case the category of smooth manifolds, there may be lots of ways to embed this in a category with lots of nice properties, such as those listed above. What makes one such enlargement better than another? This is perhaps not a good mathematical question but I would be interested your answers.

The category of Frölicher spaces is complete, cocomplete, and closed and I suspect that it is the “smallest” such category containing the category of smooth manifolds. Certainly it embeds in all of the set-based categories of smooth objects that I’ve encountered so far.

Perhaps a more leading question would be: why haven’t I managed to convince any of you that Frölicher spaces are worth considering? (I detest smileys, but if I didn’t then I would put one there which indicated a wry smile at that point (is there such a smiley? I don’t think I’ve ever seen one) just to show that I didn’t really expect to convince anyone of anything)

Anyway, as I said, the semester starts next week and I’m teaching functional analysis. Perfect opportunity to promote my point of view before they encounter any other!

Andrew

Posted by: Andrew Stacey on January 4, 2008 2:27 PM | Permalink | Reply to this

Re: Comparative Smootheology

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why haven’t I managed to convince any of you

I wouldn’t quite put it that way. As I said, I haven’t quite made up my mind yet. I do appreciate the the Frölicher definition has its charms.

To me, it seems, the most striking aspect of Frölicher spaces is that for them we have the chain rule.

For presheaves the chain rule holds only under very specific circumstances, as far as I am aware.

Posted by: Urs Schreiber on January 4, 2008 2:46 PM | Permalink | Reply to this

Re: Comparative Smootheology

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why haven’t I managed to convince any of you

I wasn’t actually trying to promote sheaf models over other possibilities; I was just putting out there some facts I know or have read. To be honest, I haven’t had a chance to study the Frölicher spaces approach.

If you look at the category of Frölicher spaces over a Frölicher space, Fröl/ $U$ , does that have good categorical properties too (in particular, is it cartesian closed)?

Since we’re not salesmen here, one should come right out and issue some warnings about the sheaf models approach. Maybe the worst is that it’s pretty hard to get one’s head wrapped around what these sheaves are really like, concretely. A lot of the experts on sheaf models seem to cheerfully counter with something like, “Don’t worry about the analytic complications of these models; it’s easier just to work with and prove theorems in the axiomatic theories which they model. You just have to be careful that your reasoning is intuitionistic: the law of excluded middle does not hold internally in these toposes.” I expect most potential customers would start to get a little nervous around talk like that; it’s sort of like saying, “Look, this is a wonderful car, you’re gonna love it, but whatever you do, don’t push this red button here.”

I should have a look at your paper.

Posted by: Todd Trimble on January 4, 2008 7:08 PM | Permalink | Reply to this

Re: Comparative Smootheology

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So you should think of the presheaf $X : S^{op} \to Set$ as a rule that assigns to any open set $U$ the set of plots from $U$ to $X$ .

Sorry, what’s a plot?

Posted by: Mike Stay on January 4, 2008 5:29 PM | Permalink | Reply to this

Re: Comparative Smootheology

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Sorry, what’s a plot?

Just a name for the elements in the set $X (U)$ which such a presheaf assigns to an object $U$ .

The point is that we think of these elements as “smooth maps from $U$ to $X$ ” only that there is a priori no smooth map, instead we are defining what a smooth map is by decree. To emphasize this, we say “plot” from $U$ to $X$ instead of “smooth map from $U$ to $X$ ”.

Or, in fact, Chen did so, back then, for the special case of his quasi-representable presheaves, where each plot is in fact a map of sets.

Posted by: Urs Schreiber on January 4, 2008 6:01 PM | Permalink | Reply to this

Re: Comparative Smootheology

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Okay, here goes.

Define the Fundamental Frölicher Category as the category with one object whose morphisms are $C^{∞} (ℝ, ℝ)$ .

A pre-Frölicher object is a triple $(C, F, ev)$ where $C$ is a covariant functor from $FFC$ to the category of sets, $F$ is a contravariant functor from $FFC$ to the category of sets, and $ev$ is a natural transformation of functors ${FFC}^{op} \times FFC \to Set$ from $F \times C$ to the hom functor. A morphism of pre-Frölicher objects is a pair of natural transformations making the obvious diagram commute.

A refinement of a pre-Frölicher object, $X_{1}$ , is a pre-Frölicher object $X_{2}$ and a morphism $X_{1} \to X_{2}$ for which both of the maps in the natural transformation are injective.

A Frölicher object is a pre-Frölicher object with no non-trivial refinements. A morphism of Frölicher objects is simply a morphism of them as pre-Frölicher objects.

Don’t know what properties this category would have because I just invented it while collecting the boxes for the Christmas decorations (I think I might be a troll - my brain works better at subzero temperatures (the boxes were in the cellar)). I’ll happily think more about it on Monday morning but just thought I’d share the idea with you all to see whether it’s worth thinking about some more.

Andrew

PS This was actually my second idea and I like it much more than my first. However if no one likes this idea I’ll try out the other on you before I climb back under my bridge.

Posted by: Andrew Stacey on January 4, 2008 9:45 PM | Permalink | Reply to this

Sheaves and Smootheology

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I’ve been thinking a bit about the sheaf versions and here are some thoughts. Hopefully they glue together to give something coherent.

I think that the inclusion of quasi-representable presheaves in the category of all presheaves has both a left and right adjoint. This should work for any of the suggested source categories (smooth manifolds, open subsets of Euclidean spaces, convex subsets of Eulidean spaces, or my FFC category described elsewhere).

The left adjoint is fairly simple. For each of the source categories we can define “constant plots”. The only one where we don’t have an object which is a singleton set is the FFC but we can still define constant curves here. Given an arbitrary contravariant functor, $P : S \to Set$ , we define the underlying set of $P$ , say $P_{c}$ , to be the set of all constant plots. Any element in $P (s)$ can be viewed as a map $s \to P_{c}$ in the obvious way. This defines an associated “quasi-representable” presheaf. Putting this together in the obvious way yields the left adjoint.

The right adjoint is more complicated. For the natural morphism $P \to P_{c}$ to fail to be an isomorphism, it must fail to be injective and so there must be two distinct plots in $P$ which define the same constant plots. What we need to do is add in extra constant plots so that they no longer do so. To be the adjoint, we need to ensure that we add in the minimum possible extra points.

Let $x$ be a constant plot. Consider the set of plots $ϕ$ such that $x$ factors through $ϕ$ . Write this as $P_{x}$ . Define a quasi-ordering on $P_{x}$ by $ϕ ⪯ ψ$ if the germ of $ϕ$ at $x$ factors through $ψ$ . That is to say, there is a neighbourhood of $x$ in the domain of $ϕ$ such that the restriction of $ϕ$ to this neighbourhood factors through $ψ$ . We then take the union over the constant plots of the sets of maximal disjoint unions of maximal chains of the $P_{x}$ ’s.

A maximal chain in $P_{x}$ corresponds to a constant plot with controversy removed; i.e., wherever there was a choice for an ambient plot we made a decision. Two such chains are disjoint if the germs don’t interact but we regard the points as the same. Such chains are independent in that controversy in one chain has no effect on the other. An example where we need this is where the point in question is the origin and the ambient space is the union of the $x$ and $y$ axes.

I think that this gives the right adjoint.

The adjoints are, respectively, ignoring controversy and resolving controversy.

As I said above, a presheaf is not quasi-representable if are two (or more) plots which look the same when probed with constant plots but which are (declared) different. As an example, consider the following presheaf. Start with two copies of $ℝ$ . The corresponding presheaf assigns to an object $s$ in our source category (whatever it is) the set $C^{∞} (s, ℝ) ⨿ C^{∞} (s, ℝ)$ . Now identify all constant plots in one part with their corresponding plot in the other part (depending on one’s definition of “constant plot” this may necessitate identification of some other plots; i.e., any that factor through a constant plot). The resulting presheaf is not quasi-representable.

So far, so good. The above applies to any of the theories. Now let us add in the Frolicher requirement that input and output should be somehow linked. I gave a suggestion elsewhere in this discussion as to how one might do this (that suggestion doesn’t work, but no one has pointed out the flaw so I’m guessing no one has looked at it in great detail; that’s not important here, though) but the actual implementation isn’t important. What we need is the general philosophy that “functionals and curves determine each other in some fashion”.

We certainly need to start with a presheaf $C$ (“plots”) and a copresheaf $F$ (“coplots”) and a composition $C \times F \to C^{∞} (-, -)$ (I’m not going to assume a particular source category here for maximum generality). One aspect of the Frolicher viewpoint says that this composition is my only source of pertinent information.

Suppose I have two plots, $ϕ$ and $ψ$ , on the same domain, say $s$ , which yield the same constant plots. Let $f$ be a coplot with codomain $t$ . The compositions $f \circ ϕ$ and $f \circ ψ$ are both in $C^{∞} (s, t)$ . Now as this is simply regular smooth maps between two standard smooth objects, such a map is completely specified by its restrictions to points, i.e. constant maps. Hence $f \circ ϕ$ and $f \circ ψ$ are the same map. Thus $ϕ$ and $ψ$ are Frolicher indistinguishable so why do we formally distinguish between them?

Notice that we haven’t used the “saturation” part of Frolicher’s philosophy. If we did, things would get even more bizarre! For then we could take an arbitrary subset of the domain of $ϕ$ , say $a$ , and define a new plot $θ$ by declaring $θ$ to be $ϕ$ on $a$ and $ψ$ on $s ∖ a$ . This plot would be Frolicher-indistinguishable from $ϕ$ and $ψ$ and so, by saturation (however that is interpreted), would have to be a plot.

I don’t know about you lot but that seems to be a little bit strange.

Strange or not, what it does say is that if one takes the map from an arbitrary presheaf to the nearest quasi-representable one and looks at the fibres of this then those fibres have no interesting structure whatsoever. So one may as well just consider quasi-representable presheaves (i.e., Frolicher spaces) and be done with it.

Of course, without the saturation requirement then there may be “interesting” structure on the fibres but in effect it is only interesting because it has been declared to be interesting and not because it is intrinsically interesting.

Andrew

Posted by: Andrew Stacey on January 11, 2008 10:09 AM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Andrew Stacey wrote:

what it does say is that if one takes the map from an arbitrary presheaf to the nearest quasi-representable one and looks at the fibres of this then those fibres have no interesting structure whatsoever. So one may as well just consider quasi-representable presheaves (i.e., Frolicher spaces) and be done with it.

Hm, that’s interesting.

I was distracted in the middle of this discussion and am now coming back to it. I want to understand this argument by Andrew better.

One remark: don’t these conclusions all follow from the assumption that I can put the composition operation $C \times F \to C^{∞} (-, -)$ on my pair (presheaf,copresheaf)?

In any case, I would like to see a concrete example. Can we do that?

I would like to start with the apparently not quasi-representable presheaf

$X_{g}$ for any Lie algebra $g$ , which assigns to each test domain $U$ the collection of flat $g$ -valued 1-forms on $U$ :

$X_{g} : U \mapsto {Hom}_{dg - algebras} (CE (g), Ω^{•} (U))$

i.e. the set of all elements $A \in Ω^{1} (U) \otimes g$ which satisfy $d A + [A \land A] = 0,$ where $[., .]$ is the Lie bracket on $g$ .

This presheaf looks like it is not quasi-representable.

Can you find me a

nearest quasi-representable presheaf and look at the fibers of this [showing that] those fibers have no interesting structure whatsoever

That would be helpful.

Posted by: Urs Schreiber on January 23, 2008 5:39 PM | Permalink | Reply to this

Re: Sheaves and Smootheology

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I think that there are two themes to explore from Urs’ comment above (is it worth splitting them into two threads?).

Firstly,

One remark: don’t these conclusions all follow from the assumption that I can put the composition operation $C \times F \to C^{∞} (-, -)$ on my pair (presheaf,copresheaf)?

Yes. However, earlier Urs said:

I am thinking that maybe this is telling us that we eventually may want to consider things that are pairs consisting of a presheaf and a co-presheaf on manifolds, compatible in some way.

My natural transformation was an attempt to make sense of the compatibility requirement. The idea is that if $C$ is meant to represent maps into the object and $F$ maps out then at the very least one ought to be able to “compose” a map in with a map out. Moreover, as we are trying to pin-point the notion of smoothness, the composition of a map in with a map out ought to be a smooth map between our test spaces.

(Incidentally, to return to my original point, merely specifying the maps in allows one to consider things strictly weaker than smoothness. For example, if I take the family of all continuous maps from my source objects into some manifold then I get a Chen space or diffeological space or whatever space. But by doing this I have somehow smothered the smooth structure. By having both the maps in and the maps out then I ensure that I am examining smooth, the whole smooth smooth, and nothing but the smooth.)

Given just a presheaf then I can define a compatible copresheaf simply by taking the set of all natural transformations from this presheaf to the presheaf $C^{∞} (-, ℝ)$ .

So my point is that this composition should be considered to be part of the initial structure and not something tacked on afterwards. At least, for a Frölicher-like definition.

Okay, so to Urs’ next point. I would very much like to understand this example. I’m not so bothered about where it came from (at least, as far as this discussion goes) but I think it would be useful to know how Urs thinks of this as a “smooth object”.

In my mind, when someone says “Here’s a presheaf, say $C$ , on the category of stuff. This is a ‘smooth object’.” then I want to think of $C (X)$ as being ‘maps from $X$ into … something’. The ‘something’ may not even be a set, but nonetheless we can probe it with $C (X)$ . Even if $C (X)$ has some other meaning, when someone says, “This is a smooth object.”, I want that to mean, “These are (generalised) plots.”.

So I think of an element of $C (X)$ as a map $c : X \to ?$ . I can maybe reconstruct $?$ by patching together the images of these maps. Namely, by starting with

(1)

\underset{X}{∐} \underset{c \in C (X)}{∐} X

and then quotienting out by some suitable equivalence relation.

I guess the idea is that I have an invisible object in the room and I want to see what it is. I can shove bits of paper up against it and trace its outline. Problem is, it’s quite big, so I need to use lots of bits of paper and these will overlap. But if I use enough paper I will eventually see the shape of the whole thing.

So my question regarding Urs’ example is this: how am I to think of elements of ${Hom}_{dg - algebras} (CE (𝔤), Ω^{•} (U))$ as maps from $U$ ?

I want the definition of a “smooth object” to be like a cheap airline flight: no baggage allowed.

I’m going to think a little more about this example, this is just my initial thoughts and question.

Here’s another random thought. For each object in my test category I get an obvious presheaf: $C^{∞} (-, X)$ . For a presheaf $C$ to be a “smooth object” then the ‘plots’ in $C (X)$ should be precisely the natural transformations from $C^{∞} (-, X)$ to $C$ . I suspect that this is not automatic and so would need to be an additional assumption.

Finally, here’s a confession: I am not a fully paid up member of the category party (despite Bruce’s efforts over the last couple of years). I accept that it may well be a good thing to say, “Let’s find the definition that is simplest in categorical terms.” (I dispute the assertion that the last three words are redundant, but that’s irrelevant), in which case “presheaf on the category of smooth manifolds” seems to fit the bill. However, I feel that the designation of “category of generalised smooth objects” is something that needs to be justified a little more strenuously than “look, I can write down the definition in seven words!” (paraphrasing Urs here a little!). In other words, I am quite happy with the statement

Presheaves on the category of smooth manifold (or open subsets of Euclidean spaces, or convex subsets of Euclidean spaces) behave a little like smooth manifolds so let’s see what we can generalise from the one to the other.

What I am uncomfortable with (and let me put it no stronger than that) is the statement

The category of presheaves on the category of manifolds (or whatever) is the category of generalised smooth objects.

Posted by: Andrew Stacey on January 25, 2008 10:26 AM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Andrew,

thanks for this comment. I very much agree with what you say, including that there is an issue of pure taste here, on which we may differ right this moment (but tastes may change), but which shouldn’t prevent us from making progress on understanding some interesting cross-relationships between various ideas.

I very much appreciate your insistance on the Frölicher idea. I did mention the idea of having compatible pre-sheaf and co-presheaves for this reason, and am glad that you picked that idea up.

So what I am trying to understand here is: are those presheaves which made me pass from quasi-representables to general presheaves, are they

- possibly naturally equipped with a co-presheaf structure satisfying some Frölicher-like compatibility condition?

- possibly then, following your argument, actually “close” to quasi-representable presheaves in some useful sense.

That would be good!

So let me say more about these presheaves, since you are asking how I am thinking about them as smooth objects probed by ordinary smooth subsets.

These things should be smooth versions of classifying spaces of Lie $n$ -groups.

Consider the simplest one which I mentioend: for $g$ an ordinary Lie algebra, the presheaf

$X_{g} : U \mapsto flat g - valued 1 - forms on U .$

How is that the classifying space $B G$ of the simply connected Lie group $G$ which integrates $G$ ?

In this sense (what I am saying now I think is true, but still needs to be written out cleanly and properly, so handle with a bit of care):

we can form the fundamental path groupoid of $X_{g}$ , i.e. we can “loop” it.

To do so, we look at all smooth maps of generalized smooth spaces:

$[0,1] \to X_{g}$

and divide out homotopies between them. In the obvious manner: two such maps are homotopic, if there is a smooth map

$[0,1]^{2} \to X_{g}$

which interpolates between them, in the obvious manner.

The claim is, forming the fundamental groupoid of $X_{g}$ this way amounts to doing nothing but performing the “integration without integration” of the Lie algebra $g$ to the Lie group $G$ which we discussed for instance here.

You can see this as follows:

by looking at the definitions, you find that smooth maps

$[0,1] \to X_{g}$

are nothing but choices of a $g$ -valued 1-form on the interval.

A homotopy between two such smooth maps is then nothing but a flat (here is where that condition comes in) smooth 1-form on $[0,1]^{2}$ , which restricts to the two given ones on the boundary.

By the nonabelian Stokes theorem, we know that two $g$ -valued 1-forms on $[0,1]$ have the same $G$ -valued parallel transport (“holonomy”) over the interval if they are related by a flat $g$ -valued 1-form on $[0,1]^{2}$ this way.

So we find: a homotopy class of smooth maps from the interval into $X_{g}$ is nothing but an element of the group $G$ .

So $X_{g}$ is, apparently, a smooth model for the classifying space $B G$ of $G$ .

And that, actually, suggests that you are right, and that there might be a quasi-represenabtle presheaf isomorphic or otherwise closely related to $X_{g}$ : namely one which comes from a set-version of $B G$ which is equipped with a smooth structure somehow.

Let’s think about that! That would be interesting.

Posted by: Urs Schreiber on January 25, 2008 1:10 PM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Let me say one more thing about the generalized smooth space given by the presheaf $X_{g} : U \mapsto {Hom}_{dg - alg} (CE (g), Ω^{•} (U)),$ which I said is like a smooth version of the classifying space $B G$ of the simply connected Lie group $G$ integrating the Lie algebra $g$ .

One striking aspect is, that is has just a single point ${pt} \to X_{g}$ since there is just a single flat $g$ -valued one form on the point: the vanishing one.

Still, there are many paths $[0,1] \to X_{g}$ in this space, one for each $g$ -valued 1-form on the interval.

This makes it quite vivid how $X_{g}$ fails to be quasi-represenable: by the axiom that every constant map is a plot, a quasi-representable presheaf has one point for each element of its underlying set.

The fact that $X_{g}$ has a single point also says that there is an operation of composition defined globally on all paths in $X_{g}$ : all paths have the same start- and endpoint.

That’s how we find that the fundamental groupoid of $X_{g}$ is indeed just a group! The fundamental groupoid has a single object, and is still a nontrivial groupoid. (That’s not true for fundamental groupoids of ordinary manifolds, of course.)

The fundamental groupoid of $X_{g}$ , which has a single object, is indeed the one-object groupoid version of the group $G$ . Which I write $B G$ .

Posted by: Urs Schreiber on January 28, 2008 7:09 PM | Permalink | Reply to this

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I’d like to simplify this example a little. I think that the Lie algebra bit is extraneous for the purposes of this discussion. So are the general forms. I propose looking at the presheaf:

(1)

X \mapsto Ω^{1} (X)

We could define this on any reasonable category of genuine smooth objects (i.e. manifolds, open subsets of Euclidean spaces, Euclidean spaces themselves, or (locally) convex subsets of Euclidean spaces), I don’t think that that is important at this stage. It’s a bit simpler than Urs’ example above but, I think, contains the kernel of the matter: the elements of $Ω^{1} (X)$ are maps from $X$ , but the target varies with $X$ and so we may not be able to paste it together to make it quasi-representable.

So, two things that I would like to explore for this example.

Can we make this into something a little more Frolicher-like? Namely, can we add in a copresheaf and suitable compatibility relations? Although we can use the fact that we know that this presheaf is 1-forms to get a form of an answer, the answer should in the end be independent of this fact. Also, can we reduce the domain somewhat?
Can we find, essentially, a representing space? If so, how does it related to the original presheaf?

How does this sound, Urs? Is this a reasonable simplification or har jeg slått barnet ut med badevannet? Any other questions that you can think of?

Posted by: Andrew Stacey on January 30, 2008 9:23 AM | Permalink | Reply to this

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I’d like to simplify this example a little.

Good idea. I was thinking of mentioning this, too, and in fact just did over in the other thread, over there. But let’s discuss it here.

I propose looking at the presheaf $U \mapsto Ω^{1} (U) .$

May I change this to $U \mapsto Ω_{closed}^{1} (U) .$ ?

That’s the presheaf I am calling $X_{CE (u (1))} .$ It comes from the Lie algebra $u (1)$ .

We could define this on any reasonable category of genuine smooth objects (i.e. manifolds, open subsets of Euclidean spaces, Euclidean spaces themselves, or (locally) convex subsets of Euclidean spaces), I don’t think that that is important at this stage.

Yes, exactly. I kept changing my mind a bit lately about which kind of test domains would be most convenient. Currently I am thinking that open subsets of Euclidean spaces is a convenient choice. It has a couple of computational advantages over admitting all manifolds. But it shouldn’t really matter much.

are maps from $X$ , but the target varies with $X$

I think I see where you are coming from here, but personally I wouldn’t put it like this. I gave a heuristic interpretation of this presheaf here.

But in any case, it is not quasi-representable, yes. Another question is if maybe it is isomorphic, in some suitable sense (for instance possibly after passing to cohomologies) to a quasi-representable one.

Can we make this into something a little more Frolicher-like? Namely, can we add in a copresheaf and suitable compatibility relations? Although we can use the fact that we know that this presheaf is 1-forms to get a form of an answer, the answer should in the end be independent of this fact. Also, can we reduce the domain somewhat?

I was thinking about that, too. This is what I could say:

One way to think of this presheaf is that it assigns to each test domain $U$ the set of algebroid morphisms from the tangent algebroid of $U$ to the Lie algebra of $U (1)$ $T X \to u (1) .$ Which is the same as the set of dg-algebra morphisms between the corresponding Chevalley-Eilenberg algebras $Ω^{•} (X) \leftarrow CE (u (1)) .$

So there is an obvious co-presheaf waiting to be identified here, namely not

$X_{CE (u (1))} : U \mapsto {Hom}_{algebroids} (T U, u (1)) = {Hom}_{DGCAs} (CE (u (1)), Ω^{•} (U)) = Ω_{closed}^{1} (U)$

but

${coX}_{CE (u (1))} : U \mapsto {Hom}_{algebroids} (u (1), T U) = {Hom}_{DGCAs} (Ω^{•} (U), CE (u (1))) .$ On general grounds, this seems to be like the thing to do.

So what are the dg-morphisms from $Ω_{closed}^{1} (U)$ to $CE (u (1))$ ? The latter contains just a single degree 1 generator which is closed. So such a morphism is just an algebra homomorphism from $Ω^{1} (U)$ to the dg-algebra on a single degree 1-generator. But that’s just a point in $U$ .

I am not entirely sure what to make of that at the moment…

Can we find, essentially, a representing space? If so, how does it related to the original presheaf?

Yes, that’s something we should try to figure out. I am thinking that $X_{CE (u (1))}$ should be a model for $K (U (1),1)$ . If so, there should be a quasi-representable smooth model of $K (U (1),1)$ , along the lines John talks about here to which it is closely related.

Posted by: Urs Schreiber on January 30, 2008 11:16 AM | Permalink | Reply to this

Re: Sheaves and Smootheology

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A small comment on the above:

Currently I am thinking that open subsets of Euclidean spaces is a convenient choice. It has a couple of computational advantages over admitting all manifolds. But it shouldn’t really matter much.

There’s a sense in which it doesn’t matter at all.

Lawvere once remarked that the category of (finite-dimensional) smooth manifolds is the idempotent-splitting completion of the full subcategory of open subsets of Euclidean space. This means that every smooth manifold $M$ is the image of some smooth idempotent map $p : U \to U$ operating on an open set $U$ , so that $p$ splits as

U \overset{r}{\to} M \overset{i}{\to} U

where $(r, i)$ is a retraction-inclusion pair. (Conversely, every idempotent $p$ factors in this way in the category of smooth manifolds.)

The thrust of his remark is that whenever you want to define some functor on the category of smooth manifolds, say for example De Rham cohomology

H^{n} (-) : {Man}^{op} \to Vect,

it suffices to do it just on the open sets $U$ , provided that in the receiver category (in this case $Vect$ ), all idempotents split. The extension of the functor (defined on the category of open sets) to all manifolds will be unique up to isomorphism.

For the same reason, every functor

{Open}^{op} \to Set

extends uniquely (up to isomorphism) to a functor

{Man}^{op} \to Set

so that there is an equivalence of toposes

{Set}^{i^{op}} : {Set}^{{Man}^{op}} \tilde{\to} {Set}^{{Open}^{op}}

induced by the inclusion $i : Open ↪ Man$ .

This remark fits generally in the context of Morita theory, or the theory of Cauchy completions, as we have talked about elsewhere on this blog.

Posted by: Todd Trimble on January 30, 2008 2:31 PM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Todd,

thanks a lot. That’s helpful.

What can be said in this regard with respect to open contractible and/or open convex subsets of Euclidean spaces, the other two popular kinds of test domains in this busines?

Posted by: Urs Schreiber on January 30, 2008 2:54 PM | Permalink | Reply to this

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I think for the category of open contractibles, the Cauchy completion is just the category of contractible manifolds (without boundary). At any rate, a retract of a contractible space is contractible, so Cauchy completion doesn’t take you outside the world of contractible spaces.

From the standpoint of Cauchy (or idempotent-splitting) completion, I’m having a hard time telling apart open contractibles and open convex sets. Is every open contractible diffeomorphic to an open convex set? Are they even all diffeomorphic to standard open balls?

Posted by: Todd Trimble on January 30, 2008 4:05 PM | Permalink | Reply to this

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But can we say anything about the relation between presheaves on open subsets versus presheaves on contractible open subsets?

We don’t have your above argument to show that they are equivalent. But does it mean they will be inequivalent?

I am hoping they are in fact equivalent. Maybe only if we restrict attention to actual sheaves (as opposed to mere presheaves)?

Posted by: Urs Schreiber on January 30, 2008 5:29 PM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Sorry, Urs, I should have been clearer. Those two toposes are inequivalent.

The Morita theorem which lies in the background is that there is an equivalence

{Set}^{C^{op}} ≃ {Set}^{D^{op}}

iff $C$ and $D$ are Morita equivalent, meaning that their idempotent-splitting completions are equivalent categories. Speaking somewhat loosely (although it can be made precise), that would mean that retracts of objects of $C$ are isomorphic to retracts of objects of $D$ .

But retracts of contractible opens are still contractible, whereas retracts of general opens obviously need not be contractible. So those two presheaf categories are not equivalent.

Posted by: Todd Trimble on January 30, 2008 6:01 PM | Permalink | Reply to this

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You also asked a companion question about whether we get an equivalence when we restrict to sheaves on opens and sheaves on contractible opens. This I don’t know.

First I’m not sure if you have specific topologies (on the sites) in mind. But second, I don’t know, off the bat, nice general methods for detecting ‘Morita equivalence of sites’ (i.e., when sheaf toposes are equivalent). I’d really have to think hard about that one (and no guarantee I’d come up with anything useful).

Posted by: Todd Trimble on January 30, 2008 7:31 PM | Permalink | Reply to this

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First I’m not sure if you have specific topologies (on the sites) in mind.

Hm, apparently I am missing something then: aren’t there “obvious” “canonical” Grothendieck topologies on these categories of subsets of Euclidean spaces?

I was tacitly assuming there are, without having really thought much about it.

Posted by: Urs Schreiber on January 30, 2008 8:00 PM | Permalink | Reply to this

Re: Sheaves and Smootheology

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Oh sure! There are ‘canonical’ topologies (with a technical meaning of ‘canonical’), and there are often ‘obvious’ topologies (as in the cases we’re looking at). There are typically many topologies; to play it safe, one should specify which one. In fact, it probably wouldn’t hurt to talk a little about topologies.

‘Canonical topology’ in the technical sense has a somewhat dry definition: a topology on a small category $C$ is subcanonical if every representable $hom (-, c)$ is a sheaf for that topology. The canonical topology is then by definition the largest subcanonical topology. It takes some practice to get a sense of what it may look like in actual examples.

There is also, as you say, an ‘obvious’ topology for each of the categories we’re looking at here, namely the ‘open covering’ topology. And, as it happens, I think the (informally) ‘obvious’ topology coincides with the canonical one.

Maybe it’s a good idea if I recall some definitions (for anyone who might need them). Let $C$ be a small category.

A sieve on an object $c$ of $C$ is a subfunctor of a representable ${hom}_{C} (-, c) : C^{op} \to Set$ .
A Grothendieck topology on a small category $C$ assigns to each object $c$ a collection of sieves on $c$ which are called covering sieves for that topology. These must satisfy certain axioms which I won’t go into right now, but which can be found in any book on topos theory. A site is a small category equipped with a Grothendieck topology.
A sheaf with respect to a Grothendieck topology is a presheaf which ‘thinks’ that covering sieves really are coverings. ;-)

This last definition obviously requires some amplification, which I’ll explain with an example. Let’s take the open covering topology (on the category of open sets in Euclidean space, and smooth mappings between them). Here, a covering sieve on an open $U$ is by definition a subfunctor (an embedding $j$ ) of the form

⋃_{i} hom (-, U_{i}) \overset{j}{↪} hom (-, U)

where the $U_{i}$ are open subsets which cover $U$ in the usual sense. The union here is a coequalizer of the form

\sum_{i, j} hom (-, U_{i}) \times_{hom (-, U)} hom (-, U_{j}) \vec{\to} \sum_{i} hom (-, U_{i}) \to ⋃_{i} hom (-, U_{i})

which in our case simplifies to a coequalizer of the form

\sum_{i, j} hom (-, U_{i} \cap U_{j}) \vec{\to} \sum_{i} hom (-, U_{i}) \to ⋃_{i} hom (-, U_{i}) .

A sheaf, then, is a presheaf $X$ which can’t tell the difference between the left and right sides of the inclusion $j$ which marks a covering sieve, or in other words ‘thinks’ $j$ is an isomorphism, in the sense that ‘probing’ maps

hom (-, U) \to X

are in natural bijection with maps

⋃_{i} hom (-, U_{i}) \to X

by restriction along $j$ . Now our good friend Yoneda tells us that the set of maps $hom (-$

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January 3, 2008