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March 6, 2008

Space and Quantity

Posted by Urs Schreiber

Am preparing some notes which are supposed to wrap up the discussion on smooth spaces, smooth function algebras, smooth algebras of differential forms, etc, which we had in

Transgression, Comaparative Smootheology, Question on Smooth Functions

and further develop it. Here is the current status:

Spaces and Differential Forms
(pdf).

I haven’t indicated any author names at this point. I have typed this so far, but, as you all know, this draws heavily on plenty of remarks by Todd Trimle and Andrew Stacey. Most of the proofs currently appearing are simply transcripts of proofs Todd described. (Of course all mistakes in the document are mine.) I also benefitted from discussing this stuff with Bruce Bartlett in person.

Posted at March 6, 2008 8:40 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1624

14 Comments & 4 Trackbacks

Re: Space and Quantity

I have restructured a bit and added a section (number 2 at the moment) on the general abstract nonsense of space and quantity. I followed Lawvere’s Taking categories seriously, but varied slightly. Please check.

(I haven’t said anything about the dotted arrows that Lawvere has on p. 17. I guess these are supposed to allude to the kind of “saturation” or “completion” operation which we keep talking about?)

I have added a statement, right now proposition 5, which says that for all C DGCAsC^\infty DGCAs AA and gg any L L_\infty-algebra we have

CE(g) ACE(g)A. CE(g)\otimes_\infty A \simeq CE(g) \otimes A \,.

I think this is essentially obvious, using the fact that the GCA underlying CE(g)CE(g) is freely generated in positive degree. But please check.

Posted by: Urs Schreiber on March 7, 2008 3:14 PM | Permalink | Reply to this

Re: Space and Quantity

space.pdf

I have tuned and expanded a bit the intro on the general abstract nonsense about conjugation and ambi-duality.

Have discarded the notion of C DGCAC^\infty DGCA alltogether, since it turned out to be less obvious than I thought it would – and since I realize that I don’t actually need it!

Added two quick examples for the integration theory of L L_\infty-algebras.

Stated the definition of the integral ΣA \int_\Sigma A

of a general L L_\infty-algebra valued form AA on the smooth space XX over smooth space Σ\Sigma.

(Compare the notes on Integration without integration #. More to come later.)

Posted by: Urs Schreiber on March 11, 2008 3:41 PM | Permalink | Reply to this
Read the post Slides: On Nonabelian Differential Cohomology
Weblog: The n-Category Café
Excerpt: On the notion of nonabelian differential cohomology.
Tracked: March 14, 2008 4:01 PM

Re: Space and Quantity

I was asked by private email how to define vector fields and curves integrating them on smooth spaces conceived in terms of sheaves.

I think this way:

using the internal hom, both the space of forms as well as the space of functions on our smooth space XX is again a smooth space.

Then I say that a vector field on X is a smooth C (X)C^\infty(X)-linear map

v:Ω 1(X)C (X). v : \Omega^1(X) \to C^\infty(X) \,.

A curve

γ:X \gamma : \mathbb{R} \to X

is parallel to this vector field if the diagram

Ω 1(X) v C (X) γ * γ * Ω 1() t C () \array{ \Omega^1(X) &\stackrel{v}{\to}& C^\infty(X) \\ \downarrow^{\gamma^*} && \downarrow^{\gamma^*} \\ \Omega^1(\mathbb{R}) &\stackrel{\partial_t}{\to}& C^\infty(\mathbb{R}) }

commutes, where t\partial_t denotes the canonical vector field on the real line.

Posted by: Urs Schreiber on March 18, 2008 3:17 PM | Permalink | Reply to this

Re: Space and Quantity

Urs, this is really clever. I knew you’d have a nice way to do it!

Let me see if I can fill in the details. If XX and YY are smooth spaces, then Hom(X,Y)Hom(X,Y) is a smooth space, by setting

Hom(X,Y)(U)=...um...theset{θ U:X(U)Y(U),θisanaturaltransformationfromXtoY} Hom(X,Y)(U) = ...um ... the set \{ \theta_U : X(U) \to Y(U), \theta is a natural transformation from X to Y\} ?

Also, C(X)C(X) acts on Omega 1(X)Omega^1(X) via

Hom(X,)×Hom(X,Ω 1)Hom(X,Ω 1) Hom (X, \mathbb{R}) \times Hom(X, \Omega^1) \to Hom(X, \Omega^1) um….?

Can you help me out here?

Posted by: Bruce on March 18, 2008 3:28 PM | Permalink | Reply to this

Re: Space and Quantity

Okay, you are asking for how the canonical closed monoidal structure on (pre)sheaves works.

Here is how:

for X and Y two presheaves, their tensor product is

X×Y:UX(U)× SetY(U) X \times Y : U \mapsto X(U) \times_{Set} Y(U)

and their hom thing is

hom(X,Y):UHom(X×U,Y). hom(X,Y) : U \mapsto Hom(X \times U, Y) \,.

The structure of a differential graded commutative algebra on the sheaf

Ω (X):=hom(X,Ω ) \Omega^\bullet(X) := hom(X,\Omega^\bullet)

is induced from the DGCA structure on each test domain (all operations defined on test domains in the obvious sense).

The wedge product in particular restricts to the product of 0-forms on 1_forms.

Typing is hard for me on this French layout keyboard here, so I’ll be that brief, sorry. But let me know what you think.

Posted by: Urs Schreiber on March 18, 2008 3:43 PM | Permalink | Reply to this

Re: Space and Quantity

Abject apologies for being a bit late to this party; had a bit of trouble getting a pumpkin (can’t be a Norwegian myth).

All of the suggested starting categories have a separator and this has serious consequences for Isbell conjugation. (To throw a link to the discussion over there, after working some of this out on paper I tried a few MathSciNet searches to see if anyone else had come up with it; several of the searches I tried used MSC codes to try to restrict the results to something useful. For some reason, the words “conjugation” and “dual” are fairly common in mathematics.)

So let 𝒞\mathcal{C} be a category with a separator, say SS. Let XX be a presheaf on 𝒞\mathcal{C} (oh how I’d love to be able to do macros in iTeX); that is, as I’ve learnt, a contravariant functor from 𝒞\mathcal{C} to SetSet.

For objects A,BA, B in 𝒞\mathcal{C} we therefore get a set map

𝒞(A,B)Set(X(B),X(A)) \mathcal{C}(A,B) \to Set(X(B), X(A))

functorial in both AA and BB. We can turn this around slightly to get a set map

X(B)Set(𝒞(A,B),X(A)) X(B) \to Set(\mathcal{C}(A,B), X(A))

still functorial in both AA and BB. In terms of elements (gosh, did I really use that word?), this is

x(fX(f)x) x \mapsto \big( f \mapsto X(f)x \big)

In particular, for a 𝒞\mathcal{C}-morphism g:BCg : B \to C we get a commutative diagram

X(C) Set(𝒞(S,C),X(S)) X(g) X(B) Set(𝒞(S,B),X(S)) \begin{aligned} X(C) & \overset{\quad\quad}{\to} && Set(\mathcal{C}(S,C), X(S)) \\ X(g) \downarrow & && \downarrow \\ X(B) & \overset{\quad\quad}{\to} && Set(\mathcal{C}(S,B), X(S)) \end{aligned}

where the right-hand vertical arrow is the obvious one. Hence we can define a new functor Y:𝒞 opSetY : \mathcal{C}^{op} \to Set by

Y(B)=Im(X(B)Set(𝒞(S,B),X(B))) Y(B) = Im\big( X(B) \to Set(\mathcal{C}(S,B), X(B)) \big)

There is an obvious natural transformation XYX \to Y which consists of surjections.

Let us denote Isbell conjugation by primes. From the natural transformation XYX \to Y we obtain a natural transformation YXY' \to X'. I claim that this is a natural isomorphism.

Let us start with injectivity.

Let BB be an object of 𝒞\mathcal{C} and let fX(B)f \in X'(B). This is a natural transformation XB *X \to B^* so for an object AA of 𝒞\mathcal{C} we have a map of sets

f A:X(A)B *(A)=𝒞(A,B). f_A : X(A) \to B^*(A) = \mathcal{C}(A,B). In particular, we get f S:X(S)𝒞(S,B)f_S : X(S) \to \mathcal{C}(S,B).

Suppose that f,gX(B)f, g \in X'(B) are such that fgf \ne g. Then there is some object AA of 𝒞\mathcal{C} such that f Ag A:X(A)𝒞(A,B)f_A \ne g_A : X(A) \to \mathcal{C}(A,B). As these are set maps, there is some xX(A)x \in X(A) such that f A(x)g A(x)𝒞(A,B)f_A(x) \ne g_A(x) \in \mathcal{C}(A,B). These are now morphisms in 𝒞\mathcal{C}. As SS is a separator, there is some morphism a:SAa : S \to A such that f A(x)ag A(x)af_A(x)a \ne g_A(x)a. Now as ff and gg are natural transformations,

f A(x)a=𝒞(a,B)f A(x)=f S(X(a)x) f_A(x)a = \mathcal{C}(a, B) f_A(x) = f_S (X(a)x)

and similarly for gg. Hence the element X(a)xX(S)X(a)x \in X(S) distinguishes f Sf_S and g Sg_S. Thus the map

X(B)Set(X(S),𝒞(S,B)),ff S X'(B) \to Set(X(S), \mathcal{C}(S,B)), \quad f \mapsto f_S

is injective.

This holds with YY in place of XX and we have a commuting diagram.

X(B) Set(X(S),𝒞(S,B)) Y(B) Set(Y(S),𝒞(S,B)) \begin{aligned} X'(B) & \overset{\quad\quad}{\to} && Set(X(S), \mathcal{C}(S,B)) \\ \uparrow &&& \uparrow \\ Y'(B) & \overset{\quad\quad}{\to} && Set(Y(S), \mathcal{C}(S,B)) \end{aligned}

with both horizontal maps being inclusions.

Now

Y(S)=Im(X(S)Set(𝒞(S,S),X(S))) Y(S) = Im\big(X(S) \to Set(\mathcal{C}(S,S), X(S))\big)

where the map is x(fX(f)x)x \mapsto (f \mapsto X(f)x). There is also a map

Set(𝒞(S,S),X(S))X(S) Set(\mathcal{C}(S,S), X(S)) \to X(S)

given by evaluation on the identity. The composition is the identity on X(S)X(S) and thus Y(S)X(S)Y(S) \cong X(S). Moreover, this isomorphism is that appearing in the natural transformation XYX \to Y. Thus in the above diagram, the right-hand vertical map is an isomorphism. The left-hand map is thus injective.

Now let us consider surjectivity.

Let fX(B)f \in X'(B). This is a natural transformation XB *X \to B^*. We need to show that it factors through YY. Let AA be an object in 𝒞\mathcal{C} and suppose that x,yX(A)x, y \in X(A) are such that f A(x)f A(y)f_A(x) \ne f_A(y) in 𝒞(A,B)\mathcal{C}(A,B). Then there is a morphism a:SAa : S \to A such that f A(x)af A(y)af_A(x)a \ne f_A(y)a. Using again the fact that ff is a natural transformation, f A(x)a=f S(X(a)x)f_A(x)a = f_S(X(a)x) and similarly for yy. Thus X(a)xX(a)yX(a)x \ne X(a)y in X(S)X(S). Hence the maps 𝒞(S,A)X(S)\mathcal{C}(S,A) \to X(S) defined by xx and yy differ. That is, the images of xx and yy in Y(A)Y(A) are distinct.

This shows that f Af_A factors through Y(A)Y(A) and hence, as XYX \to Y consists of surjections, we can build a natural transformation f Y:YB *f^Y : Y \to B^*, such that under YXY' \to X', f Yf^Y maps to ff. Thus Y(B)X(B)Y'(B) \to X'(B) is surjective.

We therefore conclude that the natural transformation YXY' \to X' is a natural isomorphism.


We have

Y(A) Set(𝒞(S,A),X(S)) Y(A) Set(X(S),𝒞(S,B)) \begin{aligned} Y(A) &\subseteq Set(\mathcal{C}(S,A), X(S)) \\ Y'(A) &\subseteq Set(X(S), \mathcal{C}(S,B)) \end{aligned}

we also have a pairing compatible with these inclusions; that is (on objects)

Y(A)×Y(B)𝒞(A,B)Set(𝒞(S,A),𝒞(S,B)) Y(A) \times Y'(B) \to \mathcal{C}(A,B) \subseteq Set\big(\mathcal{C}(S,A), \mathcal{C}(S,B)\big)

(inclusion as SS is a separator).

From this it is clear that YY'' is given by

Y(A)={c:𝒞(S,A)X(A)fc𝒞(A,B)B𝒞,fY(B)} Y''(A) = \big\{ c : \mathcal{C}(S,A) \to X(A) \mid f c \in \mathcal{C}(A,B) \forall B \in \mathcal{C}, f \in Y'(B)\big\}

Lo and behold! A Frölicher space! Or rather, A Frölicher 𝒞\mathcal{C}-object.


Thus stability under Isbell conjugation is a very strong condition indeed; it forces quasi-representability and also the saturation condition. This suggests to me that there is a qualitative difference between quasi-representability and non-quasi-representable. As I conjectured, there is a functor from all presheaves to quasi-representable ones, and of course then one can (may I say “should”?) saturate with respect to duality. So every presheaf has an underlying Frölicher object, but this assignment is by no means injective. In fact, contrary to what I thought, the fibres of this functor do have some interesting stuff in them.

For example, if we take the simplified version of Urs’ favourite functor; namely 1-forms, then the associated Frölicher space is the single point (since Ω 1(pt)={0}\Omega^1(pt) = \{0\}).

It turns out, therefore, that Urs and I are looking at orthogonal concepts!


One possible area for further study is to see what happens if you vary the separator. It doesn’t have to be a singleton point in our categories of interest. For example, if we enlarge our category of “known smooth stuff” to include \mathbb{R}^{\infty}, the direct limit of the n\mathbb{R}^ns, and X=Ω 1X = \Omega^1 to be 1-forms, then by taking \mathbb{R}^{\infty} as the separator we find that

Ω 1(M)Set(C ( ,M),Ω 1( )) \Omega^1(M) \to Set\big(C^\infty(\mathbb{R}^{\infty},M), \Omega^1(\mathbb{R}^\infty)\big)

is injective, and so Ω 1\Omega^1 is quasi-representable. Of course, it is no longer quasi-representable in the strictest sense because we do not have an injection

Ω 1(M)Set(|M|,X) \Omega^1(M) \subseteq Set(\vert M \vert, X)

for any set XX. We only get this type of quasi-representability if our separator is a singleton point.

So perhaps we should define quasi-SS-representable where SS is a separator in the category 𝒞\mathcal{C}, by which we mean that the natural transformation which on objects is

X(B)Set(𝒞(S,B),X(S)) X(B) \to Set\big(\mathcal{C}(S,B), X(S)\big)

consists of injections.


However, varying the singleton won’t change the fact that Isbell conjugation (now thinking of one of our smoothish categories) results in strictly quasi-representable presheaves since the argument above made no assumptions on the separator SS so it certainly does work for SS the singleton point.

Right, it’s lunchtime so I’m stopping here.

Andrew

Posted by: Andrew Stacey on April 2, 2008 11:27 AM | Permalink | Reply to this

Re: Space and Quantity

Thanks for the detailed computations!

Let me see if I am following:

The construction of YY you give is supposed to establish comparison of any presheaf XX with quasi-representable ones.

For SCS \in C the point ( 0\mathbb{R}^0), X(S)X(S) is the set of points of XX and the presheaf

Points(X):CSet(C(S,C),X(S)) Points(X) : C \mapsto Set(C(S,C),X(S))

is the quasi-representable presheaf which is X(S)X(S) with the trivial smooth structure: all maps of sets from the points of CC to the points of XX are considered smooth by this presheaf.

Next, you trim down the plots again to those that actually come from XX: you say, let’s pick in Points(X)Points(X) those plots which actually come from smooth maps into XX.

So you build YPoints(X) Y \hookrightarrow Points(X) by setting Y:BIm(X(B)Points(X)(B)) Y : B \mapsto Im(X(B) \to Points(X)(B))

(by the way, I hope the last BB in your version of this formula should be an SS. Right?)

I tend to get confused by all the letters here. Let me write

SmoothPoints(X):=Y SmoothPoints(X) := Y

to remind myself that YY assigns to BB the smooth maps into the set of points of XX.

Now I will take the liberty of skipping over your detailed analysis and jump to your conclusion: the Isbell conjugate (SmoothPoints(X)). (SmoothPoints(X))'' \,.

You say that the plots of this smooth space over test domain AA are precisely those maps of sets from the points of AA to the plots of XX over AA (is that a typo? do you rather mean c:C(S,A)X(S)c : C(S,A) \to X(S)), such that for any smooth function from SmoothPoints(X)SmoothPoints(X) into BB, the composite is a smooth map of test domains.

That’s a nice statement.

All right, good. If I am right about where I think you have two typos, then I it seems I am following.

It turns out, therefore, that Urs and I are looking at orthogonal concepts!

I have thought a bit about this problem. It might be that we are not in fact so much looking at orthogonal problem, but at problems of different dimension. Or something like that.

The problem you keep pointing out is that:

an ordinary space is something which is determined by how it is probed with very low dimensional things (lines at most) – whereas on the other hand my examples are spaces which are determined only when probed with sufficiently high dimensional probes.

With that in mind, think about the remark (hopefully correct) which I make somewhere in the notes:

Isbell conjugation is just a special case of duality with respect to an ambimorphic object, where the ambimorphic object in question is

C ()=Hom(,), C^\infty(-) = Hom(-,-) \,, regarded as a C C^\infty-algebra valued presheaf.

Let’s think of it just as an algebra valued presheaf for a second.

Then notice that C ()=Ω 0(). C^\infty(-) = \Omega^0(-) \,.

And suddenly it looks less surprising that ambimorphic duality with respect to C ()C^\infty(-) is to duality with respect to Ω ()\Omega^\bullet(-) as low dimensions are to higher ones.

To make this long story short, I am proposing the following:

to unify your point of view with my point of view, let’s look at stability not under the duality induced by C ()=Ω 0()C^\infty(-) = \Omega^0(-) but by Ω ()\Omega^\bullet(-).

That is, let’s look at spaces XX for which S(Ω (X)) S(\Omega^\bullet(X)) is close to being XX again.

The dual of this statement is something I have already been talking about a bit here on the Café, without maybe making much progress:

Given a DGCA AA, we can form the DGCA Ω (S(A))\Omega^\bullet(S(A)) of forms on the space defined by the DGCA. I am thinking (mainly because that’s a big theorem in the similar (but different) context of rational homotopy theory) that there is an isomorphism in cohomology (a quasi-isomorphism) AΩ (S(A)). A \simeq \Omega^\bullet(S(A)) \,. This just to indicate that the Frölicher duality you emphasize so much is something one runs into here, but in a context where functions are generalized to forms.

I don’t know if you feel appalled by that step. One standard fact which might help become friends with this idea (though I am not (yet) claiming to know how precisely it might be related in detail) is this:

Take an ordinary manifold XX and write HH (Ω (X),Ω (X))HH^\bullet(\Omega^\bullet(X),\Omega^\bullet(X)) for the Hochschild cohomology of the dg-algebra of forms on XX with values in the module being itself. Then this is isomrophic to the homology of the loop space LXL X of XX. This does not work if we restrict to 0-forms here.

There is a nice story behind this about how Hochschild cohomology is the cohomology of something like simplicial forms on maps from the standard simplicial model of the circle into XX. Hence there is some beautiful higher category theory at work in the background. And in that context the higher degree of the forms appearing corresponds to higher morphism kind of aspects.

Well, I don’t know if that convinces you of anything, vague as it is, but maybe we may regard it as an(other) indication that there are aspects of smooth spaces which are not usefully thought of as being “probeable by only low dimensional things”.

Posted by: Urs Schreiber on April 3, 2008 12:54 PM | Permalink | Reply to this

Re: Space and Quantity

Right, quite a lot to reply to there.

First off, you are right about my two typos. Those two should have been SS’s.

Next, a conceptual issue. You wrote:

The problem you keep pointing out is that:

an ordinary space is something which is determined by how it is probed with very low dimensional things (lines at most) – whereas on the other hand my examples are spaces which are determined only when probed with sufficiently high dimensional probes.

There are two things that I keep wittering on about. One is about using lines rather than open subsets of Euclidean space; the other is the saturation issue. In this thread I was concentrating on the saturation issue, and leaving the line vs other for my other comment below (or above if you’re ordering the comments chronologically). The two are somewhat separate and I view the saturation as the primary one; indeed, one can only restrict to lines when one is in a saturated context. The restriction is all about finding the smallest subcategory that still completely specifies a given type of presheaf and whilst this is a useful thing to do, it is not absolutely necessary. However, it does seem like something that you would like to do (otherwise why not just use smooth manifolds throught) so I keep harping on about it. But this thread was all about saturation and not restriction.


Now I start to get a little lost in separating the general principle from the specific example. You say

Isbell conjugation is just a special case of duality with respect to an ambimorphic object, where the ambimorphic object in question is

C ()=Hom(,), C^\infty(−)=Hom(−,−),

regarded as a C C^\infty-algebra valued presheaf.

Hmm. I’m not convinced that I truly grok this. I would regard Hom(,)Hom(-,-) as a copresheaf object in presheaves and as a presheaf object in copresheaves for the Isbell duality. I’m not sure how to regard it as an algebra valued presheaf. The algebra valued presheaf that I can see is

C ()=Hom(,) C^\infty(-) = Hom(-, \mathbb{R})

and this relies on \mathbb{R} being the armadillo (according to Rudyard Kipling, an armadillo is both a hedgehog and a tortoise; thus being a natural example of an ambimorphic object).

So it seems that you want me to consider three dualities. One between preseheaves and copresheaves given by Isbell conjugation (using Hom(,)Hom(-,-)). One between presheaves and algebras given by

X Nat(X,C ()) A (BHom(A,C (B))) \begin{aligned} X &\mapsto Nat\big(X, C^\infty(-)\big) \\ A &\mapsto \Big(B \mapsto Hom(A, C^\infty(B))\Big) \end{aligned}

And then a third in which we replace C ()C^\infty(-) with Ω ()\Omega^\bullet(-).

Right, so far I think I understand all the pieces. What I’m missing is how they all fit together. I thought that the program was as follows.

  1. Generalise smooth spaces to smooth objects in such a way that the Ω (U)\Omega^\bullet(U) are the plots of a smooth object.

  2. Study Ω ()\Omega^\bullet(-) as a smooth object.

But now it seems that you are proposing to use Ω ()\Omega^\bullet(-) to define the smooth objects. I’d like to separate out the roles a little please. Otherwise I’m fearful that we might fall into the trap of saying that “the elephant is special because it is the maximum in the poset of all animals that look like elephants”.

See, I was under the impression that you had some presheaf closely related to Ω ()\Omega^\bullet(-) that you wanted to regard as a “nice” version of C (,BG)C^\infty(-, BG). So the discussion seemed to be about what “nice” meant and how to make it sensible.

Perhaps you still want to do that. Perhaps I’m the one forcing Ω ()\Omega^\bullet(-) into too many roles. It could be that you want to use Ω ()\Omega^\bullet(-) to set up the duality and then the refined version is the object that you wish to study.

As you can probably tell, I’m a little confused here!

I want to focus on smooth spaces (and their generalisations) as that is where I feel I can contribute most. So, assuming you reply to this, could you concentrate on that aspect (not to the exclusion of all else, but primarily on this).

Thanks

Posted by: Andrew Stacey on April 4, 2008 11:08 AM | Permalink | Reply to this

Re: Space and Quantity

In re:
an ordinary space is something which is determined by how it is probed with (very low) dimensional things

that’s e.g. homotopy or homology

but an ordianry space is also something that is determined by functions from it to antiprobes

Posted by: jim stasheff on April 4, 2008 2:28 PM | Permalink | Reply to this

Re: Space and Quantity

Jim dipped a toe into the water:

In re:

an ordinary space is something which is determined by how it is probed with (very low) dimensional things

that’s e.g. homotopy or homology

but an ordianry space is also something that is determined by functions from it to antiprobes

That’ll be cohomotopy and cohomology then.

This is one of the advantages of the Frölicher, or saturated, viewpoint: that you can meet with incoming and outgoing “stuff” and treat those two impostors just the same.

I’d quite like to remove Urs’ “(very low)” from this quote as I think it’s a bit of a blue halibut. What we want is a space to be “something that can be probed and coprobed with equal aplomb”. What we’re stumbling around is figuring out how general the “something” can be. The “very low” bit comes from deciding what the probes are from (and coprobes are to) but what I think is more important is deciding what the compatibility conditions on the probes and coprobes should be. That’s what we’re after here. At least, I think that’s what we’re after but I could be wrong!

Posted by: Andrew Stacey on April 4, 2008 4:06 PM | Permalink | Reply to this

Re: Space and Quantity

Andrew,

one more comment:

I was trying to avoid it, because it did not “feel right” to me for some funny reason – even though everybody else enjoys doing it – , but since differential forms are just superfunctions on odd tangent bundles, we can, by replacing our site SS of smooth Euclidean spaces

Obj(S)=Obj(S) = \mathbb{N}

S(n,m)=Hom smoothmanifolds( n, n)S(n,m) = Hom_{smooth manifolds}(\mathbb{R}^n, \mathbb{R}^n)

by the analogous site of smooth Euclidean superspaces n|m\mathbb{R}^{n|m}, conceive them as just ordinary functions, “quantities” in Lawvere’s sense.

If we then embed ordinary smooth spaces into smooth superspaces by sending them to their odd tangent bundles, Isbell conjugation in that super context becomes essentially duality with respect to Ω ()\Omega^\bullet(-). Then there is a super-Frölicher condition and possibly that’s what I am after.

Well, except that i have been trying to avoid doing it this way.

Posted by: Urs Schreiber on April 6, 2008 1:25 PM | Permalink | Reply to this

Re: Space and Quantity

I’m not sure how to regard [Hom(,)Hom(-,-)] as an algebra valued presheaf.

It’s a C C^\infty-algebra valued presheaf!

I am thinking here of the site CC over which we are working as the category whose objects are integers and where Hom C(n,m)=Hom smoothmanifolds( n, m). Hom_C(n,m) = Hom_{smooth manifolds}(\mathbb{R}^n,\mathbb{R}^m) \,. This site has a monoidal structure induced from addition of numbers and cartesian product of Euclidean spaces.

Recall that a C C^\infty-algebra is a copresheaf on this CC which respects the monoidal structure, i.e. a monoidal functor CSet. C \to Set \,. The idea is that any such monoidal functor FF has an underlying algebra, namely F()F(\mathbb{R}), on which the algebra product is that induced from the smooth map × \mathbb{R}\times \mathbb{R} \stackrel{\cdot}{\to} \mathbb{R} which comes from the product of real numbers.

But since Hom smoothmanifolds( l, m× n)Hom smoothmanifolds( l, m)×Hom smoothmanifolds( l, n) Hom_{smooth manifolds}(\mathbb{R}^l, \mathbb{R}^m \times \mathbb{R}^n) \simeq Hom_{smooth manifolds}(\mathbb{R}^l,\mathbb{R}^m) \times Hom_{smooth manifolds}(\mathbb{R}^l,\mathbb{R}^n) we have that Hom C(,)Hom_C(-,-) is a C C^\infty-algebra valued presheaf.

I would like to generalize C C^\infty-algebras to C C^\infty-DGCAs. But after some failed attempts, I had given up on finding a good definition for that.

So therefore I just argued here that at least if we forget the full C C^\infty-algebra structure on Hom C(A,)Hom_C(A,-) and just concentrate on the “underlying algebra” Hom C(A,)Hom_C(A,\mathbb{R}) we are just talking about 0-forms.

So until somebody finds a good definition of C C^\infty-algebra, it’s just a plausibility argument. But I still think it’s worthwhile.

There are two things that I keep wittering on about. One is about using lines rather than open subsets of Euclidean space; the other is the saturation issue.

In this thread I was concentrating on the saturation issue

Right, but the point I was trying to make is that there seems to be a relation between the fact that we are dualizing with respect to 0-forms and the fact that the things stable under this dualization are completely probed already by lines.

Posted by: Urs Schreiber on April 4, 2008 9:59 PM | Permalink | Reply to this

Re: Space and Quantity

I want to focus on smooth spaces (and their generalisations) as that is where I feel I can contribute most. So, assuming you reply to this, could you concentrate on that aspect (not to the exclusion of all else, but primarily on this).

Okay, let me try to say what I am after in a coherent fashion again:

I have these sheaves here (not just those coming from forms, by the way: there is also a sheaf which assigns transport functors to each test domain, that classifies bundles with connection, for instance (and that generalizes to higher bundles), and it is not quasi-representable either) which in my applications absolutely deserve to be addressed as smooth spaces, because doing so allows us to say “We have a smooth classifying space for XYZ” in all of the situations where we will wish to do so. It would be awkward and seem to be misguided to say each time:

“we have this classifying sheaf, which however does not satisfy some axiom of those sheaves which we agreed to address as smooth spaces, so we have to address our classifying maps as maps from smooth spaces to sheaves that are not smooth spaces”.

At the same time, I am very sympathetic with your general philosophy and attitude towards the idea of probing and coprobing etc.

So I was just wondering if we might be missing something that would resolve the apparent conflict between these nice applications on one hand and the nice philosophy on the other.

My suggestion was:

while stability under dualization seems to be a very good idea indeed, it might be worthwhile to think a bit more about the available freedom to choose with respect to what exactly we are dualizing.

Posted by: Urs Schreiber on April 4, 2008 10:14 PM | Permalink | Reply to this

Re: Space and Quantity

Just a thought on a few definitions. In a sense, the discussion on smooth spaces is about finding the right home for things like Ω (M)\Omega^\bullet(M). The problem is that by having a specific example in mind, the temptation is always to find the one in which the example sits in a “nice” fashion.

The principle I would like to apply is: only trim things down for a reason.

So at first sight, Ω (M)\Omega^\bullet(M) is a presheaf on the category of all smooth manifolds (maybe even manifolds with corners). So we start with that category. In that, we can define the “nice spaces” and “very nice spaces” as you do; namely, as sheafs and as presheaves stable under Isbell conjugation.

Then we look at our subcategories. First, sheaves. As these are sheaves, they are completely determined by their effect on open subsets. As we are dealing with manifolds, these are open subsets of Euclidean spaces and in fact we can restrict to simply Euclidean spaces themselves (though if we have corners we need quadrants, I suppose).

Then we look at “very nice spaces”. By my previous comment, these are Frölicher spaces. These, as we know, are completely specified by their restriction to the real line.

So in each case we do end up with being able to define the category of “generalised smooth objects” on a smaller category than that of all smooth manifolds. But just because we can do it with the smaller categories doesn’t mean that we should do it with the largest category. After all, if we took the line that we should use the smallest throughout then we would use the full subcategory with one object which is the real line.

I guess that this is more style than substance, though.

Andrew

Posted by: Andrew Stacey on April 2, 2008 12:59 PM | Permalink | Reply to this
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