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December 30, 2007

Transgression of n-Transport and n-Connections

Posted by Urs Schreiber

Since it will play a role both for what is currently indicated in section 5 of the article on Lie \infty-algebra connections and their application to String- and Chern-Simons nn-transport as well as for the next followup of my work with Konrad Waldorf, I am thinking again in more detail about


Trangression of nn-transport and nn-connections

Abstract. After going through some ground work concerning generalized smooth spaces and their differential graded commutative algebras of forms, I talk about the issue of transgression of transport nn-functors and Lie \infty-valued nn-connections to smooth mapping spaces.

This builds on the general idea of nn-functorial transgression as the image of an internal hom as first voiced in this old comment and then later incorporated in the discussion of The charged nn-particle and detailed a bit more in the entry Multiplicative Structure of Transgressed nn-Bundles.

Currently only a few sketches are present in the above pdf, as I am going to develop this as we go along.

One important aspect, emphasized in the above abstract, is that the discussion greatly profits from a good general understanding of the relation between generalized smooth spaces and their differential graded-commutative algebras of differential forms. I started making comments on that here and now Todd Trimble thankfully chimed in by providing this detailed reply, which I will reproduce below.

But first, I’ll reproduce the introductory remarks from my notes to set the stage.

Introduction

I want to better understand the
\;\;\bullet general systematics

and the

\;\;\bullet specific details

of what it means to transgress

\;\;\bullet transport nn-functors [1, 2, 3 4]

and

\;\;\bullet Lie nn-algebra valued connections [5]

to mapping spaces.

This is essentially about understanding the pull-push operation of nn-transport and nn-connections on a “target space” tar\mathrm{tar} from right to left through a span hom(par,tar)par p 1 ev hom(par,tar) tar \array{ &&& hom(par,tar) \otimes par \\ && \multiscripts{^{p_1}}{\swarrow}{} && \searrow^{ev} \\ & hom(par,tar) &&&& tar } to obtain an nn-transport and nn-connection on the “configuration space” of maps conf:=hom(par,tar) \mathrm{conf} := \mathrm{hom}(\mathrm{par},\mathrm{tar}) from some “parameter space” par\mathrm{par} to tar\mathrm{tar}. But in fact it turns out that the “good” answer does apparently not quite involve the naïve push-forward along p 1p_1, but a slight variant, which then amounts to simply defining the transgression of the nn-transport or nn-connection tra\mathrm{tra} to be hom(Id par,tra). \mathrm{hom}(\mathrm{Id}_{\mathrm{par}}, \mathrm{tra}) \,.

This difference to the naive definition of transgression as direct push-pull through the above span actually takes care of a fact neglected in standard discussions that do not make the nn-categorical nature of nn-transport manifest: namely that under transgression not only the domain, but also the codomain of nn-transport and nn-connections changes.

For instance, in the simplest kind of example, an ordinary abelian 2-connection is not really something taking values in U(1)U(1), but rather something taking values in U(1)\mathcal{B}U(1). Transgressing it to loop spaces by setting par=S 1\mathrm{par} = S^1 in the above turns it into a 1-connection with values in hom(S 1,U(1))\mathrm{hom}(S^1, \mathcal{B}U(1)), which is indeed ΛU(1)=U(1) \Lambda \mathcal{B}U(1) = U(1) as it should be.

So this general notion of transgression is what shall be discussed here.

Before getting into the issue of transgression proper, I try to lay some necessary groundwork on the general concept of generalized smooth spaces and the differential graded-commutative algebras of differential forms on them.

Here I take “generalized smooth spaces” simply to be presheaves over manifolds. This is clearly the right ambient topos, in general, for any discussion of smooth parallel nn-transport and smooth nn-connections.

My tentative discussion of differential forms on such generalized smooth spaces, and the relation to general differential graded commutative algebras, is included here because I am not aware of a discussion of the necessary points in the literature. This may, however, well be – in parts or possibly even in total – just be due to my woeful ignorance.

Hopefully much of what I am trying to say concerning the general issue of smooth spaces versus differential graded algebras is actually well known, possibly in slightly different guise, in rational homtopy theory.

In any case, after having dealt to some extent with this groundwork, I’ll define in more detail the problem of transgression to be discussed here, and then start looking at concrete questions and specific examples.


Todd’s first comment

Here is a copy of Todd’s comment on the relation between DGCAs and generalized smooth spaces, the discussion of which we should move to the comment section below.

This is what Todd wrote:

Hi Urs,

Took a look at your comment and notes; here are some initial reactions.

First, you asked whether the formula

Uhom(U×X,Y)U \mapsto hom(U \times X, Y)

gives the ‘correct’ internal hom for smooth spaces; in this context I assume you’re asking whether this gives the correct exponential for cartesian closedness.

Yes, this formula is correct and works for general presheaf toposes, by an application of the Yoneda lemma. The result is standard and proved in many books on topos theory – the ones by Johnstone, by Mac Lane and Moerdijk, and by Freyd and Scedrov come to mind. I’m happy to go into more detail if you want.

Second: there is as you say a contravariant adjunction

S (X,DGCA(A,Ω ()))DGCA(A,S (X,Ω ())).S^{\infty}(X, DGCA(A, \Omega^{\bullet}(-))) \cong DGCA(A, S^{\infty}(X, \Omega^{\bullet}(-))).

This can be proved with the help of (again) the Yoneda lemma: since XX is a colimit of representables, one can easily reduce to the case where XX is a representable, and then check that case with the help of Yoneda. Notice that this type of adjunction has the general flavor of one coming from a Janusian/ambimorphic object (Ω ()\Omega^{\bullet}(-) having a kind of dual existence, one as a smooth space and another as a DGCA).

But, I don’t think this adjunction can be an equivalence, even if we restrict to locally quasi-free DGCA’s. The question seems to be whether DGCA opDGCA^{op} (or something like it) is equivalent to the topos S S^{\infty}, and it’s sort of an interesting question because DGCA opDGCA^{op} does partake of some of the exactness properties satisfied by a topos. For one, it’s a lextensive category (it has finite pullbacks and finite coproducts which are disjoint and preserved under pullback) – lextensive categories are a very interesting and much-studied class of categories.

But I sort of doubt DGCA opDGCA^{op} or some easily identified full subcategory is locally cartesian closed (which it would be if it were a topos). This would mean that general colimits in this category are preserved under pulling back (one says “colimits are universal”), or that limits in DGCADGCA are preserved under pushing out. The pushout of a pair of morphisms

BACB \leftarrow A \to C

in DGCADGCA is given by B ACB \otimes_A C; the question is whether B AB \otimes_A - preserves limits of AA-modules. Preservation of equalizers may be no big deal under some condition like “locally quasi-free” (although there one would have to watch out that objects satisfying that condition give a complete and cocomplete category – I’m not so sure about that), but B AB \otimes_A - preserving arbitrary products, not just finite ones, looks like a much taller order.

(Another exactness condition to check has to do with whether there is an exact correspondence between epimorphisms and equivalence relations: whether every epi is the quotient of its kernel pair, and whether every equivalence relation is the kernel pair of its quotient. Never mind that for now.)

This is pretty much a gut reaction, and I feel it will probably read like a wet blanket reaction as well, which I really don’t mean. It’s possible that the desiderata of ‘internal homs’ could be relaxed a bit to stop short of actual cartesian closedness and still be interesting, but I just don’t know off hand. (By the way, I wanted to get back to you some time on calculations of internal homs of differential graded cocommutative coalgebras, but my first two attempts were obliterated by my 3-year-old daughter, and somehow I haven’t found much time for math during this holiday season.)

I would very much like to follow up some time on what you’re saying about universal bundles with connections.

Posted at December 30, 2007 6:11 PM UTC

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76 Comments & 17 Trackbacks

Re: Transgression of n-Transport and n-Connections

Todd,

I have now incorporated your statement that Ω \Omega^\bullet and Hom(,Ω ())\mathrm{Hom}(-,\Omega^\bullet(-)) form an adjunction.

What you say about how to put this into perspective, Ω ()\Omega^\bullet(-) being Janusian or ambimorphic etc, I find very intriguing, but haven’t yet incorporated into my notes, since I am not quite sure yet what to make of that. Clearly, I have a lot to learn here.

If and when you might be interested, I’d very much enjoy if you could have a look at the beginning of section 4, where I talk about transgression.

I am pretty sure I am onto something there, but it is also clear that there should be a much better abstract nonsense way to say what I am trying to say.

So here is the point:

usually, transgression is defined as pull-push from right to left through spans of the form hom(par,tar)par p 1 ev hom(par,tar) tar. \array{ &&& hom(par,tar) \otimes par \\ && {}^{p_1}\swarrow && \searrow^{ev} \\ & hom(par,tar) &&&& tar } \,.

I am claiming that if what we want to transgress is actually an nn-functorial thing whose domain is tar\mathrm{tar}, then what we “really want” to regard as its transgression is instead its image under hom(par,). \mathrm{hom}(\mathrm{par},-) \,.

I know this is what we “really want” by looking at rather large classes of examples.

I might just be content with taking hom(par,)\mathrm{hom}(\mathrm{par},-) by definition to be “my” notion of “good” transgression. But I want to clarify what’s going on, how this relates to the pull-push operation people usually consider.

At the beginning of that section 4 I present a long remark where I give my best attemt at clarifying the situation.

If what I do there is of any value at all, then certainly it is only scratching the surface of something. I have the suspicion you might maybe recognize some abstract nonsense more elegant and more useful which refines my discussion there.

As I said, if and when you are interested, this would be something I’d very much enjoy hearing your comment on.

Posted by: Urs Schreiber on December 31, 2007 12:51 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Todd wrote:

I don’t think this adjunction [between DGCAs and smooth spaces] can be an equivalence,

Yes, I can easily believe that. My vaguely expressed hope that it yields some kind of equivalence was at best too vague.

But let’s see: shouldn’t we be able to obtain some kind of equivalence after passing to cohomologies?

That’s at least what, as far as I understand, the point of Sullivan models is.

For instance theorem 8.1 on page 301 of Sullivan’s old paper states that the cohomology of any DGCA AA is isomorphic to the rational polynomial deRham cohomology of the space, which he denotes A\langle A \rangle, defined by this algebra, H (A,)H pol.deRham (A,). H^\bullet(A,\mathbb{Q}) \simeq H^\bullet_{pol. deRham}(\langle A \rangle, \mathbb{Q}) \,.

And I gather the whole point of the study of Sullivan models is that, in some sense, every space is modeled by a DGCA in rational cohomology in this sense.

I understand that an equivalence in cohomology is much weaker that the full equivalence I seemed to be talking about initially, but I would still be very interested in understanding how this equivalence in cohomology studied in terms of Sullivan models carries over to the context of presheaves over manifolds which we are discussing here.

I wanted to get back to you some time on calculations of internal homs of differential graded cocommutative coalgebras

That would be immensely appreciated!

I haven’t made any progress on this since my attempts on it here and here.

Worse, since I stopped thinking about this, I will have to go back to these discussions to remind myself.

That would be great if we could obtain a good understanding of the internal hom in codifferential coalgebras and/or differential algebras for the relavent cases.

By the way: the fact that, as you told me, codiff. coalgebras are closed, why diff. algebras are not I assume is due to finiteness issues?

Posted by: Urs Schreiber on December 31, 2007 2:14 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Right: I had meant (but forgot) to acknowledge in my previous reply that you may have had quasi-equivalence rather than full equivalence in mind, or something like that. I think I was going to add that I wouldn’t expect the derived category to inherit closed structure from say S S^{\infty}, but that perhaps that doesn’t matter.

I don’t have the technical background to address this issue of equivalence up to cohomology, but it’s not hard for me to believe that some patrons of the Café would. Maybe I could learn a few things myself.

As for why (what I denoted) CoDGACoDGA is closed and DGCA opDGCA^{op} not. I hadn’t thought of it this way until you just said it, but I guess you’re right: that it boils down to finiteness issues! In a long comment which I was going to send you on calculating homs in CoDGACoDGA (the one that was obliterated), I was emphasizing a Very Useful Observation which I picked up from reading Getzler and Goerss. It says

  • Every (dg) cocommutative coalgebra is the filtered colimit of its finite-dimensional subcoalgebras.

Cocommutative coalgebras may be awkward to work with in some respects, but they do have the remarkable saving grace expressed by this VUA – something which dgca’s don’t have, and which makes all the difference in the world!

Getzler and Goerss note a consequence of this fact: because every coalgebra is the filtered colimit (union if you prefer) of its finite-dimensional coalgebras, and because finite-dimensional coalgebras are opposite to finite-dimensional algebras, it follows that the opposite of the category of coalgebras is the category of pro-finite algebras (i.e., algebras which are the cofiltered limits of their finite-dimensional quotients). This can also be expressed in topological language: pro-finite algebras are certain topological completions; the category of pro-finite algebras and continuous algebra maps is opposite to the category of coalgebras. All this carries over to the dg world. Thus, they are saying:

CoDGADGCA prof op.CoDGA \cong DGCA_{prof}^{op}.

They use this to construct the all-important cofree coalgebra construction (and from there one may construct internal homs in CoDGACoDGA, as discussed in my earlier comment), but for now I’ll just note this: while DGCA opDGCA^{op} may not be cartesian closed (or so my gut tells me!), DGCA prof opDGCA_{prof}^{op} is.

I have no idea whether this observation would be useful to you; following it up might entail a long goose chase of course. Caveat lector.

More to come, I think.

Posted by: Todd Trimble on December 31, 2007 3:37 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

[codifferential coalgebras are equivalent to profinite differential algebras]

Ah, I see. Very interesting. I should take a closer look at the Getzler & Goerss.

But I am running out of time here. New year is in about four hours here, and there is a party about to start which expects my attendance.

All I manged to do so far is to polish my notes slightly more.

In a long comment which I was going to send you on calculating homs in CoDGA (the one that was obliterated)

Oh dear, I didn’t know you wrote a comment to that effect and lost it. Darn. What a pity.

Maybe you still find the energy to at least very quickly point out the bottom line, maybe by telling me if my previous remarks on that topic had at all been on the right track?

Posted by: Urs Schreiber on December 31, 2007 6:43 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

not sure what
this issue of equivalence up to cohomology,
is


however I do detect a wiff of:

John Moore long since taught me that coalgebras are more natural than algebras

also similar issues with regard to Lie algebras and coalgebras
and application of the universal envelopings

whihc reminds me that the cofree coalgebra on a vector space is not what some people expect

Posted by: jim stasheff on January 1, 2008 3:27 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Probably Urs should be the one to answer, but I suppose equivalence up to cohomology means that the unit and counit of the adjunction induce isomorphisms in cohomology.

Yes, the cofree coalgebra construction was one of the main things I was going to discuss in that “lost comment” I’ve been referring to. Would you like to say a few words about that?

Posted by: Todd Trimble on January 1, 2008 1:59 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I learned late in life (from my favorite expert Walter Michaelis) that the tensor coalgebra TV = \oplus V^{\otimes n}
on and R module V with V^0=R is cofree
only for a restricted category of associative coalgebras

Todd,
Say more?

Posted by: jim stasheff on January 2, 2008 1:22 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Todd said to Jim

the cofree coalgebra construction was one of the main things I was going to discuss in that “lost comment” I’ve been referring to. Would you like to say a few words about that?

Jim said

Todd, say more?

The suspense is killing me. Really.

[Those readers following this remember: we would like to figure out how to explicitly compute the inner hom in codifferential coalgebras, the abstract construction of which Todd presented (following Jim Dolan), here]

I wish I knew how I could catalyze the information flow.

So in graded vector spaces, for VV any graded vector space, the cofree coalgebra over it is, as a vector space, the space TV= nV n T V = \oplus_{n \in \mathbb{N}} V^{\otimes n} that Jim mentions with the coproduct ΔTVTVTV \Delta T V \to T V \otimes T V acting on homogeneous elements (x 1x 2x n)(x_1 \otimes x_2 \otimes \cdots \otimes x_n) as k=0 n(x 1x k)(x k+1x n). \sum_{k = 0}^n (x_1 \otimes \cdots x_k) \otimes (x_{k+1} \otimes \cdots x_n) \,.

Okay, let me see. Suppose for starters we just ask for the internal hom in coalgebras internal to graded vector spaces (as opposed to internal to chain complexes).

Suppose I have two such graded coalgebras C 1C_1 and C 2C_2. Then there is the graded vector space Hom coalgebras(C 1,C 2) \mathrm{Hom}_{coalgebras}(C_1,C_2) of coalgebra morphisms from one to the other, whose elements in degree kk are coalgebra homomorphisms that shift the degree by kk.

A naïve candidate for the inner hom is the cofree coalgebra over this graded vector space THom coalgebras(C 1,C 2). T \mathrm{Hom}_{coalgebras}(C_1,C_2) \,.

Hm…

Posted by: Urs Schreiber on January 2, 2008 8:16 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I’ll respond as soon as I can. I’ve just gotten back from various doctors; my little daughter has childhood shingles (of all things), and I need to take care of some things first. (Apparently not quite so painful as shingles is for adults, but pretty darned uncomfortable nevertheless.)

My response is probably best served up in installments, and of course I will want to ruminate on some other things so far brought up in this thread. First installment should be on cofree coalgebras. (Cofree reminds me of coffee, which I could use about now. :-) )

Posted by: Todd Trimble on January 2, 2008 9:17 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I’ll respond as soon as I can.

Please don’t feel rushed by me.

And all my best wishes for your daughter!

Posted by: Urs Schreiber on January 2, 2008 9:22 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Here I thought childhood shingles was called “chicken pox”. Live and learn.

Wish her well.

Posted by: John Armstrong on January 2, 2008 9:57 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Thanks for the well-wishing, guys. With acyclovir, it should clear up within a week or ten days (and on the upside, another reactivation of the virus, e.g., shingles as an adult, is a rarity after an episode of childhood shingles).

As I understand it, you’re right that it’s essentially the same virus at work, just as shingles in adults is. In fact, all it is (for children or adults) is a reactivation of the virus after an initial exposure, in her case probably a chicken pox vaccination. In such cases, the virus remains dormant in the nerve cells, typically along the spine, but one day (usually after many years) reactivates and manifests itself as herpes zoster, forming blisters on the skin which creep out starting from the nerve site. (Herpes from the Greek meaning ‘creeping’; cf. herpetology = the study of reptiles, also the Latin serpere ‘to creep’, whence ‘serpent’.) The nerves are highly irritated and the skin is painfully tender to the touch.

The word “shingles” itself derives from the Latin cingula, ‘girdle’ (from the way it typically girdles the waist in adults). (Cf. ‘cinching’ a belt.) The Greek zoster also means ‘girdle’. (Fascinating stuff, word origins.)

Posted by: Todd Trimble on January 2, 2008 11:27 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Before I go to bed, I’ll share the following construction. I am not entirely sure what it’s interpretation is, but it vaguely looks like it corresponds to the internal hom of Lie \infty-groupoids transported to the world of Lie \infty-algebroids. Or something like that, I don’t know yet.

For BB and CC qDGCAs, consider the presheaf on manifolds X [B,C]X_{[B,C]} given by UHom DGCAs(B,CΩ (U)). U \mapsto \mathrm{Hom}_{DGCAs}(B, C \otimes \Omega^\bullet(U)) \,. Notice that this corresponds to the internal hom in presheaves, which would be UHom S (U×X C,X B). U \mapsto \mathrm{Hom}_{S^\infty}(U \times X_C, X_B) \,.

We know that for fixed UU, an element in Hom DGCAs(B,CΩ (U))\mathrm{Hom}_{DGCAs}(B, C \otimes \Omega^\bullet(U)) is a UU-parameterized family of DGCA morphisms BCB \to C together with the chain homotopies and homotopies of homotopies etc. describing how that family changes as we move along UU. As described in our proposition 3 on p. 11 of part B.

I am thinking that the DGCA of forms on X [C,B]X_{[C,B]}, Ω (X [C,B]) \Omega^\bullet(X_{[C,B]}) should be something not unrelated to the internal hom DGCA from CC to BB.

Let’s see what differential forms on X [B,C]X_{[B,C]} are like. Recall that they are defined to be presheaf morphisms ω:X [B,C]Ω \omega : X_{[B,C]} \to \Omega^\bullet hence for each UU a morphism of sets ω U:Hom DGCAs(B,CΩ (U))Ω (U) \omega_U : \mathrm{Hom}_{DGCAs}(B, C \otimes \Omega^\bullet(U)) \to \Omega^\bullet(U) conatural in UU.

Let V BV_B be the vector space underlying the qDGCA BB. Then for each element f ωV BC * f_\omega \in V_B \otimes C^* of degree n-n we obtain a differential form of degree nn on X [B,C]X_{[B,C]} by sending any ev U *hom(B,CΩ (U))\mathrm{ev}_U^* \in \mathrm{hom}(B,C \otimes \Omega^\bullet(U)) to f ω,ev U *Ω (U) \langle f_\omega , ev_U^*\rangle \in \Omega^\bullet(U) in the hopefully obvious sense.

That’s looking kind of promising, because elements of V B *CV_B^* \otimes C indeed describe morphisms from BB to CC and their homotopies, so elements of V BC *V_B \otimes C^* are like functions on that, which is what we want.

Better yet, we know, also from proposition 3 that acting with the deRham differential on such forms corresponds to commuting the corresponding components of Hom(B,CΩ (U))\mathrm{Hom}(B,C \otimes \Omega^\bullet(U)) with the differentials on the two sides.

So it seems that the DGCAs of the form Ω (X [C,B]) \Omega^\bullet(X_{[C,B]}) are important for something. But I need to think more about it.

Posted by: Urs Schreiber on January 2, 2008 11:23 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Above I made a comment on something like internal homs in DGCAs. I am now coming back to this, and would like to straighten this out.

Let me amplify the two constructions and the two questions hidden in what I said above:

A) We have an adjunction between smooth spaces Set S op Set^{S^{op}} and DGCAs. Smooth spaces are closed, while DGCAs is not. But still, we can “pull back the internal hom ” construction from smooth spaces to DGCAs using the adjunction:

meaning we get a map h(,):DGCAs×DGCAs opDGCAs h(-,-) : DGCAs \times DGCAs^{op} \to DGCAs by first sending each DGCA to a space, then forming the internal hom of these spaces, and then transferring that back to a DGCA DGCAs×DGCAs opHom(,Ω ())×Hom(,Ω ()) op(S ) op×S hom S S Ω DGCAs. DGCAs \times DGCAs^{op} \stackrel{ Hom(-,\Omega^\bullet(-)) \times Hom(-,\Omega^\bullet(-))^{op} }{\to} (S^\infty)^{op} \times S^\infty \stackrel{hom_{S^\infty}}{\to} S^\infty \stackrel{\Omega^\bullet}{\to} DGCAs \,.

Question: Can we say anything useful about the pairing DGCAs×DGCAs opDGCAsDGCAs \times DGCAs^{op} \to DGCAs obtained this way? Is there a good sense in which we can say in some generality that h(,)h(-,-) is a “good approximation” to the non-existant internal hom in DGCAsDGCAs?

B) There is a simpler formalization of the “idea” underlying h(,,)h(-,-,): given two algebras AA and BB, we can directly form the presheaf UHom DGCAs(B,AΩ (U)). U \mapsto Hom_{DGCAs}(B , A \otimes \Omega^\bullet(U)) \,. One can see that this presheaf is a subobject of the presheaf that appears towards the right in the construction of h(,)h(-,-).

So, I think, we have an inclusion Ω (Hom DGCAs(B,AΩ ()))h(B,A). \Omega^\bullet(Hom_{DGCAs}(B , A \otimes \Omega^\bullet(-)) ) \hookrightarrow h(B,A) \,.

Question: Is there anything useful we can say about this inclusion?

The background to all this is: by looking at some applications I find that forming from 2 DGCAs AA and BB the DGCA Ω (Hom DGCAs(B,AΩ ()) \Omega^\bullet(Hom_{DGCAs}(B , A \otimes \Omega^\bullet(-)) does pretty much what one expects to see in these applications. So I am growing fond of using that construction. But I would like to know what that means I am really doing.

Clearly, it amounts to constructing an “approximation” to a non-existant internal hom in some sense. I would like to know if one can say in a precise way what “approximation” means here.

Posted by: Urs Schreiber on January 15, 2008 2:40 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Very sorry to hear about your daughter.
So not just chicken pox but in fact shingles?
with or without the pox?

Look forwqrd to your cofree help
but only when you are free of fatherly duties.

First things first!

Posted by: jim stasheff on January 3, 2008 12:52 AM | Permalink | Reply to this

Chicken Pox

No pox, just shingles.

After a person is exposed to the chicken pox virus, even after a bout of chicken pox has run its course, some of the virus remains in the body (in the nerve roots), in a dormant state. It can be reactivated when the immune system is relatively weak, for example during a time of stress or sickness or in old age. When it is thus reactivated, it takes the form of herpes zoster (shingles). Same virus, but under different conditions.

It’s believed that most children who develop herpes zoster were exposed to the virus during their first year, when levels of protective antibodies are low. There are some cases where it is due to reactivation of attenuated virus in the vaccine (I’m guessing that’s what happened in Lydia’s case).

It’s a nasty disease, even for those who are physically tough. I remember when Ross Street (unquestionably a robust fellow) had a bout of it; the pain can be pretty incapacitating. Incidence in children is relatively rare (.75/1000), but the episode typically passes quickly and with less pain; the prognosis in my daughter’s case is very good (she’s generally very healthy).

I’ll quit talking about this and get to the math in my next post. Thanks for your thoughts.

Posted by: Todd Trimble on January 3, 2008 3:38 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

URS WROTE:
So in graded vector spaces, for V any graded vector space, the cofree coalgebra over it is, as a vector space, the space
TV=⊕ n∈ℕV ⊗n
that Jim mentions
with the coproduct
ΔTV→TV⊗TV
acting on homogeneous elements
(x 1⊗x 2⊗⋯⊗x n) as
∑ k=0 n(x 1⊗⋯x k)⊗(x k+1⊗⋯x n).

My point was:
NOT IN GENERAL
It is cofree for the category of _____
coalgebras, where I think that blank needs at least connected? That’s where I hoped Todd would weigh in.

Also watch out if V is only a module over a ring R.

Posted by: jim stasheff on January 3, 2008 12:50 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I suppose equivalence up to cohomology means that the unit and counit of the adjunction induce isomorphisms in cohomology.

Yes!

As in the basic idea of rational homotopy theory, like in theorem 1.24 on p. 10 of Kathryn Hess’ review.

The general question I keep trying to address here is this:

in rational homotopy theory people build a simplicial set from a DGCA AA, by taking the collection of nn-simplices to be Hom DGCAs(A,Ω (Δ n)) \mathrm{Hom}_{DGCAs}(A, \Omega^\bullet(\Delta^n))

(def 1.22 in Hess’ review). Then taking nerves yields a topological space.

What I would like to better understand is: if instead of rational homotopy theory, I am just interested in real homotopy theory (how much does that really loose??) then: what happens if instead of simplicial sets we go through (pre)sheaves on manifolds, using the same kind of idea.

So for every DGCA AA we obtain the presheaf X AX_A on manifolds defined by X A:UHom DGCAs(A,Ω (U)), X_A : U \mapsto \mathrm{Hom}_{DGCAs}(A,\Omega^\bullet(U)) \,, for all manifolds UU.

To every presheaf XX on manifolds we can associate its DGCA Ω (X)\Omega^\bullet(X) of differential forms. The cohomology H (Ω (X)) H^\bullet(\Omega^\bullet(X)) of this I can address as the real cohomology of XX.

So the question is, whether or not, or under which conditions, the cohomology of the DGCA AA coincides with the cohomology of the presheaf it defines: i.e. whether or not we have

H (A)=H (Ω (X A)). H^\bullet(A) = H^\bullet(\Omega^\bullet(X_A)) \,.

I am suspecting that something like this should be true, and I am thinking that using presheaves on manifolds instead of realizing simplicial sets is actually useful for many purposes.

Posted by: Urs Schreiber on January 2, 2008 7:14 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

The main advantage of rational homotopy theory is that the models are `just’ qDGCAs
i.e. polynomial algebras tensored with Grassman algebras

tensor that with reals or complex numbers and very little changes except for some artihmetic

but if you allow functions on R^n rather than polynomials, that’s a different ball game

Posted by: jim stasheff on January 3, 2008 12:58 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

but if you allow functions on n\mathbb{R}^n rather than polynomials, that’s a different ball game

Okay, thanks, I get it. I need to think about this. Maybe in everything I have been sainy here about differential forms, I should restrict to polynomial differential forms.

Hm…

Posted by: Urs Schreiber on January 3, 2008 12:37 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

X_A is defined in terms of all manifolds U
Then what is X? and how does Omega^bullet(X)
differ from Omega^bullet (X_A)?
or am I once more just notationally challenged?

Posted by: jim stasheff on January 3, 2008 1:06 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Jim wrote:

X AX_A is defined in terms of all manifolds UU Then what is XX? and how does Ω (X)\Omega^\bullet(X) differ from Omega (X A)Omega^\bullet (X_A)? or am I once more just notationally challenged?

I did introduce some notation here which I made up myself, but I think it is not too unorthodox.

So let me recall:

Objects denoted UU are manifolds, living in the category (a site, actually) of all manifolds, with morphisms of manifolds as morphisms. This category I called SS.

A “generalized smooth space” XX is a presheaf on SS, hence a functor X:S opSet. X : S^{\mathrm{op}} \to Set \,.

The set X(U)X(U) it assigns to a manifold UU should be thought of as the set of “smooth maps from UU into XX”.

In words: a “generalized smooth space” is something which may be probed by manifolds.

I wrote S :=Set S op S^\infty := Set^{S^{op}} for ther category of presheaves over manifolds, hence for the category of “generalized smooth spaces”.

One such smooth space plays a spcial role: the presheaf Ω \Omega^\bullet which assigns to each manifold UU the set of differential forms on UU: Ω :UΩ (U). \Omega^\bullet : U \mapsto \Omega^\bullet(U) \,.

The existence of this generalized smooth space allows us to define the dg-algebra of differential forms on any generalized smooth space.

For XX a smooth space, we define the dg-algebra of forms on XX, Ω (X)\Omega^\bullet(X) to be the set of smooth space morphisms (morphisms of presheaves on manifolds) Ω (X):=Hom S (X,Ω ). \Omega^\bullet(X) := \mathrm{Hom}_{S^\infty}(X, \Omega^\bullet) \,.

One checks that the operation of exterior diufferential andf wedge product on each Ω (U)\Omega^\bullet(U) for each manifold UU naturally induces the structure of a DGCA on this set.

This construction gives us a contravariant functor from smooth spaces to their DGCAs of forms:

Ω :S DGCAs. \Omega^\bullet : S^\infty \to DGCAs \,.

This functor happens to have an adjoint going the other way round:

For each DGCA AA, we can form the presheaf on manifolds which I called X A X_A and which is defined by X A:UHom DGCAs(A,Ω (U)). X_A : U \mapsto \mathrm{Hom}_{DGCAs}(A,\Omega^\bullet(U)) \,.

This defines a contravariant funcotr which I denoted Hom(,Ω ())\mathrm{Hom}(-,\Omega^\bullet(-)): S DGCAs:Hom(,Ω ()). S^\infty \leftarrow DGCAs : \mathrm{Hom}(-,\Omega^\bullet(-)) \,.

So now coming to your question:

Then what is XX? and how does Ω (X)\Omega^\bullet(X) differ from Omega (X A)Omega^\bullet (X_A)?

By the symbol XX a generic generlaized smooth space is denoted, i.e. any presheaf on manifolds. The symbol X AX_A denotes a generalized smooth space which comes from a DGCA AA in the way just described, recall: X A:UHom DGCAs(A,Ω (U)). X_A : U \mapsto \mathrm{Hom}_{DGCAs}(A,\Omega^\bullet(U)) \,.

The general definition of the DGCA of differential forms on a genralized smooth space is of course the same in both cases, generally Ω (X):=Hom S (X,Ω ) \Omega^\bullet(X) := \mathrm{Hom}_{S^\infty}(X, \Omega^\bullet)

We can unwrap this definition for the special case that we have a smooth space X A X_A coming from a DGCA AA. In that case a differential form on X AX_A is a morphism of presheaves ω:X AΩ \omega : X_A \to \Omega^\bullet which for each manifold UU is a morphisms of sets omega U:Hom DGCAs(A,Ω (U))Ω (U). omega_U : \mathrm{Hom}_{DGCAs}(A,\Omega^\bullet(U)) \to \Omega^\bullet(U) \,.

Staring at this formula, one realizes that every element of AA naturally yields a differential form on X AX_A. We have an canonical inclusion of DGCAs AΩ (X A). A \hookrightarrow \Omega^\bullet(X_A) \,.

In words: every DGCA is a sub-DGCA of the DGCA of forms on some smooth space.

But more might be true. This inclusion should actually be a quasi-isomorphism: the cohomology of AA should actually equal the deRham cohomology of X AX_A, i.e. the cohomology of Ω (X A)\Omega^\bullet(X_A). I haven’t proved that yet, it’s one thing I am trying figure out. It is true in the closely related context of rational homotopy theory.

I should maybe add one more remark about how I think this adjunction between DGCAs and smooth spaces that I am talking about here (well, it was Todd who taught me that it actually is an adjunction!) relates to rational homotopy theory.

Let XX be any generalized smooth space. Then there is an obvious notion of the following gadgets:

\;\; \bullet nn-simplices in XX

\;\; \bullet the fundamental \infty-groupoid of XX

An nn-simpley in XX is simply a morphism of generalized smooth spaces from the standard nn-simplex Δ n n\Delta^n \subset \mathbb{R}^n into XX. The collection of all nn-simplices in XX is therefore Hom S (Δ n,X). \mathrm{Hom}_{S^\infty}(\Delta^n, X) \,.

Now let XX here come from a DGCA AA, X=X AX = X_A. Then, I think, the collection of nn-simplices Hom S (Δ n,X A) \mathrm{Hom}_{S^\infty}(\Delta^n, X_A) in X AX_A is precisely what is considered in rational homotopy theory, namely the set Hom DGCAs(A,Ω (Δ n)). \mathrm{Hom}_{DGCAs}(A, \Omega^\bullet(\Delta^n)) \,.

(I am not entirely sure about this, though. Clearly the latter set sits inside the former. That it also exhausts the former is a little more subtle, if true. Maybe somebody can help me here.)

So I am thinking: forming a smooth space X AX_A from a DGCA AA is possibly the more fundamental operation.

I made the same remark already before, here and here, in the discussion of integration of Lie \infty-algebras: the integration of an L L_\infty algebra gg given dually by the DGCA CE(g)CE(g) seems to be the same as saying that

The \infty-groupoid integrating gg is the fundamental \infty-groupoid of the smooth space X CE(g)X_{CE(g)}.

Then, if you like to think of \infty-groupoids as Kan complexes, this very definition should, I think, reproduce the Lie \infty-algebra integration method described by Getzler and Henriques.

But if I rather model my \infty-groupoids in a globular fashion, the same general idea still works just as well: I simply consider globular nn-paths in X CE(g)X_{CE(g)}.

Do you see what I mean?

So this approach of creating from a DGCA a presheaf on manifolds seems to be rather useful to me. In particular if and when we are tallking about \infty-connections and their parallel transport.

Please let me know what you think!

Posted by: Urs Schreiber on January 3, 2008 12:34 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

In words: a “generalized smooth space” is something which may be probed by manifolds.

That’s great way to put it.
If you said that in the earlier version, I misssed it.

Posted by: jim stasheff on January 3, 2008 1:38 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I wrote:

In words: a “generalized smooth space” is something which may be probed by manifolds.

Jim said:

That’s great way to put it. If you said that in the earlier version, I misssed it.

I may not have said it earlier. But at some point this statement appeared at the beginning of section 3.

It’s genrally a way to think of presheaf toposes: presheaves over a category SS are “all those things which may be probed by the objects of SS”.

For the special case of presheaves over manifolds, it nicely highlights what is going on: instead of demanding that a general smooth space has to locally look like a nice space (that’s the strategy, instead, used for instance in Fréchet spaces and the like), we just demand that we may map nice spaces into it.

Posted by: Urs Schreiber on January 3, 2008 1:50 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

My belief is that a weak nn-category is something that may be probed by strict nn-categories.

More precisely, the (n+1)(n + 1)-category of strict nn-categories and strict maps, transformations etc. between them should be dense in the (n+1)(n + 1)-category of weak nn-categories and weak functors, transformations etc. between them. Of course, this depends on having a higher-dimensional notion of density. What this means is that a weak nn-category can be regarded as a ‘nice’ presheaf on strict nn-categories.

Of course, no one would want to say that a weak nn-category is locally strict.

Posted by: Tom Leinster on January 3, 2008 2:38 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

My belief is that a weak nn-category is something that may be probed by strict nn-categories.

[…]

What this means is that a weak nn-category can be regarded as a ‘nice’ presheaf on strict nn-categories.

Hm, that’s interesting. If I had heard this before I did forget about it.

Are we really talking about presheaves here, or about pre-nn-stacks?

What does the category of presheaves over ωCat\omega\mathrm{Cat} resemble? What are are prestacks over ωCat\omega\mathrm{Cat}?

Posted by: Urs Schreiber on January 3, 2008 2:49 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I meant presheaves. About half of the proposed definitions of weak nn-category are of the form ‘a weak nn-category is a presheaf on 𝒟\mathcal{D} satisfying certain conditions’, where 𝒟\mathcal{D} is a subcategory of StrnCat\mathbf{Str}n\mathbf{Cat}. (For example, when n=1n = 1 you might take 𝒟=Δ\mathcal{D} = \Delta.) The idea behind this is that a weak nn-category A\mathbf{A} corresponds to the presheaf XX whose value at D𝒟\mathbf{D} \in \mathcal{D} is the set of weak functors DA\mathbf{D} \to \mathbf{A} — although at this point you don’t know what ‘weak nn-category’ and ‘weak functor’ mean.

I’m suggesting taking C\mathbf{C} to be all of StrnCat\mathbf{Str}n\mathbf{Cat}.

Posted by: Tom Leinster on January 3, 2008 4:43 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Tom Leinster wrote:

The idea behind this is that a weak nn-category A\mathbf{A} corresponds to the presheaf XX whose value at D𝒟\mathbf{D} \in \mathcal{D} is the set of weak functors DA\mathbf{D} \to \mathbf{A} — although at this point you don’t know what ‘weak nn-category’ and ‘weak functor’ mean.

I see. Cool. That’s obvious enough now that you say it, though I must admit this had not quite occurred to me before.

(For example, when n=1n = 1 you might take 𝒟=Δ\mathcal{D} = \Delta.)

Let me see. That yields simplicial sets, some of which are Kan complexes.

I assume you’d be able to say that Kan complexes are precisely the “nice” presheaves on Δ\Delta, as in your previous comment.

But now I am confused about the counting: Kan complexes correspond to \infty-groupoids, so what is the “n=1n=1” doing here?

I’m suggesting taking C\mathbf{C} to be all of StrnCat\mathbf{Str}n\mathbf{Cat}.

Interesting. At the end of my previous comment I was asking about considering presheaves on ωCat\omega\mathrm{Cat} (meaning strict infinity categories). Would you consider that?

I guess that’s related to my first question: how does the counting work in this approach?

Posted by: Urs Schreiber on January 3, 2008 5:36 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Urs asked about Kan complexes. But I wasn’t saying anything about groupoids — just categories. So when I mentioned the example 𝒟=Δ\mathcal{D} = \Delta, I had in mind that one would impose any of the various well-known conditions on a simplicial set equivalent to it being the nerve of a category (not necessarily a groupoid).

I was asking about considering presheaves on ωCat\omega Cat (meaning strict infinity categories). Would you consider that?

Certainly. My ‘nn’ was meant to range over natural numbers and ω\omega.

Posted by: Tom Leinster on January 3, 2008 6:57 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I was asking about considering presheaves on ωCat\omega Cat (meaning strict infinity categories). Would you consider that?

Certainly. My ‘nn’ was meant to range over natural numbers and ω\omega.

How do you characterize the “nice” presheaves on ωCat\omega Cat?

Posted by: Urs Schreiber on January 3, 2008 7:04 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Well, that’s the challenge! I assume that by a ‘nice’ presheaf you mean one corresponding to a weak ω\omega-category (in some reasonable sense). I’ve made several attempts to produce a slick definition of weak ω\omega-category of this form, but was never very successful.

(If by ‘nice’ you mean corresponding to a strict ω\omega-category, then ‘having a left adjoint’ would do, and ‘limit-preserving’ would probably be equivalent.)

Posted by: Tom Leinster on January 3, 2008 7:31 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I do mean “nice” in, I think, the sense of your original comment: those presheaves coprresponding to weak nn-categories.

I’ve made several attempts to produce a slick definition of weak ω\omega-category of this form, but was never very successful.

But for some n>1n \gt 1 you do have a working definition? Maybe even for all nn \in \mathbb{N}?

I am just trying to understand what you already know and what part you are proposing as a promising avenue.

Posted by: Urs Schreiber on January 3, 2008 7:41 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I’m afraid I didn’t even do it for n=2n = 2

Posted by: Tom Leinster on January 3, 2008 7:56 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I’m afraid I didn’t even do it for n=2n=2

Okay, now I get it. You were just proposing a certain possible approach.

Well, in a way it would be nicest if it turned out that we’d want all presheaves on ωCat\omega Cat.

By the way, since ωCat\omega Cat is in fact closed using the generalized Gray tensor product for ω\omega-categories, do we maybe want to look at ωCat (ωCat op) \omega Cat^{(\omega Cat^{op})} rather than at Set (ωCat op) Set^{(\omega Cat^{op})} ??

Posted by: Urs Schreiber on January 3, 2008 9:19 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

How about functors from weak to strict
There’s likely to be more of those.

Posted by: jim stasheff on January 3, 2008 7:36 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Hmm, that’s not something I’ve thought about. Maybe I should have. Why are there likely to be more?

Posted by: Tom Leinster on January 3, 2008 8:01 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I was thinking by analogy. Associative to A_\infty or vice versa. A strict morphism from assoc to A_\infty can hit only the strictly assoc part of the A_\infty

Posted by: jim stasheff on January 4, 2008 1:47 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Right, agreed. Thanks. But it was the weak functors I was probing with.

Posted by: Tom Leinster on January 4, 2008 4:34 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

cf. Chen’s plots or at least the philosophy

the issue is the class of probes

the alternative is to observe functions out of

cf. respectively homotopy groups and cohomology groups (pov I learned from Bott)

Posted by: jim stasheff on January 3, 2008 7:35 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Anyone care to write a post on ‘probing’?

I seem to recall Lawvere writing somewhere about sampling and classifying, as maps to and from objects.

Posted by: David Corfield on January 4, 2008 4:03 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Urs wrote:

What I would like to better understand is: if instead of rational homotopy theory, I am just interested in real homotopy theory (how much does that really lose??)

I remember worrying about this kind of thing a lot at some point. Physicists always work over the reals, even in places where mathematicians work over the rationals. What difference does it make?

For example, why do mathematicians often use rational cohomology where physicists prefer real? Of course physicists like the real version since they like deRham cohomology, and algebraists like the rational version since the rational numbers are logically simpler. But: when does it really make a significant difference?

It was hard to get an answer, since nobody seemed to write much about the question. But, I gradually got a sense of what was going on…

Anyway, for your question, my feeling is that you don’t lose anything by using reals instead of rationals. But, unless you’re dividing by 2π2 \pi in some formula somewhere you also don’t gain much — except the illusion that you’re doing physics.

You lose a lot going from integral to rational homotopy (or homology, or whatever). Why? When you take a \mathbb{Z}-module (otherwise known as an abelian group) and turn it into a \mathbb{Q}-module (otherwise known as a vector space over the rationals), this process is highly destructive! All the abelian groups that are ‘pure torsion’ — like /n\mathbb{Z}/n — just die! They all get sent to the zero-dimensional vector space over \mathbb{Q}.

In other words, the functor

ModMod \mathbb{Z}Mod \to \mathbb{Q}Mod

called ‘tensoring with \mathbb{Q}’ is not ‘essentially injective’: it can send nonisomorphic objects to isomorphic objects.

I guess it also doesn’t ‘reflect isomorphisms’. ‘Reflect isomorphisms’ is a piece of jargon that category theorists use to keep other people terrified: it means ‘preserve non-isomorphisms’.

In other words: F:CDF: C \to D reflects isomorphisms if whenever F(f):F(x)F(y)F(f): F(x) \to F(y) is an isomorphism, f:xyf : x \to y had to be an isomorphism. Or in still other words, if f:xyf : x \to y is a non-isomorphism, so is F(f)F(f).

On the other hand, the functor

ModMod \mathbb{Q}Mod \to \mathbb{R}Mod

called ‘tensoring with \mathbb{R}’ is a lot less drastic: it’s essentially injective, and it reflects isomorphisms.

There may be other things you’re worrying about, but I urge that you express your worries as worries about the functor

ModMod \mathbb{Q}Mod \to \mathbb{R}Mod

and how close or far it is to being an equivalence of categories. It’s not an equivalence of categories, since it’s not full — there are more n×mn \times m matrices with entries in \mathbb{R} than with entries in \mathbb{Q}. But, it’s faithful and essentially surjective and essentially injective and it reflects isomorphisms…

Personally what I found cool about rational homotopy theory was that it uses a concept of ‘differential form’ that makes sense for simplicial sets and gives a version of deRham cohomology defined over \mathbb{Q}. Very interesting if you like to investigate the tension between the ‘discrete’ and the ‘continuous’.

Posted by: John Baez on January 2, 2008 11:06 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

my feeling is that you don’t lose anything by using reals instead of rationals. But […] you also don’t gain much

Good, that’s what I thought!

Maybe differential forms are handled a little less awkwardly over the reals? But maybe I am just being ignorant.

Posted by: Urs Schreiber on January 2, 2008 11:28 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

“Maybe differential forms are handled a little less awkwardly over the reals? ”

Not unless you consider smooth functions less awkward than polynomials ! Even integration for polynomials is not awkward.

Posted by: jim stasheff on January 3, 2008 1:02 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Urs wrote:

Maybe differential forms are handled a little less awkwardly over the reals?

On a simplicial set? — no.

On a smooth manifold? — sure.

On an algebraic variety defined over \mathbb{Q}? — no.

On an algebraic variety defined over \mathbb{R}? — sure.

On a smooth space a la Chen? — sure.

So, it depends what kind of you want to study. If your goal is to do homotopy theory, simplicial sets are enough to model all homotopy types, so rational differential forms are enough. If you want lots of solutions to interesting PDEs, they’re probably not.

Posted by: John Baez on January 3, 2008 4:17 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

For more (than you wanat to know), there is

MR2035107 (2005b:16070) Michaelis, Walter Coassociative coalgebras. Handbook of algebra, Vol. 3, 587–788, North-Holland, Amsterdam, 2003. (Reviewer: E. J. Taft) 16W30 (00A20)

where the issue of cofree coassoc coalgs is described defintinitively around p. 720

The author promises me the pdf file so as soon as I have it will ask for it to be posted

Posted by: jim stasheff on January 3, 2008 5:26 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Re Todd’s comment above concerning the importance of being pro-finite:

the deRham complex Ω (U)\Omega^\bullet(U) for UU manifold with a non-finite number of points is not a pro-finite DGCA, right?

Posted by: Urs Schreiber on January 2, 2008 7:27 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I’m almost positive that’s right.

Maybe some slight disambiguation is in order. I guess in a lot of contexts, “pro-finite” means a (cofiltered) limit of literally finite objects. For example, in Grothendieck’s reworking of Galois theory, I think the algebraic fundamental group of a field is supposed to be the inverse limit of finite Galois groups over that field. Topologically, profinite objects would behave something like Cantor space (compact, Hausdorff, totally disconnected – is that the definition of a Stone space? I think maybe; I have to look this stuff up).

Here in our context, ‘pro-finite’ means an inverse limit of finite-dimensional structures (here, commutative algebras). So, it’s a little different I guess.

Even though I doubt that Ω (X)\Omega^{\bullet}(X) is pro-finite in this sense, it’s possibly worth keeping in mind that for DCGA’s there’s a process of passing to their pro-finite completions. (I’m not really sure this will turn out to be useful; more like something to keep in the back of one’s mind for now.)

Posted by: Todd Trimble on January 3, 2008 12:11 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Unfortunately for me, southern hemisphere December holidays are the long ones…

just a brief suggestion on terminology to get myself into the hang of commenting again: how about using pro-finite-type instead of profinite?

D

Posted by: David Roberts on January 4, 2008 2:58 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Jim Stasheff kindly points me to

M. Mostow, The differential space structrures of Milnor classifying spaces, simplicial complexes, and geometric realizations

which looks like it is very relevant. But I haven’t yet found the time to really study it

Posted by: Urs Schreiber on January 3, 2008 6:59 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I wrote:

Jim Stasheff kindly points me to

M. Mostow, The differential space structrures of Milnor classifying spaces, simplicial complexes, and geometric realizations

I may be too tired, but right now I am somewhat confused about some statements Mostow makes. He defines a differentiable space to be a topological space equppied with a sub-sheaf of its sheaf of continuous functions, to be thought of as the sub-sheaf of smooth functions (by fiat).

What I find confusing is that he says, section 4, that this is close to what Chen has been doing.

But is it? Chen is considering quasi-representable sheaves on an abstract site of open subsets of 2\mathbb{R} \cup \mathbb{R}^2 \cup \cdots, whose value on any open set is to be thought of as the collection of smooth maps into the smooth space to be defined.

But in Mostow’s definition, it’t the smooth maps out of the smooth space to be defined which appear.

So Mostow’s definition seems to be actually a little closer to Frölicher spaces.

By the way, Andrew Stacey has since recently a piece of comparative smootheology which I meant to write an entry about here.

You can find it on his website:

On Categories of Smooth Objects.

Posted by: Urs Schreiber on January 3, 2008 10:09 PM | Permalink | Reply to this

Cofree coalgebras

For issues of cofree coalgebras, the manual to read might be:
MR2035107 (2005b:16070) Michaelis, Walter Coassociative coalgebras. Handbook of algebra, Vol. 3, 587–788, North-Holland, Amsterdam, 2003. (Reviewer: E. J. Taft) 16W30 (00A20)

I have a pdf copy if anyone is interested.
Both the graded and ungraded cases are considered and quite different or, rather,
if the vector space in degree 0 is not the ground field, watch out.

Caveat: The author unfortunately DEFINES tensor coagebra to be the cofree thing, which in the general case is NOT the tensor space with the deconcantenation diagonal.

Posted by: jim stasheff on January 3, 2008 7:47 PM | Permalink | Reply to this
Read the post Comparative Smootheology
Weblog: The n-Category Café
Excerpt: A survey and comparison of the various notions of generalized smooth spaces, by Andrew Stacey.
Tracked: January 3, 2008 11:02 PM
Read the post Dijkgraaf-Witten and its Categorification by Martins and Porter
Weblog: The n-Category Café
Excerpt: On Dijkgraaf-Witten theory as a sigma mode, and its categorification by Martns and porter.
Tracked: January 5, 2008 3:23 AM
Read the post The Concept of a Space of States, and the Space of States of the Charged n-Particle
Weblog: The n-Category Café
Excerpt: On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
Tracked: January 10, 2008 10:32 AM

Re: Transgression of n-Transport and n-Connections

Urs,

A couple of weeks ago (eons in Café-blogging time!), I promised you I’d say a little more about internal homs of cocommutative coalgebras. Now that you’re returning your attention to some related issues (e.g., possible internal homs in DGCA opDGCA^{op}, as for example here), I thought I’d make good on that promise.

To review: back here, I gave a construction for internal homs of (dg) cocommutative coalgebras (the category of which I had, regrettably, denoted CoDGACoDGA – I’ll use CocCoc instead). This depended crucially on cofree cocommutative coalgebras, which are a little trickier to construct than one might think at first. I began to say something about that here, but then there were various interruptions. So now let me pick up where I left off.

The upshot of my previous comment was that general cocommutative coalgebras form a category dual not to commutative algebras, but rather to the category of commutative algebras of pro-finite-type (to adopt the terminology suggested by David Roberts). This rests on two observations:

  • The category of finite-dimensional cocommutative coalgebras is dual to the category of finite-dimensional commutative algebras (by taking linear duals, which is a contravariant equivalence).
  • A general cococommutative coalgebra is a canonical filtered colimit (viz., the union) of its finite-dimensional subcoalgebras.

The linear dual is a contravariant functor

() :CocDGCA(-)^\prime: Coc \to DGCA

which takes colimits in CocCoc to limits in DGCADGCA. Hence, by the two observations above, each cocommutative coalgebra is taken to a commutative algebra which is the cofiltered limit of its finite-dimensional quotients, that is to say, a pro-finite-type commutative algebra.

Pro-finite-type algebras may be regarded as having TVS (topological vector space) structures, as inverse limits of finite-dimensional spaces with their usual topologies, and the linear dual or transpose of a coalgebra map is a continuous algebra map. It is asserted by Getzler and Goerss that the linear dual

() :CocProfAlg cont(-)^\prime: Coc \to ProfAlg_{cont}

is a contravariant equivalence; a quasi-inverse map going the other way is gotten by taking the continuous dual.

(Urs also surmised, and I was inclined to agree, that the algebras Ω (X)\Omega^\bullet(X) were not themselves of profinite type. I seem to remember that the usual TVS structure on Ω (X)\Omega^\bullet(X) is a structure of Montel space.)

According to Getzler and Goerss, the cofree cocommutative coalgebra generated by a finite-dimensional chain complex VV may be constructed as the unique coalgebra whose linear dual is the profinite symmetric algebra generated by the the linear dual V V^\prime (i.e., take the algebra of polynomial functions on VV, and pass to the inverse limit of its finite-dimensional quotient algebras to get the profinite symmetric algebra). The cofree cocommutative coalgebra is therefore the continuous dual of the profinite completion of the symmetric algebra.

This description looks somewhat abstract and forbidding; it would be nice to have a more user-friendly description. A natural first instinct is that the cofree cocommutative coalgebra is just the product of S nS_n-invariants

C(V)= n(V n) S n,C(V) = \prod_n (V^{\otimes n})^{S_n},

but recall that’s not right because this product isn’t a coalgebra in any obvious way (it would be correct if tensor products respected countable products, but they don’t). But, the cofree coalgebra is embedded inside this product, and can be considered the best coalgebra approximation to this guess: it’s the union of all finite-dimensional coalgebras “naturally embedded” inside C(V)C(V).

By “naturally embedded”, I mean this. The symmetric algebra S(V )S(V^\prime) is dense in its profinite-type completion S prof(V )S_{prof}(V^\prime), so a continuous algebra map S prof(V )S_{prof}(V^\prime) \to \mathbb{R} is uniquely determined by its restriction to S(V )S(V^\prime). Now the space of linear functionals S(V )S(V^\prime) \to \mathbb{R} is C(V)C(V) if VV is finite-dimensional. So the cofree coalgebra must be embedded in the ordinary linear dual S(V ) =C(V)S(V^\prime)^\prime = C(V). By a “natural embedding” of a finite-dimensional coalgebra i:CC(V)i: C \hookrightarrow C(V), I just mean the linear transpose of an algebra quotient q:S(V )C q: S(V^\prime) \to C^\prime.

This still doesn’t seem particularly useful, but perhaps we can develop some intuition. We are taking the continuous linear dual of an algebra of polynomial functions (with a TVS structure specified by a uniformity defined by ideals of finite codimension). It is natural to think of the elements of the dual as measures of some sort, and (therefore) the cofree coalgebra as a coalgebra of measures. For example, if VV is 1-dimensional (and here I’m ignoring dg structure), we can think of C(V)= nC(V) = \prod_n \mathbb{R} as the space of formal power series [[x]]\mathbb{R}[[x]], or (better) as a space of formal linear combinations

na nδ 0 (n)n!\sum_n \frac{a_n \delta_{0}^{(n)}}{n!}

where δ 0\delta_0 is the Dirac functional supported at 00 and δ 0 (n)\delta_{0}^{(n)} is its n thn^{th} distributional derivative. We are trying to cut back on this space of measures supported at 00 to get the cofree coalgebra, where the comultiplication Δ\Delta on the “good” measures mm should satisfy

Δ(m),fg=m,fg.\langle \Delta(m), f \otimes g \rangle = \langle m, f g \rangle.

(Urs, I’m going to stop here for the time being; this is to be continued, but I want/need to do some calculations first, and also think about what Jim Stasheff said here, which may be very useful. Also, Jim has mentioned the name Walter Michaelis a few times, who has apparently written a handbook on coalgebras with some information on cofree stuff – with some luck, I can get access to that.)

PS: I haven’t forgotten that you are contemplating internal homs for DCGA’s, based on passing back and forth between this category and the topos of presheaves on smooth manifolds. Hopefully I can think about what you are saying here.

Posted by: Todd Trimble on January 17, 2008 3:27 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Todd,

thanks a whole lot! I appreciate it. I’ll read this carefully and then get back to you.

What I did meanwhile was working my way backwards: I have this class of examples here which I know involve some kind of internal hom in some category, and so I made some educated guesses and compared the results with what I knew the answer should be.

This way I found that the following construction is “good”. It clearly has a smell of internal hom to it, but I am not sure exactly what to make of it:

the construction is: given two DGCAs AA and BB, form the DGCA of differential forms on the presheaf UHom DGCA(A,BΩ (U)), U \mapsto Hom_{DGCA}(A,B \otimes \Omega^\bullet(U)) \,, i.e. form Ω (Hom DGCA(A,BΩ ())). \Omega^\bullet(Hom_{DGCA}(A,B \otimes \Omega^\bullet(--))) \,.

From my examples I know that this is a “right answer”. But I am not entirely sure to which precise question! :-)

I’ll try to think about if it can possibly be related to the construction you describe, by somehow extracting profinite “models” for AA and BB.

Thanks again. More later.

Posted by: Urs Schreiber on January 17, 2008 5:52 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

I think it’s right in that it does what you need to do.
Whether or not it’s internal hom is interesting but not needed. Cf. vaarious topologies on maps(X,Y) whihc each of usefulness without being internal homs

Posted by: jim stasheff on January 17, 2008 7:19 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Jim Stasheff advises:

I think it’s right in that it does what you need to do. Whether or not it’s internal hom is interesting but not needed. Cf. vaarious topologies on maps(X,Y) whihc each of usefulness without being internal homs

Yes, sorry, I keep going on about this. (I had agreed in private email discussion with Jim already that I don’t overly emphasize this internal hom issue.)

I can certainly proceed with just accepting the definition

maps(A,B)=Ω (Hom DGCAs(A,BΩ ())) maps(A,B) = \Omega^\bullet(Hom_{DGCAs}(A,B \otimes \Omega^\bullet(--)))

(where “maps” is really to be read as “forms on maps”)

but eventually I want to determine its collection of nice properties.

And actually, there is something even more important to be understood:

currently in my discussion here I actually consider only elements of Ω (Hom DGCAs(A,BΩ ()))\Omega^\bullet(Hom_{DGCAs}(A,B \otimes \Omega^\bullet(--))) which come from pairs (a,c)(a,c) with aa an element of AA and cc a “current” (linear functional) on BB.

I haven’t tried to show yet whether the elements obtained this way actually exhaust all of Omega (Hom DGCAs(A,BΩ ()))Omega^\bullet(Hom_{DGCAs}(A,B \otimes \Omega^\bullet(--))).

Hm, something is going on here. What I just said really means that after all there is something like AB *A \otimes B^* lurking here, which might bring us back all the way to our discussion of internal homs in cochain complexes.

But while just forming AB *A \otimes B^* seems to be what AKSZ/Roytenberg essentially do #, I just couldn’t understand how their construction actually works (for one: I couldn’t see the algebra structure on their “inner hom”-complex). I am stronly expecting that once the dust has settled and I emerge with some definite result or other, Dmitry will just tell me that this is what he’s been saying all along. But maybe I will have learned something in the process.

Posted by: Urs Schreiber on January 17, 2008 8:40 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

MY `wait,wait’ comment was to brief. your Hom does mean morphisms of DGCAs but does the algebrastructure you see on Hom use the algebra of the source at all?

As for something like A tensor B^*, shouldn’t that be A^* tensor B unless you
are think of the additional dual when you pas to forms?

Posted by: jim stasheff on January 18, 2008 12:50 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

does the algebrastructure you see on Hom use the algebra of the source at all?

It’s maybe not the algebra structure on the forms on the “Hom space” itself which depends on the algebra structure of AA and BB, but the differential which does.

I think that makes sense: “points” in the Hom-space are algebra homomorphism, and the product of functions on this space should be just the product of functions.

But then when you act with the differential on these functions, it encodes information about how to deform on algebra homomorphism ABA \to B to another one, since it knows about tangents to the space of algebra homomorphisms.

And indeed, we see that the differential on forms on the “Hom space” comes precisely from the DGCA-homotopies you introduced, wich I liked to call “concordances”.

In fact, one can maybe see the construction Ω (Hom DGCA(A,BΩ ()))\Omega^\bullet(Hom_{DGCA}(A,B\otimes\Omega^\bullet(--))) as a unified way to subsume all these higher DGCA homotopies into one single structure:

for any given UU of dimension nn, an element in Hom DGCA(A,BΩ (U)Hom_{DGCA}(A,B\otimes\Omega^\bullet(U) is an nn-fold homotopy of algebra homomorphisms from AA to BB.

So it’s here that the crucial information about AA and BB is picked up, I would say.

Posted by: Urs Schreiber on January 18, 2008 10:23 AM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Urs wrote:

But then when you act with the differential on these functions, it encodes information about how to deform on algebra homomorphism A→B to another one, since it knows about tangents to the space of algebra homomorphisms.

So your important algebra structure is NOT on Hom itself but on the forms on Hom

If you are looking at deformations of algebra maps, you might want to consider the standard machinery of deformation theory (pace Gerstenhaber) jazzed up to include internal differentials (which has been done). N.B. no need to involve smoothness

Posted by: jim stasheff on January 18, 2008 12:45 PM | Permalink | Reply to this

Internal homs

Thanks, Urs. By the way, I’m not trying to push a particular point of view here; I actually don’t have one! That is: I’m far from convinced that the whole business of internal homs of cocommutative coalgebras is something you really need here, so don’t let me distract you – if your approach smells right to you, please continue full throttle and I’ll follow what I can. Meanwhile, I may continue submitting comments on this topic, if only because I’m having some fun with it myself!

Posted by: Todd Trimble on January 17, 2008 8:59 PM | Permalink | Reply to this

Re: Internal homs

Todd,

conversely, please don’t let my experimenting distract you from what we had been discussing. It’s not at all that I am no longer interested in that, quite on the contrary.

Somehow there is a bigger insight to be gained here, it seems, and the various puzzle piece we are being assembling will probably all find their place.

Let me indicate one interesting thing I learned from what has happened so far, just in case IT rings any bell with what you have been thinking about:

recall how one of my original puzzlements was this:

- For Lie \infty-groupoids, taking the internal hom will send two Lie \infty-groupoids to another Lie \infty-groupoid.

- Then we noticed that in some applications it seems that we actually want the internal hom to be something like a \mathbb{Z}-category, instead.

- It seemed to be clear how this came about by switching to the Lie algebraic picture, where the internal hom in cochain complexes has the property that forming it from two non-negatively graded things will yields some arbitrarily-graded thing.

-But we could’t see

a) how this operation on cochain complexes extended to dg-algebras (probably because it doesn’t! as you emphasized)

b) how this matches with the Lie \infty-groupoid picture, of which it is supposed to be the differential version.

- So then you indicated that coalgebras might come to the rescue. I need to understand which resolution these issues will find there.

- But in parallel, I noticed this other construction. Notice how that points to some unification:

the “maps” construction I provided does send two non-negatively graded dg-algebras to another non-negatively graded dg-algebra — BUT that resulting dg-algebra is, as I emphasized – two copies of what one might have expected (it’s the Weil algebra of, mapping cone of the identity on, weak cokernel of the identity on, etc. one might naively have expected).

On top of that, I am seeing that we get that other result with stuff in negative degree from that by dualizing the unexpected shifted copy appearing here.

So it’s beginning to appear to all make sense.

To amplify, for those mathematical physicists out there:

the BV dg-algebra is in supergeometry imagery the functions on the cotangent bundle of configuration space.

The “co” is responsible for the negative degrees.

AKSZ/R try to interpret that, and in particular the appearance of negative degrees here, with the inner hom in cochain complexes (or something like that). But I fail to see the algebra structure on that.

But, I find that doing that “quasi”-inner hom of dg-algebras which does behave nicely, the one that produces the naively unexpected shifted copy, corresponds to looking at the tangent bundle to configuration space.

After dualizing this “by hand” I arrive at the BV dg-algebra.

So something is going on.

Posted by: Urs Schreiber on January 17, 2008 9:28 PM | Permalink | Reply to this

Re: Transgression of n-Transport and n-Connections

Wait, wait! Did you quasi-internal hom use the algebra structure of the source? how?

Posted by: jim stasheff on January 18, 2008 12:44 AM | Permalink | Reply to this

Calculating cofree coalgebras

For such interest as it may have, I’ll continue with the theme on cofree cocommutative coalgebras, as discussed last time here. This time I’ll calculate a basic example: working within the category of vector spaces over a field (say \mathbb{C}), I’ll compute the cofree cocommutative coalgebra CC generated by a 1-dimensional space. It’s simpler than I thought it would be!

(And though this may seem very much to be a ‘toy’ example, I believe it helps me see my way clear to the more general case, and also to the dg case; more on this later.)

As explained last time, the cofree coalgebra CC on one cogenerator xx is the best coalgebra approximation to the dual of the free algebra,

C[x] [[x]].C \hookrightarrow \mathbb{C}[x]^\prime \cong \mathbb{C}[[x]].

More precisely, CC is formed as the union or filtered colimit (in VectVect) of subspace inclusions

i=q :A [x] i = q^\prime: A^\prime \hookrightarrow \mathbb{C}[x]^\prime

that are linear transposes of projections to finite-dimensional quotient algebras AA:

q:[x][x]/I=A.q: \mathbb{C}[x] \to \mathbb{C}[x]/I = A.

The A A^\prime here carry coalgebra structures dual to the algebra structures AA, and since the underlying functor

CocVectCoc \to Vect

is easily seen to create colimits (even if we don’t yet assume the theorem that it is comonadic), we get an induced coalgebra structure CC on the filtered colimit as calculated in VectVect. This CC is the cofree coalgebra.

By a kind of “calculus of symbols”, I will show that

  • CC is naturally isomorphic to the localization of [x]\mathbb{C}[x] with respect to the ideal (x)(x)

where the localization [x] (x)\mathbb{C}[x]_{(x)} is naturally embedded in the local completion [[x]]\mathbb{C}[[x]]. I will also compute the coalgebra structure on [x] (x)\mathbb{C}[x]_{(x)}.

First, let us recall the duality between [x]\mathbb{C}[x] and [[x]]\mathbb{C}[[x]]. In one sense, it’s perfectly obvious: the polynomial algebra as a direct sum of copies of \mathbb{C},

[x] nx n,\mathbb{C}[x] \cong \sum_n \mathbb{C} \cdot x^n,

is dual to the direct product

[[x]] nx n,\mathbb{C}[[x]] \cong \prod_n \mathbb{C} \cdot x^n,

where we view elements of the latter as formal power series

n=0 a nx n.\sum_{n = 0}^{\infty} a_n x^n.

In this set-up, we think of the linear basis x nx^n of [x]\mathbb{C}[x] as dual to the TVS basis x nx^n of [[x]]\mathbb{C}[[x]]:

x n,x m=δ m,n,\langle x^n, x^m \rangle = \delta_{m, n},

although it may be wiser to interpret formal power series na nx n\sum_n a_n x^n as symbols for measures which are functionals on the polynomial algebra of functions. For example, x 0x^0 would be the symbol for the Dirac functional

δ 0=eval 0:[x]\delta_0 = eval_0: \mathbb{C}[x] \to \mathbb{C}

and x 1x^1 would be the symbol for the functional which assigns to each polynomial ff its linear coefficient. This last example can also be interpreted in terms of the derivative of the Dirac functional, where we have

f ,δ 0=f,(δ 0) \langle f^\prime, \delta_0 \rangle = - \langle f, (\delta_0)^\prime \rangle

(the annoying minus sign is an artifact of a “integration by parts” heuristic often used in the theory of distributions).

Following this further, x mx^m can be interpreted as the symbol for the measure or functional

x m:[x]x_m: \mathbb{C}[x] \to \mathbb{C}

which returns the coefficient of x mx^m of a polynomial f[x]f \in \mathbb{C}[x]; we can also write it in terms of the m thm^{th} derivative of the Dirac functional:

x m=(1) mδ 0 (m)m!.x_m = (-1)^m \frac{\delta_{0}^{(m)}}{m!}.

Now let us consider a typical element

ξ= ma mx m:[x]\xi = \sum_m a_m x_m: \mathbb{C}[x] \to \mathbb{C}

of the cofree coalgebra CC. By our description of CC, it belongs to the dual of a finite-dimensional algebra quotient q:[x][x]/Iq: \mathbb{C}[x] \to \mathbb{C}[x]/I; therefore it vanishes on the ideal II, generated by a polynomial g(x)=b 0++b nx ng(x) = b_0 + \ldots + b_n x^n. This ideal II is the span of elements x jg(x)x^j g(x); therefore

x jg(x), ma mx m=0\langle x^j g(x), \sum_m a_m x_m \rangle = 0

which after a short calculation yields linear recurrence relations, one for each jj:

a jb 0+a j+1b 1++a j+nb n=0.a_j b_0 + a_{j+1} b_1 + \ldots + a_{j+n} b_n = 0.

This enables us to recursively solve for all the a ma_m once we know initial values a 0,,a n1a_0, \ldots, a_{n-1}.

It may be illuminating to work through a simple example. Let g(x)=1x 2g(x) = 1 - x^2. The linear recurrence relations say

a ja j+2=0a_j - a_{j+2} = 0

for all jj. Setting a 0=1a_0 = 1 and a 1=1a_1 = 1, we have 1=a 0=a 2=1 = a_0 = a_2 = \ldots and 0=a 1=a 3=0 = a_1 = a_3 = \ldots, so the corresponding functional becomes in this case

x 0+x 2+x 4+x_0 + x_2 + x_4 + \ldots

whose symbol is

x 0+x 2+x 4+=11x 2.x^0 + x^2 + x^4 + \ldots = \frac{1}{1-x^2}.

If we set a 0=0a_0 = 0 and a 1=1a_1 = 1, we are similarly led to the operator

x 1+x 3+x 5+x_1 + x_3 + x_5 + \ldots

with symbol

x 1+x 3+x 5+=x1x 2.x^1 + x^3 + x^5 + \ldots = \frac{x}{1-x^2}.

Somehow this reminds me a bit of fundamental solutions or Green’s functions for differential operators. The ‘xx’ is something like the Fourier transform of ‘taking the derivative at 0’, up to some possible finagling with signs, and the expression x1x 2\frac{x}{1-x^2} is something like the formal Fourier transform of a fundamental solution uu to (1D 2)(u)=δ 0 (1 - D^2)(u) = \delta_{0}^{\prime}. I’m not claiming too much should be made of this connection – this sort of thing is not exactly my métier in the first place. I just find it sort of suggestive.

A little further thought shows this is no coincidence: the linear recurrence relations required to compute the dual of [x]/(g(x))\mathbb{C}[x]/(g(x)) as embedded in [[x]]\mathbb{C}[[x]] are the same as the linear recurrence realtions required to compute g(x) 1g(x)^{-1}. Thus, the coalgebra dual to [x]/(x jg(x))\mathbb{C}[x]/(x^j g(x)), where g(x)g(x) is an n thn^{th} degree polynomial with nonzero constant coefficient, is the coalgebra of symbols spanned by formal power series expressions which represent

1g(x),xg(x),,x n+j1g(x).\frac{1}{g(x)}, \frac{x}{g(x)}, \ldots, \frac{x^{n+j-1}}{g(x)}.

Since every ξ\xi belonging to the cofree coalgebra CC is contained in the dual of some finite-dimensional algebra of the form [x]/(x jg(x))\mathbb{C}[x]/(x^j g(x)) where g(0)0g(0) \neq 0, it follows that its symbol belongs to [x]1g(x)\mathbb{C}[x] \cdot \frac{1}{g(x)}. Hence the space of symbols for the cofree coalgebra is the localization of [x]\mathbb{R}[x] obtained by inverting all polynomials g(x)g(x) with nonzero constant coefficient. That is, we may identify CC with the localization [x] (x)\mathbb{C}[x]_{(x)}.

Finally, a few words on the coalgebra structure on [x] (x)\mathbb{C}[x]_{(x)}. The comultiplication Δ\Delta is adjoint to multiplication on the algebra [x]\mathbb{C}[x], and so we have

x mx n,ξ=x mx n,Δ(ξ).\langle x^m \cdot x^n, \xi \rangle = \langle x^m \otimes x^n, \Delta(\xi) \rangle.

For ξ= na nx n\xi = \sum_n a_n x_n, this forces the familiar rule

Δ(ξ)= na n( i+j=nx ix j)\Delta(\xi) = \sum_n a_n (\sum_{i+j = n} x_i \otimes x_j)

(although this doesn’t give a finite linear combination, i.e., an element of [[x]][[x]]\mathbb{C}[[x]] \otimes \mathbb{C}[[x]], unless the a na_n satisfy linear recurrence relations coming from the fact that ξ\xi belongs to [x] (x)\mathbb{C}[x]_{(x)}).

The only reasonable explicit formulas I have so far for Δ\Delta exploit the theory of partial fraction decompositions. If g(x)g(x) is a nonconstant polynomial with nonzero constant coefficient, then (working over \mathbb{C}) there is a unique finite decomposition of the form

1g(x)= iA i(1r ix) n i\frac{1}{g(x)} = \sum_i \frac{A_i}{(1 - r_i x)^{n_i}}

where the A iA_i are constants and the pairs (r i,n i)(r_i, n_i) are distinct. So it’s enough to determine Δ((1rx) n)\Delta((1 - r x)^{-n}). For n=1n = 1, this is easy: a short calculation yields

Δ(11rx)=11rx11rx\Delta(\frac{1}{1 - r x}) = \frac{1}{1 - r x} \otimes \frac{1}{1 - r x}

so that (1rx) 1(1 - r x)^{-1} is a grouplike element. For nn > 11, the formula is more complicated; I seem to get

Δ(1(1rx) n+1)= i+j=n1(1rx) i+11(1rx) j+1 i+j=n11(1rx) i+11(1rx) j+1,\Delta(\frac{1}{(1 - r x)^{n+1}}) = \sum_{i + j = n} \frac{1}{(1 - r x)^{i+1}} \otimes \frac{1}{(1 - r x)^{j+1}} - \sum_{i + j = n-1} \frac{1}{(1 - r x)^{i+1}} \otimes \frac{1}{(1 - r x)^{j+1}},

although I don’t guarantee a mistake hasn’t slipped in somewhere!

Posted by: Todd Trimble on January 20, 2008 4:20 PM | Permalink | Reply to this

Constructing cofree cocommutative coalgebras

I’d like to continue with my monologue on cofree cocommutative coalgebras, on the off-chance that it might actually turn out to be useful one of these days, and also because these days I’m enjoying contemplating them.

One reason I’m enjoying them is that now they don’t seem nearly so scary as this description (essentially what I read in Getzler and Goerss) might suggest:

The cofree cocommutative coalgebra Cof(V)Cof(V) cogenerated by a finite-dimensional vector space VV is the continuous linear dual of the pro-completion of the free commutative algebra on the linear dual V V^\prime, where the pro-completion of an algebra is formed by taking the inverse limit of its finite-dimensional quotient algebras (with their standard topologies).

Fine, as abstract descriptions go, unless you want to do calculations with the darned things – then it becomes a headache! The point of this comment is to make the cofree construction much more transparent and amenable to calculation.

The cofree cocommutative coalgebra on one cogenerator.

In the case where VV is 1-dimensional, I indicated last time how Cof(V)Cof(V) could be described as a certain coalgebra structure on the localization of [x]\mathbb{C}[x] with respect to the prime ideal (x)(x). Let me now make this a little more explicit. In the first place, by “cofreeness” we mean there is a universal arrow

π:Cof(V)V\pi: Cof(V) \to V

which is universal among linear maps AVA \to V out of coalgebras AA. In the 1-dimensional case V=V = \mathbb{C}, it is the composite

[x] (x)i nx nπx 1=\mathbb{C}[x]_{(x)} \stackrel{i}{\hookrightarrow} \prod_{n} \mathbb{C} \cdot x^n \stackrel{\pi}{\to} \mathbb{C} \cdot x^1 = \mathbb{C}

where the first arrow interprets rational functions p(x)g(x)\frac{p(x)}{g(x)} in [x] (x)\mathbb{C}[x]_{(x)} as formal power series, and the second projects power series onto their linear coefficients.

Now suppose given a linear functional f:Af: A \to \mathbb{C} on a coalgebra AA. We are trying to lift this functional to a coalgebra map f^:ACof()\hat{f}: A \to Cof(\mathbb{C}). In the first place, there is this Very Useful ObservationTM that every coalgebra AA is the filtered colimit (union) of its finite-dimensional subcoalgebras, and that the forgetful functor from coalgebras to vector spaces preserves colimits. This implies that once we know how to obtain the lift f^\hat{f} in the case where AA is finite-dimensional, then we can do it in general.

So suppose that AA is finite-dimensional. The functional f:Af: A \to \mathbb{C} can be interpreted as an element of the linear dual A A^\prime, with its induced algebra structure. Consider the algebra map [x]A \mathbb{C}[x] \to A^\prime which sends xx to ff. The kernel of this map is an ideal II of finite codimension, and the linear transpose of this map,

A[[x]],A \to \mathbb{C}[[x]],

factors as the coalgebra map A([x]/I) A \to (\mathbb{C}[x]/I)^\prime followed by the mono j:([x]/I) ([x]) [[x]]j: (\mathbb{C}[x]/I)^\prime \to (\mathbb{C}[x])^\prime \cong \mathbb{C}[[x]] which is the transpose of the projection q:[x][x]/Iq: \mathbb{C}[x] \to \mathbb{C}[x]/I. Now the analysis given last time showed that every such jj factors through a mono ([x]/I) [x] (x)(\mathbb{C}[x]/I)^\prime \to \mathbb{C}[x]_{(x)} which is a coalgebra map. The composite coalgebra map

A([x]/I) [x] (x)A \to (\mathbb{C}[x]/I)^\prime \to \mathbb{C}[x]_{(x)}

then gives the desired lift f^:ACof()\hat{f}: A \to Cof(\mathbb{C}) of f:Af: A \to \mathbb{C}.

To connect this with the description given by Getzler and Goerss: the coalgebra [x] (x)\mathbb{C}[x]_{(x)} is uniquely determined up to isomorphism as the coalgebra whose linear dual is the algebra obtained by profinite-type completion, i.e.,

hom Vect([x] (x),)lim I[x]/Ihom_{Vect}(\mathbb{C}[x]_{(x)}, \mathbb{C}) \cong \lim_I \mathbb{C}[x]/I

where the limit is over nonzero ideals II. In other words, by taking duals, the localization [x] (x)\mathbb{C}[x]_{(x)} (which, like every ring localization, is a certain filtered colimit) is taken over to the cofiltered limit taken over the system of nonzero ideals (cofiltered, since we can take the intersection of ideals). In some sense which is not yet quite clear to me, this isomorphism feels very closely connected to the theory of partial fraction decompositions.

Aside on partial fraction decompositions.

Partial fraction decomposition is something we teach calculus students as part of their repetoire in computing antiderivatives (in this case of rational functions), and it’s something I’ve always enjoyed thinking about from first principles – there’s some real algebra there! I’m not sure whether that topic is standard in a graduate algebra course; I don’t think it is, even though I never took such a course. So maybe it won’t be taken amiss if I give my own take on this.

Let OO be a principal ideal domain, and let KK be its field of fractions. Partial fraction decomposition really has to do with the OO-module structure of K/OK/O:

Theorem (partial fraction decomposition): The canonical map

pO[1p]/OK/O,\sum_p O[\frac{1}{p}]/O \to K/O,

where the sum ranges over generators of prime ideals pp, is an isomorphism.

The standard example we teach our calculus students is the case where O=[x]O = \mathbb{R}[x], where K=(x)K = \mathbb{R}(x) is the field of rational functions. It says that any rational function f(x)/g(x)f(x)/g(x), considered modulo polynomials, has a unique expression as a linear combination of quotients a i(x)/p i(x) r ia_i(x)/p_i(x)^{r_i} where the p ip_i are irreducible factors of g(x)g(x). Of course, this is the function field case; it applies equally well to the structure of say /\mathbb{Q}/\mathbb{Z} – whenever I have taught calculus, this is the first case I would discuss before embarking on the application to rational functions (on the assumption that students are more comfortable with integers than with polynomials).

There are some funny little spin-offs of this theorem. In the case O=O = \mathbb{Z}, the summand [1p]/\mathbb{Z}[\frac{1}{p}]/\mathbb{Z} is an inductive limit (a colimit) along the evident sequence

/p j/p j+1\ldots \hookrightarrow \mathbb{Z}/p^j \hookrightarrow \mathbb{Z}/p^{j+1} \hookrightarrow \ldots

of pp-torsion groups. The dual of this torsion group,

hom([1p]/,/),hom(\mathbb{Z}[\frac{1}{p}]/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}),

is the projective limit of the sequence of quotient maps /p j+1/p j\mathbb{Z}/p^{j+1} \to \mathbb{Z}/p^j (using the fact that finite abelian groups are self-dual), and this gives a standard presentation of the pp-adic integers, denoted here as ^ p\hat{\mathbb{Z}}_p. By partial fraction decomposition, we therefore have

hom(/,/)hom( p[1p]/,/) p^ p.hom(\mathbb{Q}/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) \cong hom(\sum_p \mathbb{Z}[\frac{1}{p}]/\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) \cong \prod_p \hat{\mathbb{Z}}_p.

This looks like the maximal compact subring of the (non-archimedean part of the) adeles, and sure enough, multiplication in this ring matches composition of endomorphisms on /\mathbb{Q}/\mathbb{Z}! Continuing this thought, one can also identify hom(,/)hom(\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) with the non-archimedean part of the adeles. I somehow think something interesting could be made of this, but I’m not sure what.

The cofree construction Cof(V)Cof(V) for finite-dimensional VV.

Now that we have in hand a concrete description of Cof(V)Cof(V) when VV is 1-dimensional, we can easily get Cof(V)Cof(V) for any finite-dimensional VV. This rests on three observations:

  • The cofree construction Cof:VectCocCof: Vect \to Coc is a right adjoint, and hence preserves products;
  • Any finite-dimensional vector space VV is a product of 1-dimensional spaces;
  • The product of cocommutative coalgebras is given by the tensor product of underlying vector spaces.

So, the cofree cocommutative coalgebra Cof( n)Cof(\mathbb{C}^n) on nn cogenerators is an nn-fold tensor product of copies of [x] (x)\mathbb{C}[x]_{(x)}. Since we have

[x] (x)colim g:g(0)0[x]1g(x)\mathbb{C}[x]_{(x)} \cong colim_{g: g(0) \neq 0} \mathbb{C}[x] \cdot \frac{1}{g(x)}

where the colimit is along inclusions [x]1g(x)[x]1h(x)\mathbb{C}[x] \cdot \frac{1}{g(x)} \hookrightarrow \mathbb{C}[x] \cdot \frac{1}{h(x)} whenever gg divides hh, and since the tensor product of nn copies of [x]\mathbb{C}[x] is isomorphic to [x 1,,x n]\mathbb{C}[x_1, \ldots, x_n], we quickly conclude that Cof( n)Cof(\mathbb{C}^n) is the localization of [x 1,,x n]\mathbb{C}[x_1, \ldots, x_n] where we invert products of the form p 1(x 1)p 2(x 2)p n(x n)p_1(x_1)p_2(x_2) \ldots p_n(x_n) (p i(0)0p_i(0) \neq 0 for all ii). The coalgebra structure is easily deduced from the way it works in the 1-cogenerator case.

The cofree construction Cof(V)Cof(V) for general VV.

The Very Useful ObservationTM (that coalgebras are filtered colimits of their finite-dimensional subcoalgebras, preserved and reflected by taking underlying vector spaces) gives a quick description of Cof(V)Cof(V) for general VV, in terms of Cof(V )Cof(V^\prime) for finite-dimensional V V^\prime:

Cof(V):=colim V VCof(V )Cof(V) := colim_{V^\prime \hookrightarrow V} Cof(V^\prime)

where the colimit is along inclusions between finite-dimensional subspaces V V^\prime. For with this definition, we indeed get a universal arrow Cof(V)VCof(V) \to V from this colimit to VV, induced by the maps

Cof(V )πV V.Cof(V^\prime) \stackrel{\pi}{\to} V^\prime \hookrightarrow V.

Moreover, given a linear map f:AVf: A \to V where AA is a coalgebra, we get for each finite-dimensional subcoalgebra A AA^\prime \hookrightarrow A a linear map A f(A )A^\prime \to f(A^\prime) to a finite-dimensional space, and thence a coalgebra map A Cof(f(A ))A^\prime \to Cof(f(A^\prime)). Hence we get a coalgebra map

A Cof(f(A ))Cof(V)A^\prime \to Cof(f(A^\prime)) \hookrightarrow Cof(V)

with the definition of Cof(V)Cof(V) above, and then, since AA is a colimit of the A A^\prime, we get an induced coalgebra map f^:ACof(V)\hat{f}: A \to Cof(V) which lifts ff, as desired.

Next time: I’ll carry the discussion over to the dg world (where actually some things simplify!). Over time, I’d like to work my way toward the explicit calculation of internal homs of dg cocommutative coalgebras.

Posted by: Todd Trimble on January 23, 2008 5:20 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

Over time, I’d like to work my way toward the explicit calculation of internal homs of dg cocommutative coalgebras.

Hm, this journey is going to be longer than I thought it would be. But I am following. Somewhere behind you…

Posted by: Urs Schreiber on January 23, 2008 5:27 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

It’s only taking a long time because I’m slow, and actually inexperienced in these matters, so I’m deducing things from scratch.

Plus, as a mathematician I often lack the guts just to take an inspired guess what the answer should look like in advance. What’s that expression physicists use? Something like, “never calculate something you don’t already know the answer to,” or something like that? I never could live life like that! :-P

Posted by: Todd Trimble on January 23, 2008 5:40 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

I often lack the guts just to take an inspired guess what the answer should look like in advance.

Interesting. It feels to me like it would take less guts to embark on a complicated calculation if one has a rough idea where it will lead.

I always found the history of the ascent of Mount Everest remarkable: before the first human managed to get to the top, a couple of very serious and very experienced expeditions had failed just a little bit below the summit.

Nowadays, every grandmother and her internet sattelite phone is being carried to the summit and leaves it on ski.

So it seems to be not just a matter of modern equipment. I could imagine that the members of the very first expeditions, those who failed just beneath the summit, had, deep inside, unconciously, doubts that it might not be possible in principle to reach the summit.

Then later it was found that at least in principle, it is possible for mere mortals to reach the tip of Mount Everest.

That may make all the difference. If you are feeling completely exhausted 100 meters from the summit today, you just keep going, because you know that’s what the people before you did, too, and they survived (or 70% or so did, at least :-/ ).

If you felt completely exhausted 100 meters below the summit back then, before anyone had been there, you would have felt a huge doubt that it is possible to get there at all, and much more likely felt like giving up.

This is how I feel with big tedious time-consuming computations: if I have a good guess for where the summit is, what it looks like and that it is reachable at least in principle, then I tend to feel much less scared.

:-)

Posted by: Urs Schreiber on January 23, 2008 6:07 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

They exist? why?

Posted by: jim stasheff on January 24, 2008 1:33 AM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

Do what exist? Internal homs of cocommutative coalgebras?

Back here I gave an argument that the category of cocommutative coalgebras is cartesian closed.

Posted by: Todd Trimble on January 24, 2008 1:43 AM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

Yes,
Argument as opposed to proof?

Posted by: jim stasheff on January 24, 2008 12:13 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

No, I meant proof. I didn’t cross every last ‘t’, but all the ingredients are there I believe. (How followable it is is for others to judge, but I tried to be fairly explicit and non-handwavy.)

I don’t know where this result is to be found in the literature. It’s probably categorical folklore; I first learned of it in conversation with Jim Dolan. The proof is neither trivial nor very difficult; really just a matter of technique.

Posted by: Todd Trimble on January 24, 2008 1:16 PM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

Conceptually
why should that cat have an internal hom?

Posted by: jim stasheff on January 25, 2008 2:10 AM | Permalink | Reply to this

Re: Constructing cofree cocommutative coalgebras

Here’s a rough argument for why it’s plausible. Let CocCoc denote the category of cocommutative coalgebras. We’re trying to establish an adjunction

hom Coc(XY,Z)hom Coc(X,[Y,Z])hom_{Coc}(X \otimes Y, Z) \cong hom_{Coc}(X, [Y, Z])

where [Y,Z][Y, Z] is the alleged internal hom; in other words, we’re trying to show that Y- \otimes Y is a left adjoint of some functor [Y,][Y, -]. (Remark that \otimes is the cartesian product for the category of cocommutative coalgebras.)

Under suitable hypotheses given by an adjoint functor theorem, a functor F:CDF: C \to D is a left adjoint iff it preserves arbitrary colimits (technically, the hypotheses involve something like cocompleteness conditions and solution set conditions – these are satisfied in our situation). So the main thing to check is that

Y:CocCoc- \otimes Y: Coc \to Coc

preserves colimits in the category of cocommutative coalgebras.

Now the nice fact is that colimits in CocCoc are computed as the colimits on their underlying vector spaces (or chain complexes, if we’re in the dg world). More precisely, the forgetful functor

U:CocVectU: Coc \to Vect

preserves and reflects colimits, for a reason essentially dual to the familiar fact that the forgetful functor from commutative algebras to VectVect preserves and reflects limits.

So to check that Y- \otimes Y preserves colimits in CocCoc, it suffices to check that it preserves colimits of the underlying vector spaces. But of course tensor products do that! “QED”

(The proof that I linked to bypasses the technical hypotheses of an adjoint functor theorem, in favor of an explicit formula. I want that in view of eventually doing explicit calculations of internal homs.)

Posted by: Todd Trimble on January 25, 2008 4:59 AM | Permalink | Reply to this
Read the post Integration without Integration
Weblog: The n-Category Café
Excerpt: On how integration and transgression of differential forms is realized in terms of inner homs applied to transport n-functors and their corresponding Lie oo-algebraic connection data.
Tracked: January 24, 2008 9:20 PM

Elementary qestion about presheaves

Here is a basic question about presheaves, just to reveal the unboundedness of my ignorance:

for SS any site and sSs \in S any of its objects and XX any presheaf on XX, I am thinking that it should generally be true that

Hom Set S op(s,X)=X(s), Hom_{Set^{S^{op}}}( s, X) = X(s)\,,

i.e. that presheaf morphisms from a representable presheaf into an arbitrary one are in bijection with the set that this arbitrary presheaf assigns to that representing object.

If not true in full generality, this gotta be true under some mild conditions. Which ones?

Posted by: Urs Schreiber on January 28, 2008 7:26 PM | Permalink | Reply to this

Re: Elementary qestion about presheaves

Looks like the Yoneda lemma (no extra assumptions needed).

Posted by: Todd Trimble on January 28, 2008 8:01 PM | Permalink | Reply to this

Re: Elementary qestion about presheaves

See the section “Topological presheaves” on pages 3-4 of this paper by Tom Leinster.

(Also, your unbounded ignorance does NOT cost the world billions of dollars in losses. Thankfully, the three hosts of this blog are intellectual geniuses compared to most people who work in finance.)

Posted by: Charlie Stromeyer Jr on January 28, 2008 8:20 PM | Permalink | Reply to this

Re: Elementary qestion about presheaves

Great. Thanks. With that public embarrassment out of the way, I can proceed with what I wanted to do. More later…

Posted by: Urs Schreiber on January 28, 2008 8:47 PM | Permalink | Reply to this
Read the post Differential Forms and Smooth Spaces
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Excerpt: On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
Tracked: January 30, 2008 10:16 AM
Read the post L-infinity Associated Bundles, Sections and Covariant Derivatives
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Excerpt: Associated L-infinity structures are obtained from Lie action infinity-algebroids, leading to a concept of sections and covariant derivatives in this context.
Tracked: January 30, 2008 9:10 PM
Read the post Chern-Simons States from L-infinity Bundles, III: States over the Circle
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Excerpt: On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
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Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
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Excerpt: On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
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Excerpt: Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-pa...
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Excerpt: Thoughts on infinity Lie theory.
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Read the post Charges and Twisted n-Bundles, II
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Excerpt: Rephrasing Freed's action functional for differential cohomology in terms of L-oo connections in a simple toy example.
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Read the post Sections of Bundles and Question on Inner Homs in Comma Categories
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Excerpt: On inner homs in comma categories, motivated from a description of spaces of sections of bundles in terms of such.
Tracked: March 4, 2008 10:19 PM
Read the post Space and Quantity
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Excerpt: Notes on spaces and smooth spaces, function algebras and smooth function algebras.
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Read the post What has happened so far
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Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
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Read the post An Exercise in Groupoidification: The Path Integral
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Excerpt: A remark on the path integral in view of groupoidification and Sigma-model quantization.
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Read the post Teleman on Topological Construction of Chern-Simons Theory
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Excerpt: A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
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