Transgression of n-Transport and n-Connections

December 30, 2007

Transgression of n-Transport and n-Connections

Posted by Urs Schreiber

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

Trangression of $n$ -transport and $n$ -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 $n$ -functors and Lie $∞$ -valued $n$ -connections to smooth mapping spaces.

This builds on the general idea of $n$ -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 $n$ -particle and detailed a bit more in the entry Multiplicative Structure of Transgressed $n$ -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.

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Introduction

I want to better understand the
$•$ general systematics

and the

$•$ specific details

of what it means to transgress

$•$ transport $n$ -functors [1, 2, 3 4]

and

$•$ Lie $n$ -algebra valued connections [5]

to mapping spaces.

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

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 $n$ -categorical nature of $n$ -transport manifest: namely that under transgression not only the domain, but also the codomain of $n$ -transport and $n$ -connections changes.

For instance, in the simplest kind of example, an ordinary abelian 2-connection is not really something taking values in $U (1)$ , but rather something taking values in $ℬ U (1)$ . Transgressing it to loop spaces by setting $par = S^{1}$ in the above turns it into a 1-connection with values in $hom (S^{1}, ℬ U (1))$ , which is indeed $Λ ℬ U (1) = U (1)$ 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 $n$ -transport and smooth $n$ -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

$U \mapsto hom (U \times X, Y)$

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

$S^{∞} (X, DGCA (A, Ω^{•} (-))) ≅ DGCA (A, S^{∞} (X, Ω^{•} (-))) .$

This can be proved with the help of (again) the Yoneda lemma: since $X$ is a colimit of representables, one can easily reduce to the case where $X$ 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 ( $Ω^{•} (-)$ 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 ${DGCA}^{op}$ (or something like it) is equivalent to the topos $S^{∞}$ , and it’s sort of an interesting question because ${DGCA}^{op}$ 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 ${DGCA}^{op}$ 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 $DGCA$ are preserved under pushing out. The pushout of a pair of morphisms

$B \leftarrow A \to C$

in $DGCA$ is given by $B \otimes_{A} C$ ; the question is whether $B \otimes_{A} -$ preserves limits of $A$ -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 $B \otimes_{A} -$ 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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Re: Transgression of n-Transport and n-Connections

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Todd,

I have now incorporated your statement that $Ω^{•}$ and $Hom (-, Ω^{•} (-))$ form an adjunction.

What you say about how to put this into perspective, $Ω^{•} (-)$ 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 $\begin{matrix} hom (par, tar) \otimes par \\ ^{p_{1}} ↙ & ↘^{ev} \\ hom (par, tar) & tar \end{matrix} .$

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

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

I might just be content with taking $hom (par, -)$ 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

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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 $A$ is isomorphic to the rational polynomial deRham cohomology of the space, which he denotes $〈 A 〉$ , defined by this algebra, $H^{•} (A, ℚ) ≃ H_{pol . deRham}^{•} (〈 A 〉, ℚ) .$

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

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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 $S^{∞}$ , 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) $CoDGA$ is closed and ${DGCA}^{op}$ 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 $CoDGA$ (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:

CoDGA ≅ {DGCA}_{prof}^{op} .

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

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[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

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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 $V$ any graded vector space, the cofree coalgebra over it is, as a vector space, the space $T V = \oplus_{n \in ℕ} V^{\otimes n}$ that Jim mentions with the coproduct $Δ T V \to T V \otimes T V$ acting on homogeneous elements $(x_{1} \otimes x_{2} \otimes \dots \otimes x_{n})$ as $\sum_{k = 0}^{n} (x_{1} \otimes \dots x_{k}) \otimes (x_{k + 1} \otimes \dots x_{n}) .$

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 $C_{1}$ and $C_{2}$ . Then there is the graded vector space ${Hom}_{coalgebras} (C_{1}, C_{2})$ of coalgebra morphisms from one to the other, whose elements in degree $k$ are coalgebra homomorphisms that shift the degree by $k$ .

A naïve candidate for the inner hom is the cofree coalgebra over this graded vector space $T {Hom}_{coalgebras} (C_{1}, C_{2}) .$

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

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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

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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 $∞$ -groupoids transported to the world of Lie $∞$ -algebroids. Or something like that, I don’t know yet.

For $B$ and $C$ qDGCAs, consider the presheaf on manifolds $X_{[B, C]}$ given by $U \mapsto {Hom}_{DGCAs} (B, C \otimes Ω^{•} (U)) .$ Notice that this corresponds to the internal hom in presheaves, which would be $U \mapsto {Hom}_{S^{∞}} (U \times X_{C}, X_{B}) .$

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

I am thinking that the DGCA of forms on $X_{[C, B]}$ , $Ω^{•} (X_{[C, B]})$ should be something not unrelated to the internal hom DGCA from $C$ to $B$ .

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

Let $V_{B}$ be the vector space underlying the qDGCA $B$ . Then for each element $f_{ω} \in V_{B} \otimes C^{*}$ of degree $- n$ we obtain a differential form of degree $n$ on $X_{[B, C]}$ by sending any ${ev}_{U}^{*} \in hom (B, C \otimes Ω^{•} (U))$ to $〈 f_{ω}, {ev}_{U}^{*} 〉 \in Ω^{•} (U)$ in the hopefully obvious sense.

That’s looking kind of promising, because elements of $V_{B}^{*} \otimes C$ indeed describe morphisms from $B$ to $C$ and their homotopies, so elements of $V_{B} \otimes C^{*}$ 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 $Hom (B, C \otimes Ω^{•} (U))$ with the differentials on the two sides.

So it seems that the DGCAs of the form $Ω^{•} (X_{[C, B]})$ 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

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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 ${Set}^{S^{op}}$ 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 $h (-, -) : DGCAs \times {DGCAs}^{op} \to DGCAs$ by first sending each DGCA to a space, then forming the internal hom of these spaces, and then transferring that back to a DGCA $DGCAs \times {DGCAs}^{op} \overset{Hom (-, Ω^{•} (-)) \times Hom (-, Ω^{•} (-))^{op}}{\to} (S^{∞})^{op} \times S^{∞} \overset{{hom}_{S^{∞}}}{\to} S^{∞} \overset{Ω^{•}}{\to} DGCAs .$

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

B) There is a simpler formalization of the “idea” underlying $h (-, -,)$ : given two algebras $A$ and $B$ , we can directly form the presheaf $U \mapsto {Hom}_{DGCAs} (B, A \otimes Ω^{•} (U)) .$ One can see that this presheaf is a subobject of the presheaf that appears towards the right in the construction of $h (-, -)$ .

So, I think, we have an inclusion $Ω^{•} ({Hom}_{DGCAs} (B, A \otimes Ω^{•} (-))) ↪ h (B, A) .$

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 $A$ and $B$ the DGCA $Ω^{•} ({Hom}_{DGCAs} (B, A \otimes Ω^{•} (-))$ 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

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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 $A$ , by taking the collection of $n$ -simplices to be ${Hom}_{DGCAs} (A, Ω^{•} (Δ^{n}))$

(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 $A$ we obtain the presheaf $X_{A}$ on manifolds defined by $X_{A} : U \mapsto {Hom}_{DGCAs} (A, Ω^{•} (U)),$ for all manifolds $U$ .

To every presheaf $X$ on manifolds we can associate its DGCA $Ω^{•} (X)$ of differential forms. The cohomology $H^{•} (Ω^{•} (X))$ of this I can address as the real cohomology of $X$ .

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

$H^{•} (A) = H^{•} (Ω^{•} (X_{A})) .$

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

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but if you allow functions on $ℝ^{n}$ 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

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Jim wrote:

$X_{A}$ is defined in terms of all manifolds $U$ Then what is $X$ ? and how does $Ω^{•} (X)$ differ from ${Omega}^{•} (X_{A})$ ? 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 $U$ are manifolds, living in the category (a site, actually) of all manifolds, with morphisms of manifolds as morphisms. This category I called $S$ .

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

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

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

I wrote $S^{∞} : = {Set}^{S^{op}}$ 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 $Ω^{•}$ which assigns to each manifold $U$ the set of differential forms on $U$ : $Ω^{•} : U \mapsto Ω^{•} (U) .$

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

For $X$ a smooth space, we define the dg-algebra of forms on $X$ , $Ω^{•} (X)$ to be the set of smooth space morphisms (morphisms of presheaves on manifolds) $Ω^{•} (X) : = {Hom}_{S^{∞}} (X, Ω^{•}) .$

One checks that the operation of exterior diufferential andf wedge product on each $Ω^{•} (U)$ for each manifold $U$ 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:

$Ω^{•} : S^{∞} \to DGCAs .$

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

For each DGCA $A$ , we can form the presheaf on manifolds which I called $X_{A}$ and which is defined by $X_{A} : U \mapsto {Hom}_{DGCAs} (A, Ω^{•} (U)) .$

This defines a contravariant funcotr which I denoted $Hom (-, Ω^{•} (-))$ : $S^{∞} \leftarrow DGCAs : Hom (-, Ω^{•} (-)) .$

So now coming to your question:

Then what is $X$ ? and how does $Ω^{•} (X)$ differ from ${Omega}^{•} (X_{A})$ ?

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

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

We can unwrap this definition for the special case that we have a smooth space $X_{A}$ coming from a DGCA $A$ . In that case a differential form on $X_{A}$ is a morphism of presheaves $ω : X_{A} \to Ω^{•}$ which for each manifold $U$ is a morphisms of sets ${omega}_{U} : {Hom}_{DGCAs} (A, Ω^{•} (U)) \to Ω^{•} (U) .$

Staring at this formula, one realizes that every element of $A$ naturally yields a differential form on $X_{A}$ . We have an canonical inclusion of DGCAs $A ↪ Ω^{•} (X_{A}) .$

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 $A$ should actually equal the deRham cohomology of $X_{A}$ , i.e. the cohomology of $Ω^{•} (X_{A})$ . 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 $X$ be any generalized smooth space. Then there is an obvious notion of the following gadgets:

$•$ $n$ -simplices in $X$

$•$ the fundamental $∞$ -groupoid of $X$

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

Now let $X$ here come from a DGCA $A$ , $X = X_{A}$ . Then, I think, the collection of $n$ -simplices ${Hom}_{S^{∞}} (Δ^{n}, X_{A})$ in $X_{A}$ is precisely what is considered in rational homotopy theory, namely the set ${Hom}_{DGCAs} (A, Ω^{•} (Δ^{n})) .$

(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 $X_{A}$ from a DGCA $A$ is possibly the more fundamental operation.

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

The $∞$ -groupoid integrating $g$ is the fundamental $∞$ -groupoid of the smooth space $X_{CE (g)}$ .

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

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

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 $∞$ -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

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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 $S$ are “all those things which may be probed by the objects of $S$ ”.

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

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

More precisely, the $(n + 1)$ -category of strict $n$ -categories and strict maps, transformations etc. between them should be dense in the $(n + 1)$ -category of weak $n$ -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 $n$ -category can be regarded as a ‘nice’ presheaf on strict $n$ -categories.

Of course, no one would want to say that a weak $n$ -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

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

[…]

What this means is that a weak $n$ -category can be regarded as a ‘nice’ presheaf on strict $n$ -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- $n$ -stacks?

What does the category of presheaves over $ω Cat$ resemble? What are are prestacks over $ω Cat$ ?

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

Re: Transgression of n-Transport and n-Connections

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

I’m suggesting taking $C$ to be all of $Str n Cat$ .

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

Re: Transgression of n-Transport and n-Connections

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Tom Leinster wrote:

The idea behind this is that a weak $n$ -category $A$ corresponds to the presheaf $X$ whose value at $D \in 𝒟$ is the set of weak functors $D \to A$ — although at this point you don’t know what ‘weak $n$ -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 $n = 1$ you might take $𝒟 = Δ$ .)

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 $Δ$ , as in your previous comment.

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

I’m suggesting taking $C$ to be all of $Str n Cat$ .

Interesting. At the end of my previous comment I was asking about considering presheaves on $ω Cat$ (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

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Urs asked about Kan complexes. But I wasn’t saying anything about groupoids — just categories. So when I mentioned the example $𝒟 = Δ$ , 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 $ω Cat$ (meaning strict infinity categories). Would you consider that?

Certainly. My ‘ $n$ ’ was meant to range over natural numbers and $ω$ .

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

Re: Transgression of n-Transport and n-Connections

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

Certainly. My ‘ $n$ ’ was meant to range over natural numbers and $ω$ .

How do you characterize the “nice” presheaves on $ω Cat$ ?

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

Re: Transgression of n-Transport and n-Connections

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

(If by ‘nice’ you mean corresponding to a strict $ω$ -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

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

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

But for some $n > 1$ you do have a working definition? Maybe even for all $n \in ℕ$ ?

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

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