The n-Category Café

but it really should be $$(\wedge^\infty g^*) \otimes B$$ instead of $$\wedge^\infty (g^* \otimes B )$$

Thanks for mentioning this! I tried to discuss this issue before, but failed to make myself clear:

I understand that (when everything is suitably finite), from $$\cdots \to hom(g \wedge g,B) \stackrel{d_{CE}}{\to} hom(g \wedge g\wedge g,B) \to \cdots$$ we get, at face value, a differential on $$\wedge^\infty (I \oplus g^*) \otimes B \,.$$

But this extends, trivially, to all of $$\wedge^\infty ( g^* \oplus B)$$ doesn’t it?

And after being puzzled about what to make of this for a while, it seemed to occur to me that what is actually going on in BV formalism is that we form the full $$\wedge^\infty ( g^* \oplus B)$$ and then put a differential on that.

So this struck me as relevant. In particular, since the expression $$\wedge^\infty ( g^* \oplus B)$$ is in a way quite more “elegant” or “natural” than $$\wedge^\infty (I \oplus g^*) \otimes B \,.$$ The reason is that, as it happens,

- $g^*$ is always of positive degree

- while $B$ is always of non-negative degree

- such that they exactly fit together to form a complex

$$g^* \oplus B$$

of arbitrary degree, with no overlap.

This seems to be good, because also when we are forming, for instance, the internal hom $$\mathrm{hom}(V,M)$$ we obtain one single thing in arbitrary degree. If we like, we can separate this as $g^* \oplus B$, and this way identify ghosts, fields and antifields. But the way it arises is as one unified thing, and it would seem unnatural (and, I think, for the general case even impossible, but this I am not sure about right now) to have the dg-structure on $\mathrm{hom}(V,W)$ equivalently encoded in one on $\wedge^\infty(I \oplus g^*)\otimes B$.

So, you see, this issue you address is one I kept wondering about when starting to think about this a while ago. Let me put it like this:

how do we say “Lie $\infty$-algebra” in an arrow-theoretic way internal to cochain complexes? We say:

there is a positively graded cochain complex $$g^*$$ together with an extension of the differential on $\wedge^\infty(I \oplus g^*)$ to a complex $$(\wedge^\infty(I \oplus g^*), d_{CE})$$ such that the canonical projection $$\array{ (\wedge^\infty(I \oplus g^*), d_{CE}) \\ \downarrow \\ g^* }$$ still exists (i.e is still a cochain map).

This is equivalent to saying that we define a list of maps $$d_0 : g^* \to g^*[-1]$$ $$d_1 : g^* \to (g^*\wedge g^*)[-1]$$ $$d_2 : g^* \to (g^*\wedge g^* \wedge g^*)[-1]$$ $$\vdots$$ of “co-arity” $0 \leq n \lt \infty$ such that $$d_0 : g^* \to g^*[-1]$$ coincides with the differential on $g^*$ and which together form a map $$d : g^* \to \wedge^\infty(I \oplus g^*)[-1]$$ which we extend to a differential on all of $d : \wedge^\infty(I \oplus g^) \to \wedge^\infty(I \oplus g^) \,.$

Sorry for boring you with this, this is just a triviality formulated in a verbose way. But I am just trying to make myself clear, which I think I failed to do before.

Namely if we now think about what

a) people do in the BV literature

b) people do in the lietarure on $L_\infty$-modules

we can think of both a) and b) as summarized neatly by just repeating the above construction, with the non-negatively graded cochain complex $I \oplus g^*$ everywhere replaced by an arbitrary cochain complex $$g^* \oplus B \,.$$

Maybe I am seeing ghosts (pun intended). But right now I am pretty fond of this (admittedly simple) idea (but also simple ideas need to be thought).

There seems to be a triangle emerging

$$\array{ && \mathbb{Z}-groupoids \\ & \nearrow && \searrow \\ &Lie-Rinehart \infty-modules & \to & BV-complexes }$$

Somehow.

It only occurs to me now that Jim’s remark above to which I gave a long-winded reply is apparently caused by another typo of mine.

Argh.

It seems here I wrote $$\wedge^\infty(g^* \otimes B)$$ instead of $$\wedge^\infty(g^* \oplus B) \,.$$

Sorry for that.

boy, did that confuse me I thought that phi(x) meant the usul functional notation but what Dmitry does is use (du)^i to avoid confusion with du^i

Actually, I need to take back what I said about how that is an example of using $V^* \otimes W$ instead of $\mathrm{hom}(V,W)$.

Actually, what Dmitry has there is rather the $\mathrm{hom}(V,W)$ version, obscured through the fact that he writes

$$\xi^a = A^a d u + \cdots$$

instead of

$$\phi^* : \xi^a \mapsto A^a d u + \cdots \,.$$

I meant to point to some other literature, where people actually do use the $V^* \otimes W$-notation. Let me try to seach for a better example…

We can talk about the free commutative monoid SX on an object X whenever C also has coequalizers and (countable) coproducts:

SX=1+X+X^⊗2/2!+⋯

You just take the symmetrized tensor powers of X and add them up.

when can you get the corresponding symmetric comonoid replacing e.g. X^⊗2/2! by the symmetric invariants?

Okay, I can take a hint

Thanks! Hope I wasn’t too obnoxious.

I’ll stop here and check to see if this is at all relevant to your interests.

Yes, it is!

So let me indicate where I want to go:

above I admitted that I was being somewhat vague when I said that AKSZ form the internal hom in the world of dg-algebras.

It is sold that way, and it is “certainly morally true” (whatever that means). But now I want to get it completely straight. Or as straight as possible at least.

So: I want to understand the (if any) closed structure on

- dg-algebras

- quasi-free dgc-algebras (free graded-commutative algebras with differentials)

- dg-coalgebras

- quasi- free dg-coalgebras

and I want to undertand how the various more or less obvious constructions that one sees appear in the literature, or that one can dream up onself, fit into the general abstract nonsense.

I was busy most of the day with teaching duties. Now I will go to Todd’s detailed description of the closed structure on coDGAs and try to absorb that (thanks Todd! where would I be without the Café??).

A first glance revealed that it might take me a while to unwrap Todd’s description to the degree that I can see whether and how the expectation that I mention at then end of my previosu comment is realized here. All help is appreciated.

But, I digress… you haven’t seen anyone iterating BV quantization, have you?

I have seen BV quantization applied to the 2-particle… and then to the second quantization of the 2-particle!

(Ordinarily called “the string” and “string field theory”.)

In fact, Zwiebach’s BV treatment of bosonic string field theory seems to be one of the two outstanding successes of the BV formalism (next to the Poisson-sigma model quantization and everything related to that.)

But of course that’s not the explicit “iteratation” that you (and I! :-) want to see.

My personal idea is this:

“quantization” of sufficiently nice setups (namely sigma-models of an $n$-dimensional thing propagating on something and coupled to an $n$-background bundle) should really be a functor which eats transport functors and spits out quantum propagation functors. But the latter should be “transport functors” again. So we may try to feed them back into the machinery.

BV-formalism is a way to describe how this machine works. So once we really understand it, we’ll iteratre it.

In the comment second quantization to the thread The $n$-Café Quantum Conjecture I made some comments on this idea of iterating this endofunctor $$transport functors \stackrel{quantization}{\to} transport functors$$ and how this does seem to have a good chance to indeed capture the idea of second quantization.

I think you answered my last question before I answered it in re: symm comonoid

should it be denoted 1/(1-x)?

but the A_\infty question about maps is
the issue some would hide in the language of derived cats

It sounds as if you hope there’s an internal $\mathrm{Hom}$, internal to the category of commutative dga’s (commutative monoids in the symmetric monoidal category of chain complexes), i.e., that there’s a closed category structure on the category of commutative dga’s.

Yes, that’s the kind of thing I am looking for.

Let’s think rational homotopy theory: DGAs are much like spaces. The morphisms between one space and another should be a space itself.

That would be too much to hope for, I think, but it reminds me of some facts which I find very interesting:

Great, thanks.

The category CoDGA of cocommutative comonoids in the category of chain complexes

I’d be just as happy, or even happier, to discuss CoDGAs instead of DGAs. When everything is suitably finite both should be equivalent (I guess in the ps that you linked to this statement is prop 1.7) and people tend to want to stick with the DGAs as long as possible (or even longer).

But if its just the issue of CoDGAs versus DGAs that prevents me from getting the closed structure that I am looking for, then I won’t hesitate to stick with CoDGAs.

The category CoDGA is cartesian closed

Cool. That’s what I want!

I first learned this from Jim Dolan; I can give a proof of this and the following statement on request.

I’d be very grateful if you indeed did provide the proof. Thanks!

(So this means that this statement is not to be found in the literature??)

I’ll mostly need this statement for CoDGAs which are free as CoGCAs, i.e. for which the underlying graded coalgebra (forgetting the codifferential) is free as a graded commutative coalgebra. But the more general the statement you can provide, the better.

For this last case, I know the following:

let $\vee^\bullet V$ be the free graded commutative coalgebra generated from a graded vector space $V$. Write $(\vee^\bullet V, D)$ for the CoDGA obatined by equiupping this coalgebra with a codifferential (degree $\pm 1$, depending on your conventions) such that $D^2 = 0$.

Then a morphism $$(\vee^\bullet V, D_V) \to (\vee^\bullet W, D_W)$$ is supposed to be a linear map which is both, a morphism of graded commutative coalgebras and a chain map.

The fact that it is a morphism of free CoGCAs implies that it is entirely fixed by its value on “cogenerators”, hence by maps $$f : \vee^\bullet V \to W$$ (when beginning to think about this it helps to consider the same statement after dualizing everything, where it is more familiar).

Now, for the external hom these maps would be required to respect the differentials $[D,f] = d_W f - d_V F = 0$. But for the internal hom we don’t put any restriction on $f$ and instead have the commutator with the codifferentials act as a new codifferential.

This way we should get a new CoDGA $$(\mathrm{hom}_{Vect}(\vee^\bullet,W), [D,-])$$

and I am thinking that this, or something closely related, should be the internal hom.

But there must be some things missing here. Not sure if the above will in general be free, for instance.

Anyway, this is roughly my present understanding. Now I would like to understand this more deeply. Thanks for whatever help you can offer.

Todd’s comment
The last part means that if A and B are two commutative dgas, there is a cocommutative comonoid Hom(A,B) in Ch

reminded me to emphasize the importance of
the algebra structure (convolution product) on Hom(C,B) where
B is a dga and
C is a dg coalg

Is the categorical version of this written?

First of all, the bijection is between
equivalence classes of G-bundles over X
and
homotopy classes of maps [X, BG].

The appropriate notion of equivalence of connections on equivalent bundles is??
and the universal bundle with connection
is proved to exist
but I’m not sure what it looks like,
(netiher of the proofs - Quillen or Narasimhan and…
do I find pleasing)
nor how much further it is well behaved

If I’d known a really satisfactory answer to this, I might never have developed that ‘principal $G$-bundles with connection over $X$ are smooth anafunctors $P X \to G$’ baloney.

I got the vague impression from what some people mumbled that using something like smooth homotopy classes of maps $X \to B G$, we can classify $G$-bundles with connection. But don’t take my word for it. I don’t know.

And, worse, I don’t care. :-) Because my impression is that even if it is possible, it is not the way to go.

Instead, when I made my statement that gauge theories are $\sigma$-models with target $\mathbf{B}G$ I did mean it in the groupoid picture!

The problem is that you all convinced me not to write $\Sigma G$ for the 1-object groupoid obtained from $G$, but $\mathbf{B}G$. And, sure enough, already the second time I do so it leads to misunderstandings: you think I am talking about the huge space $B G$, where I am really talking about the nice and small groupoid $\{\bullet \stackrel{g}{\to} \bullet | g \in G\}$.

I think:

we should conceive QFTs of sigma-model kind as being about morphisms of suitably smooth $\infty$-groupoids.

Under certain circumstances it may be worthwhile to take geometric realizations everywhere and send $\{\bullet \stackrel{g}{\to} \bullet | g \in G\}$ to the space $B G$. But under many other circumstances it may not. This happens usually when a connection enters the game.

Because, as the name suggests: where bundles give rise to cohomology classes, bundles with connection give rise to differential cohomology classes, and all things differential will be hard to see once we divide out continuous homotopies.

And of course, for the BV exercise which we are talking about here, it is rather backwards to start by thinking about maps to the space $B G$. Rather, we want to think of morphisms of Lie groupoids to the Lie groupoid $\{\bullet \stackrel{g}{\to} \bullet | g \in G\}$ and then hit that with the Lie-functor which sends it to the world of $L_\infty$-algebras etc.

John wrote:

I know that, at least when G is a compact Lie group, the universal G-bundle over BG is equipped with a connection, so a map f:X→BG gives a principal G-bundle over X with connection. And, I vaguely recall that someone showed we can get any connection this way, as long as the dimension of X is finite.

This sounds related to Mark Mostow’s work on smooth spaces in

MR0587553 (84e:57030)
Mostow, Mark A.
The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations.
J. Differential Geom. 14 (1979), no. 2, 255–293.

I won’t get a chance to look at that paper for a couple of weeks, but I thought I’d mention it in case it rings a bell for you.

Urs wrote:

I got the vague impression from what some people mumbled that using something like smooth homotopy classes of maps $X \to B G$, we can classify $G$-bundles with connection. But don’t take my word for it. I don’t know.

It’s been a long time since we had this conversation, but just for the record, let me add:

This ain’t true, at least not in any sense I enjoy. But maybe I should be precise, since Jim wondered what I mean between bundles with connection. Here’s what I mean. Given two principal $G$-bundles $P \to X$ and $P^' \to X$ with connections $A$, resp. $A^'$, I say they’re isomorphic if there’s a $G$-bundle isomorphism $f: P \to P^'$, lifting the identity map on $X$, such that:

$$f^* A^' = A$$

This really says our $G$-bundles with connection are the same in every practical sense.

This equivalence relation on principal $G$-bundles with connection is bound to be much less coarse than ‘coming from smoothly homotopic maps to $B G$’. In fact, if the smooth structure on $B G$ is nice, it’s quite likely that smooth maps from a manifold into $B G$ are smoothly homotopic iff they’re homotopic. If so, two principal $G$-bundles with connection come from smoothly homotopic maps to $B G$ iff their underlying principal $G$-bundles are isomorphic! The connection becomes completely irrelevant!

And, worse, I don’t care. :-)

Well, then it’s okay that you don’t know. :-)

I guess the problem is that for many years, I wanted to think of a principal $G$ bundle with connection over $X$ as a map from $X$ to something.

By now, I know that a principal $G$ bundle with connection over $X$ is the same as a map from $P X$ to something. Here $P X$ is the smooth path groupoid of $X$.

Theorem: isomorphism classes of principal $G$-bundles with connection over $X$ are in 1-1 correspondence with isomorphism classes of smooth anafunctors from $P X$ to $G$.

This is great and probably just what we need. A map from $P X$ to $G$ really is a kind of map from $X$ to something.

But, I still keep wondering if can delete the phrase ‘path groupoid of’, basically by using an adjoint functor:

$$hom(P X , G) \cong hom(X , B G) ???$$

Here $P$ is the ‘path groupoid’ functor going from smooth spaces to smooth groupoids, and $B$ is my hoped-for, but apparently nonexistent, adjoint functor going back from smooth groupoids to smooth spaces. On the left ‘hom’ means ‘smooth anafunctors’, while on the right it means ‘smooth maps’. Or something like that.

Note: no taking homotopy classes anywhere!

I’ve sort of given up on this idea, but it still haunts me…