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August 18, 2007

On BV Quantization, Part II

Posted by Urs Schreiber

A review of some basics of classical BV formalism, with an eye towards my claim (motivated in part I) that this is secretly about quantization on an nn-category (or rather on a Lie nn-groupoid – or, rather, on the corresponding Lie nn-algebroid).

Table of Contents

Part 1. Basic classical mechanics, in which the concepts are briefly recalled, essentially symplectic geometry, which are to be generalized in the following.

Part 2. Something is missing, in which the failure to capture gauge symmetries is highlighted – and related to the need for isomorphisms.

Part 3. The plan from here on, in which the following line of attack is outlined.

Part 4. BV formalism: mechanics on graded vector spaces, in which finally some elements of the BV formalism are mentioned.

Part 5 Some literature, in which the author fails to summarize the relevant literature.

Part 6 — but why?, in which the relation between Lie nn-algebroids and differential graded algebras is highlighted.

Part 7 Running out of steam, in which the author notices that he only made it through what was intended as a mere introductory remark and postpones further discussion to an unspecified point in the future.


Basic classical mechanics

In ordinary classical mechanics we are studying spaces PP, called phase spaces. These are manifolds that come equipped with a 2-form ωΩ 2*(P)\omega \in \Omega^2*(P) that is closed and everywhere non-degenerate. Hence PP is a symplectic space.

Since ω\omega is assumed to be non-degenerate, it induces an antisymmetric product on the space of functions {,}:C (P)C (P)C (P), \{\cdot , \cdot\} : C^\infty(P) \otimes C^\infty(P) \to C^\infty(P) \,, sometimes called the space of classical observables (but that term is more strictly reserved for a subset of all such functions which are invariant under certain group actions; we’ll come to that) the Poisson bracket defined by {F,G}:=ω 1(dF,dG). \{F,G\} := \omega^{-1}(d F, d G) \,. The fact that ω\omega is closed makes this a Lie bracket and hence makes C inft(P)C^{\inft}(P) Poisson algebra.

Locally these spaces look like cotangent bundles T *XT^* X. The points of XX are the configurations of the classical system in question. When we are dealing with a field theory, then the points of XX are the field configurations.

The points in the fibers of T *XT^* X are called the canonical momenta associated to these configurations. In the context of the BV formalism, people like to call these not momenta, but antifields.

When one assumes that XX may be covered by a single coordinate chart, this chart is conventionally called q:X n q : X \to \mathbb{R}^n and the induced chart on T *XT^* X is then by usual convention called (q,p):T *X n. (q,p) : T^* X \to \mathbb{R}^n \,.

In these coordinates the symplectic form ω\omega reads, locally ω X=d( ip idq i) \omega_{X} = d( \sum_i p^i d q_i) and hence the Poisson bracket reads locally {F,G}| X= i(Fq iGp iFp iGq i). \{F , G\}|_X = \sum_i \left( \frac{\partial F}{\partial q^i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q^i} \right) \,.

The main reason to emphasize these local formulas here is that in the BV formalism, to be described in a moment, the existing literature tends to make heavily recourse to local coordinate formulas. And not entirely with lack of good reason, actually.


Something is missing

The inclined nn-Café reader may notice the following:

the above language for describing the physical world

a) is very explicit concerning the concept of a physical configuration, which is an object (I mean: element) in configuration space XX;

b) is already slightly more coy about the notion of a physical process – but of course this is at least partially encoded in the cotangent space T *XT^* X

c) says nothing about isomorphisms between physical configurations.

To the nn-categorical eye this may seem like an oddity. (You know, Space and State, Spacetime and Process). And indeed, while they didn’t think of it from this point of view, physicist went through decade-long intensive trouble with figuring out how to best add the information of isomorphisms between physical configurations to the above classical description of classical mechanics.

The length of these struggles is reflected to some degree in the plethora of terms, often weird-sounding, that have been introduced and considered over the years: when you hear somebody ponder things like the master action for the ghosts of ghosts and their antifields you know he or she is on some major quest. And indeed – that of trying to come to grips with the notion of isomorphism of physical configurations, otherwise known as the issue of gauge theory.

The current endpoint of these developments is known as

Batalin-Vilkovisky (BV) formalism.


The plan from here on

I shall now give a quick impression of what BV-formalism is and give some pointers to useful literature. Then I make a remark on why and how this should secretly be nothing but the Lie version of physics on Lie nn-groupoids. After that ideosyncratic detour I come back to the BV-formalism and talk about more of its details.


BV-formalism: mechanics on graded spaces

The quick answer to what BV formalism is about is this:

(Classical) BV theory is the internalization of the formulation of classical mechanics in terms of symplectic spaces and Poisson algebras from the world of smooth manifolds into that of graded and differential graded (“dG”) manifolds.

Sometimes, in the BV-literature, people talk about “supermanifolds” instead of graded manifolds. I shall try not to do that, since I think it is slightly misleading, or at least not helpful. (This becomes particularly manifest when one starts discussing supersymmetric physics using the BV formalism.)

For my purposes here (and indeed for the bigger part of existing literature), a graded manifold is nothing but the dual to a \mathbb{Z}-graded-commutative algebra, whose degree-0 part is an ordinary algebra of (smooth) functions on some ordinary space.

The crucial example to keep in mind is the “graded space” denoted ΠE \Pi E for EXE \to X any vector bundle over XX. This is really notation for the space which we imagine has the “algebra of smooth functions” which is the exterior algebra sΓ(E *) \Wedge^\bullet s\Gamma(E^*) of sections of the dual of this vector bundle, taken to be of degree 1.

The most important special case of this is ΠTX, \Pi T X \,, the “odd tangent bundle” of an ordinary space XX. This simply corresponds to the graded commutative algebra Ω (X) \Omega^\bullet(X) of differential forms on XX.

But we just as well have the odd tangent bundle ΠTX \Pi T X of an arbitrary graded space XX: this is given by the algebra which is freely generated as a graded-commutative algebra from the algebra “of functions” on XX itself, together with the duals of the graded derivations of this algebra, each of which is regarded as as a generator of degree one higher than the derivation it corresponds to.

For instance for X= 0|1 X = \mathbb{R}^{0|1} the “odd line” (namely the (rather boring) exterior algebra over just \mathbb{R}), we have that its shifted tangent space ΠTX \Pi T X is the free graded commutative algebra (s *ss *) \wedge^\bullet (s \mathbb{R}^* \oplus s s \mathbb{R}^*) which is generated (over our ground field, usually \mathbb{R} in the context of classical mechanics) from two generators, one of them of degree 1, the other of degree 2.

Given such a graded space, there is an obvious way how to define an analogue of a symplectic form ω\omega on it.

The graded Poisson bracket coming from that graded symplectic form is now conventionally written (,). (\cdot, \cdot) \,. And – just to confuse everybody – it is called the antibracket. Don’t let that confuse you. This is our Poisson bracket! (Just as the word “antifield” shouldn’t trick you in forgetting that these are nothing but canonical momenta).

These “graded symplectic spaces” are, essentially, what classical BV formalism is concerned with.

Up to two further details:

first, instead of just blindly internalizing classical symplectic theory into the graded world, BV-formalism takes care of the following point:

once we are in the graded world in the first place, we should admit that the canonical momenta which we had encountered before really ought to be of one degree higher than the canonical coordinates that they come from.

In other words, if XX is the graded configuration space, then we now take the graded phase space to be ΠT *X \Pi T^* X instead of just T *XT^* X.

Remark.This simple observation is of major importance. Below I will point out that nnth degree in the world of graded manifolds correspoonds to nn-morphisms in the world of Lie nn-groupoids. Canonical momenta encode physical processes. Hence morphisms. Hence they need to be in degree one higher than the object whose process they describe.

This then leads people to invent all the following funny terminology:

- a field is an element in XX

- an antifield is a canonical momentum associated to that, i.e. an element in the fibers of ΠT *X\Pi T^*X

- a ghost field is a degree-1 field

- a ghost-of-ghost field is a degree-2 field.

and so on.

Hence it remains to identify the physical meaning of these ghosts and higher ghosts. Being of higher degree, they should correspond, by the above remark, to gauge transformations and higher gauge transformations, respectively. And indeed, that’s how it is: in the physics literature these ghosts are the “gauge transformation parameters”.

Finally, the second further ingredient to the mere idea of internalizing classical mechanics in graded manifolds is to give a new characterization of the classical action functional:

in BV formalism, the ordinary classical action functional (I haven’t even talked about these yet) is essentially something like a functional on phase space (but there is a subtlety here – more on that elsewhere) extended to the graded phase space, where it is called S S and required to satisfy the condition that it has vanishing graded Poisson bracket (called “antibracket”, remember) (S,S)=0 (S,S) = 0 with itself.

This conditiion is called the classical master equation in this context. I should talk about that in more detail elsewhere.

Some literature

I certainly cannot give a canonical reading list, and would in fact be grateful for any further pointers from those more expert than me. Also, I will restrict to material which is freely available online.

In fact, for the moment I just give an extremely unbalanced and very short list of references. Maybe I’ll extend that later on..

Anyway, among all the sophisticated material which one finds, this paper here stood out as one which describes in great and careful detail the basic ideas and in particular their concrete application to the description of the two most important gauge theoretic systems which one encounters: the free relativistic partice (as the archetype of a gravity-like gauge theory) and Yang-Mills theory (as the gauge theory archetype per se):

Joaquim Gomis, Jordi París, Stuart Samuel
Antibracket, Antifields and Gauge-Theory Quantization
hep-th/9412228

The BV literature splits into two parts: in one part of the literature, people start with a gauge theoretic physical system, and then go through the process of finding the right “ghosts” such as to construct a suitable graded extension of the ordinary classical configuration and phase space.

In the other part of the literature, people go the other way round: accepting the premise that physics should be done on graded spaces, these authors start with their favorite examples of a graded space, and then turn the BV crank to study the physics described by them.

To this second part belong the two articles (AKSZ and Roytenberg) which I already mentioned last time.

Another text that I found helpful is section 2 and 3 of

Noriaki Ikeda
Deformation of Batalin-Vilkovisky Structures math/0604157

Generally very inspiring are the articles by Jim Stasheff on this topic. Jim Stasheff very much cares about reflecting on all these algebraic structures which are being found here. For instance, the end of his hep-th/9712157 is, so far, the only place I found where what looks to me like the crucial question concerning the BV master equation is mentioned:

[…] but why?

But here I am not talking about the master equation much yet, I have barely mentioned it. But I do shall now ask “…but why?” with respect to the appearance of graded manifolds themselves.


…but why?

Certainly the most profound message of

J. Baez, A. Crans, Lie 2-Algebras is the following slogan:

Lie nn-groups differentially have Lie nn-algebras. Lie nn-algebras are the same as L inftL_\inft-algebras.

Beware that this is indeed just that, a slogan. A couple of qualifications are needed to make this a precise statement. But for the moment I just want the slogan.

Notice that there is the following helpful way to understand the technology at work behind this statement:

By forming all the tangent spaces to the space of kk-morphisms starting at some identity (k1)(k-1)-morphisms in some Lie nn-groupoid, we obtain a bunch of vector spaces which encode what the Lie nn-groupoid looks like in linear approximation. These arrange into one big graded vector space V, V \,, the \mathbb{N}-grading being simply the order kk of these kk-morphisms.

By some kind of magic (but, surely, it must be the general geometry dao at work here, somehow, I am still not completely sure if anyone really understands this, truly), it turns out that all the structure which we have on the morphisms of the Lie nn-groupoid, all the compositions, all the source, target and identity-assigning morphisms, are encoded, equivalently, in a – take a deep breath – degree -1 graded codifferential D:S csVS csV D : S^c s V \to S^c s V from the cofree graded-co-commutative coalgebra on sVs V (that is, VV with all degrees shifted by one) which squares to 0 D 2=0. D^2 = 0 \,. That’s the important fact here. We should pay great attention to this fact: this is the bridge from the world of nn-groupoids to that of differential algebra. Such cofree graded co-comuutative coalgebras with nilpotent degree -1 differentials are called L L_\infty-algebras (or strongly homotopy Lie algebras).

So, suppose for a moment that you’d follow the claim that nn-particles (points, strings, membranes, etc.), i.e. certain kinds of nn-dimensional quantum field theories, classically come from setups where:

a field configuration (in precisely the sense I have been talking about above) is an nn-functor from an nn-categorical “parameter space” to an nn-categorical “target space”.

Just trick yourself for a moment into believing this. Then, what follows? It follows that if all these nn-categories are actually smooth nn-groupoids, then also the “space of field configurations”, namely the nn-functor nn-category conf:=Hom nLieGrpd(par,tar) \mathrm{conf} := \mathrm{Hom}_{n\mathrm{LieGrpd}}(\mathrm{par},\mathrm{tar}) is a Lie nn-groupoid.

Since it is a Lie nn-groupoid, we would be tempted to simplify our lives a little by studying not this full thing, but just the morphism of the underlying Lie nn-algebroids which it induces. This way we expect to find our configuration space to be turned into the corresponding Lie nn-algebroid.

I haven’t really talked about Lie nn-algebroids here, just about Lie nn-algebras. That’s because I don’t want to get into a long technical discussion which clarifies the heuristic picture that I am drawing only slightly but would distract us a whole lot. So let’s just think Lie nn-algebras for the moment and keep our fingers crossed that whatever we understand this way still applies to the Lie algebroid case.

Okay, now here is the important observation: assuming that our configuration space is a Lie nn-algebroid, hence an L L_\infty-algebra (“L L_\infty algebroid”, really, but never mind for the moment), the space of functions (physical observables!) on this is a graded-commutative differential algebra.

Conversely, every graded-commutative differential algebra is the “algebra of functions” on a Lie nn-algebra.

This is simple vector space duality in the L inftL_\inft-picture:

from the graded commutative co-free coalgebra S csVS^c s V we obtain, dually, the free graded commutative exterior algebra (sV *). \wedge^\bullet (s V^*) \,. Moreover, from the nilpotent graded codifferential DD we obtain the nilpotent graded differential d: (sV *) (sV *) d : \wedge^\bullet (s V^*) \to \wedge^\bullet (s V^*) which simply acts dually as dω:=ω(D()). d \omega := \omega(D(\cdot)) \,.

For a concise but comprehensive review of the details on all this (well known to the experts), including references to all the relevant literature, see section 3 of our preliminary article with the working title Structure of Lie nn-algebra.

It should be quite clear now what I am getting at: the important point is that whenever we see a quasi-free (meaning the underlyin graded commutative algebra is free) differential graded-algebra, with generators concentrated in degrees n\leq n, we are really seeing the “algebra of functions” (in a very concrete sense! – much more concrete than in the case of “graded manifolds”, which really don’t have a independent definition apart from that in terms of the “algebra of functions” on them) on a Lie nn-algebra.

And it is pretty clear how more general graded-commutative differential algebras correspond to Lie nn-algebroids. (But the details of that require more discussion.) For instance it is very well known that the graded-commutative differential algebra (Ω (X),d), (\Omega^\bullet(X), d) \,, namely the deRham complex of differential forms on some manifold XX, is the “algebra of functions” on the Lie 1-algebroid of the pair groupoid of XX (the codiscrete groupoid over XX). And this pattern continues.

Finally, to finish the argument which I am sketching here, recall that degree kk-generators of these graded algebras correspond to (tangents to) kk-morphisms of a Lie nn-groupoid.

With this picture in mind, the entire BV formalism suddenly appears in a very clear light:

if we have a “configuration space” of some physical system where various configurations may be related by various ismomorphisms, then we would, in the nn-categorical picture, consider not the space of configurations, but the category of configurations: its objects are the former configurations, its morphisms are these isomorphisms.

So this “configuration category” conf \mathrm{conf} would be a Lie groupoid. Hence the “algebra of functions on configuration space” would be – a graded-commutative algebra generated from

a) the ordinary algebra of functions on ordinary configuration space

together with

b) the odd generators which correspond to the (duals of the tangents to) the morphisms in conf\mathrm{conf}: this are the ghosts

I should add that what I am saying here, while I really think it deserves to be emphasized, isn’t supposed to be anything but a tautology once one thinks about it. In fact, one would expect to see this statement mentioned here and there in the literature. But if that is the case, I msut have had bad luck. At the moment I am aware of one place in the literature where the statement is explicitly mentioned, and that is

Dmitry Roytenberg
On the structure of graded symplectic supermanifolds and Courant algebroids
math/0203110

where it says somewhere in the middle of p. 6:

[…] it is natural to call NQ-manifolds of degree n n-algebroids. […]

Here an NQ-manifold is what I was referring to as a differential-graded manifold (the many-object version of the Koszul dual to a Lie n-algebra), and “nn-algebroids” is really to be read as the sloppy version of “Lie nn-algebroids”.


Running out of steam

“Hm”, I hear you saying, “while, in this picture, all the information about the space of (0kn)(0 \leq k \leq n)-morphism of the configuration space nn-groupoid is in the graded-commutative algebra, all the information about source, target and composition is in the differential. But in your review of BV formalism above we had just graded algebras, not graded differential algebras.”

Right. And – lo and behold – it turns out that graded algebras alone are not the end of the story: there is crucially, in BV formalism, an odd-graded nilpotent differential around, famously related to what is known as the BRST operator. And that’s our differential.

But I am running out of steam. The BRST operator in BV formalism will have to wait until next time.

Posted at August 18, 2007 1:01 PM UTC

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Re: On BV Quantization, Part II

I should add that my statement – whose maybe remarkable implications for interpreting the BV-formalism I tried to highlight above – that

a differential graded manifold with local generators concentrated in degree 0kn0 \leq k \leq n should be regarded as (the Koszul dual incarnation of) a Lie nn-algebroid,

while probably obvious to anyone who ever thought about it, appeared explicitly in

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids

where it says somewhere in the middle of p. 6:

[…] it is natural to call NQ-manifolds of degree nn nn-algebroids. […]

Here an NQNQ manifold is what I was referring to as a dG-manifold (the many-object version of the Koszul dual to a Lie nn-algebra), and “nn-algebroids” is really to be read as the sloppy version of “Lie nn-algebroids”.

Posted by: Urs Schreiber on August 19, 2007 11:22 AM | Permalink | Reply to this

Re: On BV Quantization, Part II

If I understand you correctly, you claim that the Batalin-Vilkovski is really about treating systems whose configuration space / phase space / history space is an n-category rather than a set.

Why is it, then, that Batalin-Vilkovski is especially useful when the gauge symmetry only works on-shell (see e.g. [1])?

I never had a clear understanding of why the quantum theory works at all in these cases. Naively, if a symmetry is only present on-shell, it is not a symmetry of the path integral, since the later involves off-shell configurations. Batalin-Vilkovski is supposed to be a way to remedy such problems, but it always seemed like black magic to me. Btw, in this capacity, Batalin-Vilkovski was applied to global supersymmetry (which is not a gauge symmetry), see [2]

[1] “The Quantum Theory of Fields” volume II by Steven Weinberg, p. 42
[2] Classical and Quantum Gravity 9 (1992) 413-443: “Analysis of the superconformal cohomology structure of N=4 super Yang Mills” by P.L.White

Posted by: Squark on September 8, 2007 5:52 PM | Permalink | Reply to this

Re: On BV Quantization, Part II

I never had a clear understanding of why the quantum theory works at all in these cases. Naively, if a symmetry is only present on-shell, it is not a symmetry of the path integral, since the later involves off-shell configurations. Batalin-Vilkovski is supposed to be a way to remedy such problems, but it always seemed like black magic to me.

The phase space (the thing we are quantizing) is nothing other than the space of classical solutions. So, a-priori, there’s nothing bad about having a symmetry of the phase space (the space of classical solutions) that does not extend to a symmetry of the full space of “off-shell” field configurations.

The real issue, in quantum field theory, is whether the symmetry survives regularization. It’s the ability to write down a regularized version of the quantum master equation that is the key to the success of the BRST/BV formalism.

Posted by: Jacques Distler on September 8, 2007 6:37 PM | Permalink | PGP Sig | Reply to this

Re: On BV Quantization, Part II

“The phase space (the thing we are quantizing) is nothing other than the space of classical solutions. So, a-priori, there’s nothing bad about having a symmetry of the phase space (the space of classical solutions) that does not extend to a symmetry of the full space of off-shell field configurations.”

I understand this. But how can we understand this from the path-integral viewpoint?

Not all symmetries survive quantization. Some are ruined by anomalies. From the path-integral viewpoint, an anomaly occurs when the “Lebesgue measure” D(phi) of path-integration is not invariant under the symmetry. However, when the symmetry is on-shell even the exp(iS[phi]) factor is not invariant. How can the symmetry survive? Is it the the product exp(iS[phi]) D(phi) is invariant, whereas each factor isn’t? How can we see this directly?

Posted by: Squark on September 8, 2007 8:09 PM | Permalink | Reply to this

Re: On BV Quantization, Part II

Not all symmetries survive quantization. Some are ruined by anomalies.

That’s exactly the point! (Local) anomalies show up in the inability to write down a regularized version of the quantum master equation.

Of course, even here, we’re not capturing the full structure of anomalies. As David Ben-Zvi points out, the BRST/BV formalism is completely insensitive to diconnected components of the group of gauge transformations (the “large” gauge transformations). So, even if there is no “local” anomaly, there remains the possibility of a “global” one, that the BRST/BV formalism (as far as I understand it) is not going to tell you about.

Posted by: Jacques Distler on September 9, 2007 3:53 AM | Permalink | PGP Sig | Reply to this

Re: On BV Quantization, Part II

“That’s exactly the point! (Local) anomalies show up in the inability to write down a regularized version of the quantum master equation.”

I understand this. What I would like is a geometrical description of what is going on, in terms of the path integral. Perhaps the n-categorical approach can provide it.

Geometrical description or not, I still don’t understand how Urs’ interpretation of BV as “n-categoricity in disguise” explains the later’s relation to anomalies.

Posted by: Squark on September 9, 2007 2:48 PM | Permalink | Reply to this

Re: On BV Quantization, Part II

I still don’t understand how Urs’ interpretation of BV as “nn-categoricity in disguise” explains the later’s relation to anomalies.

Right, good point. My discussion intentionally focused only on the kinematics: I did find it useful to emphasize that using the translation between “differential graded manifolds” and Lie nn-algebroids and the Lie (n+x)(n+x)-groupoids integrating them, we may regard the graded differential structure which BV imposes on phase space as a differential version of a higher categorical structure on phase space.

What I did not discuss was the BV master equation, classical or quantum, which then imposes a certain dynamics on this kinematical setup.

And the reason I did not was because I haven’t yet found a way to understand this equation in an analogous “nn-geometric” way. Which is, I think, what you are wondering about, too.

While I cannot offer an answer yet, let me point out that we did at least discuss this question here: it is closely related to the issue of symplectic groupoids that John Baez mentions in week 256.

My humble contribution to that discussion was to express the vaguely motivated impression that somehow the symplectic structure on the graded manifold might be regarded as intrinsically part of the weak skew-symmetry of the underlying Lie nn-algebroid. Which would translate into the structure morphisms for weak inverses in the corresponding Lie nn-groupoid.

If true (or at least roughly true) this would connect also the dynamics tightly with the nn-categorical structure.

But why this would help for regularizing the path integral, I can’t tell. But maybe in this context the close similarity between QFT regularization and categorical Euler characteristic regularization plays a role.

Posted by: Urs Schreiber on September 10, 2007 7:45 PM | Permalink | Reply to this

Re: On BV Quantization, Part II

This stuff is great.

For those of you not in Vienna: the only reason I’m not posting more about what Urs is saying is that we’re also talking in person about it! And Roytenberg, Severa and others are also here in Vienna, so I’m hoping Urs and I can talk to them and unlock the secret nn-categorical meaning of BV quantization.

Posted by: John Baez on August 20, 2007 9:13 AM | Permalink | Reply to this

Re: On BV Quantization, Part II

unlock the secret nn-categorical meaning of BV quantization.

Right now you made me spend all my time thinking harder on how to turn into a crisp precise formal statement what I mean to say regarding the nn-categorical concept of curvature.

But luckily that’s not unrelated (and this relation is indeed the reason why I keep thinking understanding what curvature really is is so important): roughly, where I keep going on about forming “inn()\mathrm{inn}(\cdots)”, in the BV world we crucially see the appearance of the shifted tangent bundle. This is related.

(Phew, I wish I were less tired. Hope I am not getting ill or something.)

Posted by: Urs Schreiber on August 20, 2007 12:02 PM | Permalink | Reply to this

Re: On BV Quantization, Part II

This comment belongs elsewhere, but SEARCH produced only this post.

For more (than you wnaat 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 fiel
so as soon as I have it
will ask for it to be posted


Posted by: jim stasheff on January 3, 2008 3:28 PM | Permalink | Reply to this
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