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I have edited at HQFT, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.
Are HQFTs taking place in some slice or context from the HoTT point of view?
I think the good abstract way to think about them is given by that version of the cobordism hypothesis which I was alluding to above: $Bord^{fr}_n(X)$ is essentially the free $(\infty,n)$-category with duals not on the point, but on the $\infty$-groupoid $\Pi(X)$.
Having worked on HQFTs with Turaev, and having checked Segal’s Seminar Bourbaki article, I do not think it is quite right to say HQFTs were ’introduced’ by Graeme Segal in 1988. Segal’s very vague sketch at that time does look at a slightly similar idea and I know that Ulrike Tillman, later, in work related to the Galatius et al theorem did look at the homotopy type of a cobordism category, as did Brightwell and Turner (The homotopy surface category), yet the definition in detail is not given there and the ’devil is in the details’, so to attribute it to Segal is a bit strange. I would like to hear Dimitri’s view on this but would suggest rewording so as to give most of the credit to Vladimir Turaev.
Another point is that Segal’s sketch seems to concentrate on the space X, whilst Turaev is thinking of the manifolds with structure encoded by the map to the base, which might be a classical form of classifying space, but need not be, a viewpoint taken further by Lurie.
Tim, I just went and checked Segal’s p. 14 and it has exactly all the detail that, at least, our entry claims there is to an HQFT.
Generally, it’s not right to try to trump attribution by adding details to somebody else’s definition, or by claiming that one now “thinks differently” about it.
What you might want to say is that other people went and worked out consequences/properties of the definition in more detail. And that’s exactly how Dmitri worded it in rev 20.
I disagree as to the detail, and to some of the substance, but do not worry about the difference unduly. From my viewpoint Segal did not define HQFTs. He sketched in that paper a neat and important idea that he and other worked out later, but it was not HQFT although related to it.
Re #5: I added a new section to the article, where the content of Section 6 from Segal’s article is transcribed.
Segal writes:
Now let us define an elliptic object of level $k$ on $X$ as a projective functor $E\colon\mathcal{C}_X\to \mathcal{V}$ of level $k$ which is holomorphic and satisfies the contraction condition.
The category V was previously defined as follows:
It is appropriate to consider functors from $\mathcal{C}$ to the category $\mathcal{V}$ of topological vector spaces and trace-class maps.
The category C_X is defined as follows:
For any space $X$ one can now define a category $\mathcal{C}_X$. Its objects are pairs $(S,s)$, where $S$ is an object of $\mathcal{C}$ and $s \colon S \to X$ is a map. Its morphisms from $(S_0,s_0)$ to $(S_1,s_1)$ are triples $(\Sigma,\alpha,\sigma)$, where $(\Sigma,\alpha)\colon S_0 \to S_1$ is a morphism in $\mathcal{C}$, and $\sigma\colon\Sigma\to X$ is a map compatible with $(s_O,s_1)$.
And the category C is defined as follows:
I have described elsewhere [23] a category $\mathcal{C}$ whose objects are all compact oriented one-dimensional manifolds, and whose morphisms from $S_0$ to $S_1$ are pairs $(\Sigma,\alpha)$, where $\Sigma$ is a Riemann surface with boundary $\partial\Sigma$, and $\alpha$ is an isomorphism between $\partial X$ and $S_1-S_0$. Two pairs $(\Sigma,\alpha)$, $(\Sigma',\alpha')$ are identified if they are isomorphic.
So Segal’s definition of an HQFT as a functor C_X→V appears to be exactly the same as Turaev’s.
Concerning this remark:
Another point is that Segal’s sketch seems to concentrate on the space X, whilst Turaev is thinking of the manifolds with structure encoded by the map to the base, which might be a classical form of classifying space, but need not be, a viewpoint taken further by Lurie.
I am not sure I understand this statement. Segal is considering bordisms equipped with a map to a topological space X, exactly like Turaev and Lurie. As far as I am aware there is no difference per se between their bordisms, only in what kind of categorical structure they are organized: Lurie organizes his into a symmetric monoidal (∞,d)-category, Segal uses ordinary categories and functors, but with restrictions like continuity.
In conversations I had with Turaev, it seemed he was not that interested in the base X as such, as that was supposed known and typically was a K(G,1) in his examples. He was interested more in the (geometric topology of the) manifolds with structure given by the characteristic map to X. What I meant in that paragraph was that in Segal and in later work by others, the object sometimes seems partially to probe X by (structured) surfaces and to interpret that in terms of 1-dimensional manifolds with structure, whilst Turaev’s is to understand the general idea of ‘$n$-manifold with characteristic map’ in an analogous way to TQFTs. The mechanisms are very closely related, of course, but the object of the study is slightly different and in addition Turaev’s definition is very much more general. I am not convinced the two approaches amount to the same thing.
I am afraid that to say the Graeme Segal gave the ’original definition’ of HQFTs to me seems untrue. He defined ‘eliptic objects’. Turaev defined HQFTs which are very general (and benefit from 10 years of development of TQFTs as well) and then it is noticable that Segal’s elliptic objects seem to be precursors of Turaev’s general definition. (I note that Turaev’s book on HQFTs does not have any reference to Segal’s Bourbaki talk. I do not know if he knew of Segal’s ’definition’. He made his definition for its own sake, generalising from the definition of TQFTs, and it so happens that Segal’s elliptic objects are a precursor of a very special case of that definition.)
I have not gone into the subtleties of Segal’s definition but the conditions that make the HQFTs a homotopy QFT I think relate to the map F for a cobordism to the base X. Segal says
and $\sigma\colon\Sigma\to X$ is a map compatible with $(s_O,s_1)$.
In the original form of HQFTs $F:\Sigma\to X$ is a homotopy class of maps agreeing with the maps on the two ends of the cobordism. In his book, Turaev changes to demanding that the linear map ‘$\tau(W,g)$ is preserved under homotopies of $g$’. Is that true in Segal’s approach? He does mention towards the end:
’related to flat elliptic objects, i.e. ones such that the operator associated to $(\Sigma,\alpha,\sigma)$ depends on $\sigma$ only up to homotopy, and is therefore a homomorphism $\pi_1(\Sigma)\to G$.
He does not include details.
Re #10:
I am afraid that to say the Graeme Segal gave the ’original definition’ of HQFTs to me seems untrue. He defined ‘eliptic objects’.
Segal’s flat elliptic objects defined in his article coincide with Turaev’s HQFTs in dimension 2.
in addition Turaev’s definition is very much more general. I am not convinced the two approaches amount to the same thing.
The only difference I see is that Turaev replaces “dimension 2” by “dimension d≥0”, a generalization proposed long before Turaev, in any case.
You mentioned the homotopy invariance condition, which is stated in exactly the same way in Segal’s article and in Turaev’s book.
What is present in Turaev’s definition that is not present in Segal’s definition?
He does not include details.
I agree. But the article does not say that Segal studied it in any detail, only that he gives a definition.
I feel that Turaev does the hard work. Segal does not. He gives a feasible definition of something with some details.
Segal defines something he calls ’elliptic objects’. He does not define Homotopy Quantum Field Theories. The n-lab article says ’HQFTs were introduced by Graeme Segal as early as 1988’, and that is factually untrue in the usual use of the English language, so gives the wrong impression. ’HQFTs are an extension of an idea that was introduced by Graeme Segal as early as 1988’.’ would be correct.
I am not nit picking as to me it seems a bad precedent to give the credit to someone who has an idea and does not work out and publish the details. Otherwise someone could give a definition, do nothing with it, and yet when 10 years later a similar idea is put-forward, the first person (who did not go to the trouble of development and publication) gets the credit and not the researcher who has checked all the details, ironed out the glitches, etc. (I know that this has happened many times in the past but that is no excuse.) By all means say that Segal put forward the idea, and that later it was rediscovered by Turaev and pushed forward / generalised and refined.
It is the wording that I do not agree with, not the essence of the mathematics, although I do not fully understand Segal’s terminology.
Many of these ideas were knocking around during the 1980s and 1990s, so it is not surprising that others came up with similar ideas. It is useful to have as many of these as possible.
I like the rewording. Thanks.
Segal defines something he calls ’elliptic objects’. He does not define Homotopy Quantum Field Theories.
Is this a statement about mathematics or about terminology?
Here are some statements, which I think are true:
Segal’s flat elliptic objects are exactly the same as Turaev’s 2-dimensional HQFTs; the two definitions are not merely equivalent, but in fact coincide almost word-for-word.
Segal does not use the term “HQFT”, which was introduced by Turaev in 1999.
Segal does not develop the notion that he defined beyond the definition itself. Starting from 1991, Freed develops it in several papers. Starting from 1999, Turaev develops it even more and publishes a book eventually.
does not work out and publish the details.
Segal’s manuscript “The definition of a conformal field theory” contained many details and did circulate very widely (it was eventually published in 2004).
Many details were worked out by Freed in a series of papers starting from 1991, which originated from Segal’s work. In particular, Freed also uses HQFTs (not under this name) in several of his papers, starting from 1991.
rediscovered by Turaev
Segal’s two papers from 1987/1988 (“The definition of conformal field theory” and “Elliptic cohomology”) created the entire area of functorial field theory, including both topological and conformal field theory, and the second paper does contain the definition of HQFTs.
Several of Freed’s papers use the same notion, starting from 1991, and these papers circulated widely and were cited a lot.
So “rediscovered” may not be fully appropriate for a notion that already appeared in one of the two foundational papers and was further developed in papers by Freed that were widely circulated.
Re #15: Since we both like it, I think this formulation can be kept.
I have looked further in Turaev’s book, and he does cite Freed and Quinn’s paper, albeit in a different context.
I agree with your timeline as far as I know it. My one disagreement might be that Segal only seems to have defined the 1+1 HQFT. Although once you have n+1 TQFTs as functors and the 1+1 case of HQFTs the rest may be thought of as being an obvious direction to go in.
As I said I am happy with your rewording, (and thanks for the Freed Quinn paper which although I knew of it and probably have a hard copy, in a box file somewhere, I had forgotten and did not have an electronic copy of.)
Just to be completely clear on Dmitri’s point, I have slightly reworded “introduced as” to “introduced under the name”.
Thanks for the above exchange. I have taken screenshots to preserve it for posterity. I knew this happened behind the scenes, but never saw it being admitted openly.
I think the point is that the wording in an entry that someone thinks fits their view may, to another person reading it, seem awkward or perhaps unfair in attribution of ’credit’ for the development of a piece of the theory. The best way to resolve the misunderstanding / disagreement is to discuss it in the Forum so that the different viewpoints can hopefully be merged into something that reflects a concensus view. It can also happen that no concensus is reached, but thankfully that seems to happen rarely here.
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