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A homotopy quantum field theory is a symmetric monoidal functor $\begin{multline} \begin{matrix} \mathrm{Cobord}(n) &\rightleftarrows\space \\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following diagram (see page 15 of this ArXiv article):
A homotopy QFT restricted to domain $\pi_1\mathrm{Map}(S^n,X)$ is a contravariant functor $\pi_1\mathrm{Map}(S^n,X)\to\mathrm{Vect}_{\mathbb{K}}\hookrightarrow\mathrm{Set}$, which is a locally constant sheaf on $\pi_1\mathrm{Map}(S^n,X)$, i.e., a sheaf of sections of a covering space of $\pi_1\mathrm{Map}(S^n,X)$. Let us denote by $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)$ the category of locally constant sheaves on $\pi_1\mathrm{Map}(S^n,X)$. Then, $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{Sh}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{Map}(S^n,X)$. We know that $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{HQFT}(n,X)$, and hence, my first question is:
Is the following statement true: $\mathrm{HQFT}(n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$? Or is it $\mathrm{HQFT}(n,X)\hookleftarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$ (which I think is unlikely, and most probably wrong)?
We know that loops on $\mathrm{Map}(S^n,X)$ are $X$-cobordisms, since $\pi_1\mathrm{Map}(S^n,X)\hookrightarrow\mathrm{HCobord}(n,X)$.
In the following article, it is written that a string connection assigns to a loop a vector space and assigns to cobordisms between loops, linear transformations. Hence, my second and third questions are:
Does it follow that string connections defined on the associated vector bundle over $\pi_1\mathrm{Map}(S^n,X)$ (what is the associated vector bundle over $\pi_1\mathrm{Map}(S^n,X)$? I haven’t been able to find any literature on it - so that would be helpful.) as defined in the article above are all homotopy quantum field theories restricted to domain $\pi_1\mathrm{Map}(S^n,X)$? What exactly would it mean for a string connection to be a HQFT?
What exactly would it mean for a string connection to be a HQFT?
I don’t know if you looked in the paper I wrote with Ulrich Bunke and Paul Turner Gerbes and homotopy quantum field theories but that might be of some use. [Sorry for the rather rushed response.]
Hi Sanath,
there is a bit of confusion in the statement of your question. Below I’ll walk through it in small steps, and I would like to ask you to follow through these steps.
But before I get into that, I have a general request: you tend to write lots of symbols which typically do not really parse. That makes it hard to tell what you really understand and what you are really after. Therefore my suggestion: before indulging in pseudo-formal mathematics, maybe try to state your statements/questions in plain English words.
In plain English words: what is the question that you are wondering about here?
Now on those symbols you give. You start with:
A homotopy QFT restricted to domain $\pi_1\mathrm{Map}(S^n,X)$ is a contravariant functor $\pi_1\mathrm{Map}(S^n,X)\to\mathrm{Vect}_{\mathbb{K}}\hookrightarrow\mathrm{Set}$, which is a locally constant sheaf on $\pi_1\mathrm{Map}(S^n,X)$,
First the bit “$\mathrm{Vect}_{\mathbb{K}}\hookrightarrow\mathrm{Set}$” is bad notation and maybe caused by a misunderstanding, so let me clarify: there is a forgetful functor seding vector spaces to their underlying sets. This is far from being a full and faithful functor which are the functors which usually one denotes by an inclusion sign $\hookrightarrow$.
Then, in case it is unclear: here $\mathrm{Map}(S^n,X)$ denotes a topological space, namely the mapping space from the topological space called $S^n$ (the $n$-sphere) to the chosen topological space $X$ (and both are regarded as pointed topological space in fact and the mapping space is that of pointed topological space, but at the point at which we are at the moment this is a detail we might maybe not want to dwell on too much for the moment).
Moreover, what Rodrigues writes $\pi_1(-)$ – and what maybe more commonly is written $\Pi_1(-)$ (not to confuse it with the fundamental group!) – is the fundamental groupoid-construction. So your $\pi_1\mathrm{Map}(S^n,X)$ is a groupoid.
Once one is sufficiently high-brow, then one may speak about presheaves etc. on groupoids and other categories, but for the moment I’d urge you to concentrate on learning first the default concept of a (pre)sheaf on a topological space. In particular that is what Rodrigues is actually talking about: locally constant presheaves on the topological space $\mathrm{Map}(S^n,X)$, not on the groupoid $\pi_1(\mathrm{Map}(S^n,X))$ as you write above.
He is alluding to, without saying it very explicitly, the following classical theorem:
For $S$ a locally simply connected topological space, then locally constant sheaves on $S$ are equivalent to functors from the fundamental groupoid of $S$ to the category of sets: $LConst(S) \simeq Func(\Pi_1(S), Set)$.
This is a fundamental fact which is really just the tip of a huge ice berg called Galois theory I am not sure if the $n$Lab has sufficiently pedagogic discussion of this fact at the moment. My reflex would have been to point to local system, but what we have there currently is certainly not the pedagogic introduction that one should point you to. Maybe somebody here has the time to write up a helpful paragraph or else to at least point to page and verse in some suitable lecture text.
In any case, in your question you seem to be mixing up the two sides of this equivalence, for instance when next you write:
i.e., a sheaf of sections of a covering space of $\pi_1\mathrm{Map}(S^n,X)$.
You see, it is the covering space of the topological space $\mathrm{Map}(S^n,X)$ that a locally constant sheaf is the sheaf of sections of, not its fundamental groupoid.
(Unfortunately for me, and I hope I don’t cause harm by saying this now and in any case you are requested to ignore for the moment what comes now —– in a sufficiently highbrow perspective of course also $\pi_1\mathrm{Map}(S^n,X)$ is a “space” and may be equipped with sheaves. On the other hand, since $\pi_1\mathrm{Map}(S^n,X)$ is a discrete space then, all its sheaves are locally constant, so it again makes no real sense to talk about locally constant sheaves on this. In any case, for the purpose of a first encounter with Rodrigues, we certainly should not.)
Then you write:
We know that $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{HQFT}(n,X)$,
No, this is not true, even if we interpret your symbols benevolently. Being benevolent here means to assume that on the left you mean $Func(\pi_1\mathrm{Map}(S^n,X), Vect)$. But then what we have is instead a forgetful functor the other way around $\mathrm{HQFT}(n,X) \to Func(\pi_1\mathrm{Map}(S^n,X), Vect)$. This is the functor that takes an HQFT and restricts it to just those cobordisms which are cylinders on $n$-spheres. This functor in turn is induced under mapping into $Vect$ from the functor $\pi_1\mathrm{Map}(S^n,X)\to HCobord(n,X)$ that Rodrigues discusses explicitly.
I would tend to want to urge you to take the time to write out and check this last sentence of mine in lots of detail on a page. If you are interested in these matters, it is paramount that you get a “foot on the floor” and know what exactly you are talking about, because at the moment you are floating around a bit in a cloud of symbols.
[ to be continued in the next comment ]
[ continued from previous comment ]
Finally your question:
and hence, my first question is: Is the following statement true: $\mathrm{HQFT}(n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$? Or is it $\mathrm{HQFT}(n,X)\hookleftarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$ (which I think is unlikely, and most probably wrong)?
As stated, this does not quite parse. If I use a benevolent interpretation of the symbols you give, then the answer is: No! to both.
What do you want to mean by “$\pi_1\mathrm{HCobord}(n,X)$” and then by “$\mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$”?
A default interpretation of these symbols would be that $\pi_1\mathrm{HCobord}(n,X)$ is to denote the core of the category $\mathrm{HCobord}(n,X)$. Do you mean that? This is of course a rather different use from the $\pi_1$ in all the rest of your message, so I doubt that this is what you really mean.
I’ll stop here. You should not try get into String-connections before some of the above basics (and a bit more) has been securely sorted out, lest you’ll be floating around with no ground under your feet.
Dear Sir (@Urs),
Thank you for the detailed response. I was not able to respond immediately - forgive me for that. As you know, I have learnt most of this by myself, and so am not completely familiar with notation. Thank you for clarifying the notational issues in the question. Thank you also for explaining to me where I did not understand a few ideas/concepts properly. I will read the nlab pages you have linked to in the comments above.
Warm Regards,
Sanath Devalapurkar
P.S. By $\pi_1 HCobord(n,X)$ I meant $\Pi_1 HCobord(n,X)$, and by $PSh\pi_1 HCobord(n,X)$ I meant $PSh\Pi_1 HCobord(n,X)$.
Sanath, I would suggest looking back in some detail at Turaev’s HQFT papers which are available in preprint form. The n-Lab pages are a very ‘skimpy’ summary. I know as I wrote one of the early drafts! Also look at Turaev’s book if you can get a copy and my papers on HQFTs as they give a different POV. Once you have absorbed some more of that then look at Lurie’s extended QFT stuff that is mentioned on the Lab at various places.
Like Urs, I do not see what you are trying to do and some of the ideas might be not that far off target but you are killing structure indiscriminately, then trying to rebuild it, rather than trying to retain it for as long as possible from the start. Try to explain what your aim is.
Hi Sanath,
I know. And I am willing to help you learn.
by $\pi_1 HCobord(n,X)$ I meant $\Pi_1 HCobord(n,X)$
And what do you want to mean by $\Pi_1 HCobord(n,X)$? Say it in words, not in symbols.
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