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Let us define a (co-)homology $X$-cobordism, where $X$ is a path connected space with basepoint $*$:
Definition: A (co-)homology $X$ cobordism $M:\Lambda_0\to\Lambda_1$ is a cobordism $M$ such that $H^\bullet(M)=H^\bullet(X)$ for cohomology and $H_\bullet(M)=H_\bullet(X)$ for homology, where $H^\bullet(A)=\bigoplus^{\dim A}_{k\in\mathbb{N}}H^k(A)$ and $H_\bullet(A)=\bigoplus^{\dim A}_{k\in\mathbb{N}}H_k(A)$.
Definition: A $\mathcal{C}$ valued (co-)homology QFT is a symmetric monoidal functor $(Co)HomCob(n,X)\to\mathcal{C}$, where the morphisms in $(Co)HomCob(n,X)$ are (co-)homology$X$-isomorphisms of (co-)homology $X$-cobordisms, defined as an isomorphism $\Phi:M\to M^\prime$ such that $\Phi(\partial_+ M)=M^\prime_+$ and $\Phi(\partial_- M)=M^\prime_-$.
What could possible uses of such a QFT be? Can this be related to Homotopy QFTs by the Hurewicz homomorphism $\pi_k(M)\to H_k(M)$?
I remember attending a talk on an idea of Homological QFT at some conference. I think this was in Spain at the Almeria workshop, but cannot find the reference.
Sanath, wait a moment.
What you write does again not really parse. Where are you taking this from?
I am a bit unwilling to give you detailed feedback on this question, because you have two pending tasks on which I already gave you fairly detailed feedback and on which I asked you to react (to show that you read the comments, thought about them, and either learned something or else voice need for specific further clarification).
Your first pending task, since a few weeks by now, is to correct the first paragraphs (to start with), of the nLab page which you created
As it stands, this page contains mostly stuff that does not quite parse. I wrote two fairly long emails to you walking you through what needs to be done about just the first paragraphs (to start with). If you don’t have interest in reacting to that, then we will have to clear that page.
The other pending assignment is that you should get back to me on my detailed reply to your previous question here, which you had titled
I would want to see some indication on your part that you thought about and understood (or else ask for clarification) the comments I made there, before we open the third construction site here. In particular I won’t get into discussion of cohomological field theory here with you before I have at least a little bit of indication that the time I spend has any effect on you.
Dear Sir (@Urs),
I tried to correct the first few paragraphs of Cobordisms as Presheaves. Please do tell me if it’s correct now.
I also responded to your comment on the page Homotopy quantum field theories and string connections. I am sorry for not being able to respond immediately.
Regarding the question on where I got this from - I thought about it while reading the nlab page on HQFTs. (I was thinking about maybe replacing homotopy with homology/cohomology, and creating a (co-)homology QFT with background $X$, as defined above.) I do not understand where it is incoherent, and how I can correct it.
Warm Regards,
Sanath Devalapurkar
Okay, I’ll look through that page further later when I have some time.
Meanwhile, let me repeat a suggestion: start by first becoming comfortable with 2d TQFT.
Specifically, start by reading – and absorbing in detail – this book here:
The first lines of the abstract seem to me to excellently match your needs:
This book, written for undergraduate math students, describes a striking connection between topology and algebra, expressed by the theorem that 2D topological quantum field theories are the same as commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics.
Let me know by email if you have trouble getting hold of a copy of that book.
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