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Say I have an "ordinary" TQFT $F$ of dimension $n$, assigning groups or vector spaces to closed $(n-1)$-manifolds and linear maps to cobordisms. Consider the different ways $F$ can be obtained from a TQFT "extended one step" which assigns categories to manifolds of dimension $n-2$ (often derived categories of algebras or dg / $A_{\infty}$ algebras).

Is there expected to be any uniqueness to these extensions of $F$? For example, are there cases where you can extend the same $F$ two ways, but the corresponding (derived) categories associated to a codimension-2 manifold aren't equivalent?

This question is motivated partly by the situation in bordered Heegaard Floer homology; the dg algebra associated to a surface depends on a choice of parametrization, but these distinct algebras end up having equivalent derived categories (of type D or type A modules).

I'd be happy with a toy example in lower dimensions or anything illustrating this uniqueness holding or not holding- maybe in some restricted context. I'd also be curious to know if there's an $F$ which doesn't extend at all (maybe this is basic knowledge for the experts...)

Thanks!

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For your $n=2$ case, with vector spaces assigned to 1-manifolds, the extension to 0-manifolds is unique up to Morita equivalence (of linear 1-categories). This assumes that the extensions assign a semisimple 1-category to a 0-manifold, which might be implied by the extended TQFT axioms depending on which version of those axioms you choose.

For $n-1>1$, there is a categorified version of Morita equivalence for $(n-1)$-categories, and Morita equivalent $(n-1)$-categories lead to TQFTs which are isomorphic at the $(n-1,n)$ level. More generally, the $k$-categories assigned by the two TQFTs to closed $(n-1-k)$-manifolds are Morita equivalent. Morita equivalent $(n-1)$-categories can look somewhat different. For example, Morita equivalent tensor categories (2-categories with one 0-morphism) can have different numbers of (isomorphism classes of) simple objects.

(Warning: There is more than one version of categorified Morita equivalence in the literature. I have in mind the version that uses bimodules rather than functors "all the way up", as described here.)

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Cool! So, to make sure I understand what you're saying, if I have an $(n-1, n)$ TQFT $F$ and two extensions of $F$ all the way down to points, and furthermore if the extensions assign Morita equivalent $(n-1)$-categories to the point, then they assign equivalent categories at every level, right? What happens if you have two fully-extended TQFTs which assign non-equivalent $(n-1)$-categories to a point (and more generally, non-equivalent categories to codim-2 manifolds). Do they have a chance of still being isomorphic at the $(n-1,n)$-level? –  Andy Manion Apr 22 '13 at 18:38
    
In answer to your first question: yes, right. I'm not sure about the second question (non-Morita-equivalent $(n-1)$-categories giving rise to same $(n-1, n)$ structure). I'm not even sure which way I would bet, if I had to bet. –  Kevin Walker Apr 22 '13 at 21:25
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