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For any commutative Frobenius algebra $A$ there is an associated window element $\omega \in A$. If $\mu: A \otimes A \to A$ denotes the multiplication, $1 \in A$ the unit, $b: A \otimes A \to k$ the n …

This MO answer by Arun Debray gives an example in the unoriented case where two specific homeomorphic manifolds can be distinguished by a specific TFT of this kind. In general all these constructions …

Given any finite dimensional algebra $A$, consider the linear dual $\hat{A}= \hom(A, k)$ as an $A$-$A$-bimodule. Then $R = A \oplus \hat{A}$ may be equipped with an algebra structure as follows: $$(a, …

The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored. We can already see this with $(\i …

Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an invertible …

We will use the fact that $M$ is invertible. Let ${}_BN_A$ be an inverse to $M$. Thus we have isomorphisms $${}_AM \otimes_B N_A \cong {}_AA_A$$ and $${}_BN \otimes_A M_B \cong {}_BB_B$$ If we make th …

In the paper Sato-Wakui "COMPUTATIONS OF TURAEV-VIRO-OCNEANU INVARIANTS OF 3-MANIFOLDS FROM SUBFACTORS" they compute certain Turaev-Viro-Ocneanu invariants of certain lens spaces. One of the results i …

I am trying to track down a copy of Graeme Segal's 1999 lecture notes on topological field theory. These are sometimes referred to as the "Stanford lectures" or something similar. For many years the …

The question of which tqfts extend is a very interesting one. To make the question more mathematically precise, we can fix the target n-categories and ask for the tqfts to extend with respect to those …

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am goi …

I might be confused about your question. Are you asking... How is trivializing the $O(n)$-action the same as giving an $O(n)$-equivariant non-degenerate trace? (as per Lurie's theorem 3.1.8). How ca …

This is covered in Moore and Segal "D-branes and K-theory in 2D topological field theory". In particular on around page 16 there is a characterization analogous to "1+1 TQFTs = Commutative Frobenius a …

These 2D TQFTs do not come from extended theories (unless X is discrete). I interpret this as saying that these theories are non-local (in the 2D bordism) and so you will have trouble interpreting the …

I also wrote a sequence of blog posts explaining the Turaev-Viro construction from the point of view of planar algebras. It has pretty pictures and might be relevant. TQFTs via Planar Algebras I TQF …

Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes_R A^{\mathrm{op}})$-algebra. Suppo …