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I won't be able to give any references, so I hope some more experts can help me out, as there is much work on traces of functors. In general, there are two reasonable notions of "trace" of a functor, …

The traditional way to define Reshetikhin–Turaev's version of Chern–Simons TQFT as a 321 theory assigns to the circle the category of representations of the quantum group at fixed root of unity (depen …

My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In hin …

Yes, there is a stably-framed bordism category. Recall that tangential structures on smooth $n$-manifolds can be parameterized by maps $X \to BO(n)$; an $X$-structure on $M$ is then by definition a li …

Some good references are the papers by Dan Freed and the book The geometry and physics of knots by Michael Atiyah. But by far the best answer to your question is in Witten's paper "Quantum field theo …

I think Davydov's papers Centre of an algebra and Full centre of an H-module algebra might be what you are looking for.

I'll echo supercooldave's vote for TikZ. For applications to knot theory, Kassel's longer book is great, as is his short book with Rosso and Turaev on quantum groups and knots. Kauffman's book Knot …

The endomorphisms of the interval are, at this informal level of discussion, surfaces with $S^1$ boundary. More generally, if you want to talk about $\hom(M,N)$, where $M$ and $N$ are $k$-dimensional …

It may take a bit of extraction, but positive answers to both of your questions follow from my results joint with Gaiotto in Condensations in higher categories (arXiv:1905.09566). In that paper we bui …

To understand the possible spaces of boundary conditions for a TQFT, it is helpful to start in highest dimension. Suppose you have a $(d+1)$-dimensional nonanomalous TQFT $\mathcal Q$. (The anomalous …