Timeline for Trace of a functor (or dimension of a category) in extended 2d TQFTs
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2012 at 18:00 | vote | accept | Ryan Thorngren | ||
| Oct 16, 2012 at 2:09 | comment | added | Ryan Thorngren | It strikes me now that the monoidal structure I've had in mind is non-canonical! It just comes from the action you mention being free and transitive. | |
| Oct 16, 2012 at 1:43 | comment | added | David Ben-Zvi | Thanks Ryan! Sorry I don't know the BF story, but I'm afraid I don't see in generality how bulk operators give local boundary conditions, the kind of things that define objects of Z(pt).. bulk operators (say in 2d) naturally act on boundary conditions (as endomorphisms of the identity of Z(pt), i.e. Hochschild cohomology), and also give "boundary states" (still objects of Z(S^1) ). But I think it's unlikely to have a monoidal structure on Z(pt) unless secretly your theory came from one of higher dimension.. | |
| Oct 16, 2012 at 1:26 | comment | added | David Ben-Zvi | In very brief: there's a general notion of dualizable object in a symmetric monoidal ($\infty$- or regular) category. Any such has a "dimension" which is an endomorphism of the unit, generalizing the dimension of a vector space, Euler characteristic of a complex, Hochschild homology (or chains) of a category, etc. In general in any TFT crossing with a circle amounts to calculating dimension in this sense. There are numerous interesting examples. I don't know a canonical source but some examples are in my Luminy lecture notes on my webpage ma.utexas.edu/users/benzvi | |
| Oct 16, 2012 at 1:21 | comment | added | Ryan Thorngren | I agree in that morphisms should be local boundary-changing operators and fusion of those gives composition. What I meant is that for boundary conditions that arise from some codimension 1 bulk operators, these can be fused as bulk operators onto the boundary. Sometimes all boundary conditions arise this way (eg. Wilson lines for 2d BF theory), but I'm not sure when one can say this for sure. | |
| Oct 16, 2012 at 0:58 | comment | added | David Ben-Zvi | The fusion of boundary conditions (if I understand your comment correctly) is what makes Z(pt) into a category to begin with -- it's the associative composition of morphisms between different objects (or endomorphisms of a fixed object), but as Theo says it doesn't make the category monoidal. | |
| Oct 15, 2012 at 21:13 | comment | added | Ryan Thorngren | @Theo I believe that it should be the category of boundary conditions for a 2d tqft, and so fusion should give a product. There's no reason to assume this for very general 2d tqfts I suppose. Anyway it's not that pertinent to the question. And thanks for your answer! | |
| Oct 15, 2012 at 20:59 | answer | added | Theo Johnson-Freyd | timeline score: 11 | |
| Oct 15, 2012 at 20:32 | comment | added | Theo Johnson-Freyd | A priori, why do you believe that $Z(pt)$ should be monoidal? In many examples, it simply isn't. | |
| Oct 15, 2012 at 19:14 | history | asked | Ryan Thorngren | CC BY-SA 3.0 |