Timeline for 2-TQFT are to Frobenius Algebras as ??? are to Hopf Algebras
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 26, 2011 at 14:39 | comment | added | Charlie Frohman | We low dimensional topologists call these local relations "skein relations". They don't help much with homotopy theory, but they make TQFT in dimension 3 computable. :) | |
| May 26, 2011 at 4:12 | comment | added | Theo Johnson-Freyd | (cont'ation) a marked interval which will play the role of the antipode, and just impose a relation like the one mentioned at in tetrapharmakon's answer below. There, you have a "topological" category for which functors out are Hopf algebras. But is it useful? NO! Or, anyway, not as such. It is useful to remember that many types of algebraic objects can be "presented" in terms of some rewrite rules on labeled graphs --- this is one of the big ideas in the theory of operads. But it does not help you study TQFTs or homotopy theory of manifolds. | |
| May 26, 2011 at 4:10 | comment | added | Theo Johnson-Freyd | (continuation) condition like the one above to hold (which is good, since we're trying to present Hopf algebras). Ok, fine, so then you could just declare that through some magic, there is an isomorphism of these surfaces of whatever (e.g. find some distinguished cobordism-ish thing that realizes what I'm about to say, and mod out by cobordism-ish-es it generates) so declare that the "bialgebra compatibility condition" holds (this condition can be expressed as an equality of two different diagrams that can be made out of your Y-shaped generators). Now if you want, also give yourself (cont'ed) | |
| May 26, 2011 at 4:07 | comment | added | Theo Johnson-Freyd | @Sammy: Well, here's a stupid way that the answer is "yes". There is a "topological" category freely generated by two "Y" shapes (one-dimensional CW modules, or pairs of pants but with enough markings to make them behave essentially one-dimensionally), with markings or orientations or whatever you want so that one has two "inputs" and one "output", and the other is the reverse. Oh, also I want, if I'm thinking of these as surfaces, some half spheres that let me retract the Ys to tubes (or use a 1-dimensional model). Then if they have enough markings, you would not expect a (continued) | |
| May 25, 2011 at 19:24 | comment | added | Sammy Black | But perhaps there are ways to add markings to the surfaces or otherwise limit the possible isotopies so that we don't expect those two maps to be equal? | |
| May 25, 2011 at 11:51 | vote | accept | fosco | ||
| May 25, 2011 at 11:51 | comment | added | fosco | Can't I flag both to be the answer? This is algebraically clear, and that provides a useful reference... :P | |
| May 25, 2011 at 1:53 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |