Timeline for Does the Tannaka-Krein theorem come from an equivalence of 2-categories?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 7, 2010 at 18:32 | comment | added | Theo Johnson-Freyd | For the sake of completeness, I should say that I think there's something wrong with my above write-up, namely I worry I confused "equivalent" with "isomorphic", and so perhaps the equivalence of categories I've asserted must be amended on the LHS. | |
| Jul 7, 2010 at 18:30 | comment | added | Theo Johnson-Freyd | @JS Milne: Ah, great. | |
| Jul 7, 2010 at 3:38 | comment | added | JS Milne | @Theo et al. --- well, you could try looking up the definition in Catégories Tannakiennes (Saavedra 1972, Deligne 1990) or Tannakian Categories (Deligne and ... 1982, Breen 1994) or .. A Tannakian category over a field $k$ is neutral if it admits a fibre functor over $k$. In general, it only admits a fibre functor over an extension of $k$. There are various expressions of Tannaka duality in 2-category terms in Saavedra, e.g., III 2.3.2, p180. | |
| Jul 7, 2010 at 0:00 | vote | accept | Theo Johnson-Freyd | ||
| Jul 4, 2010 at 20:34 | comment | added | Evan Jenkins | @Kevin: Your definitions are historically accurate (and indeed still actively in use), but in my opinion, somewhat outmoded. The notion of Tannaka duality has expanded to include a host of structures other than Hopf algebras and algebroids. For instance, results of Hayashi and Szlachányi show that any sufficiently finite tensor category (i.e., a multi-fusion category) is a category of finite dimensional representations of a quasi-Hopf algebra. This fact, combined with the general difficulty of determining when a category is "Tannakian" in the original sense, make the term misleading at best. | |
| Jul 4, 2010 at 19:10 | comment | added | Kevin Buzzard | @Theo: let me again say that I am not an expert. I thought the idea was that a neutral Tannakian category was one with a fibre functor, and was hence equivalent to the category of representations of a group, and a Tannakian category was one that admitted a fibre functor, but you weren't going to fix any one in particular, so it's slightly weaker than the cat of reps of a group. It's like the difference between the fundamental group and the fundamental groupoid I think, the idea being that choosing a fibre functor is like choosing a point of a top space (and hence getting a stalk where pi1 acts | |
| Jul 4, 2010 at 3:24 | comment | added | Theo Johnson-Freyd | @Kevin Buzzard: Oh, maybe. Then what's a Tannakian category? Just a category with a functor to Vect? Wikipedia, in its "formal definition" section, only lists "neutral Tannakian category", but it also demands monoidal rigid, or some such, so that the reconstructed coalgebra is actually a Hopf algebra. I figured that was what the adjective "neutral" was doing. | |
| Jul 3, 2010 at 20:59 | comment | added | Kevin Buzzard | I'm not an expert, but I thought that the faithful exact functor---the fibre functor, right?---was not part of the definition of a Tannakian category, and what you have defined above is a neutral Tannakian category. | |
| Jul 3, 2010 at 20:49 | answer | added | Evan Jenkins | timeline score: 6 | |
| Jul 3, 2010 at 20:27 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
fixing formating
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| Jul 3, 2010 at 20:06 | history | asked | Theo Johnson-Freyd | CC BY-SA 2.5 |