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May 7, 2010 at 20:40 comment added Matthieu Romagny OK, I get it. I remember Niels Borne asking to me once about the possibility to cook up a theory including Galois categories and Tannakian categories under the same formalism. I have to admit that I have no idea how to do that.
May 7, 2010 at 20:25 comment added Lars It's not quite an answer: They come up with a Tannakian category that contains a lot of information about coverings and then show that the resulting Tannaka group relates to the usual etale fun. group (which is not surprising as both constructions use similar data). My question rather aimed at the underlying theories (Galois categories a la SGA1 vs Tannaka). As far as I remember the nonneutral cats show up because they are interested in nonclosed bases. It certainly is a very nice paper!
May 7, 2010 at 19:47 history answered Matthieu Romagny CC BY-SA 2.5