Is there a path'' between any two fiber functors over the same field in Tannakian formalism? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:47:06Z http://mathoverflow.net/feeds/question/71731 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71731/is-there-a-path-between-any-two-fiber-functors-over-the-same-field-in-tannaki Is there a path'' between any two fiber functors over the same field in Tannakian formalism? James D. Taylor 2011-07-31T15:46:23Z 2011-08-01T11:40:15Z <p>I will take the approach of this question: <a href="http://mathoverflow.net/questions/23860/tannaka-formalism-and-the-etale-fundamental-group" rel="nofollow">http://mathoverflow.net/questions/23860/tannaka-formalism-and-the-etale-fundamental-group</a></p> <p>and think of the etale fundamental group as Tannakian formalism for $\mathbb{F}_1$. Then our "Tannakian category" is the category of finite etale covers, and each fiber functor is a functor from this category to $Sets$ (thought of as finite dimensional spaces over $\mathbb{F}_1$).</p> <p>For the etale fundamental group, it is true that for any two fiber functors (given by two different geometric points) there is a path'' between them. Meaning: there is a natural isomorphism between these two functors.</p> <p>My question is whether this independence of the basepoint'' result applies to Tannakian formalism as well:</p> <p>Is it true that for any two fiber functors $H_1, H_2: \mathcal{C}\rightarrow Vec_K$, there is a natural isomorphism $H_1 \cong H_2$?</p> http://mathoverflow.net/questions/71731/is-there-a-path-between-any-two-fiber-functors-over-the-same-field-in-tannaki/71734#71734 Answer by Angelo for Is there a path'' between any two fiber functors over the same field in Tannakian formalism? Angelo 2011-07-31T16:36:33Z 2011-07-31T16:36:33Z <p>I don't think this is true in general. The point is that that there are non-isomorphic groups with equivalent categories of representations; since the category of representations together with the fiber functor determines the group, this gives a counterexample.</p> <p>This happens under the following circumstances; the following construction was first given, I believe, in Giraud's book on non-abelian cohomology.</p> <p>Suppose that $G$ is an affine algebraic group over $\mathop{\rm Spec} K$ and $P \to \mathop{\rm Spec}K$ is a $G$-torsor. Call $H$ the group scheme of automorphisms of $P$ as a torsor; then $P$ becomes an $(H, G)$-bitorsor, that is, admits commuting actions of $G$ on the right and of $H$ on the left, and is a torsor for both. Conversely, if $P \to \mathop{\rm Spec}K$ is an $(H, G)$-bitorsor, then $H$ is the automorphism group scheme of $P$ as a $G$-torsor.</p> <p>Then the categories of representations of $G$ and $H$ are isomorphic. This follows, essentially, from descent theory; if $V$ is a representation of $G$, then the quotient $(P \times_{\mathop{\rm Spec}K} V)/G$ is a vector space on $K$ with an action of $H$; the inverse functor is obtained by exchanging $G$ and $H$ (and right and left actions).</p> <p>My favorite example of this is the following: if $q$ and $q'$ are non-degenerate quadratic forms in $n$ variable, the orthogonal groups $\mathrm O(q)$ and $\mathrm O(q')$ have equivalent categories of representations (although they are not isomorphic, in general). The bitorsor is the functor of isometries of $q$ and $q'$.</p> <p>On the other hand, if $K$ is algebraically closed the fiber functors are indeed isomorphic; if memory serves me well, this is in Deligne's paper in Grothendieck's Festschrift, but I don't have it here and can't check right now.</p> http://mathoverflow.net/questions/71731/is-there-a-path-between-any-two-fiber-functors-over-the-same-field-in-tannaki/71783#71783 Answer by Niels for Is there a path'' between any two fiber functors over the same field in Tannakian formalism? Niels 2011-08-01T07:40:52Z 2011-08-01T11:40:15Z <p>The obstruction to the existence of such an isomorphism is a (bi)torsor, that has been studied in various real-life situations. An example extracted from </p> <p>On the relation between Nori Motives and Kontsevich Periods Annette Huber, Stefan Müller-Stach <a href="http://arxiv.org/abs/1105.0865" rel="nofollow">http://arxiv.org/abs/1105.0865</a></p> <p>"As already explained by Kontsevich, singular cohomology and algebraic de Rham cohomology are both fiber functors on the same Tannaka category of motives. By general Tannaka formalism, there is a pro-algebraic torsor of isomorphisms between them. The period pairing is nothing but a complex point of this torsor."</p> <p>Basically, by tannaka duality, you can build a counter-example out of any couple of non-isomorphic objects of a gerbe.</p>