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This is another vague question. Hope you guys don't mind.

Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to be the tensor-compatible automorphisms of $F$. This fundamental group is representable by an affine group scheme.

Can one give a meaningful definition of homotopy groups $\pi_n(T,F)$ using the Tannakian formalism?

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What would you expect from the higher homotopy groups? I think higher homotopy groups make sense if you have some kind of suspension and fibre sequences. The Tannakian formalism gives you just one group, so to get more groups you need something at least slightly different. –  Konrad Voelkel Mar 24 '12 at 14:11
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3 Answers

up vote 9 down vote accepted

Consider a connected topological space $X$ with base point, then the category of local systems on $X$ is Tannakian and in fact equivalent to representations of the fundamental group of $X$. So this category depends in no way on the higher homotopy groups of $X$, hence you can not reconstruct them. In fact the argumentation here is kind of independent of the example:

Any Tannakian category is completly encoded by its fundamental group, i.e. there is simply no additional information which could be used to define higher homotopy groups.

My guess would be in order to define higher homotopy groups, you also need "higher Tannakian categories".

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While the category of local systems depends only on the fundamental group, the `derived category of local systems' knows more about the homotopy type. For example, the Ext groups of the trivial local system give the cohomology of your space, which of course may be quite non-trivial even if your space is simply connected. Toen has developed a theory of higher Tannakian categories that take this into account. –  Chris Brav Mar 24 '12 at 17:19
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There certainly is a notion of higher Tannakian category which would have meaningful higher homotopy groups. I'm not sure how much of the theory has been worked out already, but higher Tannakian duality is formulated for example in Conjecture 5.13 in this 2003 paper by Bertrand Toën, and is maybe proved by Jacob Lurie in section 5 of DAGVIII.

The higher analogue of Jan's example is the following: local systems of $\infty$-groupoids on a space $X$ are equivalent to representations of the fundamental pro-$\infty$-groupoid of $X$ (which is the homotopy type of $X$ if $X$ is a paracompact space homotopy equivalent to a CW complex).

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In Quillen's algebraic K-theory, higher homotopy groups are defined in a simplicial manner. It's maybe used to get an algebraic version of higher homotopy groups for some special cases.

Even in such a case, it is the computation of fundamental group that the "higher homotopy groups" are applied to.

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