TITLE: Can we define homotopy groups using Tannakian categories
QUESTION [13 upvotes]: 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?

REPLY [16 votes]: 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).