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? 



Can we define homotopy groups using Tannakian categories
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".


