Can we define homotopy groups using Tannakian categories - MathOverflow

Can we define homotopy groups using Tannakian categories

2012-03-24T13:22:44Z

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?

Answer by Jan Weidner for Can we define homotopy groups using Tannakian categories

2012-03-24T14:02:08Z

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".

Answer by Feng-Wen An for Can we define homotopy groups using Tannakian categories

2012-03-24T16:07:39Z

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.

Answer by Marc Hoyois for Can we define homotopy groups using Tannakian categories

2012-03-24T17:25:30Z

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).