Can we define homotopy groups using Tannakian categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:39:28Z http://mathoverflow.net/feeds/question/92076 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92076/can-we-define-homotopy-groups-using-tannakian-categories Can we define homotopy groups using Tannakian categories Harry 2012-03-24T13:22:44Z 2012-03-24T17:25:30Z <p>This is another vague question. Hope you guys don't mind.</p> <p>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.</p> <p>Can one give a meaningful definition of homotopy groups $\pi_n(T,F)$ using the Tannakian formalism? </p> http://mathoverflow.net/questions/92076/can-we-define-homotopy-groups-using-tannakian-categories/92078#92078 Answer by Jan Weidner for Can we define homotopy groups using Tannakian categories Jan Weidner 2012-03-24T14:02:08Z 2012-03-24T14:39:38Z <p>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:</p> <p>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.</p> <p>My guess would be in order to define higher homotopy groups, you also need "higher Tannakian categories".</p> http://mathoverflow.net/questions/92076/can-we-define-homotopy-groups-using-tannakian-categories/92084#92084 Answer by Feng-Wen An for Can we define homotopy groups using Tannakian categories Feng-Wen An 2012-03-24T16:07:39Z 2012-03-24T16:07:39Z <p>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.</p> <p>Even in such a case, it is the computation of fundamental group that the "higher homotopy groups" are applied to. </p> http://mathoverflow.net/questions/92076/can-we-define-homotopy-groups-using-tannakian-categories/92090#92090 Answer by Marc Hoyois for Can we define homotopy groups using Tannakian categories Marc Hoyois 2012-03-24T17:25:30Z 2012-03-24T17:25:30Z <p>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 <a href="http://ens.math.univ-montp2.fr/~toen/hab.pdf" rel="nofollow">this 2003 paper</a> by Bertrand Toën, and is maybe proved by Jacob Lurie in section 5 of <a href="http://www.math.harvard.edu/~lurie/papers/DAG-VIII.pdf" rel="nofollow">DAGVIII</a>.</p> <p>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).</p>