# axiomatizing the abelian part of the topological fundamental groupoid functor on algebraic varieties

let Vv be the category of complex algebraic varieties defined over $\bar Q\subset C$, and let $\pi_1^{top}:Vv\longrightarrow Groupoids$ sending a variety V into its (strict) fundamental groupoid $s,t: Paths(V(C))/fixed end point homotopy==>V(C) = Ob \pi_1^{top}(V)$. Let $\pi^{ab}$ be its abelianisation: $pi^{ab}(V)$ is the "largest" "factor" groupoid of $\pi_1^{top}(V)$ with an abelian fundamental group.

Let $\pi:Vv\longrightarrow Groupoids$ be another functor such that $Ob \pi(V) = V(C) = Ob \pi^{ab}(V)$. There is a field automorphism group $Aut(\Bbb C/\Bbb Q):Vv\longrightarrow Vv$ action on Vv. I am interested to know whether for some $\sigma \in Aut(\Bbb C/\Bbb Q)$ $\pi\circ\sigma$ is naturally equivalent to $\pi^{ab}:Vv\longrightarrow Groupoids$ (as functors to strict groupoids [1]). Under what assumptions on \pi is that necessarily true? There are some obvious requirements: $\pi(V)$ is connected for a connected variety V, takes direct product of varieties into the direct product of groupoids, for an etale morphism $V_1\longrightarrow V_2$ the induced morphism $\pi(V_1)\longrightarrow \pi(V_2)$ has the expected lifting property. ( Note these properties should imply that the profinite completion of $\pi(V)$ has the properties of the algebraic fundamental groupoid of V; one may be tempted to say that $\pi^{ab}(V)$ is a choice of a Z-structure on $\pi^{alg}(V)/[\pi^{alg}(V),\pi^{alg}(V)]$...)

I think there is a model theoretic proof (roughly, given in references on the webpage of Martin Bays, in a different language) that such a $\sigma$ necessarily exists if we assume that the restrictions of $\pi$ and $pi^{ab}$ to (homotopy classes of paths between) the algebraic points coincide[2], and that $\pi$ satisfies the conditions above; at least I think we can easily prove that such a sigma exists for every subcategory of Vv finitely generated in an appropriate sense. I am not sure how to state this correctly, but perhaps some model theoretic results ("categoricity over kernel") translate to say it is enough to require that that $\pi$ and $\pi^{ab}$ coincide as functors to groups rather than groupoids....Also, probably one can consider instead of $\pi^{ab}$ something like $\pi^{nil}$ (or any other functor to groupoids with subgroup separable groups).

Is this statement interesting? well-known? expected? easy to prove via algebraic geometry? In short, find "algebraic" conditions on the functor $\pi^{ab}:Vv\longrightarrow Groupoids$ such that for any functor $\pi:Vv\longrightarrow >Groupoids$ satisfying these condions, there exists some $\sigma \in Aut(\Bbb C/\Bbb Q)$ the functor $\pi\circ\sigma$ is naturally equivalent to $\pi^{ab}:Vv\longrightarrow Groupoids$ (as functors to strict groupoids [1]) provided the restrictions of $\pi$ and $pi^{ab}$ to (homotopy classes of paths between) the algebraic points coincide.

Is there a counterexample/theorem saying it cannot possibly be true?

I am only aware of examples Galois conjugated varieties that are topologically different; (Serre's Examples de varietes projectifs conjugees non-homeomorphes. C. R. Acad. Sci Paris @58 (1964), 4194-4196, more recently http://arxiv.org/abs/math/0701115 , http://arxiv.org/abs/0706.3674 ); this, however, does not seem to be directly relevant to this question.

[1] That is, a family of bijections between $Paths( \pi(V) ) = Mor ( \pi(V)) \Rightarrow Paths(\pi_1^{ab})= Mor(pi_1^{ab})$ and bijections $Ob \pi (V) = V(C) \Rightarrow Ob pi_1^top V(C)=V(C)$

[2] that is, for every V the sets of paths in \pi(V) and pi^ab(V) between points in V(\bar Q) are equal (as sets).