PRO-ALGEBRAIC FUNDAMENTAL GROUP // NORI FRAME // GALOIS CATEGORIES // NEUTRAL FIBER
The algebraic fundamental group pi_1^alg(X, x) of a variety X with basepoint x is the Tannakian group of the category of finite vector bundles with integrable connection. It is a pro-algebraic group scheme — the inverse limit of finite-dimensional quotients.
For X = P^1 \ {0, 1, infty}, this group is free pro-algebraic on two generators. Its pro-unipotent completion captures all iterated integrals and multiple zeta values. The Galois action on this group encodes deep arithmetic.
Nori constructed a universal Tannakian category from a directed graph (the Nori diagram) of effective pairs. For a variety X over a field k, the Nori fundamental group scheme pi_N(X, x) is an affine group scheme whose representations are exactly the "essentially finite" vector bundles.
Nori's construction works in arbitrary characteristic, unlike the de Rham approach. It captures both the etale fundamental group (as maximal pro-finite quotient) and the formal group (as maximal pro-infinitesimal quotient).
A Galois category (in the sense of Grothendieck) is a category equivalent to the category of finite continuous G-sets for a profinite group G. The fiber functor is evaluation at a geometric point. Tannakian categories upgrade this from G-sets to G-representations — from combinatorial to linear algebra.
The passage from Galois to Tannakian is the passage from etale covers to local systems, from permutation representations to linear representations. Both reconstruct the group, but the Tannakian group is richer — it remembers the linear structure.
A Tannakian category is "neutral" if the fiber functor lands in Vec_k (same field as the category). Non-neutral Tannakian categories (fiber functor to Vec_K for K/k extension) correspond to gerbes — twisted forms of groups. The Brauer group controls the obstruction to neutrality.
For X over a number field k, the comparison between Betti, de Rham, and etale fiber functors creates a web of periods, Galois representations, and p-adic Hodge theory. The motivic Galois group G_mot mediates: its representations are motives, and different fiber functors give different incarnations (Hodge, etale, crystalline).