Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The fundamental groupoid scheme and applications   We define a linear structure on Grothendieck's arithmetic fundamental group
$\pi_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It
allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar
k/k)$ to $\pi_1(X, x)$ with the existence of a neutral fiber functor on the
category which linearizes it. When applied to Grothendieck section conjecture,
it allows us to find a $k$-structure on the universal covering, and
$k$-rational pro-points at finite and infinite distance which lift given
$k$-rational points.
  Changes as compared to the first version: (some) typos corrected, english
impoved (?), (2.6) better explained, Thm 3.2, 3) made more precise,
correspondingly Cor. 4.6 made more precise as well. It equates conjugacy
classes of sections with neutral fiber functors. Finally Thm 5.1 establishes a
bijection between rational points and neutral fiber functors under the
assumption that Grothendieck's conjecture is true.

32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
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The categories of this paper are: eess
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga