Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Torified varieties and their geometries over F_1   This paper invents the notion of torified varieties: A torification of a
scheme is a decomposition of the scheme into split tori. A torified variety is
a reduced scheme of finite type over $\Z$ that admits a torification. Toric
varieties, split Chevalley schemes and flag varieties are examples of this type
of scheme. Given a torified variety whose torification is compatible with an
affine open covering, we construct a gadget in the sense of Connes-Consani and
an object in the sense of Soul\'e and show that both are varieties over $\F_1$
in the corresponding notion. Since toric varieties and split Chevalley schemes
satisfy the compatibility condition, we shed new light on all examples of
varieties over $\F_1$ in the literature so far. Furthermore, we compare
Connes-Consani's geometry, Soul\'e's geometry and Deitmar's geometry, and we
discuss to what extent Chevalley groups can be realized as group objects over
$\F_1$ in the given categories.

19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin