In
arithmetic geometry, a Frobenioid is a
category with some extra structure that generalizes the theory of
line bundles on models of finite extensions of
global fields. Frobenioids were introduced by . The word "Frobenioid" is a
portmanteau of
Frobenius and
monoid, as certain Frobenius morphisms between Frobenioids are analogues of the usual
Frobenius morphism, and some of the simplest examples of Frobenioids are essentially monoids.
The Frobenioid of a monoid
If ''M'' is a
commutative monoid, it is acted on naturally by the monoid ''N'' of positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s under multiplication, with an element ''n'' of ''N'' multiplying an element of ''M'' by ''n''. The Frobenioid of ''M'' is the
semidirect product of ''M'' and ''N''. The underlying category of this Frobenioid is category of the monoid, with one object and a morphism for each element of the monoid. The standard Frobenioid is the special case of this construction when ''M'' is the additive monoid of non-negative integers.
Elementary Frobenioids
An elementary Frobenioid is a generalization of the Frobenioid of a commutative monoid, given by a sort of
semidirect product of the monoid of positive integers by a family Φ of commutative monoids over a base category ''D''. In applications the category ''D'' is sometimes the category of models of finite separable extensions of a global field, and Φ corresponds to the line bundles on these models, and the action of a positive integers ''n'' in ''N'' is given by taking the ''n''th power of a line bundle.
Frobenioids and poly-Frobenioids
A Frobenioid consists of a category ''C'' together with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global fields. One of Mochizuki's fundamental theorems states that under various conditions a Frobenioid can be reconstructed from the category ''C''. A poly-Frobenioid is an extension of a Frobenioid.
See also
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Category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
*
Anabelian geometry
*
Inter-universal Teichmüller theory
References
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* {{Citation , last1=Mochizuki , first1=Shinichi , url=http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations%20(comments).pdf, title=Comments, year=2011
External links
What is an étale theta function?
Algebraic geometry
Number theory