transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
An even number is an integer $n \in \mathbb{Z}$ that is a multiple of 2, hence such that $n = 2 k$ for $k \in \mathbb{Z}$.
An odd number is an integer that is not an even number.
(adjoint modality of even and odd integers)
Regard the integers as a preordered set $(\mathbb{Z}, \leq)$ in the canonical way, and thus as a thin category.
Consider the full subcategory inclusions
of the even and the odd integers, as well as the functor
which sends any $n$ to the floor $\lfloor n/2 \rfloor$ of $n/2$, hence to the largest integer which is smaller or equal to the rational number $n/2$.
These functors form an adjoint triple
and hence induce an adjoint modality
on $(\mathbb{Z}, \leq)$ with
$Even \coloneqq 2 \lfloor -/2 \rfloor$ sending any integer to its “even floor value”
$Odd \coloneqq 2 \lfloor -/2 \rfloor + 1$ sending any integer to its “odd ceiling value”.
Observe that for all $n \in \mathbb{Z}$ we have
where the first inequality is an equality precisely if $n$ is even, while the second is an equality precisely if $n$ is odd. Hence this provides candidate unit $\eta$ and counit.
Hence by this characterization of adjoint functors
the adjunction $\lfloor -/2 \rfloor \dashv odd$ is equivalent to the condition that
for every $n \leq 2 k + 1$ we have $2 \lfloor n/2 \rfloor + 1 \leq 2 k + 1$;
the adjunction $even \dashv \lfloor -/2 \rfloor$ is equivalent to the condition that
for every $2k \leq n$ we have $2k \leq 2 \lfloor n/2 \rfloor$,
which is readily seen to be the case
Last revised on December 2, 2020 at 10:56:45. See the history of this page for a list of all contributions to it.