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 that which is divisible into two equal parts.
An odd number is that which is not divisible into two equal parts, or that which differs by a monad [=unit] from an even number.
[Euclid, Def. 6 & 7 of Elements Book VII (~ 400-300 BC), see here]
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 August 19, 2023 at 11:38:19. See the history of this page for a list of all contributions to it.