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On a preorder the comonads are exactly the monotone contractive maps with Gp below G(Gp); on a poset they are exactly the interior operators

Statement

Let $P$ be a preorder. A comonad on $P$ is equivalently a monotone map $G : P \to P$ such that $G p \leq p$ and $G p \leq G (G p)$ for every $p$ . These inequalities force $G p$ and $G (G p)$ to be mutually comparable. If $P$ is a poset, this is equivalently an interior operator: a monotone, contractive, idempotent map.

Facts & Assumptions

Given: A preorder $P$ .

[L1]

A comonad on $P$ is a monad on $P^{o p}$ (Comonad on a category).

[L2]

Monads on a preorder are exactly its monotone extensive maps equipped with the reverse comparison from their square (On a preorder the monads are exactly the monotone extensive maps with T(Tp) below Tp; on a poset they are exactly the closure operators).

[L3]

Reversing a preorder reverses each inequality (Opposite category $C^{o p}$ ).

[L4]

In a poset, mutual comparability implies equality (Partial order and partially ordered set).

Proof

technique · direct

1.1

By [L1], regard $G$ as a monad on $P^{o p}$ , with its counit and comultiplication serving as the unit and multiplication there.

L1L3

2.1

Applying [L2] in the opposite order gives monotonicity, $G p \leq p$ , and $G p \leq G (G p)$ . Monotonicity applied to $G p \leq p$ gives $G (G p) \leq G p$ , so the two values are mutually comparable.

L2L3step 1.1

3.1

Conversely, the stated inequalities reverse to the data in [L2] on $P^{o p}$ , hence give a comonad by [L1]. If $P$ is a poset, [L4] makes the two comparisons equivalent to $G (G p) = G p$ , precisely the idempotence condition for an interior operator.

L1L2L3L4step 2.1∎

Depends on

Used by

Topological interior is a comonad on the preorder of subsets, with the open subsets as its coalgebras Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Example 5.1.7 (standard reference, not scraped)