TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Singleton and union define the covariant power-set monad

Statement

The covariant power-set functor $P : S e t \to S e t$ carries a monad whose unit is $η_{X} (x) = {x}$ and whose multiplication $μ_{X} : P P X \to P X$ is union.

Facts & Assumptions

Given: A set $X$ and functions between sets.

[L1]

The power set $P (X)$ is the set of all subsets of $X$ (The power set $P (x) = {z : z \subseteq x}$ ).

[L2]

Direct image sends a subset $A \subseteq X$ along $f : X \to Y$ to $f [A] \subseteq Y$ (The image $R [A]$ and the preimage $R^{- 1} [B]$ of a set under a relation).

[L3]

A monad requires natural unit and multiplication satisfying two unit laws and associativity (Monad on a category).

Proof

technique · direct

1.1L1L2L3

Define $P (f) (A) = f [A]$ . Direct images preserve identities and composition, so this is an endofunctor; define $η_{X} (x) = {x}$ and $μ_{X} (A) = ⋃ A$ .

2.1L2step 1.1

For $f : X \to Y$ , one has $f [⋃ A] = ⋃_{A \in A} f [A]$ , so union is natural. Also $f [{x}] = {f (x)}$ , which proves naturality of the singleton unit directly.

3.1L3step 1.1step 2.1∎

The two unit laws are $⋃ {A} = A$ and $⋃_{x \in A} {x} = A$ . Associativity is the equality obtained by removing either pair of parentheses from a union of families of families. These identities also hold when $X = \emptyset$ , so [L3] gives the claimed monad.

Depends on

Used by

Algebras for the covariant power-set monad are posets with all small suprema and their morphisms preserve every small supremum Theorem

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Example 5.1.5(i) (standard reference, not scraped)