ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The power-set functor is lax monoidal but not strong

Example

Let $P : S e t \to S e t$ be the covariant power-set functor: on a function $f : X \to Y$ , it sends $A \subseteq X$ to its direct image $f (A) \subseteq Y$ . For sets $X, Y$ , define

$P_{2} : P (X) \times P (Y) ⟶ P (X \times Y), (A, B) ⟼ A \times B,$

and let

$P_{0} : 1 ⟶ P (1)$

pick the unique subset ${*}$ of the singleton set.

Facts & Assumptions

Given: The power-set construction and the cartesian monoidal structure on $S e t$ .

[L1]

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

[L2]

Lax, strong, and strict monoidal functors are distinguished by their structure maps and whether those maps are isomorphisms (Lax, strong, and strict monoidal functors).

[L3]

$S e t$ is cartesian monoidal (Set, Cat, and every complete category are cartesian monoidal).

Verification

technique · direct

1.1L1L3algebra

Direct images preserve identities and composition, so the displayed action on functions makes $P$ a functor. By [L1] and [L3], the map $P_{2}$ is well typed: if $A \subseteq X$ and $B \subseteq Y$ , then $A \times B \subseteq X \times Y$ . For functions $f : X \to X^{'}$ and $g : Y \to Y^{'}$ , one has $(f \times g) (A \times B) = f (A) \times g (B)$ , so $P_{2}$ is natural. The unit map $P_{0}$ is also well typed because a singleton set has exactly one subset equal to itself.

2.1step 1.1L2L3

For subsets $A \subseteq X$ , $B \subseteq Y$ , and $C \subseteq Z$ , both sides of the lax associativity axiom send $((A, B), C)$ to the subset of $X \times (Y \times Z)$ consisting of all $(x, (y, z))$ with $x \in A$ , $y \in B$ , and $z \in C$ . The left and right unit axioms likewise send $(*, A)$ and $(A, *)$ back to $A$ . Hence $P$ is lax monoidal.

2.2L1step 1.1

Take $X = Y = {0, 1}$ . The diagonal subset $Δ = {(0, 0), (1, 1)} \subseteq X \times Y$ is not of the form $A \times B$ for subsets $A \subseteq X$ and $B \subseteq Y$ . So $P_{2}$ is not surjective, hence not an isomorphism.

3.1step 2.1step 2.2L2∎

Therefore the power-set functor is lax monoidal but not strong.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.