ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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.

Singletons define a natural transformation from the identity functor on sets to the covariant power-set functor

Example

Sending an element to its singleton is natural when the power-set construction acts covariantly by direct image.

Facts & Assumptions

Given: Sets $X, Y$ and a function $f : X \to Y$ .

[L1]

The power set consists of all subsets, and direct image sends subsets of $X$ to subsets of $Y$ (The power set $P (x) = {z : z \subseteq x}$ , Subset $x \subseteq y$ , proper subset $x ⊊ y$ , and the separation notation ${z \in x : φ (z)}$ , The image $R [A]$ and the preimage $R^{- 1} [B]$ of a set under a relation).

[L2]

Sets form a category, and functors and natural transformations obey identity, composition, and naturality equations (Sets and functions form the large locally small category $S e t$ , Covariant functor, identity functor, composite functor, and contravariant functor, Natural transformation and its components).

Verification

technique · direct

1.1

Define $P (X)$ to be the power set of $X$ and $P (f) (S) = f [S]$ . Direct images satisfy $1_{X} [S] = S$ and $(g f) [S] = g [f [S]]$ , so $P : S e t \to S e t$ is a functor.

L1L2

1.2

Define $η_{X} : X \to P (X)$ by $η_{X} (x) = {x}$ .

2.1

For every $x \in X$ , $(P (f) η_{X}) (x) = f [{x}] = {f (x)} = (η_{Y} f) (x)$ . Therefore $P (f) η_{X} = η_{Y} f$ .

step 1.1step 1.2L1

3.1

The equality in step 2.1 is the naturality square for every function $f$ . Hence the singleton maps are the components of a natural transformation $η : 1_{S e t} \Rightarrow P$ .

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Saunders Mac Lane, Categories for the Working Mathematician, Chapter II (standard reference, not scraped)