ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 $\mathcal{P}(x) = \{\, z : z \subseteq x \,\}$ , Subset $x \subseteq y$ , proper subset $x \subsetneq y$ , and the separation notation $\{\, z \in x : \varphi(z) \,\}$ , 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 $\mathbf{Set}$ , Covariant functor, identity functor, composite functor, and contravariant functor, Natural transformation and its components).

Verification

technique · direct

1.1

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

L1L2

1.2

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

2.1

For every $x\in X$ , $(\mathcal P(f)\eta_X)(x)=f[\{x\}]=\{f(x)\}=(\eta_Yf)(x)$ . Therefore $\mathcal P(f)\eta_X=\eta_Yf$ .

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 $\eta:1_{\mathbf{Set}}\Rightarrow\mathcal P$ .

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 12 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

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