PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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.

Initial and terminal objects are exactly the representations of the constant singleton functor

Statement

Let $C$ be a locally small category and let $1 : C \to S e t$ and $1 : C^{o p} \to S e t$ denote the constant functors at a singleton set.

An object $I$ is initial if and only if $I$ represents the covariant constant singleton functor, and an object $T$ is terminal if and only if $T$ represents the contravariant constant singleton functor. Consequently the covariant functor is representable exactly when $C$ has an initial object, and the presheaf is representable exactly when $C$ has a terminal object.

Facts & Assumptions

Given: A locally small category $C$ and the constant singleton functors in the statement.

[F1]

An object $I$ is initial when every $C (I, A)$ has exactly one morphism, and $T$ is terminal when every $C (A, T)$ has exactly one morphism (Initial object, terminal object, and zero object).

[F2]

A covariant functor is represented by $I$ through a natural isomorphism $C (I, -) ≅ F$ , and a presheaf is represented by $T$ through $C (-, T) ≅ P$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[F3]

Sets and functions form the category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

Proof

technique · direct

1.1

If $I$ is initial, each $C (I, A)$ is a singleton by [F1], so its unique function to $1 (A)$ is a bijection; these functions are natural because every map between singleton sets is the unique such map. Thus $C (I, -) ≅ 1$ .

F1F2F3

1.2

Conversely, if $C (I, -) ≅ 1$ , every $C (I, A)$ is bijective with a singleton and is therefore a singleton, so $I$ is initial.

F1F2

1.3

If $T$ is terminal, the same componentwise construction gives $C (-, T) ≅ 1$ , and any such representation makes every $C (A, T)$ a singleton; hence the terminal equivalence holds.

F1F2F3

2.1

Steps 1.1--1.3 show that a representing object exists exactly when the corresponding initial or terminal object exists. In the empty category the constant functors still exist but there is no object that could represent either one, in agreement with both equivalences.

step 1.1step 1.2step 1.3∎

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.

Sources

Emily Riehl, Category Theory in Context, Definition 2.1.3 (standard reference, not scraped)