DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Universal elements of covariant functors and presheaves

Definition

Let $C$ be locally small. For a functor $F : C \to S e t$ , a universal element is a pair $(R, u)$ with $R \in C$ and $u \in F (R)$ such that the maps

$θ_{c}^{u} : C (R, c) ⟶ F (c), θ_{c}^{u} (f) = F (f) (u)$

are the components of a natural isomorphism $θ^{u} : C (R, -) \overset{≅}{\Rightarrow} F$ .

For a presheaf $P : C^{o p} \to S e t$ , a universal element is a pair $(R, u)$ with $u \in P (R)$ such that

$θ_{c}^{u} : C (c, R) ⟶ P (c), θ_{c}^{u} (f) = P (f) (u)$

are the components of a natural isomorphism $θ^{u} : C (-, R) \overset{≅}{\Rightarrow} P$ . Thus a universal element is a representation in the sense of Presheaves, covariantly and contravariantly representable functors, and representations, with its natural isomorphism specified by a distinguished element of the representing object's value.

Depends on

Presheaves, covariantly and contravariantly representable functors, and representations

Used by

Dependency tree · two levels

6 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 2.3.3 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Corollaries 4.3.2 and 4.3.3 (standard reference, not scraped)