LemmaStatement: AI-generatedProof: AI-generated

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 mapping property of the plus construction

Statement

For a presheaf F and a sheaf G of sets on a small site, composition with $η_{F}$ gives a bijection from natural maps F+ → G to natural maps F → G. The bijection is natural in F and G.

Facts & Assumptions

[F1]

The sheaf condition and plus construction are defined in Small sites and the plus construction.

[F2]

The plus construction and its unit are natural by The plus construction is a presheaf; local equality of F-sections implies equality of their unit images by The plus presheaf is separated.

Proof

Given: A natural map u: F → G with G a sheaf.

Goal: Construct its unique extension over $η_{F}$ and prove naturality.

1.1F1F2given

For a representative (R,s) of an element of F+(X), the values $u (s_{f})$ form a matching family in G on R. The sheaf condition gives a unique section of G(X) restricting to them. Equivalent representatives give the same section because their glued sections agree on a common covering sieve. Pullback and uniqueness show that this assignment is natural and extends u.

2.1F1F2step 1.1∎

If v: F+ → G also extends u, then on each f in R its value on the class of (R,s) restricts to $u (s_{f})$ : the restricted class equals $η_{F} (s_{f})$ , because these representatives agree on $f^{*} R$ . The sheaf condition therefore forces v to equal the extension from step 1.1. Uniqueness also makes the bijection natural in F and G.

Depends on

Used by · 1

Scheme sites and sheafification

Dependency tree · 2 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.