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The plus construction is a presheaf
Statement
For every set-valued presheaf on a small site, the relation and restriction maps in the plus construction are well defined; they give a presheaf F+ and a natural map η_F: F → F+. A natural map F → G induces a natural map F+ → G+.
Facts & Assumptions
The constructions and topology axioms are those of Small sites and the plus construction.
Finite intersections of covering sieves cover by Covering sieves admit common refinements.
Proof
Given: A presheaf F on a small site.
Goal: Establish the relation, functorial restrictions, unit and functoriality in F.
Reflexivity and symmetry of the relation are immediate. For transitivity, intersect the two covering sieves on which successive pairs agree; the intersection covers and both equalities hold there. Thus the relation is an equivalence relation.
Pulling back a common covering sieve along an arrow gives a covering sieve on which the pulled-back families agree. Pullback of matching families respects identities and composition, so restriction descends to equivalence classes and makes F+ a presheaf.
Restrictions of a section along the maximal sieve define η_F, and restriction along any arrow gives the same class as applying η_F after restricting the section. Applying a natural map F → G pointwise to a matching family preserves compatibility and local equality, and commutes with restriction. These constructions prove both naturality assertions.
Depends on
Dependency tree · 1 level
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