Alphabeta MathHow statement and proof provenance work
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The plus presheaf is separated
Statement
For every set-valued presheaf F on a small site, F+ is separated. Moreover, if two sections of F agree after restriction along every arrow of a covering sieve, their images in F+ agree.
Facts & Assumptions
The site, matching families and separatedness are defined in Small sites and the plus construction.
Cover intersections cover by Covering sieves admit common refinements, and F+ is a presheaf by The plus construction is a presheaf.
Proof
Given: A covering sieve R on X and two elements of F+(X) with equal restrictions along every arrow of R.
Goal: Prove their equality and the stated property of η_F.
Represent the two elements by matching families s and t on covering sieves S and T. For each f in R, equality of the restricted classes means that they agree on a covering sieve of arrows into dom f after pulling back S and T. Let E be the sieve of arrows h in S ∩ T for which . For every f in R, contains such a covering sieve and therefore covers. Transitivity shows that E covers X.
The representatives agree on E, so their classes agree. Hence F+ is separated. If two sections of F agree on R, their restrictions on the maximal sieve agree on R, so their η_F images agree by the definition of the equivalence relation.
Depends on
Dependency tree · 1 level
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