bibkey: stacks2026finitegluing
authors: The Stacks Project Authors
year: 2026
title: Sheaves and finite nonempty inverse systems
doi: null
url: https://stacks.math.columbia.edu/tag/086J
claim: Cofiltered limits of finite nonempty sets are nonempty without a surjectivity hypothesis; compatible dependent functions glue uniquely.
strata_touched:
  - D5/S3/ObserverMemory/InverseLimits/FiniteCofilteredLimit
  - D5/S3/ConceptDynamics/Observation/HistoryPayloadFactorization
license: citation-only
triage: anchor

Finite inverse systems and assignment gluing

Lemma 4.21.7, tag 086J, states that a cofiltered diagram of finite nonempty sets has nonempty limit. The original transition maps need not be surjective. The proof constructs a minimal nonempty subsystem, proves its maps surjective, and then proves its component sets are singletons.

Section 6.7, tag 006S, defines the unique gluing condition for sheaves. Its product-of-fibers example assigns to an open set U the product of the sets A(x) over x in U. Overlap equality supplies a unique common value at every point, hence a unique section. A discrete topology allows arbitrary subsets as opens.

For assignments constrained on supports K(c), the common function is legal if each support contained in the global domain lies entirely in one member of the cover. This last condition permits each relation to be checked on a single local section; it is additional to the sheaf condition for functions.

Verified locator

• Lemma 4.21.7: https://stacks.math.columbia.edu/tag/086J
• Section 6.7, including the product-of-fibers example: https://stacks.math.columbia.edu/tag/006S