sheaf

层

Algebra & Algebraic Geometry · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A presheaf of sets on a topological space such that every compatible family of sections on an open cover, including the empty cover, is the restriction of a unique section on the union.

拓扑空间上的集合值预层，满足任意开覆盖（包括空覆盖）上的相容截面族都能唯一粘合为并集上的截面。

Example

Continuous real-valued functions on open subsets, with the usual restriction maps, form a sheaf.

Usage and distinctions

Presheaf restriction identities alone do not ensure gluing; bounded continuous functions on open subsets of the real line give a counterexample.

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Definition sources

  1. The Stacks Project — SheavesDefinition 6.7.1 and Remark 6.7.2, including the empty covering · Checked 2026-10-15

Translation sources

  1. 李文威 — 模形式初步，作者公开稿Appendix B.1 层与局部系统, Definition B.1.1 explicitly names 层 through unique gluing; printed pp.341–342 (PDF pp.347–348), including the Set-valued variant · Checked 2026-10-15
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