Sheaves and Gluing

Appendix B

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Sheaves and Gluing

A presheaf assigns data to contexts and specifies how data restricts to sub-contexts (Appendix A). A sheaf is a presheaf with a coherence guarantee: local data that agree on overlaps glue into global data, uniquely.

The Sheaf Condition

Sheaf

Let $(\mathcal{C}, J)$ be a site (Appendix A). A presheaf $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is a sheaf if for every cover $\{U_i \to U\}_{i \in I}$ in $J$ :

Locality. If $s, t \in F(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$ , then $s = t$ .

Gluing. If sections $s_i \in F(U_i)$ satisfy the matching condition:

s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j} \quad \text{for all } i, j

then there exists a unique $s \in F(U)$ with $s|_{U_i} = s_i$ for all $i$ .

Diagram form. The sheaf condition is an equalizer:

F(U) \xrightarrow{\text{restrict}} \prod_{i} F(U_i) \rightrightarrows \prod_{i,j} F(U_i \times_U U_j)

Sections over $U$ correspond exactly to matching families over the cover.

Locality prevents invisible global distinctions; gluing prevents orphaned local data. Together: global sections are completely determined by local restrictions, and compatible local data always has a global origin.

Sheafification

Given a presheaf $F$ , its sheafification $F^+$ is the closest sheaf to $F$ : any map from $F$ to a sheaf $G$ factors uniquely through $F^+$ .

Sheafification proceeds in two steps: (1) force gluing by replacing $F(U)$ with the set of compatible families, producing a separated presheaf; (2) apply the same construction again to obtain a sheaf. The construction adds exactly the global sections that should exist and identifies those indistinguishable locally.

Summary

Concept	Role in Coherence
Presheaf	Local data with restriction
Locality	Global sections determined by local restrictions
Gluing	Compatible local data assembles uniquely
Matching condition	Local sections agree on overlaps
Sheafification	Forcing coherence by excluding incompatible data

The sheaf condition formalizes A5 (Coherence Requirement) and is stated precisely as A13. When gluing fails, the failure is localized: specific overlaps where the matching condition breaks. The machinery does not just say "incoherent"—it says where coherence fails.

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