Sheaves

Builds on

Required prerequisites, all the way down: 47

You will learn to

• Define presheaves as functors from the opposite of the poset of open sets to sets, and state the sheaf condition as an equaliser condition.
• Compute stalks and sheafify simple presheaves, such as constant presheaves.
• Explain how sheaves of continuous, smooth and holomorphic functions encode local-to-global structure.

Definition

• A presheaf on X is a functor O(X)ᵒᵖ → Set, where O(X) is the poset of open subsets of X.
• A presheaf F is a sheaf if, for every open cover {Uᵢ} of an open set U, the diagram F(U) → ∏ F(Uᵢ) ⇉ ∏ F(Uᵢ ∩ Uⱼ) is an equaliser.
• The stalk F_x is the colimit of the sets F(U) over open neighbourhoods U of x.

Theorem

• Sheafification is left adjoint to the inclusion of sheaves into presheaves.
• A morphism of sheaves is an isomorphism if and only if it induces isomorphisms on all stalks.
• The category of sheaves of sets on a topological space is a Grothendieck topos.

Common misconceptions

• Thinking every presheaf is a sheaf; constant presheaves usually are not.
• Believing a sheaf is determined by its global sections.

Optional discoveries

• Elementary topos theory Sheaves on a space form a Grothendieck topos, a central example of an elementary topos.
• ← Abelian categories

Further reading

• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic (1992)
• Masaki Kashiwara and Pierre Schapira, Categories and Sheaves (2006)