First-year graduateGeometry & TopologyPlanned
Sheaves
Presheaves and sheaves on a topological space, stalks and sheafification, sheaves of functions and of sections, and a preview of sheaf cohomology.
math.geom.sheaves
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
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)