Mwaka wa kwanza wa uzamiliGeometry & TopologyImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 47 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- Thinking every presheaf is a sheaf; constant presheaves usually are not.
- Believing a sheaf is determined by its global sections.
Ugunduzi wa hiari
- Elementary topos theory Sheaves on a space form a Grothendieck topos, a central example of an elementary topos.
- ← Abelian categories
Usomaji zaidi
- Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic (1992)
- Masaki Kashiwara and Pierre Schapira, Categories and Sheaves (2006)