category of abelian sheaves

 algebraic geometry  topology
Notation Sh(X,Ab)\Sh(X,\Ab) Objects sheaves of abelian groups on a topological space XX Morphisms morphisms of sheaves Related Ab\Ab, Sh(X)\Sh(X) External nLab Link

Here, we assume that the topological space XX is neither discrete nor indiscrete, since otherwise this category is just a product of copies of Ab\Ab.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There are 9 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

Special objects

  • terminal object: trivial abelian sheaf
  • initial object: trivial abelian sheaf
  • products: section-wise defined direct product
  • coproducts: associated sheaf to the section-wise direct sum

Special morphisms

  • isomorphisms: morphisms of abelian sheaves that are bijective on every open set
  • monomorphisms: morphisms of abelian sheaves that are injective on every open subset
  • epimorphisms: morphisms of abelian sheaves f:F→Gf : F \to G that are "locally surjective": for every local section g∈G(U)g \in G(U) there is an open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i such that each g∣Ui∈G(Ui)g|_{U_i} \in G(U_i) is contained in the image of f(Ui):F(Ui)→G(Ui)f(U_i) : F(U_i) \to G(U_i).
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms

Comments

  • It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally finitely presentable, etc.) are satisfied for a generic space XX, but we need to make this precise by adding additional requirements to XX. Maybe we need to create separate entries for specific spaces XX.