nLab sheaf of abelian groups



Additive and abelian categories

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




For CC a site, a sheaf of abelian groups on CC is an abelian group object in the sheaf topos Sh(C)Sh(C).

The category Ab(Sh(C))Ab(Sh(C)) of sheaves of abelian groups is an abelian category and hence serves as a context for homological algebra “parameterized over CC”. For the case that C=*C = * is the point, this is just Ab itself.

More generally, for 𝒜\mathcal{A} an abelian category one can consider 𝒜\mathcal{A}-valued sheaves Sh(C,𝒜)Sh(C,\mathcal{A}): abelian sheaves. For this to have good properties 𝒜\mathcal{A} has to be a Grothendieck category.


A basic textbook introduction begins for instance around Definition 1.6.5 of

A detailed textbook discussion is in section 18 of

category: sheaf theory

Last revised on August 19, 2014 at 22:28:41. See the history of this page for a list of all contributions to it.