nLab
sheaf of abelian groups

Context

Additive and abelian categories

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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.

References

A basic textbook introduction begins for instance around Definition 1.5.6 of

A detailed textbook discussion is in section 18 of

category: sheaf theory

Revised on March 6, 2013 19:48:22 by Zoran Škoda (161.53.130.104)