sheaf of abelian groups in nLab

Contents

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 1.6.5 of

• Charles Weibel, An Introduction to Homological Algebra

A detailed textbook discussion is in section 18 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)