# nLab sheaf of abelian groups

### Context

#### Additive and abelian categories

additive and abelian categories

## Derived categories

#### Topos Theory

Could not include topos theory - contents

# Contents

## Idea

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

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

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

## References

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

Revised on August 19, 2014 22:28:41 by Anonymous Coward (89.0.17.112)