  We expand the toolbox of (co)homological methods in computational topology by
applying the concept of persistence to sheaf cohomology. Since sheaves (of
modules) combine topological information with algebraic information, they allow
for variation along an algebraic dimension and along a topological dimension.
Consequently, we introduce two different constructions of sheaf cohomology
(co)persistence modules. One of them can be viewed as a natural generalization
of the construction of simplicial or singular cohomology copersistence modules.
We discuss how both constructions relate to each other and show that, in some
cases, we can reduce one of them to the other. Moreover, we show that we can
combine both constructions to obtain two-dimensional (co)persistence modules
with a topological and an algebraic dimension. We also show that some classical
results and methods from persistence theory can be generalized to sheaves. Our
results open up a new perspective on persistent cohomology of filtrations of
simplicial complexes.
