  We define a notion on preadditive categories which plays a role similar to
the notion of a Grothendieck pretopology on an unenriched category. Each such
additive pretopology defines an additive Grothendieck topology and suffices to
define the sheaf category. This new notion allows us to study the noetherian
and subcanonical nature of topologies, to describe easily the meet of a family
of topologies and to identify useful universal properties of the sheaf
category. As our main application we derive in which ways one can present a
tensor category and show that such presentations lead to remarkable universal
properties.
