Take the 2-minute tour ×  By an abelian $\otimes$-category I
mean a symmetric monoidal category
$(\mathcal{A},\otimes,\mathcal{O})$, such that $\mathcal{A}$
also is an abelian category and for every $M \in \mathcal{A}$
the functor $M \otimes - $ is cocontinuous (i.e. right exact and
preserves coproducts; in particular additive). A line bundle is
defined as an object $\mathcal{L}$ of $\mathcal{A}$ such that
there is some object $\mathcal{L}'$ such that $\mathcal{L}
\otimes \mathcal{L}' \cong \mathcal{O}$. An example is the
category of (quasi-coherent) modules on a locally ringed space.
The line bundles then coincide with the modules which are
locally free of rank $1$ (see here).  Now I want to show that in
general these line bundles have similar properties as in the
case of the module category. For example it's not hard to show
that if $\mathcal{L}$ is a line bundle, then it is flat in the
sense that $\mathcal{L} \otimes -$ is exact (it is even an
automorphism of $\mathcal{A}$ with inverse $\mathcal{L}^{-1} \o