  This paper describes constructions in homological algebra that are part of a
strategy whose goal is to understand and classify symplectic mapping tori. More
precisely, given a dg category and an auto-equivalence, satisfying certain
assumptions, we introduce a category $M_\phi$-called the mapping torus category
that describes the wrapped Fukaya category of an open symplectic mapping torus.
Then we define a family of bimodules on a natural deformation of $M_\phi$,
uniquely characterize it and using this, we distinguish $M_\phi$ from the
mapping torus category of the identity. The proof of the equivalence of
$M_\phi$ with wrapped Fukaya category is proven in a different paper
(arXiv:1907.01156).
