  In this thesis, we develop the theory of bifibrations of polycategories.
  We start by studying how to express certain categorical structures as
universal properties by generalising the shape of morphism. We call this
phenomenon representability and look at different variations, namely the
correspondence between representable multicategories and monoidal categories,
birepresentable polycategories and $\ast$-autonomous categories, and
representable virtual double categories and double categories.
  We then move to introduce (bi)fibrations for these structures. We show that
it generalises representability in the sense that these structures are
(bi)representable when they are (bi)fibred over the terminal one. We show how
to use this theory to lift models of logic to more refined ones. In particular,
we illustrate it by lifting the compact closed structure of the category of
finite dimensional vector spaces and linear maps to the (non-compact)
$\ast$-autonomous structure of the category of finite dimensional Banach spaces
and contractive maps by passing to their respective polycategories. We also
give an operational reading of this example, where polylinear maps correspond
to operations between systems that can act on their inputs and whose outputs
can be measured/probed and where norms correspond to properties of the systems
that are preserved by the operations.
  Finally, we recall the B\'enabou-Grothendieck correspondence linking
fibrations to indexed categories. We show how the B-G construction can be
defined as a pullback of virtual double categories and we make use of
fibrational properties of vdcs to get properties of this pullback. Then we
provide a polycategorical version of the B-G correspondence.
