Fibre Optics [lcc-0005]

Let \mathcal {I} have all finite limits, and equip it with the Cartesian monoidal structure. Let \mathcal {C},\mathcal {M},\mathcal {D} \to \mathcal {I} be bifibrations which have the Beck-Chevalley condition. Suppose \mathcal {M} is a monoidal bifibration in the sense of Framed bicategories and monoidal fibrations, and suppose further that \mathcal {M} acts on \mathcal {C} and \mathcal {D} in a way compatible with the fibrations.

Note that we are only asking for compatibility with the fibration, not with the bifibration - that is, we expect tensors of Cartesian lifts to be again Cartesian, but not that this is the case for coCartesian lifts.

Then the objects of the category of fibre optics, \mathsf {FibOptic}^\mathcal {I}_\mathcal {M}(\mathcal {C},\mathcal {D}), are tuples (I \in \mathcal {I}, X \in \mathcal {C}_I, A \in \mathcal {D}_I) The set of morphisms (I,X,A) \to (J,Y,B) is given by the scary-looking expression \int ^{M \in \mathcal {M}_{I \times J}} \mathcal {C}_\mathcal {I}(X, \pi _J^!(M \cdot \pi _J^*Y)) \times \mathcal {D}_\mathcal {I}(\pi _I^!(M \cdot \pi _J^*B), A)