Double functorial representation of indexed monoidal structures

Abstract

The data of a logical doctrine acts in two directions: that of the substitution operation, and that of quantification. Double Category Theory provides the tools to capture these two actions and their relationship concomitantly: in a double category of spans, re-indexing and acting on predicates can be packaged as tight and loose arrows respectively, and in there tight arrows always have conjoints. By mapping these conjoints into a double category of quintets, we obtain adjunctions internal to a 2-category. Moreover, by using the notion of adequate triple, the indexing category need only have certain pullbacks. Adding a monoidal structure to the fibers (and the double pseudofunctor) allows us to capture both the Beck-Chevalley and Frobenius conditions. We will concern ourselves with the monoidal analogues of regular hyperdoctrines and similar structures (where the objects of predicates are not necessarily posets, but rather live in some 2-category), and show how they are equivalent to (lax symmetric monoidal) double pseudofunctors between spans and quintet double categories.