Title:

Pseudodistributive laws and a unified framework for variable binding

This thesis provides an indepth study of the properties of pseudodistributive laws, and as one of its applications, a unified framework to model substitution and variable binding for various different types of contexts; in particular, the construction presented in this thesis for modelling substitution unifies that for Cartesian contexts as in the work by Flore et al, and that for linear contexts by Tanaka. The main mathematical result of the thesis is the proof that, given a pseudomonad S on a 2category C, the 2category of pseudodistributive laws of S over pseudoendofunctors on C and that of liftings of pseudoendofunctors on C to the 2category of the pseudoalgebras of S are equivalent. The proof for the nonpseudo case, i.e., a version for ordinary categories and monads, is given in detail as a prelude to the proof of the pseudocase, followed by some investigation into the relation between distributive laws and Kleisli categories. Our analysis of pseudodistributive laws is then extended to pseudodistributivity over pseudoendofunctors and over pseudonatural transformations and modifications. The natural bimonoidal structures on the 2category of pseudodistributive laws and that of (pseudo)liftings are also investigated as part of the proof of the equivalence. Fiore et al., and Tanaka take the free cocartesian category on 1 and the free symmetric monoidal category on 1 respectively as a category of contexts and then consider its presheaf category to construct abstract models for binding and substitution. In this thesis a model that unifies these two and other variations is constructed by using the presheaf category on a small category with structure that models contexts. Such structures for contexts are given as pseudomonads S on Cat, and presheaf categories are given as the cocompletion (partial) pseudomonad T on Cat, therefore our analysis of pseudodistributive laws is applied to the combination of a pseudomonad for contexts with the cocompletion pseudomonad T. The existence of such pseudodistributive laws leads to a natural monoidal structure that models substitution. This follows from the second main mathematical result of the thesis, the framework for such monoidal structures, which is given in terms of pseudostrengths of pseudomonads on Cat and the monoidal structures induced by them. We first prove that a pseudodistributive law of S over T renders the composite TS to be again a pseudomonad, from which it follows that the category TS1 has a monoidal structure, which, in our examples, models the substitution.
