Commutativity of Relative Pseudomonads

Relative pseudomonads simultaneously generalise two important notions of category theory; that is, they generalise pseudomonads to non-endofunctors, and relative monads to bicategories. In this thesis, we study two aspects of the theory of relative pseudomonads: relative pseudomonads on 2-multicategories and pseudoalgebras for relative pseudomonads.

We develop the theory of relative monads on multicategories, expositing an analogue of the work of Kock on monads on monoidal categories. We define notions of strength, commutativity and idempotency for a relative monad $T$, as well as the notion of a relative multimonad. We go on to prove that idempotency implies commutativity, that a commutative relative monad is a relative multimonad, and that commutativity of $T$ implies a multicategory structure on the Kleisli category $\Kl(T)$. Later, we extend this to the setting of relative pseudomonads on 2-multicategories, defining the corresponding two-dimensional notions and proving the corresponding implications.

We also develop the theory of pseudoalgebras for relative pseudomonads, constructing for a given relative pseudomonad $T$ its Eilenberg-Moore bicategory of pseudoalgebras $\TAlg$. We then use pseudoalgebras to introduce the notion of `algebraic lax idempotency' and characterise algebraically lax-idempotent relative pseudomonads; this is the counterpart in the relative setting to Kelly and Lack's characterisation of lax-idempotent pseudomonads as `fully property-like'.

We apply our results in both cases to the presheaf relative pseudomonad $P : \Cat \to \CAT$, proving that its Eilenberg-Moore 2-category is biequivalent to the 2-category of locally-small cocomplete categories and cocontinuous functors, and leading to a proof that the bicategory of profunctors $\Prof$ has a bimulticategorical structure.