Pseudomonads, Relative Monads and Strongly Finitary Notions of Multicategory

In this thesis, we investigate two important notions of category theory: monads
and multicategories.

First, we contribute to the formal theory of pseudomonads, i.e. the analogue
for pseudomonads of the formal theory of monads. In particular, we solve a
problem posed by Lack by proving that, for every Gray-category K, there is
a Gray-category Psm(K) of pseudomonads in K. We then establish a triequivalence
between Psm(K) and the Gray-category of pseudomonads introduced by
Marmolejo and give a simpler version of his proof of the equivalence between
pseudodistributive laws and liftings of pseudomonads to 2-categories of pseudoalgebras.
Secondly, we introduce the notion of a distributive law between a relative monad
and a monad. We call this a relative distributive law and define it in any 2-
category K. In order to do that, we introduce the 2-category of relative monads
in a 2-category K. We relate our definition to the 2-category of monads in K
defined by Street. Thanks to this view we prove two theorems regarding relative
distributive laws and equivalent notions. We also describe what it means to have
Eilenberg-Moore and Kleisli objects in this context and give examples in the 2-
category of locally small categories.

Finally, we consider multicategories. It is known that monoidal categories have a
finite definition, whereas multicategories have an infinite (albeit finitary) definition.
Since monoidal categories correspond to representable multicategories, it
goes without saying that representable multicategories should also admit a finite
description. With this in mind, we give a new finite definition of a structure called a
short multicategory, which has only multimaps of dimension at most four, and show
that under certain representability conditions short multicategories correspond to
various
avours of representable multicategories. This is done in both the classical
and skew settings.