Topics in low dimensional higher category theory

The first part of this thesis concerns strictification in low-dimensional higher category theory. We review existing theory in dimensions two and three, and extend some results which are known in the two dimensional setting to the three dimensional setting, obtaining new results there. New theory includes a notion of semi-strictness for trinatural transformations, a relationship which can reasonably be called a ‘semi-strictification tetra-adjunction’, and a closed structure [−, ?] on Gray-Cat which features trinatural transformations that admit some decomposition into semi-stricts. Following this, we consider enrichment over (Gray-Cat, [−, ?] ) and the relation to tetracategories. Free constructions, and a distinction between operational and free coherence data, simplify and guide our study of semi-strictification in low dimensions. Freeness is needed to leverage semi-strictification adjunctions of (n − 1)-dimensional data towards semi-strictification for n-dimensional categories. We provide precise conjectures for semi-strict models keeping interchange weak in dimensions four and five, for which we are able to provide strong evidence and outline a promising proof strategy in dimension four. The second part of this thesis studies generalisations of the Kleisli construction to the two-dimensional setting, and the broader theory of tricategorical limits and colimits. Main examples such as Gray and Hom, the tricategory of bicategories and weak (2, k)-transfors, will be shown to have trilimits and tricolimits. Tricategorical notions are related to notions enriched over Gray as a monoidal model category, in analogy to the bicategorical setting and overcoming the challenges presented by non-cofibrant 2-categories. A particular related pair of colimits is examined. These categorify the usual Kleisli construction for monads to the level of pseudomonads, one in the stricter Gray-categorical sense and the other in the weaker tricategorical sense. The weaker ‘trikleisli’ left pseudoadjoints in Hom are precisely the biessentially surjective on objects ones, and use the fact that biessentially surjective on objects pseudofunctors are closed under composition to describe what we conjecture to be the free cocompletion of a tricategory under trikleisli objects for pseudomonads. This extends the theory of wreaths to the pseudomonad setting and has potential applications to pseudo double categories and enrichment over monoidal bicategories. We explore the first of these applications in detail, using this perspective to categorify the notion of cofunctor and natural cotransformation to the pseudo double category setting, and describe a strictification of double pseudocofunctors. We extend these ideas to a definition of pseudo triple category and outline a program aimed at strictifying them using a slice-wise version of B¨ohm’s tensor product of double categories. As a different direction in our study of the formal theory of pseudomonads, we extend the theory of abstract Kleisli structures to the pseudomonad setting and show that these correspond to pseudomonads whose units satisfy a bicategorical limit condition.