On the Coherence Conditions for Pseudo-distributive Laws

The formal theory of monads, originally introduced by Street in [22] and developed further by Lack and Street [17] provides a mathematically efficient treatment of several aspects of the theory of monads [1]. For example, it exhibits a universal property of the category of algebras for a monad and provides a clear explanation for Beck’s axioms for a distributive law [2]. Over the past few years, there has been substatial progress in the development of a formal theory of pseudo-monads [4, 15, 19, 20, 21, 23, 24], with applications to pure mathematics [7, 8] and theoretical computer science [3, 5, 25]. Our aim here is to give a survey this development, both to facilitate further applications and to provide a reference for future work. We shall focus our attention on the formulation of coherence conditions, which has proved to be one of the most delicate aspects of the theory. Just as the formal theory of monads is developed within two-dimensional category theory [14], the formal theory of pseudo-monads is developed within three-dimensional category theory [10]. Within this setting, it is convenient to work with Gray-categories, which are semistrict tricategories [10, Section 4.8]. Working with Gray-categories is easier than working with general tricategories, but does not lead to an essential loss of generality, since every tricategory is triequivalent to a Gray-category [10, Theorem 8.1]. The starting point of the formal theory of pseudo-monads is the definition, for a Gray-category K, of the Gray-category PsmK of pseudo-monads in K. As we will see, this is done following a different approach to the one taken in the formal theory of monads to define the 2-category of monads in a 2category. The change of approach allows one to avoid building into the definition of PsmK the notions of a pseudo-monad morphism, pseudo-monad transformation, and pseudo-monad modification, which involve complex coherence conditions. These notions can be introduced at a later stage and then shown to lead to a tricategory that is triequivalent to PsmK.