Monads in 2-categories

This is a condensed overview of the formal theory of monads in a 2-category.

Commutative diagrams and string diagrams are given side by side.

In the string diagrams, each concept is visualized in a way that suggests its properties using topological intuition.

For example, monads themselves appear as channels of fluid, suggesting the unit and associativity laws by topological deformation.

We cover monads, modules, (co)lax 1-cells and the 2-cells between them, algebra objects, bimodules, distributive laws, codensity monads, and pushforward monads.

In addition to the standard 2-categories of monads in a 2-category, we also define two double categories of monads in a 2-category.

For example, applied to spans, this yields the two double categories of categories, functors, and retrofunctors upon restricting to the 1-cells carried by functions.