A tour of EM(Cat#)


David Spivak


May 20, 2024

Small categories are graphs with extra structure: namely, every path “composes” to form an arrow. As such, small categories are algebras for a certain monad—the “paths” monad—on the category of graphs. This monad is familial, i.e. its underlying functor can be identified with a loose map in the double category Cat#.
But what is an appropriate sort of morphism between monads in Cat#? Shulman’s “monoids and modules” construction doesn’t apply, because Cat# doesn’t have local coequalizers. Instead we consider Lack and Street’s (2002) construction of the free completion of Cat# under Eilenberg-Moore objects, denoted EM(Cat#). Its objects are monads m in Cat# and its morphisms m→n are “quintets” satisfying expectable properties; as the name would suggest, these morphisms induce functors m-Alg → n-Alg between the associated Eilenberg-Moore categories. The 2-cells in this “free completion” are slightly surprising—in particular, they are more flexible than those of (Street 1972)—but still simple to describe.
We’ll discuss various examples of objects in EM(Cat#), i.e. monads in Cat#, e.g. those arising from cofunctors out of a category, from functors into a category, from Grothendieck topologies on a category, from multicategories, and more. Then we’ll discuss morphisms in EM(Cat#), e.g. a free-forgetful adjunction for algebras, a method for turning convex spaces into monoids, one that takes the category of elements of a copresheaf, and perhaps more.