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Stream: learning: questions

Topic: Distributive laws and Eilenberg-Moore categories

Jana Nickel (Jul 12 2026 at 10:24):

I am referring to a statement in Section 2.1 (p.8) of the paper "Enhanced 2-categories and limits for lax morphisms" by Stephen Lack and Michael Shulman. Given two monads $T$ and $S$ on a category, the statement says that the notion of a distributive law $\lambda: TS\Rightarrow ST$ is equivalent to giving a lifting of $S$ to the Eilenberg-Moore category $EM(T)$ of $T$. I understand how a distributive law lifts $S$ to $EM(T)$ but how does the other direction work?

Morgan Rogers (he/him) (Jul 12 2026 at 13:05):

Are you asking how the distributive law is constructed from the lifted monad?

Nathanael Arkor (Jul 12 2026 at 14:26):

The relationship between these concepts is the subject of the paper Distributive laws by Beck, which explains how to go back and forth between them.

Jana Nickel (Jul 12 2026 at 14:58):

I see, many thanks! I thought that you would suppose to be given a family $\lambda$ of morphisms $\lambda_X: TSX\to STX$ but $\lambda$ is constructed from the lifting. This makes sense!