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Stream: theory: category theory

Topic: liftings and distributive laws

Christian Williams (Jun 24 2021 at 17:14):

A distributive law of monads $\lambda: TS\Rightarrow ST$ is equivalent to a lifting of $S$ to $T$-algebras, given by
$\bar{S}(\alpha: TA\to A) \; = \; (T(S(A))\xrightarrow{\lambda_A} S(T(A))\xrightarrow{S(\alpha)} S(A))$
(described in Toposes, Triples, and Theories).

I've only seen this correspondence described as a bijection; has anyone written down the correspondence as a 2-functor from distributive laws to liftings? The latter should be the sub-2-category of the arrow 2-category of $\mathrm{Cat}$, consisting of the monadic "forgetful" functors.

Ralph Sarkis (Jun 24 2021 at 17:41):

Slightly closer to what you are looking for but maybe the proofs can help: In Miki Tanaka's thesis, Corollary 3.29 (p.52) states the isomorphism between the categories of distributive laws and liftings. I am not sure if it is straightforward to obtain the isomorphism of 2-categories.

Christian Williams (Jun 24 2021 at 19:28):

Thanks!

Nathanael Arkor (Jun 26 2021 at 19:32):

A more general result is proven in Lobbia's Distributive laws for relative monads (Theorem 6.14), where he attributes the result for ordinary monads to Street, though the exact place where Street proves this is not so clear.