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Stream: theory: category theory

Topic: distributive laws

Jonas Frey (Dec 09 2020 at 02:23):

Given monads $S,T : C \to C$ , I'm pretty sure that there is a theorem saying that

distributive laws $\delta: TS \to ST$
monad structures on $ST$ which are compatible with the monad structures on $S$ and $T$ in a suitable way
"liftings" of $S$ to the category $C^T$ , of $T$ -algebras
"extensions" of $T$ to the Kleisli category $C_S$

are all equivalent.

Beck's original article "Distributive Laws" and Barr/Well's "Toposes, Triples, and Theories" show the equivalence of 1--3, but don't mention 4 as far as I can see.

Does anybody know where to find a statement of the "full" theorem including 4?

Christian Williams (Dec 09 2020 at 03:09):

Hm, I thought I knew a reference but it seems like it's generally taken for granted. There's an abstract proof using the formal theory of monads, thinking of $F:C\to C_S$ as the initial left $S$ -module, and pasting with the distributive law. But the down-to-earth reasoning:

given a Kleisli morphism $f:A\to S(B)$ , what structure is necessary to apply $T$ ? We have $T(f): T(A)\to T(S(B))$ , and then we need a natural transformation $\lambda_B: T(S(B))\to S(T(B))$ to define $\bar{T}(f):T(A)\to S(T(B))$ . The laws for $\lambda$ are exactly what you need for this extension of $T$ to $C_S$ to be functorial.

I can keep looking for a reference. Surely someone has written it up.

Christian Williams (Dec 09 2020 at 03:24):

Yeah it's strange because I see works, such as https://arxiv.org/pdf/2007.12982.pdf Distributive Laws for Relative Monads, which say that the original Beck article includes the equivalence with Kleisli extensions, but I don't see it in there. However the former paper makes it seem like it's been accepted by experts that the formal theory of monads gives a fully general and sufficient theorem of this equivalence - though it might not be spelled out very explicitly there.

Jonas Frey (Dec 09 2020 at 03:35):

That's funny ! I'm also confused where I know the statement from, if not from Barr/Wells, which I read as a student. Barr and Wells have a nice triangle diagram on page 264, which visualizes the equivalence of 1--3, and notably leaves out Kleisli categories.

Ralph Sarkis (Dec 09 2020 at 08:15):

Proposition 5.2.5 in Bart Jacobs' Introduction to Coalgebra shows the correspondence between extensions of a functor $F$ to the Kleisli category $C_T$ and distributive laws of the functor $F$ over the monad $T$ (two diagrams are missing from the usual monad distributive law), then they mention in the exercises that you can make this about monads only.

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Notification Bot (Dec 09 2020 at 11:05):

This topic was moved here from #general > distributive laws by [Mod] Morgan Rogers

Nathanael Arkor (Dec 09 2020 at 12:31):

Theorem 4.26 of Tanaka's thesis Pseudo-Distributive Laws and a Unified Framework for Variable Binding contains the equivalence of 1, 3 and 4. It also follows from Corollary 6.20 of Distributive laws for relative monads. However, I don't think I've seen an explicit reference to 2 outside of Beck's paper (I have also looked for a unified statement in the past and been unable to find one).

Jonas Frey (Dec 09 2020 at 15:54):

Ralph Sarkis said:

Proposition 5.2.5 in Bart Jacobs' Introduction to Coalgebra shows the correspondence between extensions of a functor $F$ to the Kleisli category $C_T$ and distributive laws of the functor $F$ over the monad $T$

ahh that's a good point that it works w/o the monad structure on the lifted functor. at this level of generality it should be precisely the universal property of the Kleisli category viewed as a lax colimit.

Jonas Frey (Dec 09 2020 at 16:12):

Nathanael Arkor said:

Theorem 4.26 of Tanaka's thesis Pseudo-Distributive Laws and a Unified Framework for Variable Binding contains the equivalence of 1, 3 and 4.

Ahh that's pretty close, thanks.

I don't think I've seen an explicit reference to 2 outside of Beck's paper (I have also looked for a unified statement in the past and been unable to find one).

Barr and Wells reproduce precisely Beck's statement, with the advantage of better readability since they write compositions in the conventional way. I guess I'm just wondering why they didn't include the statement about Kleisli categories, since the Kleisli and EM-constructions and their relation are discussed at great length in the book.

Mike Shulman (Dec 09 2020 at 16:53):

For monads $S,T$ on an object $C$ in a 2-category $\mathcal{K}$ , an extension of $T$ to $C_S$ in $\mathcal{K}$ is the same as a lifting of $T$ to $C^S$ in $\mathcal{K}^{\mathrm{op}}$ . So since the proof is purely diagrammatic and works in any 2-category, the equivalence with (3) implies the equivalence with (4). (Note that a distributive law $TS\to ST$ in $\mathcal{K}$ becomes a distributive law $ST\to TS$ in $\mathcal{K}^{\mathrm{op}}$ .)