## December 23, 2018

#### Posted by John Baez

I have a question about the relationship between Lawvere theories and monads.

Every morphism of Lawvere theories $f \colon T \to T'$ induces a morphism of monads $M_f \colon M_T \Rightarrow M_{T^'}$ which can be calculated by using the universal property of the coend formula for $M_T$. (This can be found in Hyland and Power’s paper Lawvere theories and monads.)

On the other hand $f \colon T \to T'$ gives a functor $f^\ast \colon Mod(T') \to Mod(T)$ given by precomposition with $f$. Because everything is nice enough, $f^\ast$ always has a left adjoint $f_\ast \colon Mod(T) \to Mod(T')$. (Details of this can be found in Toposes, Triples and Theories.)

My question is the following:

What relationship is there between the left adjoint $f_\ast \colon Mod(T) \to Mod(T')$ and the morphism of monads computed using coends $M_f \colon M_T \Rightarrow M_{T^'}$?

In the examples I can think of the components of $M_f$ are given by the unit of the adjunction between $f^\ast$ and $f_\ast$ but I cannot find a reference explaining this. It doesn’t seem to be in Toposes, Triples, and Theories.

Posted at December 23, 2018 1:08 AM UTC

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### Re: Monads and Lawvere Theories

I don’t have the time or brainpower to work it out right now, but the first thing I would try is to mess around with the calculus of mates for an arbitrary morphism of monads whose induced functor on algebras has a left adjoint, not assuming that it comes from a Lawvere theory.

Posted by: Mike Shulman on December 24, 2018 12:06 PM | Permalink | Reply to this

### Re: Monads and Lawvere Theories

Hi thanks for the help. This is a very helpful suggestion. In case you were curious, Clemens Berger suggested something similar and suggested using the adjoint triangle theorem. I’m still working out the details.