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December 23, 2018

Monads and Lawvere Theories

Posted by John Baez

guest post by Jade Master

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

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

On the other hand f:TTf \colon T \to T' gives a functor f *:Mod(T)Mod(T)f^\ast \colon Mod(T') \to Mod(T) given by precomposition with ff. Because everything is nice enough, f *f^\ast always has a left adjoint f *:Mod(T)Mod(T)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 *:Mod(T)Mod(T)f_\ast \colon Mod(T) \to Mod(T') and the morphism of monads computed using coends M f:M TM T M_f \colon M_T \Rightarrow M_{T^'}?

In the examples I can think of the components of M fM_f are given by the unit of the adjunction between f *f^\ast and f *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.

Posted by: Jade Master on December 25, 2018 12:47 AM | Permalink | Reply to this

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