Lawvere theories and Monads

Lawvere theories and Monads

[Jade Master]

I have a question about the relationship between Lawvere theories and
monads. Every morphism of Lawvere theories f: T ->T' induces a morphism of
monads M_f: M_T => M_T' which can be calculated by using the universal
property of the coend formula for M_T (this can be found in Hyland's
<https://www.irif.fr/~mellies/mpri/mpri-ens/articles/hyland-power-lawvere-theories-and-monads.pdf>
paper
on Lawvere theories and monads).

On the other hand f: T->T' gives a functor f* : Mod(T') -> Mod(T) given by
composition with f. Because everything is nice enough, f* always has a left
adjoint f_* : Mod(T) -> Mod(T') (details of this can be found here
<http://web.science.mq.edu.au/~street/MitchB.pdf> or in Toposes, Triples
and Theories).

My question is the following: What relationship is there between the
adjunction

  f_* \dashv f*: Mod(T) ->Mod(T')

and the morphism of monads computed using coends

M_f : M_T => M_T'?

In the examples I can think of the components of M_f are given by the unit
of the adjunction f_* \dashv f* but I cannot find a reference explaining
this. It doesn't seem to be in Toposes, Triples, and Theories.
<http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html>

Thank you,
Jade Master


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Re: Lawvere theories and Monads

[Clemens Berger]

Hi,

    the reference you are searching comes under the title "adjoint
triangle theorem" which is due to Eduardo Dubuc (cf. the nLab entry).
The left adjoint f_* is uniquely determined by the fact that M_T and
M_T' are "nice" monads. The explicit (known) formulas for this left
adjoint imply your observation.

    All the best,
                   Clemens.

Le 2018-12-22 18:45, Jade Master a ??crit??:
> I have a question about the relationship between Lawvere theories and
> monads. Every morphism of Lawvere theories f: T ->T' induces a morphism
> of
> monads M_f: M_T => M_T' which can be calculated by using the universal
> property of the coend formula for M_T (this can be found in Hyland's
> <https://www.irif.fr/~mellies/mpri/mpri-ens/articles/hyland-power-lawvere-theories-and-monads.pdf>
> paper
> on Lawvere theories and monads).
>
> On the other hand f: T->T' gives a functor f* : Mod(T') -> Mod(T) given
> by
> composition with f. Because everything is nice enough, f* always has a
> left
> adjoint f_* : Mod(T) -> Mod(T') (details of this can be found here
> <http://web.science.mq.edu.au/~street/MitchB.pdf> or in Toposes,
> Triples
> and Theories).
>
> My question is the following: What relationship is there between the
> adjunction
>
>   f_* \dashv f*: Mod(T) ->Mod(T')
>
> and the morphism of monads computed using coends
>
> M_f : M_T => M_T'?
>
> In the examples I can think of the components of M_f are given by the
> unit
> of the adjunction f_* \dashv f* but I cannot find a reference
> explaining
> this. It doesn't seem to be in Toposes, Triples, and Theories.
> <http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html>
>
> Thank you,
> Jade Master
>

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