Re: non-unital monads

[Marek Zawadowski <zawado@mimuw.edu.pl>]

Hi,

Monads on a category C are monoids in the strict monoidal category End(C)
of endofunctors on C and natural transformations. We have the
forgetful functors

Mon( End(C) ) ---> nuMon ( End(C) ) ---> End(C)

forgetting from monoids to non-unital monoids and then to endofunctors.
These functors might have left adjoints. This answers the second question
concerning universal properties.

If C is Set, and we restrict objects in End(Set) to functors with rank
at most m
(for some cardinal m) , then it was shown in

M. Barr, Coequalizers and Free Triples, Math. Z. 116, pp. 307-322 (1970)

that the left adjoint to the composition of the above functors exists
giving rise
to a monad for monads on End(Set) with rank at most m. There are also
refinements of this result saying that the free monads on polynomial,
analytic, and semi-analytic functors are polynomial, analytic, and
semi-analytic, respectively. The first occurs in the unpunlished book
of Joachim Kock and the last two in the papers I wrote recently with
S. Szawiel

Theories of analytic monads. Math. Str. in Comp. Sci. pp. 1-33, (2014)

Monads of regular theories. Appl. Cat. Struct. pp. 9331-9364, (2013)

As Tom and Peter remarked, if a monoid has a left unit and a right unit,
they need to be equal.

Best regards,
Marek

Cytowanie Vladimir Voevodsky <vladimir@ias.edu>:

> Hello,
>
> I am trying to find some information about non-unital monads
> (gadgets with \mu but without \eta).
>
> In particular I am interested in the following two questions:
>
> 1. Given a non-unital monad can it have two different "unitality" structures?
>
> 2. Is there a concept of a free non-unital monad? For example, I can think of
> the "free" non-unital monad generated by the functor X |-> X^2 on
> sets as the monad
> that sends a set X into the set of "homogeneous" expressions made
> with one binary operation
> s such that there is s(x1,x2) and s(s(x1,x2),s(x3,x4)) but no x1
> itself and no s(x1,s(x2,x3)).
> But what is the universal characterization of it?
>
> Thanks!
> Vladimir.
>
>




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