Re: non-unital monads

[Vladimir Voevodsky <vladimir@ias.edu>]

Many thanks to everybody who answered my questions!

I understand the picture with unitality being a property and not a structure now.

As for the universal characterization I have in mind something like this:

1. For a functor F on a category with finite coproducts such that for each X0 there exists the initial
algebra I(F\coprod X0) of the functor X |-> F(X)\coprd X0, these initial algebras are functorial 
and in fact X |-> I(F\coprod X) has an obvious monad structure and this monad is the free monad
generated by F. 

This construction is what connects free monads with free algebras.

2. What can one do for a non-unital monad? It seems to me at the moment that the functor

X |-> I(F\coprod F(X)) 

may be the free non-unital monad generated by F. 

Vladimir.




> On Oct 20, 2014, at 5:47 PM, Marek Zawadowski <zawado@mimuw.edu.pl> wrote:
> 
> 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
> 

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