Re: non-unital monads

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

The answer to Vladimir's first question is no. Suppose \mu: TT --> T
has two units \eta, \zeta: 1 --> T. Then, for any A, the composite
\mu_A.T\eta_A.\zeta_A reduces to \zeta_A by one unit law; but it's
equal to \mu_A.\zeta_TA.\eta_A by naturality of \zeta, and this reduces
to \eta_A by the other unit law. (If you don't demand that the units be
`two-sided' then the answer is yes.)

Peter Johnstone

On Sat, 18 Oct 2014, Vladimir Voevodsky wrote:

> 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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