Re: non-unital monads

[Tarmo Uustalu <tarmo@cs.ioc.ee>]

Dear Vladimir,

1. The answer to the first question is no, there can only be one unit
for a given underlying functor and multiplication.

(But for a given underlying functor and unit, there can of course be
multiple multiplications.)

2. "Non-unital monads" are not difficult to find.

On Set, you can consider, for example,

- T X  =  X x S   where (S, *) is some semigroup

                           ass                 X x *
    mu_X  =  (X x S) x S -------> X x (S x S) ----- -> X x S

    The simplest special case is given by right zero semigroups: Take
    any set S and define s * s' = s'; one gets

                         fst x S
    mu_X  =  (X x S) x S -------> X x S

    (For S with 2 or more elements, there is no unit.)

- T X  =  lists over X of length at least n, for some fixed n

    mu_X  =  flattening of a list of lists into a list

    (For n \geq 2, there is no unit.)

- For an endofunctor F, the free non-unital monad on F would be

    F^+ X  =  F (F^* X)  \cong  F^* (F X)

    where F^* is the free monad on F (assuming this exists).

    So concretely you can construct F+ in terms of initial algebras by

    F^+ X  =  F (mu Z. X + F Z)  \cong  mu Z. F X + F Z

    (for comparison, F^* X  \cong  mu Z. X + F Z)

    The free non-unital monad exists precisely when the free monad does,
    as you also have

    F^* X  \cong  X + F^+ X

    For your example, F X  =  X x X, one gets that F X is the set of all
    composite terms over variables from X, for a signature with one binary
    operation.

    (And free would mean left adjoint to forgetful as usual.)

Kind regards,

Tarmo U


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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