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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