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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