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 Cytowanie Vladimir Voevodsky <vladimir@ias.edu>:
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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