With reference to Tarmo's examples---it's perhaps worth pointing out the notion of an "ideal monad" on a category with coproducts. One way of defining an ideal monad is in terms of an endofunctor T together with a natural transformation m:T(1+T)--->T satisfying two axioms: - m.T(inl) = 1: T--->T - m.T(inr.m) = m.(m.Tinr)(1+T): TT(1+T) ---> T. This is something stronger than a non-unital monad; if (T,m) is as above, then (T, m.T(inr)) is a non-unital monad. The extra strength means that from such a pair (T,m) one obtains a monad structure on 1+T in an obvious way. All of the examples Tarmo mentions possess this extra strength. Some references: P. Aczel, J. Adamek, S. Milius and J. Velebil, Infinite trees and completely iterative theories: a coalgebraic view. Theor. Comput. Sci. 300 (2003) 1–45. Neil Ghani and Tarmo Uustalu, Coproducts of ideal monads, Theoretical Informatics and Applications 38 (2004), no. 4, 321–342. Richard
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
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