Dear Paul, Thanks for your perspicuous message. Your general definition of tensor product really cuts to the heart of the matter, and agrees with the cases where I knew how to take tensors previously. Has it been written down somewhere? It seems very natural. It can actually be generalised further still. Let C be any symmetric monoidal category. In this setting, your last definition, which to me looks to be the key one, still makes sense: Now a cocone of strong monads
S -----> M <----- T
is said to "commute" when for all C-objects X and Y the two C-morphisms from SX * TY to M(X * Y) are equal.
In this generality we may _define_ the tensor product S * T to be the vertex of the universal commuting cospan, if such exists (and even if it doesn't, we still get a multicategory). To see that this agrees with your definition in the case where C is monoidal closed, we make a few observations. 1) For every X in C, there's a strong monad [X,X] such that strong monad morphisms S -> [X,X] correspond to S-algebra structures on X; it's the continuation monad A |-> (A -o X) -o X. 2) For every f : X -> Y, there's a monad {f,f} and a monomorphism of monads {f,f} --> [X,X] x [Y,Y] such that a strong monad morphism k: S --> [X,X] x [Y,Y] factors through {f,f} if and only if f is an S-algebra morphism with respect to the S-algebra structures on X and Y classified by k. Explicitly, we have {f,f}A given by the pullback of (A -o X) -o X and (A -o Y) -o Y over (A -o X) -o Y. 3) Given morphisms of monads f : S -> M, g : T -> M and j : M -> N, if jf commutes with jg and j is a monomorphism, then also f commutes with g. In other words, if the tensor product S * T exists, then the pair of maps S --> S * T <-- T are jointly strongly epic. 4) Taking (3) together with (2) we deduce that (S * T)-Alg will be a full subcategory of S-Alg x_C T-Alg. 5) It remains only to ascertain the objects which lie in this full subcategory: and these will be those (X, theta, phi) in S-Alg x_C T-Alg for which the corresponding pair of monad maps S --> [X,X] <-- T commute: which are precisely those satisfying your equivalent conditions (1), (2) and (2'). It's interesting to note the parallel between this setting and Dominique Bourn's notion of "intrinsic centrality". Using his terminology, we might say that monad maps S --> M <-- T "cooperate", rather than "commute". We would then call a monad morphism S -> M "central" if it cooperated with the identity on M: and call M "commutative" if the identity on M cooperated with itself. This last, rather fortunately, coincides with the established monad-theoretic terminology. This parallel provides a context for the following proposition about the tensor product of monads that I previously had no good explanation for: Prop: For a strong monad M, the following are equivalent: 1) M is commutative. 2) M is a commutative monoid with respect to the tensor product of monads. 3) M is a unitary magma with respect to the tensor product of monads. Finally, this also tells us how to construct the commutative reflection of a monad in an extremely compact manner: take the coequaliser of the two maps S --> S*S. More constructively: take the full subcategory D of S-Alg whose objects are algebras theta : SX -> X such that (X, theta, theta) commutes in your sense. If D -> C has a left adjoint, then the monad so generated is the commutative reflection of S. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]