Re: Tensor of monads

[Richard Garner <r.h.g.garner@gmail.com>]

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


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