Fong's paper Causal Theories: A Categorical Perspective on Bayesian Networks talks about causal theories. He describes words of random variables at the top of page 42:
For the objects of CG we take the set N V of functions from V to the natural numbers. These may be considered collections of elements of the set of variables V , allowing multiplicities, and we shall often just write these as strings w of elements of V . Here the order of the symbols in the string is irrelevant, and we write Ø for empty string, which corresponds to the zero map of N V . We view these objects as the variables of the causal theory, and we further call the objects which are collections consisting of just one instance of a single element of V the atomic variables of the causal theory.
Is he not just describing something like the List monad? A single element of V, the atomic variable, is the unit map $\mu : 1_C \rightarrow T$?
He explicitly states that objects have a comonoid map for copying. I get the feeling this causal category is a bimonad. Is this correct?
Some supporting evidence includes the following. This is taken from Coecke's paper, section 1.5.1.
Recall from [31] that the internal commutative (co)monoid structures over an object X in a monoidal category C are in one-to-one correspondence with commutative (co)monad structures on the functor X ⊗ − : C → C .
That functor is a comonad, according to the same paper.
So, Fong describes a comonoid, which has an associated comonad and I am pointing out the monad structure.
If it works, please describe the interaction of the monad and comonad (mixed distributive law?). I think Fong's examples are all on SET. So the Monad and Comonad would be on SET.
Note: A bimonad is both a Monad and a comonad.
"There's a relationship between the monad structure and the comonad structure, or more exactly a mixed distributive law between them. In the terminology that some people use, this makes the monad into a "bimonad"."
