I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check.
- The category of $\mathbb{C}$-modules is monadic over set
- The category of $\mathbb{N}$-modules is monadic over Set
- Since the monad, $\mathcal{M}_C$, which factors through the category of $\mathbb{C}$-modules is seated on Set and likewise for the monad, $\mathcal{M}_N$, that factors through the category of $\mathbb{N}$-modules, these monads compose according to functor composition. Their composition, $\mathcal{M}_P = \mathcal{M}_C \cdot \mathcal{M}_N$, is also a monad on SET.
The physical interpretation: The monad, $\mathcal{M}_P$, can be interpreted as complex combinations of collections of particle types. Thus, this monad can be used to represent quantum states for the input/output of a particle scattering experiment.
Edit:
Someone has pointed out a very important fact, which is that I have to specify a distributive law for the functor composition to become a monad. Unfortunately, I don't know how to define this. Perhaps the question can be phrased better as follows. Can we define a distributive law to make the composition into a monad? What is the distributive law? If there is a large freedom, perhaps we can select a law that captures some aspect of the physics we are trying to model.
