In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors.
I want to explore the "Co-ness" of this.
My first thoughts were that you can define a comultiplication and hence, perhaps, a comonoid on the natural numbers.
Thoughts came to mind, like "Every commutative monoid admits a comonoid structure.". I think that's surely false, it's probably not even true on naturals or integers. But then I see here that comonoid is defined in every monoidal category, but it just doesn't match the idea of factoring.
My mind then wondered about a comonad: "the factor-list functor admits a comonad structure.". Where the functor takes a set to the set of all factor lists of the elements. Again probably false, a good definition of the functor probably doesn't exist. It's tantalizing.
Since the EM category of the multiset monad on Set is commutative monoids, I want to see what I can do with the "co" of this. This starts with a conjecture: "The multiset functor on Set admits a comonad structure". We want to see this as factoring. Given a set of multisets as factors, $\mu: M \rightarrow M \cdot M$, is given by further breaking down elements into factors of factors, giving multisets of multisets. Then the EM category is a category of comonoids, given by factoring (I have no idea). Is this possible?
Has anyone ever seen anything like this in the literature?
