In a subsequent post, everything will be internalised to an arbitrary category C with enough structure to define everything. The words set and function will be replaced by object and morphism. When we do this, P will be replaced by an arbitrary commutative monad M.
In particular, we can internalise everything to the category Top. That is, we assume the option space X and the payoff R are equipped with topologies, and the tasks will be continuous functions u:X→R, and optimisers will be continuous functions u:(X→R)→M(X) where (X→R) is the function space equipped with pointwise topology, and M is a monad on Top.
In the literature, everything is done with galaxy-brained category theory, but I decided to postpone that in the sequence for pedagogical reasons.
The power set part seems sus. Have you considered something more continuous?
Yes!
In a subsequent post, everything will be internalised to an arbitrary category C with enough structure to define everything. The words set and function will be replaced by object and morphism. When we do this, P will be replaced by an arbitrary commutative monad M.
In particular, we can internalise everything to the category Top. That is, we assume the option space X and the payoff R are equipped with topologies, and the tasks will be continuous functions u:X→R, and optimisers will be continuous functions u:(X→R)→M(X) where (X→R) is the function space equipped with pointwise topology, and M is a monad on Top.
In the literature, everything is done with galaxy-brained category theory, but I decided to postpone that in the sequence for pedagogical reasons.