> One cannot interchange the order of the composing morphisms while preserving the properties of the original.
Good observation one really can't but that was never a hard requirement, right? Ordering becomes actually more interesting because you can have interesting properties like anti-commutativity (https://en.wikipedia.org/wiki/Anticommutativity) which is a lot more useful than commutativity. Lie groups are anti-commutative groups btw.
> In optimization, the concept of duality is also a much stronger idea: the primal and the dual of a problem are opposing views of the same problem that correspond exactly (not approximately) in their dual properties.
My view is more general. The difference between these spaces lies in the idea of choice and in the idea of adversarial choice. You are correct, they are opposing view, like two players playing a game.
I control my moves. I do not have control over my opponents players moves, however I do have knowledge about my opponent's potential moves. Therefore I can do some sort of min-max optimization to figure out my optimal play given my situation and knowing my opponent's options.
> One cannot interchange the order of the composing morphisms while preserving the properties of the original.
Good observation one really can't but that was never a hard requirement, right? Ordering becomes actually more interesting because you can have interesting properties like anti-commutativity (https://en.wikipedia.org/wiki/Anticommutativity) which is a lot more useful than commutativity. Lie groups are anti-commutative groups btw.
> In optimization, the concept of duality is also a much stronger idea: the primal and the dual of a problem are opposing views of the same problem that correspond exactly (not approximately) in their dual properties.
My view is more general. The difference between these spaces lies in the idea of choice and in the idea of adversarial choice. You are correct, they are opposing view, like two players playing a game.
I control my moves. I do not have control over my opponents players moves, however I do have knowledge about my opponent's potential moves. Therefore I can do some sort of min-max optimization to figure out my optimal play given my situation and knowing my opponent's options.