[Solved]: What is meant by Category theory doesn't yet know how to deal with higher-order functions?

Problem Detail: 

In reading Uday Reddy's answer to What is the relation between functors in SML and Category theory? Uday states

Category theory doesn't yet know how to deal with higher-order functions. Some day, it will.

As I thought Category theory was able to serve as a foundation for math, then it should be possible to derive all of math and higher-order functions.

So, what is meant by Category theory doesn't yet know how to deal with higher-order functions? Is it valid to consider Category theory as a foundation for math?

Asked By : Guy Coder

Answered By : Uday Reddy

The issue with higher-order functions is simple enough to state.

• A type-constructor like $T(X) = [X \to X]$ is not a functor. It should have been.

• A polymorphic function like ${\it twice}_X : T(X) \to T(X) = \lambda f.\, f \circ f$ is not a natural transformation. It should have been.

If you read Eilenberg and MacLane's original category theory paper, (PDF) the intuitions they present cover those cases. But their theory doesn't. Theirs was a great paper for 1945! But, today, we need more.

The reaction of category theorists to these issues is a bit perplexing. They act as if higher-order operations form a Computer Science idea; they are of no consequence to mathematics. If that is so, then a foundation of mathematics would not be good enough for a foundation of computer science.

But I don't seriously believe that. I believe that higher-order functions would be quite important for mathematics as well. But they have not been seriously explored. I am hopeful that, some day, they will be explored and the limitations of category theory will be realized.

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Question Source : http://cs.stackexchange.com/questions/9818