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Your example for the application of the "functor" concept is trivial enough that it doesn't need category theory. Monoids are very simple objects, and you don'

Your example for the application of the "functor" concept is trivial enough that it doesn't need category theory. Monoids are very simple objects, and you don't need to use category theory to define them. Monads might be a more meaty example. But I'm not sure if knowing CT makes it easier to use monads in a programming context.

siraben 6y ago
They're trivial constructions yes. But the difference is the shift in vocabulary, instead of understanding functor fusion in an ad-hoc way, the concept can be made more precise, also it can lead you to find other "non-trivial" functors for instance the Hom(a,-) covariant functor or the Hom(-,a), but equipped with the right vocabulary these things are easy to express. Sure, monoids are trivial, because they are, but once you're in the categorical setting you can begin to talk about the category of monoids, monoid homomorphisms and so on. This isn't to say that CT is the answer, but I'm pointing out that it provides a way to understand more complicated concepts. Functors are trivial, but if you take their fixpoint you get ADTs. Monoids are trivial, but if you apply them to the category of endofunctor you derive monads, and so on.
tome 6y ago
> Your example for the application of the "functor" concept is trivial enough that it doesn't need category theory. That's kind of the point. > "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial." That is, category theory aids in making the softer bits of mathematics look utterly natural and obvious, so that one can more easily isolate the harder nuggets and attack them with abandon ... This is a tremendous service that category theory provides" https://mathoverflow.net/questions/28788/nontrivial-theorems-with-trivial-proofs#comment172290_28797 https://mathoverflow.net/questions/28788/nontrivial-theorems... In isolation applications of category theory to programming are going to be trivial, typically. Category theory gives a language and framework for extracting trivialities so one can "attack the harder nuggets" without distraction.

siraben 6y ago

They're trivial constructions yes. But the difference is the shift in vocabulary, instead of understanding functor fusion in an ad-hoc way, the concept can be made more precise, also it can lead you to find other "non-trivial" functors for instance the Hom(a,-) covariant functor or the Hom(-,a), but equipped with the right vocabulary these things are easy to express. Sure, monoids are trivial, because they are, but once you're in the categorical setting you can begin to talk about the category of monoids, monoid homomorphisms and so on. This isn't to say that CT is the answer, but I'm pointing out that it provides a way to understand more complicated concepts. Functors are trivial, but if you take their fixpoint you get ADTs. Monoids are trivial, but if you apply them to the category of endofunctor you derive monads, and so on.

tome 6y ago

> Your example for the application of the "functor" concept is trivial enough that it doesn't need category theory. That's kind of the point. > "Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial." That is, category theory aids in making the softer bits of mathematics look utterly natural and obvious, so that one can more easily isolate the harder nuggets and attack them with abandon ... This is a tremendous service that category theory provides" https://mathoverflow.net/questions/28788/nontrivial-theorems-with-trivial-proofs#comment172290_28797 https://mathoverflow.net/questions/28788/nontrivial-theorems... In isolation applications of category theory to programming are going to be trivial, typically. Category theory gives a language and framework for extracting trivialities so one can "attack the harder nuggets" without distraction.