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> Wait, a functor is just a pure unary function? And an endofunctor is just one of those where the argument type and return type are the same? No. I don't kno

> Wait, a functor is just a pure unary function? And an endofunctor is just one of those where the argument type and return type are the same? No. I don't know why the article talks about ordinary functions on values as if they're functors. They're not (or at least, not in a nontrivial way; you could come up with a silly, useless way in which they are, but that's not what anyone's talking about). Sorry, but it's the article that's misleading you here. Roughly speaking, in the programming context, a functor is a generic type with a map operation. E.g., arrays; you can map a function over an array by applying it to each element. Since there's only one category in play here, these functors are endofunctors of that category; one doesn't normally say "endofunctor" in this context since it doesn't convey any additional information. (As opposed to "a monad is a monoid in the category of endofunctors", where you want to specify that, because that statement is true beyond just the programming context.)

contravariant 5y ago
Mathematically the main thing stopping it from being a function is that categories are typically too big to be a set. Otherwise they're pretty much the same as a function. The whole type thing is just something Haskell does, it's not something that's meaningful for general categories.
- thinkharderdev 5y ago
  Not really, mathematically a functor is really two "functions," one mapping objects to objects and one mapping morphisms to morphisms, along with some conditions that make sure the functor is "structure preserving." It is fundamentally a type level construct because, as it is applied to programming, the category in question is the category of types (which is not technically a category due to the halting problem but we sort of hand-wave that away).
  - contravariant 5y ago
    Sure a functor satisfies more conditions but that doesn't make it less of a function. Heck if you restrict it to mapping a single set of objects or a particular Homset it is a function, you just can't extend this to the whole category if the objects form a proper class (which happens fairly often). The whole distinction between value-level or type-level is meaningless in general categories. Sure there's a distinction between morphisms between objects and functors between categories but you've also got the category of categories where the morphisms are functors between categories. Edit: I'll have to concede that the article confuses them too much though, you can't just pick a random function and declare it to be a functor, especially when it's not clear between which categories.

contravariant 5y ago

Mathematically the main thing stopping it from being a function is that categories are typically too big to be a set. Otherwise they're pretty much the same as a function. The whole type thing is just something Haskell does, it's not something that's meaningful for general categories.

thinkharderdev 5y ago
Not really, mathematically a functor is really two "functions," one mapping objects to objects and one mapping morphisms to morphisms, along with some conditions that make sure the functor is "structure preserving." It is fundamentally a type level construct because, as it is applied to programming, the category in question is the category of types (which is not technically a category due to the halting problem but we sort of hand-wave that away).
- contravariant 5y ago
  Sure a functor satisfies more conditions but that doesn't make it less of a function. Heck if you restrict it to mapping a single set of objects or a particular Homset it is a function, you just can't extend this to the whole category if the objects form a proper class (which happens fairly often). The whole distinction between value-level or type-level is meaningless in general categories. Sure there's a distinction between morphisms between objects and functors between categories but you've also got the category of categories where the morphisms are functors between categories. Edit: I'll have to concede that the article confuses them too much though, you can't just pick a random function and declare it to be a functor, especially when it's not clear between which categories.