Christopher Howard
Since I received the two responses to my question, I've been trying to think deeply about this subject, and go back and understand the core ideas. I think the problem is that I really don't have a clear understanding of the basics of category theory, and even less clear idea of the connection to Haskell programming. I have been reading every link I can find, but I'm still finding the ideas of "objects" and especially "morphisms" to be quite vague.
It's vague on purpose, or in better words it's abstract. A category C is a collection of objects called ob(C) and a collection of morphisms hom(C) together with morphism composition that satisfies a few laws. That's it. It's up to you to interpret the objects and morphisms. Different categories give rise to completely different interpretations. What makes category theory interesting for programming is the set of laws, because unlike the unspecified nature of objects and morphisms, the laws are very specific. They establish a rigorous notion of soundness and locality. Let's review these laws: * For all f : A -> B, g : B -> A: id A . g = g f . id A = f This law is often understated. It means that identity morphisms are free of effects that could disturb composition. It also means that the composition does not change the nature of the component morphisms and is free of effects that could disturb the neutrality of identity morphisms. * f . (g . h) = (f . g) . h Not much to be said about this one, except that it makes composition 'lightweight' in a sense. This basically means that you don't care how compositions are grouped. These laws make morphisms isolated and composition lightweight as well as undisturbing. Now try to transfer these notions to a concrete category, for example the category of web servers: The objects are sets and a morphism f : A -> B is a function from A × Request to B.
The original link I gave http://www.haskellforall.com/2012_08_01_archive.html purposely skipped over any discussion of objects, morphisms, domains, and codomains. The author stated, in his first example, that "Haskell functions" are a category, and proceeded to describe function composition. But here I am confused: If "functions" are a category, this would seem to imply (by the phrasing) that functions are the objects of the category. However, since we compose functions, and only morphisms are composed, it would follow that functions are actually morphisms. So, in the "function" category, are functions objects or morphisms? If they are morphisms, then what are the objects of the category?
You are absolutely right there. The category is common called Hask, the category of types and functions in Haskell. It is strongly related to the category of sets and functions, because Haskell types are actually just sets, where every set has one additional member: bottom. We say that the set is 'lifted'. Greets, Ertugrul -- Not to be or to be and (not to be or to be and (not to be or to be and (not to be or to be and ... that is the list monad.