Disclaimer: This post is not accurate, nor am i a professional mathematician(not even amateur) . Feel free to correct my misunderstandings in the comments.
I was trying to understand Combinatorial species(https://en.wikipedia.org/wiki/Combinatorial_species). came across somebody’s profile at the company indix . But it had references to functors, categories and i went off reading Category theory here). Couple of hours later, i still don’t have an understanding of Combinatorial Species, but I have a (slightly) better (than my previous ) grasp of Category theory. Let me try to summarise that.
Back in High school I think 11th grade, we had a small cute, and fun chapter called modern algebra. It’s one of those chapters, that left me with wanting for more. Very small simple chapter, but it was beautiful. It defined groups, sets,operators, types and properties of a group etc..
Guess what? it turns out Category theory is just one level of abstraction higher. While as i remember that ‘modern algebra’ dealt with sets (of numbers were the examples we used) and operators, Category theory deals with objects and arrows. Note the vagueness of the definition of objects and arrows is on purpose.
If the object is a set and the arrows are operators we get a group.
If the objects are data types and the arrows are functions we get type system/theory from theoretical comp. science.
I am trying to think what if objects are distributions and the arrows are transformations? I can’t think of what area that would be, but would be surprised if it has not been studied formally.
Anyway, the fact that objects and arrows are not restricted does not mean there are no rules. there are a some rules for considering a <i don’t want to use the term set/group> collection of objects and arrows(hereafter called collection ) as a category as follows:
Ah, i missed one more thing so far . arrows are not just arrows they are mathematical morphisms/maps defined between objects. at this point am thinking why this sounds very much like set theory and not really any different from sets and functions/mapping functions. why the hell would i need yet another set of terminologies to master*
Any way, here are the rules for a collection to be considered a category.
1. the morphisms must be composable to achieve a associative relationships among the objects.
i’ll use u,v,w for morphisms and a,b,c, for objects.
i.e if u: maps a to b, v: maps b to c, then u compositing v: maps a to c.
2. That composition function between morphisms must be associative itself.
i.e: (u composits v) composits w = u composits (v composits w)
3. There’s an identity morphism I such that for any u : maps a to b, I composits u = u composits I = u.
*– Will get back to that later i need to read a lot more on the background, history, and some papers to understand why it is useful. but the wikipedia explanation is that it abstracts and unifies concepts many different branches of mathematics. While it sounds true, it’s rather vague(insert rant about perils of democratic editing/article writing), i want to see some actual theorems in Category theory being applied to some math area and helping improve it. Will write another post/update this once am done with that.
UPDATE: 11-Sep-2012: Seems i was rash commenting about utility of category theory. This link suggests a new revolutionary proof on prime numbers uses some concepts from category theory.