On Thu, 24 Jun 2010 20:33:01 +1000 Matt Andrew wrote:

I have spend some time over the last couple of days trying to get my head around category theory

i've been trying to do the same matt for a couple of days, but i have nothing by way of explanation to offer at this stage and will eagerly look at what responses you get here. i have found the following useful sources though: http://www.macs.hw.ac.uk/~dpm8/files/categories/awodey-categories.pdf (awodey's book - it's good, but i'm having trouble filling in some of the gaps presently) http://en.wikiversity.org/wiki/Introduction_to_Category_Theory (wikiuniversity course) http://en.wikiversity.org/wiki/Introduction_to_Category_Theory (computational category theory course by rydeheard & burstall) http://plato.stanford.edu/entries/category-theory/ (brief synopsis from stanford encyclopedia) (now it's just a matter of trying to figure out what they are saying :D ) -- In friendship, prad ... with you on your journey Towards Freedom http://www.towardsfreedom.com (website) Information, Inspiration, Imagination - truly a site for soaring I's

I am also on a similar quest. Specifically, I want to understand the connection between the notion of monad as defined in category theory and the application of this concept to functional programming. Even more specifically I want to grok how monads provide an elegant solution to the problem of requiring side effects in a programming style that expressly forbids them. But anyway with my very limited understanding of these things I'll take a crack at some of your questions below. ::paul On Jun 24, 2010, at 3:33 AM, Matt Andrew wrote:

Hi all,

I have spend some time over the last couple of days trying to get my head around category theory, in order to understand concepts like monads a little better, in order to understand Haskell a little better =) I have grasped, at least to a degree, the basic concepts of 'category,' 'functor' and 'natural transformation' but I am still fuzzy on a few specifics and had a couple of questions. I'm doing this for self-study and have no one to ask these questions of and so thought I'd ask them here.

My first question is this: where are functors understood to 'live,' or exist? They operate on categories, are they therefore understood to exist outside of the categories they operate on? (I understand that functors can form a category themselves, but I'm asking this question in relation to the categories they operate on. Are they inside or outside of them?)

In category theory a functor exists between two categories in much the same way that a function exists between two sets. If functors belonged to a category (as in 'inside' the category) then which category would they belong to? For example, one of my favourite functors is the forgetful functor (I can really identify with that property :) ) that leaves behind some of the structure of the source object when it picks up the target. Think of a vector space forgetting its addition and scalar multiplication and becoming just a set. Such a functor does not rightly 'belong' to the category of vector spaces because its purpose is to destroy what it means to be a vector space, and it doesn't belong to the category of sets because if you are a set why would you be expected to know what it was you forgot in order to get there when you were already there all along?

In other words when I define a functor in Haskell, are the pair of functions 'fmap,' and whatever unary type constructor is defined along with it, then part of the morphisms of the Haskell category? Or are they outside of the Haskell category (where the Haskell category I've seen defined in the articles I've read is one where its objects are types and its morphisms functions on those types)?

This leads me to my next question. It seems that all functors in Haskell (or at least all members of the functor typeclass) are covalent endofunctors. Does this mean that whenever a new functor is defined in Haskell, that the Haskell category (H) grows to accomodate the effects of that functor? In other words, for any covalent endofunctor F : H -> H, do H's morphisms then grow to include F(f) and objects to include F(a), where a is any object in H and f is any morphism in H?

Again in classical category theory all the objects and morphisms exist independently of any newly identified functor, so from that view the answer is 'no', for F to be a functor on H then for all objects a and morphisms f in H, F(a) and F(f) must already exist in H or the functor could not have been defined. However, the putative existence of an object in the Haskell category does not mean that there is an implementation of it. Note that there are infinitely many integers that have never been written down, but mathematicians still believe they exist!

I have been working off the assumption that the answer to my last question is 'yes.' The reason for this is natural transformations. Where functors seem to be able to be 'outside' of the categories they operate on (whether they are in fact 'outside' or not is my first question), natural transformations appear to have to be 'inside.' My understanding of the theory is that a natural transformation is a collection of morphisms in the target category of two parallel functors (where such morphisms operate on the functors and meet some specific properties). Surely the only way such a collection of morphisms would make sense would be if the objects they map to/from (i.e. the structures imposed by the two functors) were part of that category also?

Is it possible that there is some confusion coming from the fact that you are thinking primarily about endofunctors? In that case every object and morphism in site lives in the same category, so it is natural to think that the natural transformations do also. However, I don't think this is correct. I think it is better to think of functors as occurring as bridges _between_ categories and natural transformations as existing _between_ functors. More than just a collection of morphisms, a natural transformation actually associates objects and morphisms in the source category with morphisms and diagrams respectively in the target category. Or, if you prefer the collection of morphisms in the target category is 'indexed' by the objects in the source category in such a way that morphisms between the indexing objects set up a natural relationship between the morphisms they index - that natural relationship is given by a commutative diagram.

I hope these questions make sense. I really appreciate anyone who takes to time to read this!

Matt _______________________________________________ Beginners mailing list Beginners@haskell.org http://www.haskell.org/mailman/listinfo/beginners