In many of the articles/books on Category Theory the Category theory is explained in terms of mathematical concepts of Rings, Sets, Fields... I have been struggling to see how it all relate to functional programming. So this might be useful for others too...

In section 2.2 this Category Theory lecture notes there is an interesting explanation of explaning how functional programming forms a category.

## Friday, January 22, 2010

### A metaphor for categories and mappings

I was trying to see if Design Pattern concepts (specially compositional patterns) have been explained in terms of functors, monads,.... I ran into this post about category theory that is enlightening...

A metaphor for categories and mappings

This section in particular:

Also this in the comments

A metaphor for categories and mappings

This section in particular:

For a really simple, real world example, consider a lawnmower. When out of gas/petrol, it needs to be refueled. The category of empty lawnmowers has an arrow/mapping/morphism to the category of full lawnmowers. Here the focus is more on the _process_ of mapping one category into another, or mapping one collection of things into another collection.

It's been said that category theory places mappings/arrows on a fully-equal footing with objects/points/things.

Now back to the lawnmower example. A child walks out to watch his father mowing the lawn. The lawmower runs out of gas. The child says "Daddy, I think the lawnmower needs a drink."

What the child is doing is using his own set of mappings, from thirsty to not thirsty, to metaphorize the process of fueling the lawnmower. There's a mapping between these mappings, a functor. (I happen to think this categorical way of thinking is immensely important for how we understand the world, how we might program artificial intelligences, and so on. Even more than "design patterns," I think we and other creatures chunk the world into pieces we can understand by finding the mappings and functors and so on which allow us to grok the world. We are, I think, category theory engines.)

Also this in the comments

A more useful way to think about category theory compared to set theory is to view it as a kind of inversion of reasoning. In set theory everything is built up, and we understand objects by seeing what they are made of -- picking apart their internals. We relate two objects by comparing what they are made of. Category theory turns that on its head and takes relationships between objects as primitive. From CT's point of view objects are opaque; we understand an object not by peeking inside at it's internal structure, as we would if we were using set theoretic thinking, but rather by examining how the object relates to other objects.

## Wednesday, January 20, 2010

### Functional Programming Think

It seems that as I am learning more Haskell, I find more nuggets in paper that I had read before. For instance, John Hughes Generalising Monads to Arrows paper has some very interesting insight into functional programming mind set.

And also on Category Theory

If this were an isolated case we might simply ignore it. But Swierstra and Duponcheel’s idea is clearly much more widely applicable: to optimise a combinator library, rede ne the combinators to collect static properties of the computations they construct, and then use those static properties to optimise the dynamic computations. If we think of a library as de ning a domain speci c ‘language’, whose constructions are represented as combinators, then Swierstra and Duponcheel’s idea is to implement the language via a combination of a static analysis and an optimised dynamic semantics. We may clearly wish to do this very often indeed. But every time we do, the type of >>= will make it impossible to use a monadic interface!

And also on Category Theory

3.1 On Category Theory

Before we do so, we make a short digression on the subject of category theory. The concept of a monad was developed by category theorists long before it eventually found an application in functional programming. Some might find it surprising that something so abstract as category theory should turn out to be useful for something so concrete as programming. After all, category theory is, in a sense, so abstract as to be rather unsatisfying: it is ‘all definitions and no theorems’ , almost everything turns out to be a category if you look at it long enough, to say something is a category is actually to say very little about it. The same is true of most categorical concepts: they have very many possible instantiations, and so to say that something is, for example, a monad, is to say very little. This extreme generality is one reason why it is hard for the beginner to develop good intuitions about category theory, but it is hardly surprising: category theory was, after all, developed to be a ‘theory of everything’, a framework into which very many di erent mathematical structures would t. But why should a theory so abstract be of any use for programming? The answer is simple: as computer scientists, we value abstraction! When we design the interface to a software component, we want it to reveal as little as possible about the implementation [think algebra vs calculus]. We want to be able to replace the implementation with many alternatives, many other ‘instances’ of the same ‘concept’. When we design a generic interface to many program libraries, it is even more important that the interface we choose have a wide variety of implementations. It is the very generality of the monad concept which we value so highly, it is because category theory is so abstract that its concepts are so useful for programming.

It is hardly surprising, then, that the generalisation of monads that we present below also has a close connection to category theory. But we stress that our purpose is very practical: it is not to ‘implement category theory’, it is to find a more general way to structure combinator libraries It is simply our good fortune that mathematicians have already done much of the work for us.

## Wednesday, January 13, 2010

### Class types as "convenience functions"

I had hard time understanding the common practice in haskell of MonadXXX class relationship with XXX type (e.g. MonadReader and Reader) until I run into this line:

The MonadWriter class provides a number of convenience functions for working with Writer monads. In the Writer Monad tutorial.

Looking at the class interfaces as "convenience functions" to work with underlying type is interesting way of looking at it.

The MonadWriter class provides a number of convenience functions for working with Writer monads. In the Writer Monad tutorial.

Looking at the class interfaces as "convenience functions" to work with underlying type is interesting way of looking at it.

## Sunday, January 10, 2010

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