However! Levels 1 and 3 I am fairly happy with. With Category Theory I have struggled manfully from fairly simple discussions to the beautiful output of a MacLane or a Lawvere. I need a simple example which shows me how to connect the three levels in a happy, functionning whole. How does one program an artificial child pedalling an artificial bicycle for, say, five minutes (No – ten seconds) using our three sets of tools? How do we build in the categories to make sure she does not fall off? Good luck John

programming level where we finally get down to business
#)

Ordinarily, I’d say you should get to adjunctions as quickly as possible, and then concentrate on the various theorems relating adjunctions and limits, and on the connections between monads and adjunctions. I don’t really think that topoi would be interesting to your proposed audience, but then I know very little about topos theory. BTW, Hask lacks equalisers, so it doesn’t have all finite limits (it’s an easy theorem that one can construct all finite limits if you have finite products and equalisers, or if you have binary pullbacks). Coproducts are, as the name suggests, colimits

Jeff Egger recently taught what sounded like a really interesting basic CT course, which avoids all the tedious symbol-bashing by proving all the basics using string diagrams. It might be worth getting in touch with him to ask how it went.

I can perhaps explain the monad/triple thing. Among categorists, “triple” is a not-quite-abandoned name for “monad”, still used by a few of the older generation. There’s no semantic difference between the two. What Haskellers call a “monad” is called a “strong monad” by categorists, and that’s a slightly different notion. If John’s really saying that two concepts with different but totally equivalent definitions ought to be considered different, then I’m tempted to suggest that he hasn’t grokked the point of category theory at all – if two definitions give rise to equivalent categories of models and morphisms, one should really consider them the same.

There is an issue with all limits in Hask, though, it turns out. Having all limits is more than even the idealized Haskell category manages – and the keyword there seems to be Dependent Types.

So, what monic in the category of Haskell types mimics the “element of” relation, using [T] as power object?

I don’t have a copy myself, but I think the approach I found most sensible is in chapter IV of MacLane and Moerdijk. Reviewing my notes, they start with (binary) pullbacks, a terminal object, a subobject classifier, and a slightly different set of data to specify power objects. Since we know how power objects and exponentials will eventually relate, and that Hask has exponentials, we can use that to define the appropriate power objects once we have the subobject classifier.

Okay, so I still think that the classifying object will be the Boolean type. The terminal type should be what.. does Haskell have a type like C’s “void”? Then the (monic) computable function from void to Boolean should be “return True”.

The question (and I’m not a Haskell expert) would then be this: for every (monic) computable function $latex m:S\rightarrow B$, is there a unique computable function $latex B\rightarrow\mathbf{Bool}$ that returns “true” on instances of type B in the image of m and “false” on everything else?

I’d go with pointing at the List type as a potential power object. Hence, for the type T, the power object’d be [T].

Incidentally, what you’ve told me is that your category contains exponentials, not power objects. A power object is something like a power set. It turns out that when you’ve got a subobject classifier ? then the power object P(X) is the exponential ?^X. And a subobject classifier should be something like the Boolean type. It’s like the set of truth values in the category of sets!

As for finite limits, I’m less certain that it holds. Certainly, it has products and coproducts, but equational reasoning is often a matter of programming contracts more than of strict enforcement. However, I guess that one could argue that finite limits exist by virtue of having products, coproducts and a sensible QuickCheck suite verifying, Monte Carlo-style, any further relations.

The part I’m really iffy on is the subobject classifier (having read all of … five sentences of topoi that I understood by now…) – I don’t see how that fits.

a) All finite limits?

b) All power objects?

John: One of my main intentions with the course is to be able to discuss the Hask category of Haskell types and computable functions between them. Is this a topos?

If I were trying to bridge the gap from Haskell to category theory I’d be sure to throw in all the goodies from recursion theory (cata-, ana-, hylo-, para-,…) as well as tricks like Swierstra’s Data Types a la Carte. Both of these focus a lot on functors which are still down-to-earth enough to be easily grasped, but abstract enough to see where you’re headed: the perfect gateway drug. (Be sure to cover the laws of the various recursions, not just the recursive pattern.)

I’d also spend some time doing an aside on deforestation, build/fold and unfold/destroy fusion, generalized Church encodings, why/how hylomorphisms are an optimization, and the like. These topics help to get into parametricity which leads to free theorems; and the focus on manipulating morphisms while “ignoring” the objects and their structure helps to break away from the thinking that’s so prevalent in set theory but antithetical to the category theoretic perspective.

There’s a lot of stuff out there on just the raw CT. The problem with most of it, IMO, is that it’s just CT. Basic CT is pretty easy, but seeing how to apply it to new tasks is a lot harder. Even though there are practical applications all over the place, there isn’t a really good resource for demonstrating the applicability of CT and charting out the overall geography of the field. I think the best thing for your course would be to try to give some of that geography so your students will have a better understanding for what areas of CT to continue exploring based on their interests.

I could probably take a look at Charity – thank you for the suggestion – but it’s not going to be a major focus. I don’t want to teach a new programming language _too_ if I can avoid it.

Plus they buy you the opportunity to play with Charity (a categorical programming language) which is fun because its recursion is so ‘primitive’.

However i agree that from a categorical point of view programming is much like a distraction.