In this course, we will teach category theory from first principles with an eye towards its applications to and correspondences with Haskell and the theory of functional programming. We expect students to previously or currently be taking CS242 and to have some level of mathematical maturity. We also expect students to have had contact with linear algebra and discrete mathematics in order to follow the motivating examples behind the theory expounded.

Wednesdays at 4.15.

Online notes will appear successively on the Haskell wiki on http://haskell.org/haskellwiki/User:Michiexile/MATH198

come this fall, I shall be teaching. My first lecture course, ever.

The subject shall be on introducing Category Theory from the bottom up, in a manner digestible for Computer Science Undergraduates who have seen Haskell and been left wanting more from that contact.

And thus comes my question to you all: what would you like to see in such a course? Is there any advice you want to give me on how to make the course awesome?

The obvious bits are obvious. I shall have to discuss categories, functors, (co)products, (co)limits, monads, monoids, adjoints, natural transformations, the Curry-Howard isomorphism, the Hom-Tensor adjunction, categorical interpretation of data types. And all of it with explicit reference to how all these things influence Haskell, as well as plenty of mathematical examples.

But what ideas can you give me to make this greater than I’d make it on my own?

We pressed on with knot theory. Today, we discussed knot sums, prime knots, knot tabulation, behavior of the one invariant (n-colorability) we know so far under knot sums, Dowker codes, and we got started on Conway codes for knots. Next week, I plan for us to finish up talking about the Conway knot notation, get the connection between rational knots and continued fractions down pat, and start looking into new invariants.

Anyone have a favorite invariant that you’d like me to talk about? I’m hoping (in my wildest most bizarre dreams) to get around to the Alexander polynomial and possibly even talk about Khovanov homology, but that depends a LOT on whether they’re prepared to continue through their summer holidays or not – and even then I doubt we’ll make it up to Khovanov.

Today, we’ll abandon algebraic topology completely, and instead go into knot theory. I’ll want to discuss what we mean by a knot (embedding of in ), what we mean by a knot deformation (thus introducing isotopies while we’re at it) and the Reidemeister moves. Also we’ll discuss knot invariants – and their use analogous to topological invariants.

Later on, we’ll continue with other invariants; definitely including the Jones polynomial, and possibly even covering Khovanov homology. One possible end report would be to explain a bunch of knot invariants and show using examples how these have different coarseness.

Edited to add: I got myself some damn smart students. They figured out the Reidemeister moves on their own – as well as minimal crossing number in a projection being highly relevant – with basically no prompting from me. I’m impressed.

Next term, we’ll want to show homotopy invariance and that the singular and simplicial homology coincide when applicable. After that, we’ll change directions slightly.

The future after that holds knot theory, was decided today. We’ll want to introduce knots, look at Reidemeister moves and basic knot invariants. I use basic here in a pretty wide sense – we’ll probably do the Jones polynomial and we might even end up doing Khovanov homology if I feel particularly insane late spring.

The next couple of weeks will be spent doing homology of simplicial complexes, singular homology, equivalence of the two, neat things you can do with them; and then we’ll start moving towards a Borsuk-Ulam-y topological combinatorics direction.

I might end up pulling combinatorics papers from my old “gang” in Stockholm on graph complexes, and graph property complexes, and poke around those with them.

I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write , if there is a continuous map such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps , such that and .

Two spaces X,Y are homotopy equivalent if there are maps , such that and .

Now, if f,g are maps X→Y and f=g, then , since we can just set H(x,t)=f(x)=g(x) for all t, and get a continuous map out of it. Thus homeomorphic spaces are homotopy equivalent, since the relevant maps are equal, and thus homotopic.

There are a couple of more properties for homotopic maps we’ll want. It respects composition – so if and h:Y→Z and e:W→X then and . This can be seen by considering h(H(x,t)) and H(e(x),t) respectively.

Denote by D² the unit disc in , and by {*} the subset {(0,0)} in . Then . In one direction, the relevant map is just the embedding, and in the other direction, it collapses all of D² onto {*}. One of the two relevant compositions is trivially equal the identity map, and in the other direction, the linear homotopy H(x,t)=tx will do well. Thus the disc and the one point space are homotopy equivalent.

The fundamental groupoid

Let X be a topological space (most probably with a number of neat properties – I will not list just what properties are needed though), and consider for each pair x,y of points in X, the set [x,y] of homotopy classes of paths from the point x to the point y. A path, here, is a continuous map [0,1]→X. We can compose classes – if and , then we can consider the map
. This is a path from x to z, and so belongs to a class in [x,z]. This class is well defined from the choices of γ, γ' since homotopies and composition of maps work well together.

This gives us a composition. It is associative - on homotopy classes. What happens if we look at maps instead of homotopy classes is part of the subject of my own research. It has an identity at each point x - the constant path γ(t)=x, and for each class in [x,y] there is a class in [y,x] such that their composition is homotopic to the constant path in [x,x].

Thus, we get a groupoid. This is called the fundamental groupoid, and denoted by . If we fix a point, and consider [x,x], then this is a group, called the fundamental group with basepoint x, and denoted by .

For , a linear homotopy will make any two paths in [x,y] homotopic – and so |[x,y]|=1 in for any x,y.

For S¹ – the circle – we can choose to view it as [0,1]/(0=1). Then we can consider the paths f_m(t)=a(1-t)+bt+nt. This is a path from a to b, and it winds n times around the circle. Each path in [a,b] is homotopic to a f_m, by a linear homotopy, which just rescales the speeds through various bits and pieces, and possibly straightens out when you double back. Thus, . Furthermore, if you compose f_mf_n, you’ll get f_n+m.

If we pick out the fundamental group out of this groupoid, we’ll get the well known fundamental group .

Now, suppose we have two homotopy equivalent spaces X and Y, with the homotopy equivalence given by f:X→Y and g:Y→X. Then consider the map f_*:[x,y]_X→[f(x),f(y)]_Y given by f_*γ(t)=f(γ(t)). I claim
1) f_* is bijective.
2) f_* works well with composition of classes.

For bijectivity we start with injectivity in one direction. Consider two paths in [x,y]. We need to show . If , then . However, then

which contradicts . Thus , and so also .

The proof is symmetric in the choice of direction, and so we can just repeat the same argument to get that g_* is also an injection. Thus we can conclude that f_* is in fact a bijection.

Now, for the second part, we consider and . We need to show that . But is the path that first runs through in half the time, then runs through in the rest of the time, and just transports this path point by point to Y. And transports point by point to Y and transports point by point to Y, and just runs through the first of these in half the time, then the rest in the rest of the time.

Thus, homotopy equivalent spaces have the same fundamental groupoid.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by Jänich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.

I plan to keep updates on our progress here on the blog – with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.

For the first trip, the questions I dumped in their laps were:

• What topologies are possible on the set {0,1}?
• What topologies are possible on the set {0,1,2}?
• Which are the continuous functions between the topologies above?
• Give an example of a continuous and a discontinuous function each for the following cases
  1. with the standard topology on
  2. with the standard topology on
  3. with the discrete topology
  4. with the finite-complement topology