A recent colloquium talk at UWO was given by Rick Jardine, who is a prominent member of the department, with a lot of graduate students. I’m not sure of all the details of what he works on, but it seems to mostly have to do with homotopy theory, category theory, and related things. His talk was called “Categories, Symmetric Groups, and Spheres”. It was rather involved for me to describe here, tying together as it did a bunch of different topics. However, I thought it was interesting, so I’ll try to give a summary of at least some of what it was about.

The last of the three topics – spheres – had to do with the fact that the end result was to show that some construction turns out to be closely related to sphere spectra. Spectra are sequences of spaces, say $(X_0, X_1, X_2, \dots )$, such that there’s a map from the suspension of each space into the next, $S^1 \wedge X_n \rightarrow X_{n+1}$. A suspension is just a sort of double-cone on a space: to get $S^1 \wedge X$, add two points, and then connect each point of $X$ to each of the two new points. For example, if you start with a circle, the result is a sphere – your original circle was the “equator”, then you added two poles, and drew in the points in between. This example generalizes, so a really simple spectrum is just the sequence of spheres of increasing dimension (then the map $S^1 \wedge S_n \rightarrow S_{n+1}$ is just the identity).

These spectra are important in homotopy theory, and in particular, in stable homotopy theory. As I understand it, stable homotopy talks about those parts of homotopy groups that stay the same when you repeatedly take suspensions – so you pass from homotopy classes of maps from a circle into $X$, to maps from a sphere into the suspension $S^1 \wedge X$, to maps from a 3-sphere into the suspension $S^1 \wedge S^1 \wedge X$, and so on… the only changes that can occur is that you might lose some distinctions, so the groups could get smaller. Eventually, they stabilize – and voila!, stable homotopy groups. So anyway, spectra are important to this subject.

In particular, the theorem Rick was explaining (in, as he said, a “modern exposition”, originally due to Barratt and Priddy) has to do with a space called $QS^0$, whose homotopy groups are the same as the stable homotopy groups of spheres. The theorem says that it has the same homology as the infinite symmetric group. So the idea he was presenting is a construction involving symmetric groups. The point of it is that there’s a basically combinatorial description of everything involved – that is, a description involving just finite sets (which is where the symmetric groups come from).

How does this work? Well, first of all, it uses a construction called the “category of elements” for a functor $I \stackrel{X}{\rightarrow} Set$. This is a category $E_I X$ whose objects are pairs $(i,x)$ where $x \in X(i)$, and whose morphisms $\alpha : (i,x) \rightarrow (j,y)$ are morphisms $f i \rightarrow j \in I$ such that $X(f)(x) = y$. That is, this makes a new category from all the elements of the sets coming from objects in $I$, where the arrows are compatible with those in $I$ – each object is multiplied, and so are the morphisms.

The category of elements we’re talking about is a functor $P_X : Mon \rightarrow Sets_*$. Here, $Mon$ has finite sets for objects, and 1-1 functions (“injections”, “monomorphisms”, etc.) as morphisms, and $Sets_*$ is the category of pointed sets. This functor depends on a particular choice of pointed set $(X,x)$, or $X$ for short. The way it works is that $P_X(S)$ is the set of all functions from $S$ into $X$ – which is pointed, since the function where everything goes to $x$ is distinguished – so this is just $X^S$. Given an injection $S \rightarrow S'$, you get a map from the set of functions $P_X(S) = \{ f : S \rightarrow X\}$ to $P_X(S') = \{ f' : S' \rightarrow X \}$, which you get by extending a function so anything in $S'$ not in the image of $S$ just goes to the special point $x$ (this is why we needed pointed sets). So the category of elements in question is $E_{Mon} P_X$. The point is that it gives a nice space.

Again: how? Well, this uses the idea of a “nerve”.

Any category $C$ has a nerve: this is a simplicial set related to $C$. The way you get an $n$-simplex is to look at any chain of $n$ arrows in $C$. The vertices form the edges, the arrows give some of the edges, and the various ways of composing (some of) them give other edges. Each composition of two gives a triangle, and the higher simplices come from various equations. The different simplices are stuck together by various incidence relations that show the structure of the category $C$. This nerve is called $BC$, which is a purely combinatorial object. (Ultimately, the simplicial set that’ll show up in this story is $B(E_{Mon} P_X)$ from the category of elements above). It becomes a space when you take its geometric realization: replace abstract simplices with actual triangles, tetrahedra, and so forth, taken as topological spaces living in $\mathbb{R}^n$. This space is called $|BC|$, and it’s a topologically nice space – a $CW$-complex (being built by gluing simplices together).

Then you have this simplicial set, which can be thought of as a space, $\Gamma^t(X) = B(E_{Mon} P_X)$ – a so-called “gamma space”, which are what correspond to these spectra mentioned up above. In particular, if the pointed set $X = \{ 0 , 1 \}$, with 0 the distinguished point, then it turns out that $\Gamma^t(X) = \bigcup_{n \geq 0} B(\Sigma_n)$, the disjoint union of the spaces obtained from all the finite symmetric groups. This is because the symmetric group acts on the category of elements $E_{Mon} P_X$.

So part of the point of this part is what was, as Rick pointed out, the first adjoint pair of functors which was seriously studied – a pair of functors going between $sSet$ and $Top$ (simplicial sets and topological spaces). The geometric realization functor $| \cdot |$ is a left adjoint to a functor $S$, so that $S(Y)_n = hom ( | \Delta^n |, Y)$, giving a simplicial set for a topological space $Y$. And homotopy theory in $Top$ then has an equivalent in $sSet$ – so there’s a completely combinatorial core of homotopy theory. (Technically – and I admittedly don’t quite grok this concept yet – these two adjoint functors are giving a Quillen equivalence). Now, homotopy doesn’t tell you everything about a space – but it tells a lot, so it’s useful to get the idea that all this information about a space from something very combinatorial, like permutations of finite sets.

I have to admit I find a lot of this stuff is a bit technical for me to fully appreciate what’s clearly a very elegant fact relating spaces and combinatorics, but I find it interesting that a correlation like that exists. The apparent dichotomy between “smooth” or “continuous” things like spaces, and discrete, combinatorial things like integers, finite sets, permutations, etc. – and the various ways this dichotomy gets resolved, overcome, or bridged – is one of the really interesting cores of mathematics to my mind.