Join Me

For spaces $X$ and $Y$, recall that the join $X*Y$ is obtained by "drawing all the lines from points in $X$ to points in $Y$". Ok, that's fairly informal. You can define the join as a quotient of the product of $X$ and $Y$ and the unit interval, $I$: $X*Y = (X\times Y\times I)/\sim$ where $(x,y,0)\sim(x,y',0)$ and $(x,y,1)\sim (x',y,1)$ for all $x\in X, y\in Y$. You might visualize this as taking the cylinder on $X\times Y$ (that is, the product with $I$), and collapsing one end to $X$, and the other end to $Y$. For example, the join of $X$ with a single point is the cone on $X$, while the join of $X$ with two points ($S^0$) is the suspension of $X$.

There's another, fancier looking, way to say this: $X*Y$ is the homotopy pushout of the diagram $X\leftarrow X\times Y\rightarrow Y$. If you haven't looked at homotopy pushouts recently... you take the mapping cylinder of $X\times Y \rightarrow Y$, and the mapping cylinder of $X\times Y\rightarrow X$, and you bring together the two 'end' copies of $X\times Y$.

Ok, so, that's all well and good. Now suppose that $X$ and $Y$ are contractible CW complexes with boundaries $\partial X$ and $\partial Y$, respectively (I don't want to define the boundary here... draw a CW complex, and you can tell it's boundary :)). Then the boundary of $X\times Y$ can be realized as the strict pushout (colim) of the diagram $\partial X\times Y\leftarrow \partial X\times \partial Y\rightarrow X\times \partial Y$. The inclusion of the boundary into it's complex is a cofibration, so this strict pushout is homotopy equivalent to the homotopy pushout. Furthermore, since we assumed $X$ and $Y$ were contractible, this diagram is object-wise homotopy equivalent to the diagram $\partial X\leftarrow \partial X\times \partial Y\rightarrow \partial Y$ (and there is an obvious map of diagrams). Taking homotopy pushouts is nice with respect to (object-wise) homotopy equivalent diagrams (with maps between them making everything commute), so putting everything together, we've got

$\partial(X\times Y)\simeq \mathrm{colim}(\partial X\times Y\leftarrow \partial X\times \partial Y\rightarrow X\times \partial Y)$
$\simeq \mathrm{hocolim}(\partial X\times Y\leftarrow \partial X\times \partial Y\rightarrow X\times \partial Y)$
$\simeq \mathrm{hocolim}(\partial X\leftarrow \partial X\times \partial Y\rightarrow \partial Y)$
which we recall is just the join, $\partial X * \partial Y$.

So, that's something. Why do I care? Well, I've been looking at posets recently, and their realizations in order to get some idea about homotopy types of some particular homotopy limits. My posets tend to have initial and final objects, so their realizations are contractible. If I remove the initial and final objects, then the suspension of the resulting realization is the same as the boundary of the realization of the whole poset. Furthermore, the posets turn out to be products of other posets.