So first of all, I realized that in my sleepiness yesterday, I left off the last part of the story of why homotopy groups are groups. More precisely, we need to show that if ${X, Y}$ are any pointed spaces, then

$\displaystyle \hom_{\mathbf{PT}}(\Sigma^2 X, Y)$

is an abelian group under the cogroup law of ${\Sigma^2 X}$ (the double suspension). But this group is just

$\displaystyle \hom_{\mathbf{PT}}(\Sigma X, \Omega Y)$

and one can check that the adjointness between ${\Sigma, \Omega}$ respects the group structure. And we showed by the Eckmann-Hilton argument yesterday that this is abelian under either the group law of ${\Omega Y}$ or the cogroup law of ${\Sigma X}$; they’re also both the same. So in particular, the homotopy classes out of a higher suspension form an abelian group. Since the homotopy groups ${\pi_n, n \geq 2}$ are defined in this way, they are abelian.

Last time, we defined two functors ${\Omega}$ and ${\Sigma}$ on the category ${\mathbf{PT}}$ of pointed topological spaces and (base-point preserving) homotopy classes of base-point preserving continuous maps. We showed that they were adjoint, i.e. that there was a natural isomorphism

$\displaystyle \hom_{\mathbf{PT}}(X, \Omega Y) \simeq \hom_{\mathbf{PT}}(\Sigma X, Y).$

We also showed that ${\Omega Y}$ is naturally an H group, i.e. a group object in ${\mathbf{PT}}$, for any ${Y}$. So, given that we have a group operation ${\Omega Y \times \Omega Y \rightarrow \Omega Y}$, it follows that ${\hom_{\mathbf{PT}}(X, \Omega Y)}$ is naturally a group for each ${Y}$. (more…)

I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let $1$ be this object; then we can define two operations on $End(1)$ as follows.  The first is the tensor product: given $a,b$, define $a.b := \phi^{-1} \circ a \otimes b \circ \phi$, where $\phi: 1 \to 1 \otimes 1$ is the isomorphism.  Next, define $a \ast b := a \circ b$.  It follows that $(a \ast b) . (c \ast d) = (a . c) \ast (b. d)$ by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative. (more…)