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…)