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I am working through 'Notes on logic and set theory' by P.T. Johnstone. Below, $\Omega$ is a set of operation symbols with $\alpha \colon \Omega \to \mathbb{N}$ assigning to each symbol its arity. Also, $E$ denotes a set of equations of $\Omega$-terms so that $(\Omega, E)$ is an algebraic theory.

Exercise 1.8 reads

(i) Let $\Omega = \{e,m\}$ with $\alpha(e) = 0, \alpha(m) = 2$, and let $E$ consist of the two equations $(mex = x)$ and $(mxe=x)$. Suppose a set $A$ has two $(\Omega,E)$-model structures $(e_1, m_1)$ and $(e_2,m_2)$ such that the operations of the second structure are $\Omega$-homomorphisms $1\to A$ and $A\times A \to A$ for the first structure. Show that $A$ satisfies the equations $(e_1 = e_2)$ and $(m_1m_2xzm_2yt = m_2m_1xym_1zt)$, and deduce that $m_1 = m_2$ and that $m_1$ is commutative and associative.

(ii) Ask an algebraic topologist to explain what this has to do with the result that the fundamental group of a topological group is abelian.

I solved (i) but I have no easy access to an algebraic topologist to solve (ii) (I am on holiday). Can someone explain this to me? I know the definition of fundamental groups and of topological groups but little more.

Hernán Ibarra Mejia
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Suppose $\mathcal{X}$ is a topological group and $f,g$ are loops in $\mathcal{X}$ (so $f,g:[0,1]\rightarrow\mathcal{X}$ are continuous and have $f(0)=f(1)=g(0)=g(1)=*$ where $*$ is some fixed "base point" in $\mathcal{X}$). There are two ways to "combine" these loops:

  • Just using the topological-space-ness, we can do the usual concatenation of loops to get $$f\oplus g: t\mapsto \begin{cases} f(2t) & \mbox{ if }t\le{1\over 2}\\ g(2t) & \mbox{ if }t>{1\over 2.}\\ \end{cases}$$ This is pretty bad on the face of things (e.g. it's not associative), but it lifts to a very nice operation $\widehat{\oplus}$ on homotopy classes, and in particular gives the group operation of the fundamental group of $\mathcal{X}$.

  • On the other hand, we can also combine $f$ and $g$ "pointwise" using the group structure: let $$f\boxplus g: t\mapsto f(t)\cdot g(t)$$ where $\cdot$ is the group operation in $\mathcal{X}$. This also gives rise to a group operation $\widehat{\boxplus}$ on the set of homotopy classes.

So we have two reasonably-nice operations $\widehat{\oplus}$ and $\widehat{\boxplus}$ on the same set. Can you go from here?

Noah Schweber
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