In this post we’ll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation.

Let $f \colon U \rightarrow \mathbb C$ be a holomorphic function. A holomorphic $n$-th root of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = g(z)^n$ for all $z \in U$. A logarithm of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = e^{g(z)}$ for all $z \in U$.

The main question …

This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about $\mathsf{hPairTop}$ at all.

In algebraic topology you (for example) associate every topological space $X$ with a group, like $\pi_1(X, x_0)$ or $H_5(X)$. All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup …