(17 oct 2012: this post is obsolete since the second edition is now available from the same link below)

Recently published online, the book ‘Natural Topology’ finishes a project which Wim Couwenberg and I started already in 2004. See also my math page.

This blog is perhaps handy to post some addenda and corrections over the time to come, before creating a real revision as a second edition.

An important (but not difficult) addendum which I forgot to detail in the book itself:

The composition of path morphisms.

If $f, g$ are path morphisms from natural spaces $\mathcal{V}$ to $\mathcal{W}$ and  $\mathcal{W}$ to $\mathcal{Z}$ respectively, then how do we form the composition?

(where $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Z}$ are derived from the pre-natural spaces $V, W, Z$, with (pre-natural) path spaces $V^{\wr},$ $W^{\wr}$, $Z^{\wr}$)

The thing to notice is that $f$ is defined as a refinement morphism from $V^{\wr}$ to $W$, but can be uniquely lifted to a refinement morphism $f^{\wr}$ from $V^{\wr}$ to $W^{\wr}$.

This is straightforward, for a basic dot $a = a_0, a_1,\ldots, a_{n-1}$ in $V^{\wr}$, we put $f^{\wr}(a) = f(a_0), f(a_0, a_1), \ldots, f(a_0, a_1, \ldots, a_{n-1})$ which is a basic dot in $W^{\wr}$ since $f$ is a refinement morphism.

We now define the composition of $f$ and $g$ to be the composition $g\circ f^{\wr}$, which is a path morphism from $\mathcal{V}$ to $\mathcal{Z}$.

The composition of a refinement morphism with a path morphism.

Since any refinement morphism can be thought of as a path morphism (trivially), this has been dealt with above.

A drawback to the term path space’ only occurred to me recently, namely that this is also a term used in connection with usual topological spaces. The same applies a bit to the term unglueing’. I will give some thought to finding other terms, for an ameliorated second edition.