(17 oct 2012: this post is obsolete since the second edition is now available from the same link below)
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 are path morphisms from natural spaces to and to respectively, then how do we form the composition?
(where , , are derived from the pre-natural spaces , with (pre-natural) path spaces , )
The thing to notice is that is defined as a refinement morphism from to , but can be uniquely lifted to a refinement morphism from to .
This is straightforward, for a basic dot in , we put which is a basic dot in since is a refinement morphism.
We now define the composition of and to be the composition , which is a path morphism from to .
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.