Addendum to ‘Natural Topology’

(17 oct 2012: this post is obsolete since the second edition is now available from the same link below)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

 

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About fwaaldijk

mathematician (foundations & topology in constructive mathematics) and visual artist
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One Response to Addendum to ‘Natural Topology’

  1. fwaaldijk says:

    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.

    Such second edition might also give a little more emphasis on the reason why we focus on refinement morphisms and spraids, which is elegance and computational efficiency mainly.

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