TITLE: Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?
QUESTION [5 upvotes]: In Spivak's paper on metric realization of fuzzy simplicial sets, he sends a fuzzy $n$-simplex of strength $a$ to the set
$$
\{(x_0,x_1,\dots,x_n) \in \mathbb{R^{n+1}} |x_0+x_1+\dots+x_n = -\lg(a) \}
$$
and the realization of a general $X$ is via colimits:
$$
Re(X):= \mathrm{colim_{\Delta^n_{< a} \to X}}Re(\Delta^n_{< a} )
$$
It's a very concise self-published paper, and no justification is offered.
Why $\lg$? Is that just a nice hack to store the fuzziness, or is it preserving some nice distance properties? Secondly, this is realization, so I should be able to see it - what does a fuzzy complex look like?

REPLY [3 votes]: One possible explanation of "why $-lg$?" is that he needed a way to transform fuzzy set strengths, which range from $(0, 1]$, to metric space distances, which range from $[0, \infty)$, in a non-increasing way (larger simplex strengths = smaller distances)