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?


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)


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