# Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?

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?

## 1 Answer

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)