Metric Realization of Fuzzy Simplicial Sets

We discuss fuzzy simplicial sets, and their relationship to something like metric spaces. Namely, we present an adjunction between the categories: a metric realization functor and fuzzy singular complex functor. The following document is a rough draft and may have (substantial) errors. 1. Fuzzy simplicial sets Let I denote the Grothendieck site whose objects are initial open intervals contained in the half-open unit interval [0, 1) ∈ R, whose morphisms are inclusions of open subsets, and whose covers are open covers. In other words, as a category, I is equivalent to the partially ordered set (0, 1] under the relation ≤. A sheaf S ∈ Shv(I) on I is a functor S : I → Sets satisfying the sheaf condition. Explicitly, S consists of a set S([0, a)) for all a ∈ (0, 1], which we choose to denote by S≥a, and restriction maps ρb,a : S≥b → S≥a for all b ≥ a, such that if c ≥ b ≥ a then ρb,a ◦ ρc,b = ρc,a, and such that for all a ∈ I, one has S≥a ∼= lim a′a S≥b, and note that S≥a = colimb≥a ρb,a[S(b)]. If T is a fuzzy set, we can make this easier on the eyes: T≥a = ∐