  This note presents a presheaf theoretic approach to the construction of fuzzy
sets, which builds on Barr's description of fuzzy sets as sheaves of
monomorphisms on a locale. A presheaf-theoretic method is used to show that the
category of fuzzy sets is complete and co-complete, and to present explicit
descriptions of classical fuzzy sets that arise as limits and colimits. The
Boolean localization construction for sheaves and presheaves on a locale L
specializes to a theory of stalks if L approximates the structure of a closed
interval in the real line. The system V(X) of Vietoris-Rips complexes for a
data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this
general framework. This example is explicitly discussed in this paper, in
stages.
