This paper 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. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval.The systemV∗(X)of Vietoris-Rips complexes for a data setXis both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.