On Faded Cosheaves of Sets

Abstract

We prove that the category of faded cosheaves in Set over a sober topological space $(B,Ω)$ is equivalent to a category Sett$(B,Ω)$ having the same class of objects as Set$ / B$ has, but generally a wider class of morphisms. We also prove the converse: if each cosheaf over $T_0$-space $(B,Ω)$ is isomorphic to the cosheaf of tubes of an appropriate map ranging in $(B,Ω)$, then $(B,Ω)$ is sober.