On a definition of Spivak's fuzzy set In the paper "Metric Realization of Fuzzy Simplicial Sets"  of David Spivak it takes $I=(0,1]$ as  poset and consider it as a category. He gives it a Grothendieck topology induce it from consider $I$ as topological space with topology induced from sets of the form $(0,a)$ with $a\in I$.
In the paper, he defines the notation $S((0,a))=S^{\geq a}$ for a sheaf $S$. So he also defines:
$S(a)=S^{\geq a}-colim_{b>a}S^{\geq a}$
So my question is how to understand the definition of $S(a)$. From a cetegorial point of view the colimit is unique up to isomorphism.So diferente sets makes the same colimit, in fact the colimit is determined by its cardanility, as we are working in the category of sets. But this difference depends of the choice of the specific set used to represent the colimit.
 A: (Co)limits aren't just objects in a category: they're really (co)cones, which is an object equipped maps to or from other objects, which is universal with respect to this property.
So $\operatorname{colim}_{b > a} S^{\geq b}$ isn't just a set (which as you point out would be determined up to isomorphism by its cardinality).  It is a set such that given a compatible system of maps $S^{\geq b} \to T$ for any other object $T$, there exists a unique map $\operatorname{colim}_{b > a} S^{\geq b} \to T$.
In particular we may take $T = S^{\geq a}$, and as Spivak explained there is a compatible system of maps $S^{\geq b} \to S^{\geq a}$ for all $b > a$.  So by the universal property of the colimit there is a unique map $\phi: \operatorname{colim}_{b > a} S^{\geq b} \to S^{\geq a}$, and by $S^{\geq a} \setminus \operatorname{colim}_{b > a} S^{\geq b}$ Spivak presumably means the complement of $\operatorname{colim}_{b > a} S^{\geq b}$ in $S^{\geq a}$, where $\operatorname{colim}_{b > a} S^{\geq b}$ sits inside $S^{\geq a}$ as the image of $\phi$.
