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Unlike category theory which is in many ways a freer framework in which to do mathematics and which very nicely captures universal objects and constructions (e.g., limits and colimits), mereology is a more restrictive framework than set theory. The part/whole relation can be captured by set/subset, but set/member cannot simply be recaptured in mereology. For instance, in mereotopology a space is comprised entirely of extended parts, no points. Try reformulating the separation axioms and deriving Urysohn's theorem, for example. (Maybe it can be done. I think so. But it's not immediately clear how.) For these reasons, mereology will remain of interest to nominalistically inclined mathematical philosophers (like Tarski, not to mention Russell and Whitehead in whose work I find mereological inclinations) but is not likely to spark a major mathematical research program, in my opinion.