The number 3, when considered as a universal, abstracts the property of 3'ness from all groups of 3 objects. One supposes that this universal, by conceptual neccessity of what is understood by the idea of 'universal' is unique. The problem of universals is the ontological status of this universal which can be broadly characterised as being of three kinds - real (following Plato), nominal (following Ockham) and Ideal (following Kant).
In the usual theory of Sets, one notices, that for example the empty set by the axiom of extensionality is unique.
But what about the number 3 in theory of Sets?
If one follows a particular construction of the integers, for example by Von Neumanns nested sets, then the number 3 is determined and constructed uniquely. But one can always choose another construction. And one notes that the detailed construction of the integers do not matter - at least where one is not concerned with their ontological status - but how they function in a theory which makes use of certain of their properties.
One might say, rather than use one particular construction amongst many, use all possible constructions.
In this sense, the number 3 appears to lose its universality and appears in the world of concrete objects. That is the number 3 is thought of not as the instantiation of the abstract property of 3'ness shared by all groups of 3 objects, but as all groups of three objects but understood both indefinately and particularly. Indefinate as we choose not some represenative amongst them, and particularly because we keep hold of their distinctness.
Is this a form of nominalism?
What I've described above is how numbers are understood in Topos Theory - aka generalised Set Theory.
Can one say, at least roughly, that Set Theory attempts to ground itself on Platonic Realism, which states that universals exists. And that Category/Topos Theory attempts to ground itself on nominalism?
That grounding of Set Theory on Platonic Realism appears to be a failure for the reasons I've given above: In that the only universal it contains is the empty set, and that it doesn't provide us with the universal example of the number 3?
That the attempt to ground the the uniqueness that should be inherent in the realist position on universals, at least in set theory, is/was understood to be captured by the axiom of extensionality. One notices in particular that the uniqueness of the empty set is essentially reliant on this axiom.
Does Category/Topos Theory capture the nominalist position - or are there failures here? One notes that the theory speaks of 'universal properties' which has a precise technical definition, and one notes that they do not produce a unique object but simply characterises those objects that have those properties - and show, as one should readily note, that any two such objects, so far as these properties are under consideration, are equivalent.