When it comes to deal with structures, in particular in abstract branches of mathematics – abstract in comparison to number theory, analysis and the geometry of figures, curves and planes -, such as algebraic topology, homology and homotopy theory, universal algebra, and what have you, a vast majority of mathematicians considers Category-Theory (CT) vastly superior to set-theory. CT also is the only rival to ZFC (Zermelo–Fraenkel Choice set theory) in providing a general theory of mathematical structure and in founding the whole of mathematics. The language of CT is two-sorted: it contains object-variables and arrow-variables. An arrow sends objects to objects; an identity-arrow sends an object to itself. Simply put, structures are categories, and a category is something that has objects and arrows, such that the arrows can be composed so as to form a composition monoid, which means that: (i) every object has an identity-arrow, and (ii) arrow-composition is associative. The languages of CT (L↑) and ZFC (L∈) are inter-translatable. In CT there is the specific category Set, whose objects can be identified with sets and whose arrows are maps. In ZFC one can identify objects with sets and arrows with ordered pair-sets of type ⟨f, C⟩, consisting of a mapping f and a co-domain C.
In spite of the fact that some mathematical physicists have applied categories to physics, not a single structural realist on record has advocated replacing ZFC with CT. One of the very few critics of the use of set-theory for Structural Realism is E.M. Landry, who has argued that the set-theoretical framework does not always do the work it has been suggested to do; but even she does not openly advocate CT as the superior framework for StrR, although she does advocate it for mathematical structuralism.
The objects of CT are more general than the Ur-elements one can introduce in ZFC, because whereas primordial elements are not sets, the objects of CT can be anything, arrows, sets, functors and categories included. Similar to ZFCU is that CT does not have axioms that somehow restrict the interpretation of ‘object’. A CT-object is anything that can be sent around by an arrow, similar to the fact that a set-theoretical Ur-element is anything that can be put in a set. CT-objects obtain an ‘identity’, a ‘nature’, from the category they are in: different category, different identity. Outside categories, these objects lose whatever properties and relations they had in the category they came from and they become essentially indiscernible.
One great advantage of CT is that structures, i.e. categories, are not accompanied by all these sets that arise by iterated applications of the power-set and union-set operation. Nevertheless, the grim story we have been telling for Structural Realism in the framework of ZFC, can be repeated in the framework of CT, of course with a few appropriate adjustments.