When An Abelian Category Has Enough Flat Objects?

Can you provide me an example of two objects that are different via set theory but are equal via category theory?

• I've read the following: The characteristic aspect of a category theory is that all constructions of this theory are provided in the language of diagrams, consisting of appropriate morphisms between given objects. In this sense, the concept of a mathematical structure as a ‘set of elements equipped with some properties’ is not fundamental. The category-theoretic proofs are provided by showing the commutativity of diagrams, and usually involve such structural concepts as functors between categories, natural transformations between functors, as well as limits and adjunctions of functors, what has to be contrasted with the structure-ignorant methods of set-theoretic formalism, based on proving the equality between the elements of sets. From this, it seems that set theory ignores structures. So I presume two things with different structures in category theory could be equal in set theory.

• Answer:

  You meant it the other way around. You're looking for two objects that are the same as sets but different as objects in some other category. There are countless examples. The set {1,2,3,4} can be made into a group in two different ways (one cyclic, one not), so here are two distinct groups whose underlying sets are the same. The real line is a topological space which, as a set, is identical to the real numbers with the discrete topology. And so on. But I also think you're a little confused about the terminology. Category theory is not some alternative to set theory. The objects of various categories (not all) are sets with some extra structure, but you don't need the notion of a category or the idea of "working without elements" to see that there are multiple algebraic or geometric objects with the same underlying sets. And you wouldn't call these "objects that are the same via set theory but different via category theory". You'd say that these are two distinct objects (in some category) that are enriched versions of the same set.