Genus-Differentia and Mathematical Categories

I am a mathematician by training. Category theory has become a major subfield of mathematics --- major enough that some have tried to recast the logical foundations of mathematics in terms of categories. A category consists of a collection of objects, and then for each pair of objects, a set of morphisms between the pair. Objects and morphisms are different sorts of primitive notions in this setting.

I have been thinking about types of definitions in mathematics, and in particular how category theory might help clarify these types. We might consider the objects of a category to be a "genus", and then single out certain types of these objects through differentia. However, often morphisms refer to relationships between objects, and mathematicians often consider these relationships to be more important than the objects themselves (I guess reflecting a structuralist point of view?) Thus, unlike set theory, relationships and functions are viewed as their own primitive type rather than as something constructed in an (artificial-seeming, to some) manner from the more primitive sets.

Is there a philosophical system of definition that extends the genus-differentia idea to include relationships between the objects of a differentia to be of a different sort outside the genus-differentia model? One could consider "relationship" to be its own genus, but it feels more natural to me to view relationships between objects not as another type of object themselves but as their own primitive sort, and this strikes me as seeming more in the spirit of category theory.

• When you say that this is unlike set theory... you mean that s.t. is a logical positivism and category theory is not? I'm not sure whether I see your point here. – Tames Jun 20 '12 at 12:55
• Well, I just looked up the Wikipedia page on "logical positivism". I am not asking about mathematical ontology. My question is much more "fluffy". I just feel like the genus-differentia schema for definitions is the type of definition I see most in mathematics, but that it seems limited --- that relationships between things in the genus, "morphisms" if you will (or collections of them), are most naturally given separate status. Setting them in a separate genus, called "relationship", moves them conceptually too far away from the original genus in which the relationship occurs. – Barry Smith Jun 20 '12 at 13:32
• Perhaps you refer to my sentence, "...unlike set theory, relationships and functions are viewed as their own primitive type rather than as something constructed in an (artificial-seeming, to some) manner from the more primitive sets." I confess my mathematical training has never before led me into the foundations of mathematics; this is my understanding of the categorical approach to foundations of mathematics, but correct me if I'm wrong. I have heard these foundations described as a many-sorted logic, and I wonder if an analogous conception has been used in understanding "definition". – Barry Smith Jun 20 '12 at 13:39
• I know this is not the focus of your question, I just wanted to clarify this difference between C.T and S.T. that you indicated here – Tames Jun 20 '12 at 15:25
• @tames: Since Set Theory is based on membership (an element is a member of a set) it is not structuralist, whereas Category Theory being built on Relation is. – Mozibur Ullah Jun 22 '12 at 21:16