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.