I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is usually found in "structuralist contexts" (find some examples in the appendix) and can be circumscribed by slogans such as:
“[...] objects [are] determined by the network of relationships they enjoy with all the other objects of their species.”
Barry Mazur, When is one thing equal to some other thing?
Objects are determined by their position in their network of relationships.
Categorically inclined mathematicians tend to refer to Yoneda's lemma, and I do estimate this a lot, but it seems to neglect some facets of the idea. For example, category theorists don't seem to be very interested in conjugate objects, only in isomorphic ones (see here).
I am especially interested in the question whether and how the very concept of (position in a) network of relationships can be "entified" generally - just like many "concepts" that determine other entities are entities by themselves.
Some examples of "structuralist contexts" from Wikipedia: structuralism (in general), structuralism in the philosophy of science, structuralism in linguistics, structuralism in biology, structuralism in psychology, structuralism in sociology and last but not least: structuralism in the philosophy of mathematics.