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Let me quote Barry Mazur from his essay When is one thing equal to some other thing?:

“mathematical objects [are] determined by the network of relationships they enjoy with all the other objects of their species”

I always took this slogan literally, but recently I was overcome by doubts whether I missed something.

Have a look at the toy category of graphs over two fixed vertices and an arrow whenever there is a graph homomorphism. Compositions and identities are omitted:

enter image description here

The numbers are derived from the adjacency matrices: 0 = 00|00, 1 = 10|00, ..., 15 = 11|11.

Among others, the graphs #6 and #9 are not isomorphic (i.e. "essentially the same") but indistinguishable with respect to the network of relationships they enjoy (i.e. conjugate):

graph #6 graph #6 graph #9 graph #9

What is the grain of salt I have to take Mazur's slogan with? Or is there something wrong with my toy category?

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He is almost surely alluding to the Yoneda embedding when he says that. But the Yoneda embedding does not forget which object is which! – Zhen Lin Jan 13 '12 at 11:47
@Zhen: It's easy to (believe to) understand Mazur's slogan, and it's easy to understand my counterexample. But has it to be clear to everyone (who understands Yoneda's embedding) that it does not forget about the identity of its objects, resp. what this means exactly? – Hans Stricker Jan 14 '12 at 23:30
Indeed it is clear to anyone who understands the Yoneda embedding what I mean – so I can only advise you to go understand the Yoneda embedding! The point is that we are allowed to forget the internal structure of objects, but we are not allowed to forget the "names" of objects, and these "names" are absolute with respect to the category, rather than relative with respect to each object. – Zhen Lin Jan 16 '12 at 12:36
But what are these names? Who attached them to the objects? – Hans Stricker Jan 16 '12 at 14:08
You're thinking about this too much. It's very simple. In your example, the names of the objects are "1", "2", etc. The reason why "6" and "9" are not considered to have the same "network of relationships" is because, for example, there is a morphism from "4" to "6" but not from "4" to "9". – Zhen Lin Jan 16 '12 at 16:39

I think the point is that there are two kind of equivalences between objects in a category: the first one is the relation of isomorphism between objects in the category, the second is the relation of conjugation through a given functor (here I'm using the notation of this thread).

It seems to me that the difference between these two kind of equivalences becomes clear if you see categories from a logical point of view: you can think a category as a theory in which the objects are the sorts of the theory (which can be considered set-types for the theory or unary predicates) while the morphisms can be seen as relations (or binary predicates) between this sorts. From this point of view functors are just interpretations of one theory into another one: they send sorts of one theory in sorts of the other one and relations between two sorts of the first theory in relations between the corresponding sorts of the second theory.

(Note:) categories have to be regarded as theories without identity.

From this point of view isomorphic objects are sorts which can be proven to be equivalent in the theory, that means that exists a proof that every given sentence that holds for one sort holds also for the other sort. Because categories are theories without identity isomorphism is the only way that we have to distinguish sorts (objects) into the theory (category).

Conjugation instead is a different relation, it says that two given objects can be seen as the same via a suitable invertible interpretation of theories: that means that up to rename sorts and relations every sentence that holds for one sort also holds for the corresponding sort. Conjugation of two sorts says that we can reinterpret each sort in the other without changing the logic of the theory.

So while the first relation says something just about the theory the second one is a relation the deals with the theory and an interpretation of the theory in itself. They're both meta-relations, meaning that they say something about theories, but they says different things at logical level.

I hope this answer could help you.

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