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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:

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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!2012-01-13
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    @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?2012-01-14
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    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.2012-01-16
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    But what *are* these names? Who attached them to the objects?2012-01-16
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    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".2012-01-16

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