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In the first pages of "category theory for the working mathematician" Saunders claims that category can be introduced, without set theory, as objects and arrows without some "operations" satisfying some properties which are called axioms. (He calls these metacategories.)

I like that approach, but either I just try to understand things intuitively, either if someone claims that everything is well-defined then I want to see it. What I don't see is the definition of "operation" which "assigns an object to an arrow", ex: the "domain" operation assigns to an arrow an object called domain, same for codomain, composition etc...

More precisely, if we get rid of set theory, then what is a function, what is a pair, what is something that belongs to something?

Another question: is the need to start with other axioms then those of set theory only related to the notion of "class" of objects or is it problematic in other places?

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  • $\begingroup$ That first link is really excellent! $\endgroup$ – rschwieb Aug 28 '13 at 13:57
  • $\begingroup$ Before getting rid of set theory, you should ask, how does set theory itself handle these questions? After all, a universe of sets is a collection of things called "sets" together with a certain binary relation etc. etc. $\endgroup$ – Zhen Lin Aug 28 '13 at 14:19
  • $\begingroup$ There are more axioms in Zermelo Fraenkel and they say different things than axioms of categories. $\endgroup$ – Noix07 Aug 28 '13 at 15:25
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    $\begingroup$ If you are thinking of the first-order theory of categories then you are comparing apples and oranges: models of set theory are universes of sets, models of the theory of categories are categories – not categories of categories. $\endgroup$ – Zhen Lin Aug 29 '13 at 7:16
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I think the situation is that the primitive notions being introduced are "objects," and "arrows," and that the assigment of "domain" and "codomain," "identity," and "composition" are also primitive functions. Along with the axioms about how these interact, this constitutes a suitable environment to do category theory.

There's nothing wrong with having functions that are primitive notions. Remember that $\in$ itself is a primitive relation, and it doesn't (can't!) have any set theoretic justification for its relation-ness.

Does the edition of Categories... you are using have the appendix on these foundations? He does a pretty good job there of explaining how they do their job in the absence of regular set theory.

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  • $\begingroup$ Ok, then I'll take them as primitive notions, but there are many. $\endgroup$ – Noix07 Aug 28 '13 at 14:51
  • $\begingroup$ @user39158 I think I know what you mean :) ZFC is pretty economical with its primitive notions. $\endgroup$ – rschwieb Aug 28 '13 at 14:58
  • $\begingroup$ A question that is difficult to formulate precisely would be, to what extend thoses axioms "define" a category. Also the appendix I have is very short, and doesn't seem to say more much than on the first pages $\endgroup$ – Noix07 Aug 28 '13 at 15:01
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    $\begingroup$ @user39158 I'm not sure I understand that question. The ZFC-free version defines categories without talking about sets, and the interpretation of category theory in ZFC satisfies all the ZFC-free axioms. $\endgroup$ – rschwieb Aug 28 '13 at 15:08
  • $\begingroup$ Ex: what condition on the notion of "pair" is given in the axioms of category theory? This appears in composition, but also in product of category, and tensor product. In the case of set theory, we had a least a condiction like (a,b)=(c,d) implies a=c and b=d. Here no axioms says anything about pair, nor about a function that associates something to something, ex: a functor is a "map" that... Or taking another view point: can one with these primitive notions build what one can with set theory, ex: ordinals. it doesn't look like $\endgroup$ – Noix07 Aug 28 '13 at 15:20

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