# Foundation of category theory

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?

• – Martin Brandenburg Aug 28 '13 at 13:50
• That first link is really excellent! – rschwieb Aug 28 '13 at 13:57
• 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. – Zhen Lin Aug 28 '13 at 14:19
• There are more axioms in Zermelo Fraenkel and they say different things than axioms of categories. – Noix07 Aug 28 '13 at 15:25
• 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. – Zhen Lin Aug 29 '13 at 7:16

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.