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The following is a statement from Categories for the Working Mathematician by Saunders Mac Lane (page 10) :

A category will mean any interpretation of the category axioms within set theory.

Will someone kindly explain what the author means by the phrase 'within set theory'? Is there something beyond or in the complement of it. Thank you for your help.

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There are alternatives such as type theory. You could also simply consider the formal first-order theory of categories itself and its extensions without considering (set-theoretic) models. If you wanna get real fancy, you can consider models of that theory in any category with finite limits which leads to the notion of an internal category. From this last perspective, the category of ($V$-small) sets is but one category that has finite limits. Internal categories in the category of ($V$-small) sets are categories.

It's worth noting that the set-theoretic foundations presented in Categories for the Working Mathematician already go beyond ZFC set theory. If I remember correctly, the set theory sketched is essentially Tarski-Grothendieck set theory (though it may just assume some Grothendieck universes and not something equivalent to Tarski's axiom).

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  • $\begingroup$ I don't think that answers the question. I think the question asks about the literal meaning of the sentence, whereas you just say "Oh, we can do other things too!". $\endgroup$
    – Asaf Karagila ♦
    Jun 4, 2018 at 15:04
  • $\begingroup$ I think it respond's to the OP's question, "Is there something beyond or in the complement of it?" $\endgroup$
    – saulspatz
    Jun 4, 2018 at 15:09
  • $\begingroup$ @AsafKaragila Asking for things "beyond set theory" makes me think that the question isn't "what is 'an interpretation in set theory'", i.e. a model of an essentially algebraic theory in this case. To me, the question sounds like "what else could it possibly be but a set-theoretic model?" I could easily be wrong though. $\endgroup$
    – Derek Elkins left SE
    Jun 4, 2018 at 15:10
  • $\begingroup$ No, the point is that when I tell you that the dinner will be serve "within the kitchen", you might be wondering what is else is in the house, and why aren't we eating in the dining room? $\endgroup$
    – Asaf Karagila ♦
    Jun 4, 2018 at 15:11
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    $\begingroup$ @Temari: Are you looking to understand the foundations of category theory? of "practical category theory"? of mathematics? Your clarification to the question should not be part of a comment underneath an answer, it should be part of the question. $\endgroup$
    – Asaf Karagila ♦
    Jun 4, 2018 at 17:54

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