I think usually a model of a theory is a translation to a set ( for example, models of group theory are structured sets like $(\mathbb{R}, +)$, $(\mathbb{Z}, +)$,… etc.). But a collection of all sets, or the type Set, is not a set. Is the category of sets a model of category theory?
2026-04-03 00:04:03.1775174643
Is the category of sets a model of category theory?
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I think there is some confusion in terminology here. I would not call a group a "model of group theory". It is a "model of the (first order) theory of groups". On the other hand, the universe of all sets is a model of set theory, modulo some issues about size and expressive power. The category of all sets is, modulo the same issues, a category in particular and is a model of the (first order) theory of categories. But I would definitely not say it is a "model of category theory". That should be reserved for the 2-category of all categories.