In the book Category theory by Awodey, a category can have proper classes as objects and arrows. Then he has

Definition 1.11. A category is small if both the collection of objects and the collection of arrows are sets.

Right below the definition he claims

For example, all finite categories are clearly small, as is the category $\text{Sets}_{\text{fin}}$ of finite sets and functions. (Actually, one should stipulate that the sets are only built from other finite sets, all the way down, i.e., that they are “hereditarily finite”.)

There is no set of finite sets, since there are the finite sets $\{M\}$ for any set $M$. So the second part of the first sentence is not true, right? Is that what he means by the remark in parenthesis?

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    $\begingroup$ Maybe the meaning is finite sets up to isomorphism; or what the set theorists would call $V_\omega$, the hereditarily finite sets. $\endgroup$ – Asaf Karagila Jan 14 '16 at 15:38
  • $\begingroup$ Aha, that makes a lot of sense, especially given the next sentence below in the book. Let me update the question, then you can answer it for real. $\endgroup$ – Thomas Jan 14 '16 at 16:34

The category of finite sets is not small. However, the category of finite sets is essentially small, i.e., equivalent to a small category. This suffices, for most purposes, to apply results about small categories. Usually, category theoretic stuff is (or, should!) be invariant under equivalences of categories.

The category of hereditarily finite sets is small; Asaf has already mentioned that they all belong to the set $V_{\omega}$ in the Von Neumann hierarchy.


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