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Stream: learning: questions

Topic: mistake in Borceux??

Joshua Meyers (Dec 19 2021 at 21:48):

In Borceux's Handbook of Categorical Algebra, vol. 1, Def. 1.1.2 is that a universe is a set $\mathcal{U}$ such that

(1) $x\in y\textrm{ and } y\in\mathcal{U}\ \Rightarrow \ x \in\mathcal{U}$
(2) $I\in\mathcal{U}\text{ and }\forall i\in I\ x_i\in\mathcal{U}\ \Rightarrow\ \bigcup_{i\in I}x_i\in\mathcal{U}$
(3) $x\in\mathcal{U} \ \Rightarrow\ \mathcal{P}(x)\in\mathcal{U}$
(4) $x\in\mathcal{U}\text{ and }f:x\to y\text{ surjective}\ \Rightarrow\ y\in\mathcal{U}$
(5) $\mathbb{N}\in\mathcal{U}$

But these seem to imply that every set is in $\mathcal{U}$!. In fact, for any set $x$, there is a surjective function $\mathbb{N}\to\{x\}$, so by (4) and (5), $\{x\}\in\mathcal{U}$, so by (1), $x\in\mathcal{U}$.

What is going on here?

Zhen Lin Low (Dec 19 2021 at 22:05):

I think the missing assumption is $y \subseteq \mathcal{U}$.

Mike Shulman (Dec 19 2021 at 23:51):

That would be a missing assumption in condition (4), to be precise.