I've recently started reading Categories for the Working Mathematician and I'm a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like $\textbf{Ens}$ is supposed to be a full subcategory of $\textbf{Set}$:
If $V$ is any set of sets, we take $\textbf{Ens}_V$ to be the category with objects all sets $X \in V$, arrows all functions $f : X \to Y$, with the usual composition of functions. By $\textbf{Ens}$ we mean any one of these categories.
And on the other hand it seems like $\textbf{Ens}$ can be bigger than $\textbf{Set}$:
$D$ must have small hom-sets if these functors are to be defined [...]. For larger $D$, the Yoneda lemmas remain valid if $\textbf{Set}$ is replaced by any category $\textbf{Ens}$ [...] provided of course that $D$ has hom-sets which are objects in $\textbf{Ens}$.
I suspect my confusion is arising from a misinterpretation of what $\textbf{Set}$ is and what it is used for. Mac Lane seems to make a point of building $\textbf{Set}$ using a fixed universal set $U$, but is there any benefit to doing this instead of simply taking $\textbf{Set}$ to be his “metacategory” of all sets? Indeed, set-theoretically, can such a universal set exist? Mac Lane requires the following properties:
- $x \in u \in U$ implies $x \in U$,
- $u \in U$ and $v \in U$ implies $\{ u, v \}$, $\langle u, v \rangle$, and $u \times v \in U$.
- $x \in U$ implies $\mathscr{P} x \in U$ and $\bigcup x \in U$,
- $\omega \in U$ (here $\omega = \{ 0, 1, 2, \ldots \}$ is the set of all finite ordinals),
- if $f : a \to b$ is a surjective function with $a \in U$ and $b \subset U$, then $b \in U$.
As far as I can tell, this is essentially a transitive inner set-model, but isn't the existence of such a thing unprovable in ZFC? I haven't gotten very far through the book yet, but it seems to me that there is no loss in reading “small set” as “any set” and non-small sets as proper classes...