# The categories Set and Ens

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:

1. $x \in u \in U$ implies $x \in U$,
2. $u \in U$ and $v \in U$ implies $\{ u, v \}$, $\langle u, v \rangle$, and $u \times v \in U$.
3. $x \in U$ implies $\mathscr{P} x \in U$ and $\bigcup x \in U$,
4. $\omega \in U$ (here $\omega = \{ 0, 1, 2, \ldots \}$ is the set of all finite ordinals),
5. 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...

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Yes, the existence of universes is equivalent to the existence of strongly inaccessible cardinals and unprovable in ZFC. See the Wikipedia article on Grothendieck universes. These are discussed in detail in an article by Bourbaki in SGA 4.1. – t.b. Jun 18 '11 at 10:44

In Categories for the working mathematician, Mac Lane assumes the existence of one Grothendieck universe $U$, and $\mathbf{Set}$ is the category of sets in $U$. This device ensures the existence of functor categories like $[\mathbf{Set}, \mathbf{Set}]$.
On the other hand, $\mathbf{Ens}$ is any full subcategory of the metacategory of all sets, with the restriction that the collection of objects in $\mathbf{Ens}$ is itself a set. Note that $\mathbf{Ens}$ may fail to have the properties expected of $\mathbf{Set}$, e.g. cocompleteness, cartesian closedness, etc.