In each case, the second definition is more flexible and supports the universe polymorphism Mike wrote about at the n-café here. In fact, I think the best definition of a large category is to just take a small category with respect to a larger universe. While revising my book, I found that this definition simplifies a lot of otherwise delicate size issues throughout category theory. For example, functor categories for small categories are well-defined, and hence they are also well-defined for large categories.

As for locally smallness, there is actually a third notion. But to distinguish it I will use a different name.

A set is small when it belongs to the fixed universe $\mathcal{U}$. A set is called essentially small when it is isomorphic to a small set. A category is essentially locally small when its hom-sets are essentially small.

The property of being essentially locally small obeys the principle of equivalence, whereas locally small doesn't. A concrete example is the category of functors from $\mathbf{FinSet}_{\cong} \to \mathbf{FinSet}_{\cong}$, which is the category of combinatorial species. This category is not locally small in the usual sense, but of course it is essentially locally small, and in practice this is the only thing that matters.

We might as well redefine the notion of smallness to make it invariant under isomorphisms. This is perhaps not best for set theoretic concerns, but will align with the principle of equivalence in category theory.

In each case, the second definition is more flexible and supports the universe polymorphism Mike wrote about at the n-café here. In fact, I think the best definition of a large category is to just take a small category with respect to a larger universe. While revising my book, I found that this definition simplifies a lot of otherwise delicate size issues throughout category theory. For example, functor categories for small categories are well-defined, and hence they are also well-defined for large categories.

As for locally smallness, there is actually a third notion. But to distinguish it I will use a different name.

A set is small when it belongs to the fixed universe $\mathcal{U}$. A set is called essentially small when it is isomorphic to a small set. A category is essentially locally small when its hom-sets are essentially small.

The property of being essentially locally small obeys the principle of equivalence, whereas locally small doesn't. A concrete example appearing in practice is the category of functors from $\mathbf{FinSet}_{\cong}$ to $\mathbf{FinSet}_{\cong}$, which is the category of combinatorial species. This category is not locally small in the usual sense, but of course it is essentially locally small, and in practice this is the only thing that matters.

We might as well redefine the notion of smallness to make it invariant under isomorphisms. This is perhaps not best for set theoretic concerns, but will align with the principle of equivalence in category theory.

In each case, the second definition is more flexible and supports the universe polymorphism Mike wrote about at the n-café here. In fact, I think the best definition of a large category is just take a small category with respect to a larger universe. While revising my book, I found that this definition simplifies a lot of otherwise delicate questions throughout category theory.

Just to make this even more confusing here is a third notion of locally smallness. But to distinguish it I will use a different name.

A set is small when it belongs to the fixed universe $\mathcal{U}$. A set is called essentially small when it is isomorphic to a small set. A category is essentially locally small when its hom-sets are essentially small.

Then essentially locally small categories obey the principle of equivalence, whereas locally small categories don't. A concrete example is the category of functors from $\mathbf{FinSet}_{\cong} \to \mathbf{FinSet}_{\cong}$, which is the category of combinatorial species. This category is not locally small in the usual sense, but of course it is essentially locally small, and in practice this is the only thing that matters.

We might as well redefine the notion of smallness to make it invariant under isomorphisms. This is perhaps not best for set theoretic concerns, but will align with the principle of equivalence in category theory.

In each case, the second definition is more flexible and supports the universe polymorphism Mike wrote about at the n-café here. In fact, I think the best definition of a large category is to just take a small category with respect to a larger universe. While revising my book, I found that this definition simplifies a lot of otherwise delicate size issues throughout category theory. For example, functor categories for small categories are well-defined, and hence they are also well-defined for large categories.

As for locally smallness, there is actually a third notion. But to distinguish it I will use a different name.

A set is small when it belongs to the fixed universe $\mathcal{U}$. A set is called essentially small when it is isomorphic to a small set. A category is essentially locally small when its hom-sets are essentially small.

The property of being essentially locally small obeys the principle of equivalence, whereas locally small doesn't. A concrete example is the category of functors from $\mathbf{FinSet}_{\cong} \to \mathbf{FinSet}_{\cong}$, which is the category of combinatorial species. This category is not locally small in the usual sense, but of course it is essentially locally small, and in practice this is the only thing that matters.

We might as well redefine the notion of smallness to make it invariant under isomorphisms. This is perhaps not best for set theoretic concerns, but will align with the principle of equivalence in category theory.