Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to $\mathbf{H}_\kappa$. I am interested in properties of $\kappa$ that are expressible in terms of $\mathbf{Set}_{< \kappa}$ (regarded as an abstract category, up to equivalence). For example:
- $\kappa = 0$ if and only if $\mathbf{Set}_{< \kappa} = \emptyset$.
- $\kappa = 1$ if and only if $\mathbf{Set}_{< \kappa}$ is equivalent to the terminal category.
- $\kappa$ is either $1$ or infinite if and only if $\mathbf{Set}_{< \kappa}$ has finite limits.
- $\kappa$ is either $1$ or a strong limit cardinal if and only if $\mathbf{Set}_{< \kappa}$ is cartesian closed.
Question. Is the property "$\kappa$ is regular" of this type?
Answer. Yes. We assume $\kappa$ is infinite, so that $\mathbf{Set}_{< \kappa}$ has finite limits. For each object $X$ in $\mathbf{Set}_{< \kappa}$, let $\Gamma (X)$ be the set of morphisms $1 \to X$ in $\mathbf{Set}_{< \kappa}$, where $1$ is a fixed terminal object in $\mathbf{Set}_{< \kappa}$. Pullback then defines a functor $$(\mathbf{Set}_{< \kappa})_{/ X} \to \prod_{x \in \Gamma (X)} \mathbf{Set}_{< \kappa}$$ and $\kappa$ is regular if and only if this functor is (fully faithful and) essentially surjective on objects for every object $X$ in $\mathbf{Set}_{< \kappa}$.
However, I find the above answer unsatisfactory, for two main reasons:
- It relies on the axiom of replacement – which is perhaps unavoidable because of the nature of the definition of "regular cardinal".
- It also exploits the fact that every set is the disjoint union of its elements.
Is there a better answer?