I'm trying to pin down a notion of inductive definability in category-theoretic terms.

The sorts of inductively defined sets (and classes) I'm most interested in are those that admit of induction and recursion. So far, I've noticed that (weakly?) initial algebras of an endofunctor do a pretty good job of this.

We can capture natural number objects (and therefore $\mathbb{N}$) and other inductive types this way. But what about things like the constructive ordinals (or number classes) or even segments of the cumulative hierarchy?

I want to exclude things like the classical continuum, but not through cardinality, since the cumulative hierarchy is certainly quite large. And polynomial functors don't seem to cut it since the powerset functor isn't polynomial and doesn't have a smallest fixed point (and thus not an initial algebra). Should I restrict the powerset functor a certain way to get one? Based on inaccessible cardinals?

**Is there a unified characterization of 'inductive' in category theory that would apply to these sorts of cases?**