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?](https://mathoverflow.net/questions/127540/does-induction-for-a-functor-algebra-imply-it-is-initial)) 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?**



Inductive Definitions in Category Theory
It is indeed the case that initial algebras for functors are the category-theoretic manifestation of inductive definitions. There's a whole industry surrounding this idea.


You are more specifically asking about (pieces of) the cumulative hierarchy. On that topic you should look at [algebraic set theory](http://www.phil.cmu.edu/projects/ast/index.html) (AST). A good overview of the subject was written up by Benno van den Berg and Ieke Moerdijk in [arXiv:0710.3066](https://arxiv.org/abs/0710.3066). Do not be put off by all the talk about constructive set theories. They're just *generalizing* the ordinary (classical) set theory to natural cateogry-theoretic settings where excluded middle does not hold.


In AST we study categories of classes. Such categories have a notion of *small map*. The intuition is that a map between classes is "small" if the inverse image of every point is a set (rather than a class). This is a generalisation of the distinction "classes are large, sets are small" because a class $C$ is a set, and only if, the unique map $C \to 1$ is small. Smallness is axiomatised by some fairly straightforward axioms.


Next, to get the cumulative hierarchy, we study *ZF-algebras*. A ZF-algebra is a (large) poset $(P, {\leq})$ which is closed under small suprema and is equipped with a unary operation $s : P \to P$. The *initial ZF-algebra* $(V, {\sqsubseteq}, s)$ can be seen as the cumulative hierarchy, where $\sqsubseteq$ corresponds to subset inclusion and $s$ to the singleton map $x \mapsto \{x\}$. See section 4 of the van den Berg & Moerdijk's notes.


It should be possible to cook up chunks of $V$ by taking variations on a theme. For instance, $V\_\omega$ ought to arise if we require closure under finite suprema (I have not checked this.)


By the way, we *can* get $V$ as the initial algebra of the *small* powerset functor in the category of classes. Let $\mathcal{C}$ be the (large) category of all classes. Define $P : \mathcal{C} \to \mathcal{C}$ by $P(X) = \{S \mid S \subseteq X\}$, i.e., $P(X)$ is the class of all sub**sets** of $X$. Then $V$ is the initial $P$-algebra, isn't it?


