Stabilization of taking presheaf categories

Question

Does the process of 'constructing the category of presheaves' always/never stabilize? Does it stabilize for some special class of categories?

That is, work in a foundation that allows for multiple levels of 'categorical largeness' and let $\mathcal{C}$ be a category. Recursively define

1. $\mathcal{C}_0=\mathcal{C}$,
2. $\mathcal{C}_{\alpha+1}={\bf Set}^{\mathcal{C}_\alpha^{op}}$,
3. $\mathcal{C}_\lambda=\varinjlim_{\alpha<\lambda}\mathcal{C}_\alpha$.

Does there always/never exist some $\alpha\in O_n$ such that $$\alpha\leq\beta\implies \mathcal{C}_\alpha\simeq\mathcal{C}_\beta?$$

I tried playing around with the case $\mathcal{C}={\bf Set}$ and it seems like we don't have the above property for finite $\alpha$, since ${\bf Set}$ is locally small and ${\bf Set}^{{\bf Set}^{op}}$ is not locally small (natural transformations are proper-class sized collections of sets), and higher finite iterations begin sending proper classes to sets etc. -- I don't know if this process stabilizes at $\omega$ or some higher limit ordinal, however.

Further, if we begin with the terminal category ${\bf 1}$ then ${\bf Set}^{{\bf 1}^{op}}\cong{\bf Set}$ so the above sequence repeats itself shifted back one step, and if we begin with the initial category ${\bf 0}$ then ${\bf Set}^{{\bf 0}^{op}}\cong{\bf 1}$, so the above sequence repeats itself shifted back two steps.

Answer 1

The question is extremely dependent on how size issues are handled and many choice that can be made, so that it is very hard to give a general answer.

If you really work with general presheaves categories then I cannot think of any interesting functors $C_i \to C_{i+1}$ that would define an interesting value at limit ordinal. For example the Yoneda embedding is not defined as if $C_0 \neq \varnothing$, $C_1$ will be a large category, so $C_{2}$ will have large hom-sets, and so the Yoneda embeddings $C_2 \to C_3$ will not be defined unless "Set" also become bigger as the process go, in which case the process will obviously never stabilize.

Similarly, the covariant functoriality of the presheaf construction as suggested by Peter Lumsdain in the comment is not available either as it is no longer defined when working with large categories so there are no way to use it. The contravariant functoriality of the presheaf construction is always available, but I do not see how to use it to construct a tower.

Taking the colimit to be a disjoint union ( colimit over a discrete category) is not a viable solution either: the presheaf category are always connected categories (they have initial objects and terminal objects for example) while the coproduct are always disconnected, so the value at a limit ordinal and at a successor ordinal will never agree.

The only way I can think of to make sense of this construction, is to replace the category $Set^{C^{op}}$ by the construction $\widehat{C}$ the free co-completion of $C$. When $C$ is small then $\widehat{C} \simeq Set^{C^{op}}$, when $C$ is locally small then $\widehat{C}$ the full subcategory of $Set^{C^{op}}$ of so called "small presheaves", that is the presheaves that are small colimits of representable. When $C$ has a large hom-set then $\widehat{C}$ is no longer related to $Set^{C^{op}}$, but the advantage of iterating $\widehat{C}$ is that one never gets out of locally small categories.

Here To fix what "small" means, I chose an inaccessible cardinal $\kappa$, and "small" means $\kappa$-small. Interestingly, $\kappa$ doesn't need to be inaccessible, regular is enough.

Once we restrict to small presheaf, then there are at least two interesting way to build such tower. One can use the Yoneda embeddings, or one can as suggested by Peter Lumsdaine, or one can start with some functor $C_0 \to C_1$ and then use the co-variant functionality of $\widehat{C}$ to build functor $C_1 \to C_2$ etc...

I claim that the process "stops" at the ordinal $\kappa$, in the sense that $C_\kappa \simeq C_{\kappa+1}$ and it never stops before. (so if you only induce on small ordinal it never stops).

To be clear, the Yoneda embedding $C \to \widehat{C}$ is never an equivalence (for example, the initial object is never in the essential image), I'm only saying that there will be an equivalence of categories $C_\kappa \simeq \widehat{C_\kappa}$ other than the Yoneda embeddings. It is not completely clear to me at this point that all subsequent values of the sequence are also equivalent to $C_\kappa$ (it is clear for the $\kappa+n$ of course, but I don't quite know what happen at $\kappa+\omega$ yet).

If one use Peter Lumsdaine's version then the process also clearly stops at the ordinal $\kappa$ (and this time really stay at the value it had for $\kappa$). I haven't thought to this specific example enough, but it is likely that one can show that it never stabilize before $\kappa$.

If one really wants to keep the construction $Fun(C^{op}, Set)$ without restricting to small presheaves, and one somehow find a way to make it into an interesting tower (which I don't think is possible), then it will very probably never converge:

Indeed, it is for example easy to show that there is no category $C$ such that $Fun(C,Set) \simeq C$ using (a 2-categorical version of) Lawvere fix-point theorem and the fact that $Set$ has some endofunctors without no fix-point (for example the covariant power-set endofunctor).

I suspect one can adjust this argument to show that there is no category $C$ such that $C \simeq Fun(C^{op},Set)$ as well, but I can't quite find the argument right now. This would clearly show that no variant of the construction with $Fun(C^{op},Set)$ would ever converge.