I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops growing is a non-trivial problem or the problem of determining the ordinal in which $X_{\alpha}$ stops growing is a non-trivial problem.

In particular, I want the tower $(X_{\alpha})_{\alpha}$ to be generated by the following construction. Suppose that for each object $X$, there is a new object $C(X)$ and a morphism $e:X\rightarrow C(X)$. Then define the tower generated by $X$ by letting $X_{0}=X$, $X_{\alpha+1}=C(X_{\alpha})$ and $X_{\gamma}=\varinjlim_{\alpha<\lambda}X_{\alpha}$ for limit ordinals $\gamma$ where the direct limit is taken in the category that $X$ belongs to.

I want all of the objects $X$ and each $X_{\alpha}$ to be set sized.

**Non-Example:** The hierarchy of sets $(V_{\alpha}[X])_{\alpha}$ where
$V_{0}[X]=X,V_{\alpha+1}[X]=P(V_{\alpha}[X])$ and $V_{\gamma}[X]=\bigcup_{\alpha<\gamma}V_{\alpha}[X]$ does not count as an example of what I am looking for since the tower $(V_{\alpha}[X])_{\alpha}$ never stops growing and therefore whether $(V_{\alpha}[X])_{\alpha}$ terminates is now a trivial mathematics problem.

**Example 1:** Suppose that $G$ is a group. Let $G_{0}=G$, and let
$G_{\alpha+1}=\mathrm{Aut}(G_{\alpha})$ and let $G_{\gamma}=\varinjlim_{\alpha<\gamma}G_{\alpha}$. The transition mapping from $G_{\alpha}$ to $G_{\alpha+1}$ is the mapping $e$ where $e(g)(h)=ghg^{-1}$. Then $(G_{\alpha})_{\alpha}$ is the automorphism group tower generated by $G$. The automorphism group tower always terminates. In this case, the mapping $G\mapsto G_{\alpha}$ is not functorial.

**Example 2:** Frames are the objects that people study in point-free topology. If $L$ is a frame, then let $\mathfrak{C}(L)$ denote the lattice of congruences of the frame $L$. Then $\mathfrak{C}(L)$ is always a frame. Define a mapping $e:L\rightarrow\mathfrak{C}(L)$ by letting $(x,y)\in e(a)$ if and only if $x\vee a=y\vee a$. Then the function $e$ is a frame homomorphism.

There are frames $L$ where the congruence tower generated by $L$ never terminates. However, for ordinals $\alpha$, it is a difficult open problem to determine whether there is a frame $L$ where $e:L_{\beta}\rightarrow\mathfrak{C}(L_{\beta})$ is a surjection if and only if $\beta\geq\alpha$ since in all known examples, the congruence tower either terminates before the fourth step or so or it never terminates.

If $L$ is a frame and $e:L\rightarrow\mathfrak{C}(L)$ is an isomorphism, then $L$ is a complete Boolean algebra.

Unlike Example 1, Example 2 is functorial.

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