Examples of transfinite towers

Question

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.

Answer 1

Consider the following construction of sets of ordinals.

• $X_0=\{0\}$,

• $X_{\alpha+1}=$ the closure of $X_{\alpha}$ under $\gamma\mapsto\gamma+1$ and under countable sums,

• $X_\alpha=\bigcup_{\beta<\alpha}X_\beta$ for $\alpha$ limit.

We can consider in ZF the question of whether this tower stabilizes.

It is definitely consistent that the answer is yes, since in ZFC $\omega_1$ is regular, so $X_\beta=\omega_1$ for all $\beta\geq\omega_1$.

The question of whether this tower can be non-stabilizing is much harder, and is equivalent to asking whether it is consistent that all ordinals have countable cofinality. This was shown to be consistent by Gitik in 1980, assuming consistency of a proper class of strongly compact cardinals. Let me note that large cardinals cannot be avoided, because even making $\omega_1$ and $\omega_2$ singular has consistency at least as high as $0^\#$.

Answer 2

The following is an example where is it difficult to identify where stabilization occurs. It meets all of your technical criteria but I suppose the word "growing" is not exactly accurate.

A quasitopological group $G$ is a group with a topology $\mathcal{T}$ such that inverse $g\mapsto g^{-1}$ is continuous and multiplication $\mu:G\times G\to G$ is continuous in each variable. One can "efficiently" give the underlying group of $G$ an actual topological group structure by removing the smallest number of open sets from $\mathcal{T}$ until one arrives at a topological group $\tau(G)$. This can be understood precisely in at least two equivalent ways:

1. Give $G$ the finest group topology coarser than $\mathcal{T}$, which one can show exists abstractly (the difficulty is with the product topology on $G\times G$).
2. The inclusion $\mathbf{TopGrp}\to \mathbf{qTopGrp}$ of the full subcategory of topological groups into the category of quasitopological groups and continuous homomorphisms has a left adjoint (reflection) $\tau:\mathbf{qTopGrp}\to \mathbf{TopGrp}$, which exists by the adjoint functor theorem.

The construction of $\tau(G)$ from $G$ is purely abstract so to really get your hands on something you can work with (e.g. to prove $G$ and $\tau(G)$ in fact have the same open subgroups) you approximate it inductively by a shrinking sequence of quotient topologies.

Let $c(G)$ be the underlying group of $G$ with the quotient topology with respect to multiplication $m:G\times G\to c(G)$. While $c(G)$ is not necessarily a topological group, it is a quasitopological group and indeed, one can show that if $c$ is the identity on underlying homomorphisms, $c:\mathbf{qTopGrp}\to\mathbf{qTopGrp}$ becomes a functor.

It is clear that the identity homomorphism $G\to c(G)$ is continuous and $G=c(G)$ if and only if $G$ is already topological group. Hence one inductively defines $c_0(G)=G$, $c_{\alpha+1}(G)=c_{\alpha}(G)$ and if $\alpha$ is a limit ordinal, the topology of $c_{\alpha}(G)$ is the intersection of the topologies of the groups $c_{\beta}(G)$, $\beta<\alpha$.

By basic set-theoretic arguments, $c_{\alpha}(G)$ stabilizes to the topological group $\tau(G)$ and since quotient topologies are used one has $c_{\gamma}(G)=\lim_{\alpha<\gamma}c_{\alpha}(G)$ in $\mathbf{qTopGrp}$.

However, for a given quasitopological group $G$ it is not clear at all when the inductive sequence $c_{\alpha}(G)$ first stabilizes.

Motivation for this example:

1. Free topological groups are important in topological group theory and arise in algebraic topology as well. Given a space $X$, one might attempt to create the free topological group on $X$ viewing the free group $F(X)$ as the quotient space of the free topological semigroup $\coprod_{n\geq 1}(X\cup X^{-1})^n$ with respect to word reduction; however, without some compactness assumptions, the resulting object $F_q(X)$ is only a quasitopological group. To construct the free topological group one must apply the inductive construction above and this specific use of $c$ is sometimes called the Mal'tsev transfite process. A quote from pg. 5799 of The topology of free topological groups by O. Sipacheva:

Certainly, such an approach to constructing the free group topology looks very natural. However, it is extremely difficult to trace the change of the topology at each step or at least understand at what point the topology stabilizes.

Apparently, the only known results are those where stabilization occurs after the first step.

2. The fundamental group $\pi_1(X,x)$ with it's natural quotient topology is not a topological group (same for higher homotopy groups), but it is a quasitopological group. There are plenty of interesting things about quasitopological $\pi_1$; however, one may turn $\pi_1$ naturally into a topological group (reflecting many classical results to the topological group category) by applying $\tau$. See this paper for more on topological $\pi_1$ and a detailed treatment on the reflection functor $c$. The use of the transfinite sequence is needed in the classification of certain generalized covering maps. I would, personally, like to know when the sequence stabilizes even for simple Peano continua $X$.

Similar constructions are used to build other universal objects in topological algebra.