Is there a category for which the Cantor set is the initial object?

Question

It is well known that every compact metric space is the continuous image (or quotient) of a Cantor space (i.e. a totally disconnected perfect compact metric space). Furthermore, Cantor sets are all homeomorphic. For this reason it is tempting to think of "the" Cantor set as the initial object in the category of compact metric spaces, but the issue with this is that quotient maps from the Cantor set to a compact metric space are not unique. So my question is: is there a category where the objects are compact metric spaces, and the maps are such that the Cantor set is actually the initial object?

Some ideas:

• There is a canonical way to produce a totally disconnected space from any topological space $X$: consider the complete Boolean algebra of regular open sets $RO(X)$, and by Stone duality $RO(X)$ is the Boolean algebra of clopen sets for a totally disconnected compact Hausdorff space $S$ whose points are the ultrafilters in $RO(X)$. However, the space $S$ may not be a Cantor set since it may have isolated points, and it is not clear to me how we would put a restriction on continuous maps so that $S$ constructed this way has a unique map to $X$.

• In topological dynamical systems there is the concept of a symbolic representation, which is a certain encoding of a dynamical system as a shift space. More precisely, for a topological dynamical system $(X,\varphi)$ with $\varphi:X \to X$ a homeomorphism, and $\mathcal{P}$ a finite topological partition of $X$, we get a map $\pi : \Sigma \to X$ where $\Sigma \subseteq |\mathcal{P}|^\mathbb{Z}$ is the subset such that $\pi(s) = \bigcap_{n \in \mathbb{Z}} \varphi^{-n}(P_{s_n})$ is a well-defined map. Note that $\Sigma$ is a subspace of the Cantor set $|\mathcal{P}|^\mathbb{Z}$. For some classes of dynamical systems, such as expansive ones, this map can be made to be more or less unique. However, I am only aware of these symbolic representations working (i.e. the map $\pi$ is onto and finite-to-one) for certain expansive dynamical systems, and there are compact metric spaces such as the Hilbert cube which do not admit an expansive homeomorphism. Also this highly depends on the choice of partition (although there are nice choices).

It seems like one way we might make such a category is if the objects of the category are compact metric spaces which have attached to them some canonical self-homeomorphism (along with a partition), and the maps in the category would be continuous surjections which intertwine these self-homeomorphisms. Then there would be a unique map from the Cantor set (with the shift homeomorphism) which is the object's "symbolic representation" as described above. But it is not clear to me if there is an immediate counter-example to this approach or if it is tractable in the first place.

Answer 1

Not sure whether this is exactly what you want, but one type of solution would be the following (which I will motivate first):

Why is $\mathbb{Z}$ the initial object in the category of rings? Well, because the category of rings is the category of $\mathbb{Z}$-algebras. That means that an object in this category is given by a ring $R$ together with a structure morphism $\mathbb{Z} \rightarrow R$ and morphisms are morphisms of rings that respect the structure morphisms.

Analogously, you can define a category whose objects are topological spaces/compact metric spaces equipped with a structure morphism from the cantor space etc. Since the structure morphisms have to be respected, this will make the cantor space initial.

These types of categories are called coslice categories in case that you want to read more about them.

Edit: Here is the stuff mentioned in the comments. Denote by $C$ the cantor space and let $X$ and $Y$ be compact metric spaces. Note that the cantor space as object of the above category is the identity $\text{id}_C \colon C \rightarrow C$.

A morphism $f \colon X \rightarrow Y$ in this category is a commutative diagram $$\require{AMScd} \begin{CD} C @>{\varphi_X}>> X \\ @VV{\text{id}_C}V @VV{f}V \\ C@>{\varphi_Y}>> Y \end{CD} $$ where $\varphi_X$ and $\varphi_Y$ are the structure morphisms. If we now choose $X = C$, i.e. the identity on $C$, we get

$$\require{AMScd} \begin{CD} C @>{\text{id}_C}>> C \\ @VV{\text{id}_C}V @VV{f}V \\ C@>{\varphi_Y}>> Y \end{CD} $$

which forces $f$ to be the structure morphism $\varphi_Y$. Hence the identity $\text{id}_C \colon C \rightarrow C$ is initial.

This proof does of course not depend on our specific category, but rather work for any coslice category and the dual proof for any slice category.

Answer 2

I'm going to give you an answer not really understanding the nuances of your question, but which I think may make concrete for you some of the concepts you describe, and add some insight to the notion of morphisms between dynamical systems. I'd be really interested to hear your comments on my answer, if you read it - in particular, which of the concepts herein correspond with the concepts you write in your question, and whether this much was already obvious to you.

The short answer is... that the Cantor set is the canonical object which corresponds with the standard binary representation of the 2-adic numbers. And as such, I believe that makes it canonical and initial in the sense that it has the most simple or natural commutative algebra over its ring. It has a simple commutative ring algebra and a total order which places its unit at its centre, its zero at one end and minus one at the other.

Here's why:

It sounds from your question like you have probably encountered the Dyadic transformation, which is a canonical chaotic dynamical system that acts on the unit inverval by truncating a number's binary representation from the left.

Now if we ignore the matter of whether we write with ones or twos, the Cantor set is essentially the dyadic rationals, but without the equivalence of the binary strings $...1\overline0_2$ and $...0\overline1_2$. It should be obvious that one can create a virtually identical function which deletes digits from the Cantor set. This function is almost exactly topologically conjugate to the dyadic transformation, except that it has a duplicate copy of the dyadic transformation's eventually fixed transitive orbit to which all the dyadic rationals converge.

This additional orbit is the one terminating in all ones, rather than all zeroes, and one might think of it as the right-limits of the dyadic rationals in $[0,1]$ rather than their left-limits. Let's call this the "enhanced" dyadic transformation. The actual dyadic transformation is a lot like taking the "enhanced" dyadic transformation and glueing the ends of the unit interval together, along with the two end points of each line segment which was removed to create the Cantor set.

Now if you reverse all these binary strings and write them with ones and zeroes rather than twos, you have an exact cover of the 2-adic integers. And this string reversal topologically conjugates the "enhanced" dyadic transformation to $\Bbb Z_2$.

In this domain, the transformation is called the bit shift map and now we're deleting characters from the right instead of the left, so it looks like $x\mapsto x-2^{\nu_2(x)}$, or if you prefer $x\mapsto (x-1)/2$ for odd $x$ and $x\mapsto x/2$ for even $x$. Now it turns out this bit shift map is topologically conjugate to the Collatz conjecture over $\Bbb Z_2$. That's a theorem given in Bernstein and Lagarias, 2019.

What's more, for any pair of 2-adic units $a,b$, the bit shift map is topologically conjugate to $x\mapsto ax-b\cdot2^{\nu_2(x)}$. This means that for exery pair $a,b$ you get a different exact cover of the 2-adic numbers. And when you change $a,b$ from $1,-1$ to something else, you get an (isometric) homeomorphism on $\Bbb Z_2$ which acts by conjugacy on the affine group $\textrm{Aff}(\Bbb Z_2)$. Let's call that homeomorphism $\phi_{a,b}(x)$.

For example $\phi_{3,1}(1)=-\frac13$. The binary representation of $-\frac13$ is alternating ones and zeroes, and this is essentially stating that the number $1$ has an immediately periodic Collatz sequence which will alternate between odd and even ad infinitum.

Each of these new representations of $\Bbb Z_2$ has its own additive and multiplicative ring algebra which involves pushing the number's binary string through $\phi_{a,b}(x)$, adding or multiplying as appropriate, then pushing back through the inverse homeomorphism. But for the original representation of $\Bbb Z_2$, the one in which the Bit Shift map is topologically conjugate to the enhanced dyadic transformation, the homeomorphism $\phi_{1,1}(x)$ which you have to push your binary strings through, is the identity homeomorphism. Therefore the Cantor set conjugates by string reversal directly to the simple ring algebra we know so well on binary numbers. In that sense therefore, I would say it is an initial obect, in a sense which is a little richer than simply to say other spaces are defined by being homeomorphic to it.