This is the first in a pair of questions. For the other see here.
Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, the behavior of the usual topology and the “condensed” one, align. However, for some quotient spaces, like $\mathbb{R} / \mathbb{Q}$, the usual quotient topology is indiscrete, so every map $f : \mathbb{R} / \mathbb{Q} \to \mathbb{R} / \mathbb{Q}$ is continuous. However, as a condensed set, more structure is preserved. Indeed, the “continuous” maps are exactly those coming from continuous functions $\mathbb{R} \to \mathbb{R}$ which commute with the equivalence relation. *[Edit: I was a bit careless here. The maps actually come from continuous multifunctions. See Arno's answer below.) (See here for more motivation.)
I’m trying to understand condensed sets better and how they relate to topology as it comes up in computable analysis, which I’m quite familiar with. I think the theory of “represented spaces" in computable analysis, is essentially the study of "countably generated condensed sets", and I want to know if anyone has worked this out, or thought about this?
In computable analysis, we want to encode every point $x$ in a space $X$ with a value in Cantor space $2^\omega$. This makes it possible to do computations on the points using computations on $2^\omega$ which are well understood (say via Turing machines). But not only is it a way to do computability theory, it is also a way to talk about continuous functions. (Aside: This latter viewpoint is also a big part of descriptive set theory.)
In computable analysis, a representation of a set $X$ is a partial, but surjective map $\rho : {\subseteq}2^\omega \to X$. (See sections 2 and 3 here.) There are four types of morphisms for the category of representations: (partial/total) (computable/continuous) maps. Let’s just consider total continuous maps even though usually in computable analysis one considers computable maps. Given two sets $X$ and $Y$ with representations $\rho : {\subseteq}2^\omega \to X$ and $\sigma : {\subseteq}2^\omega \to Y$, then a total function $f : X \to Y$ is $(\rho, \sigma)$-continuous if there is a corresponding continuous function $f' : \text{dom}(\rho) \to \text{dom}(\sigma)$ such that $$\rho \circ f = f' \circ \sigma$$
Conversely, it is also not hard to see that any continuous $f' : \text{dom}(\rho) \to \text{dom}(\sigma)$ generates a $(\rho, \sigma)$-continuous function $f$ iff $$\rho(x) = \rho(y) \implies f'(\sigma(x)) = f'(\sigma(x))$$
What we would like, is that if $X$ is a topological space, then there is a representation $\rho : {\subseteq}2^\omega \to X$ such that all partial continuous maps, $f : {\subseteq}2^\omega \to X$ are also $(\text{id}_{2^\omega}, \rho)$-continuous. This is often true, but there are indiscrete quotient spaces like $\mathbb{R} / \mathbb{Q}$ or $2^\omega / \text{fin}$ where this does not hold. For this reason, it is common to only consider admissible representations which have this nice property.
However, it seems to me that actually the category of representations is very close to, if not in some sense equal to, the category of condensed sets.
Let me explain. Representations only use a partial map ${\subseteq}2^\omega \to X$ to represent a space, where as condensed sets use a family of (total) maps $S \to X$ for all profinite sets $S$. A profinite set is just a space homeomorphic to a closed subspace of $2^\kappa$ for some $\kappa$. In particular, the seperable profinite sets are homeomorphic to closed subspaces of $2^\omega$.
If one only uses separable profinite sets in the definition of condensed set in place of profinite sets, let’s call those countably generated condensed sets. I conjecture the category of representations is the same as the category of countably generated condensed sets. Has anyone worked something like this out?
[Edit: This definition of countably generated condensed sets is definitely not what I'm looking for. Restricting the size of the profinite sets doesn't restrict the size of the set $X$.]
I think I've verified the following:
- Every seperable profinite set $S$ has an admissible total representation $\sigma_S : 2^\omega \to S$.
- Using these representations, the generated category of $(\sigma_S, \sigma_{S’})$-continuous functions is equivalent to the category of separable profinite sets with continuous maps.
- For every set $X$ with representation $\rho : {\subseteq}2^\omega \to X$, and every separable profinite set $S$ with admissible representation $\sigma_S : 2^\omega \to S$, one can talk about the set of $(\sigma_S, \rho)$-continuous functions. I believe this set obeys the axioms of condensed sets (which I've included at the end), but restricted to separable profinite sets.
- Conversely, I still need to check that every countably generated condensed set has a representation.
- Also, I need to check that every countably generated condensed set is indeed also a condensed set. (Clausen and Scholze have a notion of $\kappa$-condensed sets, but it is only for uncountable limit cardinals.)
- It should be possible to replace $2^\omega$ with $2^\kappa$ for any cardinal $\kappa$, giving a way to connect $\kappa$-condensed sets and with a generalization of representations, and more formally give a representation-centric definition of all condensed sets. I think everything should carry over nicely.
Anyway, there is more to work out, but I want to see if anyone has thought about this connection before. It is very possible I'm mixing up something subtle or this connection is already well known.
On the other hand, if this does work out, it hints that computable analysis can very easily adapt to condensed sets (and condensed groups/rings/vector spaces/etc.).
Appendix: Axioms of condensed sets
For better or worse, here are the axioms for condensed sets stated in a way that it might be easier for a computable analyst or (non-categorical) logician to understand. (Warning: I may have stripped out too much category theory and forgot a necessary condition.) The "topology" on a condensed set $X$ is given by describing the class of all $T(S,X)$ of all the "continuous maps” (homomorphisms) $S \to X$ for each profinite set $S$. (Recall, again the profinite sets are exactly the closed subspaces of $2^\kappa$ for some cardinal $\kappa$. To avoid proper class issues, Scholze considered $\kappa$-condensed sets for uncountable limit cardinals $\kappa$ where we only use the profinite sets of cardinality less than $\kappa$.)
A condensed set is a set $X$ with a class of sets $T(S,X)$ for every profinite set $S$. Each $T(S,X)$ must satisfy the following axioms:
- For the empty profinite set $\varnothing$, there is exactly one map $\varnothing \to X$ in $T(\varnothing,X)$.
- For the singleton profinite set $*$, the set $T(*,X)$ are exactly the set of maps $f_x : * \to X$ such that $f_x(*) = x$ for all $x \in X$.
- For profinite sets $S_1, S_2$, and their disjoint union $S_1 \sqcup S_2$:
- For every $f_1 : S_1 \to X$ in $T(S_1,X)$ and $f_2 : S_2 \to X$ in $T(S_2,X)$, the map $f : S_1 \sqcup S_2 \to X$ where $f(x) = f_1(x)$ if $x \in S_1$, else $f(z)=f_2(x)$ if $x \in S_2$ is in $T(S_1 \sqcup S_2,X)$.
- Conversely, every map $f : S_1 \sqcup S_2 \to X$ in $T(S_1 \sqcup S_2,X)$ is of this form for maps $f_1$ in $T(S_1,X)$ and $f_2$ in $T(S_2,X)$.
- Let $g : S' \to S$ be continuous and surjective map between profinite sets. For every map $f' : S' \to X$ in $T(S',X)$ such that $g(x) = g(y)$ implies $f'(x) = f'(y)$, there is a map $f : S \to X$ in $T(S,X)$ such that $f' = f \circ g$.
- If $f : S \to X$ in $T(S,X)$ and $g : S' \to S$ is continuous, then $f \circ g$ is in $T(S',X)$.
For condensed sets $X$ and $Y$, a homomorphism from $X$ to $Y$ is a function $f : X \to Y$ such that for all profinite sets $S$ and all $g : S \to X$ in $T(S,X)$ the composition $f \circ g : S \to Y$ is in $T(S, Y)$.
