Add category of compact Hausdorff spaces

[dschepler]

Addresses: #156

Currently undecided properties:
has cartesian filtered colimits
is coaccessible
has cofiltered-limit-stable epimorphisms
is coregular
is countably distributive
has exact cofiltered limits
is infinitary distributive
is locally copresentable
has a natural numbers object

[dschepler]

Hmm, for the coregular property, I have two possible approaches:
Try to adapt the proof from Top.
Otherwise, a Google search reminded me of Gelfand duality, that the opposite category of CompHaus is equivalent to the category of commutative unital $C^$-algebras; and the nLab page on that category says the category of not necessarily unital or commutative $C^$-algebras is monadic over Set, and therefore regular. Not sure whether there's a way to get from there to regularity of the subcategory.

[ScriptRaccoon]

Thanks for the PR!

For the non-existence of NNO, and hence showing that CompHaus is not countably distributive, I suggest to use the lemma nno_distributive_criterion that I added a few hours ago (to show that SemiGrp has no NNO).

If an NNO exists, it has to be $\beta(N)$, and for every compact Hausdorff space $A$, the canonical morphism

$$\alpha : \beta(A \times N) \to A \times \beta(N)$$

needs to be a split monomorphism. But it is clearly injective and has dense image. So it would actually be an isomorphism. I don't think that this is true when $A$ is not discrete. But I have no proof, yet.

[ScriptRaccoon]

I think coregularity should be easy to prove directly. Don't go via C*-algebras here.

[dschepler]

The question you raise regarding a consequence of NNO existing also happens to be a special case of whether the category has cartesian filtered colimits, for the special case of the colimit $[n] \to \beta(N)$.

[dschepler]

Regarding the property of having cofiltered-limit-stable epimorphisms: It's interesting that the counterexample from Set and Haus fails in CompHaus. In fact, by the intersection theorem, any such example where it's asking about an intersection of a codirected family of nonempty compact subobjects to the constant 1 is doomed to failure. I don't have any ideas on the general case, though.

[dschepler]

How about this: if $I = [0, 1]$, then $I^I$ does exist as an exponential in Top with the compact-open topology, it's just not compact. So, we have $Hom(\beta(N\times I), I) \simeq Hom(N\times I, I) \simeq Hom(N, I^I) \not\simeq Hom(\beta(N), I^I) \simeq Hom(\beta(N)\times I, I)$, and hopefully by analyzing the chain and a failure example at the one point, we could come up with a proof that $\beta(N\times I) \not\simeq \beta(N)\times I$. I haven't managed to get through the details yet, however.

Maybe constructing the counterexample of a sequence in $I^I$ with no convergent subnet via recursion of multiplying by $x$ would work. Again, not quite getting through the details on that yet.

[dschepler]

I think I've got it now: if a natural numbers object $N$ existed, we could iterate the initial conditions $I \to I\times I$, $x \mapsto (x, x)$ and $I\times I \to I \times I$, $(x, y) \mapsto (x, xy)$ to get a continuous function $N \times I \to I \times I$ such that $(n, x) \mapsto (x, x^n)$ for $n \in \mathbb{N}$. The sequence $(n) \in N$ has a convergent subnet $(n_\lambda){\lambda \in \Lambda}$, say with limit $y$, and for any $x\in I$, $(n\lambda, x) \mapsto (x, x^{n_\lambda})$. So taking limits, $(y, x) \mapsto (x, 0)$ if $x \ne 1$ or to $(x, 1)$ if $x=1$. That contradicts that the composition $x \mapsto (y, x) \mapsto (x, \delta_{x,1}) \mapsto \delta_{x,1}$ would have to be continuous.

[ScriptRaccoon]

Epimorphisms might actually be stable under cofiltered limits. This is equivalent to: monomorphisms of commutative unital C*-algebras ( = injective *-homomorphisms) are stable under filtered colimits – which looks correct, even for all C*-algebras.

More generally, exact cofiltered limits are likely. Also, locally copresentable looks promising.

But let's try to find a purely topological proof.

[dschepler]

I've found a proof of cofiltered-limit-stable epis. A brief outline: First, a lemma that any cofiltered limit of nonempty compact Hausdorff spaces is nonempty. To see this, consider the product, and for $f : i \to j$ in the cofiltered category, let $F_f := \{ x \in \prod X_i \mid X_f(x_i) = x_j \}$. Each one is closed, and it's easy to see the family has the finite intersection property (in writing it up, I'd of course expand this point a bit more). Therefore, the intersection of all $F_f$ is nonempty; but that intersection is precisely the limit.

For the main statement: just apply the lemma to the cofiltered limit of inverse images of ${ y_i }$.

I'm still not sure whether this is just a special case of a more general argument in disguise, where the more general argument establishes exact cofiltered limits.

[dschepler]

I think I'm getting closer to a proof that it's $\aleph_1$-coaccessible, but I'm still fuzzy on the details.

The basic idea: first prove that $[0,1]$ is $\aleph_1$-copresentable. For that, if we have $\lim X_i \to [0,1]$, then for each point of $\lim X_i$, use continuity and filtering to find an open subset of some $X_i$ whose image is within an interval of diameter at most $1/n$. Choosing a finite subcover of the limit, we can assume that all $i$ are the same. Then if we do this for each $n$, then use filtering, that should find an $i$ such that the map factors through $X_i$.

It should then follow that all countable powers of $[0,1]$ are $\aleph_1$-copresentable. And for any compact $X$, it might then be a limit of all possible maps to such countable powers?

[ScriptRaccoon]

I found an interesting paper: https://arxiv.org/pdf/1808.09738
I haven't read it. But it proves an interesting characterization of the category of compact Hausdorff spaces. Also, it gives references for the more well-known properties of this category. Maybe we can exchange the nLab references with them. And maybe this paper also contains proofs for the properties that are currently open.

[ScriptRaccoon]

https://doi.org/10.1016/j.topol.2019.02.033 proves several results about locally copresentable categories of spaces (called: dually locally presentable categories). See Theorem 3.4 and Theorem 4.9. It appears that CompHaus is never mentioned explicitly, but I assume this is because the authors are way past that example :D. I will find a better reference.

Close catch: https://arxiv.org/pdf/1508.07750 - references in particular the result that CompHaus^op is monadic over Set. So I assume the question is if this monad is accessible.

[ScriptRaccoon]

Apparently, Isbell proved that CompHaus^op is equivalent to the category of functors $T \to Set$ preserving countable products, where $T$ is the ess. small subcategory of CompHaus consisting of all countable powers of the unit interval. So CompHaus^op is actually countable-ary algebraic. From here, it should be clear that the category is locally $\aleph_1$-presentable.

J. Isbell. Generating the algebraic theory of C(X). Algebra Universalis, 15(2):153–155, 1982

I don't have access to the paper. And tbh the papers of Isbell are never easy to understand. So I would appreciate if we can find a more direct argument for local $\aleph_1$-presentability of CompHaus^op.

[ScriptRaccoon]

The introduction of http://www.tac.mta.ca/tac/volumes/33/12/33-12.pdf gives useful references.

• in [Dus69] it is proved that the representable functor hom(−, [0, 1]): CompHaus^op →
  Set is monadic,
• the unit interval [0, 1] is shown to be a ℵ1-copresentable compact Hausdorff space
  in [GU71],
• a presentation of the algebra operations of CompHaus^op is given in [Isb82], and
• a complete description of the algebraic theory of CompHaus^op is obtained in [MR17].