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This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces

To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\text{CHaus}$ of compact Hausdorff spaces in a nice way. Every compact Hausdorff space $X$ can be covered by a profinite set (specifically the stone Czech compactification of the underlying set of $X$). $\text{Prof}$ can be characterized by a universal property without reference to the category of topological spaces: It is the free completion of the category $\text{Fin}$ of finite sets under cofiltered limits.

Since $\text{CHaus}$ is, unlike $\text{Top}$, a category that is quite nicely behaved (it is for example the category of algebras for the ultrafilter monad on sets), it seems natural to ask: Can it also be characterized as a category by a universal property similar to $\text{Prof}$?

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    $\begingroup$ $\mathbf{Prof}^{\mathrm{op}}$ is equivalent to the category of Boolean algebras and therefore locally $\aleph_0$-presentable. The category $\mathbf{CHaus}^{\mathrm{op}}$ is not locally $\aleph_0$-presentable, but it is locally $\aleph_1$-presentable (and in fact monadic over $\mathbf{Set}$) because it is equivalent to the category of commutative unital C$^*$-algebras. This can be stated as $\mathbf{CHaus}$ is the free completion of the compact metric spaces (equivalently, the closed subspaces of $[0,1]^\mathbb{N}$). Is this the kind of thing you're looking for? $\endgroup$
    – Robert Furber
    Commented Jan 26, 2023 at 13:41
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    $\begingroup$ Compact Hausdorff spaces also have a "minimal" cover by a Stonean space, which is obtained by taking the Stone space of the complete Boolean algebra of regular open sets (a construction due to Andrew Gleason). $\endgroup$
    – Robert Furber
    Commented Jan 26, 2023 at 13:42
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    $\begingroup$ Isn't the fact that it is the category of algebras for the ultrafilter monad already a universal property? $\endgroup$
    – Max New
    Commented Jan 26, 2023 at 15:24
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    $\begingroup$ See V. Marra and L. Reggio, A Characterisation of the Category of Compact Hausdorff Spaces, Theory App. Cat. 35, (2020), pp.1871–1906 (arxiv version). The authors show that $CHaus$ is, up to equivalence, "the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object". Refer to $\S4$ of the paper for their notion of filtrality. $\endgroup$
    – Tyrone
    Commented Jan 26, 2023 at 16:14
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    $\begingroup$ @Tyrone This is not a comment, it is a full answer. $\endgroup$
    – Martin Brandenburg
    Commented Jan 29, 2023 at 2:42

3 Answers 3

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I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts:

  1. The category of compact Hausdorf topological space is the category of algebras for the ultrafilter monad.

  2. the Ultrafilter monad is the codensity monad for the inclusion of finite set into sets. If I'm not mistaken, it follows that it is terminal for monads on sets such that $ S \to M(S)$ is an isomorphism for all finite sets $S$.

  3. the category of monads on $Set$ is equivalent to the opposite of category of monadic right adjoint functors $C \to Set$, or equivalent to the category of monadic left adjoint functor $Set \to C$

So that we get:

Theorem: The category of compact Hausdorff spaces, together with its forgetful functor to the category of set is initial in the category of Monadic right adjoint functor $U: C \to Set$ such that the left adjoints $L: Set \to C$ is fully faithful when restricted to finite sets.

Theorem: The category of compact Hausdorff spaces, together with the Ultrafilter space functor $\beta: Set \to C$ is terminal in the category of monadic left adjoint functor that are fully faithful when restricted to finite sets.

There are many way to twist this, for example using that $U$ is $Hom(1,\_)$, or writing the universal property of the codensity monad some other way (the way of writing I chose is maybe a bit atypical...) but it depends what kind of universal property you like

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As has already been remarked, $\mathbf{CHaus}$ is monadic over $\mathbf{Set}$. It is also a pretopos, meaning roughly that it has the finitary properties of $\mathbf{Set}$.

Another more symmetrical way of viewing these two categories is as full subcategories of locally compact locales, for which my Abstract Stone Duality programme gives an axiomatisation.

In this, sets are the overt discrete objects (in the sense of the ASD axiomatisation) and compact Hausdorff spaces are chararacterised by the ASD versions of those properties, which are the lattice duals of overtness and discreteness.

So, in the purely finitary part of the ASD axiomatisations, sets and compact Hausdorff spaces are described in a formally dual way. The difference between them appears when we add the infinitary Scott continuity axiom.

The objects $\mathbb N$ and $2^{\mathbb N}$ (Cantor space) play roughly analogous roles in the two subcategories, each having three of the four properties, except that $\mathbb N$ fails compactness, whilst $2^{\mathbb N}$ fails discreteness.

The two subcategories are more similar than one might expect, and I didn't manage to pin down the axiomatic distinction between them. If some PhD student would like to take this up I would be pleased to help.

Sorry for the lack of mathematical detail here: I am on holiday in Tierra del Fuego, but will expand this answer when I get back to England.

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    $\begingroup$ "each having three of the four properties" <-- what four properties? $\endgroup$
    – David Roberts ♦
    Commented Jan 26, 2023 at 21:24
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    $\begingroup$ This kind of math is obviously better suited to England; in Tierra del Fuego you should be doing tropical mathematics. $\endgroup$
    – LSpice
    Commented Jan 26, 2023 at 21:30
  • $\begingroup$ @LSpice Apologies if I am missing the joke, but what's tropical about TdF? $\endgroup$
    – Yemon Choi
    Commented Jan 26, 2023 at 22:29
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    $\begingroup$ Ushuaia in Tierra del Fuego is not tropical, it's the southernmost city in the world, although actually only at the negative of the latitude of Newcastle. The "four properties" are compact, Hausdorff, overt and discrete. $\endgroup$
    – Paul Taylor
    Commented Jan 26, 2023 at 23:41
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    $\begingroup$ Re, a failed joke. I thought it must be a fairly tropical climate, because my mental picture of it was in the wrong place, and I was too lazy to Google. $\endgroup$
    – LSpice
    Commented Jan 26, 2023 at 23:59
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We can describe $\mathbf{CHaus}$ with a universal property inside the $2$-category of all cocomplete categories and cocontinuous functors. Namely, $\mathbf{CHaus}$ is the universal cocomplete category which has an object $X$ and a morphism of $\mathbf{Set}$-monads $\beta \to \hom(X,(-) \otimes X)$. This can be thought of as a ultrafilter "coconvergence" on $ X$. This follows rather formally from a more general result which gives a universal property of the category of algebras of a monad, see my paper on limit sketches, Example 6.9.

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