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Is there any work on compactifications of spaces in terms of category theory? I would like to know whether there is a defined category of compactifications; I will denote it Comp.

  1. Could you describe Comp like this?
  • Objects = compactifications
  • Morphisms = homeomorphisms between compactifications?
  1. Also, intuitively a Stone-Čech compactification is a "biggest" object and an Alexandroff compactification is a "smallest" object of Comp. Would they have any special properties then? I.e. would they be initial and terminal objects respectively?

Thank you for your insights.

Lee Mosher
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Tereza Tizkova
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    I suppose you start with a fixed space $X$ and you consider as objects all compactifications of $X$. The morphisms could be continuous maps between these spaces, but it seems natural to also ask the following: if $\alpha: X \to K$ and $\beta: X \to L$ are compactifications, then we should also require that a continuous map $f: K \to L$ satisfies $f\alpha = \beta$. You can check that you still get a category in this way. With the above proposal, the Stone-Cech compactification is terminal in this category by its universal property. – J. De Ro Aug 21 '21 at 13:46
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    @QuantumSpace Did you mean that the Stone-Cech compactification is initial? – Andreas Blass Aug 21 '21 at 14:54
  • @AndreasBlass Yes I was too fast. Every continuous map $f: X \to K$ extends uniquely to a map $\beta X \to K$ so initial object because the arrows start in the Stone-Cech compactification. – J. De Ro Aug 21 '21 at 15:09
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    @QuantumSpace Why not an official answer? – Paul Frost Aug 21 '21 at 22:55
  • @Paul Frost Because I only answered part of the question. I can answer (1) and (2) but (3) is too general to give a complete answer. – J. De Ro Aug 21 '21 at 23:11
  • @QuantumSpace Thank you. Would that also mean that Alexandroff comapctification is terminal object in such category? (I suppose you meant that Stone-Čech is initial). And could you recommend some resources where to read about compactifications in terms of category theory? – Tereza Tizkova Aug 23 '21 at 15:34
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    @TerezaTizkova I don't think that the Alexandroff compactification is terminal (if it would, it would satisfy some nice universal property, but I'm not aware of such a thing). A formal argument is given here: https://math.stackexchange.com/questions/3608761/is-the-one-point-compactification-terminal-in-the-category-of-compactifications I don't know about any reference about compactifications in the context of category theory. – J. De Ro Aug 23 '21 at 17:37
  • @AndreasBlass Is it really true that Stone-Čech is initial? Would you prove it via the universal property? What construction would you use? – Tereza Tizkova Aug 29 '21 at 15:34
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    @TerezaTizkova I"d use the universal property as the definition of Stone-Cech compactification. To construct it I'd take the closure of the image of the embedding of the the given (completely regular) space $X$ in the compact product space $[0,1]^F$, where $F$ is the set of continuous maps $f:X\to[0,1]$ and the embedding of $X$ into the product has $f$-th component equal to $f$. – Andreas Blass Aug 30 '21 at 02:12

1 Answers1

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This community-wiki answer is compiled from the comments.

One way to define a category of compactifications for an individual space $X$ is to consider the objects to be all compactifications of $X$, and a morphism from one compactification $\alpha : X \to K$ to another one $\beta : X \to L$ to be a continuous map $f : K \to L$ such that $f\alpha=\beta$. This does define a category.

With an extra condition on $X$ (completely regular), the Stone-Čech compactification is indeed an initial object. It is constructed by letting $F$ be the set of continuous maps $f : X \to [0,1]$, embedding $X \hookrightarrow [0,1]^F$ so that the $f$-the component is equal to $f$, and taking the closure of the image of $X$. One proves that it satisfies the universal property defining an initial object.

Also this answer shows that under a further restriction on the definition of compactification, one can prove that the one-point compactification satisfies the universal property required to be a terminal object. As stated in that link, the extra restriction on a compactification $\alpha : X \to K$ is that $\alpha(X)$ must be open in $K$ and that every compact subset of $K$ must be closed. This condition can be attained by a further restriction on $X$, namely that $X$ be a locally compact, noncompact Hausdorff space.

Lee Mosher
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  • Exactly. Well put. :) (If only I could recall what "completely regular" meant... :) – paul garrett Apr 06 '24 at 00:29
  • And there you go! – Lee Mosher Apr 06 '24 at 00:35
  • :) ............. – paul garrett Apr 06 '24 at 00:40
  • "Also this answer shows that under another extra condition on $X$ (compact subsets are closed), one can prove that the one-point compactification satisfies the universal property to be a terminal object." Actually, this is not quite correct. The answer explicitly says "assuming $X^$ is a compactification at all by this definition--it won't always be KC". In fact, under the usual definition of compactification, $X^$ is a compactification of $X$ iff $X$ is locally compact, Hausdorff and noncompact. – Smiley1000 Apr 06 '24 at 06:18
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    Good catch @Smiley1000. I think I've fixed that issue. – Lee Mosher Apr 06 '24 at 14:17