# Can we have a category of compactifications?

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.

• 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. Commented Aug 21, 2021 at 13:46
• @QuantumSpace Did you mean that the Stone-Cech compactification is initial? Commented Aug 21, 2021 at 14:54
• @QuantumSpace Why not an official answer? Commented Aug 21, 2021 at 22:55
• @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: math.stackexchange.com/questions/3608761/… I don't know about any reference about compactifications in the context of category theory. Commented Aug 23, 2021 at 17:37
• @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 $f$. Commented Aug 30, 2021 at 2:12

## 1 Answer

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.

• Exactly. Well put. :) (If only I could recall what "completely regular" meant... :) Commented Apr 6 at 0:29
• And there you go! Commented Apr 6 at 0:35
• :) ............. Commented Apr 6 at 0:40
• "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. Commented Apr 6 at 6:18
• Good catch @Smiley1000. I think I've fixed that issue. Commented Apr 6 at 14:17