# Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of compact Hausdorff spaces, the extremely disconnected spaces are exactly the projective objects: $P$ is projective if given any morphism $f: P\rightarrow X$ and any epimorphism $g: Y\rightarrow X$, there is a morphism $h: P\rightarrow Y$ with $f = g\circ h$.

I was wondering if the zero-dimensional compact Hausdorff spaces enjoy a similar category-theoretic characterization in the category CHauss.

• What would you consider "similar"? The category CHaus is rigid (every self-equivalence is naturally isomorphic to the identity), so in some sense every object in CHaus can be characterized categorically. – Eric Wofsey Sep 1 '15 at 19:34
• In particular, "clopen sets separate points" can be encoded categorically very simply by saying "every monomorphism $2\to X$ has a left inverse", where $2$ is the coproduct of two copies of the terminal object. But I'm not sure if this is the sort of thing you're looking for... – Eric Wofsey Sep 1 '15 at 19:53
• This seems pretty close to what I'm looking for. Is there a name for this type of object? – Andy Sep 2 '15 at 14:24
• No, this is not any sort of standard notion. I also feel it's not really analogous to the characterization of extremely disconnected spaces as projective because it's just a direct translation of the definition into categorical language, rather than a nontrivial and interesting characterization. – Eric Wofsey Sep 2 '15 at 14:54