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.