Is the category of compact Hausdorff spaces cocomplete?

Question

I would assume not, mostly since I don't think that infinite coproducts exist (as the disjoint union of infinitely many compact spaces is no longer compact, although its possible that infinite coproducts exist but aren't disjoint unions). https://mathoverflow.net/questions/382348/properties-of-the-category-of-compact-hausdorff-spaces claims (in Neil Strickland's answer) that the category of compact Hausdorff spaces is cocomplete, as a consequence of a certain equivalence of categories.

So, I ask: is compact Hausdorff spaces a cocomplete category? If yes, then what are infinite coproducts?

Answer 1

Note that the category of compact Hausdorff spaces is a reflective subcategory of the category of topological spaces (due to the existence of the Stone-Cech compactification).

In general, let $C$ be a cocomplete category, and let $D$ be a reflective subcategory with inclusion $U : D \to C$ and adjoint $F : C \to D$, with unit $\eta$ and counit $\epsilon$. Note that because $D$ is a full subcategory, $U$ is fully faithful. Therefore, $\epsilon : FU \to 1_D$ is a natural isomorphism.

I claim $D$ is cocomplete. Indeed, consider a diagram $J : E \to D$. Since $C$ is cocomplete, we can take the colimit of $UJ : E \to C$. Applying $F$ to the colimit diagram and using the fact that $F$ preserves colimits gives us the colimit of $FUJ : E \to C$. Now since $\epsilon : FU \cong 1_D$, we have a natural isomorphism $\epsilon J : FUJ \to J$, and therefore we have a colimit of $J$. So we have shown $D$ is cocomplete.

To summarise: if we want the colimit of a diagram in $D$, first take the colimit of the diagram in $C$. Then, apply the functor $F$ to the colimit and push across the natural isomorphism $\epsilon$ to give the colimit in $D$. In the case of compact Hausdorff spaces, we take the Stone-Cech compactification of the colimit as topological spaces.