Is it true that the Stone-Čech compactification preserves disjoint union (even infinite disjoint unions)? That is, is it true that $\beta(\bigsqcup_\alpha X_\alpha) = \bigsqcup_\alpha \beta X_\alpha$?
This seems to me true since the Stone-Čech compactification functor is left adjoint to the inclusion functor (from the category of topological spaces into the category of compact Hausdorff topological spaces), and we know that left adjoint functors preserve all colimits (as discussed here).
Yet, if $X$ is a discrete then we can write $X=\bigsqcup_{x \in X} \{x\}$ but $\beta X \neq \bigsqcup_{x \in X} \{x\}$ (where $\beta(\{x\}) = \{x\}$)
Best Answer
Yes, it's true that the Stone-Čech functor $\beta : \mathsf{Top} \to \mathsf{CompHaus}$ is left adjoint to the inclusion $\iota : \mathsf{CompHaus} \hookrightarrow \mathsf{Top}$. Thus it preserves colimits, and in particular it's true that
$$ \beta \left ( \coprod_{x \in \mathbb{N}} {}^{\mathsf{Top}} \{ x \} \right ) = \coprod_{x \in \mathbb{N}} {}^{\mathsf{CompHaus}} \beta \{ x \} = \coprod_{x \in \mathbb{N}} {}^{\mathsf{CompHaus}} \{ x \} $$
However, as I've made explicit in the above notation, the second coproduct is computed in $\mathsf{CompHaus}$, not in $\mathsf{Top}$!
Since $\mathbb{N}$ is, among other things, not compact hausdorff, the coproduct $\coprod_{x \in \mathbb{N}}^{\mathsf{CompHaus}} \{ x \}$ cannot possibly be $\mathbb{N}$. So it's possible that this issue goes away, and indeed it does.
What your computation has secretly shown is that, in the category of compact hausdorff spaces, the coproduct of countably many singletons is $\beta \mathbb{N}$!
In general, this is one way to compute colimits in a reflective subcategory. We compute colimits in the big category (for us, $\mathsf{Top}$) then apply the reflector (for us $\beta$). You can see someone asked a similar question here.
I hope this helps ^_^