Are there useful categorical characterisations of the topological separation axioms?

Solution 1:

What a funny coincidence, I've just attended a talk by M. Gavrilovich about categorical characterizations of basic notions of general topology.

Recall the notion of orthogonality in a category, and that partial orders can be viewed as topological spaces (Alexandrov topology).

• A space $X$ is connected iff $X \to \{\bullet\}$ is left orthogonal to $\{\bullet,\bullet\} \to \{\bullet\}$
• A map $f : X \to Y$ is injective iff it is right orthogonal to $\{\bullet,\bullet\} \to \{\bullet\}$
• A map $f : X \to Y$ is surjective iff it is right orthogonal to $\emptyset \to \{\bullet\}$
• A space $X$ is Hausdorff iff every injective map $\{\bullet,\bullet\} \to X$ is left orthogonal to $\{a < b > a'\} \to \{\bullet\}$
• A space $X$ is compact iff $\emptyset \to X$ is left orthogonal to $\coprod_{\beta<\alpha} \beta \to \alpha$ for every limit ordinal $\alpha$.
• As you say, $X$ is normal iff every closed embedding $A \hookrightarrow X$ is left orthogonal to $\mathbb{R} \to \{\bullet\}$