TITLE: Are there useful categorical characterisations of the topological separation axioms?
QUESTION [15 upvotes]: Tietze's extension theorem states: If $X$ is a normal space, and $A$ a closed subspace. Then any continuous function to the reals $f:A\rightarrow R$ has an extension to $f':X\rightarrow R$ that is $f=f'i$ where $i$ is the inclusion of $A$ in $X$.
This can instead be used to characterize normal spaces. With this definition there is an easy proof of Urysohn's Lemma.
Essentially this is a categorical characterization of Normality. Are there similarly nice & useful categorical characterizations of the other separation axioms?
There is also one for Tichonov spaces - these are completely regular $T_1$ spaces (in fact it always turns out that they are $T_2$). They always have Hausdorff compactifications (a Hausdorff compact space within which they embed densely) and this characterises them. Since there is always a maximal compactification - the Stone-Čech compactification. This gives a categorical charactisation of Tichonov spaces.

REPLY [18 votes]: 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 $\mathbb{R} \to \{\bullet\}$