Characterisations of closed embeddings in $Top_1$?

Question

Let $Top_1$ be the category of topological spaces which are $T_1.$

I am curious as to whether there is a categorical definition of what a closed embedding is in this environment. With a categorical definition, I mean something along the lines that in the category $Top_2$ , consisting of topological spaces which are $T_2,$ the closed embeddings are precisely the extremal monomorphisms.

In $Top$ (and $Top_1$) the extremal monomorphisms are precisely the embeddings. In $Top$ we can use the Sierpinski space to single out the ones with closed image, but this is not possible in $Top_1$ since the Sierpinski object is not $T_1.$

Any comments or thoughts would be welcome.

Answer 1

As partial motivation, let me start with the observation that in $\mathrm{Set}$, if $i: A \to B$ is a monomorphism, then $i$ is retrieved as the pullback of the monomorphism $\ast \cong A/A \hookrightarrow B/A$ along the canonical map $q: B \to B/A$ (where $B/A$ is the pushout or cofiber product of $i$ and the unique map $A \to 1$). This is one of the exactness properties singled out by Freyd in a categories-list discussion that are common to pretoposes and abelian categories. The square

$$\begin{array}{ccc} A & \to & 1 \\ i \downarrow & & \downarrow \\ B & \stackrel{q}{\to} & B/A \end{array}$$

which is simultaneously a pushout and pullback is sometimes called, affectionately, a "dolittle square" (after the pushmi-pullyu in the Doctor Dolittle story). So not knowing quite what to call such monomorphisms in general, I'll call it a dolittle monomorphism.

In other words, a monomorphism $i$ in a category with finite limits and pushouts will be called dolittle if it is the pullback of the inclusion $1 \to B/A$ in the pushout square above. Then in $\mathrm{Top}_1$, closed embeddings $i: A \to B$ are precisely dolittle monomorphisms. (Notice that the pushout construction $B/A$ in $\mathrm{Top}_1$ is constructed as the pushout $B/\bar{A}$ in $\mathrm{Top}$.)