When does base-change in topological spaces preserve quotient maps?

Question

The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves coequalizers. It is well-known that if $X$ is locally compact and $Y \to Z$ is a quotient map, then $Y \times X \to Z \times X$ is quotient map as well. Thus, taking product with a locally compact space preserves all colimits. (Here, I define a space to be locally compact if every point has a basis of (quasi-)compact neighborhoods. No Hausdorffness is required. For example, the underlying topological space of every scheme is locally compact.) Actually, $(-)\times X$ preserves colimits if and only if $X$ is core-compact in the sense of this nlab page.

My question is:

Is there a condition (without assuming Hausdorffness!) on a map $X \to X'$ such that $(-) \times_{X'} X$ preserves colimits, or, presumably equivalently: preserves quotient maps?

If $X'$ is Hausdorff, $Z\times_{X'} X$ is a closed subset of $Z\times X$. If $X$ is core-compact and $Y\to Z$ is a quotient map, $Y\times_{X'} X \to Z\times_{X'} X$ is thus a quotient map again. But I would like to apply the potential criterion to something like $\operatorname{Spec} A \to \operatorname{Spec} B$, which is the reason why I don't want to assume Hausdorffness.

Answer 1

Thanks to the comments, I found the paper Exponentiability for maps means fibrewise core-compactness by G. Richter, which gives a more concrete (albeit still complicated) characterization of exponentiable maps, i.e. maps $p\colon X \to T$ such that $(-)\times_T X$ preserves colimits. Richter characterizes them as fibrewise core-compact maps. Here, $p$ is fibrewise core-compact if for every $x\in X$ and every neighborhood $U$ of $x$, there are neighborhoods $V\subset U$ of $x$ and $p(U)\subset W$ of $p(x)$ such that: for all $t\in W$ and every open cover $\mathcal{U}$ of $U\cap p^{-1}(t)$, there exists a finite subset $\mathcal{E}\subset \mathcal{U}$ and an open neighborhood $W'\subset W$ of $t$ such that $V\cap p^{-1}(W') \subset \bigcup \mathcal{E}$.

This is quite a mouthful. Let's first consider the case that $T = \mathrm{pt}$. Then this says that for every neighborhood $U$ of $x$, there exists a neighborhood $V$ of $x$ such that every open cover of $U$ admits a finite subcover of $V$. This is the definition of core-compact. If $X$ is locally-compact, we can choose $V$ as a compact subneighborhood of $U$.

Another special case: Let $T$ be locally compact and let $p$ be locally proper: for every $x\in X$ and a neighborhood $U$ of $x$, there exists a neighborhood $V$ of $x$ and a neighborhood $W$ of $p(x)$ containing $p(V)$ such that $V\to W$ is proper, i.e. $p$ is closed and preimages of compacts are compact. I claim that then $p\colon X\to T$ is fibrewise core-compact. Indeed: Let $x\in X$ and $U$ a neighborhood of $x$. Choose $V$ and $W$ as above. Let $t\in W$ and $\mathcal{U}$ an open cover of $U\cap p^{-1}(t)$. Then $\bigcup \mathcal{U}$ is open and thus $p(V \setminus V\cap \bigcup \mathcal{U}) \subset W$ is closed and does not contain $t$. Choose a compact subneighborhood $W'$ of $t$ inside of its complement. Then $V\cap p^{-1}(W')$ is compact and a subset of $\bigcup \mathcal{U}$. Thus, there exists a finite $\mathcal{E}\subset \mathcal{U}$ covering $V\cap p^{-1}(W')$.

Unfortunately, $p$ being locally proper in the sense above is rarely fulfilled for a the underlying map of topological spaces of a morphism between schemes. It may still be the case that these are often fibrewise core-compact, but I don't know.