Do products preserve colimits in the category of locales?

Question

Does the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ preserve small colimits for all locales $X$?

---

The reason that I'm interested in this question is that the same property fails in the category of topological spaces. If products preserve colimits in a total category then it must be cartesian closed. The category $\mathbf{Top}$ is total, but fails to be cartesian closed because products don't preserve colimits. So it would be curious if $\mathbf{Loc}$ failed to be cartesian closed for the complementary reason. It's not total, so maybe products do preserve colimits in $\mathbf{Loc}$?

Another reason to ask this question is that the definition of "locale" requires precisely that products preserve colimits in its frame of opens. So it would be interesting if this property was mirrored at the category level.

Answer 1

I think so? But I'm putting together results I don't understand in much detail. Let's try the following:

Equivalently the question is whether taking coproduct with a frame $X$ preserves limits of frames. We'll write the coproduct of frames as $\otimes$, since it turns out to coincide with the tensor product of suplattices (see this nLab page and this nLab page). The tensor product of suplattices has right adjoint given by the hom suplattice, so preserves colimits of suplattices in both variables; I think this implies that it preserves limits in both variables, because I think limits of suplattices can be computed as colimits and vice versa (e.g. products are coproducts); this is the step I'm least confident in. If that's true, then because limits of frames are computed as limits of suplattices (which in turn are computed as limits of underlying sets), tensor product preserves limits.

At the very least I'm confident that tensor product preserves products, so it remains to check if it preserves equalizers, which maybe can be done by hand.

Answer 2

No, $- \times L$ does not preserve colimits for all locales $L$. Let $\Omega : Sob \leftrightarrows Loc : pt$ be the adjunction which embeds $Sob$ as a coreflective subcategory of $Loc$. I will give an example of when this is not true for spatial locales.

Let $f,g : X \to Y$ be continuous maps, $c : Y \to Z$ their coequalizer. Suppose that $X,Y,Z$ are locally compact Hausdorff spaces and $A$ is a Hausdorff space such that $- \times A$ does not preserve this coequalizer; that is, $c \times 1_A$ is not a coequalizer of $f \times 1_A$ and $g \times 1_A$. Then applying $\Omega$, we have the following commutative diagram:

where the vertical maps are the induced maps $(\Omega proj_1, \Omega proj_2)$, which are all isomorphisms since $X$, $Y$, and $Z$ are locally compact. Since $\Omega$ is a left adjoint, we know that $\Omega c$ is a coequalizer of $\Omega f$ and $\Omega g$. By natural isomorphism then, the bottom row is a coequalizer diagram (that is, $- \times \Omega A$ preserves the coequalizer) if and only if the top row is. But $\Omega$ is fully faithful, hence it reflects (co)limits. So if the top row were a coequalizer, then $c \times 1_A$ would be a coequalizer of $f \times 1_A$ and $g \times 1_A$, contradicting our choice of $A$. Thus, the bottom row is not an equalizer, so $- \times \Omega A$ does not preserve all colimits.

Of course, it remains to produce an example which attains these conditions. It will be easiest to produce an example of a quotient map $p : Y \to Z$ between locally compact Hausdorff spaces for which $p \times 1_A$ is not a quotient map for some Hausdorff space $A$; this arises as the coequalizer of $f,1_Y : (Y, \tau_{discrete}) \to Y$ where $f$ contracts fibers of $p$ to a single point (in particular, $Y$ with the discrete topology is locally compact Hausdorff). I asked for such an example yesterday here; you can read the answer there for an example. The examples in Bi-quotient maps and cartesian products of quotient maps, also references in the MSE answer, give some other examples.