# The category of Stone spaces is regular

(Does this result have an [ordered generalisation]()?)

(We refer to the nLab for a definiton of [regular category](https://ncatlab.org/nlab/show/regular+category).)

Let $Stone$ be the category of Stone spaces and $KHaus$ the category of compact Hausdorff spaces.

**Observation:** If $f:X\to Y$ is a morphism of Stone spaces and $X\to Z\to Y$ is the factorisation in compact Hausdorff spaces, then $Z$ is compact (because it is the image of a compact space), hence also closed. But a closed subspace of a Stone space is a Stone space.

**Lemma:** $Stone\to KHaus$ is a reflective subcategory.

[*Proof*](https://hackmd.io/ZGaPgc-YT_WXUhCMFeozMg).

According to Theorem 4 of [this paper](http://www.mat.uc.pt/preprints/ps/p0660.pdf), a reflective subcategory of a regular category is regular if and only if an additional condition is satisfied. In our situation, this additional condition amounts to the following.

Let $h$ in $Stone$ and $h=bp$ its image factorisation in $KHaus$. Let $f$ in $Stone$. Following the Theorem 4 mentioned above, we need to show that the left-adjoint of the inclusion preserves the pullback in $KHaus$ of $b$ and $f$. The observation above tells us that $b$ is actually in Stone. Since $b$ and $f$ are in $Stone$, their pullback in $KHaus$ is also in $Stone$ (this is a general fact about limits in reflective subcategories).

To summarise, we have shown:

**Theorem:** The category of Stone spaces is a regular category.

For the question whether Priestley spaces are order-regular, see [here](https://hackmd.io/@alexhkurz/HkJe2KqfL).