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Let us consider the following definition: a compact po-space is a pair $(X,\leq)$ where $X$ is a compact space and $\leq$ is an order, closed on $X^2$. Then, we can consider the category $KPoSp$ whose objects are compact po-spaces and whose morphisms are increasing continuous functions.

Moreover, let us denote by $KHaus$ the category of compact Hausdorff spaces with continuous functions.

It is known that the projective objects in $KHaus$ are the compact Hausdorff spaces which are extremally disconnected, in the sense that the closure of an open set is also open. (I have it from section 7 of Stone duality and Gleason covers through de Vries duality).

I would like to know if there is somewhere a similar result for the category $KPoSp$ ?

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    $\begingroup$ Are the objects of $KPoSp$ assumed to be Hausdorff? If not, I'd think one should first ask what are the projective objects in the category of compact (not-necessarily-Hausdorff) spaces. Also, what exactly is meant by "projective"? Probably you mean "object which has the left lifting property with respect to certain maps", but which maps? Surjections? Epimorphisms? Quotient maps? Finally, is $\leq$ assumed to be a poset structure or just a preorder? $\endgroup$
    – Tim Campion ♦
    Jun 24, 2020 at 14:17
  • $\begingroup$ For the definition of "projective", I must admit I don't know the kind of map I'm looking for (surjective maps would be perfect, but I can be satisfied with epimorphisms). The objects of $KPoSp$ are indeed Hausdorff. This is a consequence of $\leq$ closed. Finally, yes we have a poset structure (the relation between compact po-spaces and Priestley spaces is similar to the one between compact Hausdorff and Stone spaces, if it can help.) $\endgroup$
    – Bijco
    Jun 24, 2020 at 15:58
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    $\begingroup$ Presumably you want to first show every compact pospace is a quotient of a Priestley space and then look at injective distributive lattices. Did you look in Johnstone's stone spaces or the compendium of continuous lattices? $\endgroup$
    – Benjamin Steinberg
    Jun 24, 2020 at 16:31
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    $\begingroup$ I suspect that surjections are not what you want -- they lead to "too few" projective objects (and epimorphisms will be, if anything, worse in this regard). For instance, for $P$ to have the left lifting property against the bijection $f: \{0,1\} \to \{0 < 1\}$ (where $\{0,1\}$ has the discrete ordering) is a very strong requirement. In particular, if $P$ is totally disconnected and has the left lifting property with respect to $f$, then I believe the order on $P$ must be discrete. So perhaps you want "projective" to mean the left lifting property with respect to quotient maps. $\endgroup$
    – Tim Campion ♦
    Jun 24, 2020 at 18:44
  • $\begingroup$ On the other hand, I'm not sure I'd expect there to be a well-behaved notion of "projective" even in the category $Pos$ of posets without a topology -- generally I expect "projective" to be well-behaved in "algebraic" categories (by which I mean roughly: monadic over $Set$ or monadic over $Set^S$ for some set $S$), which $Pos$ is not. Analogously, I don't think there's a good notion of "projective object" in the category $Cat$ of categories. $\endgroup$
    – Tim Campion ♦
    Jun 24, 2020 at 18:51

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