Read a bit about alexandrov topologies and chu spaces and stuff. Is there a good completion of the analogy

preorder (i.e. alexandrov topology) : general case of topological space :: category : ???

I wonder?

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Huh, I think I have a satisfying-enough answer, but it's a little weird and I am skeptical of whether it admits any interesting applications/examples.

To recap, the alexandrov maneuver is, you take a poset or preorder or something, and you take its downward-closed sets (the sets C such that if x is less than some y ∈ C, then also x ∈ C) and call those the closed sets of a topology.

And it turns out that this perfectly realizes preorders as topologies; it's a faithful embedding, and the morphisms work out right, and continuous maps between the embedded thingies are exactly the monotone maps. However, as noted below in the comments, this doesn't yield most topologies, just the ones for which open sets are closed under arbitrary intersections, not just the finite ones.

So if we wanted an "order-theoretic" feeling expression of all topologies, what could we do?

Well, instead of talking about one point being less than another in the specialization order, we could talk about a point being less than a whole set. This is really just duct-taping together the closure operation of the topology together with the set-elementhood relation: we say x ≤ Z just in case x belongs to the topological closure of Z.

But we can separately axiomatize what it means to be a topology with just this relation. It looks like:

1. x ≤ {x} ("reflexivity")
2. If x ≤ Z and Z ⊆ W, then x ≤ W ("monotonicity")
3. If x ≤ {x_i|i∈I} and x_i ≤ Z for all i, then x ≤ Z ("transitivity")
4. x≰{} ("the empty set is closed")
5. If x ≤ Z1 ∪ Z2, then x ≤ Z1 or x ≤ Z2 ("union of two closed sets is closed")

and if we have two such structures (X,≤), (Y,⊑) then a morphism
(X,≤)→(Y,⊑) should be a map f:X→Y such that (for any x∈X and Z⊆Y) if x≤f*(Z) then f(x)⊑Z.

All I've done above is written down an equivalent presentation of the category Top.

But the point is, from this presentation, it doesn't seem too hard to smell what the categorified version looks like: replace the less-thans with homsets, and turn them around so the multiple-things occur on the domain instead of the codomain, and it starts looking like some kind of multicategory-like structure that allows infinite arities! Just like all finite topologies are alexandrov, all finite "topological multicategories" would be not interestingly different from categories. Axioms 4 and 5 seem to make it "not all that different" from the non-multi-category case; the morphisms from a finite aggregation of contexts are totally determined by the morphisms from the contexts you're aggregating. However --- and this is exactly the weird topological character I was trying to get at --- if you have "densely enough" interesting objects in the domain of the homset you're looking at, you get information about the domain that doesn't arise from any finite set of objects in it, but topologically from the aggregate, like an accumulation point.
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