Recently I stumbled upon something that seems really intresting, however I couldn't find a proper proof, so I thought that maybe you can help me.
Suppose that $\textbf{Poset}$ is the category of $partial$ $order$ sets, how do I prove that this category is isomorphic with the category of $\textbf{Alexandrov}$ topological spaces? Moreover, does this category have inverse or direct limits?
(By Alexandrov, I mean topological spaces, where intersection of infinitely many open sets is open)
An Alexandrov space is canonically preordered by $x\leq y$ when $y$ is in all the opens $x$ is in. This is clearly reflexive and transitive, but in general need not be antisymmetric: the latter holds exactly for $T_0$ spaces. A preordered set gives rise to an Alexandrov topology with open sets the upward closed sets. It's not too hard to see this correspondence respects the appropriate maps: the claim to show is that a map of preordered sets is monotone if and only if the preimage of an upward closed set remains upward closed, which is almost a direct translation of the definition of monotone.
Preordered sets are just small categories with at most one morphism between any two objects, and they inherit both limits and colimits from those in small categories since the inclusion has a left adjoint. (For colimits, apply the left adjoint after taking the colimit in small cats; for limits, just take the limit as in categories.) This isn't a terribly explicit argument, but it gets the job done if you just want to know existence.