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Here is a fact that should be much more widely known than it is. The category of posets is isomorphic (not just equivalent) to the category of $T_0$ Alexandrof spaces. A topological space is said to be Alexandrof if arbitrary (not just finite) intersections of open sets are open. For example, every finite topological space is an Alexandrov space. Finite spaces are fascinating and the connection with algebraic topology is very close: finite spaces have associated finite simplicial complexes. Don't just look categorically: that takes the fun out of it! There are notes on my web page and there is a book by Barmak.