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.