In a [paper][1] of Doubilet, Rota and Stanley , they see the incidence algebra of a poset as a contravariant functor from some subcategory of $Pos$ to the category of algebras over a field $K$. For instance, you may uniquely recover the poset from its incidence algebra. 

I think this is quite interesting cos they then show how to arrive at some types of generating functions familiar from combinatorics.   

  [1]: http://math.mit.edu/~rstan/pubs/pubfiles/10.pdf