Categorical semantics for non-monotonic logics?
Non-monotonic logic is kind of a wide area -- do you have any particular logics in mind? Anyway, defeasibly assuming :) that 


you are interested in any logic in which the principle of monotony fails, and
you want a categorical semantics in the sense of categorical proof theory (rather than, say, a hyperdoctrine semantics), then


one answer is that you can give a reasonable categorical semantics to any logic for which a sequent calculus with cut-elimination is known. Basically, the types are objects, normal forms of the sequent calculus are morphisms, and cut-elimination tells you how to implement composition. This gives you the initial category in whatever category of models you end up using to prove soundness and completeness. 

This recipe is independent of monotonicity, and so it can work well even for non-monotonic logics. For example, one of the more important successes in categorical logic is its treatment of linear logic. This is a logic where, intuitively, propositions refer to resources, so that the linear implication $A \multimap B$ can be read as "$A$ can be consumed to produce $B$". The consequence relation of linear logic is non-monotonic (since the fact that $A$ can be consumed to produce $B$ does not mean that $A$ and $X$ can both be consumed to produce $B$). However, it has an excellent proof theory, and its categorical models are closely tied to the theory of monoidal categories.