There is nice generalization of first order logic to monoidal (categorical) logics http://www.springer.com/us/book/9783642128202 which has recently been applied extensively as replacement for deontic logic by Clayton Peterson https://www.researchgate.net/profile/Clayton_Peterson2
The question is - is it possible to extend monoidal (categorical) logics by introducing modal operators?
I am aware of coalgebraic logics which are generalization of modal logics http://comjnl.oxfordjournals.org/content/54/1/31.abstract but these are different from categorical logics and I am interested in extending especially categorical logics.