I seem to have misunderstood the nature of the duality in the question. This answer is not relevant. I'll keep it in case it has any archaeological interest.
In local quantum field theory, one does use the notion of a co-presheaf (or is it a pre-cosheaf?) of operator algebras. You can read about it (perhaps not with those words) in Streater and Wightman's PCT, spin-statistics, and all that and perhaps also in the links from the above wikipedia page. This is perhaps not surprising as it is the states of the system which form a presheaf.