The first definition of the category of Poisson algebras that comes to mind is that a morphism between Poisson algebras is an algebra homomorphism that is also a Lie algebra homomorphism with respect to the Poisson bracket. This definition does not seem to be easily compatible with how people actually use Poisson algebras (in particular rings of functions on Poisson manifolds):

- A <a href="http://en.wikipedia.org/wiki/Poisson%E2%80%93Lie_group">Poisson-Lie group</a> is not a group object in the opposite of the category of Poisson algebras because inversion negates the Poisson bracket.
- The <a href="http://en.wikipedia.org/wiki/Poisson_manifold#Product_manifold">standard choice of bracket</a> on the tensor product of two Poisson algebras is <a href="http://mathoverflow.net/questions/13454/is-a-poisson-group-a-group-object-in-the-category-of-poisson-manifolds">not a categorical coproduct</a> (if I have the correct general definition: it's defined by the requirements that it restricts to the given brackets on two Poisson algebras $A, B$ and that every element of $A$ Poisson-commutes with every element of $B$).

This suggests to me that if we used a different choice of morphisms, we might get actual group objects and an actual categorical coproduct. So are there any nice choices that do this?

I read somewhere on MO that the correct definition of a morphism between Poisson manifolds is a Lagrangian submanifold of their product. How does this generalize to Poisson algebras? Does it fix the two issues above? (I'm a little more pessimistic about the second issue, so if there's a different general principle that leads to the standard choice, I would be interested in hearing about that as well.)