  Global and local Weyl Modules were introduced via generators and relations in
the context of affine Lie algebras in a work by the first author and Pressley
and were motivated by representations of quantum affine algebras. A more
general case was considered by Feigin and Loktev by replacing the polynomial
ring with the coordinate ring of an algebraic variety. We show that there is a
natural definition of the local and global modules via homological properties.
This characterization allows us to define the Weyl functor from the category of
left modules of a commutative algebra to the category of modules for a simple
Lie algebra. As an application we are able to understand the relationships of
these functors to tensor products, generalizing previous results. Finally an
analysis of the fundamental Weyl modules proves that the functors are not left
exact in general, even for coordinate rings of affine varieties.
