category with duals (list of them)
dualizable object (what they have)
Let be a commutative ring and consider the multicategory Mod of -modules and -multilinear maps. In this case the tensor product of modules of -modules and can be constructed as the quotient of the tensor product of abelian groups underlying them by the action of ; that is,
The tensor product is the coequalizer of the two maps
given by the action of on and on .
This tensor product can be generalized to the case when is not commutative, as long as is a right -module and is a left -module. More generally yet, if is a monoid in any monoidal category (a ring being a monoid in Ab with its tensor product), we can define the tensor product of a left and a right -module in an analogous way. If is a commutative monoid in a symmetric monoidal category, so that left and right -modules coincide, then is again an -module, while if is not commutative then will no longer be an -module of any sort.
The tensor product of modules can be generalized to the tensor product of functors.
Let be a commutative ring.
For a module, the functor of tensoring with this module
A general abstract way of seeing that the functor is right exact is to notice that is a left adjoint functor, its right adjoint being the internal hom (see at Mod). By the discussion at adjoint functor this means that even preserves all colimits, in particular the finite colimits.
The functor sends this to
Here the morphism on the left is the 0-morphism: in components it is given for all by
Hence this is not a short exact sequence anymore.
One kind of module for which is always exact are free modules.
There are more modules than the free ones for which is exact. One says
For a general module, a measure of the failure of to be exact is given by the Tor-functor . See there for more details.
Detailed discussion specifically for tensor products of modules is in