This is an introductory post about some easy examples of internal Hom.
Definition. Let be a symmetric monoidal category, i.e. a category with a functor that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in is a functor
such that is a left adjoint to for any , i.e. there are functorial isomorphisms
Remark. In the easiest examples, we typically think of as ‘upgrading to an object of ‘:
Example. Let be a commutative ring, and let be the category of -modules, with the tensor product. Then with its natural -module structure is an internal Hom, by the usual tensor-Hom adjunction:
The same is true when is the category of -bimodules for a not necessarily commutative ring .
However, we cannot do this for left -modules over a noncommutative ring, because there is no natural -module structure on for left -modules and . In general, the tensor product takes an -bimodule and a -bimodule and produces an -bimodule . Taking gives a way to tensor a right -module with a left -module, but there is no standard way to tensor two left -modules, let alone equip it with the structure of a left -module.
Example. Let . Then is naturally a set, making it into an internal Hom for :
When is the categorical product , the internal (if it exists) is usually called an exponential object, in analogy with the case above.
Example. Another example of exponential objects is from topology. Let be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space the functor does not preserve colimits and hence cannot admit a right adjoint.)
Example. An example of a slightly different nature is chain complexes: let be a commutative ring, and let be the category of cochain complexes
of -modules (meaning each is an -module, and the are -linear maps satisfying ). Homomorphisms are commutative diagrams
and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.
There isn’t a natural way to ‘endow with the structure of a chain complex’, but there is an internal Hom given by
with differentials given by
Then we get for example
since a morphism is given by an element such that , i.e. , meaning that is a morphism of cochain complexes.
Example. The final example for today is presheaves and sheaves. If is a topological space, then the category of abelian sheaves on has an internal Hom given by
with the obvious transition maps for inclusions of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.