category with duals (list of them)
dualizable object (what they have)
The strategy for formalizing the idea of a closed category, that “the collection of morphisms from to can be regarded as an object of itself”, is to mimic the situation in Set where for any three objects (sets) , , we have an isomorphism
Currying can be read as a characterization of the internal hom and is the basis for the following definition.
This means that for all we have a bijection
natural in all arguments.
The object is called the internal hom of and . This is commonly also denoted by lower case (and then often underlined).
If the monoidal structure of is cartesian, then is called cartesian closed. In this case the internal hom is often called an exponential and written .
If is not symmetric, then and are different functors, and either one or both may have an adjoint. The terminology here is less standard, but many people use left closed, right closed, and biclosed.
The category of abelian groups is closed: for any two abelian groups the set of homomorphisms carries (pointwise defined) abelian group structure.
Certain nice categories of topological spaces are cartesian closed: for any two nice enough topological spaces , the set of continuous maps can be equipped with a topology to become a nice topological space itself.
Certain nice categories of based topological spaces are closed symmetric monoidal. The monoidal structure is the smash product and the internal-hom is the set of basepoint-preserving maps with topology induced from the space of unbased ones.
The category of strict 2-categories and strict 2-functors is closed symmetric monoidal under the Gray tensor product. The internal-hom is the 2-category of strict 2-functors, pseudo natural transformations, and modifications.
If is a monoidal category and is endowed with the tensor product given by the induced Day convolution product, then is biclosed monoidal.
The category of species, with the monoidal structure given by substitution product of species, is closed monoidal (each functor admits a right adjoint) but not biclosed monoidal.
The category of modules over any Hopf monoid in a closed monoidal category, or more generally algebras for any Hopf monad, is again a closed monoidal category. In particular, the category of modules over any group object in a cartesian closed category is (cartesian) closed monoidal.
This commutes because the tensor product in is pointwise (here means the family of objects in ). Since is closed, has a right adjoint. Since the vertical functors are comonadic, the (dual of the) adjoint lifting theorem implies that has a right adjoint as well.
closed monoidal category , closed monoidal (∞,1)-category
In enriched category theory the enriching category is taken to be closed monoidal. Accordingly the standard textbook on enriched category theory
has a chapter on just closed monoidal categories.
See also the article
on the concept of closed categories.