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 natural 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 is monoidal not necessarily symmetric, then left and right tensor product and may be non-equivalent 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 monoidal category to indicate, respectively, that the left tensor product, the right tensor product functors or both have right adjoints. So in particular a symmetric closed monoidal category is automatically biclosed.
The analogue of exponential objects for monoidal categories are left and right residuals.
but these isomorphisms themselves “internalize” to isomorphisms in of the form
By the external natural bijections there is for every a composite natural bijection
The tautological example is the category Set of sets with its Cartesian product: the collection of functions between any two sets is itself a set – the function set. More generally, any topos is cartesian closed monoidal.
The delooping of a commutative monoid is a closed monoidal (in fact, compact closed) category with one object, with both the tensor and internal hom defined (on morphisms) using the multiplication operation . Conversely, any closed monoidal category with one object must be isomorphic to one constructed from a commutative monoid. (See Eilenberg and Kelly (1965), IV.3, p.553.)
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 pointed/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.
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. For more on this phenomenon see at Tannaka duality.
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
Original articles studying monoidal biclosed categories are
Joachim Lambek, Deductive systems and categories, Mathematical Systems Theory 2 (1968), 287-318.
Joachim Lambek, Deductive systems and categories II, Lecture Notes in Math. 86, Springer-Verlag (1969), 76-122.
For more historical development see at linear type theory – History of linear categorical semantics.
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.