With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A closed monoidal category is a monoidal category that is also a closed category, in a compatible way:
it has for each object a functor of forming the tensor product with , as well as a functor of forming the internal-hom with , and these form a pair of adjoint functors.
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
naturally in all three arguments, where is the standard cartesian product of sets. This natural isomorphism is called currying.
Currying can be read as a characterization of the internal hom and is the basis for the following definition.
Symmetric closed monoidal category
A symmetric monoidal category is closed if for all objects the functor has a right adjoint functor .
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).
Cartesian closed monoidal category
If the monoidal structure of is cartesian (and so in particular symmetric monoidal), then is called cartesian closed. In this case the internal hom is often called an exponential object and written .
Left-, right- and bi-closed monoidal category
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.)
For a closed monoidal category with internal hom denoted , then not only are there natural bijections
but these isomorphisms themselves “internalize” to isomorphisms in of the form
By the external natural bijections there is for every a composite natural bijection
Since this holds for every , the Yoneda lemma (namely the fully faithfulness of the Yoneda embedding) implies that there is already an isomorphism
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 category Ab of abelian groups with its tensor product of abelian groups is closed: for any two abelian groups the set of homomorphisms carries (pointwise defined) abelian group structure.
A discrete monoidal category (i.e., a monoid) is left closed iff it is right closed iff every object has an inverse (i.e., it is a group).
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 Cat is cartesian closed: the internal-hom is the functor category of functors and natural transformations.
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 strict -categories is also biclosed monoidal, under the Crans-Gray tensor product.
If is a monoidal category and is endowed with the tensor product given by the induced Day convolution product, then the category of presheaves 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. For more on this phenomenon see at Tannaka duality.
Let be a complete closed monoidal category and any small category. Then the functor category is closed monoidal with the pointwise tensor product, .
Since is complete, the category is comonadic over ; the comonad is defined by right Kan extension along the inclusion . Now for any , consider the following square:
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.
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
- Max Kelly, Basic concepts of enriched category theory, section 1.5, (tac)
has a chapter on just closed monoidal categories.
See also the article
on the concept of closed categories.