nLab
closed monoidal category

Idea

A closed monoidal category $C$ is a monoidal category that is also a closed category, in a compatible way:

it has for each object $X$ a functor $(-)\otimes X:C\to C$ of forming the tensor product with $X$, as well as a functor $[X,-]:C\to C$ of forming the internal-hom with $X$, and these form a pair of adjoint functors.

The strategy for formalizing the idea of a closed category, that “the collection of morphisms from $a$ to $b$ can be regarded as an object of $C$ itself”, is to mimic the situation in Set where for any three objects (sets) $a$, $b$, $c$ we have an isomorphism

$$\mathrm{Hom}(a\otimes b,c)\simeq \mathrm{Hom}(a,\mathrm{Hom}(b,c)),$$Hom(a \otimes b, c) \simeq Hom(a, Hom(b,c)) \,,

naturally in all three arguments, where $\otimes =\times$ is the standard cartesian product of sets. This natural isomorphism is called currying.

Currying can be read as a characterization of the internal hom $\mathrm{Hom}(b,c)$ and is the basis for the following definition.

Definition

A symmetric monoidal category $C$ is closed if for all objects $b\in C_0$ the functor $-\otimes b:C\to C$ has a right adjoint functor $[b,-]:C\to C$.

This means that for all $a,b,c\in C_0$ we have a bijection

$${\mathrm{Hom}}_C(a\otimes b,c)\simeq {\mathrm{Hom}}_C(a,[b,c])$$Hom_C(a \otimes b, c) \simeq Hom_C(a, [b,c])

natural in all arguments.

The object $[b,c]$ is called the internal hom of $b$ and $c$. This is commonly also denoted by lower case $\mathrm{hom}(b,c)$ (and then often underlined).

If the monoidal structure of $C$ is cartesian, then $C$ is called cartesian closed. In this case the internal hom is often called an exponential and written $c^b$.

If $C$ is not symmetric, then $-\otimes b$ and $b\otimes -$ 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.

Examples

• The tautological example is the category Set of sets: the collection of maps between any two sets is itself a set. More generally, any topos is cartesian closed.

• The category of abelian groups is closed: for any two abelian groups $A,B$ the set of homomorphisms $A\to B$ 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 $X$, $Y$ the set of continuous maps $X\to Y$ 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 Cat is cartesian closed: the internal-hom is the functor category of functors and natural transformations.

• The category $2\mathrm{Cat}$ 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 $\omega$-categories is also biclosed monoidal, under the Crans-Gray tensor product.

• If $M$ is a monoidal category and ${\mathrm{Set}}^{M^{\mathrm{op}}}$ is endowed with the tensor product given by the induced Day convolution product, then ${\mathrm{Set}}^{M^{\mathrm{op}}}$ is biclosed monoidal.

• The category of species, with the monoidal structure given by substitution product of species, is closed monoidal (each functor $-\circ G$ admits a right adjoint) but not biclosed monoidal.

Functor categories

Theorem

Let $C$ be a complete closed monoidal category and $I$ any small category. Then the functor category $[I,C]$ is closed monoidal with the pointwise tensor product, $(F\otimes G)(x)=F(x)\otimes G(x)$.

Proof

Since $C$ is complete, the category $[I,C]$ is comonadic over $C^{\mathrm{ob}I}$; the comonad is defined by right Kan extension along the inclusion $\mathrm{ob}I\hookrightarrow I$. Now for any $F\in [I,C]$, consider the following square:

$$\begin{array}{ccc}[I,C]& \stackrel{F\otimes -}{\to }& [I,C]\\ \downarrow & & \downarrow \\ C^{\mathrm{ob}I}& \underset{F_0\otimes -}{\to }& C^{\mathrm{ob}I}\end{array}$$\array{[I,C] & \overset{F\otimes - }{\to} & [I,C] \\ \downarrow && \downarrow\\ C^{ob I}& \underset{F_0 \otimes -}{\to} & C^{ob I}}

This commutes because the tensor product in $[I,C]$ is pointwise (here $F_0$ means the family of objects $F(x)$ in $C^{\mathrm{ob}I}$). Since $C$ is closed, $F_0\otimes -$ has a right adjoint. Since the vertical functors are comonadic, the (dual of the) adjoint lifting theorem implies that $F\otimes -$ has a right adjoint as well.

Related concepts

• monoidal category, monoidal (∞,1)-category

• symmetric monoidal category, symmetric monoidal (∞,1)-category

• closed monoidal category , closed monoidal (∞,1)-category

References

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

• Samuel Eilenberg and Max Kelly, Closed categories. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).

on the concept of closed categories.