# nLab closed monoidal category

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## 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 $\left(-\right)\otimes X:C\to C$ of forming the tensor product with $X$, as well as a functor $\left[X,-\right]:C\to C$ 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

$\mathrm{Hom}\left(a\otimes b,c\right)\simeq \mathrm{Hom}\left(a,\mathrm{Hom}\left(b,c\right)\right)\phantom{\rule{thinmathspace}{0ex}},$Hom(a \otimes b, c) \simeq Hom(a, Hom(b,c)) \,,

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

Currying can be read as a characterization of the internal hom $\mathrm{Hom}\left(b,c\right)$ 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 $\left[b,-\right]:C\to C$.

This means that for all $a,b,c\in {C}_{0}$ we have a bijection

${\mathrm{Hom}}_{C}\left(a\otimes b,c\right)\simeq {\mathrm{Hom}}_{C}\left(a,\left[b,c\right]\right)$Hom_C(a \otimes b, c) \simeq Hom_C(a, [b,c])

natural in all arguments.

The object $\left[b,c\right]$ is called the internal hom of $b$ and $c$. This is commonly also denoted by lower case $\mathrm{hom}\left(b,c\right)$ (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 $\left[I,C\right]$ is closed monoidal with the pointwise tensor product, $\left(F\otimes G\right)\left(x\right)=F\left(x\right)\otimes G\left(x\right)$.

###### Proof

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

$\begin{array}{ccc}\left[I,C\right]& \stackrel{F\otimes -}{\to }& \left[I,C\right]\\ ↓& & ↓\\ {C}^{\mathrm{ob}I}& \underset{{F}_{0}\otimes -}{\to }& {C}^{\mathrm{ob}I}\end{array}$\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 $\left[I,C\right]$ is pointwise (here ${F}_{0}$ means the family of objects $F\left(x\right)$ 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.

## 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.

on the concept of closed categories.

Revised on October 12, 2012 22:48:06 by Mike Shulman (192.16.204.218)