# nLab compact closed category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Definition

A compact closed category, or simply a compact category, is a symmetric monoidal category in which every object is dualizable, hence a rigid symmetric monoidal category.

More generally, if we drop the symmetry requirement, we obtain a rigid monoidal category, a.k.a. an autonomous category. Thus a compact category may also be called a rigid symmetric monoidal category or a symmetric autonomous category. A maximally clear, but rather verbose, term would be a symmetric monoidal category with duals for objects.

A compact closed category is a special case of the notion of compact closed pseudomonoid in a monoidal bicategory, and similarly for the autonomous cases.

## Properties

### Internal hom and compact closure

A rigid symmetric monoidal category $(\mathcal{C}, \otimes)$ is in particular a closed monoidal category, with the internal hom given by

$[A,B] \;\simeq\; B \otimes A^*$

(where $A^*$ is the dual object of $A$), via the adjunction natural equivalence that defines dual objects

$\mathcal{C}\big(C,[A,B]\big) \simeq \mathcal{C}\big(C, B \otimes A^\ast\big) \simeq \mathcal{C}\big(C \otimes A, B\big) \,.$

This is what the terminology โcompact closedโ refers to.

The inclusion from the category of compact closed categories into the category of closed symmetric monoidal categories also has a left adjoint (Day 1977). Given a closed symmetric monoidal category $\mathcal{S}$, the free compact closed category $C(\mathcal{S})$ over $\mathcal{S}$ may be described as a localization of $\mathcal{S}$ by the maps

$\sigma : [A,B] \otimes C \to [A, B \otimes C]$

corresponding to the tensorial strength of the functors $[A,-] : \mathcal{S} \to \mathcal{S}$.

### Relation to traced monoidal categories

Given a traced monoidal category $\mathcal{C}$, there is a free construction completion of it to a compact closed category $Int(\mathcal{C})$ [Joyal, Street & Verity 1996]:

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and composition of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by tracing out $B^+$ and $B^-$ in the evident way.

Every compact closed category is self-dual, i.e. equivalent to its opposite.

### Relation to star-autonomous categories

A compact closed category is a star-autonomous category: the tensor unit is a dualizing object. Thus it is also an isomix category. (But note that, for example, the symmetric monoidal category of sup-lattices is star-autonomous, with dualizing object given by the unit, but not compact closed. In a compact closed category, the dualizing functor is additionally monoidal.)

### Incompatibility with distributivity

###### Theorem

If a compact closed category has binary products that distribute over binary coproducts, it is thin.

###### Proof

By Lemma 4 of [Houston 08], whose proof only requires binary products and coproducts, for any objects $A$ and $B$ the canonical morphism

$(A\times A)+(B\times B) \to (A+B)\times (A+B)$

is invertible, which we can write as

$A^2 + B^2 \to (A+B)^2.$

This map factors through

$A^2 + B^2 + 2\cdot A\times B$

via the coproduct injection and a pair of distributivity maps. Since the latter are isomorphisms, so is the former. This means that for any object $X$, if there exists a morphism $A^2+B^2 \to X$, then there exists a unique morphism $2\cdot A\times B \to X$.

Now taking $B=X=A$, we observe that there is a morphism $A^2+A^2 \to A$. Therefore, there is a unique morphism $2\cdot A^2 \to A$, and therefore a unique morphism $A^2 \to A$. In particular, the two projections $\pi_1 : A\times A\to A$ and $\pi_2 : A\times A\to A$ are equal, which is to say that $A$ is subterminal. Since $A$ was arbitrary, the category is thin.

## Examples

###### Example

(finite-dimensional vector spaces)
The category FinDimVect of finite-dimensional vector spaces is compact closed with respect to the usual tensor product of vector spaces, see there.

(It is not compact closed with the direct sum as monoidal product.)

###### Example

A compact closed discrete category is just an abelian group.

###### Example

The delooping $\mathbf{B}M$ of a commutative monoid $M$ is a compact closed category, and conversely, any compact closed category (or more generally, any closed monoidal category) with a single object must be isomorphic to the delooping of some commutative monoid.

## References

The characterization of the free compact closed category over a closed symmetric monoidal category is described in

Discussion of coherence in compact closed categories is due to:

On the relation to traced monoidal categories: