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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

Relation to symmetric monoidal closed categories

A compact closed category $(\mathcal{C}, \otimes)$ becomes a closed symmetric monoidal category if we give it the internal hom defined by

where $A^*$ is the dual object of $A$. To see this, we use the adjunction that defines dual objects:

This is what the terminology “compact closed” 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

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

\begin{remark}\label{TensorAdjunctabilityDoesNotImplyCompactClosure} (Tensor-adjunctability does not imply compact closure) \linebreak

As noted above, every compact closed category $(\mathcal{C}, \otimes)$ is symmetric monoidal closed. Moreover this symmetric monoidal closed category has an additional property: for each object $A \in \mathcal{C}$ there is an object $\widehat{A} \in \mathcal{C}$ such that the functor $\widehat{A} \otimes - : \mathcal{C} \to \mathcal{C}$ is right adjoint to $A \otimes - : \mathcal{C} \to \mathcal{C}$. (Simply take $\widehat{A} = A^*$.) However, not every symmetric monoidal closed category with this additional property is compact closed. A counterexample is indicated by Noah Snyder in math.SE:a/692318, referring to Exp. 2.20 in arXiv:1406.4204. See also this n-Café discussion. \end{remark}

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](#JoyalStreetVerity96)]:

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

\begin{theorem} If a compact closed category has binary products that distribute over binary coproducts, it is thin. \end{theorem} \begin{proof} By Lemma 4 of [[Houston 08](#Houston08)], whose proof only requires binary products and coproducts, for any objects $A$ and $B$ the canonical morphism

is invertible, which we can write as

This map factors through

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. \end{proof}

Examples

\begin{example} (finite-dimensional vector spaces) \linebreak 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.) \end{example}

\begin{example} A compact closed discrete category is just an abelian group. \end{example}

\begin{example} The delooping $\mathbf{B}M$ of a commutative monoid $M$ is a compact closed category, and conversely, any compact closed category (or with more one generally, object anyclosed monoidal category$I$ ) with and a single object must be isomorphic to the delooping of some commutative monoid. \end{example}$hom(I,I) = M$. $\mathbf{B}M$ naturally becomes monoidal with multiplication in $M$ as both composition and tensoring of morphisms, by the Eckmann-Hilton argument, and it becomes symmetric monoidal with the identity as the symmetry. This symmetric monoidal category is compact closed with $I^* = I$ and the identity as unit and counit. Conversely, any monoidal category with a single object must be isomorphic to the delooping of some commutative monoid, so any monoidal category with one object is compact closed. \end{example}

Related concepts

• compact closed 2-category

• compact closed dagger category

• linear logic

• pregroup grammar

References

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

• Brian Day, Note on compact closed categories, J. Austral. Math. Soc. (Series A) 24 309-311 (1977) [[doi:10.1017/S1446788700020334](https://doi.org/10.1017/S1446788700020334)]

Discussion of coherence in compact closed categories is due to:

• Max Kelly, M. L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra, 19 193-213 (1980) [, pdf]

On the relation to traced monoidal categories:

• André Joyal, Ross Street, Dominic Verity, Traced monoidal categories, Math. Proc. Camb. Phil. Soc. 119 (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), doi:10.1017/S0305004100074338]

See also:

• Wikipedia, Compact closed category

On the relation to quantum operations and completely positive maps:

• Peter Selinger, Dagger compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer Science 170 (2007) 139-163, [[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018)]

On biproducts:

• Robin Houston, Finite products are biproducts in a compact closed category, Journal of Pure and Applied Algebra 212 2 (2008) 394-400 [[arXiv:math/0604542](https://arxiv.org/abs/math/0604542), doi:10.1016/j.jpaa.2007.05.021]

On compact closure in homotopical algebra and relating to the Barrat-Priddy theorem:

• Amit Sharma, Compact closed categories and Γ-categories (with an appendix by André Joyal), Theory and Applications of Categories 37 37 (2021) 1222-1261 [[arXiv:2010.09216](https://arxiv.org/abs/2010.09216), tac:37-37]