nLab compact closed category

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 “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

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

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.

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.)

Examples

Example

(finite-dimensional vector spaces)
The category FinDimVect of finite-dimensional vector spaces is compact closed with resoect 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.

Related concepts

• compact closed 2-category

• compact closed dagger category

• linear logic

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]

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) [doi:10.1016/0022-4049(80)90101-2, 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, 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]

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, 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, tac:37-37]