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

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

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

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

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 96):

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

• The category FinVect of finite-dimensional vector spaces is compact closed when equipped with the tensor product as monoidal structure, see finite-dimensional vector space. It is not compact closed with the direct sum as monoidal product.

• A compact closed discrete category is just an abelian group.

• 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

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. 24 (Series A), 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.

The relation to quantum operations and completely positive maps is discussed in

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

The relation to traced monoidal categories is discussed in

• André Joyal, Ross Street, Dominic Verity, Traced monoidal categories, Math. Proc. Camb. Phil. Soc. (1996), 119, 447 (pdf), doi:10.1017/S0305004100074338

• Wikipedia, Compact closed category .