With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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 โcompact 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
corresponding to the tensorial strength of the functors $[A,-] : \mathcal{S} \to \mathcal{S}$.
(Tensor-adjunctability does not imply compact closure)
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-Cafรฉ discussion.
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.
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.)
If a compact closed category has binary products that distribute over binary coproducts, it is thin.
By Lemma 4 of [Houston 08], 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.
(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.)
A compact closed discrete category is just an abelian group.
The delooping $\mathbf{B}M$ of a commutative monoid $M$ is a category with one object $I$ and $hom(I,I) = M$. $\mathbf{B}M$ 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.
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:
See also:
On the relation to quantum operations and completely positive maps:
On biproducts:
On compact closure in homotopical algebra and relating to the Barrat-Priddy theorem:
Last revised on August 15, 2024 at 10:56:42. See the history of this page for a list of all contributions to it.