Compact Closed Category

definition example theorem proof

Let $(\mathcal{C},I,\otimes )$ be a Symmetric Monoidal Category and $c$ an object. A dual for $c$ consists of

(i) an object $c^{\ast }$, the dual of $c$; (ii) a morphism ${\eta }_c:I\to c^{\ast }\otimes c$, the unit (drawn as a cup); (iii) a morphism ${\epsilon }_c:c\otimes c^{\ast }\to I$, the counit (a cap),

satisfying the snake equations (zig-zag identities):

$$c\cong c\otimes I\stackrel{c\otimes {\eta }_c}{\to }c\otimes (c^{\ast }\otimes c)\cong (c\otimes c^{\ast })\otimes c\stackrel{{\epsilon }_c\otimes c}{\to }I\otimes c\cong c\text{is }{\mathrm{id}}_c,$$

and symmetrically $c^{\ast }\to c^{\ast }\otimes c\otimes c^{\ast }\to c^{\ast }$ is ${\mathrm{id}}_{c^{\ast }}$. If every object has a dual, $\mathcal{C}$ is compact closed.

Sources: 7 Sketches §4.5 (Definition 4.58, Eq. 4.59, Proposition 4.60, Examples 4.61, Theorem 4.63, Exercises 4.62, 4.64–4.66), §4.1 (Eq. 4.1), §6.6; DaoFP §15.1 (string diagrams: cups and caps for adjunctions), §19.1; [Sel10]; St Clere Smithe & Perin arXiv:2503.18608 (notes) Remark 8, Example 4.

Wiring diagrams with feedback

In a compact closed category wires carry a direction: a forward wire labelled $c$ equals a backward wire labelled $c^{\ast }$. The cup ${\eta }_c$ and cap ${\epsilon }_c$ let any wire reverse direction, so outputs can be bent into inputs — the feedback loops of Co-design diagrams (Eq. 4.1: the chassis carries the weight of the motor that powers it) and of Eq. 4.55–4.57 (“Person 1” emits sound and fury, “Person 2” receives sound and emits a complaint; bending the fury wire back). The snake equations say a zig-zag wire can be straightened. “This same structure shows up in quantum mechanics and dynamical systems.”

Examples

• ${\mathbf{Prof}}_{\mathcal{V}}$ and $\mathbf{Feas}$ (Theorem 4.63): $\otimes$ is the product of $\mathcal{V}$-categories, $I=1$, ${\mathcal{X}}^{\ast }={\mathcal{X}}^{\mathrm{op}}$, and ${\eta }_{\mathcal{X}}(1,x,x^{\prime })={\epsilon }_{\mathcal{X}}(x,x^{\prime },1)=\mathcal{X}(x,x^{\prime })$ — “the unit and counit look like identities” (7S Exercise 4.66). This is what “puts actual mathematics behind” co-design diagrams: a resource required can be read as a resource provided in the opposite preorder.
• $\mathbf{Corel}$ (Example 4.61): finite sets and corelations (equivalence relations on $A\bigsqcup B$), monoidal under $\bigsqcup$, every set self-dual with ${\eta }_A:\varnothing \to A\bigsqcup A$ and ${\epsilon }_A:A\bigsqcup A\to \varnothing$ both the “pairing” equivalence relation $\{(a,1),(a,2)\}$ (7S Exercise 4.62).
• ${\mathbf{FinVect}}_k$ with $V^{\ast }$ the dual space, $\eta$ the identity tensor $\sum e_i^{\ast }\otimes e_i$ and $\epsilon$ evaluation — the origin of the name; $\mathbf{Rel}$ with $\bigsqcup$ or $\times$; $\mathbf{Cosp}$ and every Hypergraph Category (each object self-dual via the Frobenius cup and cap).
• $\mathbf{Set}$ with $\times$ is not compact closed (a set with more than one element has no dual), though it is cartesian closed.

Proposition 4.60

If $\mathcal{C}$ is compact closed then (1) $\mathcal{C}$ is monoidal closed with $c⊸d:=c^{\ast }\otimes d$ and the isomorphism $\mathcal{C}(b\otimes c,d)\cong \mathcal{C}(b,c^{\ast }\otimes d)$ given by precomposing with ${\mathrm{id}}_b\otimes {\eta }_c$ (and using the snake equations for the inverse); (2) duals are unique up to isomorphism; (3) $c\cong c^{\ast \ast }$. Compact closed categories are thus a special kind of closed monoidal category, hence the name; in $\mathbf{Feas}$ the internal hom $X^{\mathrm{op}}\times Y$ is the preorder whose elements are exactly the pairs in a feasibility relation. DaoFP’s string diagrams for adjunctions use the same cups/caps: an adjunction is a “dual pair” of 1-cells in the 2-category $\mathbf{Cat}$.

Open models: cups clamp data

In the AutoBayes framework, open models with copiers, cups $1\nrightarrow \bullet A\otimes A$ and (unnormalised) caps form a self-dual compact closed bicategory (St Clere Smithe & Perin, Remark 8). A cup bends an unobserved leg into an observed one: composing $X\otimes c$ after ${\mathrm{cup}}_X$ makes both inputs and outputs observed, which is supervised learning — clamping labels to data (Example 4). Wires can be bent around, so a model no longer has a fixed input/output direction, and feedback loops (cyclic models) become expressible — at the price of unnormalised measures. Compact closure bends one wire at a time; wiring an arbitrary graph, where a variable meets $d$ factors, needs the stronger structure of a Hypergraph Category.

JuliaLeanHaskell

Docs: Theories & presentations

# Catlab: the GAT of compact closed categories, with duals, units (cups) and counits (caps)
using Catlab
@present CC(FreeCompactClosedCategory) begin
  (A, B)::Ob
  f::Hom(A, B)
end
A, B, f = CC[:A], CC[:B], CC[:f]
dual(A)                                    # A*
dunit(A)                                   # η_A : I → A* ⊗ A
dcounit(A)                                 # ε_A : A ⊗ A* → I
mate(f)                                    # the transpose B* → A*, built from η_A and ε_B

#check CategoryTheory.ExactPairing        -- (X Y : C): coevaluation η : 𝟙_C ⟶ X ⊗ Y, evaluation ε : Y ⊗ X ⟶ 𝟙_C, zig-zag laws
#check CategoryTheory.HasLeftDual
#check CategoryTheory.RightRigidCategory  -- every object has a right dual
#check CategoryTheory.RigidCategory       -- both duals: compact closed when symmetric

-- a dual pair in a monoidal category: cup and cap satisfying the snake equations (unenforced)
data DualPair c c' = DualPair
  { cup :: () -> (c', c)       -- η_c : I → c* ⊗ c   (only meaningful in a linear/ finite setting)
  , cap :: (c, c') -> ()       -- ε_c : c ⊗ c* → I
  }
-- Hask with (,) is not compact closed; FinVect-style duals live in linear-algebra libraries