Duality in a monoidal category

Let $𝑋$ and $𝑌$ be objects in a monoidal category $𝖢$ . A duality $𝑌 ⊣ 𝑋$ consists of morphisms

e v : 𝑋 \otimes 𝑋 \to 𝟙 c o ë v : 𝟙 \to 𝑋 \otimes 𝑌

called evaluation and coëvaluation satisfying the zigzag identities

(𝑌 \otimes c o ë v) \circ 𝛼 - 1 𝑌 𝑋 𝑌 \circ (e v \otimes 𝑌) = 𝜌 𝑌 \circ 𝜆 - 1 𝑌 (c o ë v \otimes 𝑋) \circ 𝛼 𝑋 𝑌 𝑋 \circ (𝑋 \otimes e v) = 𝜆 𝑋 \circ 𝜌 - 1 𝑋

String diagrams

Representing $𝑋$ as strings going upwards, $𝑌$ as strings going dowards, evaluation as a cap and coëvaluation as a cup, the zigzag identities become

and

hence the name

$𝑌$ is thence called a left dual of $𝑋$ and $𝑋$ a right dual of $𝑌$ .¹ An object with a left or right dual is called dualizable. Left and right duals are unique up to unique isomorphism, thus one sometimes speak of the left dual $𝑋 * = 𝑌$ or the right dual $* 𝑌 = 𝑋$ .

Proof of uniqueness

We prove the left case, the left case is symmetrical. Suppose $𝑌 ⊣ 𝑋$ with the string diagrammatic notation given above but also $𝑅 ⊣ 𝑋$ so that

and

Letting

and

we have

i.e. $𝜓 𝜑 = 1 𝑅$ . A symmetric argument shows $𝜑 𝜓 = 1 𝑌$ , whence $𝜑$ is an isomorphism. Moreover, $𝜑$ is uniquely determined by the compatibility conditions

and

since the zigzag identities give precisely $𝜑' = 𝜓 - 1$ and thus by uniqueness of inverses $𝜑 = 𝜑'$ .

One also calles an object a left or right dual if it has a right or left dual respectively. A monoidal category $𝖢$ is called

left rigid iff every object $𝑋$ has a canonical left dual $𝑋 *$ ;
right rigid iff every object $𝑋$ has a canonical right dual $* 𝑋$ ;
autonomous iff it is both left and right rigid;
pivotal iff there is a natural isomorphism $? * ≃ * ?$ , with the functorial action defined below.

Functoriality

If $𝑅$ and $𝐵$ are right duals, and $𝑓 : 𝑅 \to 𝐵$ is a morphism, we can define the left dual $𝑓 * : 𝐵 * \to 𝑅 *$ .

Moreover if $𝖢$ , the left dual of $𝑅 \otimes 𝐵$ is naturally isomorphic to $𝐵 * \otimes 𝑅 *$ , where evaluation and coëvaluation are

respectively.

Duality in a monoidal category

Functoriality

See also

Footnotes