Dual pair in a monoidal category ($(X,Y,e,n)$)

Objects $X,Y$ with an evaluation morphism $e:X\otimes Y\to I$ and a coevaluation morphism $n:I\to Y\otimes X$ satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

Closing an identity strand using compatible evaluation morphisms and coevaluation morphisms gives the categorical dimension in a pivotal or ribbon setting. For the standard two-dimensional quantum enveloping algebra of sl2 module, the weighted contraction has diagonal coefficients $q^{-1},q$, giving quantum dimension $q+q^{-1}=[2{]}_q$. The chosen duality and framing conventions matter.

The positive square root of the sum of the squared quantum dimensions of all simple anyons. The vacuum anyon has $d_1=1$ and antianyons have equal quantum dimensions.

The two triangular identities $(e\otimes 1_X)(1_X\otimes n)=1_X$ and $(1_Y\otimes e)(n\otimes 1_Y)=1_Y$, with canonical associators and unitors inserted, for a dual pair in a monoidal category.

The unit-valued duality map $n:I\to Y\otimes X$ of a dual pair in a monoidal category, constrained together with evaluation by both snake identities.

For a dual pair in a monoidal category, the contraction $e:X\otimes Y\to I$. In a left closed monoidal category, evaluation also denotes the adjunction map $[M,N]\otimes M\to N$.