ScalingStacks

The braided monoidal 2-category $\boldsymbol{\mathrm{KhR}}_{N}$ has duals in the sense of [BMS12]. This is a slight modification of the duality proposed by [BL03] and used by [Mac99]. Following [BMS12], instead of three dualities we only consider two dualities $\#$ and $*$ which correspond to rotations by $\pi$ in two different axes.

For an object $A$, the dual object $A^{\#}$ is obtained by reversing the word $A$ and then exchanging orientations $\uparrow\leftrightarrow\downarrow$. On identity 1-morphisms, this corresponds to the result of a $\pi$ rotation in a vertical line, followed by a change of orientation. There are unit and counit 1-morphisms $i_{A}\colon I\to A\otimes A^{\#}$ and $e_{A}\colon A^{\#}\otimes A\to I$ given by nested collections of cups and caps, as well as a triangulator 2-isomorphism $T_{A}\colon(i_{A}\otimes A)(A\otimes e_{A})\to A$ represented by the obvious string-straightening isotopy. It is clear that $A^{\#\#}=A$.

Every 1-morphism $f\colon A\to B$ in $\boldsymbol{\mathrm{KhR}}_{N}$ has a simultaneous left and right adjoint $f^{*}\colon B\to A$ which is given by the Morse data of the result of reflecting $r(f)$ by $\pi$ in a horizontal axis and then reversing orientations (previously we have suggestively drawn this as a reflected $f$ in figures). Further, there are unit and counit 2-morphisms $i_{f}\colon\mathbf{1}_{A}\to ff^{*}$ and $e_{f}\colon f^{*}f\to\mathbf{1}_{B}$, which satisfy the expected identities $(i_{f}f)(fe_{f})=\mathbf{1}_{f}$ and $(f^{*}i_{f})(e_{f}f^{*})=\mathbf{1}_{f^{*}}$. It is clear that $f^{**}=f$.

For any 2-morphism $\alpha\colon f\to g$, we denote by $\alpha^{*}\colon g^{*}\to f^{*}$ the 2-morphism obtained as the image under the isomorphism

induced by a planar anticlockwise $\pi$-rotation of the shown link diagrams. The dualities $*$ and $\#$ satisfy a host of unsurprising compatibility relations with the tensor product and the horizontal and vertical composition, which are consequences of the functoriality of $\mathrm{KhR}_{N}$. The only non-trivial relation is that for $\alpha\in\boldsymbol{\mathrm{KhR}}_{N}(f,g)$ we have $\alpha^{**}=\alpha$, which is implicit in Definition 2.5, using the fact that foams in $\boldsymbol{\mathrm{Foam}}_{N}$ are considered up to isotopy relative to the boundary.