# About symmetry, braids, and pseudo-functors.

Let $(\mathcal{C}, \otimes, I)$ a monoidal category, and let $\mathbb{B}(\mathcal{C})$ the bicategory (with only one object and $(\mathcal{C}, \otimes, I)$ as (monoidal) category of morphisms and cells). Let $\mathbb{B}(\mathcal{C})^{op}$ the dual respect morphisms (i.e. twisting $\otimes$, and inverting canonical isomorphisms, in the monoidal category $\mathcal{C}$). A pseudo-functor (see ncatlab) $(Q, \theta): \mathbb{B}(\mathcal{C})\to \mathbb{B}(\mathcal{C})^{op}$ that fixes objects, morphisms, and cells, and is unitary (i.e. the canonical isomorphisms $Q(I)\cong I$ is the identity) is a family of isomorphism $q_{A, B}^{-1}: B\otimes A= Q(B)\otimes Q(A)\cong Q(A\otimes B)= A\otimes B$ such that:

U1) $q_{A, I}\circ r_A= l_A$

U2) $q_{I, A}\circ l_A= r_A$

where $r_A: A \cong A\otimes I$, $l_A: A \cong I\otimes A$ canonically.

and with:

$a\circ q_{C\otimes B, A}^{-1}\circ A\otimes q_{C, B}^{-1}= q_{C, B\otimes A}^{-1}\circ q_{B, A}^{-1}\otimes C\circ a^{-1}: A\otimes (B\otimes C) \to C\otimes (B\otimes A)$

or equivalently:

$A\otimes q_{C, B}\circ q_{C\otimes B, A}= a\circ q_{B, A}\otimes C\circ q_{C, B\otimes A}\circ a : (C\otimes B)\otimes A) \to A\otimes (B\otimes C)$

If we further require that $Q^{op}\circ Q=1_{\mathbb{B}(\mathcal{C})}$ then: $q_{A,B}\circ q_{B, A}=1$.

1) is a such pseudofunctor $Q$ a symmetry?