More explicitly, with \(\mathcal D,\mathcal C_1,\mathcal C_2,F,g,f\) as in Definition 2.4.5, a prebraiding on \(F\) with components \(\beta_{x,y}\) is a prebraiding on \(F\) over \(\mathcal D\) if the isomorphism \[g \circ \beta_{x,y}\colon g(F(x)) \boxtimes g(F(y))\simeq g(F(x) \boxtimes F(y)) \rightarrow g(F(y) \boxtimes F(x)) \simeq g(F(y)) \boxtimes g(F(x))\] coincides with the given braiding isomorphism on \(\mathcal D\), i.e. with \(f(x) \boxtimes f(y) \rightarrow f(y)\boxtimes f(x)\) for all pairs of objects \(x,y\in\mathcal C_1\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2