[002G]
Theorem 2.6.4.
Let \(F\colon \mathcal A\rightarrow\mathcal B\) be a monoidal functor between monoidal \(1\)-categories. Then the following are equivalent:
the set \(\mathrm{PreBraid}(F)\) of prebraidings on \(F\);
the set of strict monoidal factorizations of \(F\) through \(\mathrm{ev}_{1_{\mathcal A}} \colon Z(F) \rightarrow\mathcal B\), i.e. the set of monoidal functors \(s\colon \mathcal A\rightarrow Z(F)\) such that \(\mathrm{ev}_{1_{\mathcal A}} \circ s=F\).
the 1-groupoid of weak monoidal factorizations of \(F\) through \(\mathrm{ev}_{1_{\mathcal A}} \colon Z(F) \rightarrow\mathcal B\), i.e. the groupoid whose objects are pairs \((s,\eta)\) of a monoidal functor \(s \colon \mathcal A\rightarrow Z(F)\) and a monoidal natural isomorphism \(\eta \colon \mathrm{ev}_{1_{\mathcal A}} \circ s \Rightarrow F\) and whose morphisms \((s, \eta) \rightarrow(s', \eta')\) are monoidal natural isomorphisms \(\mu \colon s\Rightarrow s'\) such that \[\left( \mathrm{ev}_{1_{\mathcal A}} \circ s \xRightarrow{\mathrm{ev}_{1_{\mathcal A}} \circ \mu} \mathrm{ev}_{1_{\mathcal A}} \circ s' \xRightarrow{\eta'} F \right)=\left( \mathrm{ev}_{1_{\mathcal A}} \circ s \xRightarrow{\eta}F\right).\]
[002K]
Proof.
For the equivalence between ([002H]) and ([002I]), note that a factorization \(s\) must send, on the level of objects, \(x\) to \((F(x),\gamma)\) for some \(\gamma\), and on the level of morphisms \(f\) to \(F(f)\). The isomorphisms used for a prebraiding \(\beta\) uniquely define the isomorphisms encoded in a possible \(\gamma\). The second hexagon axiom from prebraidings ([001L]) translates into the required properties of \(\gamma\), whereas the first hexagon translates into the monoidality of \(s\).
The equivalence between ([002I]) and ([002J]) follows from abstract-nonsense: Recall that a monoidal functor \(F:\mathcal X\rightarrow\mathcal Z\) between monoidal \(1\)-categories is called an isofibration if for all isomorphisms \(\gamma \colon z \rightarrow z'\) in \(\mathcal Z\) and \(x \in \mathcal X\) with \(F(x) = z\), there exists an isomorphism \(\mu \colon x\rightarrow x'\) with \(F(\mu) = \gamma\). It is then an exercise to show that if \(F \colon \mathcal X\rightarrow\mathcal Z\) is a monoidal functor which is a faithful isofibration and \(G\colon \mathcal Y\rightarrow\mathcal Z\) is another monoidal functor, the groupoid of weak monoidal factorizations, i.e. of pairs \((s, \eta)\) of a monoidal functor \(s \colon \mathcal Y\rightarrow\mathcal X\) and a monoidal natural isomorphism \(F \circ s \simeq G\) is equivalent to a discrete groupoid isomorphic to the set of strict monoidal factorizations, i.e. the set of monoidal functors \(s\) such that \(F \circ s = G\). The equivalence between ([002I]) and ([002J]) then follows since \(\mathrm{ev}_{1_{\mathcal A}} \colon Z(f) \rightarrow\mathcal B\) is indeed a faithful isofibration. ◻