Corollary 7.4.15 (00FA) · ScalingStacks

Let \(F: \mathcal A\rightarrow\mathcal B\) be an ordinary monoidal functor between ordinary monoidal \(1\)-categories.

A \(\mathbb T_2\otimes \mathbb E_1\)-structure on \(F\) is a prebraiding on \(F\) in the sense of definition 2.4.1. More precisely, the space \(\mathbb T_2(F)\) of \(\mathbb T_2\)-structures on \(F\) is discrete and equivalent to the set \(\mathrm{PreBraid}(F)\) of prebraidings on \(F\) defined in definition 2.4.8.
An \(\mathbb A_2 \otimes \mathbb E_1\)-structure on \(\mathcal A\) is a braiding on the monoidal category \(\mathcal A\), in the usual \(1\)-categorical sense. More precisely, the space \(\mathbb A_2(\mathcal A)\) of \(\mathbb A_2\)-structures on \(\mathcal A\) is discrete and equivalent to the set \(\mathrm{Braid}(\mathcal A)\) of braidings on \(\mathcal A\) defined in definition 2.4.8.

[00FD]

Proof.

By proposition 7.4.11, a \(\mathbb T_2 \otimes \mathbb E_1\)-structure on \(F\) is a lift in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})\): Original paper diagram Explicitly, the space of \(\mathbb T_2\otimes \mathbb E_1\)-structures on \(F\) is therefore the \(1\)-groupoid of weak lifts from theorem 2.6.4.([002J]). By theorem 2.6.4, this is equivalent to the set of prebraidings on \(F\) in the sense of definition 2.4.8. This completes the proof of part ([00FB]). Part ([00FC]) follows by applying statement ([00FB]) to the case \(F=\mathrm{id}_{\mathcal A}\). ◻