ScalingStacks

1.4.5 Prebraidings via \(\infty\)-operads[000E]

To lift our \(1\)-categorical prebraiding to braided monoidal structures on \((\infty,2)\)-categories, we first generalize prebraidings themselves to this \((\infty,2)\)-categorical setting.

We implement the notion of a prebraiding in the context of \(\infty\)-operads as indicated in the diagram \[\mathbb T_2 \otimes \mathbb E_1 \longrightarrow \mathbb A_2 \otimes \mathbb E_1 \longrightarrow \mathbb E_1 \otimes \mathbb E_1 \stackrel{\sim}{\longrightarrow} \mathbb E_2\] thereof, as we now explain. For more details see subsection A.8 and section 7.

• As indicated in Subsection 1.4.1, we formalize the notion of a braided monoidal structure via the notion of an \(\mathbb E_2\)-algebra (with \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^2)\)).

• Much simpler is the notion of an \(\mathbb E_1\)-algebra, which has \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^1)\). These spaces are discrete, and this \(\infty\)-operad is equivalent to the ordinary associative operad.

• The equivalence \(\mathbb E_1 \otimes \mathbb E_1 \xrightarrow{\sim} \mathbb E_2\) is an instance of Dunn additivity. Hence, an \(\mathbb E_2\)-algebra structure is equivalent to two compatible \(\mathbb E_1\)-algebra structures: \(\mathrm{Alg}_{\mathbb E_2}(\mathcal V) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathrm{Alg}_{\mathbb E_1}(\mathcal V))\).

• Whereas \(\mathbb E_1\) parametrizes (homotopy-coherently) associative and unital binary operations, by forgetting associativity one arrives at the operad \(\mathbb A_2\), which parametrizes binary operations that are merely unital.

• The \(\infty\)-operad \(\mathbb T_2\) is a two-colored ordinary operad, and parametrizes pairs of pointed objects \(\mathcal A,\mathcal B\in \mathrm{Alg}_{\mathbb E_0}(\mathcal V)\)⁹ together with pointed morphisms \(\mathcal A\xrightarrow{F} \mathcal B\), and \(\mathcal A\otimes \mathcal A\xrightarrow{\mu} \mathcal B\), along with pointed homotopies \(\mu(\eta_\mathcal A\otimes (-)) \simeq F \simeq \mu((-) \otimes \eta_\mathcal A)\) in \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal V)}(\mathcal A,\mathcal B)\). In particular, we view a \(\mathbb T_2\)-algebra as having not an underlying object but having an underlying morphism, namely the morphism \(F\). The map of \(\infty\)-operads \(\mathbb T_2 \rightarrow\mathbb A_2\) corepresents the operation carrying an \(\mathbb A_2\)-algebra to the evident \(\mathbb T_2\)-algebra with \(\mathcal A= \mathcal B\) and \(F= \mathrm{id}_{\mathcal A}\).

We define a prebraiding on an \(\mathbb E_1\)-algebra morphism \(F\) in a symmetric monoidal \(\infty\)-category \(\mathcal V\) to be a \(\mathbb T_2\)-algebra structure on the correponding morphisms in \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V)\), or equivalently a lift of the corresponding \([1] \otimes \mathbb E_1\)-algebra in \(\mathcal V\) to a \(\mathbb T_2 \otimes \mathbb E_1\)-algebra, see subsection 8.1. In the context of ordinary categories, we show in example 8.1.2 that this notion coincides with the one described in Subsection 1.4.4.