Idea of the monoidal structure

Construction 19.4.5 (Idea of the monoidal structure). We briefly indicate how Lurie equips \(C[M]\) with a monoidal structure. For each \(n\geq 0\), the twisted arrow poset \(\Tw ^r([n])\) from Definition 13.1.6 organizes the subintervals of \([n]\). Sending an interval \((i\leq j)\) to its length, and a subinterval inclusion \((k\leq l)\subseteq (i\leq j)\) to the dual of the translated interval inclusion \([l-k]\hookrightarrow [j-i]\), gives a diagram \[ \Tw ^r([n]) \longrightarrow \simp \catop \xrightarrow {(-,0)} \simp \catop \times [1] \xrightarrow {\LCut } \oLMod ^{\otimes }. \] A suitable lift to \(D^{\otimes }\) records a string of enriched morphisms \[ M_0\longrightarrow M_1\longrightarrow \cdots \longrightarrow M_n \] together with all their coherent composites. Relative Kan extension makes these lifts functorial in \([n]\), and restriction to the consecutive subintervals satisfies the Segal condition. Taking the fiber over the constant string \((M,\ldots ,M)\) therefore produces an \(\bbA _{\infty }\)-monoidal structure on \(C[M]\). We refer to [Lurie (2017), Definitions 4.7.1.5--4.7.1.6, Proposition 4.7.1.13] for the relative Kan extension and the verification of the Segal condition.

Under the comparison between \(\bbA _{\infty }\)-monoidal categories and monoidal \(\infty \)-categories obtained by applying Theorem 19.3.2 to \(\Cat _{\infty }\), the resulting tensor product is given on objects by \[ (A,\eta ) \otimes (A',\eta ') = \left (A \otimes A', A \otimes A' \otimes M \xrightarrow {\id _A \otimes \eta '} A \otimes M \xrightarrow {\eta } M\right ). \]