The Morita $(\mathrm{\infty },2)$-Category of a Monoidal Category as a $2$-Complicial Set

We present an explicit and elementary construction of the Morita $(\mathrm{\infty },2)$-category associated to a monoidal category ${𝒞}^{\otimes }$ satisfying minimal conditions [1]. This construction provides a self-contained approach to understanding the higher categorical structure formed by monoids, bimodules, and bimodule maps.

Our main result constructs this structure as a 3-coskeletal 2-complicial set $M^{\mathrm{♮}}({𝒞}^{\otimes })$—a model for $(\mathrm{\infty },2)$-categories based on marked simplicial sets due to Verity [2]. In this construction, vertices encode monoids, edges represent bimodules, triangles capture bimodule maps ${\phi }_{012}:M_{01}{\otimes }_{A_1}{M}_{12}\to M_{02}$ from balanced tensor products, and tetrahedra encode coherence conditions of the form

$$({\phi }_{012}{\otimes }_{A_2}{M}_{23})\bullet {\phi }_{023}={\alpha }_{M_{01}|M_{12}|M_{23}}\bullet (M_{01}{\otimes }_{A_1}{\phi }_{123})\bullet {\phi }_{013}.$$

The marking distinguishes invertible structure: marked edges correspond to invertible bimodules, while marked triangles represent bimodule isomorphisms.

The key technical requirement is that ${𝒞}^{\otimes }$ admits a calculus of balanced tensor products—conditions ensuring that coequalizers of the form

exist and that functors $M\otimes (-)$ and $(-)\otimes P$ preserve them appropriately. This framework encompasses important examples including ${\mathrm{𝐀𝐛}}^{\otimes }$, ${\mathrm{𝐒𝐞𝐭}}^{\coprod }$, and more generally, cocomplete closed monoidal categories.

Rather than directly verifying the bicategory axioms (a lengthy but well-believed result), we leverage the combinatorics of simplicial sets to reformulate these coherence conditions. We prove that $M^{\mathrm{♮}}({𝒞}^{\otimes })$ satisfies the defining properties of a 2-complicial set by establishing the appropriate lifting properties against complicial horns ${\mathrm{\Lambda }}^k[m]\hookrightarrow {\mathrm{\Delta }}^k[m]$, thinness extensions ${\mathrm{\Delta }}^k{[m]}^{\prime }\hookrightarrow {\mathrm{\Delta }}^k{[m]}^{\prime \prime }$, and saturation extensions $\mathrm{\Delta }{[3]}^{\text{eq}}\star \mathrm{\Delta }[\mathrm{\ell }]\hookrightarrow \mathrm{\Delta }{[3]}^{\mathrm{♯}}\star \mathrm{\Delta }[\mathrm{\ell }]$.

This approach offers both a proof of concept for efficiently encoding higher categorical structure and a pathway toward scaling these methods to even higher dimensions, such as when treating braided or symmetric monoidal categories. Our construction recovers known structures—such as $M^{\mathrm{♮}}({\mathrm{𝐀𝐛}}^{\otimes })\simeq N_{\text{Duskin}}({\mathrm{𝐀𝐥𝐠}}^{\mathrm{𝐛𝐢}})$ [3] and $M^{\mathrm{♮}}({\mathrm{𝐒𝐞𝐭}}^{\coprod })\simeq N_{\text{Duskin}}(\mathrm{𝐒𝐩𝐚𝐧})$—while providing an explicit, elementary foundation that avoids heavy machinery.