Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.3.1.18 (The Classifying Simplicial Set of a Monoidal Category). Let $\operatorname{\mathcal{C}}$ be a monoidal category (Definition 2.1.2.10) and let $B\operatorname{\mathcal{C}}$ denote the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example 2.2.2.4). We will denote the Duskin nerve of $B\operatorname{\mathcal{C}}$ by $B_{\bullet }\operatorname{\mathcal{C}}$ and refer to it as the classifying simplicial set of $\operatorname{\mathcal{C}}$. By virtue of Proposition 2.3.1.9, we can identify $n$-simplices of the simplicial set $B_{\bullet }\operatorname{\mathcal{C}}$ with pairs

\[ ( \{ C_{j,i} \} _{0 \leq i < j \leq n}, \{ \mu _{k,j,i} \} _{ 0 \leq i < j < k \leq n} \} \]

where each $C_{j,i}$ is an object of $\operatorname{\mathcal{C}}$ and each $\mu _{k,j,i}$ is a morphism from $C_{k,j} \otimes C_{j,i}$ to $C_{k,i}$, satisfying the following coherence condition:

  • For $0 \leq i < j < k < \ell \leq n$, the diagram

    \[ \xymatrix { C_{\ell , k} \otimes (C_{k,j} \otimes C_{j,i} ) \ar [rr]^-{\alpha _{C_{\ell ,k}, C_{k,j}, C_{j,i} } } \ar [d]_{ \operatorname{id}_{ C_{\ell ,k} } \otimes \mu _{k,j,i} } & & ( C_{\ell ,k} \otimes C_{k,j} ) \otimes C_{j,i} \ar [d]^{ \mu _{\ell ,k,j} \otimes \operatorname{id}_{ C_{j,i} }} \\ C_{\ell , k} \otimes C_{k,i} \ar [dr]_{ \mu _{\ell ,k,i} } & & C_{\ell , j} \otimes C_{j,i} \ar [dl]^{ \mu _{\ell , j, i} } \\ & C_{\ell , i} & } \]

    is commutative.