Kerodon

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

Example 2.3.1.17 (The Duskin Nerve of $\mathrm{Bimod}$). Let $\mathrm{Bimod}$ denote the $2$-category of Example 2.2.2.3. Then an $n$-simplex of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }( \mathrm{Bimod})$ can be identified with a collection of abelian groups $\{ A_{j,i} \} _{0 \leq i \leq j \leq n}$ equipped with unit elements $e_{i} \in A_{i,i}$ and bilinear multiplication maps $\cdot : A_{k,j} \times A_{j,i} \rightarrow A_{k,i}$ satisfying the identities $e_{j} \cdot x = x = x \cdot e_ i$ for $x \in A_{j,i}$ and $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ for $x \in A_{\ell ,k}$, $y = A_{k,j}$, and $z \in A_{j,i}$ (where $0 \leq i \leq j \leq k \leq \ell \leq n$). In this case, the multiplication equips each $A_{i,i}$ with the structure of an associative ring (which is an object of the $2$-category $\mathrm{Bimod}$), each $A_{j,i}$ with the structure of an $A_{j,j}$-$A_{i,i}$ bimodule (which is a $1$-morphism in the $2$-category $\mathrm{Bimod}$). For $0 \leq i \leq j \leq k \leq n$, the bilinear map $A_{k,j} \times A_{j,i} \rightarrow A_{k,i}$ can be identified with a map of bimodules $\mu _{k,j,i}: A_{k,j} \otimes _{ A_{j,j} } A_{j,i} \rightarrow A_{k,i}$, which we can regard as a $2$-morphism in the category $\mathrm{Bimod}$.