Kerodon

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

Remark 2.3.1.8. Let us make Construction 2.3.1.1 more explicit. Fix a $2$-category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we see that an element of $\operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ consists of the following data:

$(0)$

A collection of objects $\{ X_ i \} _{ 0 \leq i \leq n}$ of the $2$-category $\operatorname{\mathcal{C}}$.

$(1)$

A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i \leq j \leq n }$ in the $2$-category $\operatorname{\mathcal{C}}$

$(2)$

A collection of $2$-morphisms $\{ \mu _{k,j,i}: f_{k,j} \circ f_{j,i} \Rightarrow f_{k,i} \} _{0 \leq i \leq j \leq k \leq n}$ in the $2$-category $\operatorname{\mathcal{C}}$.

These data are required to satisfy the following conditions:

$(a)$

For $0 \leq i \leq n$, the $1$-morphism $f_{i,i}: X_ i \rightarrow X_{i}$ is the identity $1$-morphism $\operatorname{id}_{X_ i}$.

$(b)$

For $0 \leq i \leq j \leq n$, the $2$-morphisms

\[ \mu _{j,j,i}: f_{j,j} \circ f_{j,i} \Rightarrow f_{j,i} \quad \quad \mu _{j,i,i}: f_{j,i} \circ f_{i,i} \Rightarrow f_{j,i} \]

are the left unit constraints $\lambda _{f_{j,i}}$ and the right unit constraints $\rho _{f_{j,i} }$, respectively.

$(c)$

For $0 \leq i \leq j \leq k \leq \ell \leq n$, we have a commutative diagram

\[ \xymatrix { f_{\ell , k} \circ (f_{k,j} \circ f_{j,i} ) \ar@ {=>}[rr]^-{\alpha _{f_{\ell ,k}, f_{k,j}, f_{j,i} } } \ar@ {=>}[d]_{ \operatorname{id}_{ f_{\ell ,k}} \circ \mu _{k,j,i} } & & ( f_{\ell ,k} \circ f_{k,j} ) \circ f_{j,i} \ar@ {=>}[d]^{ \mu _{\ell ,k,j} \circ \operatorname{id}_{ f_{j,i} }} \\ f_{\ell , k} \circ f_{k,i} \ar@ {=>}[dr]_{ \mu _{\ell ,k,i} } & & f_{\ell , j} \circ f_{j,i} \ar@ {=>}[dl]^{ \mu _{\ell , j, i} } \\ & f_{\ell , i} & } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X_ i, X_{\ell } )$.