Kerodon
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Part 1: Higher Category Theory
Chapter 2: Examples of $\infty $-Categories
Section 2.3: The Duskin Nerve of a $2$-Category
Subsection 2.3.1: The Duskin Nerve (cite)
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2.3.1 The Duskin Nerve
In §1.2, we associated to each category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, called the nerve of $\operatorname{\mathcal{C}}$. This construction has a natural generalization to the setting of $2$-categories.
Construction 2.3.1.1 (The Duskin Nerve). Let $n$ be a nonnegative integer and let $[n]$ denote the linearly ordered set $\{ 0 < 1 < 2 < \cdots < n \} $. We will regard $[n]$ as a category, hence also as a $2$-category having only identity $2$-morphisms (Example 2.2.0.6). For any $2$-category $\operatorname{\mathcal{C}}$, we let $\operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}})$ denote the set of all strictly unitary lax functors from $[n]$ to $\operatorname{\mathcal{C}}$ (Definition 2.2.4.16). The construction $[n] \mapsto \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}})$ determines a simplicial set, given as a functor by the composition
\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \hookrightarrow \operatorname{Cat}^{\operatorname{op}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}^{\operatorname{op}} \xrightarrow { \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{ULax}} }( \bullet , \operatorname{\mathcal{C}}) } \operatorname{Set}. \]
We will denote this simplicial set by $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and refer to it as the Duskin nerve of the $2$-category $\operatorname{\mathcal{C}}$.
Remark 2.3.1.2. In the setting of strict $2$-categories, the Duskin nerve $\operatorname{\mathcal{C}}\mapsto \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ was introduced by Street in [MR920944]. The generalization to arbitrary $2$-categories was given by Duskin in [MR1897816].
Example 2.3.1.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, viewed as a $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ of $\operatorname{\mathcal{C}}$ as an ordinary category (Construction 1.2.1.1).
Remark 2.3.1.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{C}}^{\operatorname{op}}$ denote the opposite $2$-category (see Construction 2.2.3.1). Then we have a canonical isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}}^{\operatorname{op}} ) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}}$, where $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}}$ denotes the opposite of the simplicial set $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ (see Notation 1.3.2.1).
Warning 2.3.1.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{C}}^{\operatorname{c}}$ be the conjugate of $\operatorname{\mathcal{C}}$, obtained by reversing vertical composition (Construction 2.2.3.4). There is no simple relationship between Duskin nerves of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}^{\operatorname{c}}$ (since the operation $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{c}}$ is not functorial with respect to lax functors; see Warning 2.2.5.11).
Remark 2.3.1.6 (Functoriality). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ determines a functor from the category $\operatorname{2Cat}_{\operatorname{ULax}}$ of small $2$-categories (with morphisms given by strictly unitary lax functors) to the category $\operatorname{Set_{\Delta }}$ of simplicial sets. This functor fits into the general paradigm of Variant 1.1.7.6: it arises from a cosimplicial object of the category $\operatorname{2Cat}_{\operatorname{ULax}}$, given by the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}$. Beware that, unlike the usual nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$ does not admit a left adjoint: Proposition 1.1.8.22 does not apply, because the category $\operatorname{2Cat}_{\operatorname{ULax}}$ does not admit small colimits (one can address this problem by restricting to strict $2$-categories: we will return to this point in §2.3.6).
Remark 2.3.1.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\{ \mu _{g,f} \} $ be a twisting cochain for $\operatorname{\mathcal{C}}$ (Notation 2.2.6.7), and let $\operatorname{\mathcal{C}}'$ be the twist of $\operatorname{\mathcal{C}}$ with respect to $\{ \mu _{g,f} \} $ (Construction 2.2.6.8). Then the twisting cochain $\{ \mu _{g,f} \} $ defines a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}'$, and therefore induces an isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}')$. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ cannot detect the difference between $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$. This should be regarded as a feature, rather than a bug. Defining the composition law for $1$-morphisms in a $2$-category $\operatorname{\mathcal{C}}$ often requires certain arbitrary (but ultimately inessential) choices (see Example 2.2.6.13). In such cases, one can often give a more direct description of the simplicial set $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ which avoids such choices. See Example 2.3.1.17 and Remark 2.3.5.9.
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 $\operatorname{\mathcal{C}}$.
A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i \leq j \leq n }$ in the $2$-category $\operatorname{\mathcal{C}}$
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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X_ i, X_{\ell } )$.
In the description of Remark 2.3.1.8, it is possible to be more efficient by eliminating some of the "redundant" information.
Proposition 2.3.1.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $n$ be a nonnegative integer. Suppose we are given the following data:
$(1')$
A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq n }$ in the $2$-category $\operatorname{\mathcal{C}}$
A collection of $2$-morphisms $\{ \mu _{k,j,i}: f_{k,j} \circ f_{j,i} \Rightarrow f_{k,i} \} _{0 \leq i < j < k \leq n}$ in the $2$-category $\operatorname{\mathcal{C}}$.
This data can be extended uniquely to an $n$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ (as described in Remark 2.3.1.8) if and only if the following condition is satisfied:
$(c')$
For $0 \leq i < j < k < \ell \leq n$, we have a commutative diagram
Proof. We wish to show that there is a unique way to choose $1$-morphisms $f_{j,i}: X_ i \rightarrow X_ j$ for $i = j$ and $2$-morphisms $\mu _{k,j,i}: f_{k,j} \circ f_{j,i} \Rightarrow f_{k,i}$ for $i = j \leq k$ and $i \leq j = k$ so that conditions $(a)$, $(b)$, and $(c)$ of Remark 2.3.1.8 are satisfied. The uniqueness is clear: to satisfy condition $(a)$, we must have $f_{i,i} = \operatorname{id}_{X_ i}$ for $0 \leq i \leq n$, and to satisfy condition $(b)$ we must have $\mu _{k,j,i} = \rho _{ f_{j,i} }$ when $i = j$ and $\mu _{k,j,i} = \lambda _{ f_{k,j} }$ when $j = k$. To complete the proof, it will suffice to verify the following:
$(I)$
The prescription above is consistent. That is, when $i = j = k$, we have $\rho _{f_{j,i}} = \lambda _{ f_{k,j} }$ (as morphisms of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X_ i, X_ k)$).
$(II)$
The prescription above satisfies condition $(c)$ of Remark 2.3.1.8. That is, the diagram
\[ \xymatrix { f_{\ell , k} \circ (f_{k,j} \circ f_{j,i} ) \ar@ {=>}[rr]^-{\alpha _{f_{\ell ,k}, f_{k,j}, f_{j,i} }} \ar@ {=>}[d]_{ \operatorname{id}_{ f_{\ell ,k} } \circ \mu _{k,j,i} } & & ( f_{\ell ,k} \circ f_{k,j} ) \circ f_{j,i} \ar@ {=>}[d]^{ \mu _{\ell ,k,j} \circ \operatorname{id}_{ f_{j,i} } } \\ f_{\ell , k} \circ f_{k,i} \ar@ {=>}[dr]_{ \mu _{\ell ,k,i} } & & f_{\ell , j} \circ f_{j,i} \ar@ {=>}[dl]^{ \mu _{\ell , j, i} } \\ & f_{\ell , i} & } \]
commutes in the special cases $0 \leq i = j \leq k \leq \ell \leq n$, $0 \leq i \leq j = k \leq \ell \leq n$, and $0 \leq i \leq j \leq k = \ell \leq n$.
Assertion $(I)$ follows from Corollary 2.2.1.15. Assertion $(II)$ follows from the triangle identity in $\operatorname{\mathcal{C}}$ in the case $j = k$, and from Proposition 2.2.1.16 in the cases $i = j$ and $k = \ell $. $\square$
Corollary 2.3.1.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is $3$-coskeletal (Definition ). In other words, if $S_{\bullet }$ is a simplicial set, then any map from the $3$-skeleton $\operatorname{sk}_{3}( S_{\bullet } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ extends uniquely to a map $S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.
Warning 2.3.1.11. Let $\operatorname{\mathcal{C}}$ be a $2$-category. By virtue of Proposition 2.3.1.9, we can identify $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ with triples
\[ ( \{ X_ i \} _{0 \leq i \leq n}, \{ f_{j,i} \} _{0 \leq i < j \leq n}, \{ \mu _{k,j,i} \} _{0 \leq i < j < k \leq n} ) \]
satisfying condition $(c')$ of Proposition 2.3.1.9. This gives a description of $\operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ which makes no reference to the identity $1$-morphisms of $\operatorname{\mathcal{C}}$ or the left and right unit constraints of $\operatorname{\mathcal{C}}$. The resulting identification is functorial with respect to injective maps of linearly ordered sets $[m] \rightarrow [n]$. In other words, we can construct the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as a semisimplicial set (see Variant 1.1.1.6) without knowing the left and right unit constraints of $\operatorname{\mathcal{C}}$. However, the left and right unit constraints of $\operatorname{\mathcal{C}}$ are needed to define the degeneracy operators on the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.
Remark 2.3.1.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a lax functor. If $F$ is strictly unitary, then composition with $F$ induces a map of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$. However, even without the assumption that $F$ is strictly unitary, one can use the description of Proposition 2.3.1.9 to obtain a collection of maps $\operatorname{N}^{\operatorname{D}}_{n}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{D}})$ which are compatible with the face operators on the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ (though not necessarily with the degeneracy operators). In other words, if we regard the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as a semisimplicial set, then it is functorial with respect to all (lax) functors between $2$-categories.
Example 2.3.1.13 (Vertices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we can identify vertices of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ with objects of the $2$-category $\operatorname{\mathcal{C}}$.
Example 2.3.1.14 (Edges of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we can identify edges of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ with $1$-morphisms $f: X \rightarrow Y$ of the $2$-category $\operatorname{\mathcal{C}}$. Under this identification, the face and degeneracy operators
\[ d_0, d_1: \operatorname{N}^{\operatorname{D}}_{1}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{0}(\operatorname{\mathcal{C}}) \quad \quad s_0: \operatorname{N}^{\operatorname{D}}_{0}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{1}(\operatorname{\mathcal{C}}) \]
are given by $d_0(f: X \rightarrow Y) = Y$, $d_1( f: X \rightarrow Y) = X$, and $s_0(X) = \operatorname{id}_ X$.
Example 2.3.1.15 ($2$-Simplices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we see that a $2$-simplex $\sigma $ of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with the following data:
A triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$.
A triple of $1$-morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ in the $2$-category $\operatorname{\mathcal{C}}$ (corresponding to the facts $d_2(\sigma )$, $d_0(\sigma )$, and $d_1(\sigma )$, respectively).
A $2$-morphism $\mu : g \circ f \Rightarrow h$, which we depict as a diagram
\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\mu } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z. } \]
Example 2.3.1.16 ($3$-Simplices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we see that a map of simplicial sets $\partial \Delta ^3 \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with the following data:
A collection of objects $\{ X_ i \} _{0 \leq i \leq 3}$ of the $2$-category $\operatorname{\mathcal{C}}$.
A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$.
A quadruple of $2$-morphisms
\[ \mu _{2,1,0}: f_{2,1} \circ f_{1,0} \Rightarrow f_{2,0} \quad \quad \mu _{3,2,1}: f_{3,2} \circ f_{2,1} \Rightarrow f_{3,1} \]
\[ \mu _{3,1,0}: f_{3,1} \circ f_{1,0} \Rightarrow f_{3,0} \quad \quad \mu _{3,2,0}: f_{3,2} \circ f_{2,0} \Rightarrow f_{3,0}. \]
This data can be conveniently visualized as a pair of diagrams
\[ \xymatrix@C =100pt@R=50pt{ & X_1 \ar [r]^-{ f_{2,1} } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\mu _{2,1,0}} & X_2 \ar [dr]^{ f_{3,2}} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\mu _{3,2,0}} & \\ X_0 \ar [ur]^{ f_{1,0}} \ar [urr]_{ f_{2,0} } \ar [rrr]^{f_{3,0}} & & & X_3 \\ & X_1 \ar [r]^-{ f_{2,1} } \ar [drr]_{ f_{3,1} } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\mu _{3,1,0}} & X_2 \ar [dr]^{ f_{3,2} } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\mu _{3,2,1}} & \\ X_0 \ar [ur]^{f_{1,0}} \ar [rrr]^{f_{3,0}} & & & X_3, } \]
representing "front" and "back" perspectives of the boundary of a $3$-simplex. A $3$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with a map $\operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as above which satisfies an additional compatibility condition: namely, the commutativity of the diagram
\[ \xymatrix { & f_{3,2} \circ (f_{2,1} \circ f_{1,0} ) \ar@ {=>}[dl]_-{ \operatorname{id}_{ f_{3,2} } \circ \mu _{2,1,0} } \ar@ {=>}[rr]^{\alpha _{ f_{3,2}, f_{2,1}, f_{1,0} } } & & (f_{3,2} \circ f_{2,1} ) \circ f_{1,0} \ar@ {=>}[dr]^-{ \mu _{3,2,1} \circ \operatorname{id}_{f_{1,0}}} & \\ f_{3,2} \circ f_{2,0} \ar@ {=>}[drr]^{ \mu _{3,2,0} } & & & & f_{3,1} \circ f_{1,0} \ar@ {=>}[dll]_{ \mu _{3,1,0} } \\ & & f_{3,0} & & } \]
in the ordinary category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X_0, X_3)$.
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 $\mathrm{Bimod}$.
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.
Remark 2.3.1.19. Let $G$ be a monoid, regarded as a monoidal category having only identity morphisms. Then the classifying simplicial set $B_{\bullet }G$ of Example 2.3.1.18 agrees (up to canonical isomorphism) with the simplicial set $B_{\bullet }G$ given by the Milnor construction, described in Example 1.2.4.3.