Proposition 8.1.1.10. Let $\operatorname{\mathcal{C}}$ be a category. Then there is a canonical isomorphism of simplicial sets $T: \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$, which is uniquely determined by the following requirements:
- $(1)$
For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the map $T$ carries $f$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$) to itself (regarded as a vertex of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$).
- $(2)$
The diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [d] \ar [r]^-{T} & \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [d]^{ ( \lambda _{-}, \lambda _{+}) } \\ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \ar [r]^-{\sim } & \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]
commutes, where the right vertical map is given by Notation 8.1.1.6 and the left vertical map is the nerve of the forgetful functor
\[ \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\quad \quad (f: X \rightarrow Y) \mapsto (X,Y). \]
Proof.
Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$, which we identify with a diagram
\[ (f_0: X_0 \rightarrow Y_0) \xrightarrow { (u_1,v_1) } (f_1: X_1 \rightarrow Y_1) \xrightarrow { (u_2,v_2)} \cdots \xrightarrow { (u_ n, v_ n) } ( f_ n: X_ n \rightarrow Y_ n ) \]
in the category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Here each $f_ i: X_ i \rightarrow Y_ i$ denotes a morphism in $\operatorname{\mathcal{C}}$, and each $(u_ i, v_ i)$ is a pair of morphisms in $\operatorname{\mathcal{C}}$ which determine a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{i-1} \ar [d]^{f_{i-1} } & X_{i} \ar [l]_-{ u_ i } \ar [d]^{f_ i} \\ Y_{i-1} \ar [r]^-{ v_ i } & Y_{i}. } \]
In this case, we can regard the chain of morphisms
8.1
\begin{equation} \begin{gathered}\label{equation:twisted-arrow-canonical-isomorphism} \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^-{f_0} & X_1 \ar [l]_-{u_1} & X_2 \ar [l]_-{u_2} & \cdots \ar [l] & X_ n \ar [l]_-{u_ n} \\ Y_0 \ar [r]^-{v_1} & Y_1 \ar [r]^-{v_2} & Y_2 \ar [r] & \cdots \ar [r]^-{v_ n} & Y_ n } \end{gathered} \end{equation}
as a $(2n+1)$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we identify with an $n$-simplex $T(\sigma )$ of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. The construction $\sigma \mapsto T(\sigma )$ then determines a morphism of simplicial sets $T: \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$, which satisfies conditions $(1)$ and $(2)$ by construction.
We now claim that $T$ is an isomorphism of simplicial sets. Let $\tau $ be an $n$-simplex of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$; we wish to show that there is a unique $n$-simplex $\sigma $ of $\operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ satisfying $T(\sigma ) = \tau $. Let us identify $\tau $ with a diagram of the form (8.1) in the category $\operatorname{\mathcal{C}}$. We wish to show that there is a unique collection of morphisms $\{ f_{i}: X_{i} \rightarrow Y_{i} \} _{1 \leq i \leq n}$ satisfying the identities $f_{i} = v_{i} \circ f_{i-1} \circ u_{i}$, which follows immediately by induction on $i$.
We now complete the proof by establishing the uniqueness of $T$. Suppose that $T': \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$ is another morphism of simplicial sets satisfying conditions $(1)$ and $(2)$. Then $T^{-1} \circ T'$ determines a functor $F$ from the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. Because $T$ and $T'$ both satisfy condition $(1)$, the functor $F$ carries each object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. Since the forgetful functor $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ is faithful, condition $(2)$ guarantees that $F$ also carries each morphism of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. It follows that $F$ is the identity functor, so that $T' = T$.
$\square$