$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example ($2$-Simplices of the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then an element of $\operatorname{N}_{2}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is given by the following data:

  • A triple of objects $X_{0}, X_1, X_2 \in \operatorname{Ob}(\operatorname{\mathcal{C}})$.

  • A triple of $0$-cycles

    \[ f_{10} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1)_{0} \quad \quad f_{20} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2)_{0} \quad \quad f_{21} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_1, X_2)_{0}. \]
  • A $1$-chain $f_{210} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2)_{1}$ satisfying the identity

    \[ \partial (f_{210}) = f_{20} - (f_{21} \circ f_{10}). \]

Here the $1$-chain $f_{210}$ can be regarded as a witness to the assertion that the $0$-cycles $f_{20}$ and $f_{21} \circ f_{10}$ are homologous: that is, they represent the same element of the homology group $\mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2) )$. We can present this data graphically by the diagram

\[ \xymatrix@C =50pt@R=50pt{ & X_1 \ar [dr]^{f_{21}} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{f_{210}} & \\ X_0 \ar [ur]^{f_{10}} \ar [rr]_{f_{20}} & & X_2. } \]