Kerodon

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Proposition 4.6.6.10. Let $\operatorname{\mathcal{C}}$ be a differential graded category, so that the differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.5.3.10). Let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

$(2)$

The object $Y$ is final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

$(3)$

The identity morphism $\operatorname{id}_{Y}: Y \rightarrow Y$ is nullhomologous: that is, there exists a $1$-chain $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_{Y}$.

Proof. We will show that $(1) \Leftrightarrow (3)$; the proof that $(2) \Leftrightarrow (3)$ is similar. If condition $(1)$ is satisfied, then there exists a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ with boundary as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{0} & \\ Y \ar [ur]^{0} \ar [rr]^{ \operatorname{id}_ Y } & & Y, } \]

which we can identify with a $1$-chain $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_ Y$ (see Example 2.5.3.4). Conversely, suppose that there exists $e \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{1}$ satisfying $\partial (e) = \operatorname{id}_{Y}$. For every object $Z \in \operatorname{\mathcal{C}}$, $e$ determines a chain homotopy from the identity map $\operatorname{id}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast }$ to the zero map. It follows that the homology of chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast }$ vanishes, so that the Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } )$ of Construction 2.5.6.3 is a contractible Kan complex. Example 4.6.5.14 supplies an isomorphism of Kan complexes $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}}(Y,Z) \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } )$. Allowing $Z$ to vary and invoking Remark 4.6.6.6, we conclude that $Y$ is an initial object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. $\square$