Variant 4.8.2.7 (058A)

Variant 4.8.2.7. Let $\operatorname{\mathcal{C}}$ be a differential graded category. For every integer $n \geq -1$, the following conditions are equivalent:

The differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is locally $n$-truncated.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ is homologically $n$-truncated: that is, the homology groups $\mathrm{H}_{m}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ vanish for $m > n$.

Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Example 4.6.5.15 supplies an isomorphism from of the Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ with the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})}( X, Y)$. The result now follows by combining Remark 4.8.2.5 with the criterion of Example 3.5.7.10. $\square$