Let $C$ be a differential graded (abbreviated to dg) category, i.e. a category enriched by the category of chain complexes. Then $N_{dg}(C)$, the differential graded nerve of $C$ is a simplicial set whose 0-simplices are objects of $C$, and $n$-simplices are ordered pairs $$(\{X_i\}_{0\leq i\leq n},\{f_{i_0,i_1,...,i_k}\}_{0\leq i_0<i_1<...<i_k\leq n}),$$ where $X_i$ is an object of $C$, $f_{i_0,...,i_k}\in Map_C(X_{i_0},X_{i_k})_{k-1}$, and equalities $$df_{i_0,...,i_k}=\sum_{0<j<k}(-1)^j(f_{i_0,...,i_{j-1},i_{j+1},...i_k}-f_{i_j,...,i_k}\circ f_{i_0,...,i_j})$$ hold.
The dg nerve $N_{dg}(C)$ is actually an $\infty$-category, and its mapping space $$Map_{N_{dg}(C)}(X,Y)\simeq DK(\tau_{\geq 0}Map_C(X,Y)),$$ where $DK$ is the Dold-Kan correspondence and $\tau_{\geq 0}$ is the truncation. All of the above are from Lurie's Higher algebra, Section 1.3.1.
Now let's assume that mapping complexes of $C$ are all non-positive chain complexes (or equivalently non-negative cochain complexes), then $N_{dg}(C)$ becomes a trivial $\infty$-category: the 1-category $(Ob(C),\{Map_C(X,Y)_0\}_{X,Y})$, and all of homotopy data of $C$ are lost. So I wonder wether there is a modification of the definition of the dg nerve, such that $$Map_{N_{dg}(C)}(X,Y)\simeq DK(\tau_{\leq 0}Map_C(X,Y)).$$