Kerodon

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Construction 2.5.3.1. Let $\operatorname{\mathcal{C}}$ be a differential graded category. For $n \geq 0$, we let $\operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the collection of all ordered pairs $( \{ X_ i \} _{0 \leq i \leq n}, \{ f_ I \} )$, where:

• Each $X_ i$ is an object of the differential graded category $\operatorname{\mathcal{C}}$.

• For every subset $I = \{ i_0 > i_{1} > \cdots > i_ k \} \subseteq [n]$ having at least two elements, $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{i_{k}}, X_{i_0} )_{k-1}$ which satisfies the identity

\begin{eqnarray*} \partial f_{I} = \sum _{a=1}^{k-1} (-1)^{a} ( f_{ \{ i_0 > i_1 > \cdots > i_ a \} } \circ f_{ \{ i_ a > \cdots > i_ k \} } - f_{I \setminus \{ i_ a \} } ) \end{eqnarray*}