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Remark Let $\operatorname{\mathcal{C}}$ be a category and let $n \geq 1$. Elements of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$ can be identified with diagrams

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n \]

in the category $\operatorname{\mathcal{C}}$ (see Remark In other words, we can identify elements of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$ with $n$-tuples $(f_1, \ldots , f_ n)$ of morphisms of $\operatorname{\mathcal{C}}$ having the property that, for $0 < i < n$, the source of $f_{i+1}$ coincides with the target of $f_{i}$.