Example 14.33.1. Let $Y : \mathcal{C} \to \mathcal{C}$ be a functor from a category to itself and suppose given transformations of functors

Using these transformations we can construct something that looks like a simplicial object. Namely, for $n \geq 0$ we define

Observe that $X_{n + m + 1} = X_ n \circ X_ m$ for $n, m \geq 0$. Next, for $n \geq 0$ and $0 \leq j \leq n$ we define using notation as in Categories, Section 4.28

So $d^ n_ j$, resp. $s^ n_ j$ is the natural transformation using $d$, resp. $s$ on the $j$th $Y$ (counted from the left) in the composition defining $X_ n$.

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