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
d : Y \longrightarrow \text{id}_\mathcal {C} \quad \text{and}\quad s : Y \longrightarrow Y \circ Y
Using these transformations we can construct something that looks like a simplicial object. Namely, for n \geq 0 we define
X_ n = Y \circ \ldots \circ Y \quad (n + 1\text{ compositions})
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
d^ n_ j = 1_{X_{j - 1}} \star d \star 1_{X_{n - j - 1}} : X_ n \to X_{n - 1} \quad \text{and}\quad s^ n_ j = 1_{X_{j - 1}} \star s \star 1_{X_{n - j - 1}} : X_ n \to X_{n + 1}
So d^ n_ j, resp. s^ n_ j is the natural transformation using d, resp. s on the jth Y (counted from the left) in the composition defining X_ n.
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