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Remark 14.19.9. Inductive construction of coskeleta. Suppose that $\mathcal{C}$ is a category with finite limits. Suppose that $U$ is an $m$-truncated simplicial object in $\mathcal{C}$. Then we can inductively construct $n$-truncated objects $U^ n$ as follows:

  1. To start, set $U^ m = U$.

  2. Given $U^ n$ for $n \geq m$ set $U^{n + 1} = \tilde U^ n$, where $\tilde U^ n$ is constructed from $U^ n$ as in Lemma 14.19.6.

Since the construction of Lemma 14.19.6 has the property that it leaves the $n$-skeleton of $U^ n$ unchanged, we can then define $\text{cosk}_ m U$ to be the simplicial object with $(\text{cosk}_ m U)_ n = U^ n_ n = U^{n + 1}_ n = \ldots $. And it follows formally from Lemma 14.19.6 that $U^ n$ satisfies the formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ n(\mathcal{C})}(V, U^ n) = \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ m(\mathcal{C})}(\text{sk}_ mV, U) \]

for all $n \geq m$. It also then follows formally from this that

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(V, \text{cosk}_ mU) = \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ m(\mathcal{C})}(\text{sk}_ mV, U) \]

with $\text{cosk}_ mU$ chosen as above.

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