The Stacks project

Lemma 14.19.2. If the category $\mathcal{C}$ has finite limits, then $\text{cosk}_ m$ functors exist for all $m$. Moreover, for any $m$-truncated simplicial object $U$ the simplicial object $\text{cosk}_ mU$ is described by the formula

\[ (\text{cosk}_ mU)_ n = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[n])_{\leq m}^{opp}} U(n) \]

and for $\varphi : [n] \to [n']$ the map $\text{cosk}_ mU(\varphi )$ comes from the identification $U(n') \circ \overline{\varphi } = U(n)$ above via Categories, Lemma 4.14.8.

Proof. During the proof of this lemma we denote $\text{cosk}_ mU$ the simplicial object with $(\text{cosk}_ mU)_ n$ equal to $\mathop{\mathrm{lim}}\nolimits _{(\Delta /[n])_{\leq m}^{opp}} U(n)$. We will conclude at the end of the proof that it does satisfy the required mapping property.

Suppose that $V$ is a simplicial object. A morphism $\gamma : V \to \text{cosk}_ mU$ is given by a sequence of morphisms $\gamma _ n : V_ n \to (\text{cosk}_ mU)_ n$. By definition of a limit, this is given by a collection of morphisms $\gamma (\alpha ) : V_ n \to U_ k$ where $\alpha $ ranges over all $\alpha : [k] \to [n]$ with $k \leq m$. These morphisms then also satisfy the rules that

\[ \xymatrix{ V_ n \ar[r]_{\gamma (\alpha )} & U_ k \\ V_{n'} \ar[r]^{\gamma (\alpha ')} \ar[u]^{V(\varphi )} & U_{k'} \ar[u]_{U(\psi )} } \]

are commutative, given any $0 \leq k, k' \leq m$, $0 \leq n, n'$ and any $\psi : [k] \to [k']$, $\varphi : [n] \to [n']$, $\alpha : [k] \to [n]$ and $\alpha ' : [k'] \to [n']$ in $\Delta $ such that $\varphi \circ \alpha = \alpha ' \circ \psi $. Taking $n = k$, $\varphi = \alpha '$, and $\alpha = \psi = \text{id}_{[k]}$ we deduce that $\gamma (\alpha ') = \gamma (\text{id}_{[k]}) \circ V(\alpha ')$. In other words, the morphisms $\gamma (\text{id}_{[k]})$, $k \leq m$ determine the morphism $\gamma $. And it is easy to see that these morphisms form a morphism $\text{sk}_ m V \to U$.

Conversely, given a morphism $\gamma : \text{sk}_ m V \to U$, we obtain a family of morphisms $\gamma (\alpha )$ where $\alpha $ ranges over all $\alpha : [k] \to [n]$ with $k \leq m$ by setting $\gamma (\alpha ) = \gamma (\text{id}_{[k]}) \circ V(\alpha )$. These morphisms satisfy all the displayed commutativity restraints pictured above, and hence give rise to a morphism $V \to \text{cosk}_ m U$. $\square$

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