Lemma 14.19.2 (0183)—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 4.14.9.

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 $n = k = k'$, $\varphi = \alpha '$, 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 $V \to \text{cosk}_ m U$. $\square$

Comments (4)

Comment #4272 by comment_bot on June 26, 2019 at 03:35

It would be useful to stress that in the limit one can restrict to the subdiagram in which the arrows $[k] \rightarrow [n]$ are injective. Relatedly, the assumptions of the lemma seem too strict. For example, the small fppf site of a scheme does not satisfy them: the site has no fiber products (same issue that obstructs the functoriality of the small fppf topos). Yet one can still use this lemma to construct hypercovers in the small fppf site: once one restricts to the smaller diagram I've mentioned all the transition maps become fppf and the limit exists.

Comment #4438 by Johan on September 01, 2019 at 09:32

Dear comment_bot, I somewhat agree, but I am going to skip changing this for now. Personally, I try to avoid working with the small fppf site or with the lisse-etale or the flat-fppf sites we introduce in Section 103.14. The way we deal with constructing hypercoverings (not using this lemma) is explained in Section 25.12. So that kind of statement should be added there if it isn't already there.

Comment #4456 by comment_bot on September 01, 2019 at 14:46

OK, I agree with what you say about those sites, but I still think that changing to at least say that one can restrict to a subdiagram indexed by injective $[k] \rightarrow [n]$ would be very useful. For instance, how else does one see that for any hypercover $X_\bullet$ of $X$ the maps $X_n \rightarrow X$ are actually coverings? I couldn't find this standard result in the Stacks Project (or in any other "down to earth" reference that would be convenient to cite).

Comment #4457 by Johan on September 01, 2019 at 15:57

The reference for the result you want is Lemma 25.8.3 in the case the site has fibre products (and the proof is actually a little bit involved --- I am interpreting your comment that that particular result could be proven starting with a change to this lemma). Note that the Stacks project doesn't define hypercoverings of objects unless the site has fibre products, see Definition 25.3.3 so this covers all cases we allow the definition for. Interesingly, even though there is a lot of material using hypercoverings in the Stacks project, we never needed to use the result of Lemma 25.8.3. Psychologically it seems necessary to have it, but in actuality one doesn't seem to need it a lot.

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