The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

14.21 Left adjoints to the skeleton functors

In this section we construct a left adjoint $i_{m!}$ of the skeleton functor $\text{sk}_ m$ in certain cases. The adjointness formula is

\[ \mathop{Mor}\nolimits _{\text{Simp}_ m(\mathcal{C})}(U, \text{sk}_ mV) = \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(i_{m!}U, V). \]

It turns out that this left adjoint exists when the category $\mathcal{C}$ has finite colimits.

We use a similar construction as in Section 14.12. Recall the category $[n]/\Delta $ of objects under $[n]$, see Categories, Example 4.2.14. Its objects are morphisms $\alpha : [n] \to [k]$ and its morphisms are commutative triangles. We let $([n]/\Delta )_{\leq m}$ denote the full subcategory of $[n]/\Delta $ consisting of objects $[n] \to [k]$ with $k \leq m$. Given a $m$-truncated simplicial object $U$ of $\mathcal{C}$ we define a functor

\[ U(n) : ([n]/\Delta )_{\leq m}^{opp} \longrightarrow \mathcal{C} \]

by the rules

\begin{eqnarray*} ([n] \to [k]) & \longmapsto & U_ k \\ \psi : ([n] \to [k']) \to ([n] \to [k]) & \longmapsto & U(\psi ) : U_ k \to U_{k'} \end{eqnarray*}

For a given morphism $\varphi : [n] \to [n']$ of $\Delta $ we have an associated functor

\[ \underline{\varphi } : ([n']/\Delta )_{\leq m} \longrightarrow ([n]/\Delta )_{\leq m} \]

which maps $\alpha : [n'] \to [k]$ to $\varphi \circ \alpha : [n] \to [k]$. The composition $U(n) \circ \underline{\varphi }$ is equal to the functor $U(n')$.

Lemma 14.21.1. Let $\mathcal{C}$ be a category which has finite colimits. The functors $i_{m!}$ exist for all $m$. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. The simplicial object $i_{m!}U$ is described by the formula

\[ (i_{m!}U)_ n = \mathop{\mathrm{colim}}\nolimits _{([n]/\Delta )_{\leq m}^{opp}} U(n) \]

and for $\varphi : [n] \to [n']$ the map $i_{m!}U(\varphi )$ comes from the identification $U(n) \circ \underline{\varphi } = U(n')$ above via Categories, Lemma 4.14.7.

Proof. In this proof we denote $i_{m!}U$ the simplicial object whose $n$th term is given by the displayed formula of the lemma. We will show it satisfies the adjointness property.

Let $V$ be a simplicial object of $\mathcal{C}$. Let $\gamma : U \to \text{sk}_ mV$ be given. A morphism

\[ \mathop{\mathrm{colim}}\nolimits _{([n]/\Delta )_{\leq m}^{opp}} U(n) \to T \]

is given by a compatible system of morphisms $f_\alpha : U_ k \to T$ where $\alpha : [n] \to [k]$ with $k \leq m$. Certainly, we have such a system of morphisms by taking the compositions

\[ U_ k \xrightarrow {\gamma _ k} V_ k \xrightarrow {V(\alpha )} V_ n. \]

Hence we get an induced morphism $(i_{m!}U)_ n \to V_ n$. We leave it to the reader to see that these form a morphism of simplicial objects $\gamma ' : i_{m!}U \to V$.

Conversely, given a morphism $\gamma ' : i_{m!}U \to V$ we obtain a morphism $\gamma : U \to \text{sk}_ m V$ by setting $\gamma _ i : U_ i \to V_ i$ equal to the composition

\[ U_ i \xrightarrow {\text{id}_{[i]}} \mathop{\mathrm{colim}}\nolimits _{([i]/\Delta )_{\leq m}^{opp}} U(i) \xrightarrow {\gamma '_ i} V_ i \]

for $0 \leq i \leq n$. We leave it to the reader to see that this is the inverse of the construction above. $\square$

Lemma 14.21.2. Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For any $n \leq m$ the colimit

\[ \mathop{\mathrm{colim}}\nolimits _{([n]/\Delta )_{\leq m}^{opp}} U(n) \]

exists and is equal to $U_ n$.

Proof. This is so because the category $([n]/\Delta )_{\leq m}$ has an initial object, namely $\text{id} : [n] \to [n]$. $\square$

Lemma 14.21.3. Let $\mathcal{C}$ be a category which has finite colimits. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. The map $U \to \text{sk}_ m i_{m!}U$ is an isomorphism.

Lemma 14.21.4. If $U$ is an $m$-truncated simplicial set and $n > m$ then all $n$-simplices of $i_{m!}U$ are degenerate.

Proof. This can be seen from the construction of $i_{m!}U$ in Lemma 14.21.1, but we can also argue directly as follows. Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subset with $V'_ i = V_ i$ for $i \leq m$ and all $i$ simplices degenerate for $i > m$, see Lemma 14.18.4. By the adjunction formula, since $\text{sk}_ m V' = U$, there is an inverse to the injection $V' \to V$. Hence $V' = V$. $\square$

Lemma 14.21.5. Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. The morphism $i_{n!} \text{sk}_ n U \to U$ identifies $i_{n!} \text{sk}_ n U$ with the simplicial set $U' \subset U$ defined in Lemma 14.18.4.

Proof. By Lemma 14.21.4 the only nondegenerate simplices of $i_{n!} \text{sk}_ n U$ are in degrees $\leq n$. The map $i_{n!} \text{sk}_ n U \to U$ is an isomorphism in degrees $\leq n$. Combined we conclude that the map $i_{n!} \text{sk}_ n U \to U$ maps nondegenerate simplices to nondegenerate simplices and no two nondegenerate simplices have the same image. Hence Lemma 14.18.3 applies. Thus $i_{n!} \text{sk}_ n U \to U$ is injective. The result follows easily from this. $\square$

Remark 14.21.6. In some texts the composite functor

\[ \text{Simp}(\mathcal{C}) \xrightarrow {\text{sk}_ m} \text{Simp}_ m(\mathcal{C}) \xrightarrow {i_{m!}} \text{Simp}(\mathcal{C}) \]

is denoted $\text{sk}_ m$. This makes sense for simplicial sets, because then Lemma 14.21.5 says that $i_{m!} \text{sk}_ m V$ is just the sub simplicial set of $V$ consisting of all $i$-simplices of $V$, $i \leq m$ and their degeneracies. In those texts it is also customary to denote the composition

\[ \text{Simp}(\mathcal{C}) \xrightarrow {\text{sk}_ m} \text{Simp}_ m(\mathcal{C}) \xrightarrow {\text{cosk}_ m} \text{Simp}(\mathcal{C}) \]

by $\text{cosk}_ m$.

Lemma 14.21.7. Let $U \subset V$ be simplicial sets. Suppose $n \geq 0$ and $x \in V_ n$, $x \not\in U_ n$ are such that

  1. $V_ i = U_ i$ for $i < n$,

  2. $V_ n = U_ n \cup \{ x\} $,

  3. any $z \in V_ j$, $z \not\in U_ j$ for $j > n$ is degenerate.

Let $\Delta [n] \to V$ be the unique morphism mapping the nondegenerate $n$-simplex of $\Delta [n]$ to $x$. In this case the diagram

\[ \xymatrix{ \Delta [n] \ar[r] & V \\ i_{(n - 1)!} \text{sk}_{n - 1} \Delta [n] \ar[r] \ar[u] & U \ar[u] } \]

is a pushout diagram.

Proof. Let us denote $\partial \Delta [n] = i_{(n - 1)!} \text{sk}_{n - 1} \Delta [n]$ for convenience. There is a natural map $U \amalg _{\partial \Delta [n]} \Delta [n] \to V$. We have to show that it is bijective in degree $j$ for all $j$. This is clear for $j \leq n$. Let $j > n$. The third condition means that any $z \in V_ j$, $z \not\in U_ j$ is a degenerate simplex, say $z = s^{j - 1}_ i(z')$. Of course $z' \not\in U_{j - 1}$. By induction it follows that $z'$ is a degeneracy of $x$. Thus we conclude that all $j$-simplices of $V$ are either in $U$ or degeneracies of $x$. This implies that the map $U \amalg _{\partial \Delta [n]} \Delta [n] \to V$ is surjective. Note that a nondegenerate simplex of $U \amalg _{\partial \Delta [n]} \Delta [n]$ is either the image of a nondegenerate simplex of $U$, or the image of the (unique) nondegenerate $n$-simplex of $\Delta [n]$. Since clearly $x$ is nondegenerate we deduce that $U \amalg _{\partial \Delta [n]} \Delta [n] \to V$ maps nondegenerate simplices to nondegenerate simplices and is injective on nondegenerate simplices. Hence it is injective, by Lemma 14.18.3. $\square$

Lemma 14.21.8. Let $U \subset V$ be simplicial sets, with $U_ n, V_ n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then there exists a sequence of sub simplicial sets

\[ U = W^0 \subset W^1 \subset W^2 \subset \ldots W^ r = V \]

such that Lemma 14.21.7 applies to each of the inclusions $W^ i \subset W^{i + 1}$.

Proof. Let $n$ be the smallest integer such that $V$ has a nondegenerate simplex that does not belong to $U$. Let $x \in V_ n$, $x\not\in U_ n$ be such a nondegenerate simplex. Let $W \subset V$ be the set of elements which are either in $U$, or are a (repeated) degeneracy of $x$ (in other words, are of the form $V(\varphi )(x)$ with $\varphi : [m] \to [n]$ surjective). It is easy to see that $W$ is a simplicial set. The inclusion $U \subset W$ satisfies the conditions of Lemma 14.21.7. Moreover the number of nondegenerate simplices of $V$ which are not contained in $W$ is exactly one less than the number of nondegenerate simplices of $V$ which are not contained in $U$. Hence we win by induction on this number. $\square$

Lemma 14.21.9. Let $\mathcal{A}$ be an abelian category Let $U$ be an $m$-truncated simplicial object of $\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_ n = 0$.

Proof. Write $V = i_{m!}U$. Let $V' \subset V$ be the simplicial subobject of $V$ with $V'_ i = V_ i$ for $i \leq m$ and $N(V'_ i) = 0$ for $i > m$, see Lemma 14.18.9. By the adjunction formula, since $\text{sk}_ m V' = U$, there is an inverse to the injection $V' \to V$. Hence $V' = V$. $\square$

Lemma 14.21.10. Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object of $\mathcal{A}$. Let $n \geq 0$ be an integer. The morphism $i_{n!} \text{sk}_ n U \to U$ identifies $i_{n!} \text{sk}_ n U$ with the simplicial subobject $U' \subset U$ defined in Lemma 14.18.9.

Proof. By Lemma 14.21.9 we have $N(i_{n!} \text{sk}_ n U)_ i = 0$ for $i > n$. The map $i_{n!} \text{sk}_ n U \to U$ is an isomorphism in degrees $\leq n$, see Lemma 14.21.3. Combined we conclude that the map $i_{n!} \text{sk}_ n U \to U$ induces injective maps $N(i_{n!} \text{sk}_ n U)_ i \to N(U)_ i$ for all $i$. Hence Lemma 14.18.7 applies. Thus $i_{n!} \text{sk}_ n U \to U$ is injective. The result follows easily from this. $\square$

Here is another way to think about the coskeleton functor using the material above.

Lemma 14.21.11. Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $V$ be a simplicial object of $\mathcal{C}$. In this case

\[ (\text{cosk}_ n \text{sk}_ n V)_{n + 1} = \mathop{\mathrm{Hom}}\nolimits (i_{n !}\text{sk}_ n \Delta [n + 1], V)_0. \]

Proof. By Lemma 14.13.4 the object on the left represents the functor which assigns to $X$ the first set of the following equalities

\begin{eqnarray*} \mathop{Mor}\nolimits (X \times \Delta [n + 1], \text{cosk}_ n \text{sk}_ n V) & = & \mathop{Mor}\nolimits (X \times \text{sk}_ n \Delta [n + 1], \text{sk}_ n V) \\ & = & \mathop{Mor}\nolimits (X \times i_{n !} \text{sk}_ n \Delta [n + 1], V). \end{eqnarray*}

The object on the right in the formula of the lemma is represented by the functor which assigns to $X$ the last set in the sequence of equalities. This proves the result.

In the sequence of equalities we have used that $\text{sk}_ n (X \times \Delta [n + 1]) = X \times \text{sk}_ n \Delta [n + 1]$ and that $i_{n!}(X \times \text{sk}_ n \Delta [n + 1]) = X \times i_{n !} \text{sk}_ n \Delta [n + 1]$. The first equality is obvious. For any (possibly truncated) simplicial object $W$ of $\mathcal{C}$ and any object $X$ of $\mathcal{C}$ denote temporarily $\mathop{Mor}\nolimits _\mathcal {C}(X, W)$ the (possibly truncated) simplicial set $[n] \mapsto \mathop{Mor}\nolimits _\mathcal {C}(X, W_ n)$. From the definitions it follows that $\mathop{Mor}\nolimits (U \times X, W) = \mathop{Mor}\nolimits (U, \mathop{Mor}\nolimits _\mathcal {C}(X, W))$ for any (possibly truncated) simplicial set $U$. Hence

\begin{eqnarray*} \mathop{Mor}\nolimits (X \times i_{n !} \text{sk}_ n \Delta [n + 1], W) & = & \mathop{Mor}\nolimits (i_{n !} \text{sk}_ n \Delta [n + 1], \mathop{Mor}\nolimits _\mathcal {C}(X, W)) \\ & = & \mathop{Mor}\nolimits (\text{sk}_ n \Delta [n + 1], \text{sk}_ n\mathop{Mor}\nolimits _\mathcal {C}(X, W)) \\ & = & \mathop{Mor}\nolimits (X \times \text{sk}_ n \Delta [n + 1], \text{sk}_ nW) \\ & = & \mathop{Mor}\nolimits (i_{n!}(X \times \text{sk}_ n \Delta [n + 1]), W). \end{eqnarray*}

This proves the second equality used, and ends the proof of the lemma. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 018K. Beware of the difference between the letter 'O' and the digit '0'.