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*}

14.19 Coskeleton functors

Let $\mathcal{C}$ be a category. The coskeleton functor (if it exists) is a functor

\[ \text{cosk}_ n : \text{Simp}_ n(\mathcal{C}) \longrightarrow \text{Simp}(\mathcal{C}) \]

which is right adjoint to the skeleton functor. In a formula

14.19.0.1
\begin{equation} \label{simplicial-equation-cosk} \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(U, \text{cosk}_ n V) = \mathop{Mor}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ n U, V) \end{equation}

Given a $n$-truncated simplicial object $V$ we say that $\text{cosk}_ nV$ exists if there exists a $\text{cosk}_ nV \in \mathop{\mathrm{Ob}}\nolimits (\text{Simp}(\mathcal{C}))$ and a morphism $\text{sk}_ n \text{cosk}_ n V \to V$ such that the displayed formula holds, in other words if the functor $U \mapsto \mathop{Mor}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ n U, V)$ is representable. If it exists it is unique up to unique isomorphism by the Yoneda lemma. See Categories, Section 4.3.

Example 14.19.1. Suppose the category $\mathcal{C}$ has finite nonempty self products. A $0$-truncated simplicial object of $\mathcal{C}$ is the same as an object $X$ of $\mathcal{C}$. In this case we claim that $\text{cosk}_0(X)$ is the simplicial object $U$ with $U_ n = X^{n + 1}$ the $(n + 1)$-fold self product of $X$, and structure of simplicial object as in Example 14.3.5. Namely, a morphism $V \to U$ where $V$ is a simplicial object is given by morphisms $V_ n \to X^{n + 1}$, such that all the diagrams

\[ \xymatrix{ V_ n \ar[r] \ar[d]_{V([0] \to [n], 0 \mapsto i)} & X^{n + 1} \ar[d]^{\text{pr}_ i} \\ V_0 \ar[r] & X } \]

commute. Clearly this means that the map determines and is determined by a unique morphism $V_0 \to X$. This proves that formula (14.19.0.1) holds.

Recall the category $\Delta /[n]$, see Example 14.11.4. We let $(\Delta /[n])_{\leq m}$ denote the full subcategory of $\Delta /[n]$ consisting of objects $[k] \to [n]$ of $\Delta /[n]$ with $k \leq m$. In other words we have the following commutative diagram of categories and functors

\[ \xymatrix{ (\Delta /[n])_{\leq m} \ar[r] \ar[d] & \Delta /[n] \ar[d] \\ \Delta _{\leq m} \ar[r] & \Delta } \]

Given a $m$-truncated simplicial object $U$ of $\mathcal{C}$ we define a functor

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

by the rules

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

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

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

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 $V \to \text{cosk}_ m U$. $\square$

Lemma 14.19.3. Let $\mathcal{C}$ be a category. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. For $n \leq m$ the limit $\mathop{\mathrm{lim}}\nolimits _{(\Delta /[n])_{\leq m}^{opp}} U(n)$ exists and is canonically isomorphic to $U_ n$.

Proof. This is true because the category $(\Delta /[n])_{\leq m}$ has an final object in this case, namely the identity map $[n] \to [n]$. $\square$

Lemma 14.19.4. Let $\mathcal{C}$ be a category with finite limits. Let $U$ be an $n$-truncated simplicial object of $\mathcal{C}$. The morphism $\text{sk}_ n \text{cosk}_ n U \to U$ is an isomorphism.

Let us describe a particular instance of the coskeleton functor in more detail. By abuse of notation we will denote $\text{sk}_ n$ also the restriction functor $\text{Simp}_{n'}(\mathcal{C}) \to \text{Simp}_ n(\mathcal{C})$ for any $n' \geq n$. We are going to describe a right adjoint of the functor $\text{sk}_ n : \text{Simp}_{n + 1}(\mathcal{C}) \to \text{Simp}_ n(\mathcal{C})$. For $n \geq 1$, $0 \leq i < j \leq n + 1$ define $\delta ^{n + 1}_{i, j} : [n - 1] \to [n + 1]$ to be the increasing map omitting $i$ and $j$. Note that $\delta ^{n + 1}_{i, j} = \delta ^{n + 1}_ j \circ \delta ^ n_ i = \delta ^{n + 1}_ i \circ \delta ^ n_{j - 1}$, see Lemma 14.2.3. This motivates the following lemma.

Lemma 14.19.5. Let $n$ be an integer $\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\mathcal{C}$. Consider the contravariant functor from $\mathcal{C}$ to $\textit{Sets}$ which associates to an object $T$ the set

\[ \{ (f_0, \ldots , f_{n + 1}) \in \mathop{Mor}\nolimits _\mathcal {C}(T, U_ n) \mid d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j \ \forall \ 0\leq i < j\leq n + 1\} \]

If this functor is representable by some object $U_{n + 1}$ of $\mathcal{C}$, then

\[ U_{n + 1} = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[n + 1])_{\leq n}^{opp}} U(n) \]

Proof. The limit, if it exists, represents the functor that associates to an object $T$ the set

\[ \{ (f_\alpha )_{\alpha : [k] \to [n + 1], k \leq n} \mid f_{\alpha \circ \psi } = U(\psi ) \circ f_\alpha \ \forall \ \psi : [k'] \to [k], \alpha : [k] \to [n + 1] \} . \]

In fact we will show this functor is isomorphic to the one displayed in the lemma. The map in one direction is given by the rule

\[ (f_\alpha )_{\alpha } \longmapsto (f_{\delta ^{n + 1}_0}, \ldots , f_{\delta ^{n + 1}_{n + 1}}). \]

This satisfies the conditions of the lemma because

\[ d^ n_{j - 1} \circ f_{\delta ^{n + 1}_ i} = f_{\delta ^{n + 1}_ i \circ \delta ^ n_{j - 1}} = f_{\delta ^{n + 1}_ j \circ \delta ^ n_ i} = d^ n_ i \circ f_{\delta ^{n + 1}_ j} \]

by the relations we recalled above the lemma. To construct a map in the other direction we have to associate to a system $(f_0, \ldots , f_{n + 1})$ as in the displayed formula of the lemma a system of maps $f_\alpha $. Let $\alpha : [k] \to [n + 1]$ be given. Since $k \leq n$ the map $\alpha $ is not surjective. Hence we can write $\alpha = \delta ^{n + 1}_ i \circ \psi $ for some $0 \leq i \leq n + 1$ and some $\psi : [k] \to [n]$. We have no choice but to define

\[ f_\alpha = U(\psi ) \circ f_ i. \]

Of course we have to check that this is independent of the choice of the pair $(i, \psi )$. First, observe that given $i$ there is a unique $\psi $ which works. Second, suppose that $(j, \phi )$ is another pair. Then $i \not= j$ and we may assume $i < j$. Since both $i, j$ are not in the image of $\alpha $ we may actually write $\alpha = \delta ^{n + 1}_{i, j} \circ \xi $ and then we see that $\psi = \delta ^ n_{j - 1} \circ \xi $ and $\phi = \delta ^ n_ i \circ \xi $. Thus

\begin{eqnarray*} U(\psi ) \circ f_ i & = & U(\delta ^ n_{j - 1} \circ \xi ) \circ f_ i \\ & = & U(\xi ) \circ d^ n_{j - 1} \circ f_ i \\ & = & U(\xi ) \circ d^ n_ i \circ f_ j \\ & = & U(\delta ^ n_ i \circ \xi ) \circ f_ j \\ & = & U(\phi ) \circ f_ j \end{eqnarray*}

as desired. We still have to verify that the maps $f_\alpha $ so defined satisfy the rules of a system of maps $(f_\alpha )_\alpha $. To see this suppose that $\psi : [k'] \to [k]$, $\alpha : [k] \to [n + 1]$ with $k, k' \leq n$. Set $\alpha ' = \alpha \circ \psi $. Choose $i$ not in the image of $\alpha $. Then clearly $i$ is not in the image of $\alpha '$ also. Write $\alpha = \delta ^{n + 1}_ i \circ \phi $ (we cannot use the letter $\psi $ here because we've already used it). Then obviously $\alpha ' = \delta ^{n + 1}_ i \circ \phi \circ \psi $. By construction above we then have

\[ U(\psi ) \circ f_\alpha = U(\psi ) \circ U(\phi ) \circ f_ i = U(\phi \circ \psi ) \circ f_ i = f_{\alpha \circ \psi } = f_{\alpha '} \]

as desired. We leave to the reader the pleasant task of verifying that our constructions are mutually inverse bijections, and are functorial in $T$. $\square$

Lemma 14.19.6. Let $n$ be an integer $\geq 1$. Let $U$ be a $n$-truncated simplicial object of $\mathcal{C}$. Consider the contravariant functor from $\mathcal{C}$ to $\textit{Sets}$ which associates to an object $T$ the set

\[ \{ (f_0, \ldots , f_{n + 1}) \in \mathop{Mor}\nolimits _\mathcal {C}(T, U_ n) \mid d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j \ \forall \ 0\leq i < j\leq n + 1\} \]

If this functor is representable by some object $U_{n + 1}$ of $\mathcal{C}$, then there exists an $(n + 1)$-truncated simplicial object $\tilde U$, with $\text{sk}_ n \tilde U = U$ and $\tilde U_{n + 1} = U_{n + 1}$ such that the following adjointness holds

\[ \mathop{Mor}\nolimits _{\text{Simp}_{n + 1}(\mathcal{C})}(V, \tilde U) = \mathop{Mor}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ nV, U) \]

Proof. By Lemma 14.19.3 there are identifications

\[ U_ i = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[i])_{\leq n}^{opp}} U(i) \]

for $0 \leq i \leq n$. By Lemma 14.19.5 we have

\[ U_{n + 1} = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[n + 1])_{\leq n}^{opp}} U(n). \]

Thus we may define for any $\varphi : [i] \to [j]$ with $i, j \leq n + 1$ the corresponding map $\tilde U(\varphi ) : \tilde U_ j \to \tilde U_ i$ exactly as in Lemma 14.19.2. This defines an $(n + 1)$-truncated simplicial object $\tilde U$ with $\text{sk}_ n \tilde U = U$.

To see the adjointness we argue as follows. Given any element $\gamma : \text{sk}_ n V \to U$ of the right hand side of the formula consider the morphisms $f_ i = \gamma _ n \circ d^{n + 1}_ i : V_{n + 1} \to V_ n \to U_ n$. These clearly satisfy the relations $d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j$ and hence define a unique morphism $V_{n + 1} \to U_{n + 1}$ by our choice of $U_{n + 1}$. Conversely, given a morphism $\gamma ' : V \to \tilde U$ of the left hand side we can simply restrict to $\Delta _{\leq n}$ to get an element of the right hand side. We leave it to the reader to show these are mutually inverse constructions. $\square$

Remark 14.19.7. Let $U$, and $U_{n + 1}$ be as in Lemma 14.19.6. On $T$-valued points we can easily describe the face and degeneracy maps of $\tilde U$. Explicitly, the maps $d^{n + 1}_ i : U_{n + 1} \to U_ n$ are given by

\[ (f_0, \ldots , f_{n + 1}) \longmapsto f_ i. \]

And the maps $s^ n_ j : U_ n \to U_{n + 1}$ are given by

\begin{eqnarray*} f & \longmapsto & ( s^{n - 1}_{j - 1} \circ d^{n - 1}_0 \circ f, \\ & & s^{n - 1}_{j - 1} \circ d^{n - 1}_1 \circ f, \\ & & \ldots \\ & & s^{n - 1}_{j - 1} \circ d^{n - 1}_{j - 1} \circ f, \\ & & f, \\ & & f, \\ & & s^{n - 1}_ j \circ d^{n - 1}_{j + 1} \circ f, \\ & & s^{n - 1}_ j \circ d^{n - 1}_{j + 2} \circ f, \\ & & \ldots \\ & & s^{n - 1}_ j \circ d^{n - 1}_ n \circ f ) \end{eqnarray*}

where we leave it to the reader to verify that the RHS is an element of the displayed set of Lemma 14.19.6. For $n = 0$ there is one map, namely $f \mapsto (f, f)$. For $n = 1$ there are two maps, namely $f \mapsto (f, f, s_0d_1f)$ and $f \mapsto (s_0d_0f, f, f)$. For $n = 2$ there are three maps, namely $f \mapsto (f, f, s_0d_1f, s_0d_2f)$, $f \mapsto (s_0d_0f, f, f, s_1d_2f)$, and $f \mapsto (s_1d_0f, s_1d_1f, f, f)$. And so on and so forth.

Remark 14.19.8. The construction of Lemma 14.19.6 above in the case of simplicial sets is the following. Given an $n$-truncated simplicial set $U$, we make a canonical $(n + 1)$-truncated simplicial set $\tilde U$ as follows. We add a set of $(n + 1)$-simplices $U_{n + 1}$ by the formula of the lemma. Namely, an element of $U_{n + 1}$ is a numbered collection of $(f_0, \ldots , f_{n + 1})$ of $n$-simplices, with the property that they glue as they would in a $(n + 1)$-simplex. In other words, the $i$th face of $f_ j$ is the $(j-1)$st face of $f_ i$ for $i < j$. Geometrically it is obvious how to define the face and degeneracy maps for $\tilde U$. If $V$ is an $(n + 1)$-truncated simplicial set, then its $(n + 1)$-simplices give rise to compatible collections of $n$-simplices $(f_0, \ldots , f_{n + 1})$ with $f_ i \in V_ n$. Hence there is a natural map $\mathop{Mor}\nolimits (\text{sk}_ nV, U) \to \mathop{Mor}\nolimits (V, \tilde U)$ which is inverse to the canonical restriction mapping the other way.

Also, it is enough to do the combinatorics of the construction in the case of truncated simplicial sets. Namely, for any object $T$ of the category $\mathcal{C}$, and any $n$-truncated simplicial object $U$ of $\mathcal{C}$ we can consider the $n$-truncated simplicial set $\mathop{Mor}\nolimits (T, U)$. We may apply the construction to this, and take its set of $(n + 1)$-simplices, and require this to be representable. This is a good way to think about the result of Lemma 14.19.6.

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

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

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

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

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

Lemma 14.19.10. Let $\mathcal{C}$ be a category which has finite limits.

  1. For every $n$ the functor $\text{sk}_ n : \text{Simp}(\mathcal{C}) \to \text{Simp}_ n(\mathcal{C})$ has a right adjoint $\text{cosk}_ n$.

  2. For every $n' \geq n$ the functor $\text{sk}_ n : \text{Simp}_{n'}(\mathcal{C}) \to \text{Simp}_ n(\mathcal{C})$ has a right adjoint, namely $\text{sk}_{n'}\text{cosk}_ n$.

  3. For every $m \geq n \geq 0$ and every $n$-truncated simplicial object $U$ of $\mathcal{C}$ we have $\text{cosk}_ m \text{sk}_ m \text{cosk}_ n U = \text{cosk}_ n U$.

  4. If $U$ is a simplicial object of $\mathcal{C}$ such that the canonical map $U \to \text{cosk}_ n \text{sk}_ nU$ is an isomorphism for some $n \geq 0$, then the canonical map $U \to \text{cosk}_ m \text{sk}_ mU$ is an isomorphism for all $m \geq n$.

Proof. The existence in (1) follows from Lemma 14.19.2 above. Parts (2) and (3) follow from the discussion in Remark 14.19.9. After this (4) is obvious. $\square$

Remark 14.19.11. We do not need all finite limits in order to be able to define the coskeleton functors. Here are some remarks

  1. We have seen in Examples 14.19.1 that if $\mathcal{C}$ has products of pairs of objects then $\text{cosk}_0$ exists.

  2. For $k > 0$ the functor $\text{cosk}_ k$ exists if $\mathcal{C}$ has finite connected limits.

This is clear from the inductive procedure of constructing coskeleta (Remarks 14.19.8 and 14.19.9) but it also follows from the fact that the categories $(\Delta /[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ used in Lemma 14.19.2 are connected. Observe that we do not need the categories for $n \leq k$ by Lemma 14.19.3 or Lemma 14.19.4. (As $k$ gets higher the categories $(\Delta /[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ are more and more connected in a topological sense.)

Lemma 14.19.12. Let $U$, $V$ be $n$-truncated simplicial objects of a category $\mathcal{C}$. Then

\[ \text{cosk}_ n (U \times V) = \text{cosk}_ nU \times \text{cosk}_ nV \]

whenever the left and right hand sides exist.

Proof. Let $W$ be a simplicial object. We have

\begin{eqnarray*} \mathop{Mor}\nolimits (W, \text{cosk}_ n (U \times V)) & = & \mathop{Mor}\nolimits (\text{sk}_ n W, U \times V) \\ & = & \mathop{Mor}\nolimits (\text{sk}_ n W, U) \times \mathop{Mor}\nolimits (\text{sk}_ nW, V) \\ & = & \mathop{Mor}\nolimits (W, \text{cosk}_ n U) \times \mathop{Mor}\nolimits (W, \text{cosk}_ n V) \\ & = & \mathop{Mor}\nolimits (W, \text{cosk}_ n U \times \text{cosk}_ n V) \end{eqnarray*}

The lemma follows. $\square$

Lemma 14.19.13. Assume $\mathcal{C}$ has fibre products. Let $U, V, W$ be $n$-truncated simplicial objects of the category $\mathcal{C}$. Then

\[ \text{cosk}_ n (V \times _ U W) = \text{cosk}_ nU \times _{\text{cosk}_ n U} \text{cosk}_ nV \]

whenever the left and right hand side exist.

Proof. Omitted, but very similar to the proof of Lemma 14.19.12 above. $\square$

Lemma 14.19.14. Let $\mathcal{C}$ be a category with finite limits. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $\mathcal{C}/X \to \mathcal{C}$ commutes with the coskeleton functors $\text{cosk}_ k$ for $k \geq 1$.

Proof. The statement means that if $U$ is a simplicial object of $\mathcal{C}/X$ which we can think of as a simplicial object of $\mathcal{C}$ with a morphism towards the constant simplicial object $X$, then $\text{cosk}_ k U$ computed in $\mathcal{C}/X$ is the same as computed in $\mathcal{C}$. This follows for example from Categories, Lemma 4.16.2 because the categories $(\Delta /[n])_{\leq k}$ for $k \geq 1$ and $n \geq k + 1$ used in Lemma 14.19.2 are connected. Observe that we do not need the categories for $n \leq k$ by Lemma 14.19.3 or Lemma 14.19.4. $\square$

Lemma 14.19.15. The canonical map $\Delta [n] \to \text{cosk}_1 \text{sk}_1 \Delta [n]$ is an isomorphism.

Proof. Consider a simplicial set $U$ and a morphism $f : U \to \Delta [n]$. This is a rule that associates to each $u \in U_ i$ a map $f_ u : [i] \to [n]$ in $\Delta $. Furthermore, these maps should have the property that $f_ u \circ \varphi = f_{U(\varphi )(u)}$ for any $\varphi : [j] \to [i]$. Denote $\epsilon ^ i_ j : [0] \to [i]$ the map which maps $0$ to $j$. Denote $F : U_0 \to [n]$ the map $u \mapsto f_ u(0)$. Then we see that

\[ f_ u(j) = F(\epsilon ^ i_ j(u)) \]

for all $0 \leq j \leq i$ and $u \in U_ i$. In particular, if we know the function $F$ then we know the maps $f_ u$ for all $u\in U_ i$ all $i$. Conversely, given a map $F : U_0 \to [n]$, we can set for any $i$, and any $u \in U_ i$ and any $0 \leq j \leq i$

\[ f_ u(j) = F(\epsilon ^ i_ j(u)) \]

This does not in general define a morphism $f$ of simplicial sets as above. Namely, the condition is that all the maps $f_ u$ are nondecreasing. This clearly is equivalent to the condition that $F(\epsilon ^ i_ j(u)) \leq F(\epsilon ^ i_{j'}(u))$ whenever $0 \leq j \leq j' \leq i$ and $u \in U_ i$. But in this case the morphisms

\[ \epsilon ^ i_ j, \epsilon ^ i_{j'} : [0] \to [i] \]

both factor through the map $\epsilon ^ i_{j, j'} : [1] \to [i]$ defined by the rules $0 \mapsto j$, $1 \mapsto j'$. In other words, it is enough to check the inequalities for $i = 1$ and $u \in X_1$. In other words, we have

\[ \mathop{Mor}\nolimits (U, \Delta [n]) = \mathop{Mor}\nolimits (\text{sk}_1 U, \text{sk}_1 \Delta [n]) \]

as desired. $\square$


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