# Kerodon

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### 3.5.4 Weakly Coskeletal Simplicial Sets

It will be useful to consider a variant of Definition 3.5.3.1.

Definition 3.5.4.1. Let $n$ be an integer. We say that a simplicial set $X$ is weakly $n$-coskeletal if the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X)$

is a bijection for $m \geq n+2$ and and injection for $m = n+1$ (provided that $n \geq -1$).

Remark 3.5.4.2. Let $n$ be an integer and let $X$ be a simplicial set. Then:

• If $X$ is $n$-coskeletal, then it is weakly $n$-coskeletal.

• If $X$ is weakly $n$-coskeletal, then it is $(n+1)$-coskeletal.

Example 3.5.4.3. For $n \leq -2$, a simplicial set $X$ is weakly $n$-coskeletal if and only if it isomorphic to the standard $0$-simplex $\Delta ^0$.

Example 3.5.4.4. A simplicial set $X$ is weakly $(-1)$-coskeletal if and only if it is either empty or isomorphic to the standard $0$-simplex $\Delta ^0$.

Example 3.5.4.5. Let $Q$ be a partially ordered set. Then the nerve $\operatorname{N}_{\bullet }(Q)$ is weakly $0$-coskeletal. In particular, every discrete simplicial set is weakly $0$-coskeletal.

Example 3.5.4.6. Let $\operatorname{\mathcal{C}}$ be a category. Then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is weakly $1$-coskeletal. See Exercise 1.3.1.5.

Example 3.5.4.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is weakly $2$-coskeletal. See Corollary 2.3.1.10.

Exercise 3.5.4.8. Let $A_{\ast }$ be a chain complex of abelian groups and let $n \geq -1$ be an integer. Show that the Eilenberg-MacLane space $\mathrm{K}(A_{\ast } )$ is weakly $n$-coskeletal if and only if it satisfies the following conditions:

• The abelian groups $A_{m}$ vanish for $m \geq n+2$.

• The differential $\partial : A_{n+1} \rightarrow A_{n}$ is a monomorphism.

Compare with Proposition 3.5.3.9.

Remark 3.5.4.9. For every integer $n$, the collection of weakly $n$-coskeletal simplicial sets is closed under the formation of limits.

Remark 3.5.4.10. Let $n$ be a nonnegative integer. Then a simplicial set $X$ is weakly $n$-coskeletal if and only if each connected component of $X$ is weakly $n$-coskeletal.

Exercise 3.5.4.11. Let $X$ be a Kan complex and let $n \geq -1$ be an integer. Show that $X$ is weakly $n$-coskeletal if and only if, for every integer $m > n$, the restriction map $\theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X)$ is injective.

Proposition 3.5.4.12. Let $n$ be an integer. Then a simplicial set $X$ is weakly $n$-coskeletal if and only if it satisfies the following conditions:

$(1)$

For every simplicial set $S$, the restriction map

$\theta _{S}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X )$

is a monomorphism.

$(2)$

The image of $\theta _{S}$ consists of those morphisms $\operatorname{sk}_{n}(S) \rightarrow X$ which can be extended to the $(n+1)$-skeleton of $S$.

Proof. For every integer $m > n$, the $n$-skeleton $\operatorname{sk}_{n}( \Delta ^ m )$ is contained in the boundary $\operatorname{\partial \Delta }^{m}$, and therefore coincides with the $n$-skeleton of $\operatorname{\partial \Delta }^ m$. We therefore have a commutative diagram of restriction maps

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(\Delta ^ m, X ) \ar [dr]_{ \theta _{ \Delta ^ m} } \ar [rr] & & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ m, X) \ar [dl]^{ \theta _{ \operatorname{\partial \Delta }^ m} } \\ & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(\Delta ^ m), X). & }$

If condition $(1)$ is satisfied, then the vertical maps are injective, so the upper horizontal map is also injective. Moreover, if $m \geq n+2$, then $\operatorname{\partial \Delta }^{m}$ contains the $(n+1)$-skeleton of $\Delta ^ m$. In this case, condition $(2)$ guarantees that the vertical maps have the same image, so that the horizontal map is bijective. It follows that $X$ is weakly $n$-coskeletal.

We now prove the converse. Assume that $X$ is weakly $n$-coskeletal, and let $S$ be a simplicial set. Then we can identify $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(S, X)$ with the inverse limit of the tower of restriction maps

$\cdots \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n+2}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n+1}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X).$

Consequently, to prove $(1)$, it will suffice to show that the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m+1}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m}(S), X)$

is injective for $m=n$ and bijective for $m > n$. Using Proposition 1.1.4.12, we can reduce to the case $S = \Delta ^{m+1}$ is a standard simplex, in which case the desired result is immediate from the definition. $\square$

Corollary 3.5.4.13. Let $n$ be an integer and let $K$ and $X$ be simplicial sets. If $X$ is weakly $n$-coskeletal, then $\operatorname{Fun}(K,X)$ is also weakly $n$-coskeletal.

Proof. It follows from Corollary 3.5.3.12 that $\operatorname{Fun}(K,X)$ is $(n+1)$-coskeletal. It will therefore suffice to show that, if $n \geq -1$, then the restriction map $\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n+1}, \operatorname{Fun}(K,X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n+1}, \operatorname{Fun}(K, X) )$ is injective. Note that $\theta$ can be identified with the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n+1} \times K, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n+1} \times K, X ).$

Since $\operatorname{\partial \Delta }^{n+1} \times K$ contains the $n$-skeleton of $\Delta ^{n+1} \times K$, the injectivity of this map follows from Proposition 3.5.4.12. $\square$

Definition 3.5.4.14. Let $X$ be a simplicial set and let $n$ be an integer. We will say that a morphism of simplicial sets $f: X \rightarrow Y$ exhibits $Y$ as a weak $n$-coskeleton of $X$ if the following conditions are satisfied:

• The simplicial set $Y$ is weakly $n$-coskeletal.

• The morphism $f$ is bijective on simplices of dimension $\leq n$ and surjective on $(n+1)$-simplices (provided that $n \geq -1$).

Warning 3.5.4.15. The terminology of Definition 3.5.4.14 is potentially confusing. If $f: X \rightarrow Y$ is a morphism which exhibits $Y$ as an $n$-coskeleton of $X$, then it generally does not exhibit $Y$ as a weak $n$-coskeleton of $X$ (because $f$ need not be surjective on $(n+1)$-simplices).

Remark 3.5.4.16. Let $f: X \rightarrow Y$ be a morphism of simplicial sets which exhibits $Y$ as a weak $n$-coskeleton of $X$. Then $f$ is $(n+1)$-connective. See Corollary 3.5.2.2.

Proposition 3.5.4.17. Let $X$ be a simplicial set, let $n$ be an integer, and let $\operatorname{cosk}_{n}^{\circ }( X )$ denote the image of tautological map $\operatorname{cosk}_{n+1}(X) \rightarrow \operatorname{cosk}_{n}(X)$. Then the composite map

$f: X \rightarrow \operatorname{cosk}_{n+1}(X) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(X)$

exhibits $\operatorname{cosk}_{n}^{\circ }(X)$ as a weak $n$-coskeleton of $X$.

Proof. The tautological map $X \rightarrow \operatorname{cosk}_{n+1}(X)$ is bijective on $m$-simplices for $m \leq n+1$, so $f$ is surjective on $m$-simplices for $m \leq n+1$. Moreover, for $m \leq n$, the composite map $X \xrightarrow {f} \operatorname{cosk}_{n}^{\circ }(X) \hookrightarrow \operatorname{cosk}_{n}(X)$ is bijective on $m$-simplices for $m \leq n$, so that $f$ is injective on $m$-simplices. To complete the proof, it will suffice to show that $\operatorname{cosk}_{n}^{\circ }(X)$ is weakly $n$-coskeletal. Fix an integer $m > n$ and a morphism $\sigma _0: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$. Since $\operatorname{cosk}_{n}(X)$ is $n$-coskeletal, the morphism $\sigma _0$ extends uniquely to an $m$-simplex $\sigma$ of $\operatorname{cosk}_{n}(X)$. To complete the proof, it will suffice to show that if $m \geq n+2$, then $\sigma$ is contained in $\operatorname{cosk}_{n}^{\circ }(X)$: that is, that it can be lifted to an $m$-simplex of $\operatorname{cosk}_{n+1}(X)$. Using Remark 3.5.3.21, we can identify $\sigma$ with a morphism of simplicial sets $u: \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow X$; we wish to show that $u$ can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$. By virtue of Proposition 1.1.4.12, this is equivalent to the requirement that, for every nondegenerate $(n+1)$-simplex $\tau$ of $\Delta ^{m}$, the composite map

$\operatorname{\partial \Delta }^{n+1} \xrightarrow { \tau |_{ \operatorname{\partial \Delta }^{n+1} } } \operatorname{sk}_{n}( \Delta ^{m} ) \xrightarrow {u} X$

can be extended to an $(n+1)$-simplex of $X$. This follows from our assumption that $\sigma _0 \circ \tau$ factors through $\operatorname{cosk}_{n}^{\circ }(X)$. $\square$

Definition 3.5.4.14 can be reformulated as a universal mapping property.

Proposition 3.5.4.18. Let $n$ be an integer and let $f: X \rightarrow Y$ be a morphism of simplicial sets, where $Y$ is weakly $n$-coskeletal. The following conditions are equivalent:

$(1)$

The morphism $f$ exhibits $Y$ as a weak $n$-coskeleton of $X$: that is, it is bijective on $m$-simplices for $m \leq n$ and surjective on $(n+1)$-simplices (provided that $n \geq -1$).

$(2)$

For every weakly $n$-coskeletal simplicial set $Z$, composition with $f$ induces an isomorphism of simplicial sets $\operatorname{Fun}( Y, Z ) \rightarrow \operatorname{Fun}(X, Z)$.

$(3)$

For every weakly $n$-coskeletal simplicial set $Z$, composition with $f$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(Y, Z) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(X, Z)$.

Proof. Condition $(3)$ is equivalent to the requirement that, for every weakly $n$-coskeletal simplicial set $Z$ and every simplicial set $K$, composition with $f$ induces a bijection

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(Y,Z ) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(X,Z) ).$

By virtue of Corollary 3.5.4.13, we can replace $Z$ by $\operatorname{Fun}(K, Z)$ and thereby reduce to the case $K = \Delta ^0$. This proves the equivalence $(2) \Leftrightarrow (3)$.

We next show that $(1)$ implies $(3)$. Assume that condition $(1)$ is satisfied, and let $Z$ be a weakly $n$-coskeletal simplicial set. We then have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, Z ) \ar [r]^-{ \circ f} \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, Z ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(Y), Z) \ar [r]^-{\circ f} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(X), Z ). }$

Since $f$ is bijective on $m$-simplices for $m \leq n$, the lower horizontal map is a bijection. Using Proposition 3.5.4.12, we see that the vertical maps are injective. Consequently, to prove $(3)$, it will suffice to show that their images agree (under the bijection provided by the the lower horizontal map). This follows from Proposition 3.5.4.12, together with our assumption that $f$ is surjective on $(n+1)$-simplices.

We now show that $(3)$ implies $(1)$. Using Proposition 3.5.4.17, we can choose a morphism $u: X \rightarrow \operatorname{cosk}^{\circ }_{n}(X)$ which exhibits $\operatorname{cosk}^{\circ }_{n}(X)$ as a weak $n$-coskeleton of $X$. Then $u$ satisfies condition $(3)$, so $f$ factors (uniquely) as a composition $X \xrightarrow {u} \operatorname{cosk}^{\circ }_{n}(X) \xrightarrow {g} Y$. We can therefore replace $X$ by $\operatorname{cosk}^{\circ }_{n}(X)$ and thereby reduce to the case where $X$ is weakly $n$-coskeletal. Condition $(3)$ then guarantees that $f$ is an isomorphism of (weakly $n$-coskeletal) simplicial sets, so condition $(1)$ is automatic. $\square$

Notation 3.5.4.19. Let $X$ be a simplicial set and let $n$ be an integer. It follows from Proposition 3.5.4.17 that there exists a morphism of simplicial sets $f: X \rightarrow Y$ which exhibits $Y$ as a weak $n$-coskeleton of $X$. Moreover, Proposition 3.5.4.18 guarantees that $Y$ is unique up to (canonical) isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{cosk}^{\circ }_{n}(X)$ and refer to it as the weak $n$-coskeleton of $X$. More explicitly, we can take $\operatorname{cosk}^{\circ }_{n}(X)$ to be the image of the restriction map $\operatorname{cosk}_{n+1}(X) \rightarrow \operatorname{cosk}_{n}(X)$ (see the proof of Proposition 3.5.4.17).

Corollary 3.5.4.20. Let $n$ be an integer. Then the inclusion functor

$\{ \textnormal{Weakly n-coskeletal simplicial sets} \} \hookrightarrow \operatorname{Set_{\Delta }}$

admits a left adjoint, given on objects by the construction $X \mapsto \operatorname{cosk}^{\circ }_{n}(X)$.

Remark 3.5.4.21. Let $X$ be a simplicial set, let $n$ be an integer, and let $\operatorname{cosk}^{\circ }_{n}(X)$ denote the weak $n$-coskeleton of $X$. It follows from Proposition 3.5.4.12 that, for every simplicial set $S$, the restriction map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(S, \operatorname{cosk}^{\circ }_{n}(X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{cosk}^{\circ }_{n}(X) ) \xleftarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X)$

is an injection, whose image consists of those morphisms $f: \operatorname{sk}_{n}(S) \rightarrow X$ which can be extended to the $(n+1)$-skeleton of $S$.

Let $X$ be a simplicial set. For every $n$, the weak $n$-coskeleton $\operatorname{cosk}_{n}^{\circ }(X)$ is $(n+1)$-coskeletal (Remark 3.5.4.2). It follows from Proposition 3.5.3.17 that the tautological map $X \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ factors (uniquely) through the $(n+1)$-coskeleton of $X$.

Proposition 3.5.4.22. Let $X$ be a simplicial set. For every integer $n$, the tautological map $q: \operatorname{cosk}_{n+1}(X) \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a trivial Kan fibration.

Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem

3.72
$$\begin{gathered}\label{equation:strong-coskeleton-vs} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{cosk}_{n+1}(X) \ar [d]^{q} \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{cosk}_{n}^{\circ }(X) } \end{gathered}$$

admits a solution. We consider two cases:

• If $m \leq n+1$, then $\overline{\sigma }$ can be lifted to an $m$-simplex of $X$. In particular, there exists an $m$-simplex $\sigma$ of $\operatorname{cosk}_{n+1}(X)$ satisfying $q(\sigma ) = \overline{\sigma }$. Since $q$ is bijective on $k$-simplices for $k \leq n$, the commutativity of the diagram (3.72) guarantees that $\sigma |_{ \operatorname{\partial \Delta }^{m} } = \sigma _0$.

• If $m \geq n+2$, then $\sigma _0$ extends uniquely to an $m$-simplex $\sigma$ of $\operatorname{cosk}_{n+1}(X)$. The commutativity of diagram (3.72) guarantees that $q(\sigma )$ and $\overline{\sigma }$ have the same restriction to $\operatorname{\partial \Delta }^{m}$, and therefore coincide (since $\operatorname{\partial \Delta }^ m$ contains the $(n+1)$-skeleton of $\Delta ^ m$).

In either case, the $m$-simplex $\sigma$ is a solution to the lifting problem (3.72). $\square$

Corollary 3.5.4.23. Let $X$ be a simplicial set and let $n$ be an integer. If $X$ is a Kan complex, then the weak $n$-coskeleton $\operatorname{cosk}_{n}^{\circ }(X)$ is a Kan complex.

Proof. Proposition 3.5.4.22 supplies a trivial Kan fibration

$q: \operatorname{cosk}_{n+1}(X) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(X).$

Since $\operatorname{cosk}_{n+1}(X)$ is a Kan complex (Proposition 3.5.3.23), it follows that $\operatorname{cosk}_{n}^{\circ }(X)$ is also a Kan complex (Proposition 1.5.5.11). $\square$

Corollary 3.5.4.24. Let $X$ be a simplicial set and let $n \geq -2$ be an integer. The following conditions are equivalent:

$(1)$

The comparison map $f: \operatorname{cosk}_{n+2}(X) \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a trivial Kan fibration.

$(2)$

Every morphism $\operatorname{\partial \Delta }^{n+2} \rightarrow X$ can be extended to an $(n+2)$-simplex of $X$.

$(3)$

The weak $(n+1)$-coskeleton $\operatorname{cosk}_{n+1}^{\circ }(X)$ coincides with $\operatorname{cosk}_{n+1}(X)$.

Proof. The equivalence of $(2)$ and $(3)$ follows from Remark 3.5.4.21. The implication $(3) \Rightarrow (1)$ follows from the observation that $f$ factors as a composition

$\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \twoheadrightarrow \operatorname{cosk}_{n+1}^{\circ }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{cosk}_{n+1}(\operatorname{\mathcal{C}}) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}),$

where the outer maps are trivial Kan fibrations (Proposition 3.5.4.22). We will complete the proof by showing that $(2)$ implies $(3)$. Suppose we are given a morphism $\sigma _0: \operatorname{\partial \Delta }^{n+2} \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ is $(n+1)$-coskeletal we can extend $F \circ \sigma _0$ to an $(n+2)$-simplex $\overline{\sigma }$ of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$. If condition $(2)$ is satisfied, then the lifting problem

$\xymatrix { \operatorname{\partial \Delta }^{n+2} \ar [r]^{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n+2} \ar@ {-->}[ur] \ar [r]^{ \overline{\sigma } } & \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) }$

admits a solution; in particular, $\sigma _0$ can be extended to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$. $\square$

Example 3.5.4.25. Let $A_{\ast }$ be a nonnegatively graded chain complex of abelian groups and let $X = \mathrm{K}( A_{\ast } )$ denote the associated simplicial abelian group. For every integer $n \geq -1$, the weak $n$-coskeleton $\operatorname{cosk}^{\circ }_{n}( X)$ inherits the structure of a simplicial abelian group. It follows from Theorem 2.5.6.1 that $\operatorname{cosk}^{\circ }_{n}(X)$ can be identified with the Eilenberg-MacLane space $\mathrm{K}( A'_{\ast } )$, for some nonnegatively graded chain complex $A'_{\ast }$. Here $A'_{\ast }$ is universal among chain complexes which satisfy the criterion of Exercise 3.5.4.8 and are equipped with a chain map $A_{\ast } \rightarrow A'_{\ast }$. More concretely, we can identify $A'_{\ast }$ with the chain complex

$\cdots \rightarrow 0 \rightarrow A_{n+1} / Z_{n+1} \hookrightarrow A_{n} \xrightarrow {\partial } A_{n-1} \rightarrow \cdots ,$

where $Z_{n+1} \subseteq A_{n+1}$ denotes the subgroup of $(n+1)$-cycles of $A_{\ast }$.

Proposition 3.5.4.26. Let $X$ be a Kan complex. Then, for every integer $n$, the tautological map $u: X \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a Kan fibration.

Warning 3.5.4.27. Let $X$ be a Kan complex. For every integer $n$, we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [rr]^{u} \ar [dr]_{v} & & \operatorname{cosk}^{\circ }_{n}(X) \\ & \operatorname{cosk}_{n+1}(X) \ar [ur]_{q} & }$

where $u$ is a Kan fibration (Proposition 3.5.4.26) and $q$ is a trivial Kan fibration (Proposition 3.5.4.22). Beware that $v$ is usually not a Kan fibration.

Proof of Proposition 3.5.4.26. Fix a pair of integers $0 \leq i \leq m$ with $m > 0$; we wish to show that every lifting problem

3.73
$$\begin{gathered}\label{equation:fibration-to-strong-coskeleton} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & X \ar [d]^{u} \\ \Delta ^{m} \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{cosk}^{\circ }_{n}(X) } \end{gathered}$$

admits a solution. We consider two cases:

• If $m \leq n+1$, then we can choose an $m$-simplex $\sigma$ of $X$ satisfying $u(\sigma ) = \overline{\sigma }$. Since $u$ is bijective on simplices of dimension $\leq n$, the commutativity of the diagram (3.73) guarantees that $\sigma |_{ \Lambda ^{m}_{i} } = \sigma _0$.

• If $m \geq n+2$, then our assumption that $X$ is a Kan complex guarantees that $\sigma _0$ can be extended to an $m$-simplex $\sigma$ of $X$. The commutativity of the diagram (3.73) then guarantees that $u(\sigma )$ and $\overline{\sigma }$ have the same restriction to the horn $\Lambda ^{m}_{i} \subset \Delta ^{m}$. In particular, they have the same restriction to the $n$-skeleton of $\Delta ^{m}$, so $u(\sigma ) = \overline{\sigma }$.

In either case, it follows that $\sigma$ is a solution to the lifting problem (3.73). $\square$