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3.5.3 Coskeletal Simplicial Sets

Let $X$ be a simplicial set and let $n$ be an integer. Recall that $X$ has dimension $\leq n$ if every $m$-simplex of $X$ is degenerate for $m > n$ (Definition 1.1.3.1). If this condition is satisfied, then $X$ is determined by its simplices of dimension $\leq n$ in the following sense: to give a morphism of simplicial sets $f: X \rightarrow Y$, it suffices to specify the value of $f$ on $m$-simplices for $m \leq n$ (see Proposition 1.1.3.11 for a precise statement). In this section, we introduce a dual condition which instead controls the classification of morphisms $Y \rightarrow X$ (Proposition 3.5.3.10).

Definition 3.5.3.1. Let $n$ be an integer and let $X$ be a simplicial set. We say that $X$ is $n$-coskeletal if, for every nonnegative 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 a bijection: that is, every morphism of simplicial sets $\operatorname{\partial \Delta }^{m} \rightarrow X$ extends uniquely to an $m$-simplex of $X$.

Example 3.5.3.2. Let $n$ be a negative integer. Then a simplicial set $X$ is $n$-coskeletal if and only if it is a final object of $\operatorname{Set_{\Delta }}$: that is, if and only if it is isomorphic to the standard $0$-simplex $\Delta ^0$.

Example 3.5.3.3. Let $Q$ be a partially ordered set. Then the nerve $\operatorname{N}_{\bullet }(Q)$ is $1$-coskeletal. In particular, every discrete simplicial set is $1$-coskeletal.

Example 3.5.3.4. Let $\operatorname{\mathcal{C}}$ be a category. Then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is $2$-coskeletal. See Exercise 1.3.1.5.

Example 3.5.3.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is $3$-coskeletal. See Corollary 2.3.1.10.

Remark 3.5.3.6 (Monotonicity). Let $m$ be an integer and let $X$ be an $m$-coskeletal simplicial set. Then $X$ is $n$-coskeletal for every integer $n \geq m$.

Remark 3.5.3.7. Let $n$ be an integer. Then the collection of $n$-coskeletal simplicial sets is closed under the formation of limits (in the category $\operatorname{Set_{\Delta }}$).

Remark 3.5.3.8. Let $n$ be a positive integer. Then a simplicial set $X$ is $n$-coskeletal if and only if each connected component of $X$ is $n$-coskeletal (beware that this is false for $n = 0$).

Proposition 3.5.3.9. Let $(A_{\ast }, \partial )$ be a chain complex of abelian groups, let $X = \mathrm{K}( A_{\ast } )$ be the associated Eilenberg-MacLane space, and let $n$ be a nonnegative integer. Then $X$ is $n$-coskeletal if and only if it satisfies the following conditions:

$(a)$

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

$(b)$

The boundary map $\partial : A_{n+1} \rightarrow A_{n}$ is a monomorphism, whose image is the group of $n$-cycles $Z_{n} = \{ y \in A_{n}: \partial y = 0 \} $.

Proof. Fix an integer $m \geq 0$, and let $\sigma : \Delta ^ m \rightarrow \Delta ^ m$ be the identity map, which we identify with its image in the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^ m; \operatorname{\mathbf{Z}})$. Then $\partial (\sigma ) = \sum _{i=0}^{m} (-1)^{i} d^{m}_{i}(\sigma )$ is a cycle in the subcomplex $\mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}})$. Suppose we are given a morphism of simplicial sets $\tau _0: \operatorname{\partial \Delta }^{m} \rightarrow X$, which we identify with a chain map $f_0: \mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}}) \rightarrow A_{\ast }$. Then $y = f_0( \partial \sigma )$ is an $(m-1)$-cycle of $A_{\ast }$. Note that, if $m > 0$, then every $(m-1)$-cycle $y$ of $A_{\ast }$ can be obtained in this way (for example, we can take $f_0: \mathrm{N}_{\ast }( \operatorname{\partial \Delta }^{m}; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ to be the map of chain complexes which carries $d^{m}_{0}( \sigma )$ to $y$ and every other nondegenerate simplex to zero).

If $\tau : \Delta ^{m} \rightarrow X$ is an extension of $\tau _0$ corresponding to a map of chain complexes $f: \mathrm{N}_{\ast }( \Delta ^{m}; \operatorname{\mathbf{Z}}) \rightarrow A_{\ast }$, then $x = f( \sigma )$ is an $m$-chain of $A_{\ast }$ satisfying $\partial (x) = y$. This construction induces a bijection

\[ \{ \textnormal{$m$-simplices $\tau $ with $\tau |_{ \operatorname{\partial \Delta }^{m} } = \tau _0$} \} \xrightarrow {\sim } \{ \textnormal{$x \in A_{m}$ with $\partial (x) = y$} \} . \]

It follows that the simplicial set $X$ is $n$-coskeletal if and only if it satisfies the following condition for each $m > n$:

$(b_ m)$

The boundary map $\partial : A_{m} \rightarrow A_{m-1}$ is a monomorphism whose image is the group of $(m-1)$-cycles $Z_{m-1} = \{ y \in A_{m-1}: \partial y = 0 \} $.

Note that $(b_{n+1})$ is a restatement of $(b)$. Moreover, if condition $(b_ m)$ is satisfied for some integer $m$, then condition $(b_{m+1} )$ is equivalent to the requirement that the abelian group $A_{m+1}$ is trivial. In particular, $(b_ m)$ is satisfied for all $m > n$ if and only if $A_{\ast }$ satisfies conditions $(a)$ and $(b)$. $\square$

Proposition 3.5.3.10. Let $X$ be a simplicial set. For every integer $n$, the following conditions are equivalent:

$(1)$

The simplicial set $X$ is $n$-coskeletal.

$(2)$

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 bijection.

Proof. For every nonnegative integer $m > n$, the $n$-skeleton of $\Delta ^ m$ is contained in $\operatorname{\partial \Delta }^ m$, and therefore coincides with 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 $(2)$ is satisfied, then the vertical maps are bijections, so the horizontal map is a bijection as well. Allowing $m$ to vary, we deduce that $X$ is $n$-coskeletal.

We now prove the converse. For every simplicial set $S$, 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 $(2)$, it will suffice to show that the restriction map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m}(S), X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{m-1}(S), X) \]

is a bijection for $m > n$. Using Proposition 1.1.4.12, we can reduce to the case $S = \Delta ^ m$, in which case the statement reduces to the assertion that $X$ is $n$-coskeletal. $\square$

Remark 3.5.3.11. In the situation of Proposition 3.5.3.10, it suffices to consider the special case where $S = \Delta ^ m$ is a standard simplex. This follows from Corollary 1.1.4.9, since any simplicial set can be realized as a colimit of simplices (Remark 1.1.3.13).

Corollary 3.5.3.12. Let $n$ be an integer and let $K$ and $X$ be simplicial sets. If $X$ is $n$-coskeletal, then $\operatorname{Fun}(K,X)$ is also $n$-coskeletal.

Proof. By virtue of Proposition 3.5.3.10, it will suffice to show that for every simplicial set $S$, the restriction map $\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Fun}(K,X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{Fun}(K,X) )$ is a bijection. Using Proposition 1.5.3.2, we can identify $\theta $ with the horizontal map in the commutative diagram

\[ \xymatrix@R =50pt@C=30pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S \times K, X) \ar [rr] \ar [dr] & & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{k}(S) \times K, X ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S \times K), X). } \]

It will therefore suffice to show that the vertical maps in this diagram are bijections, which follows from our assumption that $X$ is $n$-coskeletal (Proposition 3.5.3.10). $\square$

Corollary 3.5.3.13. Let $X_{\bullet }: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a simplicial set and let $n$ be an integer. Then $X_{\bullet }$ is $n$-coskeletal if and only if it satisfies the following condition for each $m \geq 0$:

$(\ast _ n)$

Let $\operatorname{\mathcal{C}}= \operatorname{{\bf \Delta }}_{ \Delta ^{m} }^{\leq n}$ denote the category of simplices of $\Delta ^ m$ having dimension $\leq m$ (see Construction 1.1.3.9). Then the tautological map

\[ \theta _{m}: X_{m} \rightarrow \varprojlim _{ ([k], \sigma ) \in \operatorname{\mathcal{C}}^{\operatorname{op}} } X_{k} \]

is a bijection.

Proof. For each $n \geq 0$, we can identify $\theta _{m}$ with the restriction map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, X_{\bullet } ) \rightarrow \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( S_{\bullet }, X_{\bullet } ), \]

where $S_{\bullet }$ denotes the colimit $\varinjlim _{ ( [k], \sigma ) \in \operatorname{\mathcal{C}}} \Delta ^ k$. Using Corollary 1.1.4.8, we can identify $S_{\bullet }$ with the $n$-skeleton $\operatorname{sk}_{n}( \Delta ^ m )$, so the desired result follows from Proposition 3.5.3.10 and Remark 3.5.3.11. $\square$

Remark 3.5.3.14. Corollary 3.5.3.13 can be reformulated using the language of Kan extensions (see Definition 7.3.0.1): it asserts that a simplicial set $X_{\bullet }: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is $n$-coskeletal if and only if it is right Kan extended from the full subcategory of $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ spanned by the objects $\{ [k] \} _{k \leq n}$. Compare with Remark 1.1.3.12.

Definition 3.5.3.15. 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 an $n$-coskeleton of $X$ if it satisfies the following pair of conditions:

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

  • The morphism $f$ is bijective on $m$-simplices for $m \leq n$.

Proposition 3.5.3.16 (Existence). Let $X$ be a simplicial set. For every integer $n$, there exists a simplicial set $\operatorname{cosk}_{n}(X)$ and a morphism $f: X \rightarrow \operatorname{cosk}_{n}(X)$ which exhibits $\operatorname{cosk}_{n}(X)$ as an $n$-coskeleton of $X$.

Proof. Let $\operatorname{cosk}_{n}(X)$ denote the simplicial set given by the construction

\[ ( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ), \]

and let $f: X \rightarrow \operatorname{cosk}_{n}(X)$ be the morphism of simplicial sets given on $m$-simplices by the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^ m), X) )$. If $m \leq n$, then $\operatorname{sk}_{n}( \Delta ^ m ) = \Delta ^ m$; it follows that $f$ is bijective on $m$-simplices for $m \leq n$. We will complete the proof by showing that $\operatorname{cosk}_{n}(X)$ is $n$-coskeletal. Fix an integer $m > n$; we wish to show that the restriction map

\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, \operatorname{cosk}_{n}(X)) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{cosk}_{n}(X) ) \]

is a bijection. Writing $\operatorname{\partial \Delta }^{m}$ as a colimit of simplices (Remark 1.1.3.13) and applying Corollary 1.1.4.9, we can identify $\theta $ with the restriction map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \operatorname{\partial \Delta }^ m), X ). \]

The desired result now follows from the observation that the $n$-skeleton of $\Delta ^ m$ is contained in $\operatorname{\partial \Delta }^ m$. $\square$

Definition 3.5.3.15 can be reformulated as a universal mapping property.

Proposition 3.5.3.17 (Uniqueness). Let $n$ be an integer and let $f: X \rightarrow Y$ be a morphism of simplicial sets, where $Y$ is $n$-coskeletal. The following conditions are equivalent:

$(1)$

The morphism $f$ exhibits $Y$ as an $n$-coskeleton of $X$: that is, it is bijective on $m$-simplices for $m \leq n$.

$(2)$

For every $n$-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 $n$-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. Assertion $(3)$ is equivalent to the requirement that, for every $n$-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 3.5.3.12, we can replace $Z$ by $\operatorname{Fun}(K, Z)$ and thereby reduce to the case $K = \Delta ^0$. This proves the equivalence $(2) \Leftrightarrow (3)$.

The implication $(1) \Rightarrow (3)$ follows immediately from Proposition 3.5.3.10. We will complete the proof by showing that $(3)$ implies $(1)$. Using Proposition 3.5.3.16, we can choose a morphism $u: X \rightarrow \operatorname{cosk}_{n}(X)$ which exhibits $\operatorname{cosk}_{n}(X)$ as an $n$-coskeleton of $X$. Then $u$ satisfies condition $(3)$, so $f$ factors (uniquely) as a composition $X \xrightarrow {u} \operatorname{cosk}_{n}(X) \xrightarrow {g} Y$. We can therefore replace $X$ by $\operatorname{cosk}_{n}(X)$ and thereby reduce to the case where $X$ is $n$-coskeletal. In this case, condition $(3)$ implies that $f$ is an isomorphism of ($n$-coskeletal) simplicial sets, and therefore bijective on $m$-simplices for $m \leq n$. $\square$

Notation 3.5.3.18. Let $X$ be a simplicial set and let $n$ be an integer. It follows from Proposition 3.5.3.16 that there exists a morphism of simplicial sets $f: X \rightarrow Y$ which exhibits $Y$ as an $n$-coskeleton of $X$. Moreover, Proposition 3.5.3.17 guarantees that $Y$ is unique up to (canonical) isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $Y$ by $\operatorname{cosk}_{n}(X)$ and refer to it as the $n$-coskeleton of $X$. More explicitly, we can take $\operatorname{cosk}_{n}(X)$ to be the simplicial set constructed in the proof of Proposition 3.5.3.16, given by the construction

\[ ( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ). \]

Corollary 3.5.3.19. Let $n$ be an integer. Then the inclusion functor

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

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

Remark 3.5.3.20. For every integer $n$, the coskeleton functor

\[ \operatorname{cosk}_{n}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }} \]

preserves small limits and filtered colimits. If $n > 0$, then it also preserves coproducts.

Remark 3.5.3.21. Let $X$ be a simplicial set, let $n$ be an integer, and let $\operatorname{cosk}_{n}(X)$ denote the $n$-coskeleton of $X$. For every simplicial set $S$, we have canonical isomorphisms

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{cosk}_{n}(X) ) \xleftarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{cosk}_{n}(X) ) \]

where the map on the left is bijective because the map $X \rightarrow \operatorname{cosk}_{n}(X)$ is bijective on simplices of dimension $\leq n$, and the map on the right is bijective by virtue of Proposition 3.5.3.10. Note that this observation was implicitly used (in the special case $S = \operatorname{\partial \Delta }^{m}$) in the proof of Proposition 3.5.3.16.

Remark 3.5.3.22. Let $X$ be a simplicial set and let $n$ be an integer. Then the tautological map $u: X \rightarrow \operatorname{cosk}_{n}(X)$ is bijective on $m$-simplices for $m \leq n$. Applying Corollary 3.5.2.2, we deduce that $f$ is $n$-connective.

Proposition 3.5.3.23. Let $X$ be a Kan complex and let $n$ be an integer. Then the $n$-coskeleton $\operatorname{cosk}_{n}( X )$ is also a Kan complex.

Proof. Let $m$ be a positive integer. Fix an integer $0 \leq i \leq m$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{cosk}_{n}(X)$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{cosk}_ n(X)$. Using Remark 3.5.3.21, we can identify $\sigma _0$ with a morphism of simplicial sets $f_0: \operatorname{sk}_{n}( \Lambda ^{m}_{i} ) \rightarrow X$; we wish to show that $f_0$ can be extended to the $n$-skeleton of $\Delta ^{m}$. If $n < m-1$, then $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n}( \Delta ^{m} )$ and there is nothing to prove. We may therefore assume that $n \geq m-1$, so that $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \Lambda ^{m}_{i}$. In this case, our assumption that $X$ is a Kan complex guarantees that $f_0$ can be extended to an $n$-simplex of $X$. $\square$

Example 3.5.3.24. 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 $n \geq 0$, the coskeleton $\operatorname{cosk}_{n}( X)$ inherits the structure of a simplicial abelian group. It follows from Theorem 2.5.6.1 that $\operatorname{cosk}_{n}(X)$ 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 conditions $(a)$ and $(b)$ of Proposition 3.5.3.9 and are equipped with a chain map $A_{\ast } \rightarrow A'_{\ast }$. More concretely, we can identify $A'_{\ast }$ with the chain complex

\[ \cdots \rightarrow 0 \rightarrow Z_{n} \hookrightarrow A_{n} \xrightarrow {\partial } A_{n-1} \xrightarrow {\partial } A_{n-2} \rightarrow \cdots , \]

where $Z_{n}$ is the group of $n$-cycles of $A_{\ast }$.