10.1.3 Skeletal Simplicial Objects
Recall that every set $S$ can be regarded as a simplicial set, by identifying it with the constant functor
\[ \underline{S}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \{ S \} \hookrightarrow \operatorname{Set}. \]
This construction has a counterpart in an $\infty $-category:
Definition 10.1.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}$ denote the simplicial object of $\operatorname{\mathcal{C}}$ given by the constant functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \{ C\} \hookrightarrow \operatorname{\mathcal{C}}. \]
We say that a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is constant if it is equal to $\underline{C}$, for some object $C \in \operatorname{\mathcal{C}}$ (in this case, we must have $C = X_{0}$). We say that $X_{\bullet }$ is essentially constant it is isomorphic to a constant simplicial object $\underline{C}$, for some $C \in \operatorname{\mathcal{C}}$.
Proposition 10.1.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The simplicial object $X_{\bullet }$ is essentially constant: that is, there exists an isomorphism of simplicial objects $\alpha : \underline{C} \rightarrow X_{\bullet }$ for some object $C \in \operatorname{\mathcal{C}}$.
- $(2)$
The functor $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carries each morphism in the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(3)$
The functor $X_{\bullet }$ is left Kan extended from the full subcategory $\{ [0] \} \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$.
- $(4)$
For every simplicial object $Y_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}^{\operatorname{op}}), \operatorname{\mathcal{C}})}( X_{\bullet }, Y_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{0}, Y_0 ) \]
is a homotopy equivalence.
Proof.
The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ are special cases of Corollary 7.3.3.14, since $[0]$ is a final object of the category $\operatorname{{\bf \Delta }}$. The equivalence $(3) \Leftrightarrow (4)$ follows from Corollary 7.3.6.13.
$\square$
We now introduce a generalization of Definition 10.1.3.1.
Notation 10.1.3.3. Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.0.2). For every integer $n$, we let $\operatorname{{\bf \Delta }}^{\leq n}$ denote the full subcategory of $\operatorname{{\bf \Delta }}$ spanned by those objects $[m] = \{ 0 < 1 < \cdots < m \} $ where $0 \leq m \leq n$ (see Construction 1.1.3.9).
Definition 10.1.3.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $n$-skeletal if the functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
is left Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n})^{\operatorname{op}} \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}}$.
Example 10.1.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $0$-skeletal (in the sense of Definition 10.1.3.4) if and only if it is essentially constant (in the sense of Definition 10.1.3.1). See Proposition 10.1.3.2.
Example 10.1.3.6. Let $X_{\bullet }$ be a simplicial set and let $n$ be an integer. Then $X_{\bullet }$ is $n$-skeletal (in the sense of Definition 10.1.3.4) if and only if it has dimension $\leq n$ (in the sense of Definition 1.1.3.1) . This is a reformulation of Proposition 1.1.3.11 (see Remark 1.1.3.12).
Exercise 10.1.3.7. Let $n$ be an integer. Show that a simplicial abelian group $A_{\bullet }$ is $n$-skeletal (when regarded as a simplicial object of the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{ Ab })$) if and only if the normalized Moore complex
\[ \cdots \rightarrow \mathrm{N}_{2}(A) \xrightarrow {\partial } \mathrm{N}_{1}(A) \xrightarrow {\partial } \mathrm{N}_{0}(A) \]
is concentrated in degrees $\leq n$: that is, the abelian group $\mathrm{N}_{k}(A)$ vanishes for $k > n$. Beware that this condition does not guarantee that $A_{\bullet }$ is $n$-skeletal when regarded as a simplicial set.
To make Definition 10.1.3.4 more explicit, it will be convenient to introduce an auxiliary construction.
Construction 10.1.3.10 (Degeneracy Cubes). Fix an integer $k \geq 0$, set $K = \{ 1, 2, \cdots , k \} $, and let $\operatorname{\raise {0.1ex}{\square }}^{k} = \operatorname{\raise {0.1ex}{\square }}^{K}$ denote the simplicial cube of dimension $k$ (Notation 2.4.5.2). Recall that $\operatorname{\raise {0.1ex}{\square }}^{k}$ can be identified with the nerve of the set $P(K)$ of subsets of $K$, partially ordered with respect to inclusion.
For every subset $J \subseteq K$ having cardinality $j$, let $\alpha _{J}: [k] \twoheadrightarrow [j]$ denote the nondecreasing function function which carries an element $k' \in [k]$ to the cardinality of the intersection $J \cap \{ 1,2, \cdots , k' \} $. The construction $J \mapsto \alpha _{J}$ then determines a functor from $P(K)^{\operatorname{op}}$ to the coslice category $\operatorname{{\bf \Delta }}_{[k] / }$.
If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $X_{\bullet }$ is a simplicial object of $\operatorname{\mathcal{C}}$, we let $\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} \rightarrow \operatorname{\mathcal{C}}$ denote the map given by the composition
\[ \operatorname{\raise {0.1ex}{\square }}^{k} \simeq \operatorname{N}_{\bullet }( P(K) ) \xrightarrow {J \mapsto \alpha _{J} } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{[k]/} )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}. \]
We will refer to $\sigma _{k}$ as the $k$th degeneracy cube of the simplicial object $X_{\bullet }$.
Example 10.1.3.11. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. For small values of $k$, the degeneracy cube $\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.1.3.10 can be described more concretely:
The degeneracy cube $\sigma _0$ can be identified with the object $X_0$ of $\operatorname{\mathcal{C}}$.
The degeneracy cube $\sigma _1$ can be identified with the degeneracy operator $s^{0}_{0}: X_0 \rightarrow X_1$ of Notation 10.1.0.4.
The degeneracy cube $\sigma _{2}$ is a square diagram
\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [r]^{ s^{0}_{0} } \ar [d]^{s^{0}_{0} } & X_1 \ar [d]^{ s^{1}_{0} } \\ X_1 \ar [r]^{ s^{1}_{1} } & X_{2} } \]
which witnesses the identity $[ s^{1}_{0} ] \circ [s^{0}_{0} ] = [ s^{1}_{1} ] \circ [s^{0}_{0} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
Notation 10.1.3.12. Let $k$ be a nonnegative integer. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{ [k] / }^{\leq n}$ denote the full subcategory of the coslice category $\operatorname{{\bf \Delta }}_{ [k] / }$ spanned by objects which correspond to nondecreasing functions $[k] \rightarrow [m]$, where $m \leq n$.
Lemma 10.1.3.13. Let $k \geq 0$ and $n$ be integers, let $K = \{ 1, \cdots , k \} $, and let $P^{\leq n}(K)$ denote the partially ordered collection of all subsets $J \subseteq K$ which have cardinality $\leq n$. Then the assignment $J \mapsto \alpha _{J}$ of Construction 10.1.3.10 determines a right cofinal functor of $\infty $-categories
\[ \alpha : \operatorname{N}_{\bullet }( P^{\leq n}(K) ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ [k] / }^{\leq n})^{\operatorname{op}}. \]
Proof.
This is a special case of Corollary 7.2.3.7, since the functor $\alpha $ has a left adjoint (which carries a morphism $f: [k] \rightarrow [m]$ to the subset $J = \{ j \in K: f(j-1) < f(j) \} \in P_{\leq n}(K)$).
$\square$
Proposition 10.1.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The simplicial object $X_{\bullet }$ is $n$-skeletal, in the sense of Definition 10.1.3.4.
- $(2)$
Let $k \geq 0$ be a nonnegative integer and set $K = \{ 1, 2, \cdots , k \} $. Then the $k$th degeneracy cube
\[ \sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} = \operatorname{N}_{\bullet }( P(K) ) \rightarrow \operatorname{\mathcal{C}} \]
exhibits $X_{k}$ as a colimit of the diagram $\sigma _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) }$.
- $(3)$
For every nonnegative integer $k > n$, the $k$-degeneracy cube $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows immediately from Lemma 10.1.3.13 (together with Corollary 7.2.2.3). For each integer $k \geq 0$, set $K = \{ 1, 2, \cdots , k \} $ and consider the following conditions:
- $(2_ k)$
The degeneracy cube $\sigma _{k}: \operatorname{N}_{\bullet }( P(K) ) \rightarrow \operatorname{\mathcal{C}}$ exhibits $X_{k}$ as a colimit of the diagram $\sigma _{k}|_{ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) }$.
- $(3_ k)$
The degeneracy cube $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Note that condition $(2_ k)$ is automatic for $k \leq n$. We will complete the proof by showing that if $k > n$ and condition $(2_{\ell } )$ is satisfied for every integer $0 \leq \ell < k$, then conditions $(2_ k)$ and $(3_ k)$ are equivalent. Our hypothesis that condition $(2_{\ell } )$ is satisfied for $\ell < k$ guarantees that the functor $\sigma _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq k-1}(K) ) }$ is left Kan extended from the full subcategory $\operatorname{N}_{\bullet }( P^{\leq n}(K) ) \subseteq \operatorname{N}_{\bullet }( P^{\leq k-1}(K) )$. The equivalence of $(2_ k)$ and $(3_ k)$ is therefore a special case of Corollary 7.3.8.2.
$\square$
Corollary 10.1.3.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. Then a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $(n-1)$-skeletal if and only if it is $n$-skeletal and the degeneracy cube $\sigma _{n}: \operatorname{\raise {0.1ex}{\square }}^{n} \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Corollary 10.1.3.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n \geq 0$ be an integer. Assume that $\operatorname{\mathcal{C}}$ admits pushouts and that $F$ preserves pushouts. If $X_{\bullet }$ is an $n$-skeletal simplicial object of $\operatorname{\mathcal{C}}$, then $F( X_{\bullet } )$ is an $n$-skeletal simplicial object of $\operatorname{\mathcal{D}}$.
Proof.
Fix an integer $k > n$, and let $\sigma _{k}: \operatorname{\raise {0.1ex}{\square }}^{k} \rightarrow \operatorname{\mathcal{C}}$ be the $k$th degeneracy cube of the simplicial object $X_{\bullet }$. Set $K = \{ 1, 2, \cdots , k \} $, and let $P^{> 0}(K)$ denote the collection of all nonempty subsets of $K$. Then $\sigma _{k}$ can be identified with a functor $\sigma _{k}^{\circ }$ from $\operatorname{N}_{\bullet }( P^{> 0}(K) )$ to the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ X_0 / }$. Our assumption that $X_{\bullet }$ is $n$-skeletal guarantees that $\sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ (Proposition 10.1.3.14), or equivalently that $\sigma _{k}^{\circ }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{X_0/}$ (Remark 7.1.2.11). Since the functor $F$ preserves pushouts, the induced functor of coslice $\infty $-categories $F_{X_0/}: \operatorname{\mathcal{C}}_{X_0/} \rightarrow \operatorname{\mathcal{D}}_{ F(X_0)/ }$ preserves finite colimits (Example 7.6.3.28). In particular, $F_{X_0/} \circ \sigma _{k}^{\circ }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}_{ F(X_0) / }$, so that $F \circ \sigma _{k}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Remark 7.1.2.11). Allowing $k$ to vary, we conclude that $F( X_{\bullet } )$ is an $n$-skeletal simplicial object of $\operatorname{\mathcal{D}}$ (Proposition 10.1.3.14).
$\square$
Let $X = X_{\bullet }$ be a simplicial set. Recall that the $n$-skeleton of $X$ is the largest simplicial subset $\operatorname{sk}_{n}(X) \subseteq X$ of dimension $\leq n$ (see Construction 1.1.4.1 and Corollary 1.1.4.7). This construction has a counterpart for more general simplicial objects.
Definition 10.1.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: Y_{\bullet } \rightarrow X_{\bullet }$ be a morphism between simplicial objects of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. We will say that $u$ exhibits $Y_{\bullet }$ as an $n$-skeleton of $X_{\bullet }$ if the following conditions are satisfied:
The simplicial object $Y_{\bullet }$ is $n$-skeletal.
For $0 \leq m \leq n$, the induced map $Y_{m} \rightarrow X_{m}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
Example 10.1.3.19. Let $X = X_{\bullet }$ be a simplicial set. For every integer $n$, the inclusion of simplicial sets $\operatorname{sk}_{n}(X) \hookrightarrow X$ exhibits $\operatorname{sk}_{n}(X)$ as an $n$-skeleton of $X$, in the sense of Definition 10.1.3.18: see Proposition 1.1.4.6.
Example 10.1.3.21. Let $A_{\bullet }$ be a simplicial abelian group and let $\mathrm{N}_{\ast }(A)$ denote the normalized Moore complex of $A_{\bullet }$ (Construction 2.5.5.7). For every integer $n \geq 0$, let $\mathrm{N}_{\leq n}(A)$ denote the subcomplex of $\mathrm{N}_{\ast }(A)$ depicted in the diagram
\[ \cdots \rightarrow 0 \rightarrow \mathrm{N}_{n}(A) \xrightarrow {\partial } \mathrm{N}_{n-1}(A) \rightarrow \cdots \rightarrow \mathrm{N}_{1}(A) \xrightarrow {\partial } \mathrm{N}_{0}(A) \rightarrow 0 \rightarrow \cdots \]
Then the inclusion map
\[ \mathrm{K}( \mathrm{N}_{\leq n}(A) ) \hookrightarrow \mathrm{K}( \mathrm{N}_{\ast }(A) ) \simeq A_{\bullet } \]
exhibits the Eilenberg-MacLane space $\mathrm{K}( \mathrm{N}_{\leq n}(A) )$ as an $n$-skeleton of $A_{\bullet }$ in the category of simplicial abelian groups. Beware that the image of this inclusion is usually larger than the $n$-skeleton of $A_{\bullet }$ as a simplicial set (see Exercise 10.1.3.7).
Example 10.1.3.22 (0-Skeleta). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and set $C = X_0$. Then the constant simplicial object $\underline{C}$ is an $n$-skeleton of $X_{\bullet }$. More precisely, the identity morphism $\operatorname{id}: C \xrightarrow {\sim } X_0$ admits an (essentially unique) extension to a morphism of simplicial objects $\underline{C} \rightarrow X_{\bullet }$ which exhibits $\underline{C}$ as a $0$-skeleton of $X_{\bullet }$.
Proposition 10.1.3.23 (Existence of Skeleta). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. If $\operatorname{\mathcal{C}}$ admits pushouts, then every simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ admits an $n$-skeleton.
Proof.
We will show that the functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n} )^{\operatorname{op}} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a left Kan extension $Y_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$; Corollary 7.3.6.13 then guarantees that there is an (essentially unique) morphism of simplicial objects $u: Y_{\bullet } \rightarrow X_{\bullet }$ which is the identity when restricted to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n} )^{\operatorname{op}}$. By virtue of Corollary 7.3.5.8, it will suffice to show that for every integer $k > n$, the diagram
\[ F: \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}^{\leq n}_{ [k] / } )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a colimit. Set $K = \{ 1, 2, \cdots , k\} $, let $P^{\leq n}(K)$ denote the collection of all subsets of $K$ having cardinality $\leq n$, and let $F_0$ denote the composition of $F$ with the right cofinal functor
\[ \operatorname{N}_{\bullet }( P^{\leq n}(K) ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{ [k] / } )^{\operatorname{op}} \]
supplied by Lemma 10.1.3.13. Let $Q \subseteq P^{\leq n}(K)$ denote the collection of nonempty subsets of $K$ of cardinality $\leq n$, so that $F_0$ can be identified with a functor $G: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{C}}_{X_0/}$. Since $\operatorname{\mathcal{C}}$ admits pushouts, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X_0/}$ admits finite limits (Example 7.6.3.28). In particular, the functor $G$ admits a colimit in $\operatorname{\mathcal{C}}_{X_0/}$, which we can identify with a colimit of $F_{0}$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 7.1.2.11).
$\square$
Warning 10.1.3.24. If $X = X_{\bullet }$ is a simplicial set, then the comparison map $\operatorname{sk}_{n}(X) \rightarrow X$ is a monomorphism of simplicial sets. Beware that the analogous statement is generally false for simplicial objects of more general $\infty $-categories.