10.2.4 Coskeletal Simplicial Objects
Let $S$ be a set. For each $n \geq 0$, we let $\operatorname{\check{C}}_{n}( S ) = \operatorname{Hom}( [n], S )$ denote the collection of functions from the set $[n] = \{ 0 < 1 < \cdots < n \} $ into $S$. The construction $[n] \mapsto \operatorname{\check{C}}_{n}(S)$ determines a simplicial set $\operatorname{\check{C}}_{\bullet }(S)$, which we will refer to as the Čechnerve of $S$. In this section, we study an $\infty $-categorical counterpart of this construction.
Definition 10.2.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. We will say that $X_{\bullet }$ is a Čechnerve if, for every integer $n \geq 0$, the following condition is satisfied:
- $(\ast _ n)$
For $0 \leq i \leq n$, let $\nu _{i}: X_{n} \rightarrow X_{0}$ be the morphism of $\operatorname{\mathcal{C}}$ induced by the inclusion $[0] \simeq \{ i \} \subseteq [n]$. Then the morphisms $\{ \nu _{i} \} _{0 \leq i \leq n}$ exhibit $X_{n}$ as a product of $(n+1)$-copies of $X_0$.
Definition 10.2.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We will say that a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čechnerve of $X$ if $X_{\bullet }$ is a Čechnerve (in the sense of Definition 10.2.4.1) and $X_{0} = X$.
Notation 10.2.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $X_{\bullet }$. It follows from Corollary 7.3.6.13 that, for every simplicial object $Y_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}}) }( Y_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y_0, X_0) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y_0, X) \]
is a homotopy equivalence. In particular, the simplicial object $X_{\bullet }$ is unique up to isomorphism and depends functorially on $X$. To emphasize this dependence, we will denote $X_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X)$ and refer to it as the Čechnerve of the object $X$.
Proposition 10.2.4.5 (Existence). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X)$ if and only if, for every nonempty finite set $J$, there exists a product of $J$ copies of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
Proof.
For every integer $n \geq 0$, the category $\{ [0] \} \times _{\operatorname{{\bf \Delta }}} \operatorname{{\bf \Delta }}_{/ [n] }$ is isomorphic to the finite set $\{ 0, 1, \cdots , n \} $, regarded as a category having only identity morphisms. By virtue of Remark 10.2.4.2, the desired result is a special case of the existence criterion for Kan extensions (Corollary 7.3.5.8).
$\square$
Corollary 10.2.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products. Then every object $X \in \operatorname{\mathcal{C}}$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X)$.
Corollary 10.2.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite products. Then the evaluation functor
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} , \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad X_{\bullet } \mapsto X_{0} \]
admits a right adjoint, given on objects by the Čechnerve $X \mapsto \operatorname{\check{C}}_{\bullet }(X)$.
Proof.
Combine Corollaries 10.2.4.6 and 7.3.6.4.
$\square$
We now introduce a generalization of Definition 10.2.4.1.
Definition 10.2.4.9. 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 $n$-coskeletal if the functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}} \]
is right 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 10.2.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $0$-coskeletal if and only if it is a Čechnerve (Remark 10.2.4.2).
Example 10.2.4.11. Let $X_{\bullet }$ be a simplicial set and let $n$ be an integer. The following conditions are equivalent:
The simplicial set $X_{\bullet }$ is $n$-coskeletal in the sense of Definition 3.5.3.1: that is, the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X)$ is bijective for $m > n$.
The simplicial set $X_{\bullet }$ is $n$-coskeletal in the sense of Definition 10.2.4.9: that is, it is a right Kan extension of its restriction to (the opposite of) the subcategory $\operatorname{{\bf \Delta }}^{\leq n} \subset \operatorname{{\bf \Delta }}$.
This is a restatement of Corollary 3.5.3.13 (see Remark 3.5.3.14).
Definition 10.2.4.9 has a counterpart for semisimplicial objects.
Variant 10.2.4.14. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\leq n} = \operatorname{{\bf \Delta }}_{\operatorname{inj}} \cap \operatorname{{\bf \Delta }}^{\leq n}$ denote the category whose objects are linearly ordered sets $[m] = \{ 0 < 1 < \cdots < m \} $ for $0 \leq m \leq n$, and whose morphisms are strictly increasing functions. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we say that a semisimplicial object $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ is $n$-coskeletal if it is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\leq n} )^{\operatorname{op}}$.
Proposition 10.2.4.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. Then $X_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.2.4.9) if and only if its underlying semisimplicial object is $n$-coskeletal (in the sense of Variant 10.2.4.14).
Proof.
Fix an integer $k \geq 0$, $\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{ / [k] }$ spanned by those objects which correspond to nondecreasing functions $\alpha : [m] \rightarrow [k]$ where $m \leq n$, and let $\operatorname{\mathcal{J}}$ be the full subcategory of $\operatorname{{\bf \Delta }}_{ / [k]}^{\leq n}$ spanned by those objects where $\alpha $ is strictly increasing. Unwinding the definitions, we see that $X_{\bullet }$ is $n$-coskeletal if and only if, for every integer $k \geq 0$, the composite functor
\[ F: \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}))^{\triangleright } \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ / [k] }^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \xrightarrow { X_{\bullet } } \operatorname{\mathcal{C}}^{\operatorname{op}} \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Similarly, the underlying semisimplicial object of $X_{\bullet }$ is $n$-coskeletal if and only if, for every integer $k \geq 0$, the restriction $F|_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})^{\triangleright } }$ is a colimit diagram in $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Consequently, to show that these conditions are equivalent, it will suffice to prove that the inclusion functor $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{ / [k] }^{\leq n})$ is right cofinal (Corollary 7.2.2.3). This is a special case of Corollary 7.2.3.7, since the inclusion functor $\operatorname{\mathcal{J}}\hookrightarrow \operatorname{{\bf \Delta }}_{ / [k] }^{\leq n}$ has a left adjoint (which carries a nondecreasing function $\alpha : [m] \rightarrow [k]$ to the inclusion map $\operatorname{im}(\alpha ) \hookrightarrow [k]$; here we abuse notation by identifying $\operatorname{im}(\alpha )$ with the corresponding object of $\operatorname{{\bf \Delta }}$).
$\square$
Corollary 10.2.4.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves finite limits and let $n$ be an integer. Then:
- $(1)$
If $X_{\bullet }$ is an $n$-coskeletal simplicial object of $\operatorname{\mathcal{C}}$, then $F(X_{\bullet } )$ is an $n$-coskeletal simplicial object of $\operatorname{\mathcal{D}}$.
- $(2)$
If $X_{\bullet }$ is an $n$-coskeletal semisimplicial object of $\operatorname{\mathcal{C}}$, then $F( X_{\bullet } )$ is an $n$-coskeletal semisimplicial object of $\operatorname{\mathcal{D}}$.
Proof.
Assertion $(2)$ is immediate from the definitions (since the category $\operatorname{\mathcal{J}}$ appearing in the proof of Proposition 10.2.4.15 is a finite partially ordered set). Assertion $(1)$ follows by combining $(2)$ with Proposition 10.2.4.15.
$\square$
Recall that a morphism of simplicial sets $f: X_{\bullet } \rightarrow Y_{\bullet }$ exhibits $Y_{\bullet }$ as an $n$-coskeleton of $X_{\bullet }$ if $Y_{\bullet }$ is $n$-coskeletal and $f$ is bijective on $m$-simplices for $m \leq n$. This notion has an obvious counterpart for simplicial objects in general:
Definition 10.2.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: X_{\bullet } \rightarrow Y_{\bullet }$ 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 $n$-coskeleton of $X_{\bullet }$ if the following conditions are satisfied:
The simplicial object $Y_{\bullet }$ is $n$-coskeletal.
For $0 \leq m \leq n$, the induced map $X_{m} \rightarrow Y_{m}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
Example 10.2.4.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. If the object $X = C_0$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X)$ (Definition 10.2.4.3), then the identity map $C_0 \rightarrow X$ can be promoted to a morphism of simplicial objects $C_{\bullet } \rightarrow \operatorname{\check{C}}_{\bullet }(X)$ (see Notation 10.2.4.4) which exhibits $\operatorname{\check{C}}_{\bullet }(X)$ as a $0$-coskeleton of $C_{\bullet }$.
Proposition 10.2.4.21 (Existence of Coskeleta). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits finite limits and let $n$ be an integer. Then every (semi)simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ admits an $n$-coskeleton $\operatorname{cosk}_{n}(X)_{\bullet }$.
Proof.
We will prove the assertion for simplicial objects; the analogous statement for semisimplicial objects is similar (but easier). It will suffice to 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 right 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 $u: X_{\bullet } \rightarrow Y_{\bullet }$ 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 $k$, the diagram
\[ G: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n} )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }})^{\operatorname{op}} \xrightarrow {X_{\bullet } } \operatorname{\mathcal{C}} \]
admits a limit. As in the proof of Proposition 10.2.4.15, we observe that the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n} )$ is right cofinal, where $\operatorname{\mathcal{J}}\subseteq \operatorname{{\bf \Delta }}_{ /[k]}^{\leq n}$ is the full subcategory spanned by the injective maps $[m] \hookrightarrow [k]$. We are therefore reduced to showing that $G|_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})^{\operatorname{op}} }$ has a limit in $\operatorname{\mathcal{C}}$ (Corollary 7.2.2.10), which follows from our assumption that $\operatorname{\mathcal{C}}$ admits finite limits (since $\operatorname{\mathcal{J}}$ is the category associated to a finite partially ordered set).
$\square$