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10.1.5 The ČechNerve of a Morphism

We now consider a relative version of Definition 10.1.4.1.

Definition 10.1.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, which we identify with a simplicial object $C'_{\bullet }$ of the $\infty $-category $\operatorname{\mathcal{C}}_{ / C_{-1} }$ (see Remark 10.1.1.15). We will say that $C_{\bullet }$ is a Čechnerve if the simplicial object $C'_{\bullet }$ is a Čechnerve in the $\infty $-category $\operatorname{\mathcal{C}}_{/C_{-1} }$ (see Definition 10.1.4.1).

Remark 10.1.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Stated more informally, an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čechnerve if, for every integer $n \geq 0$, it exhibits $C_{n}$ as an iterated fiber product

\[ C_{0} \times _{ C_{-1} } C_{0} \times _{ C_{-1} } \cdots \times _{ C_{-1} } C_{0} \]

(where the factor $X_0$ appears $n+1$ times).

Remark 10.1.5.3. In the augmented simplex category $\operatorname{{\bf \Delta }}_{+}$, there is unique morphism $\delta ^{0}_{0}: [-1] \rightarrow [0]$. This morphism determines a fully faithful functor $[1] \rightarrow \operatorname{{\bf \Delta }}_{+}$, whose image is the full subcategory $\operatorname{{\bf \Delta }}_{+}^{\leq 0} \subseteq \operatorname{{\bf \Delta }}_{+}$ spanned by the objects $[0]$ and $[-1]$. Combining Remarks 10.1.4.2 and 7.3.2.4, we see that an augmented simplicial object $X_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ is a Čechnerve if and only if it is right Kan extended from the subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq 0} )^{\operatorname{op}} \subset \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )^{\operatorname{op}}$.

Definition 10.1.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čechnerve of $f$ if $C_{\bullet }$ is a Čechnerve (in the sense of Definition 10.1.5.1) and the face operator $d^{0}_{0}: C_{0} \rightarrow C_{-1}$ coincides with the morphism $f$ (so that $C_0 = X$ and $C_{-1} = Y$).

Notation 10.1.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. It follows from Remarks 10.1.5.3 and 7.3.6.6 that if $f$ admits a Čechnerve $C_{\bullet }$, then the augmented simplicial object $C_{\bullet }$ is determined up to isomorphism and depends functorially on $f$. To emphasize this dependence, we will denote $C_{\bullet }$ by $\operatorname{\check{C}}_{\bullet }(X/Y)$ and refer to it as the Čechnerve of the morphism $f: X \rightarrow Y$. Alternatively, we can identify $\operatorname{\check{C}}_{\bullet }(X/Y)$ with the simplicial object of $\operatorname{\mathcal{C}}_{/Y}$ given by the Čechnerve of $f$ (in the sense of Notation 10.1.4.4).

Proposition 10.1.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$.

Proof. Apply Corollary 10.1.4.6 to the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, which admits finite products by virtue of our assumption that $\operatorname{\mathcal{C}}$ admits pullbacks (Corollary 7.6.3.20). $\square$

Remark 10.1.5.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which preserves fiber products. Then the induced functor of augmented simplicial objects

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{D}}) \]

carries Čechnerves to Čechnerves (see Remark 10.1.4.7). In particular, if $u: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ which admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$, then the morphism $F(u): F(X) \rightarrow F(Y)$ admits a Čechnerve in the $\infty $-category $\operatorname{\mathcal{D}}$, given by $F( \operatorname{\check{C}}_{\bullet }(X/Y) )$.

Corollary 10.1.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then the forgetful functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \quad \quad C_{\bullet } \mapsto (d^{0}_{0}: C_0 \rightarrow C_{-1}) \]

admits a right adjoint, given on objects by the construction $(f: X \rightarrow Y) \mapsto \operatorname{\check{C}}_{\bullet }(X/Y)$.

It will sometimes be useful to consider a generalization of Definition 10.1.5.1.

Notation 10.1.5.9. Let $\operatorname{{\bf \Delta }}_{+}$ be the augmented simplex category (Definition 10.1.1.10). For every integer $n$, we let $\operatorname{{\bf \Delta }}_{+}^{\leq n}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{+}$ spanned by the collection of objects $\{ [m] \} _{-1 \leq m \leq n}$.

Definition 10.1.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. We say that an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $n$-coskeletal if the functor

\[ C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}} \]

is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\leq n} )^{\operatorname{op}}$.

Example 10.1.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simpicial object of $\operatorname{\mathcal{C}}$. Then $C_{\bullet }$ is $0$-coskeletal (in the sense of Definition 10.1.5.10) if and only if it is a Čechnerve (in the sense of Definition 10.1.5.1). See Remark 10.1.5.3.

Example 10.1.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For $n \leq -2$, an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $n$-coskeletal if and only if each $C_{m}$ is a final object of $\operatorname{\mathcal{C}}$.

Example 10.1.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The augmented simplicial object $C_{\bullet }$ is $(-1)$-coskeletal, in the sense of Definition 10.1.5.10.

$(2)$

The augmented simplicial object $C_{\bullet }$ is essentially constant: that is, it is isomorphic to a constant functor from $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} )$ to $\operatorname{\mathcal{C}}$.

$(3)$

The functor $C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carries each morphism in the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}_{+}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

$(4)$

For every augmented simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map

\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})}( X_{\bullet }, C_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{-1}, C_{-1} ) \]

is a homotopy equivalence.

The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ are special cases of Corollary 7.3.3.14, since $[-1]$ is an initial object of the category $\operatorname{{\bf \Delta }}_{+}$. The equivalence $(1) \Leftrightarrow (4)$ follows from Corollary 7.3.6.13.

Variant 10.1.5.14. For every integer $n$, we let $\operatorname{{\bf \Delta }}_{+,\operatorname{inj}}^{\leq n}$ denote the category whose objects are linearly ordered sets $[m] = \{ 0 < 1 < \cdots < n\} $ for $-1 \leq m \leq n$, and whose morphisms are strictly increasing functions. We say that an augmented semisimplicial object $C_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$ if the functor

\[ C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+, \operatorname{inj}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}} \]

is a right Kan extension of its restriction to $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+, \operatorname{inj}} )^{\operatorname{op}}$.

Remark 10.1.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, which we identify with a (semi)simplicial object $C'_{\bullet }$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{ / C_{-1} }$ (see Remark 10.1.1.15). Then, for $n \geq -1$, the augmented simplicial object $C_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.1.5.10) if and only if the simplicial object $C'_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.1.4.9). Moreover, the analogous statement holds for semisimplicial objects. See Remark 7.3.2.4.

Warning 10.1.5.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. It follow from Remark 10.1.5.15 that if $C_{-1}$ is a final object of $\operatorname{\mathcal{C}}$, then $C_{\bullet }$ is $n$-coskeletal if and only if its underlying simplicial object is $n$-coskeletal. Beware that neither implication holds in general if we do not assume that $C_{-1}$ is final.

Remark 10.1.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. For every integer $n$, the augmented simplicial object $C_{\bullet }$ is $n$-coskeletal (in the sense of Definition 10.1.5.10) if and only if its underlying augmented semisimplicial object is $n$-coskeletal (in the sense of Variant 10.1.5.14). For $n \leq -2$, this is trivial (see Example 10.1.5.12). For $n \geq -1$, it follows by combining Remark 10.1.5.15 with Proposition 10.1.4.15 (applied to the slice $\infty $-category $\operatorname{\mathcal{C}}_{ / C_{-1} }$).

To make Definition 10.1.5.10 more concrete, it will be convenient to introduce a dual version of Construction 10.1.3.10.

Construction 10.1.5.18 (Face Cubes). Fix an integer $k \geq -1$, and let $\operatorname{\raise {0.1ex}{\square }}^{k+1}$ be the simplicial cube of dimension $k+1$ (Notation 2.4.5.2). In what follows, we will identify $\operatorname{\raise {0.1ex}{\square }}^{k+1}$ with the opposite of the nerve of the partially ordered set $P( [k] )$ of all subsets of $[k] = \{ 0 < 1 < \cdots < k \} $. Note that there is an isomorphism of categories $P( [k] ) \rightarrow (\operatorname{{\bf \Delta }}_{+,\operatorname{inj}})_{ / [k] }$, which carries each subset $J \subseteq [k]$ of cardinality $j+1$ to the unique strictly increasing function $[j] \hookrightarrow [k]$ having image $J$. If $C_{\bullet }$ is an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, we let $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ denote the composite functor

\[ \operatorname{\raise {0.1ex}{\square }}^{k+1} \simeq \operatorname{N}_{\bullet }( (\operatorname{{\bf \Delta }}_{+,\operatorname{inj}})_{ /[k] } )^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+,\operatorname{inj}}^{\operatorname{op}} ) \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}. \]

We will refer to $\tau _{k}$ as the $k$th face cube of the augmented semisimplicial object $C_{\bullet }$.

Example 10.1.5.19. Let $C_{\bullet }$ be an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. For small values of $k$, the face cube $\tau _ k: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.1.5.18 can be described more explicitly:

  • The face cube $\tau _{-1}$ can be identified with the object $C_{-1}$ of $\operatorname{\mathcal{C}}$.

  • The face cube $\tau _0$ can be identified with the face operator $d^{0}_{0}: C_0 \rightarrow C_{-1}$.

  • The face cube $\tau _1$ is a square diagram

    \[ \xymatrix@R =50pt@C=50pt{ C_1 \ar [r]^{ d^{1}_{0} } \ar [d]^{ d^{1}_{1} } & C_0 \ar [d]^{ d^{0}_{0} } \\ C_0 \ar [r]^{ d^{0}_{0} } & C_{-1} } \]

    which witnesses the identity $[ d^{0}_{0} ] \circ [d^{1}_{0} ] = [ d^{0}_{0} ] \circ [d^{1}_{1} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Proposition 10.1.5.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $C_{\bullet }$ be an augmented semisimplicial object of $\operatorname{\mathcal{C}}$, and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The augmented semisimplicial object $C_{\bullet }$ is $(n-1)$-coskeletal, in the sense of Variant 10.1.5.14.

$(2)$

For each $k \geq n$, the face cube $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.1.5.18 is a limit diagram in $\operatorname{\mathcal{C}}$.

Proof. We proceed as in the proof of Proposition 10.1.3.14. Let us identify each $\tau _{k}$ with a functor $\operatorname{N}_{\bullet }( P([k]) )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$, where $P( [k] )$ denotes the collection of all subsets of $\{ 0 < 1 < \cdots < k \} $. Let $P^{\leq n}([k] )$ denote the subset of $P( [k] )$ consisting of subsets of cardinality $\leq n$. Unwinding the definitions, we see that $C_{\bullet }$ is $(n-1)$-coskeletal if and only if the following condition is satisfied for each $k \geq n$:

$(1_ k)$

The functor $\tau _{k}$ exhibits $C_{k}$ as a limit of its restriction to $\operatorname{N}_{\bullet }( P^{\leq n}( [k] ) )^{\operatorname{op}}$.

Similarly, $(2)$ asserts that the following condition is satisfied for each $k \geq n$:

$(2_ k)$

The face cube $\tau _{k}: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.1.5.18 is a limit diagram in $\operatorname{\mathcal{C}}$.

To complete the proof, it will suffice to show that if condition $(1_{\ell } )$ is satisfied for $n \leq \ell < k$, then conditions $(1_ k)$ and $(2_ k)$ are equivalent. Our hypothesis that condition $(2_{\ell } )$ is satisfied for $\ell < k$ guarantees that the functor $\tau _{k} |_{ \operatorname{N}_{\bullet }( P^{\leq k}( [k] ) )^{\operatorname{op}} }$ is right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( P^{\leq n}( [k]) )^{\operatorname{op}} \subseteq \operatorname{N}_{\bullet }( P^{\leq k}( [k]) )^{\operatorname{op}}$. The equivalence of $(1_ k)$ and $(2_ k)$ is therefore a special case of Corollary 7.3.8.2. $\square$

Corollary 10.1.5.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -1$ be an integer. Then an augmented (semi)simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is $(n-1)$-coskeletal if and only if it is $n$-coskeletal and the face cube $\tau _{n}: \operatorname{\raise {0.1ex}{\square }}^{n+1} \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$.

Corollary 10.1.5.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $C_{\bullet }$ be an augmented (semi)simplicial object of $\operatorname{\mathcal{C}}$, and let $n \geq -1$ be an integer. Assume that $\operatorname{\mathcal{C}}$ admits pullbacks and that $F$ preserves pullbacks. If $C_{\bullet }$ is $n$-coskeletal, then the image $F( C_{\bullet } )$ is $n$-coskeletal.