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5.4.5 The Pith of an $(\infty ,2)$-Category

Let $\operatorname{\mathcal{C}}$ be a $2$-category. Recall that the pith of $\operatorname{\mathcal{C}}$ is the subcategory $\operatorname{Pith}(\operatorname{\mathcal{C}}) \subseteq \operatorname{\mathcal{C}}$ obtained by removing the non-invertible $2$-morphisms of $\operatorname{\mathcal{C}}$ (Construction 2.3.2.10). In this section, we generalize this definition to the setting of $(\infty ,2)$-categories.

Construction 5.4.5.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. We let $\operatorname{Pith}(\operatorname{\mathcal{C}}) \subseteq \operatorname{\mathcal{C}}$ denote the simplicial subset consisting of those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ which carry every $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. We will refer to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as the pith of $\operatorname{\mathcal{C}}$.

Remark 5.4.5.2. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin. Consequently, to check that a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ belongs to the pith $\operatorname{Pith}(\operatorname{\mathcal{C}})$, it suffices to check that $\sigma $ carries every nondegenerate $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. In particular:

  • Every object of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$.

  • Every morphism of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$.

  • A $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ if and only if it is thin.

Remark 5.4.5.3. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is the largest simplicial subset of $\operatorname{\mathcal{C}}$ which does not contain any non-thin $2$-simplices of $\operatorname{\mathcal{C}}$.

Example 5.4.5.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{Pith}(\operatorname{\mathcal{C}})$ denote its pith (Construction 2.3.2.10). Then the inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\operatorname{\mathcal{C}}) ) \simeq \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) )$. This is an immediate consequence of Theorem 2.3.2.5.

Example 5.4.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{Pith}(\operatorname{\mathcal{C}}) = \operatorname{\mathcal{C}}$ (see Example 2.3.2.4).

Proposition 5.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category.

Our proof of Proposition 5.4.5.6 will make use of a closure property of the collection of thin $2$-simplices of an $(\infty ,2)$-category $\operatorname{\mathcal{C}}$.

Definition 5.4.5.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. We will say that $T$ has the inner exchange property if the following condition is satisfied:

$(\ast )$

Let $\sigma : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ be a $3$-simplex of $\operatorname{\mathcal{C}}$. For every triple of integers $0 \leq i < j < k \leq 3$, let $\sigma _{kji}$ be the face of $\sigma $ given by the restriction $\sigma |_{ \operatorname{N}_{\bullet }( \{ i < j < k \} )}$. Assume that the outer faces $\sigma _{210}$ and $\sigma _{321}$ belong to $T$. Then $\sigma _{310}$ belongs to $T$ if and only if $\sigma _{320}$ belongs to $T$.

Remark 5.4.5.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$, and let $T^{\operatorname{op}}$ denote the set $T$, regarded as a collection of simplices of the of opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then $T$ has the inner exchange property if and only if $T^{\operatorname{op}}$ has the inner exchange property.

Remark 5.4.5.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{D}}$. If $T$ has the inner exchange property, then the inverse image $F^{-1}(T)$ has the inner exchange property.

Proposition 5.4.5.10 (Inner Exchange). Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then the collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$ has the inner exchange property (Definition 5.4.5.7).

Remark 5.4.5.11. To get a feeling for the content of Proposition 5.4.5.10, let us specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{D}})$ is the Duskin nerve of a $2$-category $\operatorname{\mathcal{D}}$. In this case, we can identify a $3$-simplex $\sigma : \Delta ^{3} \rightarrow \operatorname{\mathcal{C}}$ with a collection of objects $\{ X_ i \} _{0 \leq i \leq 3}$ of $\operatorname{\mathcal{D}}$, a collection of $1$-morphisms $\{ f_{ji}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$, and a collection of $2$-morphisms $\{ \mu _{kji}: f_{kj} \circ f_{ji} \Rightarrow f_{ki} \} $ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ f_{32} \circ (f_{21} \circ f_{10} ) \ar@ {=>}[rr]^-{\alpha }_{\sim } \ar@ {=>}[d]_{ \operatorname{id}_{ f_{32}} \circ \mu _{210} } & & ( f_{32} \circ f_{21} ) \circ f_{10} \ar@ {=>}[d]^{ \mu _{321} \circ \operatorname{id}_{ f_{10} }} \\ f_{32} \circ f_{20} \ar@ {=>}[dr]_{ \mu _{320} } & & f_{31} \circ f_{10} \ar@ {=>}[dl]^{ \mu _{310} } \\ & f_{30} & } \]

is commutative, where $\alpha = \alpha _{f_{32}, f_{21}, f_{10} }$ is the associativity constraint for the composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ (Proposition 2.3.1.9). The assumption that the outer faces of $\sigma $ are thin guarantees that the $2$-morphisms $\mu _{321}$ and $\mu _{210}$ are isomorphisms. In this case, Proposition 5.4.5.10 asserts that $\mu _{320}$ is an isomorphism if and only if $\mu _{310}$ is an isomorphism, which follows by inspection.

Proof of Proposition 5.4.5.10. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $\sigma : \Delta ^{3} \rightarrow \operatorname{\mathcal{C}}$ be a $3$-simplex of $\operatorname{\mathcal{C}}$ and let $C = \sigma (3) \in \operatorname{\mathcal{C}}$ be the image of the final vertex. Let us regard the face $\sigma _{210} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )}$ as a morphism of simplicial sets from $\Delta ^2$ to $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ denote the pullback $\Delta ^{2} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the projection map $\operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration (Proposition 5.4.3.1). If $\sigma _{210}$ is thin, then the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ is also an interior fibration (Remark 5.4.2.4); since $\Delta ^2$ is an $\infty $-category, it is an inner fibration (Example 5.4.2.2). Unwinding the definitions, we can identify $\sigma $ with a $2$-simplex of $\operatorname{\mathcal{E}}$ lying over the unique nondegenerate $2$-simplex of $\Delta ^2$, which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z. } \]

If $\sigma _{321} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )}$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, then the “easy direction” of Theorem 5.4.4.1 guarantees that $g$ is $\pi $-cartesian. It follows that $f$ is $\pi $-cartesian if and only if $h$ is $\pi $-cartesian (Corollary 5.2.2.5). Equivalently, $f$ is locally $\pi $-cartesian if and only if $h$ is locally $\pi $-cartesian (see Remark 5.2.3.5). Applying the “hard direction” of Theorem 5.4.4.1, we conclude that the $2$-simplex $\sigma _{310} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3\} ) }$ is thin if and only if the $2$-simplex $\sigma _{320} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < 3 \} )}$ is thin. $\square$

Proof of Proposition 5.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Suppose we are given integers $0 < i < n$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$. If $n = 2$, then condition $(1)$ of Definition 5.4.1.3 guarantees that we can extend $\sigma _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$, which then belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ by virtue of Remark 5.4.5.2. We may therefore assume that $n \geq 3$. In this case, we observe that the composite map

\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \hookrightarrow \Lambda ^{n}_{i} \xrightarrow {\sigma _0} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, so that we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $\sigma $ carries each $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. If $n \geq 4$, this is automatic (since every $2$-simplex of $\Delta ^ n$ is contained in the horn $\Lambda ^{n}_{i}$). In the case $n=3$, it follows from our assumption that the collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$ has the inner exchange property (Proposition 5.4.5.10). $\square$

Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Heuristically, one can think of the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as obtained from $\operatorname{\mathcal{C}}$ by removing its noninvertible $2$-morphisms, just as the core $\operatorname{\mathcal{E}}^{\simeq }$ of an $\infty $-category $\operatorname{\mathcal{E}}$ is obtained by removing its noninvertible morphisms (see Construction 4.4.3.1). We now make this heuristic more precise (see Corollary 5.4.7.11 for a relative version):

Proposition 5.4.5.12. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category containing objects $X$ and $Y$. Then the inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces isomorphisms of simplicial sets

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \simeq \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( X, Y)^{\simeq } \quad \quad \operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \simeq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)^{\simeq }. \]

Proof. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$, which we view as a morphism of simplicial sets $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ whose restriction to the face $\Delta ^{n} \subseteq \Delta ^{n+1}$ equal to the constant map $\Delta ^{n} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. Then $\sigma $ belongs to the simplicial subset $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \subseteq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ if and only if, for every $2$-simplex $\rho : \Delta ^2 \rightarrow \Delta ^{n+1}$, the composition $\tau \circ \rho $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Note that this condition is automatically satisfied if $\rho $ is degenerate, or takes values in the subset $\Delta ^{n} \subseteq \Delta ^{n+1}$ (since every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin). Consequently, it suffices to verify this condition in the case where $\rho $ is the right cone of a map $\rho _0: \Delta ^1 \rightarrow \Delta ^{n}$. In this case, $\tau \circ \rho $ is thin if and only if the edge $\Delta ^{1} \xrightarrow {\rho _0} \Delta ^{n} \xrightarrow {\sigma } \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ (Theorem 5.4.4.1). Allowing $\tau _0$ to vary, we obtain the identification $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \simeq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)^{\simeq }$; the proof of the analogous statement for left-pinched morphism spaces is similar. $\square$

Proposition 5.4.5.13. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then:

$(1)$

The projection map $\pi : \operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cartesian fibration of $\infty $-categories. Moreover, a morphism $u$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\pi $-cartesian if and only if, for every vertex $z \in K$, the composite map

\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(2)$

The projection map $\pi ': \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories. Moreover, a morphism $v$ of $\operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\pi '$-cocartesian if and only if, for every vertex $x \in K$, the composite map

\[ \Delta ^2 \simeq \{ x\} \star \Delta ^1 \hookrightarrow K \star \Delta ^1 \xrightarrow {v} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. It follows from Remark 5.4.2.4 that $\pi $ is an interior fibration. Since $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Proposition 5.4.5.6), it is an inner fibration of $\infty $-categories (Example 5.4.2.2). Let us say that a morphism $u$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is special if, for every vertex $z \in K$, the composite map

\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Let $\overline{\pi }: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. It follows from Corollary 5.4.4.2 every special morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore also $\pi $-cartesian (Remark 5.2.1.9). Conversely, any $\pi $-cartesian morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is locally $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore special (again by Corollary 5.4.4.2). To complete the proof, it will suffice to show that if $Y$ is an object of $\operatorname{\mathcal{C}}_{/f}$, then any morphism $\overline{u}: \overline{X} \rightarrow q( \overline{Y} )$ in $\operatorname{Pith}( \operatorname{\mathcal{C}})$ can be lifted to a special morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$, which follows from Proposition 5.4.3.9. $\square$