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5.4.1 Definitions

We begin by introducing some terminology.

Definition 5.4.1.1. Let $X$ be a simplicial set and let $\sigma : \Delta ^2 \rightarrow X$ be a $2$-simplex of $X$. We will say that $\sigma $ is left-degenerate if it factors through the map $\sigma ^{0}: \Delta ^2 \rightarrow \Delta ^1$ given on vertices by $\sigma ^{0}(0) = 0 = \sigma ^{0}(1)$ and $\sigma ^{0}(2) = 1$ (Notation 1.1.1.9). We say that $\sigma $ is right-degenerate if it factors through the map $\sigma ^{1}: \Delta ^2 \rightarrow \Delta ^1$ given on vertices $\sigma ^{1}(0) = 0$ and $\sigma ^{1}(1) = 1 = \sigma ^{1}(2)$.

Remark 5.4.1.2. Let $X$ be a simplicial set. Then:

  • A $2$-simplex $\sigma $ of $X$ is degenerate (in the sense of Definition 1.1.3.2) if and only if it is either left-degenerate or right-degenerate.

  • A $2$-simplex $\sigma $ of $X$ is constant (that is, factors through the projection map $\Delta ^2 \rightarrow \Delta ^0$) if and only if it is both left-degenerate and right-degenerate.

  • A $2$-simplex $\sigma $ of $X$ is left-degenerate if and only if it is right-degenerate when viewed as a $2$-simplex of the opposite simplicial set $X^{\operatorname{op}}$.

Definition 5.4.1.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We will say that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category if it satisfies the following axioms:

$(1)$

Every morphism of simplicial sets $\Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(2)$

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

$(3)$

Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1 < n\} ) }$ is left-degenerate. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.

$(4)$

Let $n \geq 3$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0< n-1 < n\} ) }$ is right-degenerate. Then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$.

Proposition 5.4.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category.

Proof. Our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that every $2$-simplex of $\operatorname{\mathcal{C}}$ is thin (Example 2.3.2.4). Consequently, condition $(2)$ of Definition 5.4.1.3 is automatic, and condition $(1)$ follows immediately from the definition. Conditions $(3)$ and $(4)$ follow from Theorem 4.4.2.5 (since every degenerate edge of $\operatorname{\mathcal{C}}$ is an isomorphism). $\square$

Remark 5.4.1.5. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. We will refer to vertices of $\operatorname{\mathcal{C}}$ as objects, and to the edges of $\operatorname{\mathcal{C}}$ as morphisms. If $f$ is an edge of $\operatorname{\mathcal{C}}$ satisfying $d_1(f) = X$ and $d_0(f) = Y$, then we say that $f$ is a morphism from $X$ to $Y$ and write $f: X \rightarrow Y$.

Suppose we are given morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$. We will say that a $2$-simplex $\sigma $ witnesses $h$ as a composition of $f$ and $g$ if it is thin and satisfies $d_0(\sigma ) = g$, $d_1(\sigma ) = h$, and $d_2(\sigma ) = f$, as indicated in the diagram

\[ \xymatrix { & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z. } \]

Note that:

  • When $\operatorname{\mathcal{C}}$ is an $\infty $-category, this recovers the terminology of Definition 1.3.4.1 (since the $2$-simplex $\sigma $ is automatically thin).

  • If $\operatorname{\mathcal{C}}$ is the Duskin nerve of a $2$-category $\operatorname{\mathcal{E}}$, the $2$-simplex $\sigma $ can be identified with a $2$-morphism $\gamma : g \circ f \Rightarrow h$ of $\operatorname{\mathcal{E}}$, which is invertible if and only if $\sigma $ is thin. In other words, $\sigma $ witnesses $h$ as a composition of $f$ and $g$ if and only if it encodes the datum of an isomorphism $g \circ f \xRightarrow {\sim } h$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(X,Z)$.

  • Axiom $(1)$ of Definition 5.4.1.3 asserts that the composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ is defined (albeit not uniquely). More precisely, it asserts that for every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, there exists a $1$-morphism $h: X \rightarrow Z$ and a $2$-simplex which witnesses $h$ as a composition of $f$ and $g$.

Remark 5.4.1.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category if and only if the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is an $(\infty ,2)$-category.

Proposition 5.4.1.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category.

Proof. Condition $(1)$ of Definition 5.4.1.3 follows immediately from Theorem 2.3.2.5, and condition $(2)$ from Corollary 2.3.2.7. We will verify $(4)$; the proof of $(3)$ is similar. Suppose we are given an integer $n \geq 3$ and a map $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. for which the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n\} ) }$ is right-degenerate. We wish to show that $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. We now consider three cases:

  • Suppose that $n = 3$. Then $\sigma _0$ can be identified with a collection of objects $\{ X_ i \} _{ 0 \leq i \leq 3}$, $1$-morphisms $\{ f_{ji}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$, and $2$-morphisms

    \[ \mu _{321}: f_{32} \circ f_{21} \Rightarrow f_{31} \quad \quad \mu _{320}: f_{32} \circ f_{20} \Rightarrow f_{30} \quad \quad \mu _{310}: f_{31} \circ f_{10} \Rightarrow f_{30} \]

    in the $2$-category $\operatorname{\mathcal{C}}$. The assumption that $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n\} ) }$ is right-degenerate guarantees that $X_2 = X_3$, that $f_{20} = f_{30}$, that the $1$-morphism $f_{32}$ is the identity $\operatorname{id}_{X_2}$, and that $\mu _{320}$ is the left unit constraint $\lambda _{ f_{20} }$. To extend $\sigma _0$ to a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}$, we must show that there exists a $2$-morphism $\mu _{210}: f_{21} \circ f_{10} \Rightarrow f_{20}$ for which the diagram

    5.12
    \begin{equation} \begin{gathered}\label{equation:3-simplex-of-Dusk} \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} & } \end{gathered} \end{equation}

    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). This commutativity can be rewritten as an equation

    \[ \mu _{320}(\operatorname{id}_{ f_{32} } \circ \mu _{210}) = \mu _{310} (\mu _{321} \circ \operatorname{id}_{ f_{10}} ) \alpha . \]

    This equation has a unique solution, because $\mu _{320}$ is invertible and horizontal composition with $f_{32}$ induces an equivalence of categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X_0, X_2) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X_0, X_3 )$.

  • Suppose that $n=4$. The restriction of $\sigma _0$ to the $2$-skeleton of $\Delta ^4$ can be identified with a collection of objects $\{ X_ i \} _{0 \leq i \leq 4}$, $1$-morphisms $\{ f_{ji}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 4}$, and $2$-morphisms $\{ \mu _{kji}: f_{kj} \circ f_{ji} \Rightarrow f_{ki} \} _{0 \leq i < j < k \leq 4}$ in the $2$-category $\operatorname{\mathcal{C}}$. The assumption that $\sigma _0|_{ \operatorname{N}_{\bullet }(\{ 0 < n-1 < n\} ) }$ is right-degenerate guarantees that $X_3 = X_4$, that $f_{30} = f_{40}$, that the $1$-morphism $f_{43}$ is the identity $\operatorname{id}_{X_3}$, and that $\mu _{430}$ is the left unit constraint $\lambda _{ f_{30} }$. Consider the diagram

    \[ \xymatrix@C =0pt{ f_{43} (f_{31} f_{10} ) \ar@ {=>}[rrrr]^{\sim } \ar@ {=>}[ddddd]^{ \mu _{310} } & & & & (f_{43} f_{31}) f_{10} \ar@ {=>}[ddddd]^{\mu _{431} } \\ & f_{43}( (f_{32} f_{21} ) f_{10} ) \ar@ {=>}[ul]_{\mu _{321}} \ar@ {=>}[rr]^{\sim } & & (f_{43} (f_{32} f_{21})) f_{10} \ar@ {=>}[ur]^{\mu _{321}} \ar@ {=>}[d]^{\sim } & \\ & f_{43} ( f_{32} (f_{21} f_{10} ) ) \ar@ {=>}[u]^{\sim } \ar@ {=>}[dr]^{ \sim } \ar@ {=>}[d]^{\mu _{210}} & & (( f_{43} f_{32}) f_{21}) f_{10} \ar@ {=>}[d]^{\mu _{432} } & \\ & f_{43} ( f_{32} f_{20} ) \ar@ {=>}[d]^{\sim } \ar@ {=>}[ddl]_{ \mu _{320} } & (f_{43} f_{32} ) (f_{21} f_{10}) \ar@ {=>}[dl]_{\mu _{210} } \ar@ {=>}[dr]^{ \mu _{432} } \ar@ {=>}[ur]^{\sim } & ( f_{42} f_{21} ) f_{10} \ar@ {=>}[ddr]^{ \mu _{421} } \ar@ {=>}[d]_{\sim } & \\ & (f_{43} f_{32} ) f_{20} \ar@ {=>}[r]_-{\mu _{432}} & f_{42} f_{20} \ar@ {=>}[d]^{\mu _{420}} & f_{42} (f_{21} f_{10} ) \ar@ {=>}[l]^-{\mu _{210}} & \\ f_{43} f_{30} \ar@ {=>}[rr]^{\mu _{430}}_{\sim } & & f_{04} & & f_{41} f_{10}; \ar@ {=>}[ll]_{ \mu _{410} } } \]

    in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X_0, X_4)$, where the unlabeled $2$-morphisms are given by the associativity constraints. Note that the $4$-cycles in this diagram commute by functoriality, and the central $5$-cycle commutes by the pentagon identity of $\operatorname{\mathcal{C}}$. Our assumption that $\sigma _0$ is defined on the horn $\Lambda ^{4}_{4}$ guarantees that pentagonal cycles on the right and bottom of the diagram are commutative and that the outer cycle commutes. Since the $2$-morphism $\mu _{430}$ is invertible, a diagram chase shows that the pentagonal cycle on the left of the diagram also commutes. Since $f_{43}$ is an identity $1$-morphism, horizontal composition with $f_{43}$ is isomorphic to the identity (via the left unit constraint of Construction 2.2.1.12) and is therefore faithful. It follows that the diagram (5.12) is commutative, so that $\sigma _0$ extends (uniquely) to a $4$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.

  • If $n \geq 5$, then the horn $\Lambda ^{n}_{n}$ contains the $3$-skeleton of $\Delta ^ n$. In this case, the morphism $\sigma _0: \Lambda ^ n_ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ extends uniquely to an $n$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ by virtue of Corollary 2.3.1.10.

$\square$