Remark 5.4.0.1. The ideas presented in this section are closely related to the work of Verity, who has proposed a simplicial framework for studying higher categories with noninvertible morphisms at all levels. We refer the reader to [MR2399898], [MR2450607], and [MR2342841] for Verity's work, and to [gagna2020equivalence] for a discussion of its relationship to the theory of $(\infty ,2)$-categories presented here.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
5.4 $(\infty ,2)$-Categories
In §1.4, we defined an $\infty $-category to be a simplicial set $\operatorname{\mathcal{C}}$ which satisfies the weak Kan extension condition: for $0 < i < n$, every morphism of simplicial sets $\Lambda ^{n}_{i} \hookrightarrow \operatorname{\mathcal{C}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$ (Definition 1.4.0.1). Beware that this terminology is potentially confusing, because the theory of $\infty $-categories does not generalize the classical theory of $2$-categories. For every $2$-category $\operatorname{\mathcal{E}}$, the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ is a simplicial set which determines $\operatorname{\mathcal{E}}$ up to isomorphism (Theorem 2.3.4.1). However, the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ is an $\infty $-category if and only if $\operatorname{\mathcal{E}}$ is a $(2,1)$-category: that is, every $2$-morphism in $\operatorname{\mathcal{E}}$ is invertible (Theorem 2.3.2.1). Consequently, one can view the notions of $2$-category and $\infty $-category as mutually incomparable extensions of the notion of $(2,1)$-category. Our goal in this section is to show that these extensions admit a common generalization: a class of simplicial sets which we will refer to as $(\infty ,2)$-categories.
Our starting point is the notion of a thin $2$-simplex, which was introduced in §2.3.2. Recall that if $\operatorname{\mathcal{C}}$ is a simplicial set, then a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ is thin if every morphism of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$, provided that $0 < i < n$, $n \geq 3$, and the $2$-simplex $\tau _0|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )}$ is equal to $\sigma $ (Definition 2.3.2.3). By virtue of Example 2.3.2.4, $\operatorname{\mathcal{C}}$ is an $\infty $-category if and only if it satisfies the following pair of conditions:
- $(1)$
Every morphism of simplicial sets $\Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a $2$-simplex of $\operatorname{\mathcal{C}}$.
- $(2)$
Every $2$-simplex of $\operatorname{\mathcal{C}}$ is thin.
We will obtain the notion of $(\infty ,2)$-category by weakening $(2)$ to the requirement that degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ are thin, but strengthening $(1)$ to require that every map $\Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. We will also add additional axioms that guarantee the ability to fill outer horns of $\operatorname{\mathcal{C}}$ in certain special circumstances (see Definition 5.4.1.1).
Every $\infty $-category is an $(\infty ,2)$-category (Proposition 5.4.1.2), and every $2$-category can be regarded as an $(\infty ,2)$-category by passing to its Duskin nerve (Proposition 5.4.1.5). The situation is summarized in the following diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Groupoids} \} \ar@ {}[r]|{\subset } \ar@ {}[d]|{\cap } & \{ \textnormal{Categories} \} \ar@ {}[d]|{\cap } & \\ \{ \textnormal{$2$-Groupoids} \} \ar@ {}[r]|{\subset } \ar@ {^{(}->}[d]^{\operatorname{N}_{\bullet }^{\operatorname{D}} } & \{ \textnormal{$(2,1)$-Categories} \} \ar@ {}[r]|{\subset } \ar@ {^{(}->}[d]^{\operatorname{N}_{\bullet }^{\operatorname{D}}} & \{ \textnormal{$2$-Categories} \} \ar@ {^{(}->}[d]^{\operatorname{N}_{\bullet }^{\operatorname{D}}} \\ \{ \textnormal{Kan Complexes} \} \ar@ {}[r]|{\subset } & \{ \textnormal{$\infty $-Categories} \} \ar@ {}[r]|{\subset } & \{ \textnormal{$(\infty ,2)$-Categories} \} , } \]
where none of the inclusions is reversible.
Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a pair of objects $X$ and $Y$, and let $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ denote the pinched morphism spaces of Construction 4.6.5.1. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the simplicial sets $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ are Kan complexes (Proposition 4.6.5.5). In §5.4.3, we prove an analogous result: if $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, then the simplicial sets $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ are $\infty $-categories (Corollary 5.4.3.5). Recall that $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ is defined as the fiber over $Y$ of the projection map $q: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$, and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ is defined as the fiber over $X$ of the projection map $q': \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$. When $\operatorname{\mathcal{C}}$ is an $\infty $-category, the morphism $q$ is a left fibration of simplicial sets and the morphism $q'$ is a right fibration of simplicial sets (Corollary 4.3.6.12). Beware that, in the case where $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, the morphisms $q$ and $q'$ are generally not inner fibrations. Nevertheless, we will deduce that the fibers of $q$ and $q'$ are $\infty $-categories by showing that $q$ and $q'$ are interior fibrations (Definition 5.4.2.1), a class of morphisms which we introduce and study in §5.4.2. From this we deduce also that the simplicial sets $\operatorname{\mathcal{C}}_{X/}$ and $\operatorname{\mathcal{C}}_{/Y}$ are $(\infty ,2)$-categories; moreover, an analogous result holds more generally for the slice and coslice constructions associated to any diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ (Corollary 5.4.3.4).
Suppose that we are given a $2$-simplex $\sigma $ of a simplicial set $\operatorname{\mathcal{C}}$, whose $1$-skeleton we indicate in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^-{h} & & Z. } \]
Writing $q: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ for the projection map, we can identify $\sigma $ with an edge $\widetilde{g}$ of the simplicial set $\operatorname{\mathcal{C}}_{X/}$ satisfying $q(\widetilde{g}) = g$. It follows immediately from the definition that if the $2$-simplex $\sigma $ is thin, then the edge $\widetilde{g}$ is $q$-cocartesian (in the sense of Definition 5.1.1.1); in particular, it is locally $q$-cocartesian. In §5.4.4, we prove that if $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, then the converse holds: every locally $q$-cocartesian edge of $\operatorname{\mathcal{C}}_{X/}$ is thin when viewed as a $2$-simplex of $\operatorname{\mathcal{C}}$ (Theorem 5.4.4.1). Roughly speaking, one can think of $\widetilde{g}$ as encoding the datum of a morphism $\gamma $ from $g \circ f$ to $h$ in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Z)$; Theorem 5.4.4.1 confirms the heuristic that $\gamma $ is an isomorphism if and only if $\sigma $ is thin (in the case where $\operatorname{\mathcal{C}}$ is the Duskin nerve of a $2$-category, this is also the content of Theorem 2.3.2.5).
Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. We define the pith of $\operatorname{\mathcal{C}}$ to be the simplicial subset $\operatorname{Pith}(\operatorname{\mathcal{C}}) \subseteq \operatorname{\mathcal{C}}$ consisting of those simplices $\Delta ^{m} \rightarrow \operatorname{\mathcal{C}}$ which carry each $2$-simplex of $\Delta ^{m}$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (Construction 5.4.5.1). In §5.4.5, we show that $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Proposition 5.4.5.6) whose pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}(X,Y)$ can be identified with the cores of the $\infty $-categories $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$, respectively (Proposition 5.4.5.13). Roughly speaking, one can think of the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as obtained from the $(\infty ,2)$-category by “discarding” its noninvertible $2$-morphisms. In particular, when $\operatorname{\mathcal{C}}$ is the Duskin nerve of a $2$-category $\operatorname{\mathcal{E}}$, we can identify $\operatorname{Pith}(\operatorname{\mathcal{C}})$ with the Duskin nerve of the $(2,1)$-category $\operatorname{Pith}(\operatorname{\mathcal{E}})$ introduced in Construction 2.2.8.9 (Example 5.4.5.4).
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories. We define a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ to be a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carries thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$ (Definition 5.4.7.1). This definition can be somewhat cumbersome to work with in practice, because it requires us to check a condition for every thin $2$-simplex of $\operatorname{\mathcal{C}}$. In §5.4.7, we show that this is unnecessary: to verify that a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor, it suffices to show that every morphism $\sigma _0: \Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ for which $F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$ (Proposition 5.4.7.9). Here we can think of $\sigma _0$ as given by a pair of morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, and the thinness assumption on $F(\sigma )$ corresponds heuristically to the requirement that $F$ “preserves” the composition of $f$ and $g$ (up to isomorphism). Our proof will make use of a certain closure property enjoyed by the thin $2$-simplices of an $(\infty ,2)$-category which we refer to as the four-out-of-five property, which we formulate and study in §5.4.6 (see Definition 5.4.6.8 and Proposition 5.4.6.11).
Recall that a $2$-category $\operatorname{\mathcal{E}}$ is strict if its unit and associativity constraints are identity morphisms (Example 2.2.1.4); in this case, we can view $\operatorname{\mathcal{E}}$ as an ordinary category which is enriched over $\operatorname{Cat}$ (see Definition 2.2.0.1). This notion has a counterpart in the setting of $(\infty ,2)$-categories. Let $\operatorname{Set_{\Delta }}$ denote the ordinary category of simplicial sets, and let $\operatorname{\mathbf{QCat}}$ denote the full subcategory of $\operatorname{Set_{\Delta }}$ whose objects are $\infty $-categories. Let $\operatorname{\mathcal{E}}$ be a $\operatorname{\mathbf{QCat}}$-enriched category: that is, a simplicial category with the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. In §5.4.8, we show that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $(\infty ,2)$-category (Theorem 5.4.8.1). The construction $\operatorname{\mathcal{E}}\mapsto \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ can be regarded as a generalization of the inclusion from strict $2$-categories into general $2$-categories (recall that if $\operatorname{\mathcal{E}}$ is a strict $2$-category, then its Duskin nerve can be identified with the homotopy coherent nerve of the associated simplicial category; see Example 2.4.3.11). Beware that not every $(\infty ,2)$-category $\operatorname{\mathcal{C}}$ is isomorphic to the homotopy coherent nerve of a $\operatorname{\mathbf{QCat}}$-enriched category. Nevertheless, we will later prove a coherence theorem which guarantees that $\operatorname{\mathcal{C}}$ is equivalent to the homotopy coherent nerve of a $\operatorname{\mathbf{QCat}}$-enriched category: see Theorem 
.
Structure
- Subsection 5.4.1: Definitions
- Subsection 5.4.2: Interior Fibrations
- Subsection 5.4.3: Slices of $(\infty ,2)$-Categories
- Subsection 5.4.4: The Local Thinness Criterion
- Subsection 5.4.5: The Pith of an $(\infty ,2)$-Category
- Subsection 5.4.6: The Four-out-of-Five Property
- Subsection 5.4.7: Functors of $(\infty ,2)$-Categories
- Subsection 5.4.8: Strict $(\infty ,2)$-Categories
- Subsection 5.4.9: Comparison of Homotopy Transport Representations