2.3 The Duskin Nerve of a Bicategory
In §1.4, we defined an $\infty $-category to be a simplicial set $X_{\bullet }$ which satisfies the weak Kan extension condition. Beware that this terminology is potentially misleading. Roughly speaking, an $\infty $-category (in the sense of Definition 1.4.0.1) should be viewed as a higher category $\operatorname{\mathcal{C}}$ with the property that every $k$-morphism in $\operatorname{\mathcal{C}}$ is invertible for $k \geq 2$. The framework of weak Kan complexes does not capture the entirety of higher category theory, or even the entirety of the theory of bicategories (as described in §2.2). Nevertheless, we will show in this section that the theory of $\infty $-categories can be viewed as a generalization of the theory of $2$-categories (that is, bicategories in which every $2$-morphism is invertible). Recall that, to every category $\operatorname{\mathcal{C}}$, one can associate a simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ called the nerve of $\operatorname{\mathcal{C}}$ (Construction 1.3.1.1). We proved in Chapter 1 that $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding from the category $\operatorname{Cat}$ of small categories to the category $\operatorname{Set_{\Delta }}$ of simplicial sets (Proposition 1.3.3.1), and that every simplicial set of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category (Example 1.4.0.4). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ has a generalization to the setting of bicategories. In §2.3.1, we associate to each bicategory $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{C}})$ called the Duskin nerve of $\operatorname{\mathcal{C}}$ (introduced by Duskin and Street; see [MR1897816] and [MR920944]). This construction has the following features (both established by Duskin in [MR1897816]):
If $\operatorname{\mathcal{C}}$ is a $2$-category, then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). We prove this in §2.3.2 as a consequence of a more general result which applies to the Duskin nerve of any bicategory (Theorem 2.3.2.5), whose proof we defer to §2.3.3.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be bicategories. In §2.3.4, we show that passage to the Duskin nerve induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Strictly unital lax functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Maps of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$} \} ; } \]see Theorem 2.3.4.1. In other words, the formation of Duskin nerves induces a fully faithful embedding from the category $\operatorname{2Cat}_{\operatorname{ULax}}$ of Definition 2.2.5.5 to the category of simplicial sets.
By virtue of Theorem 2.3.4.1, it is mostly harmless to abuse terminology by identifying a bicategory $\operatorname{\mathcal{C}}$ with the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ (each can be recovered from the other, up to canonical isomorphism). Theorem 2.3.2.1 then asserts that, under this identification, every $2$-category can be regarded as an $\infty $-category (see Remark 2.3.4.2 for a more precise statement).
In §2.3.5, we study the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ in the case where $\operatorname{\mathcal{C}}$ is a strict bicategory. In this case, we show that $n$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with strict functors $\operatorname{Path}_{(2)}[n] \rightarrow \operatorname{\mathcal{C}}$ (Corollary 2.3.5.7). Here $\operatorname{Path}_{(2)}[n]$ denotes a certain bicategorical variant of the path category introduced in §1.3.7, which will play an important role in our discussion of the homotopy coherent nerve of a simplicial category (see §2.4.3).