2.3 The Duskin Nerve of a $2$-Category
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 $2$-categories (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 $(2,1)$-categories. 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 $2$-categories. In §2.3.1, we associate to each $2$-category $\operatorname{\mathcal{C}}$ 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 $(2,1)$-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 $2$-category (Theorem 2.3.2.5), whose proof we defer to §2.3.3.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. In §2.3.4, we show that passage to the Duskin nerve induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Strictly unitary lax functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Maps of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$} \} ; } \]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 $2$-category $\operatorname{\mathcal{C}}$ 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 $(2,1)$-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 $2$-category. 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 $2$-categorical 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).