Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

2.3 The Duskin Nerve of a $2$-Category

In §1.3, 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 1.3.0.1) 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). To address this point, it is convenient to restrict our attention to a special class of $2$-categories.

Definition 2.3.0.1. A $(2,1)$-category is a $2$-category $\operatorname{\mathcal{C}}$ with the property that every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible.

Remark 2.3.0.2. The terminology of Definition 2.3.0.1 fits into a general paradigm. Given $0 \leq m \leq n \leq \infty $, let us informally use the term $(n,m)$-category to refer to an $n$-category $\operatorname{\mathcal{C}}$ having the property that every $k$-morphism of $\operatorname{\mathcal{C}}$ is invertible for $k > m$. Following this convention, the $\infty $-categories of Definition 1.3.0.1 should really be called $(\infty ,1)$-categories.

Remark 2.3.0.3. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category. Then every lax functor of $2$-categories $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is automatically a functor. Consequently, there is no need to distinguish between functors and lax functors when working in the setting of $(2,1)$-categories.

Our goal in this section is to show 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 1.2.1.1). 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 1.2.2.1), and that every simplicial set of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category (Example 1.3.0.4). 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

    \[ \xymatrix { \{ \text{Strictly unitary lax functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Maps of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$} \} ; } \]

    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).

The remainder of this section is devoted to giving a more concrete description of the Duskin nerve in two examples:

  • Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products, let $\operatorname{Corr}(\operatorname{\mathcal{E}})$ be the $2$-category of correspondences in $\operatorname{\mathcal{E}}$ (Example 2.2.2.1). In §2.3.5, we give an explicit description of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$: roughly speaking, $n$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ can be identified with diagrams in $\operatorname{\mathcal{E}}$ indexed by the “pyramid” $\{ (i,j) \in [n] \times [n]^{\operatorname{op}}: i \leq j \} $ (see Remark 2.3.5.9). We prove this by establishing a universal property of the $2$-category $\operatorname{Corr}(\operatorname{\mathcal{E}})$ related to the formation of twisted arrow categories (Theorem 2.3.5.7).

  • Let $\operatorname{\mathcal{C}}$ be a strict $2$-category. In §2.3.6, we show that $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with strict functors $\operatorname{String}([n])^{\operatorname{c}} \rightarrow \operatorname{\mathcal{C}}$ (Corollary 2.3.6.6). Here $\operatorname{String}([n])$ denotes a certain strict $2$-category which will play a central role in our discussion of the homotopy coherent nerve of a simplicially enriched category (see §).

Structure

  • Subsection 2.3.1: The Duskin Nerve
  • Subsection 2.3.2: From $2$-Categories to $\infty $-Categories
  • Subsection 2.3.3: Thin $2$-Simplices of a Duskin Nerve
  • Subsection 2.3.4: Recovering a $2$-Category from its Duskin Nerve
  • Subsection 2.3.5: Twisted Arrows and the Nerve of $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{c}}$
  • Subsection 2.3.6: The Duskin Nerve of a Strict $2$-Category