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2.2.2 From $2$-Categories to $\infty $-Categories

We now use Construction 2.2.1.1 to connect the theory of $2$-categories (in the sense of Definition 2.1.1.1) to the theory of $\infty $-categories (in the sense of Definition 1.3.0.1).

Theorem 2.2.2.1 (Duskin [MR1897816]). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{\mathcal{C}}$ is $(2,1)$-category if and only if the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category.

We will deduce Theorem 2.2.2.1 from a more general statement (Theorem 2.2.2.4), which gives a filling criterion for inner horns in the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ for an arbitrary $2$-category $\operatorname{\mathcal{C}}$. First, we need a bit of terminology.

Definition 2.2.2.2. Let $X_{\bullet }$ be a simplicial set. We will say that a $2$-simplex $\sigma $ of $X_{\bullet }$ is thin if it satisfies the following condition:

$(\ast )$

Let $n \geq 3$, let $0 < i < n$, and let $\tau $ denote the $2$-simplex of $\Lambda ^{n}_{i}$ given by the map

\[ [2] \simeq \{ i-1, i, i+1 \} \subseteq [n]. \]

Then any map of simplicial sets $f_0: \Lambda ^{n}_{i} \rightarrow X_{\bullet }$ satisfying $f_0( \tau ) = \sigma $ can be extended to an $n$-simplex of $X_{\bullet }$.

Example 2.2.2.3. Let $X_{\bullet }$ be a simplicial set. If $X_{\bullet }$ is an $\infty $-category (in the sense of Definition 1.3.0.1), then every $2$-simplex of $X_{\bullet }$ is thin. Conversely, if every $2$-simplex of $X_{\bullet }$ is thin, then $X_{\bullet }$ is an $\infty $-category if and only if every map of simplicial sets $f_0: \Lambda ^{2}_{1} \rightarrow X_{\bullet }$ can be extended to a $2$-simplex of $X_{\bullet }$.

We will deduce Theorem 2.2.2.1 from the following result, whose proof will be given in §2.2.3:

Theorem 2.2.2.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\sigma $ be a $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]

(see Example 2.2.1.15). Then $\sigma $ is thin if and only if $\gamma : g \circ f \Rightarrow h$ is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$.

Proof of Theorem 2.2.2.1 from Theorem 2.2.2.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category. If the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ is an $\infty $-category, then every $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ is thin (Example 2.2.2.3), so that every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible by virtue of Theorem 2.2.2.4. Conversely, if $\operatorname{\mathcal{C}}$ is a $(2,1)$-category, then every $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is thin (Theorem 2.2.2.4). Consequently, to show that $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category, it will suffice to show that every map of simplicial sets $u_0: \Lambda ^2_1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Note that we can identify $u_0$ with a composable pair of $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{C}}$. To extend this to a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, it suffices to choose a $1$-morphism $h: X \rightarrow Z$ and a $2$-morphism $\gamma : g \circ f \Rightarrow h$. This is always possible: for example, we can take $h = g \circ f$ and $\gamma $ to be the identity $2$-morphism. $\square$

Remark 2.2.2.5. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category. Then:

  • Objects of the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with objects of the $2$-category $\operatorname{\mathcal{C}}$.

  • If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with $1$-morphisms from $X$ to $Y$ in the $2$-category $\operatorname{\mathcal{C}}$.

  • If $f,g: X \rightarrow Y$ are $1$-morphisms in $\operatorname{\mathcal{C}}$ having the same domain and codomain, then $f$ and $g$ are homotopic when regarded as morphisms of the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ (Definition 1.3.3.1) if and only if they are isomorphic when viewed as objects of the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. More precisely, vertical composition with the left unit constraint $\lambda _{f}: \operatorname{id}_{Y} \circ f \xRightarrow {\sim } f$ induces a bijection

    \[ \xymatrix { \{ \text{Isomorphisms from $f$ to $g$ in the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} \ar [d]^{\sim } \\ \{ \text{Homotopies from $f$ to $g$ in the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$} \} . } \]

Let us now collect some other consequences of Theorem 2.2.2.4.

Corollary 2.2.2.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then every degenerate $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is thin.

Proof. Combine Theorem 2.2.2.4 with the observation that, for every $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the left and right unit constraints

\[ \lambda _{f}: \operatorname{id}_ Y \circ f \Rightarrow f \quad \quad \rho _{f}: f \circ \operatorname{id}_ X \Rightarrow f \]

are isomorphisms (in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Y)$). $\square$

Corollary 2.2.2.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strictly unitary lax functor. Then $F$ is a functor if and only if the induced map of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \Rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ carries thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ to thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$.

Proof. Let $\sigma $ be a $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]

in $\operatorname{\mathcal{C}}$. Let $\sigma '$ denote the image of $\sigma $ in $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$, corresponding to the diagram

\[ \xymatrix { & F(Y) \ar [dr]^{F(g)} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^{\gamma '} & \\ F(X) \ar [ur]^{F(f)} \ar [rr]_{F(h)} & & F(Z) } \]

where $\gamma '$ is given by the (vertical) composition

\[ F(g) \circ F(f) \xRightarrow { \mu _{g,f} } F(g \circ f) \xRightarrow {F(\gamma )} F(h). \]

Since $\sigma $ is thin, the $2$-morphism $\gamma $ is an isomorphism (Theorem 2.2.2.4). It follows that $\sigma '$ is an isomorphism if and only if $\mu _{g,f}$ is an isomorphism. In particular, the strictly unitary lax functor $F$ preserves thin $2$-simplices if and only if $\mu _{g,f}$ is an isomorphism for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ of $\operatorname{\mathcal{C}}$: that is, if and only if $F$ is a functor. $\square$

Warning 2.2.2.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Let us say that a $2$-simplex $\sigma $ of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is special if it corresponds to a diagram

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z, } \]

where $h = g \circ f$ and $\gamma = \operatorname{id}_{g \circ f}$. Arguing as in the proof of Corollary 2.2.2.7, we see that a strictly unitary lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strict if and only if it carries special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ to special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$. Beware, however, that the special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$ do not have an intrinsic description in terms of the simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$ themselves. In particular, it possible to have an isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ which does not preserve special $2$-simplices (corresponding to an isomorphism of $2$-categories which is not strict).

In general, passage from a $2$-category $\operatorname{\mathcal{C}}$ to its Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ involves a slight loss of information. From the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, we can recover the objects of $\operatorname{\mathcal{C}}$ (these can be identified with vertices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$) and the collection of $1$-morphisms $f: X \rightarrow Y$ from an object $X$ to an object $Y$ (these can be identified with edges of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ having source $X$ and target $Y$). However, the composition $g \circ f$ of a pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ cannot be recovered from the structure of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as an abstract simplicial set. The best we can do is to ask for a thin $2$-simplex $\sigma $ of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ satisfying $d_0(\sigma ) = g$ and $d_2(\sigma ) = f$. Such a simplex can be viewed as “witnessing” the presence of an isomorphism of the edge $h = d_1(\sigma )$ with the composition $g \circ f$. Put another way, the abstract simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ contains enough information to reconstruct the composition $g \circ f$ up to (unique) isomorphism, but not enough information to select a canonical representative of its isomorphism class. This can be viewed as a feature, rather than a bug: the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ often admits a more invariant description than the $2$-category $\operatorname{\mathcal{C}}$ itself (since the information lost by passing from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ depends on choices that one would prefer not make in the first place; see Remark 2.2.1.7.

If $\operatorname{\mathcal{C}}$ is a $2$-category which contains non-invertible $2$-morphisms, then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is not an $\infty $-category. However, we can extract an $\infty $-category by applying the Duskin nerve to a smaller $2$-category.

Construction 2.2.2.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ as follows:

  • The objects of $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{Core_{1}}(\operatorname{\mathcal{C}})}( X, Y)$ is the core $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.2.4.4).

  • The composition law, associativity constraints, and unit constraints of $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ are given by restricting the composition law, associativity contrasints, and unit constraints of $\operatorname{\mathcal{C}}$.

Then $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category which we will refer to as the $1$-core of $\operatorname{\mathcal{C}}$.

More informally: for any $2$-category $\operatorname{\mathcal{C}}$, the $(2,1)$-category $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ is obtained by the discarding the non-invertible $2$-morphisms of $\operatorname{\mathcal{C}}$.

Remark 2.2.2.10 (The Universal Property of the $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the $1$-core $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ is characterized (up to isomorphism) by the following properties:

  • The $1$-core $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category.

  • For every $(2,1)$-category $\operatorname{\mathcal{D}}$, every functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors (uniquely) through $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$.

Warning 2.2.2.11. In the situation of Remark 2.2.2.10, it is not true that a lax functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ factors through the $1$-core $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ (even when $\operatorname{\mathcal{D}}$ is a $(2,1)$-category): any lax functor which admits such a factorization is automatically a functor, by virtue of Remark 2.2.0.3.

Remark 2.2.2.12. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{Core_{1}}(\operatorname{\mathcal{C}})$ denote its $1$-core. Then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Core_{1}}(\operatorname{\mathcal{C}}) )$ is an $\infty $-category (Theorem 2.2.2.1). Unwinding the definitions, we see that $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Core_{1}}(\operatorname{\mathcal{C}}) )$ can be identified with the largest simplicial subset $X_{\bullet }$ of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ having the property that each $2$-simplex of $X_{\bullet }$ is thin when regarded as a $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ (so that an $n$-simplex $\sigma \in \operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ belongs to $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Core_{1}}(\operatorname{\mathcal{C}}) )$ if and only if, for every map $\Delta ^2 \rightarrow \Delta ^ n$, the composition $\Delta ^2 \rightarrow \Delta ^ n \xrightarrow {\sigma } \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is thin).