Definition 5.4.7.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories. A functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is 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 $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
5.4.7 Functors of $(\infty ,2)$-Categories
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Recall that a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (Definition 1.5.0.1). In this case, it is automatic that $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$ (Remark 1.5.1.6). Beware that the $(\infty ,2)$-categorical analogue of this statement is false: if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $(\infty ,2)$-categories, then a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ will generally not carry thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$. This motivates the following:
Example 5.4.7.2. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then every $2$-simplex of $\operatorname{\mathcal{D}}$ is thin, so every morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor. In particular, when $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, Definition 5.4.7.1 reduces to Definition 1.5.0.1.
Example 5.4.7.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. By virtue of Theorem 2.3.4.1 and Corollary 2.3.4.5, passage to the Duskin nerve induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Strictly unitary functors of $2$-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d] \\ \{ \textnormal{Functors of $(\infty ,2)$-categories $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$} \} . } \]
Remark 5.4.7.4 (Functoriality). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a functor (Definition 5.4.7.1) if and only it carries $\operatorname{Pith}(\operatorname{\mathcal{C}})$ into $\operatorname{Pith}(\operatorname{\mathcal{D}})$. If this condition is satisfied, then $\operatorname{Pith}(F) = F|_{ \operatorname{Pith}(\operatorname{\mathcal{C}}) }$ can be regarded as a functor from the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ to the $\infty $-category $\operatorname{Pith}(\operatorname{\mathcal{D}})$.
Remark 5.4.7.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. If $F$ is a functor of $(\infty ,2)$-categories and $u: X \rightarrow Y$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}$, then $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{\mathcal{D}}$ (see Definition 5.4.5.12). This follows by applying Remark 1.5.1.6 to the functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ of Remark 5.4.7.4. Beware that, if $F$ is not assumed to be a functor, then $F(u)$ need not be an isomorphism.
Remark 5.4.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category. Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ takes values in the pith $\operatorname{Pith}(\operatorname{\mathcal{D}}) \subseteq \operatorname{\mathcal{D}}$. Consequently, the inclusion $\operatorname{Pith}(\operatorname{\mathcal{D}}) \hookrightarrow \operatorname{\mathcal{D}}$ induces a bijection Note that this property (together with Proposition 5.4.5.6) characterize the simplicial set $\operatorname{Pith}(\operatorname{\mathcal{D}})$ up to unique isomorphism.
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of $\infty $-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{Pith}(\operatorname{\mathcal{D}})$} \} \ar [d] \\ \{ \textnormal{Functors of $(\infty ,2)$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$} \} . } \]
Remark 5.4.7.7. The existence of morphisms between $(\infty ,2)$-categories which do not preserve thin $2$-simplices should be viewed as a feature of our formalism, rather than a bug. Recall that, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $2$-categories, then Theorem 2.3.4.1 supplies a bijection Consequently, we can think of general morphisms of simplicial sets as providing a generalization of the notion of (strictly) unitary lax functors to the setting of $(\infty ,2)$-categories.
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Strictly unitary lax functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^-{\sim } \\ \{ \textnormal{Morphisms of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$} \} . } \]
Warning 5.4.7.8. For every pair of simplicial sets $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we let $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the simplicial set introduced in Construction 1.5.3.1. When working with $(\infty ,2)$-categories, this notation is potentially confusing. By construction, vertices of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ can be identified with morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $(\infty ,2)$-categories, then such morphisms need not carry thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$, and therefore need not correspond to functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the sense of Definition 5.4.7.1. We will return to this point in ยง
.
The following criterion is often useful for checking that a morphism of $(\infty ,2)$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor:
Proposition 5.4.7.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $(\infty ,2)$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $F$ is a functor: that is, it carries thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$.
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$, there exists a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ with $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$, as indicated in the diagram
such that $F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z, } \]
Proof. The implication $(1) \Rightarrow (2)$ is immediate. To prove the converse, let $T$ be the collection of all $2$-simplices of $\operatorname{\mathcal{C}}$ for which $F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$. Since the collection of thin $2$-simplices of $\operatorname{\mathcal{D}}$ has the four-out-of-five property (Proposition 5.4.6.11), it follows that $T$ also has the four-out-of-five property (Remark 5.4.6.10). Since the collection of thin $2$-simplices of $\operatorname{\mathcal{D}}$ has the inner exchange property (Proposition 5.4.5.10), $T$ has the inner exchange property (Remark 5.4.5.9). Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, every degenerate $2$-simplex of $\operatorname{\mathcal{D}}$ is thin, so every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$. If condition $(2)$ is satisfied, then Proposition 5.4.6.14 guarantees that every thin $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$, so that $F$ is a functor. $\square$
Proposition 5.4.7.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of $(\infty ,2)$-categories (Definition 5.4.2.1). Then:
The morphism $F$ is a functor of $(\infty ,2)$-categories: that is, it carries thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$, and therefore induces a functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$.
The diagram of simplicial sets
is a pullback square.
The functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ is an inner fibration of $\infty $-categories.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Pith}(\operatorname{\mathcal{C}}) \ar [r] \ar [d]^-{ \operatorname{Pith}(F) } \ar [r] & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \operatorname{Pith}(\operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{D}}} \]
Proof. We will prove assertion $(1)$ by showing that $F$ satisfies the criterion of Proposition 5.4.7.9. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, we can choose a thin $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ satisfying $d^{2}_0( \overline{\sigma } ) = F(g)$ and $d^{2}_2( \overline{\sigma } ) = F(f)$, which we depict as a diagram
\[ \xymatrix@R =50pt@C=50pt{ & F(Y) \ar [dr]^{ F(g) } & \\ F(X) \ar [ur]^{ F(f) } \ar [rr] & & F(Z ). } \]
Since $F$ is an interior fibration, the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r]^-{ ( g, \bullet , f) } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \Delta ^{2} \ar [r]^-{ \overline{\sigma } } \ar [ur]^{\sigma } & \operatorname{\mathcal{D}}} \]
admits a solution. Then $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (Lemma 5.4.2.6) for which the image $\overline{\sigma } = F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.
We now prove $(2)$. Let $\tau $ be an $m$-simplex of the simplicial set $\operatorname{\mathcal{C}}$, and suppose that $F(\tau )$ belongs to the pith $\operatorname{Pith}(\operatorname{\mathcal{D}})$. We wish to show that $\tau $ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$: that is, that it carries each $2$-simplex of $\Delta ^ m$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. This follows immediately from Lemma 5.4.2.6, since the composite map
\[ \Delta ^2 \rightarrow \Delta ^ m \xrightarrow {\tau } \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.
Combining $(2)$ with Remark 5.4.2.4, we conclude that the functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ is an interior fibration. Since $\operatorname{Pith}(\operatorname{\mathcal{D}})$ is an $\infty $-category (Proposition 5.4.5.6), it follows that $\operatorname{Pith}(F)$ is an inner fibration (Example 5.4.2.2). $\square$
Corollary 5.4.7.11. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let be the projection maps. Then:
The functor $\operatorname{Pith}(q): \operatorname{Pith}( \operatorname{\mathcal{C}}_{/f} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cartesian fibration of $\infty $-categories. Moreover, a morphism $u$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/f})$ is $\operatorname{Pith}(q)$-cartesian if and only if, for every vertex $z \in K$, the composite map
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
The functor $\operatorname{Pith}(q'): \operatorname{Pith}( \operatorname{\mathcal{C}}_{f/} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories. Moreover, a morphism $v$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{f/})$ is $\operatorname{Pith}(q')$-cocartesian if and only if, for every vertex $x \in K$, the composite map
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
\[ q': \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}\quad \quad q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}} \]
\[ \Delta ^2 \simeq \Delta ^1 \star \{ z\} \hookrightarrow \Delta ^1 \star K \xrightarrow {u} \operatorname{\mathcal{C}} \]
\[ \Delta ^2 \simeq \{ x\} \star \Delta ^1 \hookrightarrow K \star \Delta ^1 \xrightarrow {v} \operatorname{\mathcal{C}} \]
Proof. Combine Propositions 5.4.7.10 and 5.4.5.14. $\square$
Specializing Corollary 5.4.7.11 to the case $K = \Delta ^0$, we obtain the following:
Corollary 5.4.7.12. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $Z$ be an object of $\operatorname{\mathcal{C}}$. Then:
The projection map $\pi : \operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ induces a cartesian fibration of $\infty $-categories $\operatorname{Pith}(\pi ): \operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$.
A morphism $u$ of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z})$ is $\operatorname{Pith}(\pi )$-cartesian if and only if it corresponds to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (in this case, it is also $\pi $-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/Z}$).
The inclusion $\operatorname{Pith}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism from $\operatorname{Pith}(\operatorname{\mathcal{C}})_{/Z}$ to the (non-full) subcategory of $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/Z} )$ spanned by the $\pi $-cartesian morphisms.
Proof. Assertions $(1)$ and $(2)$ follow from Corollary 5.4.7.11, and assertion $(3)$ is an immediate consequence of $(2)$. $\square$
Remark 5.4.7.13. Recall that every cartesian fibration of simplicial sets $\pi : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ has an underlying right fibration $\pi ': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{D}}$, given by restricting $\pi $ to the simplicial subset $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ spanned by those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $\Delta ^ n$ to $\pi $-cartesian edge of $\operatorname{\mathcal{E}}$. Corollary 5.4.7.12 asserts that, when $\pi $ is the cartesian fibration $\operatorname{Pith}( \operatorname{\mathcal{C}}_{/Z} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ associated to a choice of object $Z$ of an $(\infty ,2)$-category $\operatorname{\mathcal{C}}$, then $\pi '$ can be identified with the right fibration $\operatorname{Pith}(\operatorname{\mathcal{C}})_{/Z} \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ supplied by Corollary 4.3.6.11; compare with Proposition 5.4.5.13.
We can also use Proposition 5.4.7.10 to deduce a relative version of Proposition 5.4.3.1:
Corollary 5.4.7.14. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ denote the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then the projection maps are interior fibrations of $(\infty ,2)$-categories.
\[ \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/} \quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0} \]
Warning 5.4.7.15. In the situation of Corollary 5.4.7.14, the induced map $\operatorname{Pith}(\operatorname{\mathcal{C}}_{/f} ) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}}_{/f_0})$ is generally not a cartesian fibration, and the induced map $\operatorname{Pith}( \operatorname{\mathcal{C}}_{f/}) \rightarrow \operatorname{Pith}( \operatorname{\mathcal{C}}_{f_0/} )$ is generally not a cocartesian fibration.
Proof of Corollary 5.4.7.14. We will show that the map of slice simplicial sets $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is an interior fibration; the analogous statement for coslice simplicial sets follows by a similar argument. We first observe that $\operatorname{\mathcal{C}}_{/f_0}$ is an $(\infty ,2)$-category (Corollary 5.4.3.4). Suppose we are given an integer $n \geq 2$ and a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d]^-{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}_{/f_0}. } \]
We wish to show that this lifting problem admits a solution provided that one of the following conditions is satisfied:
- $(a)$
The integer $i$ is equal to $0$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.
- $(b)$
The integer $i$ satisfies $0 < i < n$ and the restriction $\overline{\sigma }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}_{/f_0}$.
- $(c)$
The integer $i$ is equal to $n$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ n-1 < n \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.
In cases $(a)$ and $(c)$, this follows immediately from Proposition 5.4.3.8. In case $(b)$, it suffices (by virtue of Proposition 5.4.3.8) to verify that the composite map
\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{n} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. This follows from our hypothesis, since the projection map $\operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}}$ preserves thin $2$-simplices (Proposition 5.4.7.10). $\square$