Definition 9.1.4.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is weakly right cofinal if, for every object $D \in \operatorname{\mathcal{D}}$, there exists an object $C \in \operatorname{\mathcal{C}}$ and a morphism $v: D \rightarrow F(C)$ in the $\infty $-category $\operatorname{\mathcal{D}}$.
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9.1.4 Cofinal Functors of Filtered $\infty $-Categories
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Recall that $F$ is right cofinal if and only if, for every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is weakly contractible (Theorem 7.2.3.1). Our first goal in this section is to show that, if the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are filtered, then this criterion can be substantially simplified (Theorem 9.1.4.5). We begin by introducing some terminology.
Example 9.1.4.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
The functor $F$ is weakly right cofinal if and only if, for each $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is nonempty.
The functor $F$ is right cofinal if and only if, for each $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is weakly contractible (Theorem 7.2.3.1).
In particular, if $F$ is right cofinal, then it is weakly right cofinal.
Example 9.1.4.3. Recall that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially surjective if, for every object $D \in \operatorname{\mathcal{D}}$, there exists an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $v: D \xrightarrow {\sim } F(C)$ (Definition 4.6.2.12). It follows that every essentially surjective functor is weakly right cofinal.
Remark 9.1.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. If $F$ and $G$ are weakly right cofinal, then the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is weakly right cofinal.
Our first main result can be stated as follows:
Theorem 9.1.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are filtered. Then $F$ is right cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}$, the induced functor $F_{C/}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{ F(C) / }$ is weakly right cofinal.
Remark 9.1.4.6. Theorem 9.1.4.5 asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between filtered $\infty $-categories is right cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}$ and every morphism $u: F(C) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$, there exists a morphism $w: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ and a $2$-simplex of the $\infty $-category $\operatorname{\mathcal{D}}$. This is equivalent to the requirement that $F([w])$ factors through $[u]: F(C) \rightarrow D$ when regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$. In particular, the condition that $F$ is right cofinal depends only on the induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$.
\[ \xymatrix@R =50pt@C=50pt{ & D \ar [dr]_{v} & \\ F(C) \ar [ur]^{u} \ar [rr]^{ F(w) } & & F(C'). } \]
Corollary 9.1.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between filtered $\infty $-categories. Then $F$ is right cofinal if and only if (the nerve of) the functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is right cofinal.
Corollary 9.1.4.8. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category. Then the tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is right cofinal.
The following is a generalization of Corollary 7.2.3.4:
Corollary 9.1.4.9. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category, let $(Q, \leq )$ be a directed partially ordered set, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ be a functor. Then $F$ is right cofinal if and only if it is weakly right cofinal.
Proof. Assume that $F$ is weakly right cofinal; we will show that $F$ satisfies the criterion of Remark 9.1.4.6, and is therefore right cofinal (the converse follows from Example 9.1.4.2). Fix an object $X \in \operatorname{\mathcal{C}}$ and an element $q \in Q$ satisfying $F(X) \leq q$. We wish to show that there exists a morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ such that $q \leq F(Z)$. Applying our assumption that $F$ is weakly right cofinal, we can choose an object $Y \in \operatorname{\mathcal{C}}$ satisfying $q \leq F( Y )$. We complete the proof by observing that there is another object $Z \in \operatorname{\mathcal{C}}$ with morphisms $f: X \rightarrow Z$ and $g: Y \rightarrow Z$, by virtue of our assumption that $\operatorname{\mathcal{C}}$ is filtered. $\square$
Example 9.1.4.10. Let $(P, \leq )$ and $(Q, \leq )$ be directed partially ordered sets and let $f: P \rightarrow Q$ be a nondecreasing function. Corollary 9.1.4.9 implies that the functor $\operatorname{N}_{\bullet }(f): \operatorname{N}_{\bullet }(P) \rightarrow \operatorname{N}_{\bullet }(Q)$ is right cofinal if and only if, for each $q \in Q$, there exists $p \in P$ satisfying $q \leq f(p)$.
Warning 9.1.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be filtered $\infty $-categories. In general, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is weakly right cofinal need not be right cofinal (Exercise 9.1.4.12). That is, the coslice construction appearing in the formulation of Theorem 9.1.4.5 cannot be omitted.
Exercise 9.1.4.12. Let $\operatorname{\mathcal{C}}$ be the category described as follows:
The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}})$ is given by $\{ X_ n \} _{n \in \operatorname{\mathbf{Z}}}$.
Morphisms in $\operatorname{\mathcal{C}}$ are given by
with composition given by multiplication on the set $\{ 0, 1 \} $.
\[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X_ m, X_ n ) = \begin{cases} \{ 0, 1 \} & \text{ if $m \leq n$ } \\ \emptyset & \text{ otherwise, } \end{cases} \]
Then the diagram
\[ \cdots \rightarrow X_{-2} \xrightarrow {1} X_{-1} \xrightarrow {1} X_0 \xrightarrow {1} X_1 \xrightarrow {1} X_2 \xrightarrow {1} \cdots \]
determines a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}, \leq ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which is essentially surjective, and therefore weakly right cofinal (Example 9.1.4.3). Show the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ filtered but $F$ is not right cofinal.
We now turn to the proof of Theorem 9.1.4.5. We begin by considering the special case where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration of $\infty $-categories. Then, for every object $C \in \operatorname{\mathcal{C}}$ and every morphism $u: F(C) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$, we can choose a $F$-cocartesian morphism $\widetilde{u}: C \rightarrow \widetilde{D}$ of $\operatorname{\mathcal{C}}$ satisfying $F( \widetilde{u} ) = u$. It follows that the map of coslice $\infty $-categories $F_{/C}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{F(C)/}$ is essentially surjective, and therefore automatically weakly right cofinal (Example 9.1.4.3). In this case, Theorem 9.1.4.5 reduces to the following:
Proposition 9.1.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are filtered, then $F$ is right cofinal.
Proof. For each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{D}}_{D/}$ is filtered (Proposition 9.1.1.17) and the projection map $\operatorname{\mathcal{D}}_{D/} \rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration (Proposition 4.3.6.1). Applying Corollary 9.1.3.14, we conclude that the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is a filtered $\infty $-category, and is therefore weakly contractible (Proposition 9.1.1.18). Allowing the object $D$ to vary, we conclude that the functor $F$ is right cofinal (Theorem 7.2.3.1). $\square$
Corollary 9.1.4.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a left fibration of $\infty $-categories, where the $\infty $-category $\operatorname{\mathcal{D}}$ is filtered. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.
The functor $F$ is right cofinal.
The $\infty $-category $\operatorname{\mathcal{C}}$ is weakly contractible.
Proof. The equivalence $(1) \Leftrightarrow (3)$ is Theorem 9.1.3.2 and the implication $(1) \Rightarrow (2)$ follows from Corollary 9.1.4.13. The implication $(2) \Rightarrow (3)$ follows from the weak contractibility of the filtered $\infty $-category $\operatorname{\mathcal{D}}$ (Proposition 9.1.1.18), since every right cofinal functor is a weak homotopy equivalence (Proposition 7.2.1.5). $\square$
Corollary 9.1.4.15. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category. For every $\kappa $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}$, the forgetful functor $U: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{C}}$ is right cofinal.
Proof. It follows from Proposition 9.1.1.17 that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{q/}$ is $\kappa $-filtered, so that $U$ is right cofinal by virtue of Corollary 9.1.4.14. $\square$
Corollary 9.1.4.16. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category. For every object $C \in \operatorname{\mathcal{C}}$, the forgetful functor $\operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{C}}$ is right cofinal.
Proof. Apply Corollary 9.1.4.15 in the special case $\kappa = \aleph _0$ and $K = \Delta ^0$. $\square$
Proposition 9.1.4.17. Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be filtered $\infty $-categories. Suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, where $G$ is a left fibration. Then $F$ is right cofinal if and only if $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is right cofinal.
Proof. Assume that $G \circ F$ is right cofinal; we will prove that $F$ is right cofinal (the converse follows from Proposition 7.2.1.6, since $G$ is right cofinal by virtue of Corollary 9.1.4.14). Note that $F$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {V} \operatorname{\mathcal{D}}, \]
where $F'$ is given given by the pair of functors $(\operatorname{id}_{\operatorname{\mathcal{C}}}, F)$ and $V$ is given by projection onto the second factor. Here $V$ is the pullback of the right cofinal functor $(G \circ F)$ along the left fibration $G$, and is therefore right cofinal by virtue of Proposition 7.2.3.10. It will therefore suffice to show that $F'$ is right cofinal (Proposition 7.2.1.6).
Note that $V$ is a weak homotopy equivalence (Proposition 7.2.1.5). Since $\operatorname{\mathcal{D}}$ is weakly contractible (Proposition 9.1.1.18), it follows that $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also weakly contractible. Let $G': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map onto the second factor. Then $G'$ is a pullback of $G$, and is therefore a left fibration. Applying Corollary 9.1.4.14, we conclude that $G'$ is right cofinal. Since the composition $G' \circ F' = \operatorname{id}_{\operatorname{\mathcal{C}}}$ is also right cofinal, Proposition 7.2.1.6 guarantees that $F'$ is right cofinal as desired. $\square$
Corollary 9.1.4.18. Let $\kappa $ be a regular cardinal, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\kappa $-filtered $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right cofinal functor. Then, for every $\kappa $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}$, the functor of coslice $\infty $-categories $F_{q/}: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{D}}_{ (F \circ q)/}$ is also right cofinal.
Proof. Let $U: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{D}}_{ (F \circ q)/} \rightarrow \operatorname{\mathcal{D}}$ be the forgetful functors. Since $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, the left fibration $U$ is right cofinal (Corollary 9.1.4.15). Applying Proposition 9.1.4.17, we conclude that $F \circ U = V \circ F_{q/}$ is also right cofinal. Since $V$ is also left fibration, Proposition 9.1.4.17 guarantees that $F_{q/}$ is right cofinal. $\square$
Corollary 9.1.4.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right cofinal functor between filtered $\infty $-categories. Then, for every object $C \in \operatorname{\mathcal{C}}$, the induced map of coslice $\infty $-categories $F_{C/}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{ F(C)/ }$ is right cofinal.
Proof. Apply Corollary 9.1.4.18 in the special case $\kappa = \aleph _0$ and $K = \Delta ^0$. $\square$
Warning 9.1.4.20. In the statement of Corollary 9.1.4.19, the hypothesis that $\operatorname{\mathcal{C}}$ is filtered cannot be omitted. For example, let $\operatorname{\mathcal{C}}$ denote the $\infty $-category given by the left cone $( \operatorname{\partial \Delta }^1 )^{\triangleleft }$. The inclusion map $\operatorname{\partial \Delta }^1 \hookrightarrow \Delta ^1$ then extends uniquely to a functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^1$, carrying the cone point of $\operatorname{\mathcal{C}}$ to the vertex $0 \in \Delta ^1$. The functor $F$ is right cofinal, but the induced map $F_{0/}: \operatorname{\mathcal{C}}_{0/} \rightarrow ( \Delta ^1 )_{0/} \simeq \Delta ^1$ is not.
Proof of Theorem 9.1.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between filtered $\infty $-categories. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the functor $F_{C/}: \operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{D}}_{ F(C) / }$ is weakly right cofinal; we wish to show that $F$ is right cofinal (the converse follows Corollary 9.1.4.19 and Example 9.1.4.2). By virtue of Theorem 7.2.3.1, it will suffice to show that for each object $X \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ X/ }$ is weakly contractible. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the first factor. We will complete the proof by showing that the left fibration $U$ satisfies the criterion of Proposition 9.1.3.8, so that $\widetilde{\operatorname{\mathcal{C}}}$ is filtered. Fix an object $C \in \operatorname{\mathcal{C}}$ and a finite diagram $f: K \rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$. We wish to show that there is a morphism $w: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ such that the composite map
\[ K \xrightarrow {f} \{ C\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \xrightarrow { w_{!} } \{ C'\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \]
is nullhomotopic, where $v_{!}$ is the covariant transport functor associated to the left fibration $U$. Note that, since the $\infty $-category $\operatorname{\mathcal{D}}_{X/}$ is filtered (Proposition 9.1.1.17), the analogous condition is satisfied for the left fibration $V: \operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}$; that is, we can choose a morphism $u: F(C) \rightarrow D$ in $\operatorname{\mathcal{D}}$ for which the composite map
\[ K \xrightarrow {f} \{ F(C) \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \xrightarrow { u_{!} } \{ D \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}. \]
We now complete the proof by taking $w: C \rightarrow C'$ to be any morphism of $\operatorname{\mathcal{C}}$ having the property that $F(w)$ factors through $u$ (the existence of which is guaranteed by our assumption that $F_{C/}$ is weakly right cofinal; see Remark 9.1.4.6. $\square$