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7.2.2 Cofinality and Limits

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In ยง7.1.4, we introduced the notion of a limit $\varprojlim (G)$ and colimit $\varinjlim (G)$ for a diagram $G: B \rightarrow \operatorname{\mathcal{C}}$ (Definition 7.1.1.10). Our goal in this section is to show that, if $F: A \rightarrow B$ is a left cofinal morphism of simplicial sets, then the limit $\varprojlim (G)$ (if it exists) can be identified with the limit $\varprojlim (G \circ F)$. Similarly, if $F: A \rightarrow B$ is right cofinal, then the colimit $\varinjlim (G)$ (if it exists) can be identified with the colimit $\varinjlim (G \circ F)$. Our proof is based on the following characterization of (left) cofinality:

Proposition 7.2.2.1. Let $F: A \rightarrow B$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $F$ is left cofinal (in the sense of Definition 7.2.1.1).

$(2)$

For every $\infty $-category $\operatorname{\mathcal{C}}$ and every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, the restriction map $\operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{C}}_{ / (G \circ F) }$ is an equivalence of $\infty $-categories.

$(3)$

For every $\infty $-category $\operatorname{\mathcal{C}}$ and every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, composition with $F$ induces an equivalence of $\infty $-categories

\[ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} . \]
$(4)$

For every $\infty $-category $\operatorname{\mathcal{C}}$, every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, and every object $X \in \operatorname{\mathcal{C}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{\operatorname{Fun}(B,\operatorname{\mathcal{C}})}( \underline{X}, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}})}( \underline{X} \circ F, G \circ F); \]

here $\underline{X}: B \rightarrow \operatorname{\mathcal{C}}$ denotes the constant diagram taking the value $X$.

$(5)$

For every $\infty $-category $\operatorname{\mathcal{C}}$, every diagram $G: B \rightarrow \operatorname{\mathcal{C}}$, and every object $X \in \operatorname{\mathcal{C}}$, precomposition with $F$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}} (A, \operatorname{\mathcal{C}}_{X/} ). \]

Proof. We first show that $(1)$ implies $(2)$. Let $F: A \rightarrow B$ be a left cofinal morphism of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $G: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram; we wish to show that precomposition with $F$ induces an equivalence of $\infty $-categories $\theta : \operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{C}}_{ / (G \circ F) }$. By virtue of Corollary 7.2.1.13, we may assume without loss of generality that $F$ is either left anodyne or a trivial Kan fibration. In the first case, the functor $\theta $ is a trivial Kan fibration (Corollary 4.3.6.13). In the second case, the diagram $F$ is a categorical equivalence of simplicial sets (Proposition 4.5.2.10), so that $\theta $ is an equivalence of $\infty $-categories by virtue of Corollary 4.6.4.18.

We next prove the equivalences $(2) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5)$. Let $G: B \rightarrow \operatorname{\mathcal{C}}$ be as above. Applying Construction 4.6.4.12, we obtain a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/G} \ar [r] \ar [d]^{\theta } & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \ar [d]^{\theta '} \\ \operatorname{\mathcal{C}}_{ / (G \circ F)} \ar [r] & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} , } \]

where the horizontal maps are equivalences of $\infty $-categories (Theorem 4.6.4.16). It follows that $\theta $ is an equivalence of $\infty $-categories if and only if $\theta '$ is an equivalence of $\infty $-categories. This proves the equivalence $(2) \Leftrightarrow (3)$. Note that the functor $\theta '$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \ar [dr] \ar [rr]^{\theta '} & & \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} \ar [dl] \\ & \operatorname{\mathcal{C}}, & } \]

where the vertical maps are right fibrations (Corollary 4.6.4.11). Applying Corollary 5.1.6.15 and Proposition 5.1.6.5, we see that $\theta '$ is an equivalence of $\infty $-categories if and only if it induces a homotopy equivalence

\[ \theta '_{X}: \{ \underline{X} \} \operatorname{\vec{\times }}_{ \operatorname{Fun}(B, \operatorname{\mathcal{C}}) } \{ G\} \rightarrow \{ \underline{X} \circ F \} \operatorname{\vec{\times }}_{ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) } \{ G \circ F \} \]

for each object $X \in \operatorname{\mathcal{C}}$, which proves the equivalence $(3) \Leftrightarrow (4)$. Unwinding the definitions, we can identify $\theta '$ with the lower horizontal map appearing in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \operatorname{\mathcal{C}}_{X/} ) \ar [r]^-{\theta ''_{X}} \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( A, \operatorname{\mathcal{C}}_{X/} ) \ar [d] \\ \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( B, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \ar [r]^-{\theta '_{X}} & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( A, \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}), } \]

where the vertical maps are given by postcomposition with the coslice diagonal morphism $\rho : \operatorname{\mathcal{C}}_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. Theorem 4.6.4.16 guarantees that $\rho $ is an equivalence of $\infty $-categories. It is therefore also an an equivalence of left fibrations over $\operatorname{\mathcal{C}}$ (Proposition 5.1.6.5), so that the vertical maps are homotopy equivalences. It follows that $\theta '_{X}$ is a homotopy equivalence if and only if $\theta ''_{X}$ is a homotopy equivalence, which proves the equivalence $(4) \Leftrightarrow (5)$.

We now complete the proof by showing that $(5)$ implies $(1)$. Assume that condition $(5)$ is satisfied; we wish to show that $F$ is left cofinal. Let $q: \widetilde{B} \rightarrow B$ be a left fibration; we must show that composition with $F$ induces a homotopy equivalence $\operatorname{Fun}_{/B}( B, \widetilde{B} ) \rightarrow \operatorname{Fun}_{/B}(A, \widetilde{B} )$. To prove this, we are free to replace $q: \widetilde{B} \rightarrow B$ by any other left fibration which is equivalent to it (in the sense of Definition 5.1.6.1). We may therefore assume without loss of generality that there exists a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{B} \ar [r] \ar [d]^{q} & \operatorname{\mathcal{S}}_{\ast } \ar [d]^{q_{\mathrm{univ}}} \\ B \ar [r]^-{G} & \operatorname{\mathcal{S}}, } \]

where $q_{\mathrm{univ}}: \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is the universal left fibration of Corollary 5.6.0.6. We are then reduced to proving that $F$ induces a homotopy equivalence $\operatorname{Fun}_{ / \operatorname{\mathcal{S}}}( B, \operatorname{\mathcal{S}}_{\ast } ) \rightarrow \operatorname{Fun}_{ \operatorname{\mathcal{S}}}( A, \operatorname{\mathcal{S}}_{\ast } )$, which is a special case of $(5)$ (applied to the $\infty $-category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ and the object $X = \Delta ^0$). $\square$

Corollary 7.2.2.2. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is a $U$-limit diagram.

Proof. Set $f = \overline{f}|_{B}$ and apply Remark 7.1.6.8 to the commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / f } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ / (f \circ e) } \ar [d] \\ \operatorname{\mathcal{D}}_{ / (U \circ f) } \ar [r] & \operatorname{\mathcal{D}}_{ / (U \circ f \circ e) }, } \]

noting that the horizontal maps are equivalences by virtue of Proposition 7.2.2.1. $\square$

Corollary 7.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

is a limit diagram.

Proof. Apply Corollary 7.2.2.2 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (see Example 7.1.7.3). $\square$

Remark 7.2.2.4. We will see later that the converse of Corollary 7.2.2.3 is also true: if $e: A \rightarrow B$ is a morphism of simplicial sets having the property that precomposition with the induced map $e^{\triangleleft }: A^{\triangleleft } \rightarrow B^{\triangleleft }$ carries limit diagrams to limit diagrams, then $e$ is left cofinal (Proposition ).

Corollary 7.2.2.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $C \in \operatorname{\mathcal{C}}$ and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Suppose we are given a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \underline{C} \rightarrow f$, where $\underline{C} \in \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ denotes the constant diagram taking the value $C$. Then $\alpha $ exhibits $C$ as a limit of $f$ (in the sense of Definition 7.1.1.1 if and only if the induced natural transformation $\alpha |_{A}: \underline{C}|_{A} \rightarrow f|_{A}$ exhibits $C$ as a limit of the diagram $f|_{A}$.

Proof. By virtue of Remark 7.1.1.6, we are free to modify the natural transformation $\alpha $ by a homotopy and may therefore assume that it corresponds to a morphism of simplicial sets $\Delta ^0 \diamond B \rightarrow \operatorname{\mathcal{C}}$ which factors through the categorical equivalence $\Delta ^0 \diamond B \twoheadrightarrow \Delta ^0 \star B$ of Theorem 4.5.5.8. In this case, the desired result follows from Corollary 7.2.2.3 and Remark 7.1.4.6. $\square$

Corollary 7.2.2.6. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $e$ is left cofinal and the functor $F$ preserves $A$-indexed limits, then $F$ preserves $B$-indexed limits. If $e$ is right cofinal and the functor $F$ preserves $A$-indexed colimits, then the functor $F$ preserves $B$-indexed colimits.

Proposition 7.2.2.7. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ B \ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B^{\triangleleft } \ar [r]^-{ \overline{g} } & \operatorname{\mathcal{D}}, } \]

where $U$ is an inner fibration of $\infty $-categories. Let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. The following conditions are equivalent:

$(1)$

There exists a $U$-limit diagram $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{B} = f$ and $U \circ \overline{f} = \overline{g}$.

$(2)$

There exists a $U$-limit diagram $\overline{f}_0: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}_0|_{A} = f \circ e$ and $U \circ \overline{f}_0 = \overline{g} \circ e^{\triangleleft }$.

Proof. The implication $(1) \Rightarrow (2)$ follows by observing that if $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram, then the left cofinality of $e$ guarantees that $\overline{f} \circ e^{\triangleleft }$ is also a $U$-limit diagram (Corollary 7.2.2.2). We will complete the proof by showing that $(2)$ implies $(1)$. By virtue of Corollary 7.2.1.13, we can assume that the morphism $e$ is either left anodyne or a trivial Kan fibration. We first treat the case where $e$ is a trivial Kan fibration. Let $s: B \rightarrow A$ be a section of $e$, and let $\overline{f}_0: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfy the requirements of $(2)$. Let $\overline{f}$ denote the composite map

\[ B^{\triangleleft } \xrightarrow { s^{\triangleleft } } A^{\triangleleft } \xrightarrow { \overline{f}_0 } \operatorname{\mathcal{C}}. \]

It follows immediately from the construction that $\overline{f}|_{B} = f$ and $U \circ \overline{f} = \overline{g}$. Moreover, the composition $\overline{f} \circ e^{\triangleleft }$ is isomorphic to $\overline{f}_0$ (as an object of the $\infty $-category $\operatorname{Fun}( B^{\triangleleft }, \operatorname{\mathcal{C}})$), and is therefore also a $U$-limit diagram (Proposition 7.1.7.13). Since $e$ is left cofinal, it follows that $\overline{f}$ is also a $U$-limit diagram (Corollary 7.2.2.2).

We now treat the case where $e$ is left anodyne. In this case, the induced map $A^{\triangleleft } \coprod _{A} B \hookrightarrow B^{\triangleleft }$ is inner anodyne. Since $U$ is an inner fibration, we can extend $f$ to a morphism $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ \overline{f} = \overline{g}$ and $\overline{f} \circ e^{\triangleleft } = \overline{f}_0$. Since $e$ is left cofinal, the morphism $\overline{f}$ is automatically a $U$-limit diagram (Corollary 7.2.2.2). $\square$

Corollary 7.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ has a limit if and only if the composite diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$ has a limit.

Proof. If $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f$, then Corollary 7.2.2.3 guarantees that $\overline{f} \circ e^{\triangleleft }: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f \circ e$. Conversely, if $f \circ e$ can be extended to a colimit diagram, then Proposition 7.2.2.7 (applied in the special case $\operatorname{\mathcal{D}}= \Delta ^0$) guarantees that $f$ can also be extended to a colimit diagram. $\square$

Corollary 7.2.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets, and let $f: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$ if and only if it is a limit of the diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$.

Proof. If an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$, then we can choose a limit diagram $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point of $f^{\triangleleft }$ to the object $X$. Applying Corollary 7.2.2.8, we deduce that $\overline{f} \circ e^{\triangleleft }$ exhibits $X$ as a limit of the diagram $f \circ e$. Conversely, if $X$ is a limit of the diagram $f \circ e$, then Corollary 7.2.2.8 guarantees that the diagram $f$ admits a limit $Y \in \operatorname{\mathcal{C}}$. The preceding argument shows that $Y$ is also a limit of the diagram $f \circ e$. Applying Proposition 7.1.1.11, we deduce that $Y$ is isomorphic to $X$, so that $X$ is also a limit of the diagram $f$. $\square$

Corollary 7.2.2.10. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $e$ is left cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed limits. If $e$ is right cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed colimits.

Corollary 7.2.2.11. Let $e: A \rightarrow B$ be a morphism of simplicial sets and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $e$ is left cofinal and the functor $F$ creates $A$-indexed limits, then $F$ creates $B$-indexed limits. If $e$ is right cofinal and the functor $F$ creates $A$-indexed colimits, then the functor $F$ creates $B$-indexed colimits.