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7.2.2 Cofinality and Limits

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In ยง7.1.3, we introduced the notion of a limit $\varprojlim (G)$ and colimit $\varinjlim (G)$ for a diagram $G: B \rightarrow \operatorname{\mathcal{C}}$ (Definition 7.1.3.1). 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 $\varprojlim (G \circ F)$ (Corollary 7.2.2.2). Similarly, if $F: A \rightarrow B$ is right cofinal, then the colimit $\varinjlim (G)$ (if it exists) can be identified with the colimit $\varinjlim (G \circ F)$. The proof will make use of the following cofinality criterion:

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 4.3.6.11). In the second case, the diagram $F$ is a categorical equivalence of simplicial sets (Proposition 4.5.2.9), so that $\theta $ is an equivalence of $\infty $-categories by virtue of Corollary 4.6.4.17.

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 5.6.2.13 and Proposition 5.6.2.5, 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 5.6.2.5), 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 5.6.2.1). 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 5.6.3.14. 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 $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $G: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $F: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then:

$(1)$

The diagram $G$ admits a limit if and only if the diagram $(G \circ F): A \rightarrow \operatorname{\mathcal{C}}$ admits a limit.

$(2)$

Let $\overline{G}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be an extension of $G$. Then $\overline{G}$ is a limit diagram if and only if the composite map

\[ A^{\triangleleft } \xrightarrow { F^{\triangleleft } } B^{\triangleleft } \xrightarrow { \overline{G} } \operatorname{\mathcal{C}} \]

is a limit diagram.

$(3)$

An object $X \in \operatorname{\mathcal{C}}$ is a limit of the diagram $G$ if and only if it is a limit of the diagram $(G \circ F): A \rightarrow \operatorname{\mathcal{C}}$.

Proof. Since $F$ is left cofinal, Proposition 7.2.2.1 guarantees that the induced map $\operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{C}}_{/(G \circ F)}$ is an equivalence of $\infty $-categories. Assertions $(1)$ and $(2)$ now follow by applying Corollaries 7.1.1.22 and 7.1.1.21, respectively. It follows immediately from assertion $(2)$ that if $X \in \operatorname{\mathcal{C}}$ is a limit of the diagram $G$, then it is also a limit of the diagram $G \circ F$. Conversely, if $X$ is a limit of the diagram $G \circ F$, then assertion $(1)$ guarantees that the diagram $G$ admits a limit $Y$. The preceding argument shows that $Y$ is also a limit of the diagram $G \circ F$. Applying Proposition 7.1.3.7, we deduce that $Y$ is isomorphic to $X$, so that $X$ is also a limit of the diagram $G$. $\square$

Remark 7.2.2.3. We will see later that the converse of Corollary 7.2.2.2 is also true: if $F: A \rightarrow B$ is a morphism of simplicial sets having the property that precomposition with the induced map $F^{\triangleleft }: A^{\triangleleft } \rightarrow B^{\triangleleft }$ carries limit diagrams to limit diagrams, then $F$ is left cofinal (Proposition ).

Corollary 7.2.2.4. Let $F: A \rightarrow B$ be a morphism of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $F$ is left cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed limits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed limits. If $F$ is right cofinal and $\operatorname{\mathcal{C}}$ admits $A$-indexed colimits, then $\operatorname{\mathcal{C}}$ also admits $B$-indexed colimits.

Corollary 7.2.2.5. Let $F: A \rightarrow B$ be a morphism of simplicial sets and let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is left cofinal and the functor $G$ preserves $A$-indexed limits, then $G$ preserves $B$-indexed limits. If $F$ is right cofinal and the functor $G$ preserves $A$-indexed colimits, then the functor $G$ preserves $B$-indexed colimits.

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