# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 7.4.5 Limits and Colimits of Spaces

Let $\operatorname{\mathcal{S}}$ denote the $\infty$-category of spaces (Construction 5.5.1.1), which we regard as a full subcategory of the $\infty$-category $\operatorname{\mathcal{QC}}$ (Remark 5.5.4.8). Our goal in this section is to describe limits and colimits in the $\infty$-category $\operatorname{\mathcal{S}}$. Given the results of §7.4.1 and §7.4.3, this is a relatively formal exercise. We begin with an elementary observation:

Proposition 7.4.5.1. Let $f: K \rightarrow \operatorname{\mathcal{S}}$ be a diagram. Then:

• An extension $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram if and only if it is a limit diagram in the $\infty$-category $\operatorname{\mathcal{QC}}$.

• An extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{S}}$ is a colimit diagram if and only if it is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{QC}}$.

Proof. It follows immediately from the definitions that a diagram in $\operatorname{\mathcal{S}}$ which is a limit (or colimit) diagram in the larger $\infty$-category $\operatorname{\mathcal{QC}}$, then it is already a limit (or colimit) diagram in $\operatorname{\mathcal{S}}$ (see Variant 7.1.3.10). To prove the converse implications, we must show that the inclusion functor $\iota : \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{QC}}$ preserves all limits and colimits. This follows from Corollary 7.1.3.21, since the functor $\iota$ admits both left and right adjoints (Example 6.2.2.16). $\square$

Corollary 7.4.5.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration between small simplicial sets and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Then the simplicial set $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of sections of $U$ is a Kan complex, which is a limit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the $\infty$-category $\operatorname{\mathcal{S}}$.

Proof. Since $U$ is a left fibration, Corollary 4.4.2.4 guarantees that the simplicial set $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a Kan complex. Note that every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian (Example 5.1.1.3), so that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ coincides with the $\infty$-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$. Applying Corollary 7.4.1.9, we see that the Kan complex $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a limit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the $\infty$-category $\operatorname{\mathcal{QC}}$, and therefore also in the full subcategory $\operatorname{\mathcal{S}}\subseteq \operatorname{\mathcal{QC}}$ (Proposition 7.4.5.1). $\square$

Corollary 7.4.5.3. Let $\operatorname{\mathcal{C}}$ be a small simplicial set. Then any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits a limit in the $\infty$-category $\operatorname{\mathcal{S}}$, given by the $\infty$-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$.

Proof. Apply Corollary 7.4.5.2 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.6.2.9. $\square$

Corollary 7.4.5.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration between small simplicial sets and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Then the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits a colimit in the $\infty$-category $\operatorname{\mathcal{S}}$. Moreover, a Kan complex $X$ is a colimit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.

Proof. Since $U$ is a left fibration, every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian (Example 5.1.1.3). Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. By virtue of Corollary 7.4.3.11, an $\infty$-category $X$ is a colimit of $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the $\infty$-category $\operatorname{\mathcal{QC}}$ if and only if there exists a functor $f: \operatorname{\mathcal{E}}\rightarrow X$ which exhibits $X$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. By virtue of Proposition 6.3.1.20, this is equivalent to the requirement that $X$ is a Kan complex and that $f$ is a weak homotopy equivalence. In this case, $X$ is also a colimit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ in the full subcategory $\operatorname{\mathcal{S}}\subseteq \operatorname{\mathcal{QC}}$ (Proposition 7.4.5.1). $\square$

Corollary 7.4.5.5. Let $\operatorname{\mathcal{C}}$ be a small simplicial set. Then any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits a colimit in the $\infty$-category $\operatorname{\mathcal{S}}$. Moreover, a Kan complex $X$ is a colimit of the diagram $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow X$.

Proof. Apply Corollary 7.4.5.4 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.6.2.9. $\square$

Corollary 7.4.5.6. The $\infty$-category $\operatorname{\mathcal{S}}$ is complete and cocomplete.

Remark 7.4.5.7 (Size Estimates for Colimits). Let $\lambda$ be an uncountable cardinal and let $\kappa = \mathrm{cf}(\lambda )$ be its cofinality. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, where $\operatorname{\mathcal{C}}$ is a $\kappa$-small simplicial set, and that the Kan complex $\mathscr {F}(C)$ is essentially $\lambda$-small for each $C \in \operatorname{\mathcal{C}}$. Then the colimit $\varinjlim (\mathscr {F})$ is also essentially $\lambda$-small. This follows from Corollary 7.4.3.15 and Proposition 7.4.5.1.

Variant 7.4.5.8 (Size Estimates for Limits). Let $\lambda$ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, where $\operatorname{\mathcal{C}}$ is a $\kappa$-small simplicial set, and that the Kan complex $\mathscr {F}(C)$ is essentially $\lambda$-small for each $C \in \operatorname{\mathcal{C}}$. Then the limit $\varprojlim (\mathscr {F})$ is also essentially $\lambda$-small. This follows from Corollary 7.4.1.13 and Proposition 7.4.5.1.

Remark 7.4.5.9 (Limits of Truncated Spaces). Let $n$ be an integer. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ such that, for every vertex $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is $n$-truncated. Then the limit $\varprojlim (\mathscr {F} )$ is $n$-truncated. For $n \geq -1$, this follows by combining Corollary 7.4.1.12 with Example 4.8.2.4. For $n \leq -2$, our assumption ensures that each of the Kan complexes $\mathscr {F}(C)$ is contractible, and we wish to show that the Kan complex $\varprojlim (\mathscr {F} )$ is also contractible. This follows from the description given in Corollary 7.4.5.3, since the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (see Proposition 4.4.2.14).

Corollary 7.4.5.10. Let $n$ be an integer, let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a diagram. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is $n$-truncated. Then the limit $\varprojlim ( \mathscr {F} )$ is an $n$-truncated Kan complex.$\infty$-category.

Proposition 7.4.5.11. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is right cofinal (Definition 7.2.1.1).

$(2)$

For every corepresentable functor $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the composite map $K \xrightarrow {f} \operatorname{\mathcal{C}}\xrightarrow {h} \operatorname{\mathcal{S}}$ has a contractible colimit.

Proof. Fix an object $X \in \operatorname{\mathcal{C}}$, and let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor corepresented by $X$ (Theorem 5.6.6.13). Using Proposition 5.6.6.21, we see that $f \circ h^{X}$ is a covariant transport representation for the left fibration $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow K$. Using Corollary 7.4.5.4, we can reformulate condition $(2)$ as follows:

$(2')$

For each object $X \in \operatorname{\mathcal{C}}$, the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ is weakly contractible.

The equivalence $(1) \Leftrightarrow (2')$ follows from Theorem 7.2.3.1. $\square$

For strictly commutative diagrams, we can use the results of §5.3 to give an alternative construction.

Corollary 7.4.5.12. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a (strictly commutative) diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Then a Kan complex $X$ is a colimit of the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{S}}$ if and only if it is weakly homotopy equivalent to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Definition 5.3.3.1.

For many applications, it will be useful to have more precise versions of the preceding results, which characterize limit and colimit diagrams in the $\infty$-category $\operatorname{\mathcal{S}}$.

Corollary 7.4.5.13. Suppose we are given a pullback diagram of small simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, }$

where $U$ and $\overline{U}$ are left fibrations. The following conditions are equivalent:

$(1)$

The restriction map

$\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$

is a homotopy equivalence of Kan complexes.

$(2)$

The covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{S}}$.

Proof. Since $\overline{U}$ is a left fibration, every edge of $\overline{\operatorname{\mathcal{E}}}$ is $\overline{U}$-cocartesian (Example 5.1.1.3). We can therefore identify $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} )$ and $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ with the $\infty$-categories $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} )$ and $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, respectively. The desired result now follows by combining Theorem 7.4.1.1 with Proposition 7.4.5.1. $\square$

As an application, we prove a converse of Corollary 7.2.2.3:

Corollary 7.4.5.14. Let $e: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $e$ is left cofinal if and only if it satisfies the following condition:

$(\ast )$

For every limit diagram $\overline{F}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$, the composition $(\overline{F} \circ e^{\triangleleft }): \operatorname{\mathcal{C}}'^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ is also a limit diagram.

Proof. Assume that condition $(\ast )$ is satisfied; we will show that $e$ is left cofinal (the reverse implication is a special case of Corollary 7.2.2.3). Fix a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$; we wish to show that the restriction map

$e^{\ast }: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}})$

is a homotopy equivalence of Kan complexes. Using Proposition 7.4.1.6 (together with Remark 7.4.1.8), we can extend $U$ to a left fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$ for which the restriction map

$T: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$

is a homotopy equivalence.

Form a pullback diagram of simplicial sets

$\xymatrix@C =50pt@R=50pt{ \overline{\operatorname{\mathcal{E}}}' \ar [r] \ar [d]^{ \overline{U}' } & \overline{\operatorname{\mathcal{E}}} \ar [d]^{U} \\ \operatorname{\mathcal{C}}'^{\triangleleft } \ar [r]^-{e} & \operatorname{\mathcal{C}}^{\triangleleft }. }$

Let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for the left fibration $\overline{U}$, so that $\overline{F} \circ e^{\triangleleft }$ is a covariant transport representation for the left fibration $\overline{U}'$. It follows from the criterion of Corollary 7.4.5.13 that $\overline{F}$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{S}}$. Applying assumption $(\ast )$, we see that $\overline{F} \circ e^{\triangleleft }$ is also a limit diagram in the $\infty$-category $\operatorname{\mathcal{S}}$. We therefore have a commutative diagram of restriction maps

$\xymatrix@C =50pt@R=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}'^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \ar [r]^-{T'} \ar [d]^{(e^{\triangleleft })^{\ast }} & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [d]^{ e^{\ast } } \\ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \ar [r]^-{T} & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}), }$

where the horizontal maps are homotopy equivalences (Corollary 7.4.5.13). Consequently, to show that $e^{\ast }$ is a homotopy equivalence, it will suffice to show that $( e^{\triangleleft } )^{\ast }$ is a homotopy equivalence. We now observe that $(e^{\triangleleft } )^{\ast }$ fits into a commutative diagram

$\xymatrix@C =50pt@R=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}'^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \ar [dd]^{(e^{\triangleleft } )^{\ast } } \ar [dr] & \\ & \{ {\bf 0} \} \times _{ \operatorname{\mathcal{C}}^{\triangleleft } } \overline{\operatorname{\mathcal{E}}} \\ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ), \ar [ur] & }$

where the horizontal maps are given by evaluation at the cone points of the simplicial sets $\operatorname{\mathcal{C}}^{\triangleleft }$ and $\operatorname{\mathcal{C}}'^{\triangleleft }$ are are therefore trivial Kan fibrations (Corollary 5.3.1.23). $\square$

Corollary 7.4.5.15. Suppose we are given a pullback diagram of small simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, }$

where $U$ and $\overline{U}$ are left fibrations. The following conditions are equivalent:

$(1)$

The inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence of simplicial sets.

$(2)$

The inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is left cofinal.

$(3)$

The covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{S}}$ is a colimit diagram in the $\infty$-category $\operatorname{\mathcal{S}}$.

Proof. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$, and let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } = \{ {\bf 1} \} \times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$ denote the corresponding fiber of $\overline{\operatorname{\mathcal{E}}}$. Since the inclusion map $\{ {\bf 1} \} \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is right anodyne (Example 4.3.7.11), the inclusion $\iota : \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } \hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is also right anodyne (Corollary 7.2.3.13). In particular, $\iota$ is a weak homotopy equivalence of simplicial sets. Let $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ be a covariant refraction diagram (Proposition 7.4.3.3), so that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is homotopic to the composition $\iota \circ \mathrm{Rf}$. It follows that condition $(1)$ can be reformulated as follows:

$(1')$

The covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a weak homotopy equivalence.

The equivalence $(1') \Leftrightarrow (3)$ follows by combining Proposition 7.4.5.1, Theorem 7.4.3.6, and Proposition 6.3.1.20.

The implication $(2) \Rightarrow (1)$ follows from Proposition 7.2.1.5. We will complete the proof by showing that $(1')$ implies $(2)$. Choose an inner anodyne monomorphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $K'$ is an $\infty$-category. Then the induced map $\operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}'^{\triangleright }$ is also inner anodyne (Corollary 4.3.6.6); in particular, it is a categorical equivalence. Using Proposition 5.6.7.2 (and Remark 5.6.7.4), we can assume that $\overline{U}$ is the pullback of a left fibration $\overline{U}': \overline{\operatorname{\mathcal{E}}}' \rightarrow \operatorname{\mathcal{C}}'^{\triangleright }$. Setting $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{ \operatorname{\mathcal{C}}'^{\triangleright } } \overline{\operatorname{\mathcal{E}}}'$, we have a commutative diagram of inclusion maps

$\xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d] & \overline{\operatorname{\mathcal{E}}} \ar [d] \\ \operatorname{\mathcal{E}}' \ar [r] & \overline{\operatorname{\mathcal{E}}'}, }$

where the vertical maps are categorical equivalences (Corollary 5.6.7.6). Consequently, to prove that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is left cofinal, it will suffice to show that the inclusion $\operatorname{\mathcal{E}}' \hookrightarrow \overline{\operatorname{\mathcal{E}}}'$ is left cofinal (Corollary 7.2.1.22). We may therefore replace $\overline{U}$ by the left fibration $\overline{U}': \overline{\operatorname{\mathcal{E}}}' \rightarrow \operatorname{\mathcal{C}}'^{\triangleright }$, and thereby reduce to proving the implication $(1') \Rightarrow (2)$ under the assumption that $\operatorname{\mathcal{C}}$ is an $\infty$-category.

Let $\operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ denote the oriented fiber product of Definition 4.6.4.1, and consider the projection maps

$\operatorname{\mathcal{E}}\xleftarrow {\pi } \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } \xrightarrow {\pi '} \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }.$

The functor $\pi$ is a trivial Kan fibration, and the refraction functor $\mathrm{Rf}$ is obtained by composing $\pi '$ with a choice of section of $\pi$. Consequently, assumption $(1')$ guarantees that $\pi '$ is a weak homotopy equivalence of simplicial sets. For each vertex $X \in \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$, we have a pullback diagram of simplicial sets

7.51
$$\begin{gathered}\label{equation:diagram-for-left-cofinality} \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \{ X \} \ar [r] \ar [d] & \{ X\} \ar [d] \\ \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } \ar [r]^-{ \pi } & \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }. } \end{gathered}$$

Since $\pi '$ is an isofibration of $\infty$-categories (Corollary 5.3.7.3), the diagram( 7.51 ) is a categorical pullback square (Corollary 4.5.2.27). Because $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a Kan complex, the diagram ( 7.51) is also a homotopy pullback square (Variant 4.5.2.11). Our assumption that $\pi '$ is a weak homotopy equivalence guarantees that the upper horizontal map is also a weak homotopy equivalence: that is, the simplicial set $\operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \{ X\}$ is weakly contractible (Corollary 3.4.1.5). Condition $(2)$ now follows by allowing the object $X$ to vary and applying the criterion of of Theorem 7.2.3.1 (together with Remark 7.2.3.2). $\square$

Corollary 7.4.5.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty$-categories, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and set $\operatorname{\mathcal{E}}^{0} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}$. The following conditions are equivalent:

$(1)$

The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

The inclusion functor $\operatorname{\mathcal{E}}^{0} \hookrightarrow \operatorname{\mathcal{E}}$ is left cofinal.

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{C}}^{0}_{/C} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the inclusion map $\operatorname{\mathcal{C}}^{0}_{/C} \rightarrow \operatorname{\mathcal{C}}_{/C}$ factors as a composition

$\operatorname{\mathcal{C}}^{0}_{/C} \hookrightarrow ( \operatorname{\mathcal{C}}^{0}_{ / C } )^{\triangleright } \xrightarrow {\lambda } \operatorname{\mathcal{C}}_{/C},$

where $\lambda$ carries the cone point $(\operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright }$ to the final object $\operatorname{id}_{C} \in \operatorname{\mathcal{C}}_{/C}$. In particular, $\lambda$ is right cofinal (Corollary 7.2.1.9). It follows that the induced map $\lambda _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is also right cofinal (Proposition 7.2.3.12); in particular, $\lambda _{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence (Proposition 7.2.1.5). Combining this observation with Corollary 7.4.5.15, we see that the following conditions are equivalent:

$(1_ C)$

The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at the object $C \in \operatorname{\mathcal{C}}$.

$(1'_ C)$

The inclusion map $\iota _{C}: \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}_{/C} \hookrightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is a weak homotopy equivalence.

Choose an object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = C$ and set $\operatorname{\mathcal{E}}^{0}_{/X} = \operatorname{\mathcal{E}}^{0} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{/X}$, so that we have a pullback diagram of simplicial sets

7.52
$$\begin{gathered}\label{equation:left-fibration-Kan-extension-characterization} \xymatrix { \operatorname{\mathcal{E}}^{0}_{/X} \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{/X} \ar [d] \\ \operatorname{\mathcal{E}}\times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}_{/C} \ar [r]^{\iota _{C}} & \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}. } \end{gathered}$$

Since $U$ is a left fibration, the vertical maps in this diagram are Kan fibrations; it follows that (7.52) is also a homotopy pullback square (Example 3.4.1.3). In particular, if condition $(1'_ C)$ is satisfied, then the inclusion map $\operatorname{\mathcal{E}}^{0}_{/X} \hookrightarrow \operatorname{\mathcal{E}}_{/X}$ is a weak homotopy equivalence (Corollary 3.4.1.5), so that the $\infty$-category $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible. Conversely, if $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible for every object $X \in \operatorname{\mathcal{C}}$ satisfying $U(X) = C$, then $\iota _{C}$ is a weak homotopy equivalence: this follows from the observation that every connected component of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ has nonempty intersection with the fiber $\operatorname{\mathcal{E}}_{C} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ C\}$. It follows that $(1'_ C)$ can be reformulated as follows:

$(2_ C)$

For every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = C$, the $\infty$-category $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible.

The equivalence of $(1)$ and $(2)$ now follows from the equivalence $(1_ C) \Leftrightarrow (2_ C)$ by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary (and applying Theorem 7.2.3.1). $\square$

We conclude this section with an application of Corollary 7.4.5.13.

Proposition 7.4.5.17. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty$-category and let $K$ be a small simplicial set. Then a morphism $F: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the composition

$K^{\triangleleft } \xrightarrow { F } \operatorname{\mathcal{C}}\xrightarrow {h^{X} } \operatorname{\mathcal{S}}$

is a limit diagram in the $\infty$-category of spaces; here $h^{X}$ denotes the functor corepresented by $X$ (Notation 5.6.6.14).

Proof. Applying Proposition 7.1.5.12, we see that $F$ is a limit diagram if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the restriction map

$\theta _{X}: \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \underline{X}, F) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{C}}) }( \underline{X}|_{K}, F|_{K} )$

is a homotopy equivalence of Kan complexes. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the second factor. Note that $U$ is a left fibration (Proposition 4.6.4.11) and that $\theta _{X}$ can be identified with the restriction map

$\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( K^{\triangleleft }, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( K, \operatorname{\mathcal{E}}).$

The identity morphism $\operatorname{id}_{X}$ can be viewed as an initial object of $\operatorname{\mathcal{E}}$ satisfying $U( \operatorname{id}_{X} ) = X$ (Proposition 4.6.7.22), so the corepresentable functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a covariant transport representation for $U$ (Proposition 5.6.6.21). Applying Corollary 7.4.5.13, we see that $\theta _{X}$ is a homotopy equivalence if and only if $h^{X} \circ F$ is a limit diagram in the $\infty$-category $\operatorname{\mathcal{S}}$. $\square$

Corollary 7.4.5.18. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty$-category. For every object $X \in \operatorname{\mathcal{C}}$, the functors

$h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$

preserve $K$-indexed limits, for every small simplicial set $K$.

Remark 7.4.5.19. Let $\lambda$ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty$-category which is locally $\lambda$-small. Let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda$ and let $K$ be a $\kappa$-small simplicial set. Then, in the statements of Proposition 7.4.5.17 and Corollary 7.4.5.18, we can replace $\operatorname{\mathcal{S}}$ by the $\infty$-category $\operatorname{\mathcal{S}}^{< \lambda }$ of $\lambda$-small spaces (see Variant 7.4.5.8).