Proposition 7.4.3.1. Let $\operatorname{\mathcal{C}}$ be a small simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be the covariant transport representation of a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 5.6.5.1). Then an object $X \in \operatorname{\mathcal{S}}$ is a colimit of $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
7.4.3 Colimits of Spaces
Let $\operatorname{\mathcal{S}}$ be the $\infty $-category of spaces (Construction 5.5.1.1). Our goal in this section is to show that every small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits a colimit $X = \varinjlim ( \mathscr {F} )$. More precisely, we will prove the following:
Remark 7.4.3.2. The proof of Proposition 7.4.3.1 given in this section will proceed by reduction to the special case where $\mathscr {F}$ is the constant functor $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}$, using the theory of Kan extensions developed in §7.3. In §7.4.5, we will formulate a more general result which describes the colimit of the covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ for a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is not assumed to be a left fibration (see Proposition 7.4.5.1 and Remark 7.4.5.4). For this, we give a different proof which is independent of the results of this section (and does not use the theory of Kan extensions).
Corollary 7.4.3.3. 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 Proposition 7.4.3.1 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.6.2.9. $\square$
Corollary 7.4.3.4. 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.
Proof. Combine Proposition7.4.3.1 with Example 5.6.5.6. $\square$
Example 7.4.3.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be the constant diagram $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}$ taking the value $\Delta ^0 \in \operatorname{\mathcal{S}}$. Then $\mathscr {F}$ is a covariant transport representation for the left fibration $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. In this case, Proposition 7.4.3.1 asserts that a Kan complex $X$ is a colimit of $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{C}}\rightarrow X$, which is a special case of Example 7.1.2.10.
Following the convention of Remark 4.7.0.5, we can regard Proposition 7.4.3.1 as a special case of the following more general result:
Proposition 7.4.3.6. Let $\kappa $ be an uncountable cardinal and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be the covariant transport representation of a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Then a Kan complex $X$ is a colimit of $\mathscr {F}$ (in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$) if and only if there exists a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$.
Proof. Using Proposition 5.6.7.2, we can reduce to the situation where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, Corollary 7.4.2.15 guarantees that $\mathscr {F}$ is a left Kan extension of the constant functor $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$ along $U$. Applying Corollary 7.3.8.20, we see that an object $X \in \operatorname{\mathcal{S}}^{< \kappa }$ is a colimit of $\mathscr {F}$ if and only if it is a colimit of the constant diagram $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$. By virtue of Variant 7.1.2.11, this is equivalent to the existence of a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$. $\square$
Example 7.4.3.7 (Colimits of Corepresentable Functors). Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$. Then the colimit $\varinjlim (\mathscr {F} )$ is contractible. To prove this, we observe that $\mathscr {F}$ is a covariant transport representation for the left fibration $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ (Proposition 5.6.6.21). By virtue of Proposition 7.4.3.6, it will suffice to show that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ is weakly contractible, which follows from Corollary 4.6.7.25.
Corollary 7.4.3.8 (Size Estimates for Colimits). Let $\lambda $ be an uncountable regular cardinal and let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\kappa $. Then the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small colimits, which are preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for any $\mu \geq \lambda $.
Proof. Let $\operatorname{\mathcal{C}}$ be a $\kappa $-small simplicial set; we wish to show that every diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ admits a colimit (which is preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for $\mu \geq \lambda $). Choose a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having $\mathscr {F}$ as its covariant transport representation (for example, we can take $\operatorname{\mathcal{E}}$ to be the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Definition 5.6.2.1). Since $U$ is essentially $\lambda $-small and $\operatorname{\mathcal{C}}$ is $\kappa $-small, the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small (Proposition 4.7.9.10). We can therefore choose a weak homotopy equivalence $\operatorname{\mathcal{E}}\rightarrow X$, where $X$ is a $\lambda $-small Kan complex. Applying Proposition 7.4.3.6, we conclude that $X$ is a colimit of $\mathscr {F}$. $\square$
Warning 7.4.3.9 (Fake Colimits). Let $\lambda $ be an uncountable cardinal and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ indexed by a simplicial set $\operatorname{\mathcal{C}}$. It follows from Variant 7.4.1.4 that if $\mathscr {F}$ admits a limit in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$, then that limit is independent of $\lambda $: that is, it is automatically preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \mu }$ for every cardinal $\lambda \geq \kappa $. It follows from Corollary 7.4.3.8 that the same result holds for colimits, provided that the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small for $\kappa = \mathrm{cf}(\lambda )$. Beware that this cardinality assumption cannot be omitted: if the simplicial set $\operatorname{\mathcal{C}}$ is too large, then it is possible for a $X$ to be a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$, which does not remain a colimit in $\operatorname{\mathcal{S}}^{< \mu }$ for $\mu \gg \lambda $ (see Exercise 7.1.2.12). In this case, we will generally reserve the notation $\varinjlim (\mathscr {F} )$ for the “true” colimit of $\mathscr {F}$ (computed in the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$ for $\mu $ sufficiently large), characterized by the requirement that it is weakly homotopy equivalent to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$.
Corollary 7.4.3.10. Let $\kappa $ be an uncountable regular cardinal. Then the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ admits $\kappa $-small colimits, which are preserved by the inclusion functors $\operatorname{\mathcal{S}}^{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda }$ for $\lambda \geq \kappa $.
Proposition 7.4.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
The morphism $F$ is right cofinal (Definition 7.2.1.1).
For every functor of $\infty $-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admits a colimit $X = \varinjlim (G)$, the object $X \in \operatorname{\mathcal{D}}$ is also a colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$.
For every uncountable regular cardinal $\kappa $ and every diagram $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which admits a colimit $X = \varinjlim (G)$, the object $X \in \operatorname{\mathcal{S}}^{< \kappa }$ is also a colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$.
For every uncountable regular cardinal $\kappa $ and every corepresentable functor $G: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the colimit $\varinjlim ( G \circ F)$ is contractible.
Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 7.2.2.11, the implication $(2) \Rightarrow (3)$ is immediate, and the implication $(3) \Rightarrow (4)$ follows from Example 7.4.3.7. We will complete the proof by showing that $(4)$ implies $(1)$. Choose an uncountable regular cardinal $\kappa $ such that $K$ is $\kappa $-small and $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. 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 $h^{X} \circ F$ is a covariant transport representation for the left fibration $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow K$. If condition $(4)$ is satisfied, then Proposition 7.4.3.6 guarantees that the simplicial set $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ is weakly contractible. Allowing $X$ to vary and applying the criterion of Theorem 7.2.3.1, we conclude that $F$ is right cofinal. $\square$
Corollary 7.4.3.12. Let $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, let $\kappa $ be an uncountable regular cardinal for which $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, and let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $. The following conditions are equivalent:
The morphism $F$ is left cofinal.
For every $\infty $-category $\operatorname{\mathcal{D}}$ and every limit diagram $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$, the composite map $( \overline{G} \circ F^{\triangleleft } ): \operatorname{\mathcal{C}}'^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is also a limit diagram.
For every limit diagram $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composite map $( \overline{G} \circ F^{\triangleleft } ): \operatorname{\mathcal{C}}'^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ is also a limit diagram.
Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 7.2.2.3 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied; we wish to show that the map $F^{\operatorname{op}}: \operatorname{\mathcal{C}}'^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is right cofinal. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is a $\kappa $-small $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, let $h_ X: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be the functor represented by $X$, set $G = h_{X}^{\operatorname{op}}$, and let $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow ( \operatorname{\mathcal{S}}^{< \kappa } )$ be an extension of $G$ to a limit diagram (so that $\overline{G}$ carries the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$ to a contractible Kan complex, by virtue of Example 7.4.3.7). By virtue of Proposition 7.4.3.11, it will suffice to show that the composite map
\[ \operatorname{\mathcal{C}}'^{\triangleleft } \xrightarrow { F^{\triangleleft } } \operatorname{\mathcal{C}}^{\triangleleft } \xrightarrow { \overline{G} } (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \]
is also a limit diagram. Our assumption that $\lambda \geq \mathrm{ecf}(\kappa )$ guarantees that the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ is locally $\lambda $-small (Remark 5.5.4.14). Applying the criterion of Proposition 7.4.1.18, we are reduced to showing that for every representable functor $H: (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composite map
\[ \operatorname{\mathcal{C}}'^{\triangleleft } \xrightarrow { F^{\triangleleft } } \operatorname{\mathcal{C}}^{\triangleleft } \xrightarrow { \overline{G} } (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \xrightarrow {H} \operatorname{\mathcal{S}}^{< \lambda } \]
is a limit diagram, which is a special case of assumption $(3)$. $\square$
For some applications, we will need a more precise version of Proposition 7.4.3.6.
Proposition 7.4.3.13. Let $\kappa $ be an uncountable cardinal, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a diagram, and let $\beta : \mathscr {F} \rightarrow \underline{X}_{\operatorname{\mathcal{C}}}$ be a natural transformation. Suppose we are given a left fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and a commutative diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Assume that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small and that $\alpha $ exhibits $\mathscr {F}$ as a covariant transport representation for $U$ (Definition 7.4.1.8). The following conditions are equivalent:
The natural transformation $\beta $ exhibits $X$ as a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$.
The natural transformation $\gamma $ determines a weak homotopy equivalence of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}^{< \kappa } }( \Delta ^0, X )$.
\[ \xymatrix { & \mathscr {F}|_{ \operatorname{\mathcal{E}}} \ar [dr]^{ \beta |_{\operatorname{\mathcal{E}}} } & \\ \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \ar [ur]^{ \alpha } \ar [rr]^{\gamma } & & \underline{X}_{\operatorname{\mathcal{E}}} } \]
Proof. Using Variant 7.4.2.14, we see that $\alpha $ exhibits $\mathscr {F}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}}$ along $U$. By virtue of Corollary 7.3.8.20, this is equivalent to the assertion that $\gamma $ exhibits $X$ as a colimit of the diagram $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$. The equivalence of $(1)$ and $(2)$ now follows from Variant 7.1.2.11. $\square$
Corollary 7.4.3.14. Suppose we are given a pullback diagram of small simplicial sets where $U$ and $\overline{U}$ are left fibrations. Choose an uncountable regular cardinal $\kappa $ such that $U$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a covariant transport representation for $\overline{U}$. The following conditions are equivalent:
The inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence of simplicial sets.
The inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is left cofinal.
The covariant transport representation $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a colimit diagram.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
Proof. Let $v$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright } = \operatorname{\mathcal{C}}\star \{ v\} $ and let $\overline{\operatorname{\mathcal{E}}}_{v} = \{ v\} \times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$ denote the corresponding fiber of $\overline{\operatorname{\mathcal{E}}}$. Since the inclusion map $\{ v \} \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is right anodyne (Example 4.3.7.11), the inclusion $\iota : \overline{\operatorname{\mathcal{E}}}_{ v } \hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is also right anodyne (Corollary 7.2.3.13). In particular, $\iota $ is a weak homotopy equivalence of simplicial sets.
Let $q: \Delta ^1 \times \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ be the morphism of simplicial sets characterized by the requirement that $q|_{ \{ 0\} \times \operatorname{\mathcal{C}}^{\triangleright } }$ is the identity morphism and $q|_{ \{ 1\} \times \operatorname{\mathcal{C}}^{\triangleright } }$ is the constant morphism taking the value $v$. Set $X = \overline{\mathscr {F}}(v)$, so that the composition
\[ \Delta ^1 \times \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow {q} \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow { \overline{\mathscr {F}} } \operatorname{\mathcal{S}}^{< \kappa } \]
can be identified with a natural transformation $\beta : \overline{\mathscr {F}} \rightarrow \underline{X}|_{\operatorname{\mathcal{C}}^{\triangleright }}$.
Choose a natural transformation $\alpha : \underline{ \Delta ^{0} }_{ \overline{\operatorname{\mathcal{E}}} } \rightarrow \overline{\mathscr {F}}|_{ \overline{\operatorname{\mathcal{E}}} }$ which exhibits $\overline{\mathscr {F}}$ as a covariant transport representation for $\overline{U}$, and let $\gamma : \underline{ \Delta ^{0} }|_{ \overline{\operatorname{\mathcal{E}}} } \rightarrow \underline{X}_{ \overline{\operatorname{\mathcal{E}}} }$ be a composition of $\beta |_{ \overline{\operatorname{\mathcal{E}}} }$ with $\alpha $. Then $\gamma $ can be identified with a morphism of simplicial sets $T: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}^{< \kappa } }( \Delta ^0, X)$. Our assumption on $\alpha $ guarantees that the the composite map
\[ \overline{\operatorname{\mathcal{E}}}_{v} \xrightarrow {\iota } \overline{\operatorname{\mathcal{E}}} \xrightarrow {T} X \]
is a weak homotopy equivalence of simplicial sets. We can therefore reformulate condition $(1)$ as follows:
- $(1')$
The restriction $T|_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}^{< \kappa } }( \Delta ^0, X)$ is a weak homotopy equivalence.
Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$. Using Proposition 7.4.3.13, we see that $(1')$ is equivalent to the requirement that the natural transformation $\beta |_{\operatorname{\mathcal{C}}}: \mathscr {F} \rightarrow \underline{X}_{\operatorname{\mathcal{C}}}$ exhibits $X$ as a colimit of $\mathscr {F}$. The equivalence $(1') \Leftrightarrow (3)$ now follows from Remark 7.1.3.6.
The implication $(2) \Rightarrow (1)$ follows from Proposition 7.2.1.5. We will complete the proof by showing that $(1)$ implies $(2)$. monomorphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ 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 $\operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{v}$ denote the oriented fiber product of Definition 4.6.4.1, so that the diagram of simplicial sets
\[ \xymatrix { \operatorname{\mathcal{E}}\operatorname{\vec{\times }}_{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{v} \ar [r]^{ \pi } \ar [r]^{ \pi ' } & \overline{\operatorname{\mathcal{E}}}_{v} \ar [d]^{\iota } \\ \operatorname{\mathcal{E}}\ar [r] & \overline{\operatorname{\mathcal{E}}} } \]
commutes up to homotopy. Assume that condition $(1)$ is satisfied: that is, the lower horizontal map is a weak homotopy equivalence. Since $\pi $ is a trivial Kan fibration and $\iota $ is a weak homotopy equivalence, it follows that $\pi '$ is also a weak homotopy equivalence. For each vertex $X \in \overline{\operatorname{\mathcal{E}}}_{v}$, we have a pullback diagram of simplicial sets
7.51
\begin{equation} \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}}}_{v} \ar [r]^-{ \pi ' } & \overline{\operatorname{\mathcal{E}}}_{v}. } \end{gathered} \end{equation}
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 $\overline{\operatorname{\mathcal{E}}}_{v}$ is a Kan complex, the diagram (7.51) is also a homotopy pullback square (Variant 4.5.2.11). Since $\pi '$ is a weak homotopy equivalence, it follows that the upper horizontal map is also a weak homotopy equivalence: that is, the oriented fiber product $\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 7.4.3.15. 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 $\operatorname{\mathcal{E}}^{0} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}$. Let $\kappa $ be an uncountable regular cardinal for which $U$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small. The following conditions are equivalent:
The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
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 7.4.3.14, 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{equation} \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} \end{equation}
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). $\square$