Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1), which we regard as a full subcategory of the $\infty $-category $\operatorname{\mathcal{QC}}$ (Remark 5.6.4.8). 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 6.2.2.13). $\square$
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 5.7.2.9. $\square$
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 5.7.2.9. $\square$
Corollary 7.4.5.6. The $\infty $-category $\operatorname{\mathcal{S}}$ admits small limits and colimits.
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 $\varinjlim (\mathscr {F})$ is also essentially $\lambda $-small. This follows from Corollary 7.4.1.12 and Proposition 7.4.5.1.
For strictly commutative diagrams, we can use the results of §5.3 to give an alternative construction.
Corollary 7.4.5.9. 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 Corollary 7.4.5.4 with Example 5.7.5.6. $\square$
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 7.4.5.10. Suppose we are given a pullback diagram of small simplicial sets where $U$ and $\overline{U}$ are left fibrations. The following conditions are equivalent:
The restriction map
is a homotopy equivalence of Kan complexes.
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 $\operatorname{\mathcal{S}}$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
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 7.4.5.11. 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:
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
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
is a homotopy equivalence.
Form a pullback diagram of simplicial sets
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 7.4.5.10 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
where the horizontal maps are homotopy equivalences (Corollary 7.4.5.10). 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
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 7.4.5.12. Suppose we are given a pullback diagram of small simplicial sets where $U$ and $\overline{U}$ are left fibrations. 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 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 $\operatorname{\mathcal{S}}$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
Proof. Let ${\bf 1}$ 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 7.2.3.14). 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:
The covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a weak homotopy equivalence.
The equivalence $(1') \Leftrightarrow (2)$ follows by combining Proposition 7.4.5.1, Theorem 7.4.3.6, and Proposition 6.3.1.20. $\square$
We conclude this section with an application of Corollary 7.4.5.10.
Proposition 7.4.5.13. 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 is a limit diagram in the $\infty $-category of spaces; here $h^{X}$ denotes the functor corepresented by $X$ (Notation 5.7.6.14).
\[ K^{\triangleleft } \xrightarrow { F } \operatorname{\mathcal{C}}\xrightarrow {h^{X} } \operatorname{\mathcal{S}} \]
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
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
The identity morphism $\operatorname{id}_{X}$ can be viewed as an initial object of $\operatorname{\mathcal{E}}$ satisfying $U( \operatorname{id}_{X} ) = X$ (Proposition 4.6.6.23), so the corepresentable functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a covariant transport representation for $U$ (Proposition 5.7.6.21). Applying Corollary 7.4.5.10, 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 7.4.5.14. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, the functors preserve $K$-indexed limits, for every small simplicial set $K$.
\[ h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}} \]
Remark 7.4.5.15. 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 7.4.5.13 and Corollary 7.4.5.14, we can replace $\operatorname{\mathcal{S}}$ by the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ of $\lambda $-small spaces (see Variant 7.4.5.8).