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7.3.6 Limit and Colimits of Spaces

Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.4.1.1), which we regard as a full subcategory of the $\infty $-category $\operatorname{\mathcal{QC}}$ (Remark 5.4.4.8). Our goal in this section is to describe limits and colimits in the $\infty $-category $\operatorname{\mathcal{S}}$. Given the results of §7.3.1 and §7.3.4, this is a relatively formal exercise. We begin with an elementary observation:

Proposition 7.3.6.1. 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 7.1.5.6). 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 7.1.5.20, since the functor $\iota $ admits both left and right adjoints (Example 6.2.2.12). $\square$

Corollary 7.3.6.2. 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 7.3.1.18, 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 7.3.6.1). $\square$

Corollary 7.3.6.3. 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 7.3.6.2 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.5.4.9. $\square$

Corollary 7.3.6.4. 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 7.3.4.11, 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 6.3.1.19, 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 7.3.6.1). $\square$

Corollary 7.3.6.5. 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 7.3.6.4 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.5.4.9. $\square$

Corollary 7.3.6.6. The $\infty $-category $\operatorname{\mathcal{S}}$ admits small limits and colimits.

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 7.3.6.7. 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 7.3.1.8 with Proposition 7.3.6.1. $\square$

Corollary 7.3.6.8. 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 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 7.2.3.14). 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 7.3.4.3), 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 $(1') \Leftrightarrow (2)$ follows by combining Proposition 7.3.6.1, Theorem 7.3.4.6, and Proposition 6.3.1.19. $\square$