Remark 7.4.0.1. In the outline above, we have implicitly suggested that $\operatorname{\mathcal{C}}$ is an $\infty $-category. This is not important: all of the results of this section can be applied to diagrams $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ indexed by an arbitrary (small) simplicial set $\operatorname{\mathcal{C}}$.
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7.4 Limits and Colimits of Spaces
Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.5.1.1). In the first part of this section, we show that every small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits both a limit $\varprojlim (\mathscr {F} )$ colimit $\varinjlim (\mathscr {F} )$, both of which can be described explicitly in terms of the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ introduced in Definition 5.6.2.1:
- $(1)$
Let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the forgetful functor. Since $U$ is a left fibration (Example 5.6.2.9), the collection of sections of $U$ can be organized into a Kan complex $Y = \operatorname{Hom}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$. In §7.4.1, we show that $Y$ is a limit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.4.1.7).
- $(2)$
Let $X$ be a fibrant replacement for the $\infty $-category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$: that is, a Kan complex equipped with a weak homotopy equivalence $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow X$. In §7.4.3, we show that $X$ is a colimit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.4.3.3).
Recall that the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ can be regarded as a full subcategory of the $\infty $-category $\operatorname{\mathcal{QC}}$ of small $\infty $-categories (Remark 5.5.4.8). In the second half of this section, we extend the preceding results to the situation where $\mathscr {F}$ is a diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$. In this case, the forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration, which is generally not a left fibration. However, we have the following variants of $(1)$ and $(2)$:
- $(1^{+})$
Let $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ denote the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ spanned by those sections of $U$ which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. In §7.4.4, we show that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ is a limit of the diagram $\mathscr {F}$ (Corollary 7.4.4.2).
- $(2^{+})$
Let $W$ be the collection of all $U$-cocartesian morphisms of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, and let $( \int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$ denote a localization of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ with respect to $W$ (Definition 6.3.1.9). In §7.4.5, we show that $(\int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$ is a colimit of the diagram $\mathscr {F}$ (Corollary 7.4.5.2).
For many applications, it is not enough to describe the limit $\varprojlim (\mathscr {F})$ and colimit $\varinjlim (\mathscr {F})$ as abstract $\infty $-categories: we also need to understand their relationship to the diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. In other words, we would like to have criteria which can be used to detect when an extension $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is a limit diagram, and when an extension $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ is a colimit diagram. To formulate these criteria, it will be convenient to slightly shift our perspective. Fix a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having covariant transport representation $\mathscr {F}$ (that is, a cocartesian fibration which is equivalent to the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$).
Suppose $U$ is obtained as the pullback of a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$, and let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ denote the fiber of $\overline{U}$ over the cone point ${\bf 0} \in \operatorname{\mathcal{C}}^{\triangleleft }$. In §7.4.4, we introduce a map
\[ \mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ \bf 0} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}), \]which we will refer to as the covariant diffraction functor (Construction 7.4.4.9). Roughly speaking, it is characterized by the requirement that for every object $X \in \overline{\operatorname{\mathcal{E}}}_{\bf 0}$ and every object $C \in \operatorname{\mathcal{C}}$, there is a $\overline{U}$-cocartesian morphism $X \rightarrow \mathrm{Df}(X)(C)$ (depending functorially on $X$ and $C$).
Suppose $U$ is obtained as the pullback of a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$, and let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ denote the fiber of $\overline{U}$ over the cone point ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$. In §7.4.5, we introduce a map
\[ \mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{\bf 1}, \]which we will refer to as the covariant refraction functor (Definition 7.4.5.6). Roughly speaking, it is characterized by the requirement that for every object $X \in \operatorname{\mathcal{E}}$, there is a $\overline{U}$-cocartesian morphism $X \rightarrow \mathrm{Rf}(X)$ (depending functorially on $X$).
We will deduce $(1^{+})$ and $(2^{+})$ from the following more precise assertions:
- Diffraction Criterion:
Suppose we are given a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]where $U$ and $\overline{U}$ are cocartesian fibrations. Then the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{E}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is a limit diagram (in the $\infty $-category $\operatorname{\mathcal{QC}}$) if and only if the covariant diffraction functor $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ \bf 0} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a fully faithful embedding, whose essential image is the the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian sections of $U$ (Theorem 7.4.4.6 and Remark 7.4.4.11).
- Refraction Criterion:
Suppose we are given a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]where $U$ and $\overline{U}$ are cocartesian fibrations. Then the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{E}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ is a colimit diagram (in the $\infty $-category $\operatorname{\mathcal{QC}}$) if and only if the covariant refraction functor $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ \bf 1}$ exhibits $\overline{\operatorname{\mathcal{E}}}_{ \bf 1}$ as a localization of $\operatorname{\mathcal{E}}$ with respect to the collection of $U$-cocartesian morphisms (Theorem 7.4.5.11).
Remark 7.4.0.2. For any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$, the associated covariant diffraction functor $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ \bf 0} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ automatically factors through the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (see Construction 7.4.4.9). Similarly, for any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$, the covariant refraction functor $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{\bf 1}$ automatically carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to isomorphisms in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{\bf 1}$ (Remark 7.4.5.10).