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7.3 Limits and Colimits of $\infty $-Categories

Recall that the collection of (small) $\infty $-categories can be organized into a (large) $\infty $-category $\operatorname{\mathcal{QC}}$ (see Construction 5.4.4.1). Our goal in this section is to study limits and colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$. Fix a small $\infty $-category $\operatorname{\mathcal{C}}$, and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. We will show that the diagram $\mathscr {F}$ admits both a limit $\varprojlim (\mathscr {F})$ and a colimit $\varinjlim (\mathscr {F})$, which can be described explicitly in terms of the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ introduced in Definition 5.5.4.1:

$(1)$

Let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor, and 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 functors $F: \operatorname{\mathcal{C}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which satisfy $U \circ F = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. In §7.3.1, 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 7.3.1.19).

$(2)$

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 §7.3.4, we show that $(\int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$ is a colimit of the diagram $\mathscr {F}$ (Corollary 7.3.4.12).

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 §7.3.1, 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 7.3.1.13). 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 §7.3.4, 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 7.3.4.1). 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 $U$ (see Theorem 7.3.1.8 and Remark 7.3.1.15).

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.

We will establish the diffraction and refraction criteria in §7.3.3 and §7.3.4, respectively. In the former case, our argument will make use of a general stability property of (co)cartesian fibrations with respect to direct images (Proposition 7.3.2.20), which we formulate and prove in §7.3.2. In §7.3.6, we restrict our attention to the special case where $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration, and apply the results described above to describe limits and colimits in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces.

Remark 7.3.0.1. 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 $\operatorname{\mathcal{C}}$.

Remark 7.3.0.2. 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 7.3.1.13). 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 7.3.4.5).

Structure

  • Subsection 7.3.1: Limits of $\infty $-Categories
  • Subsection 7.3.2: Digression: Direct Image Fibrations
  • Subsection 7.3.3: Proof of the Diffraction Criterion
  • Subsection 7.3.4: Colimits of $\infty $-Categories
  • Subsection 7.3.5: Proof of the Refraction Criterion
  • Subsection 7.3.6: Limit and Colimits of Spaces