9.5.3 The $\infty $-Category of Presentable $\infty $-Categories
Let $\operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ denote the $\infty $-category whose objects are accessible $\infty $-categories and whose morphisms are accessible functors (Notation 9.4.7.11).
Construction 9.5.3.1. We define a (non-full) subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ as follows:
An accessible $\infty $-category $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ if and only if $\operatorname{\mathcal{C}}$ is is presentable: that is, it admits small colimits.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then an accessible functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ if and only if $F$ preserves small colimits.
Variant 9.5.3.2. We define a (non-full) subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ as follows:
An accessible $\infty $-category $\operatorname{\mathcal{C}}\in \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ belongs to the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ if and only if it is presentable: that is, it admits small limits.
Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Then an accessible functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ if and only if $F$ preserves small limits.
Warning 9.5.3.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Every cocontinuous functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is automatically accessible, and can therefore be regarded as a morphism in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. However, a continuous functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ need not be accessible (see Warning 9.5.1.10), and therefore need not be a morphism in $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$.
It follows from Proposition 8.6.8.22 that the $\infty $-categories $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ and $\operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ are anti-equivalent (that is, each is equivalent to the opposite of the other). To formulate a more precise statement, it will be convenient to introduce some terminology.
Definition 9.5.3.5 (Presentable Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is a presentable fibration if it satisfies the following conditions:
- $(a)$
The morphism $U$ is a locally cocartesian fibration of simplicial sets.
- $(b)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a presentable $\infty $-category.
- $(c)$
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is cocontinuous.
Proposition 9.5.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a presentable $\infty $-category. The following conditions are equivalent:
- $(1)$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a presentable fibration (Definition 9.5.3.5). That is, it is a locally cocartesian fibration having the property that, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ is cocontinuous.
- $(2)$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is accessible and continuous.
Proof.
Using Theorem 9.5.2.1, we obtain the following reformulations of conditions $(1)$ and $(2)$:
- $(1')$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$ admits a right adjoint.
- $(2')$
The morphism $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration. Moreover, for each edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ admits a left adjoint.
The equivalence of $(1')$ and $(2')$ is a special case of Corollary 6.2.5.5.
$\square$
Corollary 9.5.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $U$ is a presentable cocartesian fibration.
- $(2)$
The morphism $U$ is a cocartesian fibration and the covariant transport representation of $U$ takes values in the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ of Construction 9.5.3.1.
- $(3)$
The morphism $U$ is a presentable cartesian fibration.
- $(4)$
The morphism $U$ is a cartesian fibration and the contravariant transport representation of $U$ factors through the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ of Variant 9.5.3.2.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ follows immediately from the definition of presentable fibration and the equivalence $(3) \Leftrightarrow (4)$ follows from the reformulation provided by Proposition 9.5.3.7. Since every presentable fibration is both locally cartesian and locally cocartesian (Proposition 9.5.3.7), the equivalence $(1) \Leftrightarrow (3)$ is a special case of Proposition 6.2.5.7.
$\square$
Corollary 9.5.3.9. There is an equivalence of $\infty $-categories
\[ \Psi : (\operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{\operatorname{RPr}}, \]
which is characterized up to isomorphism by the following requirement:
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable cocartesian fibration with covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. Then the composition
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}} } ( \operatorname{\mathcal{QC}}^{\operatorname{LPr}} )^{\operatorname{op}} \xrightarrow {\Psi } \operatorname{\mathcal{QC}}^{\operatorname{RPr}} \]
is a contravariant transport representation of $U$ (regarded as a cartesian fibration).
Proof.
Combine Corollary 9.5.3.8 with Proposition 8.6.8.22.
$\square$
More precisely, this description characterizes the equivalence of homotopy categories induced by the functor $\Psi $ (see Remark 8.6.8.20).