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6.2.3 Correspondences

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories. To every morphism $e: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, Proposition 5.2.2.8 supplies a covariant transport functor

$e_{!}: \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ D \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}_{D},$

which is well-defined up to isomorphism. Our goal in this section is to show that $U$ is a cartesian fibration if and only if each of the functors $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ admits a right adjoint (Proposition 6.2.3.5). Moreover, if this condition is satisfied, then the right adjoint to $e_{!}$ is given by the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ of Proposition 5.2.2.17. We begin by analyzing the special case $\operatorname{\mathcal{C}}= \Delta ^1$.

Lemma 6.2.3.1. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category equipped with a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. Then:

• The morphism $f$ exhibits $X$ as a $\operatorname{\mathcal{E}}_0$-coreflection of $Y$ (in the sense of Definition 6.2.2.1) if and only if $X$ belongs to $\operatorname{\mathcal{E}}_0$ and $f$ is $\pi$-cartesian.

• The morphism $f$ exhibits $Y$ as a $\operatorname{\mathcal{E}}_1$-reflection of $X$ if and only if $Y$ belongs to $\operatorname{\mathcal{E}}_1$ and $f$ is $\pi$-cocartesian.

Proof. This is a special case of Corollary 5.1.2.3. $\square$

Corollary 6.2.3.2. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category equipped with a functor $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$. Then:

• The functor $U$ is a cartesian fibration if and only if the full subcategory $\{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{E}}$ is coreflective.

• The functor $U$ is a cocartesian fibration if and only if the full subcategory $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{E}}$ is reflective.

Remark 6.2.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be a functor of $\infty$-categories having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Suppose that $U$ is a cocartesian fibration, so that the full subcategory $\operatorname{\mathcal{E}}_1 \subseteq \operatorname{\mathcal{E}}$ is reflective (Corollary 6.2.3.2). By virtue of Lemma 6.2.2.14, there exists a $\operatorname{\mathcal{E}}_1$-reflection functor $L: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_1$. Then the restriction $L|_{\operatorname{\mathcal{E}}_0}: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ is given by covariant transport along the unique nondegenerate edge $e$ of $\Delta ^1$ (in the sense of Definition 5.2.2.4). More precisely, if $\eta : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow L$ is a natural transformation which exhibits $L$ as a $\operatorname{\mathcal{E}}_1$-reflection functor, then $\eta$ carries each object $X \in \operatorname{\mathcal{E}}$ to a $U$-cocartesian morphism $\eta _ X: X \rightarrow L(X)$, so that $\eta$ restricts to a natural transformation $\operatorname{id}_{\operatorname{\mathcal{E}}_0} \rightarrow L|_{\operatorname{\mathcal{E}}_0}$ which witnesses $L|_{\operatorname{\mathcal{E}}_0}$ as given by covariant transport along $e$.

Similarly, if $U$ is a cartesian fibration, then the full subcategory $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ is coreflective; if $L': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_0$ is a $\operatorname{\mathcal{E}}_0$-coreflection functor, then the restriction $L'|_{\operatorname{\mathcal{E}}_1}: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$ is given by contravariant transport along $e$, in the sense of Definition 5.2.2.15.

Proposition 6.2.3.4. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category equipped with a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_1 = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Let $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ be a functor given by covariant transport along the nondegenerate edge $e$ of $\Delta ^1$. Then the functor $F$ admits a right adjoint if and only if $U$ is a cartesian fibration. In this case, the right adjoint to $F$ is given by contravariant transport along $e$.

Proof. Let $\iota _0: \operatorname{\mathcal{E}}_0 \hookrightarrow \operatorname{\mathcal{E}}$ and $\iota _1: \operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{E}}$ denote the inclusion maps. Since $U$ is a cocartesian fibration, $\operatorname{\mathcal{E}}_1$ is a reflective subcategory of $\operatorname{\mathcal{E}}$ (Corollary 6.2.3.2). Let $L: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_1$ be a $\operatorname{\mathcal{E}}_1$-reflection functor (Lemma 6.2.2.14). Without loss of generality, we may assume that the functor $F: \operatorname{\mathcal{E}}_{0} \rightarrow \operatorname{\mathcal{E}}_1$ factors as a composition $\operatorname{\mathcal{E}}_0 \xrightarrow {\iota _0} \operatorname{\mathcal{E}}\xrightarrow {L} \operatorname{\mathcal{E}}_1$ (Remark 6.2.3.3). Note that $L$ is a left adjoint to the inclusion $\iota _1: \operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{E}}$ (Proposition 6.2.2.15).

Suppose that $U$ is also a cartesian fibration, so that the subcategory $\operatorname{\mathcal{E}}_0 \subseteq \operatorname{\mathcal{E}}$ is coreflective (Corollary 6.2.3.2). Let $L': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}_0$ be a $\operatorname{\mathcal{E}}_0$-coreflection functor (Corollary 6.2.3.2), so that $L'$ can be regarded as a right adjoint to $\iota _0$ (Proposition 6.2.2.15). Invoking Remark 6.2.1.8, we conclude that the composite functor $F = L \circ \iota _0$ has a right adjoint $G$, given by the composition $L' \circ \iota _1 = L'|_{\operatorname{\mathcal{E}}_1}$. Moreover, Remark 6.2.3.3 guarantees that $G: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$ is given by contravariant transport along $e$.

We now prove the converse. Suppose that the functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ admits a right adjoint $G: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}_0$. Fix an object $Z \in \operatorname{\mathcal{E}}_1$; we wish to show that there exists an object $Y \in \operatorname{\mathcal{E}}_0$ and a $U$-cartesian morphism $f: Y \rightarrow Z$. Let $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{E}}_1}$ be the counit of an adjunction between $F$ and $G$. Set $Y = G(Z)$, so that $\epsilon$ determines a morphism $\epsilon _{Z}: F(Y) \rightarrow Z$ in the $\infty$-category $\operatorname{\mathcal{E}}_1$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow L$ be a natural transformation which exhibits $L$ as a $\operatorname{\mathcal{E}}_1$-reflection functor, so that $\eta$ determines a morphism $\eta _{Y}: Y \rightarrow F(Y)$. Let $f: Y \rightarrow Z$ be a composition of $\eta _{Y}$ with $\epsilon _{Z}$. We will complete the proof by showing that $f$ is $U$-cartesian. To prove this, it will suffice to show that for every object $X \in \operatorname{\mathcal{E}}_0$, the composite map

$\operatorname{Hom}_{\operatorname{\mathcal{E}}_0}( X, Y) \xrightarrow { [\eta _ Y] \circ } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, F(Y) ) = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( X, (F \circ G)(Z)) \xrightarrow { [\epsilon _ Z] \circ } \operatorname{Map}_{\operatorname{\mathcal{E}}}( X, Z)$

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Corollary 5.1.2.3). Unwinding the definitions, we see that this map factors as a composition

$\operatorname{Hom}_{\operatorname{\mathcal{E}}_0}( X, G(Z) ) \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{E}}_1}( F(X), (F \circ G)(Z) ) \xrightarrow { [ \epsilon _ Z ] \circ } \operatorname{Hom}_{\operatorname{\mathcal{E}}_1}( F(X), Z ) \xrightarrow { \circ [\eta _ X] } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z),$

where the composition of the first two maps is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$ because $\epsilon$ is the counit of an adjunction (see Proposition 6.2.1.17), and third is an isomorphism because $\eta _ X$ exhibits $F(X)$ as a $\operatorname{\mathcal{E}}_1$-reflection of $X$. $\square$

Proposition 6.2.3.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ is a cartesian fibration of simplicial sets.

$(2)$

For every edge $e: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ of Notation 5.2.2.9 admits a right adjoint.

Moreover, if these conditions are satisfied and $e: C \rightarrow D$ is an edge of $\operatorname{\mathcal{C}}$, then the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ of Notation 5.2.2.18 is right adjoint to $e_{!}$.

Proof. Assume first that condition $(1)$ is satisfied and let $e: C \rightarrow D$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$, which we identify with a morphism $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Applying Proposition 6.2.3.4 to the projection map $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, we deduce that the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is right adjoint to the contravariant transport functor $e^{\ast } \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$, which proves $(2)$.

We now show that $(2)$ implies $(1)$. By virtue of Proposition 5.1.4.7, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. For $0 \leq i \leq n$, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $\{ i\} \times _{\Delta ^ n} \operatorname{\mathcal{E}}$, which we regard as a full subcategory of $\operatorname{\mathcal{E}}$. We wish to show that, for every pair of integers $0 \leq j < k \leq n$ and every object $Z \in \operatorname{\mathcal{E}}_{k}$, there exists an object $Y \in \operatorname{\mathcal{E}}_{j}$ and a $U$-cartesian morphism $g: Y \rightarrow Z$ in $\operatorname{\mathcal{E}}$. Proposition 6.2.3.4 implies that the projection map $\operatorname{N}_{\bullet }( \{ j < k \} ) \times _{\Delta ^ n} \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \{ j < k \} )$ is a cartesian fibration, so we can choose an object $Y \in \operatorname{\mathcal{E}}_{j}$ and a morphism $g: Y \rightarrow Z$ which is locally $U$-cartesian. We will complete the proof by showing that $g$ is $U$-cartesian. To prove this, we must show that for each integer $0 \leq i \leq j$ and each object $W \in \operatorname{\mathcal{E}}_{i}$, composition with the homotopy class $[g]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{E}}}( W, Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,Z)$ in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (see Corollary 5.1.2.3). Since $U$ is a cocartesian fibration, we can choose a $U$-cocartesian morphism $f: W \rightarrow X$, where $X$ belongs to $\operatorname{\mathcal{E}}_{j}$. We conclude by observing that there is a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X, Y) \ar [r]^-{[g] \circ }_{\sim } \ar [d]^{ \circ [f]}_{\sim }& \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z) \ar [d]^{\circ [f]}_{\sim } \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( W, Y) \ar [r]^-{ [g] \circ } & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W, Z) }$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the upper horizontal map is an isomorphism by virtue of our assumption that $g$ is locally $U$-cartesian, and the vertical maps are isomorphisms by virtue of our assumption that $f$ is $U$-cocartesian (Corollary 5.1.2.3). $\square$