Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

The notion of adjunction between two (∞,1)-functors generalizes the notion of adjoint functors from category theory to (∞,1)-category theory.

There are many equivalent definitions of the ordinary notion of adjoint functor. Some of them have more evident generalizations to some parts of higher category theory than others.

• One definition of ordinary adjoint functors says that a pair of functors $C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ is an adjunction if there is a natural isomorphism

$Hom_C(L(-),(-) \simeq Hom_D(-,R(-)) \,.$

The analog of this definition makes sense very generally in (∞,1)-category theory, where $Hom_C(-,-) : C^{op} \times C \to \infty Grpd$ is the $(\infty,1)$-categorical hom-object.

• One other characterization of adjoint functors in terms of their cographs/heteromorphisms: the Cartesian fibrations to which the functor is associated. At cograph of a functor it is discussed how two functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if the cograph of $L$ coincides with the cograph of $R$ up to the obvious reversal of arrows

$(L \dashv R) \Leftrightarrow (cograph(L) \simeq cograph(R^{op})^{op}) \,.$

Using the (∞,1)-Grothendieck construction the notion of cograph of a functor has an evident generalization to $(\infty,1)$-categories.

## Definition

### In terms of hom-equivalences

###### Definition

(in terms of hom equivalence induced by unit map)

A pair of (∞,1)-functors

$C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$

is an adjunction, if there exists a unit transformation $\epsilon : Id_D \to R \circ L$ – a morphism in the (∞,1)-category of (∞,1)-functors $Func(D,D)$ – such that for all $d \in D$ and $c \in C$ the induced morphism

$Hom_C(L(d),c) \stackrel{R_{L(d), c}}{\to} Hom_D(R(L(d)), R(c)) \stackrel{Hom_D(\epsilon, R(c))}{\to} Hom_D(d,R(c))$

In terms of the concrete incarnation of the notion of $(\infty,1)$-category by the notion of quasi-category, we have that $Hom_C(L(d),c)$ and $Hom_D(d,R(c))$ are incarnated as hom-objects in quasi-categories, which are Kan complexes, and the above equivalence is a homotopy equivalence of Kan complexes.

In this form is due to Lurie 09, Def. 5.2.2.7.

Streamlined discussion is in Riehl & Verity 15, 4.4.2-4.4.4 and Riehl & Verity 20, 3.3.3-3.5.1 and Riehl & Verity “Elements”, Prop. 4.1.1.

### In terms of cographs/heteromorphisms

We discuss here the quasi-category theoretic analog of Adjoint functors in terms of cographs (heteromorphisms).

We make use here of the explicit realization of the (∞,1)-Grothendieck construction in its incarnation for quasi-categories: here an (∞,1)-functors $L : D \to C$ may be regarded as a map $\Delta^{op} \to$ (∞,1)Cat, which corresponds under the Grothendieck construction to a Cartesian fibration of simplicial sets $coGraph(L) \to \Delta$.

###### Definition

(in terms of Cartesian/coCartesian fibrations)

Let $C$ and $D$ be quasi-categories. An adjunction between $C$ and $D$ is

• a morphism $K \to \Delta$ of simplicial sets, which is both a Cartesian fibration as well as a coCartesian fibration.

• together with equivalence of quasi-categories $C \stackrel{\simeq}{\to} K_{\{0\}}$ and $D \stackrel{\simeq}{\to} K_{\{1\}}$.

Two (∞,1)-functors $L : C \to D$ and $R : D \to C$ are called adjoint – with $L$ left adjoint to $R$ and $R$ right adjoint to $L$ if

• there exists an adjunction $K \to I$ in the above sense

• and $L$ and $K$ are the associated functors to the Cartesian fibation $p \colon K \to \Delta$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta^{op}$, respectively.

### In the homotopy 2-category

###### Definition

(in terms of the homotopy 2-category)

Say that a 2-categorical pair of adjoint (∞,1)-functors is an adjunction in the homotopy 2-category of (∞,1)-categories.

This concept, in the spirit of formal $\infty$-category theory, was mentioned, briefly, in Joyal 2008, p. 159 (11 of 348) and then expanded on in Riehl-Verity 15, Def. 4.0.1.

Such a 2-categorical adjunctions (Def. ) determines an adjoint pair of $\infty$-functors in the sense of Lurie 2009 (Riehl-Verity 15, Rem. 4.4.5):

###### Proposition

An anti-parallal pair of morphisms in $Cat_\infty$ is a pair of adjoint $\infty$-functors in the sense of Lurie 2009, Sec. 5.2 if and only its image in the homotopy 2-category $Ho_2\big(Cat_\infty\big)$ forms an adjunction in the classical sense of 2-category theory (Def. ).

(Riehl & Verity 2022, Sec. F.5, Prop. F.5.6)

The conceptual content of Prop. may be made manifest as follows:

###### Proposition

Every 2-categorical pair of adjoint $(\infty,1)$-functors in the sense of Def. extends to a “homotopy coherent adjunction” in an essentially unique way.

## Properties

###### Proposition

For $C$ and $D$ quasi-categories, the two definitions of adjunction,

1. in terms of Hom-equivalence induced by unit maps (Def. )

2. in terms of Cartesian/coCartesian fibrations (Def. )

are equivalent.

This is HTT, prop 5.2.2.8.

###### Proof

First we discuss how to produce the unit for an adjunction from the data of a correspondence $K \to \Delta$ that encodes an $\infty$-adjunction $(f \dashv g)$.

For that, define a morphism $F' : \Lambda_2 \times C \to K$ as follows:

• on $\{0,2\}$ it is the morphism $F : C \times \Delta \to K$ that exhibits $f$ as associated to $K$, being $Id_C$ on $C \times \{0\}$ and $f$ on $C \times \{2\}$;

• on $\{1,2\}$ it is the morphism $C \times \Delta \stackrel{f \times Id}{\to} D \times \Delta \stackrel{G}{\to} K$, where $G$ is the morphism that exhibits $g$ as associated to $K$;

Now observe that $F'$ in particular sends $\{1,2\}$ to Cartesian morphisms in $K$ (by definition of functor associated to $K$). By one of the equivalent characterizations of Cartesian morphisms, this means that the lift in the diagram

$\array{ \Lambda_2 &\stackrel{F'}{\to}& K \\ \downarrow &{}^{F''}\nearrow& \downarrow \\ \Delta \times C &\to & \Delta }$

exists. This defines a morphism $C \times \{0,1\} \to K$ whose components may be regarded as forming a natural transformation $u : d_C \to g \circ f$.

To show that this is indeed a unit transformation, we need to show that the maps of hom-object in a quasi-category for all $c \in C$ and $d \in D$

$Hom_D(f(f), d) \to Hom_C(g(f(c)), g(d)) \to Hom_C(c, g(d))$

is an equivalence, hence an isomorphism in the homotopy category. Once checks that this fits into a commuting diagram

$\array{ Hom_D(f(c), d) &\to& Hom_C(g(f(c)), g(d)) &\to& Hom_C(c, g(d)) \\ \downarrow &&&& \downarrow \\ Hom_K(C,D) &&=&& Hom_K(C,D) } \,.$

For illustration, chasing a morphism $f(c) \to d$ through this diagram yields

$\array{ (f(c) \to d) &\mapsto& (g(f(c)) \to g(d)) &\mapsto& (c \to g(f(c)) \to g(d)) \\ \downarrow && && \downarrow \\ (c \to g(f(c)) \to f(c) \to d) &&=&& (c \to g(f(c)) \to g(d) \to d) } \,,$

where on the left we precomposed with the Cartesian morphism

$\array{ && g(f(c)) \\ & \nearrow &\Downarrow^{\simeq}& \searrow \\ c &&\to&& f(c) }$

given by $F''|_{c} : \Delta \to K$, by …

The adjoint of a functor is, if it exists, essentially unique:

###### Proposition

If the $(\infty,1)$-functor between quasi-categories $L : D \to C$ admits a right adjoint $R : C \to D$, then this is unique up to homotopy.

Moreover, even the choice of homotopy is unique, up to ever higher homotopy, i.e. the collection of all right adjoints to $L$ forms a contractible ∞-groupoid, in the following sense:

Let $Func^L(C,D), Func^R(C,D) \subset Func(C,D)$ be the full sub-quasi-categories on the (∞,1)-category of (∞,1)-functors between $C$ and $D$ on those functors that are left adjoint and those that are right adjoints, respectively. Then there is a canonical equivalence of quasi-categories

$Func^L(C,D) \stackrel{\simeq}{\to} Func^R(D,C)^{op}$

(to the opposite quasi-category), which takes every left adjoint functor to a corresponding right adjoint.

###### Proof

This is HTT, prop 5.2.1.3 (also remark 5.2.2.2), and HTT, prop. 5.2.6.2.

The idea is to construct the category of right adjoints as an intersection of full subcategories

$\array{ Func^R(C,D) &\to& C^D \\ \downarrow & & \downarrow \\ (D^C)^{op} &\to& \infty Gpd^{C^{op} \times D} }$

where the inclusions are given by the yoneda embedding. An element of $Func^R(C,D)$ corresponds to a functor $p : C^{op} \times D \to \infty Gpd$ for which there exists a pair of functors $g : D \to C$ and $f : C \to D$ such that $p \simeq D(f-,-) \simeq C(-,g-)$.

### Uniqueness of unit and counit

Given functors $f : C \to D$ and $g : D \to C$, we can use the (∞,1)-end to determine compute a chain of equivalences

\begin{aligned} C^C(id, gf) &\simeq \int_{c \in C} C(c, gf(c)) \\ &\simeq \int_{c \in C} \infty Gpd^D(D(f(c), -), C(c, g-)) \\ &\simeq Gpd^{C^{\op} \times D}(D(f-, -), C(-, g-)) \end{aligned}

dually, we can identify the space of counits as

$D^D(fg, id) \simeq Gpd^{C^{\op} \times D}(C(-, g-), D(f-, -))$

So each half of the equivalence $D(f-,-) \simeq C(-,g-)$ corresponds essentially uniquely to a choice of unit and counit transformation.

### Preservation of limits and colimits

Recall that for $(L \dashv R)$ an ordinary pair of adjoint functors, the fact that $L$ preserves colimits (and that $R$ preserves limits) is a formal consequence of

1. the hom-isomorphism $Hom_C(L(-),-) \simeq Hom_D(-,R(-))$;

2. the fact that $Hom_C(-,-) : C^{op} \times C \to Set$ preserves all limits in both arguments;

3. the Yoneda lemma, which says that two objects are isomorphic if all homs out of (into them) are.

Using this one computes for all $c \in C$ and diagram $d : I \to D$

\begin{aligned} Hom_C(L(\lim_{\to} d_i), c) & \simeq Hom_D(\lim_\to d_i, R(c)) \\ & \simeq \lim_{\leftarrow} Hom_D(d_i, R(c)) \\ & \simeq \lim_{\leftarrow} Hom_C(L(d_i), c) \\ & \simeq Hom_C(\lim_{\to} L(d_i), c) \,, \end{aligned}

which implies that $L(\lim_\to d_i) \simeq \lim_\to L(d_i)$.

Now to see this in $(\infty,1)$-category theory (…) HTT Proposition 5.2.3.5

###### Proposition

For $(L \dashv R) : C \stackrel{\leftarrow}{\to} D$ an $(\infty,1)$-adjunction, its image under decategorifying to homotopy categories is a pair of ordinary adjoint functors

$(Ho(L) \dashv Ho(R)) : Ho(C) \stackrel{\leftarrow}{\to} Ho(D) \,.$
###### Proof

This is HTT, prop 5.2.2.9.

This follows from that fact that for $\epsilon : Id_C \to R \circ L$ a unit of the $(\infty,1)$-adjunction, its image $Ho(\epsilon)$ is a unit for an ordinary adjunction.

###### Remark

The converse statement is in general false. A near converse is given by HTT, prop 5.2.2.12 if one instead considers $Ho$-enriched homotopy categories: if $Ho(L)$ has a right adjoint, then so does $L$.

It is important to consider the $Ho$-enriched homotopy category rather than the ordinary one. For a counterexample, when $Ho$ is considered as an ordinary category, $\pi_0 : Ho \to Set$ is both left and right adjoint to the inclusion $Set \subseteq Ho$. However, $\pi_0 : \infty Gpd \to Set$ does not have a left adjoint.

One way to find that an ordinary adjunction of homotopy categories lifts to an $(\infty,1)$-adjunction is to exhibit it as a Quillen adjunction between simplicial model category-structures. This is discussed in the Examples-section Simplicial and derived adjunction below.

As for ordinary adjoint functors we have the following relations between full and faithful adjoints and idempotent monads.

###### Proposition

Given an $(\infty,1)$-adjunction $(L \dashv R) : C \to D$

###### Proposition

Let

$\mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}$

be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).

Then:

1. For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form

(1)$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}$

where:

• $L_{/b}$ is the evident induced functor (applying $L$ to the entire triangle diagrams in $\mathcal{C}$ which represent the morphisms in $\mathcal{C}_{/b}$);

• $R_{/b}$ is the composite

$R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $R$;

2. the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).

2. For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form

(2)$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}$

where:

• $R_{/b}$ is the evident induced functor (applying $R$ to the entire triangle diagrams in $\mathcal{D}$ which represent the morphisms in $\mathcal{D}_{/b}$);

• $L_{/b}$ is the composite

$L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $L$;

2. the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. 5.2.5.1. For discussion in model category theory see at sliced Quillen adjunctions.
###### Proof

Recall that (this Prop.) the hom-isomorphism that defines an adjunction of functors (this Def.) is equivalently given in terms of composition with

• the adjunction unit $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$

• the adjunction counit $\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$

as follows:

Using this, consider the following transformations of morphisms in slice categories, for the first case:

(1a)

(2a)

(2b)

(1b)

Here:

• (1a) and (1b) are equivalent expressions of the same morphism $f$ in $\mathcal{D}_{/L(b)}$, by (at the top of the diagrams) the above expression of adjuncts between $\mathcal{C}$ and $\mathcal{D}$ and (at the bottom) by the triangle identity.

• (2a) and (2b) are equivalent expression of the same morphism $\tilde f$ in $\mathcal{C}_{/b}$, by the universal property of the pullback.

Hence:

• starting with a morphism as in (1a) and transforming it to $(2)$ and then to (1b) is the identity operation;

• starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the first claimed form (1).

The second case follows analogously, but a little more directly since no pullback is involved:

(1a)

(2)

(1b)

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the second claimed form (2).

###### Remark

The sliced adjunction (Prop. ) in the second form (2) is such that the sliced left adjoint sends slicing morphism $\tau$ to their adjuncts $\widetilde{\tau}$, in that (again by this Prop.):

$L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b}$

### In terms of universal arrows

###### Proposition

An $(\infty,1)$-functor $G:D\to C$ admits a left adjoint if and only if for each $X\in C$, the comma (infinity,1)-category? $(X \downarrow G)$ has an initial object, i.e. every object $X\in C$ admits a universal arrow $X\to G F X$ to $G$.

This is stated explicitly as Riehl-Verity, Corollary 16.2.7, and can be extracted with some work from HTT, Proposition 5.2.4.2.

### Preservation by exponentiation

###### Proposition

Let $f : C \to D$ be left adjoint to $g : D \to C$. Then for any $A$, $f^A$ is left adjoint to $g^A$ and $A^g$ is left adjoint to $A^f$.

###### Proof

Let $\eta : id_C \Rightarrow gf$ be a unit transformation. The property of being a unit transformation can be detected at the level of enriched homotopy categories, so $A^\eta: id_{A^C} \Rightarrow A^f A^g$ and $\eta^A : id_{C^A} \Rightarrow g^A f^A$ are also unit transformations.

The functorality of adjunctions can be organized into the existence of two wide subcategories $LAdj \subseteq (\infty,1)Cat$ and $RAdj \subseteq (\infty,1)Cat$ whose functors are the left adjoints and the right adjoints respectively.

We can then define the functor categories

• $Func^L : LAdj^{op} \times LAdj \to (\infty,1)Cat$ is defined by taking $Func^L(C,D) \subseteq Func(C, D)$ to be the full subcategory spanned by $LAdj(C, D)$.

• $Func^R : RAdj^{op} \times RAdj \to (\infty,1)Cat$ is defined by taking $Func^R(C,D) \subseteq Func(C, D)$ to be the full subcategory spanned by $RAdj(C, D)$.

Lurie defines an adjunction to be a functor $X \to $ that is both a cartesian and a cocartesian fibration. We can generalize this to

###### Definition

A functor $p : X \to S$ is an adjunct fibration iff it is both a cartesian fibration and a cocartesian fibration

By the (∞,1)-Grothendieck construction construction, adjunct fibrations over $S$ correspond to category-valued functors on $S$ that send arrows of $S$ to adjoint pairs of categories.

###### Lemma

For a functor $p : X \to S$ of (∞,1)-categories with small fibers.

• If $p$ is a cartesian fibration classified by $\chi : S^\op \to (\infty,1)Cat$, $\chi$ factors through $RAdj$ iff $p$ is an adjunct fibration

• If $p$ is a cocartesian fibration classified by $\chi : S \to (\infty,1)Cat$, $\chi$ factors through $LAdj$ iff $p$ is an adjunct fibration

###### Proof

This is a restatement of HTT, corr. 5.2.2.5.

###### Lemma

There are anti-equivalences $ladj : RAdj^{op} \to LAdj$ and $radj : LAdj^{op} \to RAdj$ that are the identity on objects and the action on homspaces $LAdj(C, D) \simeq RAdj(D,C)$ is the equivalence sending a functor to its adjoint.

###### Proof

By the covariant Grothendieck construction, for any (∞,1)-category C, $Map(C, LAdj)$ can be identified with the core of the category of adjunct fibrations over $C$ with small fibers. The same is true of $Map(C^{\op}, RAdj)$.

Since the Grothendieck construction is natural in the base category, we obtain the asserted equivalence between $LAdj$ and $RAdj^{op}$. Taking $C = $, this establishes the correspondence between an adjunction and its associated adjoint pair of functors.

As discussed at Uniqueness of Adjoints, this anti-equivalence extends to the (∞,2)-enrichment, in the sense they induce anti-equivalences $radj : Func^L(C, D)^{op} \to Func^R(D, C)$ and $ladj : Func^R(C, D)^{op} \to Func^L(D, C)$.

The preservation of adjunctions by products and exponentials implies

###### Lemma

The product and exponential on $(\infty,1)Cat$ restrict to functors

• $- \times - : LAdj \times LAdj \to LAdj$ and $- \times - : RAdj \times RAdj \to RAdj$
• $Func(-,-) : RAdj^{op} \times LAdj \to LAdj$ and $Func(-,-) : LAdj^{op} \times RAdj \to RAdj$

## Examples

A large class of examples of $(\infty,1)$-adjunctions arises from Quillen adjunctions of model categories, or adjunctions in sSet-enriched category theory.

Any Quillen adjunction induces an adjunction of (infinity,1)-categories on the simplicial localizations. See Hinich 14 or Mazel-Gee 15.

We want to produce Cartesian/coCartesian fibration $K \to \Delta$ from a given sSet-enriched adjunction. For that first consider the following characterization

###### Lemma

Let $K$ be a simplicially enriched category whose hom-objects are all Kan complexes, regard the interval category $\Delta := \{0 \to 1\}$ as an $sSet$-category in the obvious way using the embedding $const : Set \hookrightarrow sSet$ and consider an $sSet$-enriched functor $K \to \Delta$. Let $C := K_0$ and $D := K_1$ be the $sSet$-enriched categories that are the fibers of this. Then under the homotopy coherent nerve $N : sSet Cat \to sSet$ the morphism

$N(p) : N(K) \to \Delta$

is a Cartesian fibration precisely if for all objects $d \in D$ there exists a morphism $f : c \to d$ in $K$ such that postcomposition with this morphism

$C(c',f ) : C(c',c) = K(c',c) \to K(c',d)$

is a homotopy equivalence of Kan complexes for all objects $c' \in C'$.

This appears as HTT, prop. 5.2.2.4.

###### Proof

The statement follows from the characterization of Cartesian morphisms under homotopy coherent nerves (HTT, prop. 2.4.1.10), which says that for an $sSet$-enriched functor $p : C \to D$ between Kan-complex enriched categories that is hom-object-wise a Kan fibration, a morphim $f : c' \to c''$ in $C$ is an $N(p)$-Cartesian morphism if for all objects $c \in C$ the diagram

$\array{ C(c,c') &\stackrel{C(c,f)}{\to}& C(c,c'') \\ \downarrow^{\mathrlap{p_{c,c'}}} && \downarrow^{\mathrlap{p_{c,c''}}} \\ D(p(c),p(c')) &\stackrel{D(p(c),p(f))}{\to}& D(p(c), p(c'')) }$

For the case under consideration the functor in question is $p : K \to \Delta$ and the above diagram becomes

$\array{ K(c,c') &\stackrel{K(c,f)}{\to}& K(c,c'') \\ \downarrow && \downarrow \\ * &\to& * } \,.$

This is clearly a homotopy pullback precisely if the top morphism is an equivalence.

Using this, we get the following.

###### Proposition

For $C$ and $D$ sSet-enriched categories whose hom-objects are all Kan complexes, the image

$N(C) \underoverset {\underset{N(R)}{\longleftarrow}} {\overset{N(L)}{\longrightarrow}} {\bot} N(D)$

under the homotopy coherent nerve of an sSet-enriched adjunction between $sSet$-enriched categories

$C \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} D$

Moreover, if $C$ and $D$ are equipped with the structure of a simplicial model category then the quasi-categorically derived functors

$N(C^\circ) \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} N(D^\circ)$

###### Proof

The first part is HTT, cor. 5.2.4.5, the second HTT, prop. 5.2.4.6.

To get the first part, let $K$ be the $sSet$-category which is the join of $C$ and $D$: its set of objects is the disjoint union of the sets of objects of $C$ and $D$, and the hom-objects are

• for $c,c' \in C$: $K(c,c') := C(c,c')$;

• for $d,d' \in D$: $K(d,d') := D(d,d')$;

• for $c \in C$ and $d \in D$: $K(c,d) := C(L(c),d) = D(c,R(d))$;

and

$K(d,c) = \emptyset$

and equipped with the evident composition operation.

Then for every $d \in D$ there is the morphism $Id_{R(d)} \in K(R(d),d)$, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence $N(K) \to \Delta$ is a Cartesian fibration.

By the analogous dual argument, we find that it is also a coCartesian fibration and hence an adjunction.

For the second statement, we need to refine the above argument just slightly to pass to the full $sSet$-subcategories on fibrant cofibrant objects:

let $K$ be as before and let $K^\circ$ be the full $sSet$-subcategory on objects that are fibrant-cofibrant (in $C$ or in $D$, respectively). Then for any fibrant cofibrant $d \in D$, we cannot just use the identity morphism $Id_{R(d)} \in K(R(d),d)$ since the right Quillen functor $R$ is only guaranteed to respect fibrations, not cofibrations, and so $R(d)$ might not be in $K^\circ$. But we can use the small object argument to obtain a functorial cofibrant replacement functor $Q : C \to C$, such that $Q(R(d))$ is cofibrant and there is an acyclic fibration $Q(R(d)) \to R(d)$. Take this to be the morphism in $K(Q(R(d)), d)$ that we pick for a given $d$. Then this does induce a homotopy equivalence

$C(c', Q(R(d))) \to C(c',R(d)) = K(c',d)$

because in an enriched model category the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.

### Localizations

A pair of adjoint $(\infty,1)$-functors $(L \dashv R) : C \stackrel{\leftarrow}{\hookrightarrow} D$ where $R$ is a full and faithful (∞,1)-functor exhibits $C$ as a reflective (∞,1)-subcategory of $D$. This subcategory and the composite $R \circ L : D \to D$ are a localization of $D$.

• adjoint $(\infty,1)$-functor

The suggestion that a pair of adjoint $\infty$-functors should just be an adjunction in the homotopy 2-category of $\infty$-categories was originally stated, briefly, in:

The definition as an isofibration of quasicategories over $\Delta$ is due to:

The original suggestion of Joyal 2008 was then much expanded on (and generalized to ∞-cosmoi), in the spirit of formal $\infty$-category theory:

That the two definitions (of Joyal 2008 and Lurie 2009) are in fact equivalent is first indicated in Riehl-Verity 15, Rem. 4.4.5 and then made fully explicit in:

A proof that a Quillen adjunction of model categories induces an adjunction between (∞,1)-categories (in the sense of Lurie 2009) is recorded in:

and also in