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 this definition appears as HTT, def. 5.2.2.7.

### 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.

## Properties

The two different definition above are indeed equivalent:

###### Proposition

For $C$ and $D$ quasi-categories, the two definitions of adjunction, in terms of Hom-equivalence induced by unit maps and in terms of Cartesian/coCartesian fibrations are equivalent.

###### Proof

This is HTT, prop 5.2.2.8.

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.

### 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.

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 $(L \dashv R) : C \to D$

### On over-$(\infty,1)$-categories

###### Proposition

Let

$(L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} C$

be a pair of adjoint $(\infty,1)$-functors where the $(\infty,1)$-category $C$ has all (∞,1)-pullbacks.

Then for every object $X \in C$ there is induced a pair of adjoint $(\infty,1)$-functors between the over-(∞,1)-categories

$(L/X \dashv R/X) : D/(L X) \stackrel{\overset{L/X}{\leftarrow}}{\underset{R/X}{\to}} C/X$

where

• $L/X$ is the evident induced functor;

• $R/X$ is the composite

$R/X : D/{L X} \stackrel{R}{\to} C/{(R L X)} \stackrel{i_{X}^*}{\to} C/X$

of the evident functor induced by $R$ with the (∞,1)-pullback along the $(L \dashv R)$-unit at $X$.

This is HTT, prop. 5.2.5.1.

### 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.

## 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

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

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

Section 5.2 in

A study of adjoint functors between quasi-categories is given in

and further discussion, including also that of (infinity,1)-monads is in

A textbook development in the context of infinity-cosmoi? can be found in

The proof that a Quillen adjunction of model categories induces an adjunction of (∞,1)-categories is recorded in

and also in