### Context

#### $\left(\infty ,1\right)$-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 $C\stackrel{\stackrel{L}{←}}{\underset{R}{\to }}D$ is an adjunction if there is a natural isomorphism

${\mathrm{Hom}}_{C}\left(L\left(-\right),\left(-\right)\simeq {\mathrm{Hom}}_{D}\left(-,R\left(-\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Hom_C(L(-),(-) \simeq Hom_D(-,R(-)) \,.

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

• One other characterization of adjoint functors in terms of their cographs: 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

$\left(L⊣R\right)⇔\left(\mathrm{cograph}\left(L\right)\simeq \mathrm{cograph}\left({R}^{\mathrm{op}}{\right)}^{\mathrm{op}}\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $\left(\infty ,1\right)$-categories.

## Definition

### In terms of hom-equivalences

###### Definition

(in terms of hom equivalence induced by unit map)

A pair of (∞,1)-functors

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

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

${\mathrm{Hom}}_{C}\left(L\left(c\right),d\right)\stackrel{{R}_{L\left(c\right),d}}{\to }{\mathrm{Hom}}_{D}\left(R\left(L\left(c\right)\right),R\left(d\right)\right)\stackrel{{\mathrm{Hom}}_{D}\left(ϵ,R\left(d\right)\right)}{\to }{\mathrm{Hom}}_{D}\left(c,R\left(d\right)\right)$Hom_C(L(c),d) \stackrel{R_{L(c), d}}{\to} Hom_D(R(L(c)), R(d)) \stackrel{Hom_D(\epsilon, R(d))}{\to} Hom_D(c,R(d))

In terms of the concrete incarnation of the notion of $\left(\infty ,1\right)$-category by the notion of quasi-category, we have that ${\mathrm{Hom}}_{\left(}C\right)\left(L\left(c\right),d\right)$ and ${\mathrm{Hom}}_{D}\left(c,R\left(d\right)\right)$ 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

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 $\Delta \left[1{\right]}^{\mathrm{op}}\to$ (∞,1)Cat, which corresponds under the Grothendieck construction to a Cartesian fibration of simplicial sets $\mathrm{coGraph}\left(L\right)\to \Delta \left[1\right]$.

###### Definition

(in terms of Cartesian/coCartesian fibrations)

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

• a morphism $K\to \Delta \left[1\right]$ of simplicial sets, which is both a Cartesian fibration as well as a coCartesian fibration.

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

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 $p:K\mathrm{to}\Delta \left[1\right]$ and the Cartesian fibration ${p}^{\mathrm{op}}:{K}^{\mathrm{op}}\to \Delta \left[1{\right]}^{\mathrm{op}}$, 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 $K\to \Delta \left[1\right]$ that encodes an $\infty$-adjunction $\left(f⊣g\right)$.

For that, define a morphism $F\prime :\Lambda \left[2{\right]}_{2}×C\to K$ as follows:

• on $\left\{0,2\right\}$ it is the morphism $F:C×\Delta \left[1\right]\to K$ that exhibits $f$ as associated to $K$, being ${\mathrm{Id}}_{C}$ on $C×\left\{0\right\}$ and $f$ on $C×\left\{2\right\}$;

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

Now observe that $F\prime$ in particular sends $\left\{1,2\right\}$ 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

$\begin{array}{ccc}\Lambda \left[2{\right]}_{2}& \stackrel{F\prime }{\to }& K\\ ↓& {}^{F″}↗& ↓\\ \Delta \left[2\right]×C& \to & \Delta \left[1\right]\end{array}$\array{ \Lambda[2]_2 &\stackrel{F'}{\to}& K \\ \downarrow &{}^{F''}\nearrow& \downarrow \\ \Delta[2] \times C &\to & \Delta[1] }

exists. This defines a morphism $C×\left\{0,1\right\}\to K$ 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 $d\in D$

${\mathrm{Hom}}_{D}\left(f\left(f\right),d\right)\to {\mathrm{Hom}}_{C}\left(g\left(f\left(c\right)\right),g\left(d\right)\right)\to {\mathrm{Hom}}_{C}\left(c,g\left(d\right)\right)$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

$\begin{array}{ccccc}{\mathrm{Hom}}_{D}\left(f\left(c\right),d\right)& \to & {\mathrm{Hom}}_{C}\left(g\left(f\left(c\right)\right),g\left(d\right)\right)& \to & {\mathrm{Hom}}_{C}\left(c,g\left(d\right)\right)\\ ↓& & & & ↓\\ {\mathrm{Hom}}_{K}\left(C,D\right)& & =& & {\mathrm{Hom}}_{K}\left(C,D\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $f\left(c\right)\to d$ through this diagram yields

$\begin{array}{ccccc}\left(f\left(c\right)\to d\right)& ↦& \left(g\left(f\left(c\right)\right)\to g\left(d\right)\right)& ↦& \left(c\to g\left(f\left(c\right)\right)\to g\left(d\right)\right)\\ ↓& & & & ↓\\ \left(c\to g\left(f\left(c\right)\right)\to f\left(c\right)\to d\right)& & =& & \left(c\to g\left(f\left(c\right)\right)\to g\left(d\right)\to d\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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

$\begin{array}{ccc}& & g\left(f\left(c\right)\right)\\ & ↗& {⇓}^{\simeq }& ↘\\ c& & \to & & f\left(c\right)\end{array}$\array{ && g(f(c)) \\ & \nearrow &\Downarrow^{\simeq}& \searrow \\ c &&\to&& f(c) }

given by $F″{\mid }_{c}:\Delta \left[2\right]\to K$, by …

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

###### Proposition

If the $\left(\infty ,1\right)$-functor between quasi-categoris $L:D\to C$ 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 ${\mathrm{Func}}^{L}\left(C,D\right),{\mathrm{Func}}^{R}\left(C,D\right)\subset \mathrm{Func}\left(C,D\right)$ 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

${\mathrm{Func}}^{L}\left(C,D\right)\stackrel{\simeq }{\to }{\mathrm{Func}}^{R}\left(C,D{\right)}^{\mathrm{op}}$Func^L(C,D) \stackrel{\simeq}{\to} Func^R(C,D)^{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 $\left(L⊣R\right)$ 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 ${\mathrm{Hom}}_{C}\left(L\left(-\right),-\right)\simeq {\mathrm{Hom}}_{D}\left(-,R\left(-\right)\right)$;

2. the fact that ${\mathrm{Hom}}_{C}\left(-,-\right):{C}^{\mathrm{op}}×C\to \mathrm{Set}$ 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 $d:I\to D$

$\begin{array}{rl}{\mathrm{Hom}}_{C}\left(L\left(\underset{\to }{\mathrm{lim}}{d}_{i}\right),c\right)& \simeq {\mathrm{Hom}}_{D}\left(\underset{\to }{\mathrm{lim}}{d}_{i},R\left(c\right)\right)\\ & \simeq \underset{←}{\mathrm{lim}}{\mathrm{Hom}}_{D}\left({d}_{i},R\left(c\right)\right)\\ & \simeq \underset{←}{\mathrm{lim}}{\mathrm{Hom}}_{C}\left(L\left({d}_{i}\right),c\right)\\ & \simeq {\mathrm{Hom}}_{C}\left(\underset{\to }{\mathrm{lim}}L\left({d}_{i}\right),c\right)\phantom{\rule{thinmathspace}{0ex}},\end{array}$\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 $L\left({\mathrm{lim}}_{\to }{d}_{i}\right)\simeq {\mathrm{lim}}_{\to }L\left({d}_{i}\right)$.

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

###### Proposition

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

$\left(\mathrm{Ho}\left(L\right)⊣\mathrm{Ho}\left(R\right)\right):\mathrm{Ho}\left(C\right)\stackrel{←}{\to }\mathrm{Ho}\left(D\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $ϵ:{\mathrm{Id}}_{C}\to R\circ L$ a unit of the $\left(\infty ,1\right)$-adjunction, its image $\mathrm{Ho}\left(ϵ\right)$ 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 $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-adjunction $\left(L⊣R\right):C\to D$

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

###### Proposition

Let

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

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

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

$\left(L/X⊣R/X\right):D/\left(LX\right)\stackrel{\stackrel{L/X}{←}}{\underset{R/X}{\to }}C/X$(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

$R/X:D/LX\stackrel{R}{\to }C/\left(RLX\right)\stackrel{{i}_{X}^{*}}{\to }C/X$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 $\left(L⊣R\right)$-unit at $X$.

This is HTT, prop. 5.2.5.1.

## Examples

A large class of examples of $\left(\infty ,1\right)$-adjunctions arises from adjunctions in sSet-enriched category theory, and in particular from enriched Quillen adjunctions between simplicial model categories.

We want to produce Cartesian/coCartesian fibration $K\to \Delta \left[1\right]$ 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 $\Delta \left[1\right]:=\left\{0\to 1\right\}$ as an $\mathrm{sSet}$-category in the obvious way using the embedding $\mathrm{const}:\mathrm{Set}↪\mathrm{sSet}$ and consider an $\mathrm{sSet}$-enriched functor $K\to \Delta \left[1\right]$. Let $C:={K}_{0}$ and $D:={K}_{1}$ be the $\mathrm{sSet}$-enriched categories that are the fibers of this. Then under the homotopy coherent nerve $N:\mathrm{sSet}\mathrm{Cat}\to \mathrm{sSet}$ the morphism

$N\left(p\right):N\left(K\right)\to \Delta \left[1\right]$N(p) : N(K) \to \Delta[1]

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

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

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

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 $\mathrm{sSet}$-enriched functor $p:C\to D$ between Kan-complex enriched categories that is hom-object-wise a Kan fibration, a morphim $f:c\prime \to c″$ in $C$ is an $N\left(p\right)$-Cartesian morphism if for all objects $c\in C$ the diagram

$\begin{array}{ccc}C\left(c,c\prime \right)& \stackrel{C\left(c,f\right)}{\to }& C\left(c,c″\right)\\ {↓}^{{p}_{c,c\prime }}& & {↓}^{{p}_{c,c″}}\\ D\left(p\left(c\right),p\left(c\prime \right)\right)& \stackrel{D\left(p\left(c\right),p\left(f\right)\right)}{\to }& D\left(p\left(c\right),p\left(c″\right)\right)\end{array}$\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 $p:K\to \Delta \left[1\right]$ and the above diagram becomes

$\begin{array}{ccc}K\left(c,c\prime \right)& \stackrel{K\left(c,f\right)}{\to }& K\left(c,c″\right)\\ ↓& & ↓\\ *& \to & *\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

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

under the homotopy coherent nerve of an sSet-enriched adjunction between $\mathrm{sSet}$-enriched categories

$C\stackrel{\stackrel{L}{\to }}{\underset{R}{←}}D$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

$N\left({C}^{\circ }\right)\stackrel{\stackrel{L}{\to }}{\underset{R}{←}}N\left({D}^{\circ }\right)$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 $\mathrm{sSet}$-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 $c,c\prime \in C$: $K\left(c,c\prime \right):=C\left(c,c\prime \right)$;

• for $d,d\prime \in D$: $K\left(d,d\prime \right):=D\left(d,d\prime \right)$;

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

and

$K\left(d,c\right)=\varnothing$

and equipped with the evident composition operation.

Then for every $d\in D$ there is the morphism ${\mathrm{Id}}_{R\left(d\right)}\in K\left(R\left(d\right),d\right)$, composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence $N\left(K\right)\to \Delta \left[1\right]$ 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 $\mathrm{sSet}$-subcategories on fibrant cofibrant objects:

let $K$ be as before and let ${K}^{\circ }$ be the full $\mathrm{sSet}$-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 ${\mathrm{Id}}_{R\left(d\right)}\in K\left(R\left(d\right),d\right)$ since the right Quillen functor $R$ is only guaranteed to respect fibrations, not cofibrations, and so $R\left(d\right)$ 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 $Q\left(R\left(d\right)\right)$ is cofibrant and there is an acyclic fibration $Q\left(R\left(d\right)\right)\to R\left(d\right)$. Take this to be the morphism in $K\left(Q\left(R\left(d\right)\right),d\right)$ that we pick for a given $d$. Then this does induce a homotopy equivalence

$C\left(c\prime ,Q\left(R\left(d\right)\right)\right)\to C\left(c\prime ,R\left(d\right)\right)=K\left(c\prime ,d\right)$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 $\left(\infty ,1\right)$-functors $\left(L⊣R\right):C\stackrel{←}{↪}D$ 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 $\left(\infty ,1\right)$-functor