equivalences in/of $(\infty,1)$-categories
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
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: 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
Using the (∞,1)-Grothendieck construction the notion of cograph of a functor has an evident generalization to $(\infty,1)$-categories.
(in terms of hom equivalence induced by unit map)
A pair of (∞,1)-functors
is an adjunction, if there exists a unit transformation $\epsilon : Id_C \to R \circ L$ – a morphism in the (∞,1)-category of (∞,1)-functors $Func(C,D)$ – such that for all $c \in C$ and $d \in D$ the induced morphism
is an equivalence of ∞-groupoids.
In terms of the concrete incarnation of the notion of $(\infty,1)$-category by the notion of quasi-category, we have that $Hom_(C)(L(c),d)$ and $Hom_D(c,R(d))$ 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.
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 $\Delta[1]^{op} \to$ (∞,1)Cat, which corresponds under the Grothendieck construction to a Cartesian fibration of simplicial sets $coGraph(L) \to \Delta[1]$.
(in terms of Cartesian/coCartesian fibrations)
Let $C$ and $D$ be quasi-categories. An adjunction between $C$ and $D$ is
a morphism $K \to \Delta[1]$ 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 $p : K to \Delta[1]$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta[1]^{op}$, respectively.
The two different definition above are indeed equivalent:
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.
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 $K \to \Delta[1]$ that encodes an $\infty$-adjunction $(f \dashv g)$.
For that, define a morphism $F' : \Lambda[2]_2 \times C \to K$ as follows:
on $\{0,2\}$ it is the morphism $F : C \times \Delta[1] \to K$ 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 $C \times \Delta[1] \stackrel{f \times Id}{\to} D \times \Delta[1] \stackrel{G}{\to} K$, 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
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 $d \in D$
is an equivalence, hence an isomorphism in the homotopy category. Once checks that this fits into a commuting diagram
For illustration, chasing a morphism $f(c) \to d$ through this diagram yields
where on the left we precomposed with the Cartesian morphism
given by $F''|_{c} : \Delta[2] \to K$, by …
The adjoint of a functor is, if it exists, essentially unique:
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
(to the opposite quasi-category), which takes every left adjoint functor to a corresponding right adjoint.
This is HTT, prop 5.2.1.3 (also remark 5.2.2.2), and HTT, prop. 5.2.6.2.
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
the hom-isomorphism $Hom_C(L(-),-) \simeq Hom_D(-,R(-))$;
the fact that $Hom_C(-,-) : C^{op} \times C \to Set$ preserves all limits in both arguments;
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 $d : I \to D$
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
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
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.
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.
Given an $(\infty,1)$-adjunction $(L \dashv R) : C \to D$
$R$ is a full and faithful (∞,1)-functor precisely is the counit $L R \stackrel{}{\to} Id$ is an equivalence of (∞,1)-functor
In this case $C$ is a reflective (∞,1)-subcategory of $D$.
$L$ is a full and faithful (∞,1)-functor precisely is the unit $Id \to L R$ is an equivalence of (∞,1)-functors.
Lurie, prop. 5.2.7.4, See also top of p. 308.
Let
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
where
$L/X$ is the evident induced functor;
$R/X$ is the composite
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.
A large class of examples of $(\infty,1)$-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 $K \to \Delta[1]$ from a given sSet-enriched adjunction. For that first consider the following characterization
Let $K$ be a simplicially enriched category whose hom-objects are all Kan complexes, regard the interval category $\Delta[1] := \{0 \to 1\}$ as an $sSet$-category in the obvious way using the embedding $const : Set \hookrightarrow sSet$ and consider an $sSet$-enriched functor $K \to \Delta[1]$. 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
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
is a homotopy equivalence of Kan complexes for all objects $c' \in C'$.
This appears as HTT, prop. 5.2.2.4.
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
is a homotopy pullback in the model structure on sSet-categories.
For the case under consideration the functor in question is $p : K \to \Delta[1]$ and the above diagram becomes
This is clearly a homotopy pullback precisely if the top morphism is an equivalence.
Using this, we get the following.
For $C$ and $D$ sSet-enriched categories whose hom-objects are all Kan complexes, the image
under the homotopy coherent nerve of an sSet-enriched adjunction between $sSet$-enriched categories
is an adjunction of quasi-categories.
Moreover, if $C$ and $D$ are equipped with the structure of a simplicial model category then the quasi-categorically derived functors
form an adjunction of quasi-categories.
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 $N(K) \to \Delta[1]$ 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
because in an enriched model category the enriched hom out of a cofibrant object preserves weak equivalences between fibrant objects.
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