One definition of ordinary adjoint functors says that a pair of functors is an adjunction if there is a natural isomorphism
The analog of this definition makes sense very generally in (∞,1)-category theory, where is the -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 and are adjoint precisely if the cograph of coincides with the cograph of up to the obvious reversal of arrows
Using the (∞,1)-Grothendieck construction the notion of cograph of a functor has an evident generalization to -categories.
(in terms of hom equivalence induced by unit map)
A pair of (∞,1)-functors
is an adjunction, if there exists a unit transformation – a morphism in the (∞,1)-category of (∞,1)-functors – such that for all and the induced morphism
is an equivalence of ∞-groupoids.
In terms of the concrete incarnation of the notion of -category by the notion of quasi-category, we have that and 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. 22.214.171.124.
We make use here of the explicit realization of the (∞,1)-Grothendieck construction in its incarnation for quasi-categories: here an (∞,1)-functors may be regarded as a map (∞,1)Cat, which corresponds under the Grothendieck construction to a Cartesian fibration of simplicial sets .
(in terms of Cartesian/coCartesian fibrations)
Let and be quasi-categories. An adjunction between and is
together with equivalence of quasi-categories and .
Two (∞,1)-functors and are called adjoint – with left adjoint to and right adjoint to if
there exists an adjunction in the above sense
and and are the associated functors to the Cartesian fibation and the Cartesian fibration , respectively.
The two different definition above are indeed equivalent:
For and 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 126.96.36.199.
First we discuss how to produce the unit for an adjunction from the data of a correspondence that encodes an -adjunction .
For that, define a morphism as follows:
on it is the morphism that exhibits as associated to , being on and on ;
on it is the morphism , where is the morphism that exhibits as associated to ;
Now observe that in particular sends to Cartesian morphisms in (by definition of functor associated to ). By one of the equivalent characterizations of Cartesian morphisms, this means that the lift in the diagram
exists. This defines a morphism whose components may be regarded as forming a natural transformation .
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 and
is an equivalence, hence an isomorphism in the homotopy category. Once checks that this fits into a commuting diagram
For illustration, chasing a morphism through this diagram yields
where on the left we precomposed with the Cartesian morphism
given by , by …
The adjoint of a functor is, if it exists, essentially unique:
If the -functor between quasi-categories admits a right adjoint , then this is unique up to homotopy.
Let be the full sub-quasi-categories on the (∞,1)-category of (∞,1)-functors between and 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.
the hom-isomorphism ;
the fact that 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 and diagram
which implies that .
Now to see this in -category theory (…) HTT Proposition 188.8.131.52
This is HTT, prop 184.108.40.206.
This follows from that fact that for a unit of the -adjunction, its image 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 -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 -adjunction
Lurie, prop. 220.127.116.11, See also top of p. 308.
be a pair of adjoint -functors where the -category has all (∞,1)-pullbacks.
Then for every object there is induced a pair of adjoint -functors between the over-(∞,1)-categories
This is HTT, prop. 18.104.22.168.
Let be a simplicially enriched category whose hom-objects are all Kan complexes, regard the interval category as an -category in the obvious way using the embedding and consider an -enriched functor . Let and be the -enriched categories that are the fibers of this. Then under the homotopy coherent nerve the morphism
is a Cartesian fibration precisely if for all objects there exists a morphism in such that postcomposition with this morphism
This appears as HTT, prop. 22.214.171.124.
The statement follows from the characterization of Cartesian morphisms under homotopy coherent nerves (HTT, prop. 126.96.36.199), which says that for an -enriched functor between Kan-complex enriched categories that is hom-object-wise a Kan fibration, a morphim in is an -Cartesian morphism if for all objects the diagram
For the case under consideration the functor in question is 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.
is an adjunction of quasi-categories.
form an adjunction of quasi-categories.
To get the first part, let be the -category which is the join of and : its set of objects is the disjoint union of the sets of objects of and , and the hom-objects are
for : ;
for : ;
for and : ;
and equipped with the evident composition operation.
Then for every there is the morphism , composition with which induced an isomorphism and hence an equivalence. Therefore the conditions of the above lemma are satisfied and hence 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 -subcategories on fibrant cofibrant objects:
let be as before and let be the full -subcategory on objects that are fibrant-cofibrant (in or in , respectively). Then for any fibrant cofibrant , we cannot just use the identity morphism since the right Quillen functor is only guaranteed to respect fibrations, not cofibrations, and so might not be in . But we can use the small object argument to obtain a functorial cofibrant replacement functor , such that is cofibrant and there is an acyclic fibration . Take this to be the morphism in that we pick for a given . 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.
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
and also in