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/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
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 $\epsilon : Id_D \to R \circ L$ – 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
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 $Hom_C(L(d),c)$ 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.
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 $\Delta[1]^{op} \to$ (∞,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 $p \colon K \to \Delta[1]$ and the Cartesian fibration $p^{op} : K^{op} \to \Delta[1]^{op}$, respectively.
(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):
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. ).
The conceptual content of Prop. may be made manifest as follows:
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.
(Riehl & Verity 2016, Thm. 4.3.11, 4.4.11)
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.
The idea is to construct the category of right adjoints as an intersection of full subcategories
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 $p \simeq D(f-,-) \simeq C(-,g-)$.
Given functors $f : C \to D$ and $g : D \to C$, we can use the (∞,1)-end to determine compute a chain of equivalences
dually, we can identify the space of counits as
So each half of the equivalence $D(f-,-) \simeq C(-,g-)$ corresponds essentially uniquely to a choice of unit and counit transformation.
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. 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.
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 $Id \to R L$ is an equivalence of (∞,1)-functors.
Lurie, prop. 5.2.7.4, See also top of p. 308.
(sliced adjoints)
Let
be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).
Then:
For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form
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
of
the evident functor induced by $R$;
the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).
For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form
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
of
the evident functor induced by $L$;
the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).
(in 1-category theory)
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).
(left adjoint of sliced adjunction forms adjuncts)
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.):
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.
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 $A^f$.
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 $X \to [1]$ that is both a cartesian and a cocartesian fibration. We can generalize this to
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.
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
This is a restatement of HTT, corr. 5.2.2.5.
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.
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 $C = [1]$, 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
The product and exponential on $(\infty,1)Cat$ restrict to functors
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 $K \to \Delta[1]$ 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
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 $\Delta[1]$ 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:
Emily Riehl, Dominic Verity, The 2-category theory of quasi-categories, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 (arXiv:1306.5144, doi:10.1016/j.aim.2015.04.021),
Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 (arXiv:1310.8279, doi:10.1016/j.aim.2015.09.011)
Emily Riehl, Dominic Verity, Def. 1.1.2 in: Infinity category theory from scratch, Higher Structures Vol 4, No 1 (2020) (arXiv:1608.05314, pdf)
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
Last revised on December 5, 2021 at 05:56:31. See the history of this page for a list of all contributions to it.