aa r X i v : . [ m a t h . C T ] J a n CARTESIAN EXPONENTIATION AND MONADICITY
EMILY RIEHL AND DOMINIC VERITY
Abstract.
An important result in quasi-category theory due to Lurie is the that co-cartesian fibrations are exponentiable , in the sense that pullback along a cocartesian fi-bration admits a right Quillen right adjoint that moreover preserves cartesian fibrations;the same is true with the cartesian and cocartesian fibrations interchanged. To expli-cate this classical result, we prove that the pullback along a cocartesian fibration betweenquasi-categories forms the oplax colimit of its “straightening,” a homotopy coherent dia-gram valued in quasi-categories, recovering a result first observed by Gepner, Haugseng,and Nikolaus. As an application of the exponentiation operation of a cartesian fibrationby a cocartesian one, we use the Yoneda lemma to construct left and right adjoints tothe forgetful functor that carries a cartesian fibration over B to its ob B -indexed family offibers, and prove that this forgetful functor is monadic and comonadic. This monadicity isthen applied to construct the reflection of a cartesian fibration into a groupoidal cartesianfibration, whose fibers are Kan complexes rather than quasi-categories. Contents
1. Introduction 21.1. Acknowledgments 82. Weighted colimits in simplicial categories 82.1. Weighted colimits and collages 92.2. Flexible weights as projective cell complexes 112.3. Oplax colimits 133. Cocartesian fibrations and quasi-categorical collages 163.1. Cocartesian fibrations of quasi-categories 163.2. The quasi-categorically enriched category of cocartesian fibrations 183.3. Collages for quasi-categories 204. The comprehension construction 244.1. Cocartesian transformations between lax cocones 244.2. The comprehension construction 275. Pullback along a cocartesian fibration as an oplax colimit 295.1. A replacement for pullback along a cocartesian fibration 305.2. Comparison with the strict pullback 346. Pushforward along a cocartesian fibration 40
Date : January 26, 2021.2010
Mathematics Subject Classification.
Primary 18G55, 55U35, 55U40; Secondary 18A05, 18D20,18G30, 55U10.
Introduction
Famously the category Cat of small categories is not a topos because, among other things,it fails to be locally cartesian closed. A finitely complete category E is locally cartesianclosed just when each slice category E /B is cartesian closed, or equivalently, which thepullback functor associated to any morphism f : A → B admits a right adjoint (as well asa left adjoint given by composition with f ): E /B f ∗ / / E /A Σ f ⊥ z z Π f ⊥ e e In the case E = Cat, those functors f for which the pullback functor f ∗ does admit aright adjoint Π f are called exponentiable and have been characterized by Conduché [4].Famously,(i) All cartesian and cocartesian fibrations p : E → B of 1-categories are exponen-tiable , in the sense that the pullback along p functor p ∗ has a right adjoint called pushforward along p : Cat /E Cat /B Π p ⊥ p ∗ (ii) If p : E → B is a cocartesian fibration and q : F → E is a cartesian fibration thenthe pushforward Π p ( q : F → E ) is also a cartesian fibration, and the dual resultholds when the cartesian and cocartesian fibrations are interchanged.In [12] Lurie established ∞ -categorical analogues of these results for quasi-categories. Subsequent authors, for instance Barwick and Shah [2], have stressed the importance of A general characterization of the exponentiable functors between quasi-categories, while not the focusof our interest here, can be found in [12, §B.3] or [1].
ARTESIAN EXPONENTIATION AND MONADICITY 3 these results both to the foundations of ∞ -category theory and in applications, and wehave further applications in mind. In [16, §12.3], we use this result to prove that modules between quasi-categories admit all right and left extensions. It follows that the questionof existence of pointwise right and left Kan extensions can be reduced to the existence ofcertain limits and colimits. For those results, it is useful to have a somewhat more refinedversion of these results than is easily found in the literature—see especially Theorem 6.2.9and Corollary 6.2.10—which is the motivation for the present expositionA cocartesian fibration of quasi-categories is an isofibration p : E ։ B whose fibersdepend covariantly functorially on B . In the simplest non-trivial case, when B = ∆ , thedata is given by a pair of quasi-categories E and E together with a functor E → E . Ingeneral, the comprehension construction of [23] “straightens” p : E ։ B into a simplicialfunctor c p : C B → Q Cat that sends each vertex b ∈ B to the fiber E b . The domain categoryappearing here is the homotopy coherent realization of the quasi-category B , a cofibrantsimplicial category that indexes B -shaped homotopy coherent diagrams. At the level ofobjects and 1-arrows f : a → b in B , the comprehension construction is defined by liftingthe 1-arrow f to a p - cocartesian 1-arrow with codomain E : B ◆◆◆◆◆◆◆◆◆◆◆◆◆◆ b a ) ) f w (cid:127) ①①①①①① E b E a E ℓ E b ℓ E a ) ) p b (cid:15) (cid:15) (cid:15) (cid:15) p a (cid:15) (cid:15) (cid:15) (cid:15) p (cid:15) (cid:15) (cid:15) (cid:15) E f & & ℓ E f v ~ ✉✉✉✉✉✉✉✉ Together, this data defines a lax cocone ℓ E under the comprehension functor c p with nadir E the data of which is given by a functor ℓ E : C [ B ⋆ ∆ ] → Q Cat that restricts along C B ֒ → C [ B ⋆ ∆ ] to c p . In fact, as we shall discover that ℓ E is a colimit cocone:5.2.8. Corollary.
The domain of a cocartesian fibration p : E ։ B is equivalent to the oplaxcolimit of the associated comprehension functor c p : C B → Q Cat , with + C B jJ x x ♣♣♣♣♣♣♣♣ h E ,c p i % % ❑❑❑❑❑❑❑❑ C [ B ⋆ ∆ ] ℓ E / / Q Cat as the associated colimit cocone. In the Joyal model structure on simplicial sets, we refer to the fibrations between fibrant objects (thequasi-categories) as isofibrations because they have a lifting property for isomorphisms analogous to thatfor the isofibrations in classical category theory. The simplicial categories that are cofibrant in the Bergner model structure are precisely the simplicialcomputads that are freely generated by their non-degenerate “atomic” n -arrows for each n ≥ , admittingno non-trivial factorizations; see Definition 2.2.3. RIEHL AND VERITY
In particular the domain E of the cocartesian fibration p can be recovered up to equiv-alence as the oplax colimit of a the comprehension functor c p : C B → Q Cat . Gepner,Haugseng, and Nikolaus, who obtain a similar result to Corollary 5.2.8 as one of the maintheorems of [9], interpret this result as a proof that “Lurie’s unstraightening functor is amodel for the ∞ -categorical analogue of the Grothendieck construction.” Their methodol-ogy is quite different from ours, constructing oplax colimits directly at the quasi-categoricallevel, whereas our comprehension construction enables us to work at the level of simplicialcategories and functors. The comprehension functor c p : C B → Q Cat can be used to definea “straightening” of the pullback of p along any generalized element b : X → B , even in thecase where X is not a quasi-category simply by restricting the comprehension functor (andits lax cocone) along b . We derive Corollary 5.2.8 as a special case of our first main theo-rem, which proves that the fiber E b is equivalent to the oplax colimit of this straighteneddiagram.5.1.3. Theorem.
For any cocartesian fibration p : E ։ B and any b : X → B , the compre-hension cocone induces a canonical map over E from the oplax colimit of the diagram C X C b −→ C B c p −→ Q Cat to the fiber E b (cid:15) (cid:15) / / E p (cid:15) (cid:15) (cid:15) (cid:15) X b / / B and this map is a natural weak equivalence in the Joyal model structure. The canonical natural transformation of Theorem 5.1.3 defines a natural Joyal equiva-lence relating the pullback functor p ∗ to a functor ˜ p ∗ defined by forming oplax colimits ofrestrictions of the comprehension cocone:sSet / B ˜ p ∗ $ $ p ∗ ⇓ γ sSet / E Both functors p ∗ and ˜ p ∗ are left Quillen with respect to the sliced Joyal model structures,admitting right Quillen adjoints: Unfortunately, the assignment of the terms “oplax colimit” and “lax colimit” given in [9, 2.8] is theopposite of the one used here. There is a standard convention in 2-category theory that the 2-cell componentof an oplax natural transformation is parallel to its 1-cell components, while for a lax natural transformationthese 2-cells are reversed. A lax cocone is then a lax natural transformation whose codomain is a constantdiagram. Confusingly, due to the principle that a W -weighted colimit in an enriched category shouldcoincide with a W -weighted limit in the opposite category, oplax colimits represent lax cocones under adiagram. ARTESIAN EXPONENTIATION AND MONADICITY 5
Proposition. If p : E ։ B is a cocartesian fibration, the adjunctions sSet / E sSet / B and sSet / E sSet / B Π p ⊥ p ∗ ˜Π p ⊥ ˜ p ∗ are Quillen with respect to the sliced Joyal model structures. By taking mates, it follows that there is a canonical natural transformation ˆ γ : Π p ⇒ ˜Π p whose component at any isofibration q : F ։ E is an equivalence. In this way we obtainan alternate model ˜Π p for the pushforward functor that is more easily understood. Atan isofibration q : F ։ E the right adjoint Π p is equivalent to the pullback along thecomprehension cocone of the induced map between lax slices induced by whiskering with q : ˜Π p ( F q −− ։ E ) (cid:15) (cid:15) (cid:15) (cid:15) / / qCat // F q ◦− (cid:15) (cid:15) (cid:15) (cid:15) B ℓ E / / qCat // E To prove Proposition 6.2.5, we show that the “whiskering with q ” map is an isofibration.This establishes the quasi-categorical analogue of desiderata (i) above. We then showfurther that if q : F ։ E is a cartesian fibration, then the “whiskering with q ” map hasa certain right horn lifting property, thereby proving the quasi-categorical analogue ofdesiderata (ii):6.2.8. Corollary. If p : E ։ B is a cocartesian fibration and q : F ։ E is a cartesianfibration between quasi-categories, then the pushforward Π p ( q : F ։ E ) ։ B is a cartesian fibration between quasi-categories. We show also that the pullback and pushforward functors along a cocartesian fibrationpreserve the accompanying class of cartesian functors between cartesian fibrations. Theseresults are summarized in the following theorem: The precise meaning of this notation, involving slices of the homotopy coherent nerve of Q Cat regardedas a 2-complicial set, is explained in Lemma 6.2.1.
RIEHL AND VERITY
Theorem.
For any cocartesian fibration p : E ։ B between quasi-categories, thepullback-pushforward adjunction restricts to define an adjunction Q Cat / E Q Cat / B Cart ( Q Cat ) / E Cart ( Q Cat ) / B Π p ⊥ p ∗ Π p ⊥ p ∗ As an immediate corollary, we construct “exponentials” whose exponents are either carte-sian or cocartesian, justifying the appelation “exponentiable” for these maps, and prove:6.3.3.
Proposition. If p : E ։ B is a cocartesian fibration and q : F ։ B is a cartesianfibration, then ( q : F ։ B ) p : E ։ B is a cartesian fibration. Dually, if p : E ։ B is a cartesian fibration and q : F ։ B is acocartesian fibration, then the exponential is a cocartesian fibration. The final two sections of this paper supply some first applications of these results. As thecomprehension construction reveals, cartesian fibrations over B encode functors B op → qCat valued in the (large) quasi-category of small quasi-categories. In ordinary category theory itis well-known that for any small category B and complete and cocomplete category C , theforgetful functor C B → C ob B that carries a diagram to the ob B -indexed family of objectsin its image admits both left and right adjoints, given by left and right Kan extension,and is moreover monadic and comonadic . Informally, this means that B -indexed diagramscan be understood as “algebras” or as “coalgebras” for a monad or comonad acting on thecategory of ob B -indexed families of objects.The corresponding result for quasi-categories will be proven in a sequel to this paper[25], but here we demonstrate that the analogous result holds for cartesian fibrations, usinga version of Beck’s monadicity theorem for quasi-categories proven in [20]. Writing Cart / B for the large quasi-category of cartesian fibrations and cartesian functors over B , we prove:7.2.6. Theorem.
The forgetful functor u : Cart / B −→ Cart / ob B ∼ = Y ob B qCat is comonadic and hence also monadic. In a further sequel, we will use this monadicity to establish an equivalence between
Cart / B and qCat B op , both quasi-categories being monadic over Q ob B qCat .Here we include another application of the monadicity of Theorem 7.2.6. Using ouranalysis of limits and colimits in quasi-categories defined as homotopy coherent nerves in[24], we prove: ARTESIAN EXPONENTIATION AND MONADICITY 7
Theorem.
The inclusion
Kan ֒ → qCat admits both left and right adjoints Kan (cid:31) (cid:127) ⊥⊥ / / qCat invert y y core c c and is monadic and comonadic. The right adjoint here is the familiar functor that takes a quasi-category to its maximalKan complex core, while the left adjoint is a somewhat more delicate “groupoidal reflec-tion” functor. Our final result establishes an analogous groupoidal reflection for cartesianfibrations into the subcategory of groupoidal cartesian fibrations , whose fibers are Kancomplexes rather than quasi-categories.8.3.6.
Theorem.
There is a left adjoint to the inclusion
Cart gr / B Cart / B ⊥ invert defining the reflection of a cartesian fibration into a groupoidal cartesian fibration. All of the results mentioned above have duals with cocartesian and cartesian fibrationsinterchanged. The comprehension functor associated to a cartesian fibration is contravari-ant and its domain is recovered as the lax colimit of this diagram. It is to avoid thiscontravariance that we choose to focus the bulk of our presentation on the case of cocarte-sian fibrations.This paper is organized as follows. In §2, we introduce oplax colimits through the generalmechanism of weighted colimits. We prove that oplax weights are flexible , which impliesthat the oplax weighted colimit functor is equivalence-invariant. We also review the collageconstruction, which allows us to construct flexible weights by instead specifying the shapeof their corresponding cocones. In particular, a lax cocone of shape X is indexed by thehomotopy coherent realization of the join X ⋆ ∆ .In §3, we review some basic aspects of the theory of cocartesian fibrations betweenquasi-categories. We introduce a quasi-categorical version of the collage construction justdiscussed, and prove that the quasi-categorical collage of a functor f : A → B defines acocartesian fibration over ∆ that models the oplax colimit of f .In §4, we review the comprehension construction from [23], devoting somewhat moreattention to the lax cocones that are the focus of much of the work here. Then in §5, weprove that pullback along a cocartesian fibration can be modeled as a oplax colimit of arestriction of the comprehension functor.The corresponding results for the pushforward functor, including in particular (i) and(ii), are then proven in §6. The oplax colimits defining the functor ˜ p ∗ in §5 are properlyunderstood as a variety of ( ∞ , -categorical colimits. Consequently, the description ofthe corresponding right adjoint ˜Π p involves an ( ∞ , -categorical cocone construction, in-stantiated by forming the slice of 2-complicial set over a vertex. As we explain in §6.1, RIEHL AND VERITY
Cart ( Q Cat ) / B → Cart ( Q Cat ) / ob B and constructleft and right biadjoints : quasi-categorically enriched functors equipped with a naturalequivalence of function complexes encoding the adjoint transpose relation. Such datadescends to an adjunction between the quasi-categorical cores of these quasi-categoricallyenriched categories. We then review the monadicity theorem from [20] and apply it toprove that this forgetful functor is monadic and comonad as a map between large quasi-categories.To say that the functor Cart / B → Cart / ob B ∼ = Q ob B qCat is monadic is to say that Cart / B may be recovered as the quasi-category of homotopy coherent algebras for a homotopycoherent monad acting on Q ob B qCat . In §8, we show that Cart gr / B is similarly the quasi-category of homotopy coherent algebras for the restriction of this homotopy coherent monadalong the inclusion Q ob B Kan ֒ → Q ob B qCat . We then show that this characterizationallows us to construct the groupoidal reflection functor as a lift of the groupoidal reflectionfunctor qCat → Kan .This paper is a continuation of a series of papers that redevelop the foundations of ( ∞ , -category theory [18, 20, 19, 21, 22, 23, 26, 24], the results of which are referencedas I.x.x.x, . . . , and VIII.x.x.x respectively. However, we deploy relatively few of the toolsdeveloped in our previous work to prove the theorems appearing here, and when we doreference prior results, we typically restate them in considerably less generality. Many ofthe results from previous work recalled here — for instance Theorem 4.2.1 — are provenin the more abstract setting of any ∞ - cosmos , while in the present manuscript we consideronly a single example: the quasi-categorically enriched category Q Cat of quasi-categories.As we do not need this notion, we do not recall any specifics here. Acknowledgments.
The authors are grateful for support from the National ScienceFoundation (DMS-1551129) and from the Australian Research Council (DP160101519).This work was commenced when the second-named author was visiting the first at Harvardand then at Johns Hopkins and completed while the first-named author was visiting thesecond at Macquarie. We thank all three institutions for their assistance in procuring thenecessary visas as well as for their hospitality.2.
Weighted colimits in simplicial categories
Our aim in this section is to define the oplax colimit of a homotopy coherent diagram C X → S Set indexed by the homotopy coherent realization of a simplicial set X , our namefor the left adjoint of the homotopy coherent nerve functor N : sSet-Cat → sSet. Oplaxcolimits are introduced as particular weighted colimits , where the weights in question aresimplicial functors that describe the shape of lax cocones. In §2.1, we review the generalrubric of weighted colimits and explain how these cocone shapes may be presented as This said, however, we cannot resist appealing to ∞ -cosmic techniques in a handful of our proofs. ARTESIAN EXPONENTIATION AND MONADICITY 9 simplicial collages . In §2.2, we highlight a special class of flexible weights that have usefulhomotopical properties. Finally, in §2.3, we define the weights for oplax colimits as collagesand observe that such weights are flexible, a fact that will be exploited in our future work.Some of this material was previously discussed in §VII.4, where the “oplax” weightswere called “pseudo” weights. See Remark 2.3.7 for an explanation of this contrast innomenclature. Collages for weights for limits made an appearance in §VII.5.2 and werefer the proofs of a few of the results appearing below to there, but we reintroduce thisconstruction here to clarify the details in the dual case and because we will require a moreextensive analysis of collages than we did in [26].2.1.
Weighted colimits and collages.
In a simplicially enriched category, the appropri-ately general notion of colimit allows for the specification of any particular “shape” of coneunder the diagrams being considered. This specification is given by a simplicial functorreferred to as a weight for the colimit.2.1.1.
Definition (weights for simplicial colimits) . Suppose D is a small simplicial category,which we think of as a diagram shape. Then a weight on D is a simplicial functor W : D op →S Set . For any diagram F : D → K of shape D valued in a simplicial category K , a W -cocone with nadir an object e ∈ K is a simplicial natural transformation ι : W → Fun K ( F ( − ) , e ) .We say that the W -cocone ι displays e as a W -colimit of F if and only if for all objects e ′ ∈ K the simplicial map Fun K ( e, e ′ ) ∼ = / / Fun S Set D op ( W, Fun K ( F ( − ) , e ′ )) given by post-composition with ι is an isomorphism.Many notations are common for the nadir of a weighted colimit cone; here we write colim W F for the colimit of F weighted by W . When these objects exist for all weights anddiagrams in K then we may extend colim to a simplicial bifunctor: S Set D op × K D colim / / K that is cocontinuous in both variables.A simplicial functor W : D op → S Set may otherwise be described as comprising a familyof simplicial sets { W d } d ∈ obj( D ) along with right actions W d ′ × Fun D ( d, d ′ ) ∗ / / W d (2.1.2)of the hom-spaces of D which must collectively satisfy the customary axioms with respectto the identities and composition of D . This description leads us to define a simpliciallyenriched category coll( W ) , called the collage of W .2.1.3. Definition (collages) . For any weight W : D op → S Set , the collage of W is a simpli-cial category coll( W ) that contains D as a full simplicial subcategory along with preciselyone extra object ⊤ whose endomorphism space is the point. The function complexes Fun coll( W ) ( ⊤ , d ) are all taken to be empty and we define: Fun coll( W ) ( d, ⊤ ) := W d for objects d ∈ D . The composition operations between hom-spaces in D and those with codomain ⊤ are givenby the actions depicted in (2.1.2).In the statement of the following result, sSet D op denotes the underlying category of thesimplicially enriched category S Set D op .2.1.4. Proposition (collage adjunction, VII.5.2.3) . (i) The collage construction defines a fully faithful functor sSet D op coll / / + D / sSet - Cat from the category of D -indexed weights to the category of simplicial categories under + D whose essential image is comprised of those h e, F i : + D → K that arebijective on objects, fully faithful when restricted to D and , and have the propertythat there are no maps in K from e to the image of F .(ii) The collage functor admits a right adjoint + D / sSet - Cat wgt ⊥ sSet D op coll r r which carries a pair h e, F i : + D → K to the weight
Fun K ( F ( − ) , e ) : D op → S Set . This adjunction has a useful and important interpretation:2.1.5.
Corollary (VII.5.2.4) . The collage coll( W ) of a weight realises the shape of W -cocones, in the sense that simplicial functors G : coll( W ) −→ K stand in bijection to W -cocones under the diagram G | D with nadir G ( ⊤ ) . (cid:3) Lemma.
For any simplicial functor I : D → C , weight W : D op → S Set , and diagram G : C → K , we have an isomorphism colim W GI ∼ = colim lan I W G where the colimit on one side exists if and only if the one on the other does.Proof. Simplicial left Kan extension provides an adjunction S Set C op −◦ I ⊥ S Set D op lan I r r In particular
Fun S Set C op (lan I W, Fun K ( G ( − ) , e )) ∼ = Fun S Set D op ( W, Fun K ( GI ( − ) , e )) which shows that colim lan I W G and colim W GI have the same defining universal property. (cid:3) ARTESIAN EXPONENTIATION AND MONADICITY 11
Lemma.
Left Kan extension of W : D op → S Set along a simplicial functor I : D → C gives rise to a pushout square + D + I / / (cid:127) _ (cid:15) (cid:15) + C (cid:127) _ (cid:15) (cid:15) coll( W ) / / coll(lan I W ) in the category of simplicial categories and simplicial functors.Proof. By the defining universal property, a simplicial functor whose domain is the pushoutof + D ֒ → coll( W ) along + I and whose codomain is K is given by a pair of functors γ : coll( W ) → K and h e, G i : + C → K . By Corollary 2.1.5, the simplicial functor γ represents a W -cocone in K with nadir e underthe diagram GI : D → S
Set . By Lemma 2.1.6, such data equivalently describes a lan( W ) -shaped cocone under the diagram G with nadir e . Applying Corollary 2.1.5 again, weconclude that this pushout is given by the simplicial category coll(lan I W ) , as claimed. (cid:3) In ordinary unenriched category theory, the colimit cone under a D -shaped diagram maybe formed as the left Kan extension along the inclusion D ֒ → D ⋆ into the category D ⋆ formed by freely adjoining a terminal object “ ⊤ ” to D . The following lemma reveals thatthe collage plays the roll of the category D ⋆ for weighted colimits, a perspective whichwe will return to in §2.3.2.1.8. Lemma.
The pointwise left Kan extension of any simplicial functor F : D → K along I : D ֒ → coll( W ) exists if and only if the colimit colim W F exists in K , and then lan I F ( ⊤ ) ∼ = colim W F .Proof. Since D ֒ → coll( W ) is fully faithful, when the pointwise left Kan extension of anysimplicial functor F : D → K along
D ∈ coll( W ) exists, it is displayed by an isomorphism: D lL I z z ✈✈✈✈✈✈✈ ∼ = F (cid:30) (cid:30) ❂❂❂❂❂❂ coll( W ) lan I F / / K By Corollary 2.1.5, this data defines a W -cocone under F ∼ = lan I F ◦ I with nadir lan I F ( ⊤ ) ∈K . It is easy to verify that the universal property of the left Kan extension specializes todescribe the universal property of the colimit cocone for colim W F , and conversely that theuniversal property of the weighted limit cocone implies the universal property of the leftKan extension. (cid:3) Flexible weights as projective cell complexes.
In order to understand the sensein which certain weighted colimits, including in particular the oplax colimits to be in-troduced below, are homotopically well behaved, we recall some facts about weights andsimplicial computads from §II.5.3:
Definition (flexible weights and projective cell complexes) . Suppose that D is asimplicial category. Then a simplicial natural transformation of the form ∂ ∆ n × Fun D ( − , d ) ֒ → ∆ n × Fun D ( − , d ) , for some [ n ] ∈ ∆ and object d ∈ D , is said to be the projective n -cell associated with d . Anatural transformation α : W → V in S Set D op is a relative projective cell complex if it maybe constructed as a countable composite of pushouts of coproducts of projective cells. Aweight W in S Set D op is a flexible weight if the map ! : ∅ → W is a relative projective cellcomplex, i.e., if W is a projective cell complex.Our interest in colimits weighted by flexible weights is due to the fact that they arehomotopically well behaved. We state the following result for diagrams valued in simplicialsets, but its proof extends without change to pointwise cofibrant diagrams valued in anymodel category enriched over the Joyal model structure on simplicial sets.2.2.2. Proposition (II.5.2.6, VII.4.1.5) . (i) For a flexible weight W : D op → S Set and any diagram F : D → S
Set , colim W F may be expressed as a countable composite of pushouts of coproducts of maps ∂ ∆ n × F d ֒ → ∆ n × F d. (ii) If α : F → G is a simplicial natural transformation between two such diagramswhose components are weak equivalences in the Joyal model structure, then forany flexible weight W the map colim W α : colim W F → colim W G is a weak equivalence in the Joyal model structure. The collage construction defines a correspondence between flexible weights and simpli-cial computads , a class of “freely generated” simplicial categories that define precisely thecofibrant objects [14, §16.2] in the model structure due to Bergner [3].2.2.3.
Definition (simplicial computad) . A simplicial category A , regarded as a simplicialobject [ n ]
7→ A n in the category of categories with a common set of objects ob A andidentity-on-objects functors, is a simplicial computad if and only if: • each category A n of n - arrows is freely generated by the reflexive directed graph of atomic n -arrows, these being those arrows that admit no non-trivial factorizations, andif • if f is an atomic n -arrow in A n and α : [ m ] → [ n ] is a degeneracy operator in ∆ thenthe degenerated m -arrow f · α is atomic in A m .A simplicial category A is a simplicial computad if and only if all of its non-identity arrows f can be expressed uniquely as a composite f = ( f · α ) ◦ ( f · α ) ◦ · · · ◦ ( f ℓ · α ℓ ) (2.2.4)in which each f i is non-degenerate and atomic and each α i ∈ ∆ is a degeneracy operator. ARTESIAN EXPONENTIATION AND MONADICITY 13
We have the following recognition principle for flexible weights on simplicial computads,a mild variant of Proposition II.5.3.5, proven in §VII.5.2.2.2.5.
Proposition (relating flexible weights and simplicial computads, VII.5.2.6) . Supposethat D is a simplicial computad and that W : D op → S Set is a weight. Then W is a flexibleweight if and only if its collage coll( W ) is a simplicial computad. Oplax colimits.
Oplax colimits represent particular cones under a homotopy coher-ent diagram C X → K indexed by a simplicial set X . In Definition 2.3.6, we first presentthe collage construction that describes the shape of a lax cocone and then use Proposition2.1.4 to extract the corresponding weight. To give a concise description of the collagethat defines the oplax weight, we make use of the simplicial computad structure on the homotopy coherent realization C X of a simplicial set X , our term for the left adjoint tothe homotopy coherent nervesSet-Cat N ⊥ sSet C r r We briefly review this material from §VI.4, which gives a more leisurely presentation withconsiderably more details.2.3.1.
Example (homotopy coherent simplices as simplicial computads; §VI.4.2) . Recallthe simplicial category C ∆ n whose objects are integers , , . . . , n and whose function com-plexes are the cubes Fun C ∆ n ( i, j ) = ⊓⊔ j − i − i < j ∆ i = j ∅ i > j Here we write ⊓⊔ k := (∆ ) k . For i < j , the vertices of Fun C ∆ n ( i, j ) are naturally identifiedwith subsets of the closed interval [ i, j ] = { i ≤ t ≤ j } containing both endpoints, a setwhose cardinality is j − i − ; more precisely, Fun C ∆ n ( i, j ) is the nerve of the poset withthese elements, ordered by inclusion. Under this isomorphism, the composition operationcorresponds to the simplicial map Fun C ∆ n ( i, j ) × Fun C ∆ n ( j, k ) ◦ / / ∼ = Fun C ∆ n ( i, j ) ∼ = ⊓⊔ × ( j − i − × ⊓⊔ × ( k − j − / / ⊓⊔ × ( k − i − which maps the pair of vertices T ⊂ [ i, j ] and S ⊂ [ j, k ] to T ∪ S ⊂ [ i, k ] .Again for i < j , an r -arrow T • in Fun C ∆ n ( i, j ) corresponds to a sequence T ⊂ T ⊂ · · · ⊂ T r of subsets of [ i, j ] = { i ≤ t ≤ j } and is non-degenerate if and only if each of these inclusionsare proper. The composite of a pair of r -arrows T • : i → j and S • : j → k is the levelwiseunion T • ∪ S • : i → k of these sequences. From this description, it is easy to see that the simplicial category C ∆ n is a simplicialcomputad (Lemma VI.4.2.5), in which an r -arrow T • from i to j is atomic if and only ifthe set T = { i, j } ; the only atomic r -arrows from j to j are identities. Geometrically, theatomic arrows in each function complex Fun C ∆ n ( i, j ) ∼ = ⊓⊔ j − i − are precisely those simplicesthat contain the initial vertex in the poset whose nerve defines the simplicial cube.If X is a simplicial subset of ∆ n , then Lemma VI.4.3.9 tells us that its homotopy coherentrealisation C X is a simplicial subcomputad of C ∆ n .2.3.2. Example (homotopy coherent nerves of subsimplices VI.4.4.3) . In particular:(i) boundaries:
The inclusion C ∂ ∆ n ֒ → C ∆ n is the identity on objects and full onall function complexes except for the one from to n . The inclusion Fun C ∂ ∆ n (0 , n ) (cid:31) (cid:127) / / ∼ = Fun C ∆ n (0 , n ) ∼ = ∂ ⊓⊔ n − (cid:31) (cid:127) / / ⊓⊔ n − is isomorphic to the cubical boundary inclusion, where ∂ ⊓⊔ k is the domain of theiterated Leibniz product ( ∂ ∆ ⊂ ∆ ) b × k . (ii) inner horns: The inclusion C Λ n,k ֒ → C ∆ n is identity on objects and full on allfunction complexes except for the one from to n . The inclusion Fun C Λ n,k (0 , n ) (cid:31) (cid:127) / / ∼ = Fun C ∆ n (0 , n ) ∼ = ⊓⊓ n − ,k (cid:31) (cid:127) / / ⊓⊔ n − is isomorphic to the cubical horn inclusion, defined by the the following Leibnizproduct: ( ∂ ∆ ⊂ ∆ ) b × ( j − b × (∆ { } ⊂ ∆ ) b × ( ∂ ∆ ⊂ ∆ ) b × ( k − j ) Definition (bead shapes) . We shall call those atomic arrows T • : 0 → n of C ∆ n whichare not members of C ∂ ∆ n bead shapes . By Examples 2.3.1 and 2.3.2, an r -dimensional beadshape T • : 0 → n is given by a sequence of subsets { , n } = T ⊂ T ⊂ · · · ⊂ T r = [0 , n ] with T = { , n } and T r equal to the full interval [0 , n ] = { ≤ t ≤ n } .More generally, any simplicial category C X arising as the homotopy coherent realizationof a simplicial set X defines a simplicial computad whose atomic arrows ( x, T • ) , describedin Proposition 2.3.4, are called beads in X . As a consequence of this result we find that r -simplices of C X correspond to sequences of abutting beads, structures which are called necklaces in the work of Dugger and Spivak [7] and Riehl [13]. In this terminology, C X is a simplicial computad in which the atomic arrows are those necklaces that consist of asingle bead with non-degenerate image. For more details about the Leibniz or “pushout-product” construction see [17, §4].
ARTESIAN EXPONENTIATION AND MONADICITY 15
Proposition ( C X as a simplicial computad; VI.4.4.7) . The homotopy coherent re-alization C X of a simplicial set X is a simplicial computad with • objects the vertices of X and • non-degenerate atomic r -arrows given by pairs ( x, T • ) , wherein x is a non-degenerate n -simplex of X for some n > r and T • : 0 → n is an r -dimensional bead shape.The domain of ( x, T • ) is the initial vertex x of x while the codomain is the terminal vertex x n . The point of this review is to permit us to define weights for oplax colimits of homo-topy coherent diagrams valued in a simplicially (or frequently quasi-categorically) enrichedcategory. In a homotopy coherent diagram, the indexing shape is given by the homotopycoherent realization of a simplicial set X . In this context, the join operation X ⋆ ∆ pro-duces another simplicial set with a freely adjoined cocone vertex. We shall argue thatthe its homotopy coherent realization defines a collage that presents the weight for oplaxcolimits.2.3.5. Recall.
For any simplicial set X , there is a canonical inclusion X ֒ → X ⋆ ∆ into itsjoin with the point. The join X ⋆ ∆ has a single vertex of X ⋆ ∆ that is not also a vertexof its subset X , which we shall denote by “ ⊤ .” Now for each non-degenerate n -simplex x ∈ X the join X ⋆ ∆ has two corresponding non-degenerate simplices: • a simplex of dimension n identified with x itself and • a simplex ( x, ⊤ ) of dimension n + 1 ,and these two cases enumerate all of the non-degenerate simplices of X ⋆ ∆ with theexception of ⊤ .2.3.6. Definition (weights for oplax colimits) . Applying homotopy coherent realisation tothe canonical inclusion
X ֒ → X ⋆ ∆ , we obtain a simplicial subcomputad I X : C X ֒ → C [ X ⋆ ∆ ] for any simplicial set X .Now from Proposition VI.4.4.7 we know that C [ X ⋆ ∆ ] may be built from C X byadjoining atomic arrows corresponding to beads (( x, ⊤ ) , T • ) and these all have codomain ⊤ . It is clear that the conditions discussed in Proposition 2.1.4(i) hold for the inclusion h⊤ , I X i : + C X ֒ → C [ X ⋆ ∆ ] . Hence, via the counit isomorphism of the collage adjunction,this simplicial category is isomorphic to the collage of the corresponding weight C X op L X / / S Set given by L X ( x ) := Fun C [ X⋆ ∆ ] ( x, ⊤ ) . We refer to L X as the weight for oplax colimits of diagrams of shape C X . When F : C X →K is a homotopy coherent diagram of shape X , then its oplax colimit is defined to be theweighted colimit colim oplax F := colim L X F, should this exist in K .2.3.7. Remark.
The oplax weights being defined here are precisely the “pseudo” weightsintroduced in Definition VII.5.2.8. The reason for the difference in nomenclature is that inthat paper the diagrams considered in [26] are valued in Kan complex enriched categories, whereas here the diagrams are valued in quasi-categorically (or simplicially) enriched cat-egories. In a Kan complex, the 1-simplex ∆ represents an invertible morphism, while ina quasi-category it models a non-invertible morphism.Immediately from Proposition 2.2.5:2.3.8. Lemma (VII.5.2.9) . For all simplicial sets X the weight L X : C X op → S Set for oplaxcolimits of diagrams of shape C X is a flexible weight. Cocartesian fibrations and quasi-categorical collages
In this section, we construct an explicit example of an oplax colimit of diagram of quasi-categories via the quasi-categorical collage construction , which we introduce in §3.3. In animportant special case, the quasi-categorical collage defines a cocartesian fibration over the1-simplex, so we begin in §3.1 with a review of this notion, since we shall need it anyways.In §3.2, we introduce the quasi-categorically enriched category of cocartesian fibrations andcartesian functors and observe that pullback defines a functor between such categories.3.1.
Cocartesian fibrations of quasi-categories.
Of the many equivalent definitionsof cocartesian fibration (see §IV.4 and §VI.3), the following will be the most convenient forthis paper:3.1.1.
Definition (IV.4.1.24) . Let p : E ։ B be an isofibration between quasi-categories.(i) A 1-arrow χ : e → e ′ of E is p - cocartesian if and only if any lifting problem ∆ { , } χ * * / / Λ n, (cid:15) (cid:15) / / E p (cid:15) (cid:15) (cid:15) (cid:15) ∆ n / / : : ✈✈✈✈✈ B has a solution.(ii) An isofibration p : E ։ B is a cocartesian fibration of quasi-categories preciselywhen any arrow α : pe → b in B admits a lift to an arrow χ : e → e ′ in E whichenjoys the lifting property of (i).For efficiency of exposition, we focus largely on the cocartesian fibrations, and leave it tothe reader to formulate the dual statements for cartesian fibrations, obtained by replacingeach simplicial set by its opposite.3.1.2. Example.
The product projection π : A × B ։ B defines a bifibration , that is, botha cocartesian and a cartesian fibration. A 1-arrow of A × B is π -cocartesian (and also π -cartesian) just when its component in A is an isomorphism.3.1.3. Example (IV.4.1.16, IV.5.2.3) . For any quasi-category B , we write B := B ∆ forits cotensor with the 1-simplex and p , p : B ։ B for the evaluation maps at the vertices , ∈ ∆ respectively. The codomain functor p : B ։ B is a cocartesian fibration, inwhich the p -cocartesian arrows are those whose projections along p are invertible. Dually, ARTESIAN EXPONENTIATION AND MONADICITY 17 the domain functor p : B ։ B is a cartesian fibration, in which the p -cartesian arrowsare those whose codomain components are invertible.More generally, for any cospan of quasi-categories f : B → A and g : C → A , the commaquasi-category is defined by the pullback f ↓ g ( p ,p ) (cid:15) (cid:15) (cid:15) (cid:15) / / A ( p ,p ) (cid:15) (cid:15) (cid:15) (cid:15) C × B g × f / / A × A and once more p : f ↓ g ։ C is a cocartesian fibration and p : f ↓ g ։ B is a cartesianfibration.In the special case where one of the functors in the cospan is taken to be the identity, wewrite f ↓ A and A ↓ f for what we call the contravariant and covariant representable commaquasi-categories respectively. In the special case where the functor a : 1 → A identifiesa vertex of A , the codomain projection p : a ↓ A ։ A is a cocartesian fibration thatencodes the covariant representable functor associated to a , while the domain projection p : a ↓ A ։ A is a cartesian fibration encoding the contravariant representable functor.These define the images of a in the co- and contravariant Yoneda embeddings of §VI.6.2.3.1.4. Lemma (VI.3.2.4) . If p : E ։ B is a cocartesian fibration and X is a simplicialset, then p X : E X ։ B X is a cocartesian fibration in which a 1-arrow e : ∆ → E X is p X -cocartesian just when for each vertex x ∈ X its component e ( x · σ , id [1] ) : ∆ → E is p -cocartesian. In §VI.3, a p X -cocartesian 1-arrow is called a pointwise p -cocartesian cylinder .3.1.5. Lemma (VI.3.2.5) . Let
X ֒ → Y be a simplicial subset of a simplicial set Y .(i) Any lifting problem X × ∆ ∪ Y × ∆ { } e / / (cid:127) _ (cid:15) (cid:15) E p (cid:15) (cid:15) (cid:15) (cid:15) Y × ∆ b / / ¯ e B in which the cylinder X × ∆ ⊆ X × ∆ ∪ Y × ∆ { } e −→ E is pointwise p -cocartesianadmits a solution ¯ e which is also pointwise p -cocartesian.(ii) Any lifting problem ( n > ) X × ∆ n ∪ Y × Λ n,n e / / (cid:127) _ (cid:15) (cid:15) E p (cid:15) (cid:15) (cid:15) (cid:15) Y × ∆ n b / / ¯ e B in which the cylinder Y × ∆ { n − ,n } ⊆ X × ∆ n ∪ Y × Λ n,n e −→ E is pointwise p -cocartesian admits a solution ¯ e . Definition. If p and q are cocartesian fibrations over B then a functor E p (cid:31) (cid:31) (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ g / / F q (cid:127) (cid:127) (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) B is a cartesian functor just when it carries p -cocartesian 1-arrows to q -cocartesian 1-arrows.As one illustration of the importance of this notion, we have the following importantproposition:3.1.7. Proposition (VIII.5.1.3) . A cartesian functor E g / / p (cid:31) (cid:31) (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ F q (cid:127) (cid:127) (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) B between cocartesian fibrations of quasi-categories is an equivalence if and only if it is afiberwise equivalence: for each b ∈ ob B , the induced functor g b : E b → F b is an equivalence. The quasi-categorically enriched category of cocartesian fibrations. If B isa quasi-category, then we adopt the notation Q Cat / B for the quasi-categoricaly enrichedcategory of isofibrations over B defined as follows.3.2.1. Definition.
For a quasi-category B , let Q Cat / B denote the category whose: • objects are isofibrations p : E ։ B with codomain B and • whose function complexes Fun B ( p : E ։ B , q : F ։ B ) are defined by the pullbacks Fun B ( p : E ։ B , q : F ։ B ) (cid:15) (cid:15) (cid:15) (cid:15) / / Fun ( E , F ) q ◦− (cid:15) (cid:15) ∆ p / / Fun ( E , B ) where Fun ( E , F ) ∼ = F E denotes the usual internal hom in Q Cat .3.2.2.
Definition.
For a quasi-category B , let coCart ( Q Cat ) / B denote the category whose: • objects are cocartesian fibrations p : E ։ B with codomain B and • whose function complexes Fun c B ( p : E ։ B , q : F ։ B ) are defined to be the full subquasi-categories of the function complexes Fun B ( p : E ։ B , q : F ։ B ) of Q Cat / B definedby restricting the 0-arrows to be cartesian functors over B .The quasi-categorically enriched category Cart ( Q Cat ) / B of cartesian fibrations and cartesianfunctors is defined similarly. ARTESIAN EXPONENTIATION AND MONADICITY 19
Proposition IV.5.2.1 proves that the pullback of a cocartesian fibration is a cocartesianfibration F q (cid:15) (cid:15) (cid:15) (cid:15) g / / E p (cid:15) (cid:15) (cid:15) (cid:15) A f / / B in which an arrow χ is q -cocartesian if and only gχ is p -cocartesian. It follows that pullbackalso preserves cartesian functors. Hence:3.2.3. Proposition.
Pullback along any f : A → B defines a quasi-categorically enrichedfunctor coCart ( Q Cat ) / B f ∗ / / ⊃ coCart ( Q Cat ) / A ⊃ Q Cat / B f ∗ / / Q Cat / A We now argue that the pullback functor preserves simplicial tensors. This will be usedin §6 to show that its right adjoint is simplicially enriched, when this functor exists.3.2.4.
Observation (tensors and pullback) . Let X ∈ sSet be a simplicial set. The tensor ofan isofibration p : E ։ B with X is the right-hand vertical composite, which pulls back tothe right-hand vertical composite F × X π (cid:15) (cid:15) (cid:15) (cid:15) / / E × X π (cid:15) (cid:15) (cid:15) (cid:15) F f ∗ ( p ) (cid:15) (cid:15) (cid:15) (cid:15) / / E p (cid:15) (cid:15) (cid:15) (cid:15) A f / / B which defines the tensor of f ∗ ( p ) : F ։ A with X .The following lemma tells us that this tensor construction respects cartesian functors.3.2.5. Lemma.
For any simplicial set X and cocartesian fibrations p : E ։ B and q : F ։ B ,the isomorphism Fun B ( E × X, F ) ∼ = Fun B ( E , F ) X restricts to an isomorphism Fun c B ( E × X, F ) ∼ = Fun c B ( E , F ) X . Proof.
We make use of Theorem IV.5.1.4 which provides the following characterization ofthe sub quasi-category
Fun c B ( E , F ) ⊂ Fun B ( E , F ) . Any functor f : E → F over B induces acommutative square over B E f / / (cid:15) (cid:15) F (cid:15) (cid:15) p ↓ B ( f, id B ) / / ℓ ⊣ D D ✴✤ ✎ q ↓ B ℓ ⊢ Z Z ✎ ✤✴ whose vertical functors are the canonical ones induced by p : E ։ B and q : F ։ B . Because p and q are cocartesian, Theorem IV.4.1.10 proves the vertical functors admit left adjointsover B . Theorem IV.5.1.4 proves that f is cartesian if and only if the mate of this canonicalisomorphism is an isomorphism.The mate that detects whether f is a cartesian functor lives as a 1-simplex in thesimplicial set Sq B ( p ↓ B → E , q ↓ B → F ) := Fun B ( E , F ) × Fun B ( p ↓ B , F ) Fun B ( p ↓ B , q ↓ B ) . of commutative squares from ℓ : p ↓ B → E to ℓ : q ↓ B → F . The adjunction over B associatedto the cocartesian fibration E × X π −− ։ E p −− ։ B is E × X ⊥ / / pπ " " ❋❋❋❋❋❋❋❋❋ p ↓ B × X p π z z ✉✉✉✉✉✉✉✉✉✉ ℓ s s B the product of the adjunction for p with X . In particular, Sq B ( p ↓ B × X → E × X, q ↓ B → F ) ∼ = Sq B ( p ↓ B → E , q ↓ B → F ) X . Now a 1-simplex C X is an isomorphism if and only if it is a pointwise isomorphism, whichproves that Fun c B ( E × X, F ) ∼ = Fun c B ( E , F ) X . (cid:3) Collages for quasi-categories.
We conclude this section with an example of anoplax colimit. When X = ∆ a homotopy coherent diagram C ∆ → Q Cat is just a functor f : A → B between quasi-categories. The oplax colimit in simplicial sets is given by thepushout A f / / id × δ (cid:15) (cid:15) B (cid:15) (cid:15) A × ∆ / / colim oplax f Our aim is to prove Proposition 3.3.5 which demonstrates that this oplax colimit is modeledup to equivalence by the quasi-categorical collage construction that we now introduce.3.3.1.
Definition (the quasi-categorical collage construction) . Consider any cospan f : A → C and g : B → C , with A , B , and C all quasi-categories. Define a new simplicial set coll( f, g ) by declaring that coll( f, g ) n = n(cid:16) ∆ i a −→ A , ∆ j b −→ B , ∆ n c −→ C (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) c | { ,...,i } = f ( a ) ,c | { n − j,...,n } = g ( b ) , i, j ≥ − ,i + j = n − . o ARTESIAN EXPONENTIATION AND MONADICITY 21 with the convention that conditions indexed by ∆ − are empty (or that each simplicial setis terminally augmented). There are simplicial maps B coll( f, g ) A { } ∆ { } y ρ x the top ones being the evident inclusions. The map ρ sends an n -simplex ( a : ∆ i → A , b : ∆ j → B , c : ∆ n → C ) to the n -simplex [ n ] → [1] that carries , . . . , i to and i +1 , . . . , n to . Note that the fiber of ρ over is isomorphic to A while the fiber of ρ over is isomorphic to B .3.3.2. Remark (on right and left) . As with simplicial collages, we customarily write B + A ֒ → coll( f, g ) for the inclusions of the fibers over and — with the fiber over 1 on the left andthe fiber over 0 on the right. As with our convention for quasi-categories in Example 3.1.3,this positions the covariantly-acting quasi-category on the “left” and the contravariantly-acting quasi-category on the “right.”3.3.3. Lemma.
The map ρ : coll( f, g ) → ∆ is an inner fibration. In particular, thesimplicial set coll( f, g ) is a quasi-category.Proof. Since the fibers of ρ over and are the quasi-categories A and B , it suffices toconsider inner horns Λ n,k / / (cid:15) (cid:15) coll( f, g ) ρ (cid:15) (cid:15) ∆ n α / / : : ttttt ∆ for which α : [ n ] → [1] is a surjection. Suppose α carries , . . . , i to and i + 1 , . . . , n to . Note that for any < k < n , the faces { , . . . , i } and { i + 1 , . . . , n } of ∆ n belong tothe horn Λ n,k . In particular, the map Λ n,k → coll( f, g ) identifies simplices a : ∆ i → A and ∆ n − i − → B together with a horn Λ n,k → C whose initial and final faces are the imagesof these simplices under f : A → C and g : B → C . Since C is a quasi-category this hornadmits a filler c : ∆ n → C and the triple ( a, b, c ) defines an n -simplex in coll( f, g ) thatsolves the lifting problem. (cid:3) We write coll( f, B ) for the collage of f : A → B with the identity on B .3.3.4. Lemma.
For any f : A → B , the map ρ : coll( f, B ) → ∆ is a cocartesian fibration.Proof. To prove the claim, we need only specify cocartesian lifts of the non-degenerate 1-simplex of ∆ and demonstrate that these edges have the corresponding universal property.To that end, for any vertex a ∈ A , let χ a : ∆ → coll( f, B ) be the 1-simplex χ a := ( a : ∆ → A, f a : ∆ → B, f a · σ : ∆ → B ) , defined by the degenerate edge at f a ∈ B lying over the 1-simplex in ∆ . To show that χ a is ρ -cocartesian, we must construct fillers for any left horn ∆ { , } χ a * * / / Λ n, (cid:15) (cid:15) / / coll( f, B ) ρ (cid:15) (cid:15) ∆ n : : ttttt β / / ∆ whose initial edge is χ a . Note that this condition implies that the bottom map β : [ n ] → [1] carries to and the remaining vertices to . The map Λ n, → coll( f, B ) defines a horn Λ n, → B in the quasi-category B whose first edge is degenerate. By Joyal’s lemma aboutfilling “special outer horns,” such horns admit a filler b : ∆ n → B and the triple ( a : ∆ → A , b · δ : ∆ n − → B , b : ∆ n → B ) defines an n -simplex in coll( f, B ) that solves the lifting problem. (cid:3) Proposition.
For any f : A → B between quasi-categories, the collage coll( f, B ) defines the oplax colimit of f in Q Cat . That is coll( f, B ) defines a cone under the pushoutdiagram A f / / (cid:127) _ id × δ (cid:15) (cid:15) B (cid:127) _ (cid:15) (cid:15) (cid:15) o (cid:22) (cid:22) A × ∆ / / h + + P k ❍❍❍❍❍ coll( f, B ) so that the induced map k is inner anodyne, and in particular a weak equivalence in theJoyal model structure.Proof. The map k is a quotient of the map h , which has the following explicit description.For each n -simplex ( a, α ) : ∆ n → A × ∆ define i := | α − (0) | − , so that − ≤ i ≤ n .Then h carries ( a, α ) to the n -simplex of coll( f, B ) corresponding to the triple ( a | { ,...,i } : ∆ i → A , f a | { i +1 ,...,n } : ∆ n − i − → B , f a : ∆ n → B ) . Note that the composite ρh : A × ∆ → ∆ is the projection.It remains to present k as a sequential composite of pushouts of coproducts of inner horninclusions. To do so, first note that coll( f, B ) n = A n a A n − × B n − B n a · · · a A × B B n a B n where each map B n → B i is the initial face map corresponding to ∆ { ,...,i } ֒ → ∆ n . From theperspective of this decomposition, P n is the subset containing the sets A n and B n and thesubset of A i × B i B n whose component in B n is in the image of f . The n -simplices of coll( f, B ) that remain to be attached correspond to elements of A i × B i B n , for ≤ i < n , that are not ARTESIAN EXPONENTIATION AND MONADICITY 23 in the image of f in the sense just discussed. Note in particular that k : P ֒ → coll( f, B ) is an isomorphism and k : P n ֒ → coll( f, B ) n is an injection for all n ≥ .To enumerate our attaching maps, we start with the collection of non-degenerate n -simplices of coll( f, B ) for n ≥ that are not in the image of f and remove also thoseelements of A i × B i B n whose components b ∈ B n are in the image of the degeneracy map σ i : B n − → B n . Partially order this set of simplices first in the order of increasing n andthe in order of increasing index i ; that is we lexicographically order the collection of pairs ( n, i ) with n ≥ and ≤ i < n . We will filter the inclusion P ֒ → coll( f, B ) as P ֒ → P (1 , ֒ → P (2 , ֒ → P (2 , ֒ → P (3 , ֒ → · · · ֒ → P ( n,i ) ֒ → · · · ֒ → colim ∼ = coll( f, B ) where the simplicial set P ( n,i ) is built from the previous one by a pushout of a coproductof inner horns indexed by the set of n -simplices ( a, b ) ∈ A i × B i B n with b not in the imageof f or σ i . The filler for the horn indexed by ( a, b ) will attach this n simplex to B n as themissing face of the horn and also the n + 1 simplex ( a, bσ i ) ∈ A i × B i B n +1 .Consider a simplex ( a, b ) ∈ A i × B i B n with b not in the image of f or σ i . Define a horn Λ n +1 ,i +1 / / (cid:127) _ (cid:15) (cid:15) P ( n,i ) (cid:127) _ (cid:15) (cid:15) ∆ n +1 ( a,bσ i ) / / coll( f, B ) For each ≤ j < i + 1 , the δ j -face of the n + 1 simplex ( a, bσ i ) is the n -simplex ( aδ j , bσ i δ j ) ,which lies in P ( n,i − or in B ֒ → P in the case i = 0 . For each i + 1 < j ≤ n + 1 , the δ j -faceof the n + 1 simplex ( a, bσ i ) is the n -simplex ( a, bσ i δ j ) = ( a, bδ j − σ i ) ∈ A i × B i B n , whichwas previously attached to P ( n − ,i ) . So the Λ n +1 ,i +1 indeed maps to P ( n,i ) , permitting aninductive construction of the next simplicial set in this sequence as the pushout ` ∼ Λ n +1 ,i +1 / / (cid:127) _ (cid:15) (cid:15) P ( n,i ) (cid:15) (cid:15) ` ∼ ∆ n +1 / / P ( n,i )+1 where P ( n,i )+1 equals P ( n +1 , in the case i = n − and P ( n,i +1) otherwise. (cid:3) Corollary.
Consider a pair of functors between quasi-categories f : A → B and u : B → A . Then f is left adjoint to u if and only if the collages coll( f, B ) and coll( A , u ) are equivalent under B + A and over ∆ .Proof. First suppose that coll( f, B ) ≃ coll( A , u ) under B + A and over ∆ . By Lemma 3.3.4this means that the map coll( f, B ) → ∆ is both a cocartesian and a cartesian fibration, a bifibration in the terminology of §IV.4. By Proposition IV.4.1.20 it follows that the 1-arrowin ∆ from to induces an adjunction between the fibers A and B . By inspection of thatproof, the left adjoint functor so-constructed in the case of the bifibration coll( f, B ) → ∆ is f ; substituting the equivalent bifibration coll( A , u ) → ∆ , we see that the right adjointis equivalent to u .For the converse, we work in the opposite ∞ -cosmos Q Cat op , an ∞ -cosmos in which “notall objects are cofibrant,” as described in Observation IV.2.2.2. In that context, Proposition3.3.5 proves that coll( f, B ) and coll( A , u ) construct the contravariant and covariant commaobjects associated to the functors f and u . If f ⊣ u in Q Cat then these functors arealso adjoint in Q Cat op and Proposition I.4.4.2 then proves that the commas coll( f, B ) and coll( A , u ) are equivalent under B + A . By construction, this equivalence also lies over ∆ . (cid:3) The comprehension construction
In this section we review the comprehension construction from [23] in considerably lessgenerality than given in that source. It constructs, for any cocartesian fibration p : E ։ B of quasi-categories, a “straightening,” which has the form of a simplicial functor c p : C B →Q Cat that sends each vertex b ∈ B to the fiber E b . It also constructs a canonical lax cocone ℓ E : C [ B ⋆ ∆ ] → Q Cat of shape B under this diagram with nadir E , the lax colimits ofrestrictions of which will be used in §5 to model pullbacks along the functor p .The underlying mechanics of the comprehension construction are reviewed in §4.1 andthe comprehension construction itself is given in §4.2.4.1. Cocartesian transformations between lax cocones.
Corollary 2.1.5 tells us thatthe collage of a weight W realizes the shape of W -cocones. Applying this result to theweights for oplax colimits introduced in Definition 2.3.6, we obtain the following definitionof a lax cocone .4.1.1. Definition (lax cocones VI.5.2.4) . Suppose that X is a simplicial set. Then a laxcocone of shape X in S Set is defined to be a simplicial functor ℓ B : C [ X ⋆ ∆ ] → S Set + C X jJ w w ♦♦♦♦♦♦♦♦♦ h B,B • i % % ❑❑❑❑❑❑❑❑ C [ X ⋆ ∆ ] ℓ B / / S Set
The restriction of a lax cocone ℓ B : C [ X ⋆ ∆ ] → K to a functor B • : C X → S Set is called its base . We say that ℓ B is a lax cocone under the diagram B • ; the object B ∈ sSet obtainedby evaluating ℓ B at the object ⊤ is called the nadir of that lax cocone.4.1.2. Remark.
In the original Definition VI.5.2.4, the target was required to be a quasi-categorically enriched category K and the base of a lax cocone was required to factorthrough through the inclusion g ∗ K ⊆ K of the maximal Kan complex enriched subcategory.The point of this requirement was so that the transpose of the base diagram defined adiagram X → N g ∗ K valued in the large quasi-category of objects and morphisms in K .But in this paper we will frequently consider lax cocones whose codomain is S Set , in whichcase the requirement that the base diagram factor through the local groupoidal core won’t
ARTESIAN EXPONENTIATION AND MONADICITY 25 necessarily make sense and should be dropped. We shall also be more interested in thetotality of the data of a lax cocone than in studying the base diagram in isolation.4.1.3.
Definition (canonical lax cocones VI.6.1.6) . For any simplicial set X , there existsa lax cocone + C X jJ w w ♦♦♦♦♦♦♦♦♦ h X, i % % ❑❑❑❑❑❑❑❑ C [ X ⋆ ∆ ] k X / / S Set whose base is constant at the terminal quasi-category and whose nadir is X that we referto as the canonical X -shaped lax cocone .To define k X : C [ X ⋆ ∆ ] → S Set , it remains to define the images in
Fun (1 , X ) ∼ = X of the atomic r -arrows with domain a vertex x in X and with codomain ⊤ . ApplyingProposition 2.3.4, each atomic r -arrow from x to ⊤ corresponds to a bead (( x, ⊤ ) , T • ) represented by some non-degenerate n -simplex x ∈ X whose initial vertex is x and atomic r -arrow T • : 0 → n + 1 in C ∆ n +1 defined by { , n + 1 } = T ⊂ T ⊂ · · · ⊂ T r = [0 , n + 1] . Define µ : ∆ r → ∆ n by µ ( i ) = max( T i \{ n + 1 } ) . Then the image of the bead (( x, ⊤ ) , T • ) is the r -simplex x · µ ∈ X in the hom-quasi-category X ∼ = Fun (1 , X ) from the image of thedomain vertex x of x .4.1.4. Observation (whiskering lax cocones VI.5.2.6) . Let ℓ A : C [ X ⋆ ∆ ] → S Set be a laxcocone with base diagram A • : C X → S Set and nadir ℓ A ⊤ = A , and let f : A → B be anymap of simplicial sets. Then there is a whiskered lax cocone f · ℓ A : C [ X ⋆ ∆ ] → S Set with the same base diagram A • : C X → S Set and with nadir B , whose components from avertex x ∈ X to ⊤ are defined by whiskering with f : Fun C [ X⋆ ∆ ] ( x, ⊤ ) ℓ Xx, ⊤ −−→ Fun ( A x , A ) f ◦− −−→ Fun ( A x , B ) Lemma.
For any map of simplicial sets f : Y → X , the canonical lax cocone ofshape X restricts along C [ f ⋆ id] : C [ Y ⋆ ∆ ] → C [ X ⋆ ∆ ] to the whiskered composite + C Y + C X + C Y C [ Y ⋆ ∆ ] C [ X ⋆ ∆ ] S Set C [ Y ⋆ ∆ ] S Set + C f h X, i = h X, i C [ f⋆ id] k X f · k Y of the canonical lax cocone of shape Y with f : Y → X .Proof. By direct verification from Definition 4.1.3 and Observation 4.1.4. (cid:3) §VI.5 introduces a mechanism for producing new lax cocones from given ones: namelyas domain components of cocartesian cocones over a given codomain lax cocone.
Definition (cocartesian cocones VI.5.3.1) . Suppose we are given a simplicial set X and lax cocones ℓ E , ℓ B : C [ X ⋆ ∆ ] → S Set of shape X with bases E • and B • respectively.Suppose also that we are given a simplicial natural transformation C [ X ⋆ ∆ ] ℓ B ℓ E + + ⇓ p S Set . Then we say that the triple ( ℓ E , ℓ B , p ) is a cocartesian cocone if and only if(i) the nadir of the natural transformation p , that being its component p : E ։ B atthe object ⊤ , is a cocartesian fibration between quasi-categories(ii) for all -simplices x ∈ X the naturality square E xp x (cid:15) (cid:15) (cid:15) (cid:15) ℓ Ex / / E p (cid:15) (cid:15) (cid:15) (cid:15) B x ℓ Bx / / B is a pullback, and(iii) for all non-degenerate -simplices f : x → y ∈ X the -arrow E xE f (cid:15) (cid:15) ℓ Ex ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ⇓ ℓ Ef E E y ℓ Ey ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ is p -cocartesian.In this situation we also say that the pair ( ℓ E , p ) defines a cocartesian cocone over ℓ B .4.1.7. Lemma (pullbacks of cocartesian cocones VI.5.3.3) . Suppose given: • a pullback diagram of quasi-categories F q (cid:15) (cid:15) (cid:15) (cid:15) g / / E p (cid:15) (cid:15) (cid:15) (cid:15) A f / / B (4.1.8) in which p and q are cocartesian fibrations; • a lax cocone ℓ A : C [ X ⋆ ∆ ] → K with nadir A ; and • a cocartesian cocone ( ℓ E , ℓ B , p ) whose nadir component is p : E ։ B and whose codomaincocone ℓ B = f · ℓ A is obtained from the lax cocone ℓ A by whiskering with f : A → B asin Observation 4.1.4. ARTESIAN EXPONENTIATION AND MONADICITY 27
Then there is a cocartesian cone ( ℓ F , ℓ A , q ) whose codomain is ℓ A , whose nadir componentis q : F ։ A , and whose domain component is a lax cocone ℓ F that whiskers with g to thelax cone ℓ E = g · ℓ F . Conversely, a cocartesian cocone ( ℓ F , ℓ A , q ) with nadir component q : F ։ A can bewhiskered with a pullback square (4.1.8) to define a cocartesian cocone ( g · ℓ F , f · ℓ A , p ) with nadir p : E ։ B and whose domain and codomain are whiskered lax cocones as definedin Observation 4.1.4.4.1.9. Remark.
If the map f of Lemma 4.1.7 is replaced by any map of simplicial sets f : X → B , whose domain is not necessarily a quasi-category, it is still possible to pullback the data of a cocartesian cocone ( ℓ E , ℓ B , p ) whose codomain lax cocone ℓ B = f · ℓ X is obtained by whiskering a lax cocone with nadir X . This constructs a simplicial naturaltransformation ( ℓ F , ℓ X , q ) whose nadir component is the pullback q : F → X of p along f .Since this is not a map between quasi-categories, it doesn’t really make sense to call it acocartesian fibration. Nonetheless, this construction produces a lax cocone ℓ F of shape X ,which will have some utility. See Remark 4.2.5.4.2. The comprehension construction.
The comprehension functor associated to acocartesian fibration p : E ։ B between quasi-categories is the base diagram of the domainof a cartesian cocone over the canonical B -shaped lax cocone.4.2.1. Theorem (VI.6.1.7) . For any cocartesian fibration p : E ։ B of quasi-categories,there is a cocartesian cocone C [ B ⋆ ∆ ] k B ℓ E + + ⇓ p Q Cat . of shape B in Q Cat with nadir component p : E → B over the canonical lax cocone k B ofDefinition 4.1.3 whose base is constant at . The base of the domain component defines thecomprehension functor c p , which acts on an object b : 1 → B of C B by forming the pullback E b ℓ B b / / p b (cid:15) (cid:15) (cid:15) (cid:15) E p (cid:15) (cid:15) (cid:15) (cid:15) b / / B and acts on 1-arrows f : a → b of B by factoring the codomain of a p -cocartesian lift ℓ E f of f through the pullback at the front of the diagram: B ◆◆◆◆◆◆◆◆◆◆◆◆◆◆ b a ) ) f w (cid:127) ①①①①①① E b E a E ℓ E b ℓ E a ) ) p b (cid:15) (cid:15) (cid:15) (cid:15) p a (cid:15) (cid:15) (cid:15) (cid:15) p (cid:15) (cid:15) (cid:15) (cid:15) E f & & ℓ E f v ~ ✉✉✉✉✉✉✉✉ (4.2.2) These cocartesian lifts define components of the lax cocone + C B jJ x x ♣♣♣♣♣♣♣♣ h E ,c p i % % ❑❑❑❑❑❑❑❑ C [ B ⋆ ∆ ] ℓ E / / Q Cat with nadir E under the comprehension functor. Remark.
The comprehension functor c p : C B → Q Cat is a “straightening” of thecocartesian fibration p : E ։ B , a homotopy coherent diagram of shape B that sends eachvertex b to the fiber E b of p over b .By Observation VI.6.1.9, any pair of lax cocones that arise as the domain of a cocartesiancocone over a common codomain define vertices in a contractible Kan complex and arein particular equivalent as diagrams. This is used to prove the following result relatingcomprehension functors with pullbacks.4.2.4. Proposition (comprehension and change of base VI.6.1.11) . Suppose that we aregiven a pullback F q (cid:15) (cid:15) (cid:15) (cid:15) g / / E p (cid:15) (cid:15) (cid:15) (cid:15) A f / / B of quasi-categories in which p and thus q are cocartesian fibrations. Then the diagrams C A c q −→ Q Cat and C A C f −→ C B c p −→ Q Cat are connected by a homotopy coherent natural isomorphism.
Remark. If A is not a quasi-category, it is not possible to directly construct thecomprehension functor for the pullback of p along f . However, by Lemmas 4.1.5 andRemark 4.1.9, the cocartesian cocone over the canonical B -shaped lax cocone can be pulledback along any map of simplicial sets f : X → B to define a cocartesian cocone over the ARTESIAN EXPONENTIATION AND MONADICITY 29 canonical X -shaped lax cocone. Thus, a posteriori, we can think of the base of the laxcocone C [ X ⋆ ∆ ] C [ f⋆ id] −−−−→ C [ B ⋆ ∆ ] ℓ E −→ Q Cat as defining a comprehension functor for the pullback of p : E ։ B along f : X → B .5. Pullback along a cocartesian fibration as an oplax colimit
Our aim in this section is to provide an equivalent model of the pullback functor p ∗ : sSet / B −→ sSet / E along a cocartesian fibration p : E ։ B between quasi-categories. In the next section, wewill use this to construct an equivalent model of its right adjoint, the pushforward Π p ,whose homotopical properties are more easily established. Before commencing with ourwork, we briefly sketch the connection between the pullback and pushforward.The category of simplicial sets, as a presheaf topos, is locally cartesian closed: in otherwords any simplicial map p : E → B is exponentiable in the sense that pullback along p admits a right adjoint: sSet / E sSet / B Π p ⊥ p ∗ By Observation 3.2.4, pullback along p is a simplicially enriched functor that preservestensors with simplicial sets, so by Theorem 4.85 of Kelly [11] it follows that the adjunction p ∗ ⊣ Π p is simplicially enriched.The right adjoint may be described explicitly:5.0.1. Lemma.
The n -simplices of the pushforward Π p ( q : F → E ) correspond to paircomprised of an n -simplex b : ∆ n → B together with a map E b → F in sSet / E , whosedomain is defined by the pullback E b e b / / p b (cid:15) (cid:15) E p (cid:15) (cid:15) ∆ n b / / B Moreover, under the representation Π p ( q : F → E ) n ∼ = a b : ∆ n → B Hom E E be b (cid:15) (cid:15) E , F q (cid:15) (cid:15) E ! a simplicial operator α : [ m ] → [ n ] acts on an n -simplex by pre-composition with E b · α E α / / p b · α (cid:15) (cid:15) E bp b (cid:15) (cid:15) ∆ m α / / ∆ n Proof.
A generalised element g : X → Π p ( q : F → E ) gives a map X b (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ g / / Π p ( F, q ) { { ✇✇✇✇✇✇✇✇✇ B in sSet / B in which we define b := g ¯ q . Transposing this under the adjunction p ∗ ⊣ Π p , weget a corresponding map E b f / / e b ❆❆❆❆❆❆❆ F q (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ E where E b e b / / p b (cid:15) (cid:15) E p (cid:15) (cid:15) X b / / B in sSet / E . The claim follows by specializing to the case X = ∆ n . (cid:3) To study the pushforward construction along a cocartesian fibration p : E ։ B , wewill replace the test objects e b : E b → E involved the description of Π p given in Lemma5.0.1 by weakly equivalent test objects in the Joyal model structure, whose form will bemore amenable to computations. These new test objects ˜ e b : ˜ E b → E will arise as certainweighted colimits of a fixed diagram c p : C B → Q Cat ⊆ S
Set , namely the straightening ofthe cocartesian fibration p defined using the comprehension construction.The construction of the replacement to the pullback functor is given in §5.1, and theproof that the pullback replacement is equivalent to the strict pullback is given in §5.2.5.1. A replacement for pullback along a cocartesian fibration.
Notation.
For the remainder of this section shall fix a cocartesian fibration of quasi-categories p : E ։ B as well as a corresponding comprehension functor C B c p / / Q Cat , the straightening of the cocartesian fibration p ; see Remark 4.2.3. We also fix the associatedlifted lax cocone + C B jJ x x ♣♣♣♣♣♣♣♣ h E ,c p i % % ❑❑❑❑❑❑❑❑ C [ B ⋆ ∆ ] ℓ E / / Q Cat (5.1.2)with nadir E described in Theorem 4.2.1.Our aim is to prove that this lax cocone is a colimit cocone. We will achieve this as the b = id B special case of our first main theorem:5.1.3. Theorem.
For any cocartesian fibration p : E ։ B and any b : X → B , the compre-hension cocone induces a canonical map over E from the oplax colimit of the diagram C X C b −→ C B c p −→ Q Cat
ARTESIAN EXPONENTIATION AND MONADICITY 31 to the fiber E b (cid:15) (cid:15) / / E p (cid:15) (cid:15) (cid:15) (cid:15) X b / / B and this map is a natural weak equivalence in the Joyal model structure. Before proving this result, we tighten up its statement. As we explain presently, thereis a functor ˜ p ∗ : S Set / B → S Set / E that acts on objects by carrying a generalized element b : X → B to a canonical map colim oplax ( c p ◦ C b ) → E . After defining this more formally,we construct a comparison natural transformation S Set / B S Set / E ˜ p ∗ p ∗ ⇓ γ (5.1.4)Theorem 5.1.3 asserts that this map is a componentwise Joyal equivalence. Our first taskis to give a precise definition of the functor ˜ p ∗ : S Set / B → S Set / E . For ease of exposition, wefirst describe its action on objects before establishing the functoriality of this construction.Recall from Remark 4.2.5 that the comprehension functor c p : C B → Q Cat can be used todefine a “straightening” of its pullbacks: C X C b / / C B c p / / S Set even in the case where X is not a quasi-category.5.1.5. Definition.
Given an generalized element b : X → B in S Set / B , define a simplicialset ˜ E b := colim oplax (cid:16) C X C B S Set C b c p (cid:17) . The oplax colimit ˜ E b is the nadir of the universal lax cocone under the diagram c p ◦ C X .This is the weighted colimit weighted by the weight for oplax colimits L X introduced inDefinition 2.3.6.The simplicial functor + C X + C B C [ X ⋆ ∆ ] C [ B ⋆ ∆ ] S Set + C b h E, E i C [ b⋆ id] ℓ E (5.1.6)defines a lax cocone ℓ E | b : C [ X ⋆ ∆ ] → S Set under the diagram c p ◦ C b with nadir E ,inducing a unique simplicial map ˜ e b : ˜ E b → E from the oplax colimit. This constructs anobject of S Set / E .To establish the functoriality of the construction of Definition 5.1.5, it will be convenientto re-express the oplax colimits of Definition 5.1.5. Lemma. (i) For any simplicial map b : X → B define a weight L b : C B op → S Set by taking theleft Kan extension along C b : C X op → C B op of the weight for oplax colimits. Thenfor any diagram F : C B → S Set , there is an isomorphism colim oplax ( F ◦ C b ) : − colim L X ( F ◦ C b ) ∼ = colim L b F. (ii) The weight L b : C B op → S Set is flexible and its collage is given by the pushout + C X + C B C [ X ⋆ ∆ ] C (( X ⋆ ∆ ) ∪ X +∆ ( B + ∆ )) ∼ = coll L b p + C b Proof.
By Lemma 2.1.6, the weighted colimit of a restricted diagram is isomorphic to thecolimit of the original diagram weighted by the left Kan extension of the weight. Thisspecializes to prove (i).By Lemma 2.1.7 the collage of the left Kan extended weight L b is computed by thepushout of (ii). Since coll( L b ) is the homotopy coherent realization of the pushout ofsimplicial sets, Proposition 2.3.4 tells us that it is a simplicial computad and thus, byProposition 2.2.5, L b is a flexible weight. (cid:3) Observation.
The utility of Lemma 5.1.7 is as follows. Suppose now that we have amap
X YB b u c in the slice category sSet / B . This gives rise to a commutative diagram of simplicial com-putads C [ X ⋆ ∆ ] + C X + C B C [ Y ⋆ ∆ ] + C Y + C B C u ⊲ + C u + C b + C c + (5.1.9)inducing a simplicial computad morphism coll( L b ) → coll( L c ) in the category + C B / sSet-Cptd.This construction is functorial, defining the horizontal functor in the following squaresSet / B + C B / sSet-CptdsSet C B op + C B / sSet-Cat coll L • L • coll By the description of the essential image of the collage functor given in Proposition 2.1.4,we see that coll L • factors as indicated defining a functor L • : sSet / B → + C B / sSet-Cat. ARTESIAN EXPONENTIATION AND MONADICITY 33
Finally note that the collage coll L B op ∼ = C B ⋆ ∆ for the weight for oplax colimits ofshape B defines a cone under the pushout diagram of Lemma 5.1.7(ii). Thus the codomainof the functor coll( L • ) lifts to the slice categorysSet / B (cid:0) + C B / sSet-Cptd (cid:1) / C B ⋆ ∆ coll L • Correspondingly, by the fully faithfulness of the collage construction, we can equally regard L • as a functor sSet / B (cid:0) sSet C B op (cid:1) /L B L • landing in the full subcategory spanned by the flexible weights.Observation 5.1.8 allows us to extend Definition 5.1.5 to a functor.5.1.10. Definition.
Define ˜ p ∗ : sSet / B → sSet / E to be the composite functor ˜ p ∗ : − sSet / B (cid:0) sSet C B op (cid:1) /L B op sSet / ˜ E B sSet / E L • colim − c p ˜ e B where ˜ E B : − colim oplax c p and ˜ e B : ˜ E B → E is the map induced by the lax cocone (5.1.2).For later use, we record a few properties of the functor just constructed.5.1.11. Lemma.
The functor ˜ p ∗ : sSet / B → sSet / E preserves colimits.Proof. In Definition 5.1.10 the functor under consideration is defined as a composite ofthree functors, the latter two of which manifestly preserve colimits. Since colimits ina slice category over an object are created by the forgetful functor, it remains only toprove that the functor L • : sSet / B → sSet C B op preserves colimits. Since Proposition 2.1.4demonstrates that the inclusion sSet C B op ֒ → + C B / sSet-Cat is full and coreflective, to showthat this functor preserves colimits, it suffices to show thatsSet / B + C B / sSet-Cat coll L • preserves them.By Observation 5.1.8, the action of this functor on objects and morphisms is defined bythe pushout of Lemma 5.1.7(ii), which we regard as a diagram in / sSet-Cat. The functorssSet / B / sSet-Cat + C ( − ) and sSet / B / sSet-Cat C ( − ) ⋆ ∆ both preserve colimits. Thus, the functor from sSet / B to the category of pushout diagramsin / sSet-Cat with one vertex fixed at + C B preserves colimits. The pushout preservescolimits as well so we conclude that coll( L • ) and hence ˜ p ∗ : sSet / B → sSet / E preservescolimits, as desired. (cid:3) Lemma.
The functor ˜ p ∗ : sSet / B → sSet / E preserves monomorphisms. Proof.
From the Definition 5.1.10, to see that ˜ p ∗ preserves monomorphisms X Y B ub c over B it suffices to show that the composite of the functors L • and colim C B ( − , c p ) preservemonomorphisms. To do so, we’ll prove that the comparison functor L u : L b → L c betweenweights in sSet C B op is a projective cell complex, as defined in 2.2.1. A theorem of Gam-bino [8] implies that colim C B ( − , c p ) carries projective cell complexes to monomorphisms insimplicial sets; see also [14, 11.5.1]Recall that the weight L b is constructed as a collage defined by a pushout, which is thehomotopy coherent realization of a pushout of simplicial sets. The natural transformation L u is encoded by the map between collages constructed as the pushout (5.1.9); again thismap is the homotopy coherent realization of a map of simplicial sets. Since the left-handhorizontal inclusions are also simplicial subcomputad inclusions, it follows from the stan-dard argument that the induced map coll( L u ) : coll( L b ) ֒ → coll( L c ) between the pushoutsis a simplicial subcomputad inclusion and by the relative analogue Proposition II.5.3.5 ofProposition 2.2.5, L u : L b → L c is a projective cell complex. This is what we needed toshow. (cid:3) Comparison with the strict pullback.
Now that we’ve precisely defined a functor ˜ p ∗ : sSet / B → sSet / E that carries a generalized element to the oplax colimit of the restrictedcomprehension functor, our next task is to define the natural transformation (5.1.4) alludedto in the statement of Theorem 5.1.3. To explain the existence of the natural map γ b : ˜ E b → E b for b : X → B , recall that the p -cocartesian lifts with codomain E used to define theaction of c p : C B → Q Cat on arrows in the image of C b : C X → C B lie over arrows withcodomain B which have a given factorisation through b : X → B . This is depicted in thefollowing diagram by the arrow bγ and its p -cocartesian lift ℓ E bγ : ∆ ∆ X BE bx ′ E bx ❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂ x ′ : : x % % b / / γ (cid:5) (cid:13) ✓✓✓✓✓✓✓✓✓✓ E b E ℓ E b / / p (cid:15) (cid:15) (cid:15) (cid:15) ℓ E bx ′ - - ℓ E bx ! ! p bx ′ (cid:15) (cid:15) (cid:15) (cid:15) p bx (cid:15) (cid:15) (cid:15) (cid:15) p b (cid:15) (cid:15) (cid:15) (cid:15) e bγ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ ℓ E bγ (cid:4) (cid:12) ✑✑✑✑✑✑✑✑✑✑ * * (cid:25) (cid:25) (cid:4) (cid:12) From the diagram, it is clear that such p -cocartesian arrows factor through e b : E b → E to give the dotted arrows with codomain E b as drawn; this is the main component of theproof of Lemma 4.1.7. This idea is formalized as follows: ARTESIAN EXPONENTIATION AND MONADICITY 35
Lemma.
For any b : X → B , the diagram c p ◦ C b is the base of a lax cocone withnadir E b + C X C [ X ⋆ ∆ ] S Set h E b ,c p ◦ C b i ℓ E Hence, by the universal property of the oplax colimit, there is a natural map ˜ E b → E b over E .Proof. Apply Lemma 4.1.7 to the cocartesian cone of Theorem 4.2.1 as described in Remark4.1.9. (cid:3)
To prove Theorem 5.1.3 we must verify that (5.1.4) is a componentwise Joyal weakequivalence. We first demonstrate this for generalized elements b : ∆ n → B whose domainsare simplices and then use the results of Lemmas 5.1.11 and 5.1.12 to extend these resultsto the general case.5.2.2. Example.
By definition ˜ p ∗ ( b : ∆ → B ) is the oplax colimit of the diagram C ∆ C B Q Cat C b c p that sends the unique object to the fiber E b of p : E fib B over b : ∆ → B . The weight forlax cocones of shape ∆ is the terminal weight so the weighted colimit is just the ordinarycolimit of this one object diagram. Thus ˜ p ∗ : sSet / B → sSet / E sends b : ∆[ b ] → B to E b → E ,which is isomorphic to the strict pullback p ∗ ( b : ∆ → B ) .For b : ∆ → B , ˜ p ∗ ( b : ∆ → B ) is the oplax colimit of the diagram C ∆ C B Q Cat C b c p whose image is diagram e b : E b → E b of quasi-categories constructed in (5.2.4). In thiscase, the oplax colimit has a simple description: it is given by an “mapping cylinder” formedby attaching E b along the codomain edge of the cylinder E b × ∆ via the map e b . We nowshow this simplicial set is Joyal weak equivalent to the strict fiber E b .5.2.3. Proposition.
The data formed by applying the comprehension construction E b E b EE b B ℓ E e b p p b y pℓ E χ y b p κ (5.2.4) to the pullback p b : E b → ∆ of p : E fib B along b : ∆ → B induces a Joyal weak equivalence E b E b E b × ∆ ˜ E b E be b id × δ p ℓ E χ γ b ∼ Proof.
By Proposition 3.3.5, the oplax colimit ˜ E b is weakly equivalent to the quasi-categoricalcollage coll( e b , E b ) , introduced in Definition 3.3.1. Moreover, Proposition 3.3.5 demon-strates that the equivalence k : ˜ E b ∼ −−→ coll( e b , E b ) is inner anodyne. In particular, thereexists a lift ˜ E b E b coll( e b , E b ) ∆ ∼ k γ b p b ρℓ defining a direct comparison map ℓ : coll( e b , E b ) → E b over ∆ .To prove that ℓ is an equivalence, observe by Lemma 3.3.4 and Proposition 3.2.3 that ρ and p b are both cocartesian fibrations. Hence, Proposition 3.1.7 tells us that if ℓ is acartesian functor, then to demonstrate that ℓ is an equivalence, we need only show that itrestricts to an equivalence on the fibers over 0 and 1. Indeed, ℓ is an isomorphism on bothfibers, so now our only remaining task is to demonstrate that it is a cartesian functor.The proof of Lemma 3.3.4 reveals that the non-degenerate cocartesian edges of coll( e b , E b ) are those represented by the degenerate edge of some vertex lying over the non-degenerate1-simplex in ∆ . Such edges lie in the image of the functor E b × ∆ → E b used to definethe map ˜ E b → E b , and this functor in turn was defined to be a representative for the co-cartesian lift of the 1-arrow between the two objects of ∆ to a map with domain E b → E b .In particular, it defines a cocartesian cylinder in the sense of Lemma 3.1.4, which tells usthat its components indexed by vertices of E b are p -cocartesian 1-arrows. This proves that ℓ carries ρ -cocartesian arrows to p -cocartesian arrows, completing the proof that ℓ is anequivalence. (cid:3) We argue inductively that γ b : ˜ E b → E b is an equivalence for any n -simplex b : ∆ n → B under the assumption that this is true for simplices of lower dimension. Our strategymirrors that adopted for the 1-simplex: we construct a quasi-categorical model for theoplax colimit of a homotopy coherent diagram c p ◦ C b : C ∆ n → Q Cat , i.e., a quasi-categoryequivalent to the simplicial set ˜ E b defined as the oplax colimit of c p ◦ C b , and then show thatthis is equivalent to the strict pullback E b . The inductive step makes use of the followingweights. ARTESIAN EXPONENTIATION AND MONADICITY 37
Notation (weights for the inductive comparison) . To compare the weights L ∆ n − and L ∆ n for oplax colimits of a homotopy coherent n − -simplex and n -simplex, we left Kan ex-tend the former along the inclusion δ n : ( C ∆ n − ) op ֒ → ( C ∆ n ) op , writing L ∆ n − : ( C ∆ n ) op → sSet for the left Kan extension of L ∆ n − . Explicitly, this weight is defined by L ∆ n − : ( C ∆ n ) op / / S Set i ( Fun C ∆ n ( i, n ) i < n ∅ i = n Let Y n denote the representable weight Y n : ( C ∆ n ) op / / S Set i Fun C ∆ n ( i, n ) Note there is a natural inclusion L ∆ n − ֒ → Y n that is the identity in all components exceptthe one indexed by the object n ∈ C ∆ n .5.2.6. Lemma. (i) The following diagram defines a pushout of weights in sSet ( C ∆ n ) op : L ∆ n − Y n L ∆ n − × ∆ L ∆ n id × δ p (ii) Let F : C ∆ n → sSet be a homotopy coherent diagram whose L ∆ n − -weighted colimitis Joyal weakly equivalent to the simplicial set E n − . Then the oplax colimit of F is Joyal weakly equivalent to the pushout E n − F n E n − × ∆ E nι n id × δ p along a canonical map ι n induced by the diagram F .Proof. The pushout in (i) can be verified componentwise at each i ∈ C ∆ n at which pointthis relationship is evident from the definitions.The pushout of (ii) follows. If E n − is isomorphic to the L ∆ n − -weighted colimit of F ,then the pushout diagram of (ii) is obtained by applying the cocontinuous functor ⊛ − F to the pushout diagram of (i). In this case, the map ι n has a natural explicit description.By Lemma 2.1.6, the L ∆ n − -weighted colimit of F coincides with the oplax colimit of therestricted diagram F ◦ C ( δ n ) op . The functor F itself defines a canonical lax cocone underthis restricted diagram with nadir F n . Hence there is a natural comparison ι n from the L ∆ n − -weighted colimit to F n . Observe from Proposition 2.2.5 and Lemma 2.3.8 that all of the weights appearing in(i) are flexible. Proposition 2.2.2 then demonstrates that the pushout being constructed isequivalence-invariant. (cid:3)
This lemma provides the inductive step in the following computation:5.2.7.
Proposition.
For any simplex b : ∆ n → B , the component γ b : ˜ E b → E b from theoplax colimit to the strict pullback is a Joyal weak equivalence.Proof. The base cases for n = 0 and n = 1 appear as Example 5.2.2 and Proposition 5.2.3.For the induction step, suppose we have shown this componentwise weak equivalence forall n − -simplices in B . By Lemma 2.1.6 and Notation 5.2.5, the L ∆ n − -weighted colimitof the diagram C ∆ n C b −→ C B c p −→ sSet is isomorphic to the oplax weighted colimit of therestricted diagram C ∆ n − C ∆ n C B sSet . C δ n C b c p By the inductive hypothesis, this weighted colimit ˜ E b · δ n is weakly equivalent to the pullback E b · δ n . By Lemma 5.2.6, the diagram E b · δ n E b n E b · δ n × ∆ ˜ E bι n id × δ is then a pushout up to Joyal weak equivalence. So it follows from Proposition 3.3.5 that ˜ E b is equivalent to the quasi-categorical collage coll( ι n , E b n ) , and as in the proof of Proposition5.2.3, the map γ b factors to define a map coll( ι n , E b n ) E b ∆ ρ ℓ π ℓ · b in this case involving the map π ℓ : ∆ n → ∆ that carries every element but the last oneto . Observe that π ℓ a cocartesian fibration, and indeed a bifibration, as it is covariantlyrepresented by the functor ! : [ n − → [0] , which admits both left and right adjoints; seeCorollary 3.3.6.Our task, again, is to show that ℓ is an equivalence. By Lemma 3.3.4 and the fact thatcocartesian fibrations compose, it is a functor between cocartesian fibrations. Moreover, ℓ is bijective on the fibers over , ∈ ∆ , the latter being E b n in both cases and the formerbeing E b · δ n . As in the proof of Proposition 5.2.3, ℓ is a cartesian functor, so Proposition3.1.7 implies that ℓ is an equivalence, as desired. (cid:3) Combining the work in this section, we can finally prove our main result.
ARTESIAN EXPONENTIATION AND MONADICITY 39
Proof of Theorem 5.1.3.
Our task is to demonstrate that the canonical natural transfor-mation sSet / B sSet / E ˜ p ∗ p ∗ ⇓ γ is a componentwise Joyal weak equivalence using the result of Proposition 5.2.7, whichdemonstrates that this is the case for the simplices b : ∆ n → B of B .The category sSet / B is equivalent to the category sSet el B op of presheaves indexed by thecategory el B of simplices of B ; its objects are simplices b : ∆ n → B and a morphism from b to c : ∆[ m ] → B is a simplicial operator α : ∆ n → ∆[ m ] so that c · α = b . The representablepresheaves generate sSet el B op under colimits, and such colimits are preserved by both ofthe functors p ∗ and ˜ p ∗ , the former case because of the right adjoint Π p that exists in thelocally cartesian closed category sSet and the latter case by Lemma 5.1.11. Under theequivalence sSet el B op ∼ = sSet / B , these representables correspond to the objects b : ∆ n → B whose domain is a simplex. Proposition 5.2.7 verifies that the components of γ indexedby such objects are equivalences, which is the moral reason why γ is an equivalence at allobjects.To demonstrate this, note that b : X → B is a colimit indexed by the category el X ofits simplices ∆ n x −→ X b −→ B , i.e., ( X b −→ B ) ∼ = colim el X (∆ n → B ) . The map γ b factors as γ b : ˜ p ∗ (colim el X ∆ n → B ) ∼ = colim el X ˜ p ∗ (∆ n → B ) −→ colim el X p ∗ (∆ n → B ) ∼ = p ∗ (colim el X ∆ n → B ) , so it remains only to show that this middle map, the colimit of the equivalences γ bx indexedby the simplices of X , is itself an equivalence. The indexing category el X is a Reedycategory, so if we can show that the two el X -indexed diagrams are Reedy cofibrant andthat the category el X has fibrant constants, then the pointwise equivalence between thediagrams will induce the desired equivalence between their colimits. To say that the Reedycategory el X has fibrant constants means that for each element x : ∆ n → X , the categoryof elements of the covariant representable boundary functor ∂ el X x is either empty orconnected. This category is empty just when x is non-degenerate and has a terminalobject, and is connected in particular, when x is degenerate. So el X has fibrant constantsand the colimit functor ( S Set / B ) el X → S Set / B carries pointwise weak equivalences betweenReedy cofibrant diagrams to weak equivalences.To verify this Reedy cofibrancy, it suffices to show(i) the canonical diagram el X → sSet / B is Reedy cofibrant(ii) ˜ p ∗ and p ∗ preserve Reedy cofibrant objects.Since ˜ p ∗ and p ∗ preserve colimits, they in particular preserve latching objects, so for thissecond item it suffices to show that both functors also preserve monomorphisms. Here, the fact that the pullback functor p ∗ preserves monomorphisms is standard, and the fact thatits replacement ˜ p ∗ preserves monomorphisms was proven in Lemma 5.1.12.So it remains only to prove (i), that is, to argue that the functor el X / / sSet / B x : ∆ n → X bx : ∆ n → B is Reedy cofibrant. The latching object associated to x : ∆ n → X is the composite ∂ ∆ n ֒ → ∆ n x −→ X b −→ B and the latching map is the inclusion ∂ ∆ n ֒ → ∆ n over B , which is obviouslya monomorphism. This completes the proof. (cid:3) Specializing Theorem 5.1.3 to the identity morphism on B , we have5.2.8. Corollary.
The domain of a cocartesian fibration p : E ։ B is equivalent to the oplaxcolimit of the associated comprehension functor c p : C B → Q Cat , with + C B jJ x x ♣♣♣♣♣♣♣♣ h E ,c p i % % ❑❑❑❑❑❑❑❑ C [ B ⋆ ∆ ] ℓ E / / Q Cat as the associated lax cocone. Pushforward along a cocartesian fibration
In this section, we shall fix a cocartesian fibration p : E ։ B of quasi-categories andprove that the pushforward functor sSet / E Π p −→ sSet / B has two properties that are relevant to the development of the category of theory of quasi-categories:(i) The pushforward functor preserves isofibrations . In model categorical terminology,this implies that the adjunctionsSet / E sSet / B Π p ⊥ p ∗ is Quillen with respect to slices of the Joyal model structure.Moreover:(ii) The pushforward functor preserves cartesian fibrations and cartesian functors be-tween them. ARTESIAN EXPONENTIATION AND MONADICITY 41
In fact, both pullback and pushforward along p define cosmological functors , a property webriefly note for use in future work.Both of the properties (i) and (ii) are more easily established for an alternate modelof the pushforward functor defined as a right adjoint to the functor ˜ p ∗ : sSet / B → sSet / E introduced in §5. Theorem 5.1.3 demonstrates that the pullback E b → E of a functor b : X → B along a cocartesian fibration p : E ։ B is computed, up to equivalence, as theoplax colimit of a particular diagram C X C B Q Cat C b c p When the oplax colimit is defined strictly as a simplicial set it enjoys the universal propertyof Definition 2.1.1: maps colim oplax ( c p ◦ C b ) → F correspond to lax cocones under c p ◦ C b with nadir F . This correspondence defines a right adjointsSet / E sSet / B ˜Π p ⊥ ˜ p ∗ characterized on an object q : F → E by the bijection ∆ n ˜Π p F colim oplax ( c p ◦ C b ) FB E b ˜Π p ( q ) ˜ p ∗ ( b ):= ℓ E | b q ! That is, n -simplices in ˜Π p F over b : ∆ n → B correspond to lax cocones under the homotopycoherent n -simplex c p ◦ C b with nadir F whose whiskered composite with q recovers therestriction ℓ E | b of the lax cocone produced by the comprehension construction of Theorem4.2.1.To make this simplex level construction of ˜Π p ( q ) precise, we require a simplicial set whose n -simplices correspond to lax cocones under a homotopy coherent n -simplex in Q Cat withnadir F . One might think that the slice quasi-category qCat / F provides just such a gadget,where qCat is the quasi-category of quasi-categories defined by passing to the maximalKan complex enriched core and then applying the homotopy coherent nerve, but this isn’tquite correct: since we’ve passed to the ( ∞ , -categorical core of Q Cat before takingthe homotopy coherent nerve, simplices in qCat / F correspond to pseudo cocones ratherthan lax cocones. The solution is to drop the core functor, in which case the homotopycoherent nerve qCat := N Q Cat is not a quasi-category but rather a , atype of marked simplicial set which is introduced in §6.1. Definition 6.1.9 introduces aslice construction for marked simplicial sets, which does not commute with the functorthat forgets the markings, but this is a good thing. The marked slice qCat // F has exactlythe property we desire, in that its n -simplices correspond to lax cocones under a homotopycoherent n -simplex in Q Cat with nadir F . In §6.2, we describe ˜Π p ( q : F → E ) explicitly as the pullback of a map between lax slicesof the homotopy coherent nerve of Q Cat defined by “whiskering with q .” After establish-ing the properties (i) and (ii) for ˜Π p , we use the natural Joyal equivalence ˜ p ∗ ⇒ p ∗ totransfer these properties to the pushforward functor Π p . Having established that pullbackand pushforward along a cocartesian fibration both preserve cartesian fibrations, in §6.3,we construct a closely related exponentiation operation ( q : F ։ B ) p : E ։ B of a cartesianfibration q by a cocartesian fibration p with the same codomain. These exponentials areused in §7 to establish the comonadicity and monadicity of the quasi-category of cartesianfibrations over B over the quasi-category of ob B -indexed families of quasi-categories.6.1. We know from Cordier and Porter [5, 6] that the homotopy coher-ent nerve of a Kan complex enriched category is itself a quasi-category. But when we applythe homotopy coherent nerve to a quasi-category enriched category, such as Q Cat itself, itis not the case that these nerves are quasi-categories. The fact that the hom-spaces of aquasi-category enriched category contain -simplices that cannot be regarded as invertibleimplies that its homotopy coherent nerve contains 2-simplices which cannot be regardedas being invertible. A homotopy coherent nerve of this kind is most naturally regarded aspossessing the structure of a -complicial set [28, 29].6.1.1. Definition (marked simplicial sets) . A marked simplicial set , called a stratifiedsimplicial set in [29] or a simplicial set with thinness in [27], is a simplicial set X equippedwith a chosen subset of its simplices of dimension greater than zero called marked simplices which (at least) contains all of the degenerate simplices. Simplicial maps f : X → Y between marked simplicial sets are required to preserve the chosen markings. The categoryof all such marked simplicial sets and mark preserving simplicial maps between them isdenoted sSet • .6.1.2. Definition (marking conventions) . When working with marked simplicial sets, thefollowing conventions are commonly adopted:(i) When X is a general simplicial set we silently promote it to a marked simplicial setby marking only those simplices that are degenerate. This is called the minimalmarking of X .(ii) When A is a quasi-category we silently promote it to a marked simplicial set bymarking all simplices above dimension and marking those -simplices that areisomorphisms. This is called the natural marking , and any simplicial map of quasi-categories preserves this natural marking.(iii) We say that a marked simplicial set is n -marked when all of its simplices abovedimension n are marked. For example, under its natural marking a quasi-categoryis -marked. A quasi-category is a Kan complex if and only if it is -marked underits natural marking.(iv) When X is marked simplicial set, we let ♯ n X denote the marked simplicial setconstructed by extending the marking of X to mark all simplices above dimension n ; ♯ X is the maximal marking of a simplicial set. This operator, called n -sharp , ARTESIAN EXPONENTIATION AND MONADICITY 43 is functorial and gives rise to an adjunction:sSet • core n ⊥ sSet • ♯ n r r When X ∈ sSet • is a marked simplicial set, its n -core core n ( X ) is the subset of X comprising those simplices whose faces above dimension n are all marked.(v) The abbreviated notation ♯ ∆ n := ♯ n − ∆ n is used for the marked n -simplex , whichextends the minimal marking by also marking the top-dimensional non-degeneratesimplex.Our primary motivation for recalling these notions is to define the natural marking ofthe homotopy coherent nerve of a quasi-category enriched category.6.1.3. Definition.
Suppose that K is a quasi-category enriched category. Its homotopycoherent nerve N K has: • -simplices corresponding to the objects a of K , • -simplices corresponding to -arrows f : a → a , • -simplices corresponding to diagrams a f / / f ! ! ❈❈❈❈❈❈ a a f = = ④④④④④④ α (cid:11) (cid:19) where α is a -arrow in the hom-space Fun K ( a , a ) with source f and target f ◦ f .Now we define the natural marking of the homotopy coherent nerve, a 2-marked simplicialset we denoted by K := N K , by marking:(i) all n -simplices with n > ,(ii) those -simplices, as depicted above, for which α is an invertible arrow in thequasi-category Fun K ( a , a ) , and(iii) each -simplex f : a → a which is an equivalence , in the sense that it possessesan equivalence inverse f ′ : a → a witnessed by a pair of invertible -arrows a f ! ! ❈❈❈❈❈❈ a a f ′ = = ④④④④④④ ∼ (cid:11) (cid:19) a f ′ ! ! ❈❈❈❈❈❈ a a f = = ④④④④④④ ∼ (cid:11) (cid:19) in the quasi-categories Fun K ( a , a ) and Fun K ( a , a ) respectively.Complicial sets [29] are certain marked simplicial sets satisfying a horn filler conditiongeneralising those that characterise Kan complexes and quasi-categories. To describe thisnotion precisely we introduce a couple more standard marked simplicial sets.6.1.4. Definition (complicial sets) . Suppose that k is an integer in some [ n ] then: • The standard k -admissible n -simplex ∆ n,k is obtained from ∆ n by marking any facethat has all of the elements of { k − , k, k + 1 } ∩ [ n ] among its vertices. • The standard ( n, k ) -horn Λ n,k has the usual k -horn as its underlying simplicial set andinherits its markings from ∆ n,k . • The marked simplicial set ♮ ∆ n,k is constructed from ∆ n,k by also marking its codimension-one faces δ i : [ n − → [ n ] for i ∈ { k − , k + 1 } ∩ [ n ] and ♯ ∆ n,k is constructed from ♮ ∆ n,k by also marking the further codimension-one face δ k : [ n − → [ n ] .Using these we may make the following definitions:(i) A complicial set A is a marked simplicial set that has the right lifting propertywith respect to the horn inclusions Λ n,k ֒ → ∆ n,k and the inclusions ♮ ∆ n,k ֒ → ♯ ∆ n,k for n ≥ and k ∈ [ n ] .(ii) A marked simplicial map p : E → B between complicial sets is an isofibration ofcomplicial sets if it has the right lifting property with respect to the horn inclusions Λ n,k ֒ → ∆ n,k for n ≥ and k ∈ [ n ] .(iii) An inclusion i : X → Y of marked simplicial sets is said to be an anodyne extension if and only if it may be constructed as the union of a countable chain of pushoutsof coproducts of horns Λ n,k ֒ → ∆ n,k and inclusions ♮ ∆ n,k ֒ → ♯ ∆ n,k . By a standardargument, every isofibration of complicial sets has the right lifting property withrespect to all anodyne extensions.6.1.5. Remark (isofibrations of complicial sets) . Corollary 55 of [29] demonstrates that amap p : E → B of complicial sets is an isofibration if it: • has the right lifting property with respect to all inner horns Λ n,k ֒ → ∆ n,k , for n ≥ and < k < n • admits lifts of marked -equivalences. More precisely, this latter condition asks for p tohave the right lifting property with respect to either of the inclusions ∆ ֒ → ♯ ∆ In particular, a map between naturally marked quasi-categories is an isofibration of com-plicial sets if and only if it is an isofibration of quasi-categories. In the quasi-categoricalcontext, the outer horn lifting property of Definition 6.1.4(ii) is typically referred to as“special outer horn” lifting.6.1.6.
Definition (saturated complicial sets) . A complicial set A is saturated if “all n -equivalences in A are marked,” where the notion of n -equivalence is defined relative to thecollection of marked ( n + 1) -simplices. An n - complicial set is a saturated complicial setwhich is n -marked. A complicial set is saturated if and only if it satisfies a certain unique right liftingproperty [15, 3.2.7], so in particular any complicial set that is defined as the limit of adiagram of saturated complicial sets is again saturated. The class of saturated complicialsets is also stable under passing to n -cores and forming slices, in the sense of 6.1.9 below.6.1.7. Example (Kan complexes and quasi-categories as complicial sets) . Under theirnatural markings, Kan complexes are precisely the 0-complicial sets and quasi-categoriesare precisely the 1-complicial sets. The isofibrations between 1-complicial sets coincidewith the usual classes of isofibrations between quasi-categories. One might think of the n -complicial sets as being a model for the theory of ( ∞ , n ) -categories, althoughwe will not pursue that intuition here. ARTESIAN EXPONENTIATION AND MONADICITY 45
Example.
Suppose that K is a quasi-category enriched category, then Theorem 40of [28] tells us that its naturally marked homotopy coherent nerve K := N K is a 2-complicial set.6.1.9. Definition (joins and slices of marked simplicial sets) . The join operation extendsto marked simplicial sets as follows. Concretely, the join
X ⋆ Y of two marked augmentedsimplicial sets X and Y has as its simplices pairs ( x, y ) with x ∈ X and y ∈ Y of arbitrarydimension with dim( x, y ) = dim( x ) + dim( y ) + 1 , where the convention is to augment amarked simplicial set with a single − -simplex. We declare that a simplex ( x, y ) ∈ X ⋆ Y is marked if x is marked in X or y is marked in Y .Now consider a map of marked simplicial sets f : X → Y . The slice Y //f is the simplicalset of whose n -simplices are maps g : ∆ n ⋆ X → Y which restrict on X ⊆ ∆ n ⋆ X to thefixed map f : X → Y . Such a simplex g : ∆ n ⋆ X → Y is marked if and only if it extendsalong the inclusion ∆ n ⋆ X ⊆ ♯ ∆ n ⋆ X , and this happens exactly when g maps every simplex (id [ n ] , x ) for x ∈ X to a marked simplex in Y . A dual construction defines f// Y .Suppose that A is a complicial set and that f : X → A is any map of marked simplicialsets. As shown in [29], it is then the case that f// A and A //f are also complicial sets andthat the projections r f : f// A → A and r f : A //f → A are isofibrations of such.6.2. The right adjoint to pullback.
We now have all the tools we require to construct analternate model of the pushforward functor along a cocartesian fibration p : E ։ B whosevalue at any isofibration q : F ։ B will be equivalent to those of the strict pushforward.The alternate model for the pushforward ˜Π p ( q : F → E ) := ˜Π p ( F , q ) → B of an isofibration q : F ։ E along a cocartesian fibration p : E ։ B is defined as a pullbackof a whiskering map for slices of homotopy coherent nerves that we now introduce.. Let K be a quasi-category enriched category such as Q Cat , and write K := N K for its homotopycoherent nerve, a 2-complicial set.6.2.1. Lemma.
Let q : F → E be a 0-arrow in a quasi-category enriched category K .(i) There is a functor of slice 2-complicial sets K // F K // E q ◦− induced from the whiskering operation for lax cocones.(ii) If q : F ։ E is a representably-defined isofibration, then q ◦ − : K // F ։ K // E is aisofibration of complicial sets.Proof. By the Yoneda lemma and the natural isomorphisms arising from the slice andhomotopy coherent nerve adjunctionssSet ( X, K //F ) ∼ = sSet ⊤7→ F ( X ⋆ ∆ , K ) ∼ = sSet-Cat ⊤7→ F ( C [ X ⋆ ∆ ] , K ) , to define the map in (i), it suffices to provide a natural operation that converts a lax coconeof shape X with nadir F into a lax cocone with shape X and nadir E . The whiskering operation for lax cocones described in Observation 4.1.4 defines such a natural transforma-tion. Since whiskering preserves fibered equivalences and isomorphisms, which correspondto marked 1- and 2-simplices in K //F , this defines the desired map of 2-complicial sets.We must show that the map between the sliced complicial sets has the right liftingproperty with respect to each of the marked anodyne extensions introduced in Definition6.1.4. Λ n,k K //F ∆ n,k K //Eq ◦− By [29, Corollary 49], it suffices to consider the case < k ≤ n . Here the bottom horizontalfunctor is given by a homotopy coherent n + 1 -simplex C ∆ n +1 → K that sends the first n + 1 objects to E , . . . , E n and the final object to E ∈ K and satisfiesone additional condition forced by the markings on ∆ n,k and K //E . If k < n , then thisfunctor must be defined so that the 1-simplex E k − f k − ,k +1 / / f k − ,k " " ❊❊❊❊❊❊ E k +1 E k f k,k +1 < < ②②②②②② α (cid:11) (cid:19) α ∈ Fun ( E k − , E k +1 ) is invertible. If k = n , then the 1-simplex E n − f n − ,n / / f n − ,n − $ $ ❍❍❍❍❍❍❍ E n E n − f n − ,n < < ②②②②②② α (cid:14) (cid:22) ✪✪✪✪✪✪ α ∈ Fun ( E n − , E n ) must be invertible and f n − ,n : E n − → E n must admit an equivalenceinverse.The δ n +1 -face of this homotopy coherent simplex and the top horizontal together definea simplicial functor C Λ n +1 ,k → K that carries the n + 1 objects to E , . . . , E n , F , respectively, and has the property that foreach ≤ j ≤ n the diagram of function complexes Fun C Λ n +1 ,k ( j, n + 1) (cid:127) _ (cid:15) (cid:15) / / Fun ( E j , F ) q ◦− (cid:15) (cid:15) (cid:15) (cid:15) Fun C ∆ n +1 ( j, n + 1) / / Fun ( E j , E ) commutes. ARTESIAN EXPONENTIATION AND MONADICITY 47
Because < k < n + 1 , by Example 2.3.2, to solve the original lifting problem, it remainsonly to construct a single lift ⊓⊓ n,k ∼ = Fun C Λ n +1 ,k (0 , n + 1) (cid:127) _ (cid:15) (cid:15) / / Fun ( E , F ) q ◦− (cid:15) (cid:15) (cid:15) (cid:15) ⊓⊔ n ∼ = Fun C ∆ n +1 (0 , n + 1) / / ❥❥❥❥❥❥❥❥ Fun ( E , E ) the other inclusions being full. The left-hand side is a cubical horn and the right-hand sideis an isofibration of quasi-categories. By Remark 6.1.5, isofibrations of quasi-categoriesadmit lifts against “special outer horns” Λ m,m ֒ → ∆ m,m — those in which the image of thefinal edge is invertible. Such extensions solve this lifting problem. (cid:3) Proposition.
There is a right adjoint sSet / E sSet / B ˜Π p ⊥ ˜ p ∗ to the oplax colimit functor of Definition 5.1.10 defined at q : F → E by the pullback ˜Π p F qCat // F B qCat // E ˜Π p ( q ) y q ◦− ℓ E Moreover, when q : F ։ E is an isofibration, ˜Π p q : ˜Π p F ։ B is an isofibration betweenquasi-categories.Proof. Recall from Lemma 6.2.1 that n -simplices in qCat // F correspond to lax coconesunder homotopy coherent simplices with nadir F , and observe that the whiskering functor q ◦ − : qCat // F → qCat // E does not change the underlying homotopy coherent diagram. Bythe defining universal property, an n -simplex in the pullback over b : ∆ n → B correspondsto lax cocone under the homotopy coherent n -simplex c p ◦ C b : C ∆ n → Q Cat with nadir F that whiskers with q to the lax cocone of (5.1.2). This recovers the characterizationof the right adjoint ˜Π p given above and Lemma 5.1.11 demonstrates that this adjointcorrespondence extends to all elements of sSet / B . The action of ˜Π p on morphisms u : G → F over E is given similarly by the pullback ˜Π p G qCat // G ˜Π p F qCat // F B qCat // E y u ◦− ˜Π p q y q ◦− ℓ E By Lemma ?? and Example 6.1.7, it is immediate from the fact that B is a quasi-categoryand qCat is a 2-complicial set that ˜Π p ( F , q ) is a 2-complicial set. We argue that in fact it is1-trivial: by the defining universal property, a 2-simplex in ˜Π p ( F , q ) corresponds to a paircomprised of a 2-simplex in B and a 2-simplex in qCat // F and both of these 2-simplices aremarked. Thus, ˜Π p ( F , q ) is a 1-complicial set, which Example 6.1.7 tells us is the same thingas a quasi-category, and now the isofibration of complicial sets ˜Π p q becomes an isofibrationbetween quasi-categories. (cid:3) Corollary. If p : E ։ B is a cocartesian fibration.:(i) The functor ˜Π p : sSet / E → sSet / B carries isofibrations over E to isofibrations over B , restricting to define a functor Q Cat / E ˜Π p −→ Q Cat / B . (6.2.4) (ii) The functor (6.2.4) preserves isofibrations, now considered as morphisms in theseslice categories.Proof. Proposition 6.2.2 demonstrates that ˜Π p carries isofibrations to isofibrations, restrict-ing to define a functor ˜Π p : Q Cat / E → Q Cat / B . Moreover this functor preserves isofibra-tions, now considered as morphisms in these slice categories, since the action of ˜Π p on anisofibration u : G ։ F over E is defined by pulling back the isofibration of complicial sets u ◦ − : qCat // G ։ qCat // F . (cid:3) We now transfer the properties of the functor ˜Π p to the right adjoint Π p : sSet / E → sSet / B to the strict pullback functor p ∗ : sSet / B → sSet / E .6.2.5. Proposition. If p : E ։ B is a cocartesian fibration, then the adjunctions sSet / E sSet / B and sSet / E sSet / B Π p ⊥ p ∗ ˜Π p ⊥ ˜ p ∗ are Quillen with respect to the sliced Joyal model structure. ARTESIAN EXPONENTIATION AND MONADICITY 49
In particular, Π p preserves both fibrant objects and the fibrations between, and thushas the properties enumerated for ˜Π p in Corollary 6.2.3. Consequently, the natural Joyalequivalence γ : ˜ p ∗ ⇒ p of Theorem 5.1.3, which defines a natural isomorphism of total leftderived functors, transposes to a natural equivalence ˆ γ : Π p ⇒ ˜Π p , which defines a naturalisomorphism of total right derived functors. Proof.
By an observation of Joyal and Tierney [10, 7.15], to show that ˜ p ∗ ⊣ ˜Π p is Quillen itsuffices to show that the left adjoint preserves cofibrations and the right adjoint preservesfibrations between fibrant objects. Lemma 5.1.12 demonstrates the first of these andCorollary 6.2.3(ii) proves the second.To prove that p ∗ ⊣ Π p is Quillen, we prove that p ∗ is left Quillen. Thus functor preservescofibrations because pullbacks preserve monomorphisms. By Theorem 5.1.3, p ∗ is naturallyweakly equivalent to the left Quillen functor ˜ p ∗ . Since all objects in sSet / E are cofibrant,the left Quillen functor ˜ p ∗ preserves all Joyal weak equivalences, and hence by the 2-of-3property p ∗ does as well. (cid:3) We now consider the actions of the pushforward functors ˜Π p and Π p along a cocartesianfibration p : E ։ B when applied to a cartesian fibration q : F ։ E . As before, we demon-strate directly that ˜Π p q : ˜Π p F ։ B is a then a cartesian fibration and then use Theorem5.1.3 to conclude the same for Π p .6.2.6. Lemma.
Let q : F ։ E between quasi-categories. Then the corresponding map q ◦− : qCat // F ։ qCat // E of 2-complicial sets has the right lifting property with respect to anyouter horn inclusion ∆ { n − ,n } Λ n,n qCat // F ∆ n qCat // E χ q ◦− whose final edge defines a cartesian 1-arrow E n − FE nf n − f n − ,n − ⇓ χ f n for the cartesian fibration q ◦ − : Fun ( E n − , F ) → Fun ( E n − , E ) .Proof. As in the proof of Lemma 6.2.1, the bottom horizontal functor is given by a homo-topy coherent n + 1 -simplex C ∆ n +1 → Q Cat that sends the first n + 1 objects to E , . . . E n and the final object to E , while the δ n +1 -face of this homotopy coherent simplex and thetop horizontal functor together define a simplicial functor C Λ n +1 ,n → Q Cat that caries the n + 2 objects to E , . . . , E n , F and has the property that for each ≤ j ≤ n the diagram of function complexes C Λ n +1 ,n ( j, n + 1) Fun ( E j , E ) C ∆ n +1 ( j, n + 1) Fun ( E j , F ) q ◦− commutes.By Example 2.3.2, to solve the original lifting problem, we need only construct a singlelift ⊓ n,n ∼ = C Λ n +1 ,n (0 , n + 1) Fun ( E , E ) (cid:3) n ∼ = C ∆ n +1 (0 , n + 1) Fun ( E , F ) q ◦− This extension problem can be solved by filling inner horns and “special outer horns” Λ m,m → ∆ m , those whose final edges are complies of the 1-simplex χ ∈ Fun ( E n − , F ) pre-composed with some functor E → E n − . Such 1-simplices represent ( q ◦ − ) -cartesian cellsso these “special outer horn” lifting problems also admit solutions by Lemma 3.1.5(i). (cid:3) Proposition. If p : E ։ B is a cocartesian fibration and q : F ։ E is a cartesianfibration between quasi-categories, then ˜Π p q : ˜Π p F ։ B is a cartesian fibration between quasi-categories. Moreover, ˜Π p preserves cartesian functors,restricting to define a functor ˜Π p : Cart ( Q Cat ) / E → Cart ( Q Cat ) / B . Proof.
By Proposition 6.2.2, ˜Π p ( F , q ) ։ B defines an isofibration between quasi-categories.Lemma 6.2.6 identifies a class of cartesian 1-arrows in ˜Π p ( F , q ) in the sense of Definition3.1.1, which we now describe explicitly. Recall from the construction of Proposition 6.2.2,that a 1-simplex χ : ∆ → ˜Π p F in the fiber over b : ∆ → B corresponds to a 1-arrow E F E f e b ⇓ χ f in Fun ( E , F ) that whiskers with q : F → E to define the lax cocone that restricts thelax cocone associated to the comprehension construction along b . As observed previously,this compatibility condition tells us that the quasi-categories E and E are the fibers of p : E ։ B over the vertices in b and the functor e b : E → E is the comprehension of b .To form such a lift with codomain f : E → F , start by lifting b to the lax cocone E EE ℓ E e b ⇓ ǫ ℓ E ARTESIAN EXPONENTIATION AND MONADICITY 51 under e b : E → E with nadir E associated with the comprehension construction, asdisplayed in (5.2.4). Since f is in the fiber of ˜Π p q : ˜Π p F ։ B over the codomain of b , wemust have qf = ℓ E . Now we can lift ǫ along the cartesian fibration q to a q -cartesian cellwith codomain f ◦ e b . This defines the ( q ◦ − ) -cartesian cell χ .Now if u : G → F is a cartesian functor from r : G ։ E to q : F ։ E , then u isrepresentabily cartesian in the sense that u ◦ − : Fun ( X, G ) → Fun ( X, F ) carries ( r ◦ − ) -cartesian 1-arrows to ( q ◦ − ) -cartesian 1-arrows. Since u ◦ − : qCat // G → qCat // F preservesthe cartesian 1-arrows just identified, proving that ˜Π p carries this map to a cartesian functorbetween cartesian fibrations over B . (cid:3) Corollary. If p : E ։ B is a cocartesian fibration and q : F ։ E is a cartesianfibration between quasi-categories, then Π p q : Π p F ։ B is a cartesian fibration between quasi-categories. Moreover, Π p preserves cartesian functors,restricting to define a functor Cart ( Q Cat ) / E Cart ( Q Cat ) / B . Π p Proof.
By Proposition 6.2.5, the components Π p F ˜Π p FB ˆ γ q ∼ Π p q ˜Π p q at an isofibration q : F ։ E of the transpose ˆ γ : Π p ⇒ ˜Π p of the natural weak equivalenceof Theorem 5.1.3 are equivalences of isofibrations over B . If q is a cartesian fibration, thenProposition 6.2.7 proves that ˜Π p q is a cartesian fibration, and since the notion of cartesianfibration is equivalence-invariant, Π p q must be as well. (cid:3) Theorem.
For a cocartesian fibration p : E ։ B between quasi-categories, thepullback-pushforward adjunction restricts to define an adjunction Q Cat / E Q Cat / B Cart ( Q Cat ) / E Cart ( Q Cat ) / B Π p ⊥ p ∗ Π p ⊥ p ∗ Proof.
By Proposition 6.2.5, the adjoint functors p ∗ ⊣ Π p define an adjunction Q Cat / E Q Cat / B Π p ⊥ p ∗ By Proposition 3.2.3, the left adjoint restricts to a define a functor p ∗ : Cart ( Q Cat ) / B → Cat ( Q Cat ) / E . By Corollary 6.2.8, the right adjoint also restricts to a functor Π p : Cart ( Q Cat ) / E → Cart ( Q Cat ) / B . Since the inclusion Cart ( Q Cat ) / B ֒ → Q Cat / B is not full, this isn’t quiteenough to demonstrate adjointness of the restricted adjunction: it remains to argue thatthe adjoint transpose of a cartesian functor is a cartesian functor.To that end, let q : F ։ E and r : G ։ B be cartesian fibrations. A functor f : G → ˜Π p q over B is cartesian if and only if the square G qCat // F B qCat // E fr q ◦− ℓ E carries r -cartesian arrows to representably q -cartesian arrows in qCat // F , as described inLemma 6.2.6. Fixing an 1-arrow arrow ζ : ∆ → G over b : ∆ → B as below-left, the arrow f ζ transposes to the functor over E displayed below-right ∆ qCat // F colim oplax ( c p ◦ C b ) FB qCat // E E fζb q ◦− c fζℓ E | b qℓ E ! By Proposition 5.2.3, the oplax colimit is equivalent to the fiber E b and the functor c f ζ represents the whiskered lax cocone E b F E b E r EE b G B ℓ E e b p qp b y c fζ ˆ f y pℓ E χ y bζ r p κ Now f is a cartesian functor if and only if the whiskered composite c f ζχ is q -cartesianwhenever ζ is r -cartesian. Since Proposition 3.2.3 demonstrates that cartesian arrows arecreated by pullbacks, this proves that f is a cartesian functor if and only if the transposed ARTESIAN EXPONENTIATION AND MONADICITY 53 functor E r F E ˆ fp ∗ r q is cartesian. (cid:3) For use in sequels to this work, we note that the pushforward is a simplicially enrichedfunctor between the ∞ -cosmoi established in Proposition VIII.3.2.18.6.2.10. Corollary.
Let p : E → B be a cocartesian fibration. Then the pushforward con-struction defines cosmological functors Π p : Q Cat / E → Q Cat / B and Π p : Cart ( Q Cat ) / E → Cart ( Q Cat ) / B , which is to say that they are simplically enriched, preserve all simplicially enriched limitswith flexible weights, and preserve the isofibrations, considered as morphisms in the slicecategory.Proof. The functor Π p : Q Cat / E → Q Cat / B is the restriction of a Quillen right adjoint Π p : sSet / E → sSet / B . To prove that this defines a functor of ∞ -cosmoi, it remains only toshow that the adjunction p ∗ ⊣ Π p is simplicially enriched. This follows from Observation3.2.4, which notes that the left adjoint preserves tensors with simplicial sets. Lemma 3.2.5observes that the simplicial enrichment descends to the subcosmoi of cartesian fibrations. (cid:3) The argument given in the proof of Theorem 6.2.9 provides a characterization of the Π p q -cartesian 1-arrows in the cartesian fibration constructed from a cocartesian fibration p : E ։ B and a cartesian fibration q : F ։ E between quasi-categories that lift a specifiedarrow β : ∆ → B .6.2.11. Lemma. If p : E ։ B is a cocartesian fibration and q : F ։ E is a cartesian fibrationbetween quasi-categories, then the cartesian 1-arrows χ in Π p q : Π p F ։ B are those mapsthat transpose to define functors ∆ Π p FB β χ Π p q ! FE β E ∆ B q ˆ χℓ β p β y pβ that carry p -cocartesian lifts of β to q -cartesian lifts.Proof. By Theorem 5.1.3, E β may be identified with the oplax colimit of the canonical laxcocone formed by taking a p -cocartesian lift of β . From this perspective, the transposedfunctor ˆ χ : E β → F acts by whiskering this p -cocartesian arrow. By the construction in the proof of Theorem 6.2.9, the Π p q -cartesian lifts of β are those arrows for which thiswhiskered composite is q -cartesian, as claimed. (cid:3) Exponentiation.
As is familiar in any locally cartesian closed category, the pullbackand pushforward functors can be used to construct exponentials in Cart ( Q Cat ) / B , wherethe exponent is given by a cocartesian fibration.6.3.1. Definition (exponentials) . For p : E ։ B either a cartesian or cocartesian fibrationand q : F ։ B an isofibration, define ( q : F ։ B ) p : E ։ B ∈ Q Cat / B (6.3.2)to be the image of q under the composite functor Q Cat / B p ∗ / / Q Cat / E Π p / / Q Cat / B Note that by the adjunctions Σ p ⊣ p ∗ ⊣ Π p , if r : G ։ B is also a cartesian or c ocartesianfibration, there are natural isomorphisms Fun B ( G ։ B , ( F ։ B ) E ։ B ) ∼ = Fun B ( E × B G ։ B , F ։ B ) ∼ = Fun B ( E ։ B , ( F ։ B ) G ։ B ) , and the left-hand isomorphism still holds in the case where r : G → B is a mere functor,whose domain need not even be a quasi-category.6.3.3. Proposition. If p : E ։ B is a cocartesian fibration and q : F ։ B is a cartesianfibration, then (6.3.2) is a cartesian fibration. The cartesian 1-arrows are those maps ∆ b (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂ χ / / ( F q −− ։ B ) E p −− ։ B x x x x qqqqqqqqqqqqq B that transpose to define functors E bp b (cid:15) (cid:15) (cid:15) (cid:15) / / b χ ( ( E p (cid:15) (cid:15) (cid:15) (cid:15) F q (cid:0) (cid:0) (cid:0) (cid:0) ✁✁✁✁✁✁✁✁ ! E b p b ❆❆❆❆❆❆❆ b χ / / F bq b ~ ~ ~ ~ ⑥⑥⑥⑥⑥⑥⑥ ∆ b / / B ∆ that carry p -cocartesian lifts of b to q -cartesian lifts of b .Dually, if p : E ։ B is a cartesian fibration and q : F ։ B is a cocartesian fibration, then (6.3.2) is a cocartesian fibration whose cocartesian 1-arrows over b : ∆ → B representfunctors that carry p -cartesian lifts to q -cocartesian lifts.Proof. The first statement follows from Corollary 6.2.8, while the characterization of carte-sian cells is given in Lemma 6.2.11. (cid:3)
Recall the function complexes constructed in Definitions 3.2.1 and 3.2.2.
ARTESIAN EXPONENTIATION AND MONADICITY 55
Lemma.
Let q : F ։ B be a cartesian fibration, let p : E ։ B be a cocartesianfibration, and let π : A × B ։ B denote the projection, a bifibration. Then the isomorphism Fun B ( F q −− ։ B , ( A × B ։ B ) E p −− ։ B ) ∼ = Fun B ( E p −− ։ B , ( A × B ։ B ) F q −− ։ B ) restricts to an isomorphism Fun c B ( F q −− ։ B , ( A × B ։ B ) E p −− ։ B ) ∼ = Fun c B ( E p −− ։ B , ( A × B ։ B ) F q −− ։ B ) between the function complexes in the quasi-category enriched categories Cart ( Q Cat ) / B and coCart ( Q Cat ) / B , which is to say, cartesian functors between the cartesian fibrations on theleft transpose to define cartesian functors between the cocartesian fibrations on the right.Proof. By adjunction, the data of a map over BF ( A × B ։ B ) E p −− ։ B B q f π p is given by a single functor ˆ f : F × B E → A . By Proposition 6.3.3, f is a cartesian functorif and only if for each q -cocartesian cell χ : ∆ → F over b : ∆ → B , the induced functor E b A × B ∆ B p b ˆ χ πb carries p -cartesian cells γ : ∆ → E over b to π -cocartesian ones, these being those maps ∆ → A × B whose component along the other projection A × B → A is invertible. Insummary, the functor f is cartesian if and only if for every cocartesian lift χ and cartesianlift γ of b , the composite morphism ∆ F b × ∆ E b F × B E A h χ,γ i ˆ f (cid:3) Monadicity and comonadicity of cartesian fibrations
The 0-skeleton of a quasi-category B defines the “underlying set of objects” ob B , togetherwith a canonical inclusion ob B ֒ → B . By Proposition 3.2.3, pulling back along the inclusion ob B ֒ → B induces a forgetful functor Cart ( Q Cat ) / B / / Cart ( Q Cat ) / ob B ∼ = Q ob B Q Cat ( E p −− ։ B ) ( E b ) b ∈ ob B whose codomain is isomorphic to the product of the quasi-categorically enriched categoriesof quasi-categories, a cartesian fibration over a set being simply an indexed family of quasi-categories. Our aim in this section is to construct left and right adjoints and prove thatthis functor is monadic and comonadic in a suitable sense.The adjoint functors are constructed as what we refer to as biadjoint functors of quasi-categorically enriched categories: that is, we construct quasi-categorically enriched functors L, R : Cart ( Q Cat ) / ob B −→ Cart ( Q Cat ) / B together with natural equivalences of function complexes that encode the adjoint transposecorrespondence. The right adjoint makes use of the exponentiation construction of §6.3and the Yoneda lemma is used to prove biadjointness. These tasks occupy §7.1.Any quasi-categorically enriched category K has a (typically large) quasi-categorical core K := N g ∗ K defined by passing to the maximal Kan complex enriched core and then applyingthe homotopy coherent nerve. For example, the quasi-category qCat of quasi-categories andfunctors is the quasi-categorical core of Q Cat . In §7.1 we also prove that biadjoint functorsof quasi-categorically enriched categories descend to adjoint functors between their quasi-categorical cores.In particular, this implies that the map of large quasi-categories of cartesian fibrationsand cartesian functors
Cart / B −→ Cart / ob B ∼ = Y ob B qCat admits both left and right adjoints. In §7.2, we prove first that this forgetful functor iscomonadic and then use comonadicity to prove that it is also monadic. To do so, we appealto the comonadicity theorem proven in §II.7 and recalled as Theorem 7.2.1 below. Themonadicity of this forgetful functor will be used in §8 to construct a “groupoidal reflection”functor for cartesian fibrations.7.1. Adjoint functors.
A functor U : K → L between quasi-categorically enriched cat-egories gives rise to a functor between the large quasi-categories defined by passing tothe Kan complex enriched cores of K and L and applying the homotopy coherent nerveconstruction. We frequently find it convenient to construct adjoints to this functor ofquasi-categories “at the point-set level” by producing the structures axiomatized in thefollowing definition:7.1.1. Definition. A biadjunction of quasi-categorically enriched categories consists of: • a pair of quasi-categorically enriched categories K and L ; • a pair of simplicial functors F : L → K and U : K → L ; and • a simplicially-enriched natural equivalence Fun K ( F L, K ) ≃ Fun L ( L, U K ) of function complexes .7.1.2. Proposition.
Given quasi-categorically enriched functors F : L → K and U : K → L ,we write F : L → K and U : K → L for the induced functors between the quasi-categorical ARTESIAN EXPONENTIATION AND MONADICITY 57 cores K := N g ∗ K and L := N g ∗ L . If F and U define a biadjunction of quasi-categoricallyenriched categories, then K U ⊥ L F r r is an adjunction between the quasi-categorical cores.Proof. Reprising a construction from the proof of Theorem I.6.2.1, we define simplicialcategories coll( F, K ) and coll( L , U ) whose objects are ob K + ob L and which include K and L as full subcategories. Each of the function complexes from an object of K to an objectof L are empty, while for L ∈ L and K ∈ K , we define Fun coll( F, K ) ( L, K ) :=
Fun K ( F L, K ) and
Fun coll( L ,U ) ( L, K ) :=
Fun L ( L, U K ) . The natural equivalence
Fun K ( F L, K ) ∼ −−→ Fun L ( L, U K ) of the biadjunction gives rise toa simplicial functor coll( F, K ) → coll( L , U ) under K + L that is bijective on objects and alocal equivalence of quasi-categories.Passing to Kan complex enriched cores, this functor defines a Dwyer-Kan equivalence,and thus yields an equivalence of quasi-categories upon passing to homotopy coherentnerves. Note that the homotopy coherent nerve of the groupoid core of coll( F, K ) is iso-morphic to the quasi-categorical collage of the underlying functor F : L → K constructedin Definition 3.3.1. Now Corollary 3.3.6 demonstrates that F ⊣ U as functors between thequasi-categories K and L . (cid:3) Proposition.
The functor
Cart ( Q Cat ) / B −→ Cart ( Q Cat ) / ob B admits a quasi-cate-gorically enriched right biadjoint defined by Cart ( Q Cat ) / ob B / / Cart ( Q Cat ) / B ( E b ) b ∈ ob B Q b ( E b × B ։ B ) b ↓ B ։ B that is, an ob B -indexed family of quasi-categories ( E b ) b ∈ ob B is sent to the product in Cart ( Q Cat ) / B of the cartesian fibrations ( E b × B ։ B ) b ↓ B ։ B .Proof. Recall from Example 3.1.3 that b ↓ B ։ B is the cocartesian fibration represented bythe vertex b ∈ B . Since the product projection π : E b × B ։ B is a bifibration, Proposition6.3.3 implies that ( E b × B ։ B ) b ↓ B ։ B is a cartesian fibration.For another cartesian fibration q : F ։ B with fibers ( F b ) b ∈ ob B , we will define a naturalequivalence of function complexes Fun c B ( F q −− ։ B , Y b ∈ ob B ( E b × B ։ B ) b ↓ B ։ B ) ∼ −−→ Y b ∈ ob B Fun ( F b , E b ) and so establish the claimed adjoint correspondence. To begin, the universal property of the product together with Definition 6.3.1 providesisomorphisms
Fun B ( F q −− ։ B , Y b ∈ ob B ( E b × B ։ B ) b ↓ B ։ B ) ∼ = Y b ∈ ob B Fun B ( F q −− ։ B , ( E b × B ։ B ) b ↓ B ։ B ) ∼ = Y b ∈ ob B Fun B ( b ↓ B ։ B , ( E b × B ։ B ) F q −− ։ B ) . By Lemma 6.3.4, these isomorphisms restrict to the full sub quasi-categories spanned bythe cartesian functors. Hence,
Fun c B ( F q −− ։ B , Y b ∈ ob B ( E b × B ։ B ) b ↓ B ։ B ) ∼ = Y b ∈ ob B Fun c B ( b ↓ B ։ B , ( E b × B ։ B ) F q −− ։ B ) . By the dual of the Yoneda lemma, proven as Theorem IV.6.0.1, restriction along theelement → b ↓ B corresponding to the identity at b : 1 → B defines an equivalence Fun c B ( b ↓ B ։ B , ( E b × B ։ B ) F q −− ։ B ) ∼ −−→ Fun B (1 b −→ B , ( E b × B ։ B ) F q −− ։ B ) . The proof is finished by the isomorphisms
Fun B (1 b −→ B , ( E b × B ։ B ) F q −− ։ B ) ∼ = Fun B ( F b → b −→ B , E b × B → B ) ∼ = Fun ( F b , E b ) . (cid:3) The construction of the left adjoint to
Cart ( Q Cat ) / B −→ Cart ( Q Cat ) / ob B will make useof the tensor of a cartesian fibration, namely B ↓ b ։ B , with a quasi-category, namely E b ,described in Observation 3.2.4 and Lemma 3.2.5.7.1.4. Proposition.
The functor
Cart ( Q Cat ) / B −→ Cart ( Q Cat ) / ob B admits a quasi-cate-gorically enriched left biadjoint defined by Cart ( Q Cat ) / ob B / / Cart ( Q Cat ) / B ( E b ) b ∈ ob B ` b E b × B ↓ b ։ B that is, an ob B -indexed family of quasi-categories ( E b ) b ∈ ob B is sent to the coproduct in Cart / B of the cartesian fibrations E b × B ↓ b π −− ։ B ↓ b p −−− ։ B .Proof. To make sense of this construction, note that the coproduct of cartesian fibrationsover B is again a cartesian fibration over B : since the simplices are connected, each liftingproblem of Definition 3.1.1 is supported in a single component. It follows that a functorout of the coproduct of cartesian fibrations is cartesian if and only if each of its legs is acartesian functor.For another cartesian fibration q : F ։ B with fibers ( F b ) b ∈ ob B , we will define a naturalequivalence of function complexes Fun c B ( a b ∈ ob B E b × B ↓ b ։ B , F q −− ։ B ) ∼ −−→ Y b ∈ ob B Fun ( E b , F b ) and so establish the claimed adjoint correspondence. ARTESIAN EXPONENTIATION AND MONADICITY 59
To begin, the universal property of the coproduct provides the first isomorphism, whilethe universal property of the tensor proven as Lemma 3.2.5 provides the second
Fun c B ( a b ∈ ob B E b × B ↓ b ։ B , F q −− ։ B ) ∼ = Y b ∈ ob B Fun c B ( E b × B ↓ b ։ B , F q −− ։ B ) ∼ = Y b ∈ ob B Fun c B ( B ↓ b ։ B , F q −− ։ B ) E b . By the Yoneda lemma, proven as Theorem IV.6.0.1, restriction along the element → B ↓ b corresponding to the identity at b : 1 → B defines an equivalence Fun c B ( B ↓ b ։ B , F q −− ։ B ) ∼ −−→ Fun B (1 b −→ B , F q −− ։ B ) ∼ = F b . This equivalence is respected by the cotensor ( − ) E b and the product, so we have the desiredequivalence Fun c B ( a b ∈ ob B E b × B ↓ b ։ B , F q −− ։ B ) ∼ = Y b ∈ ob B Fun c B ( B ↓ b ։ B , F q −− ։ B ) E b ≃ Y b ∈ ob B Fun ( E b , F b ) . (cid:3) Monadicity and comonadicity.
A functor u : A → B between quasi-categories is comonadic if it is the left adjoint part of a comonadic adjunction. This means that A isequivalent to the quasi-category of coalgebras for the homotopy coherent comonad on B underlying the corresponding homotopy coherent adjunction derived from u and its rightadjoint. The quasi-category of coalgebras is defined as a particular flexible weighted limitof the homotopy coherent comonad. A few specific details of this construction are needin the proofs in §8.3 and these are reviewed there. To avoid an unnecessary digression,we refer the reader §7 for the definition of these notions and omit them from the currentpresentation.Recall Theorem II.7.2.7, presented here in the dual:7.2.1. Theorem (comonadicity II.7.2.7) . A functor u : A → B between quasi-categories iscomonadic if and only if:(i) u admits a right adjoint,(ii) A admits and u preserves limits of u -split cosimplicial objects, and(iii) u is conservative. A functor between quasi-categories is conservative if its reflects isomorphisms. For ex-ample:7.2.2.
Lemma.
Let U : K → L be a functor of quasi-categorically enriched categories thatreflects equivalences in the sense that any 0-arrow f : A → B in K whose image is anequivalence in L is an equivalence in K . Then the corresponding functor U : K → L betweenquasi-categorical cores is conservative.Proof. To prove this, we must show that an equivalence f : A → B in a quasi-categoricallyenriched category K defines an isomorphism in its quasi-categorical core. An equivalence in K is comprised of the data enumerated in 6.1.3(iii), which is contained in the subcategory g ∗ K ⊂ K . In the homotopy coherent nerve K = N g ∗ K this data gives rise to a pair ofobjects A and B , a pair of 1-simplices f : A → B and f ′ : B → A , and a pair of 2-simpliceswitnessing that f and f ′ compose to identities. This is the data that defines an isomorphismin a quasi-category. (cid:3) A rather delicate application of monadicity is given in §8 and explained there. A moreimmediate application of comonadicity is the following result:7.2.3.
Theorem (comonadicity and colimit creation III.5.7) . Let u : A → B be a comonadicfunctor between quasi-categories. Then u creates any colimits that B admits. In particular any quasi-category admits colimits of split simplicial objects : a simplicialobject ∆ op → B is split if it extends along the inclusion ∆ op ֒ → ∆ op + ֒ → ∆ ∞ that augments itwith an terminal object — note that ∆ op + ∼ = ∆ op ⋆ — and then adds an “extra degeneracy”map in each dimension (see §I.5.3).7.2.4. Theorem (split simplicial objects define colimits I.5.3.1) . Any split simplicial object ∆ op → B admits a colimit, whose colimit cone is given by the augmented diagram ∆ op + → B : ∆ op ❆❆❆❆❆❆❆❆❆ Nn | | ③③③③③③③③③ (cid:127) _ (cid:15) (cid:15) ∆ op + (cid:31) (cid:127) / / ∆ ∞ / / ❴❴❴ B Combining these results, it follows that if a functor admits both left and right adjoints,its monadicity can be leveraged to help establish its comonadicity, or conversely:7.2.5.
Proposition.
Suppose u : A → B admits both left and right adjoints A u / / B ℓ ⊥ } } r ⊥ a a Then if u is comonadic it is also monadic, and conversely if u is monadic then it is alsocomonadic.Proof. If u is comonadic, then u is conservative, verifying condition (iii) of the dualMonadicity Theorem 7.2.1. We have already assumed that the left adjoint required by(i) exists. Finally, Theorem 7.2.4 implies that B admits colimits of u -split simplicial ob-jects, and then comonadicity of u together with Theorem 7.2.3 then implies that A admitsthem as well and these are preserved by u . This verifies (ii), and Theorem 7.2.1 thenimplies that u is also monadic. A dual argument proves the converse implication. (cid:3) Theorem.
The forgetful functor u : Cart ( qCat ) / B −→ Cart ( qCat ) / ob B ∼ = Y ob B qCat Furthermore, such colimits are absolute , that is preserved by any functor.
ARTESIAN EXPONENTIATION AND MONADICITY 61 is comonadic and hence also monadic.Proof.
We use Theorem 7.2.1 to prove comonadicity and then deduce monadicity fromProposition 7.1.4 and Proposition 7.2.5. The right adjoint to u is constructed in Proposition7.1.3 proving (i). Example VIII.6.1.7 and Remark VIII.6.1.9 combine to prove that Cart / B admits and Cart / B ֒ → qCat / B preserves all limits. The functor u is the composite ofthis inclusion with the projection functor qCat / B → qCat / ob B , which also preserves alllimits, being the homotopy coherent nerve of a functor of ∞ -cosmoi Q Cat / B → Q Cat / ob B .This proves (ii). Proposition 3.1.7 and Lemma 7.2.2 assert that u : Cart / B → Cart / ob B isconservative, proving (iii). (cid:3) Groupoidal reflection
In this section, we give a first application of the monadicity and comonadicity results ofthe previous section. A cartesian fibration between quasi-categories is groupoidal if its fibersare Kan complexes, rather than quasi-categories. In this section we construct a reflectionto the fully faithful inclusion of the quasi-category of groupoidal cartesian fibrations intothe quasi-category of cartesian fibrations:
Cart gr / B ⊥ Cart / B invert r r A different proof of this result will be given in [25], making use of explicit fiberwise coin-verters.In §8.1, we study the relationship between the quasi-categorically enriched category ofcartesian fibrations and its subcategory of groupoidal cartesian fibrations and establish agroupoidal reflection functor in the “base case,” reflecting quasi-categories into Kan com-plexes. In §8.2, we prove that the monadic and comonadic adjunctions of Theorem 7.2.6restrict to define analogous monadic and comonadic adjunctions for groupoidal cartesianfibrations. It follows that the large quasi-categories
Cart gr / B and Cart / B can be understoodas quasi-categories of algebras for closely related homotopy coherent monads acting on Q b ∈ ob B Kan and Q b ∈ ob B qCat respectively. In §8.3, we exploit this presentation to con-struct an adjunction Cart gr / B ⊥ Cart / B invert r r defining the reflection of a cartesian fibration into a groupoidal cartesian fibration.8.1. Reflecting quasi-categories into Kan complexes.
Before we begin, we argue thatthe notion of groupoidal cartesian fibration of quasi-categories just defined agrees with thedefinition given in §IV.4.2, which declares that a cartesian fibration p : E ։ B is groupoidal just when the quasi-category of functors from any f : X → B to p is a Kan complex.The reader who is content to work with the present definition may safely skip this bit ofbookkeeping. Lemma.
Let p : E ։ B be an isofibration between quasi-categories. Then p isgroupoidal as an object of Q Cat / B if and only if the fibers of p are Kan complexes. Proof.
The quasi-category of functors in Q Cat / B from b : 1 → B to p : E → B is the fiber E b of p over b , so if p is groupoidal as an object of Q Cat / B , then p necessarily has Kancomplex fibers. For the converse, we must argue that the pullback Fun B ( f : X → B , p : E ։ B ) (cid:15) (cid:15) (cid:15) (cid:15) / / Fun ( X, E ) Fun ( X,p ) (cid:15) (cid:15) (cid:15) (cid:15) f / / Fun ( X, B ) is a Kan complex supposing that p has groupoidal fibers. Unraveling the definition, wemust show that any map h that lies over f in the sense of the following commutativediagram X × ∆ π (cid:15) (cid:15) h / / E p (cid:15) (cid:15) (cid:15) (cid:15) X f / / B defines an isomorphism E X . Corollary I.2.3.12 observes that such 1-simplices are invertibleif and only if each component h ( x, − ) : ∆ → E indexed by a vertex x ∈ X defines anisomorphism in E . As this 1-simplex lives in the fiber over px , it is invertible, so the resultfollows. (cid:3) At the level of simplicially-enriched categories, the subcategory of groupoidal cartesianfibrations is defined by the pullback:
Cart gr ( Q Cat ) / B (cid:31) (cid:127) / / (cid:15) (cid:15) Cart ( Q Cat ) / B (cid:15) (cid:15) Cart gr ( Q Cat ) / ob B ∼ = Q ob B K an (cid:31) (cid:127) / / Q ob B Q Cat ∼ = Cart ( Q Cat ) / ob B (8.1.2)In §8.3, we construct a groupoidal reflection functor , by which we mean a left adjoint tothe inclusion Cart gr ( qCat ) / B ֒ → Cart ( qCat ) / B as a functor between large quasi-categories. We begin by describing groupoidal reflectionin the case where B = 1 . Warning: this result does not hold for generic ∞ -cosmoi, having to do with the fact that the terminalquasi-category is a generating object in Q Cat in a suitable sense.
ARTESIAN EXPONENTIATION AND MONADICITY 63
Theorem.
The inclusion
Kan ֒ → qCat admits both left and right adjoints Kan (cid:31) (cid:127) ⊥⊥ / / qCat invert y y core c c and is monadic and comonadic.Proof. The inclusion K an ֒ → Q Cat is left adjoint to a functor core : Q Cat → K an that carriesa quasi-category to the maximal sub Kan complex spanned by the isomorphisms; this is arestriction of the 0-sharp -core adjunction of 6.1.2(iv) to the subcategories of saturated1-complicial and saturated 0-complicial sets. This adjunction is simplicial with respect tothe Kan complex enrichments of both K an and Q Cat , the latter obtained by applying the core functor to the function complexes, so this simplicially enriched adjunction descendsto provide a right adjoint to K an ֒ → Q Cat .The left adjoint can also be modeled at the point-set level. The quasi-category qCat is isomorphic to the homotopy coherent nerve of the Kan-complex enriched category ofnaturally marked quasi-categories in the sense of Example 6.1.7. There is a simplicialQuillen adjunction sSet ( − ) ♯ ⊥ sSet • U r r connecting this simplicial model structure for quasi-categories to the Quillen model struc-ture for Kan complexes on simplicial sets. Applying Theorem I.6.2.1, this provides the leftadjoint to the inclusion Kan ֒ → qCat . From this vantage point, we may apply PropositionVII.2.2.3 to see that K an is closed in Q Cat under flexible weighted limits, so we concludethat
Kan ֒ → qCat creates all limits.By the argument used to prove Lemma 7.2.2, a functor between Kan complexes is anisomorphism in Kan if and only if it is an equivalence in the Quillen model structure ifand only if it is an equivalence in the Joyal model structure, i.e., if and only if it is anequivalence in qCat . This tells us that the inclusion
Kan ֒ → qCat is conservative. NowTheorem 7.2.1 implies that Kan → qCat is comonadic, and Proposition 7.2.5 then impliesthat Kan → qCat is also monadic. (cid:3) (Co)monadicity of groupoidal cartesian fibrations. We now argue that
Cart gr ( Q Cat ) / B / / Cart gr ( Q Cat ) / ob B ∼ = Q ob B K an ( E p −− ։ B ) ( E b ) b ∈ ob B admits left and right quasi-categorically enriched biadjoints, given by restricting those fromthe non-groupoidal case, and that moreover the restricted adjunction is both monadic andcomonadic at the level of functors between underlying quasi-categories. Theorem.
The quasi-categorical biadjoints to
Cart ( Q Cat ) / B → Cart ( Q Cat ) / ob B re-strict to groupoidal cartesian fibrations Cart gr ( Q Cat ) / B (cid:31) (cid:127) / / (cid:15) (cid:15) Cart ( Q Cat ) / B (cid:15) (cid:15) Cart gr ( Q Cat ) / ob B (cid:31) (cid:127) / / R ⊣ [ [ ✟ ✤✻ L ⊣ C C ✻✤ ✟ Cart ( Q Cat ) / ob B R ⊣ [ [ L ⊣ C C and moreover these restricted adjunctions display the functor between the quasi-categoricalcores Cart gr / B → Cart gr / ob B ∼ = Q / ob B Kan as both monadic and comonadic.Proof.
Proposition 7.1.4 defines L : Cart ( Q Cat ) / ob B → Cart ( Q Cat ) / B to be the functor thatcarries a family ( E b ) b ∈ ob B to L (( E b ) b ∈ ob B ) := a b E b × B ↓ b ։ B . The fiber over x ∈ ob B is ` b E b × x ↓ b . Since x ↓ b is a Kan complex, it is clear that thisfiber is groupoidal if each E b is a Kan complex. Thus, we see immediately that L restrictsto groupoidal cartesian fibrations.Proposition 7.1.3 defines R : Cart ( Q Cat ) / ob B → Cart ( Q Cat ) / B to be the functor thatcarries a family ( E b ) b ∈ ob B to R (( E b ) b ∈ ob B ) := Y b ( E b × B ։ B ) b ↓ B ։ B . The fiber over x ∈ ob B of this product of fibrations is isomorphic to the product of thefibers of each ( E b × B ։ B ) b ↓ B ։ B over x , so it suffices to show that each of these fibers isa Kan complex if E b is a Kan complex. By the bijection of Definition 6.3.1, a 1-simplex inthe fiber of ( E b × B ։ B ) b ↓ B ։ B over x : 1 → B corresponds to the displayed dashed map ∆ × b ↓ x (cid:15) (cid:15) (cid:15) (cid:15) / / + + ❢ ❞ ❝ ❜ ❵ ❴ ❫ ❭ ❬ ❩ ❳ ❲ b ↓ x (cid:15) (cid:15) (cid:15) (cid:15) / / b ↓ B (cid:15) (cid:15) (cid:15) (cid:15) E b × B π z z z z ✉✉✉✉✉✉✉✉✉✉ ∆ / / ∆ x / / B i.e., to a map ∆ × b ↓ x → E b . If each E b is a Kan complex, this map can be extended alongthe inclusion ∆ ֒ → I from the 1-simplex into the free-living isomorphism. This provesthat every 1-simplex in the fiber of ( E b × B ։ B ) b ↓ B ։ B is an isomorphism, which tells usthat R restricts to groupoidal cartesian fibrations.By what is now a familiar line of argument, we apply Theorem 7.2.1 to prove that therestricted adjunctions are comonadic, and then deduce monadicity from Proposition 7.2.5.The required adjoints have already been constructed and Proposition 3.1.7 implies that Cart gr / B → Cart gr / ob B is conservative, so it remains only to establish condition (ii) of Theorem7.2.1. ARTESIAN EXPONENTIATION AND MONADICITY 65
The pullback of simplicial categories (8.1.2) is preserved by passing to the level of quasi-categories:
Cart gr / B (cid:31) (cid:127) / / (cid:15) (cid:15) Cart / B (cid:15) (cid:15) Q ob B Kan (cid:31) (cid:127) / / Q ob B qCat By the monadicity established in Theorems 7.2.6 and 8.1.3 and the dual of Theorem7.2.3, both the left-hand vertical and lower horizontal functors create all limits presentin Q ob B qCat , which is to say all limits, since by Proposition VII.6.2.1 qCat is complete.The lower inclusion is replete up to isomorphism, so Kan ֒ → qCat defines an isofibration oflarge quasi-categories. Thus, Lemma 8.2.2 below applies to tell us that Cart gr ( qCat ) / B isalso complete and all limits are preserved by the left-hand vertical functor.Now Theorem 7.2.1 implies that Cart gr / B → Cart gr / ob B ∼ = Q ob B Kan is comonadic, andmonadicity follows from Proposition 7.2.5. (cid:3)
Lemma.
Consider a pullback diagram of quasi-categories whose vertical morphismsare isofibrations E p (cid:15) (cid:15) (cid:15) (cid:15) f / / F q (cid:15) (cid:15) (cid:15) (cid:15) B g / / A Then p creates and f preserves any class of limits or colimits that g preserves and q creates.Proof. This can be proven directly or by appealing to Theorem III.1.1, which states thatthe subcategory of ∞ -categories admitting and functors preserving limits or colimits of aparticular variety is closed under flexible weighted limits. Since q : F ։ A is an isofibration,the strict limit of the cospan B g −→ A q և −− F is equivalent to the limit weighted by theprojective cofibrant weight ⌟ → sSet with image ∆ δ −→ I δ ←− ∆ . Theorem III.1.1 nowimplies that E admits and f and p preserve any limits or colimits present in B , A , and F and preserved by g and q .It remains only to argue that (co)limits are created by p : E ։ B . Consider a family ofdiagrams d : D → E X so that pd : D → B X has a colimit b : D → B . Then gb : D → A isa colimit for gpd = qf d : D → A X , and the fact that q creates such colimits implies thatthere is an colimit object c : D → B for f d with qc isomorphic to gb in Fun ( D , A ) . In fact,since Fun ( D , q ) : Fun ( D , F ) ։ Fun ( D , A ) is an isofibration, we may assume that qc = gb , sothis pair induces an object e : D → E with pe = b . By construction, there is a pullback of contravariant represented modules displayed below left e ↓ E p (cid:15) (cid:15) (cid:15) (cid:15) f / / c ↓ B q (cid:15) (cid:15) (cid:15) (cid:15) d ↓ ∆ f / / p (cid:15) (cid:15) (cid:15) (cid:15) f d ↓ ∆ q (cid:15) (cid:15) (cid:15) (cid:15) b ↓ B g / / gb ↓ A pd ↓ ∆ g / / gpd ↓ ∆ whose underlying cospan is equivalent to the cospan in the pullback of quasi-categories ofcones displayed above right. It follows that e ↓ E ≃ d ↓ ∆ , which says that e is a colimitfor d . (cid:3) Groupoidal reflection.
Our monadicity results, Theorems 8.2.1 and 7.2.6, tell usthat the quasi-categories
Cart / B and Cart gr / B are equivalent to the quasi-categories of algebrasassociated to closely related homotopy coherent monads acting on Cart / ob B ∼ = Q ob B qCat and Cart gr / ob B ∼ = Q ob B Kan respectively. In this section, we will use this result to liftthe reflection functor invert : qCat → Kan from quasi-categories to Kan complexes to agroupoidal reflection functor invert :
Cart / B → Cart gr / B that is left adjoint to the inclusion.To do this we make use of a convenient representation for adjoint functors that can beexpressed in any 2-category, dual to the more familiar representation of the unit of anadjunction as an absolute left extension diagram:8.3.1. Lemma (I.5.0.4) . To define a left adjoint to a functor u : A → B is to define anabsolute left lifting of id B along u : ⇑ η A u (cid:15) (cid:15) B f > > ⑦⑦⑦⑦ B in which case f ⊣ u with unit η : id B ⇒ uf . Let ⌟ denote the category indexing a cospan and write Q Cat ⌟ for the simplicially en-riched category of cospans of quasi-categories, whose objects are cospans and whose 0-arrows are natural transformations B f (cid:15) (cid:15) v ❅❅❅❅❅❅ C g / / w (cid:31) (cid:31) ❃❃❃❃❃❃ A u ❅❅❅ (cid:31) (cid:31) ❅❅ B ′ f ′ (cid:15) (cid:15) C ′ g ′ / / A ′ (8.3.2) ARTESIAN EXPONENTIATION AND MONADICITY 67
Definition.
Transformations of the kind depicted in (8.3.2) between diagrams whichadmit absolute left liftings give rise to the following diagram ⇑ λ B f (cid:15) (cid:15) v ❅❅❅❅❅❅ C ℓ ? ? ⑧⑧⑧⑧⑧⑧ g / / w (cid:31) (cid:31) ❃❃❃❃❃❃ A u ❅❅❅ (cid:31) (cid:31) ❅❅ B ′ f ′ (cid:15) (cid:15) C ′ g ′ / / A ′ = B v (cid:31) (cid:31) ❅❅❅❅❅❅ C ℓ ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) w (cid:30) (cid:30) ❃❃❃❃❃❃ ⇑ τ ⇑ λ ′ B ′ f ′ (cid:15) (cid:15) C ′ ℓ ′ ? ? ⑧⑧⑧⑧⑧⑧ g ′ / / A ′ (8.3.4)in which the triangles are absolute left liftings and the 2-cell τ is induced by the universalproperty of the triangle on the right. We say that the transformation (8.3.4) is left exact if and only if the induced 2-cell τ is an isomorphism. This left exactness condition holds ifand only if, in the diagram on the left, the whiskered 2-cell uλ displays vℓ as the absoluteleft lifting of g ′ w through f ′ .Our interest in these notions is on account of the following result:8.3.5. Proposition (III.4.9) . Consider any simplicial functor T : A → Q
Cat ⌟ and anyflexible weight W : A → S
Set . If each of the objects in the image of T admits an absoluteleft lifting and each of the 0-arrows in the image of T is left exact, then the weighted limit { W, T } ∈ Q
Cat ⌟ admits an absolute left lifting and the legs of the limit cone are left exacttransformations. By Lemma II.6.1.8, the quasi-category of algebras construction, introduced in DefinitionII.6.1.7, is an instance of a flexible weighted limit. We will use Proposition 8.3.5 applied ina larger Grothendieck universe to the quasi-categorically enriched category Q CAT of largequasi-categories to lift the absolute left lifting diagram ⇑ η Kan (cid:127) _ (cid:15) (cid:15) qCat invert ; ; ✇✇✇✇✇✇✇✇✇ qCat whose left adjoint part defines the groupoidal reflection associated to the inclusion Cart gr / ob B ∼ = Y ob B Kan ֒ → Y ob B qCat ∼ = Cart / ob B to these flexible weighted limits, defining an absolute left lifting diagram ⇑ η Cart gr / B (cid:127) _ (cid:15) (cid:15) Cart / B invert : : ✈✈✈✈✈ Cart / B the left adjoint being the desired groupoidal reflection functor. Theorem.
There is a left adjoint to the inclusion
Cart gr / B Cart / B ⊥ invert defining the reflection of a cartesian fibration into a groupoidal cartesian fibration.Proof. By Theorem II.4.3.9, the adjunction
Cart / B Q ob B qCat ⊥ L (8.3.7)defined in Proposition 7.1.4 extends to a homotopy coherent adjunction : a simplicial functorAdj → Q CAT valued in the quasi-categorically enriched category of large quasi-categorieswhose domain is the simplicial computad obtained by applying the nerve functor to thehom-categories of the free 2-category containing an adjunction; see §II.3. The two ob-jects of Adj, called “ − ” and “ + ” are mapped to the quasi-categories Cart / B and Q ob B qCat respectively. The full subcategory Mnd ֒ → Adj on the object “ + ” defines the free homo-topy coherent monad . In this case, the data of the underlying homotopy coherent monad T : Mnd ֒ → Adj → Q
CAT on the object Q ob B qCat is given by a the map of objects + Q ob B qCat and the map of function complexes T : ∆ + = Fun
Mnd (+ , +) → Fun Q CAT ( Y ob B qCat , Y ob B qCat ) , satisfying an appropriate simplicial functoriality condition. This functoriality conditionimplies that the 0-arrows of Mnd, which coincide with the objects [ n ] ∈ ∆ + , are all finitecomposites of the object [0] , whose image t := T [0] : Y ob B qCat → Y ob B qCat is the monad endofunctor defined by composing the left and right adjoints of (8.3.7).To apply Proposition 8.3.5 we must extend the homotopy coherent monad T to a ho-motopy coherent monad on Q CAT ⌟ . To do so, we argue that this homotopy coherentmonad restricts along Q ob B Kan ֒ → Q ob B qCat to define a homotopy coherent monad T gr : Mnd → Q
CAT on Q ob B Kan in such a way that this map will define the component ofa simplicial natural transformation T gr ⇒ T . To see this, note that Q ob B Kan , the nerveof the simplicially enriched category spanned by ob B -indexed families of Kan complexes,is a full sub quasi-category of Q ob B qCat , the nerve of the simplicially enriched categoryspanned by ob B -indexed families of quasi-categories, in the sense that it contains all of the n -simplices whose vertices are Kan complexes, not mere quasi-categories. So to check thatthe data of the homotopy coherent monad restricts to define a simplicial functor given by ARTESIAN EXPONENTIATION AND MONADICITY 69 + Q ob B Kan and T gr : ∆ + = Fun
Mnd (+ , +) → Fun Q CAT ( Y ob B Kan , Y ob B Kan ) , it suffices to check this at the level of vertices [ n ] ∈ ∆ + , which amounts to checking thatthe monad t of (8.3.7) restricts to define a monad t gr on groupoidal cartesian fibrations;this was done in Theorem 8.2.1.In this way we obtain a simplicial functor Mnd → Q CAT sending the object “ + ”to the arrow Q ob B Kan ֒ → Q ob B qCat . Pairing this with the identity simplicial naturaltransformation we obtain a simplicial functor Mnd → Q CAT ⌟ sending the object “ + ” tothe cospan displayed below-left Q ob B Kan (cid:127) _ (cid:15) (cid:15) ⇑ η Q ob B Kan (cid:127) _ (cid:15) (cid:15) Q ob B qCat Q ob B qCat Q ob B qCat Q ob B invert : : ✈✈✈✈✈✈✈✈✈✈ Q ob B qCat (8.3.8)This cospan admits an absolute left lifting displayed above right, defining the left adjointand unit of an adjunction whose counit is invertible. In fact the entire diagram Mnd →Q CAT ⌟ restricts to the subcategory spanned by those cospans that admit absolute leftliftings and those 0-arrows that define left exact transformations between them. To see this,we need only argue that the generating 0-arrow [0] in Mnd, the endofunctor of the monad,defines a left exact transformation. That is, we must show that the endotransformation of(8.3.8) whose components are the functors Q ob B Kan t gr / / Q ob B Kan Q ob B qCat t / / Q ob B qCat ( E b ) b ∈ B (cid:18) ` b ∈ ob B E b × x ↓ b (cid:19) x ∈ ob B ( E b ) b ∈ B (cid:18) ` b ∈ ob B E b × x ↓ b (cid:19) x ∈ ob B is left exact. This amounts to showing that the whiskered 2-cell ⇑ η Q ob B Kan (cid:127) _ (cid:15) (cid:15) t gr / / Q ob B Kan (cid:127) _ (cid:15) (cid:15) Q ob B qCat Q ob B invert : : ✈✈✈✈✈✈✈✈✈✈ Q ob B qCat t / / Q ob B qCat is invertible. This is the case because the process of freely inverting a family of quasi-categories commutes up to equivalence with forming the product with the Kan complex x ↓ b and with the coproduct ` b ∈ ob B .In this way we obtain a homotopy coherent monad Mnd → Q CAT ⌟ valued in the sub-category of cospans admitting absolute left liftings and left exact transformations between them. There is a flexible weight W − : Mnd → S
Set introduced in Definition II.6.1.7 —the precise details of which are not relevant here — so that the W − -weighted limit of ahomotopy coherent monad define its quasi-category of algebras, as characterized up toequivalence by the Monadicity Theorem 7.2.1. By Theorems 8.2.1 and 7.2.6 the W − -weighted limit of the composite diagram Mnd → Q CAT ⌟ defines the cospan displayedbelow and by Proposition 8.3.5 it therefore admits an absolute left lifting: ⇑ η Cart gr / B (cid:127) _ (cid:15) (cid:15) Cart / B invert : : ✈✈✈✈✈ Cart / B By Lemma 8.3.1, this absolute left lifting diagram defines the adjunction that constructsthe groupoidal reflection of a cartesian fibration. (cid:3)
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Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
Email address : [email protected] Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia
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