ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM FOR ARTIN STACKS

Transcription

1 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM FOR ARTIN STACKS YIFENG LIU AND WEIZHE ZHENG Abstract. In this article, we develop a theory o Grothendieck s six operations or derived categories in étale cohomology o Artin stacks. We prove several desired properties o the operations, including the base change theorem in derived categories. This extends all previous theories on this subject, including the recent one developed by Laszlo and Olsson, in which the operations are subject to more assumptions and the base change isomorphism is only constructed on the level o sheaves. Moreover, our theory works or higher Artin stacks as well. Our method diers rom all previous approaches, as we exploit the theory o stable -categories developed by Lurie. We enhance derived categories, unctors, and natural isomorphisms to the level o -categories and introduce -categorical (co)homological descent. To handle the homotopy coherence, we apply the results o our previous article [26] and develop several other -categorical techniques. Contents Introduction Results Why -categories? What do we need to enhance? Structure o the article Convention and notation Acknowledgments Preliminaries on -categories Elementary lemmas Constructing unctors via the category o simplices Multisimplicial sets Partial adjoints Symmetric monoidal -categories Enhanced operations or ringed topoi The lat model structure Enhanced operations Enhanced operations or schemes Abstract descent properties Enhanced operation map Poincaré duality and (co)homological descent DESCENT: a program Description Construction Properties Running DESCENT Quasi-separated schemes 53 Date: April 4, Mathematics Subject Classiication. 14F05 (primary), 14A20, 14F20, 18D05, 18G30 (secondary). 1

2 2 YIFENG LIU AND WEIZHE ZHENG 5.2. Algebraic spaces Artin Stacks Higher Artin stacks Higher Deligne Mumord stacks Summary and complements Recapitulation More adjointness in the inite-dimensional Noetherian case Constructible complexes Compatibility with the work o Laszlo and Olsson 71 Reerences 73 Introduction Derived categories in étale cohomology on Artin stacks and Grothendieck s six operations between such categories have been developed by many authors including [40] (or Deligne Mumord stacks), [25], [5], [31] and [24]. These theories all have some restrictions. In the most recent and general one [24] by Laszlo and Olsson on Artin stacks, a technical condition was imposed on the base scheme which excludes, or example, the spectra o certain ields 1. More importantly, the base change isomorphism was constructed only on the level o (usual) cohomology sheaves [24, 5]. The Base Change theorem is undamental in many applications. In the Geometric Langlands Program or example, the theorem has already been used on the level o perverse cohomology. It is thus necessary to construct the Base Change isomorphism not just on the level o cohomology, but also in the derived category. Another limitation o most previous works is that they dealt only with constructible sheaves. When working with morphisms locally o inite type, it is desirable to have the six operations or more general lisse-étale sheaves. In this article, we develop a theory that provides the desired extensions o previous works. Instead o the usual unbounded derived category, we work with its enhancement, which is a stable -category in the sense o Lurie [29, ]. This makes our approach dierent rom all previous ones. We construct unctors and produce relations in the world o -categories, which themselves orm an -category. We start by upgrading the known theory o six operations or (coproducts o) quasi-compact and separated schemes to -categories. The coherence o the construction is careully recorded. This enables us to apply -categorical descent to carry over the theory o six operations, including the Base Change theorem, to algebraic spaces, higher Deligne Mumord stacks and higher Artin stacks Results. In this section, we will state our results only in the classical setting o Artin stacks on the level o usual derived categories (which are homotopy categories o the derived -categories), among other simpliication. We reer the reader to Chapter 6 or a list o complete results or higher Deligne Mumord stacks and higher Artin stacks, stated on the level o stable -categories. By an algebraic space, we mean a shea in the big pp site satisying the usual axioms [4, 025Y]: its diagonal is representable (by schemes); and it admits an étale and surjective map rom a scheme (in Sch U ; see 0.5). By an Artin stack X, we mean an algebraic stack in the sense o [4, 026O]: it is a stack in (1-)groupoids over (Sch U ) pp ; its diagonal is representable by algebraic spaces; and it admits a smooth and surjective map rom a scheme. In particular, we do not assume that an Artin stack is quasi-separated. Our main results are the construction o the six operations or the derived categories o lisse-étale sheaves on Artin stacks and the expected relations among them. In what ollows, Λ is a unital commutative ring, or more generally, a ringed diagram in Deinition For example, the ield k(x1, x 2,... ) obtained by adjoining countably ininitely many variables to an algebraically closed ield k in which l is invertible.

3 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 3 Let X be an Artin stack. We denote by D(X lis-ét, Λ) the unbounded derived category o (X lis-ét, Λ)- modules, where X lis-ét is the lisse-étale topos associated to X. Recall that an (X lis-ét, Λ)-module F is equivalent to an assignment to each smooth morphism v : Y X with Y an algebraic space a (Yét, Λ)-module F v and to each 2-commutative triangle Y Y v σ X with v, v smooth and Y, Y being algebraic spaces, a morphism τ σ : F v F v which is an isomorphism i is étale such that the collection {τ σ } satisies a natural cocycle condition [25, ]. An (X lis-ét, Λ)-module F is Cartesian i in the above description, all morphisms τ σ are isomorphisms [25, 12.3]. Let D cart (X lis-ét, Λ) be the ull subcategory o D(X lis-ét, Λ) spanned by complexes whose cohomology sheaves are all Cartesian. I X is Deligne Mumord, then we have an equivalence D cart (X lis-ét, Λ) D(Xét, Λ). Let : Y X be a morphism o Artin stacks. We deine operations in 6.1: : D cart (X lis-ét, Λ) D cart (Y lis-ét, Λ), v : D cart (Y lis-ét, Λ) D cart (X lis-ét, Λ); X : D cart (X lis-ét, Λ) D cart (X lis-ét, Λ) D cart (X lis-ét, Λ), Hom X : D cart (X lis-ét, Λ) op D cart (X lis-ét, Λ) D cart (X lis-ét, Λ). The pairs (, ) and ( K, Hom(K, )) or every K D cart (X lis-ét, Λ) are pairs o adjoint unctors. We ix a nonempty set L o rational primes. A ring is L-torsion [2, IX 1.1] i each element is killed by an integer that is a product o primes in L. An Artin stack X is L-coprime i there exists a morphism X Spec Z[L 1 ]. I X is L-coprime (resp. Deligne Mumord), : Y X is locally o inite type, and Λ is L-torsion (resp. torsion), then we have another pair o adjoint unctors:! : D cart (Y lis-ét, Λ) D cart (X lis-ét, Λ),! : D cart (X lis-ét, Λ) D cart (Y lis-ét, Λ). Next we list some properties o the six operations. We reer the reader to 6.1 or a more complete list. Theorem (Base Change, Proposition 6.1.1). Let Λ be an L-torsion (resp. torsion) ring, and let q W Y g be a Cartesian square o L-coprime Artin stacks (resp. any Deligne Mumord stacks) where p is locally o inite type. Then we have a natural isomorphism o unctors: p! q! g : D cart (Z lis-ét, Λ) D cart (Y lis-ét, Λ). Theorem (Projection Formula, Proposition 6.1.2). Let Λ be an L-torsion (resp. torsion) ring, and let : Y X be a morphism locally o inite type o L-coprime Artin stacks (resp. o arbitrary Deligne Mumord stacks). Then we have a natural isomorphism o unctors:! ( Y ) (! ) X : D cart (Y lis-ét, Λ) D cart (X lis-ét, Λ) D cart (X lis-ét, Λ). Z X p

4 4 YIFENG LIU AND WEIZHE ZHENG Corollary (Künneth Formula, Proposition 6.1.3). Let Λ be an L-torsion (resp. torsion) ring, and let q 1 q 2 Y 1 Y Y p 1 p 2 X 1 X X 2 be a diagram o L-coprime Artin stacks (resp. o arbitrary Deligne Mumord stacks) that exhibits Y as the limit Y 1 X1 X X2 Y 1, where 1 and 2 are locally o inite type. Then we have a natural isomorphism o unctors:! (q 1 Y q 2 ) (p 1 1! ) X (p 2 2! ): D cart (Y 1,lis-ét, Λ) D cart (Y 2,lis-ét, Λ) D cart (X lis-ét, Λ). Theorem (Trace Map and Poincaré Duality, Proposition 6.1.9). Let Λ be an L-torsion ring, and let : Y X be a lat morphism locally o inite presentation o L-coprime Artin stacks. Then (1) There is a unctorial trace map Tr : τ 0! Λ Y d = τ 0! ( Λ X ) d Λ X, where d is an integer larger than or equal to the dimension o every geometric iber o ; Λ X and Λ Y denote the constant sheaves placed in degree 0; and d = [2d](d) is the composition o the shit by 2d and the d-th power o Tate s twist. (2) I is moreover smooth, the induced natural transormation u :! dim id X is a counit transormation, where id X is the identity unctor o D cart (X lis-ét, Λ). In other words, we have a natural isomorphism o unctors: dim! : D cart (X lis-ét, Λ) D cart (Y lis-ét, Λ). Corollary (Smooth Base Change, Corollary ). Let Λ o an L-torsion ring, and let q W Y g be a Cartesian diagram o L-coprime Artin stacks where p is smooth. Then the natural transormation o unctors p g q : D cart (Y lis-ét, Λ) D cart (Z lis-ét, Λ) is a natural isomorphism. Theorem (Descent, Proposition ). Let Λ be a ring, let : Y X be morphism o Artin stacks and let y : Y + 0 Y be a smooth surjective morphism. Let Y+ be the Čech nerve o y with the morphism y n : Y + n Y + 1 = Y. Put n = y n : Y + n X. (1) For every complex K D 0 (Y, Λ), we have a convergent spectral sequence E p,q 1 = H q ( p y pk ) H p+q K. (2) I X is L-coprime; Λ is L-torsion, and is locally o inite type, then or every complex K D 0 (Y, Λ), we have a convergent spectral sequence Z X Ẽ p,q 1 = H q ( p! y! pk ) H p+q! K. Note that even in the case o schemes Theorem (2) seems to be a new result. To state our results or constructible sheaves, we work over an L-coprime base scheme S that is either quasi-excellent inite-dimensional or regular o dimension 1. We consider only Artin stacks X that are locally o inite type over S. Let Λ be a Noetherian L-torsion ring. Recall that an (X lis-ét, Λ)-module is constructible i it is Cartesian and its pullback to every scheme, inite type over S, is constructible in the usual sense. Let D cons (X lis-ét, Λ) be the ull subcategory o D(X lis-ét, Λ) spanned by complexes whose cohomology sheaves are constructible. Let D (+) cons(x lis-ét, Λ) (resp. p

5 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 5 D ( ) cons(x lis-ét, Λ)) be the ull subcategory o D cons (X lis-ét, Λ) spanned by complexes whose cohomology sheaves are locally bounded below (resp. above). The six operations mentioned previously restrict to the ollowing reined ones as in 6.3 (see Lemma and Proposition or precise statements): : D cons (X lis-ét, Λ) D cons (Y lis-ét, Λ),! : D cons (X lis-ét, Λ) D cons (Y lis-ét, Λ); X : D ( ) cons(x lis-ét, Λ) D ( ) cons(x lis-ét, Λ) D ( ) cons(x lis-ét, Λ), Hom X : D ( ) cons(x lis-ét, Λ) op D (+) cons(x lis-ét, Λ) D (+) cons(x lis-ét, Λ). I is quasi-compact and quasi-separated, then we have : D (+) cons(y lis-ét, Λ) D (+) cons(x lis-ét, Λ),! : D ( ) cons(y lis-ét, Λ) D ( ) cons(x lis-ét, Λ). We will also show that when the base scheme, the coeicient ring, and the morphism are all in the range o [24], our operations or constructible complexes are compatible with those constructed by Laszlo and Olsson on the level o usual derived categories. In particular, our Theorem implies that their operations satisy Base Change in derived categories, which was let open in [24]. In a subsequent article [27], we will develop an adic ormalism and establish adic analogues o the above results. Let (Ξ, Λ) be a partially ordered diagram o coeicient rings, that is, Ξ is a partially ordered set and Λ is a unctor rom Ξ op to the category o commutative rings (with units). A typical example is the projective system Z/l n+1 Z Z/l n Z Z/lZ, where l is a ixed prime number and the transition maps are natural projections. Inside the category D cart (X Ξ lis-ét, Λ), there is a ull subcategory D(XΞ lis-ét, Λ) adic spanned by (Ξ, Λ)-adic complexes. The inclusion admits a right adjoint D cart (X Ξ lis-ét, Λ) D(X Ξ lis-ét, Λ) adic which exhibits D(X Ξ lis-ét, Λ) adic as a colocalization o D cart (X Ξ lis-ét, Λ). We will construct operations on D(X Ξ lis-ét, Λ) adic and establish relations between them Why -categories? The -categories in this article reer to the ones studied by A. Joyal [21, 22] (where they are called quasi-categories), J. Lurie [28], et al. Namely, an -category is a simplicial set satisying liting properties o inner horn inclusions [28, ]. In particular, they are models or (, 1)-categories, that is, higher categories whose n-morphisms are invertible or n 2. For readers who are not amiliar with this language, we recommend [17] or a brie introduction o Lurie s theory [28], [29], etc. There are also other models or (, 1)-categories such as topological categories, simplicial categories, complete Segal spaces, Segal categories, model categories, and, in a looser sense, dierential graded (DG) categories and A -categories. We address two questions in this section. First, why do we need (, 1)-categories instead o (usual) derived categories? Second, why do we choose this particular model o (, 1)-categories? To answer these questions, let us ix an Artin stack X and an atlas u: X X, that is, a s- mooth and surjective morphism with X an algebraic space. We denote by Mod(X lis-ét, Λ) (resp. Mod(Xét, Λ)) the category o (X lis-ét, Λ)-modules (resp. (Xét, Λ)-modules) which is a Grothendieck abelian category. Let p α : X X X X (α = 1, 2) be the two projections. We know that i F Mod(X lis-ét, Λ) is Cartesian, then there is a natural isomorphism σ : p 1 u F p 2 u F satisying a cocycle condition. Conversely, an object G Mod(Xét, Λ) such that there exists an isomorphism σ : p 1 G p 2 G satisying the same cocycle condition is isomorphic to u F or some F Mod(X lis-ét, Λ). This descent property can be described in the ollowing ormal way. Let Mod cart (X lis-ét, Λ) be the ull subcategory o Mod(X lis-ét, Λ) spanned by Cartesian sheaves. Then it

6 6 YIFENG LIU AND WEIZHE ZHENG is the (2-)limit o the ollowing diagram Mod(Xét, Λ) p 1 p 2 Mod((X X X)ét, Λ) Mod((X X X X X)ét, Λ) in the (2, 1)-category o abelian categories 2. Thereore, to study Mod cart (X lis-ét, Λ), we only need to study Mod(Xét, Λ) or (all) algebraic spaces X in a 2-coherent way, that is, we need to track down all the inormation o natural isomorphisms (2-cells). Such 2-coherence is not more complicated than the one in Grothendieck s theory o descent [18]. One may want to apply the same idea to derived categories. The problem is that the descent property mentioned previously, in its naïve sense, does not hold anymore, since otherwise the classiying stack BG m over an algebraically closed ield will have inite cohomological dimension which is incorrect. In act, when orming derived categories, we throw away too much inormation on the coherence o homotopy equivalences or quasi-isomorphisms, which causes the ailure o such descent. A descent theory in a weaker sense, known as cohomological descent [2, V bis] and due to Deligne, does exist partially on the level o objects. It is one o the main techniques used in Olsson [31] and Laszlo Olsson [24] or the deinition o the six operations on Artin stacks in certain cases. However, it has the ollowing restrictions. First, Deligne s cohomological descent is valid only or complexes bounded below. Although a theory o cohomological descent or unbounded complexes was developed in [24], it comes at the price o imposing urther initeness conditions and restricting to constructible complexes. Second, relevant spectral sequences suggest that cohomological descent cannot be used directly to deine!-pushorward. A more natural solution can be reached once the derived categories are enhanced. Roughly speaking (see Proposition or the precise statement), i we write X n = X X X X ((n+1)- old), then D cart (X lis-ét, Λ) is naturally equivalent to the limit o ollowing cosimplicial diagram D(X 0,ét, Λ) p 1 p 2 D(X 1,ét, Λ) D(X 2,ét, Λ) in a suitable -category o closed symmetric monoidal presentable stable -categories. This is completely parallel to the descent property or module categories. Here D cart (X lis-ét, Λ) (resp. D(X n,ét, Λ)) is a closed symmetric monoidal presentable stable -category which serves as the enhancement o D cart (X lis-ét, Λ) (resp. D(X n,ét, Λ)). Strictly speaking, the previous diagram is incomplete in the sense that we do not mark all the higher cells in the diagram, that is, all natural equivalences o unctors, equivalences between natural equivalences, etc. In act, there is an ininite hierarchy o (homotopy) equivalences hidden behind the limit o the previous diagram, not just the 2-level hierarchy in the classical case. To deal with such kind o homotopy coherence is the major diiculty o the work, that is, we need to ind a way to encode all such hierarchy simultaneously in order to make the idea o descent work. In other words, we need to work in the totality o all -categories o concern. It is possible that such a descent theory (and other relevant higher-categorical techniques introduced below) can be realized by using other models or higher categories. We have chosen the theory developed by Lurie in [28], [29] or its elegance and availability. Precisely, we will use the techniques o the (marked) straightening/unstraightening construction, Adjoint Functor Theorem, and the -categorical Barr Beck Theorem. Based on Lurie s theory, we develop urther -categorical techniques to treat the homotopy-coherence problem mentioned as above. These techniques would enable us to, or example, take partial adjoints along given directions ( 1.4); ind a coherent way to decompose morphisms ([26, 4]); 2 A (2, 1)-category is a 2-category in which all 2-cells are invertible.

7 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 7 gluing data rom Cartesian diagrams to general ones ([26, 5]); make a coherent choice o descent data ( 4.2). In the next section, we will have a chance to explain some o them. During the preparation o this article, Gaitsgory [13] studied operations or ind-coherent sheaves on DG schemes and derived stacks in the ramework o -categories. Our work bears some similarity to his. We would like to point out however that he ignored homotopy-theoretical issues (in the same sense o homotopy coherence), or example, in the proo o [13, 6.1.9], which is a key step or the entire construction. Meanwhile, a sizable portion (Chapter 1 and [26]) o our work is devoted to developing general techniques to handle homotopy coherence. We would also like to remark that Lurie s theory has already been used, or example, in [6] to study quasi-coherent sheaves on certain (derived) stacks with many applications. This work, which studies lisse-étale sheaves, is another maniestation o the power o Lurie s theory. Moreover, the -categorical enhancement o six operations and its adic version, which is studied in the subsequent article [27], are necessary in certain applications o geometric/categorical method to the Langlands program, as shown in the recent work o Bezrukavnikov, Kazhdan and Varshavsky [7] What do we need to enhance? In the previous section, we mentioned the enhancement o a single derived category. It is a stable -category (which can be thought o as an -categorical version o a triangulated category) D cart (X lis-ét, Λ) (resp. D(Xét, Λ) or X an algebraic space) whose homotopy category (which is an ordinary category) is naturally equivalent to D cart (X lis-ét, Λ) (resp. D(Xét, Λ)). The enhancement o operations is understood in the similar way. For example, the enhancement o -pullback or : Y X should be an exact unctor (0.1) such that the induced unctor : D cart (X lis-ét, Λ) D cart (Y lis-ét, Λ) h : D cart (X lis-ét, Λ) D cart (Y lis-ét, Λ) is the -pullback unctor o usual derived categories. However, such enhancement is not enough or us to do descent. The reason is that we need to put all schemes and then algebraic spaces together. Let us denote by Sch qc.sep the category o coproducts o quasi-compact and separated schemes. The enhancement o -pullback or schemes in the strong sense is a unctor: (0.2) Λ Sch qc.sepeo : N(Sch qc.sep ) op Pr L st where N denotes the nerve unctor (see the deinition preceding [28, ]) and Pr L st is certain -category o presentable stable -categories, which will be speciied later. Then (0.1) is just the image o the edge : Y X i is in Sch qc.sep. The construction o (0.2) (and its right adjoint which is the enhancement o -pushorward) is not hard, with the help o the general construction in [29]. The diiculty arises in the enhancement o!-pushorward. Namely, we need to construct a unctor: Λ Sch qc.sepeo! : N(Sch qc.sep ) F Pr L st, where N(Sch qc.sep ) F is the subcategory o N(Sch qc.sep ) only allowing morphisms that are locally o inite type. The basic idea is similar to the classical approach: using Nagata compactiication theorem. The problem is the ollowing: or a morphism : Y X in Sch qc.sep, locally o inite type, we need to choose (non-canonically!) a relative compactiication Y i Y X p I X,

8 8 YIFENG LIU AND WEIZHE ZHENG that is, i is an open immersion and is proper, and deine! = p! i! (in the derived sense). It turns out that the resulting unctor o usual derived categories is independent o the choice, up to natural isomorphism. First, we need to upgrade such natural isomorphisms to natural equivalences between -categories. Second and more importantly, we need to remember such natural equivalences or all dierent compactiications, and even equivalences among natural equivalences. We immediately ind ourselves in the same scenario o an ininity hierarchy o homotopy equivalences again. For handling this kind o homotopy coherence, we use a technique called multisimplicial descent in [26, 4], which can be viewed as an -categorical generalization o [2, XVII 3.3]. This is not the end o the story since our goal is to prove all expected relations among six operations. To use the same idea o descent, we need to enhance not just operations, but also relations as well. To simpliy the discussion, let us temporarily ignore the two binary operations ( and Hom) and consider how to enhance the Base Change theorem which essentially involves -pullback and!-pushorward. We deine a simplicial set δ2,{2} N(Schqc.sep ) cart F,A in the ollowing way: The vertices are objects X o Sch qc.sep. The edges are Cartesian diagrams (0.3) X 01 q X 11 g with p locally o inite type, whose source is X 00 and target is X 11. Simplices o higher dimensions are deined in a similar way. Note that this is not an -category. Assuming that Λ is torsion, the enhancement o the Base Change theorem (or Sch qc.sep ) is a unctor such that it sends the edge X 00 p X 10 Λ Sch qc.sepeo! : δ2,{2} N(Schqc.sep ) cart F,A Prst L X 00 id X 00 (resp. X 11 X 00 ) p X 11 id X 11 to p! : D(X 00,ét, Λ) D(X 11,ét, Λ) (resp. : D(X 11,ét, Λ) D(X 00,ét, Λ)). The upshot is that the image o the edge (0.3) is a unctor D(X 11,ét, Λ) D(X 00,ét, Λ) which is naturally equivalent to both p! and q! g. In other words, this unctor has already encoded the Base Change theorem (or Sch qc.sep ) in a homotopy coherent way. This allows us to apply the descent method to construct the enhancement o the Base Change theorem or Artin stacks, which itsel includes the enhancement o the our operations,,! and! by restriction and adjunction. To deal Λ with the homotopy coherence involved in the construction o Sch qc.sepeo!, we use another technique called Cartesian gluing in [26, 5], which can be viewed as an -categorical variant o [39, 6, 7]. We hope the discussion so ar explains the meaning o enhancement to some degree. The actual enhancement (3.3) constructed in the article is more complicated than the ones mentioned previously, since we need to include also the inormation o binary operations, the projection ormula and extension o scalars Structure o the article. The main body o the article is divided into seven chapters. Chapter 1 is a collection o preliminaries on -categories, including the technique o partial adjoints ( 1.4) and the introduction o an -operad P which will be used to encode the projection ormula coherently. Chapter 2 is the starting point o the theory, where we construct enhanced operations or ringed topoi. The irst two chapters do not involve algebraic geometry. id X 11 id X 00

9 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 9 In Chapter 3, we construct the enhanced operation map or schemes in the category Sch qc.sep. The enhanced operation map encodes even more inormation than the enhancement o the Base Change theorem we mentioned in 0.3. We also prove several properties o the map that are crucial or later constructions. In Chapter 4, we develop an abstract program which we name DESCENT. The program allows us to extend the existing theory to a larger category. It will be run recursively rom schemes to algebraic spaces, then to Artin stacks, and eventually to higher Artin or Deligne Mumord stacks. The detailed running process is described in Chapter 5. There, we also prove certain compatibility between our theory and existing ones. In Chapter 6, we write down the resulting six operations or the most general situations and summarize their properties. We also develop a theory o constructible complexes, based on initeness results o Deligne [3, Th. initude] and Gabber [32]. Finally, we show that our theory is compatible with the work o Laszlo and Olsson [24]. For more detailed descriptions o the individual chapters, we reer to the beginning o these chapters. We assume that the reader has some knowledge o Lurie s theory o -categories, especially Chapters 1 through 5 o [28], and Chapters 1, 2 and 6 o [29]. In particular, we assume that the reader is amiliar with basic concepts o simplicial sets [28, A.2.7]. However, an eort has been made to provide precise reerences or notation, concepts, constructions, and results used in this article, (at least) at their irst appearance Convention and notation. All rings are assumed to be commutative with unity. For set-theoretical issues: We ix two (Grothendieck) universes U and V such that U belongs to V. The adjective small means U-small. In particular, Grothendieck abelian categories and presentable -categories are relative to U. A topos means a U-topos. All rings are assumed to be U-small. We denote by Ring the category o rings in U. By the usual abuse o language, we call Ring the category o U-small rings. All schemes are assumed to be U-small. We denote by Sch the category o schemes belonging to U and by Sch a the ull subcategory consisting o aine schemes belonging to U. We have an equivalence o categories Spec: (Ring) op Sch a. The big pp site on Sch a is not a U-site, so that we need to consider prestacks with values in V. More precisely, or W = U or V, let S W [28, ] is the -category o spaces in W. We deine the -category o prestacks to be Fun(N(Sch a ) op, S V ) [28, ]. However, a (higher) Artin stack is assumed to be contained in the essential image o the ull subcategory Fun(N(Sch a ) op, S U ). See 5.4 or more details. The (small) étale site o an algebraic scheme and the lisse-étale site o an Artin stack are U-sites. For every V-small set I, we denote by Set I the category o I-simplicial sets in V. See also variants in 1.3. We denote by Cat the (non V-small) -category o -categories in V [28, ] 3. (Multi)simplicial sets and -categories are usually tacitly assumed to be V-small. For lower categories: Unless otherwise speciied, a category will be understood as an ordinary category. A (2, 1)- category C is a (strict) 2-category in which all 2-cells are invertible, or, equivalently, a 3 In [28], Cat denotes the category o small -categories. Thus our Cat corresponds more closely to the notation Ĉat in [28, ], where the extension o universes is tacit.

10 10 YIFENG LIU AND WEIZHE ZHENG category enriched in the category o groupoids. We regard C as a simplicial category by taking N(Map C (X, Y )) or all objects X and Y o C. Let C, D be two categories. We denote by Fun(C, D) the category o unctors rom C to D, whose objects are unctors and morphisms are natural transormations. Let A be an additive category. We denote by Ch(A) the category o cochain complexes o A. Recall that a partially ordered set P is an (ordinary) category such that there is at most one arrow (usual denoted as ) between each pair o objects. For every element p P, we identiy the overcategory P /p (resp. undercategory P p/ ) with the ull partially ordered subset o P consisting o elements p (resp. p). In particular, or p, p P, P p//p is identiied with the ull partially ordered subset o P consisting o elements both p and p, which is empty unless p p. Let [n] be the ordered set {0,..., n} or n 0 and let [ 1] =. Let us recall the category o combinatorial simplices (resp. n, +, n + ). Its objects are the linearly ordered sets [i] or i 0 (resp. 0 i n, i 1, 1 i n) and its morphisms are given by (nonstrictly) order-preserving maps. In particular, or every n 0 and 0 k n, we have the ace map d n k : [n 1] [n] that is the unique injective map with k not in the image; and the degeneration map s n k : [n + 1] [n] that is the unique surjective map such that s n k (k + 1) = sn k (k). For higher categories: As we have mentioned, the word -category reers to the one deined in [28, ]. Throughout the article, an eort has been made to keep our notation consistent with those in [28] and [29]. For C a category, a (2, 1)-category, a simplicial category, or an -category, we denote by id C the identity unctor o C. We denote by N(C) the (simplicial) nerve o a (simplicial) category C [28, ]. We identiy Ar(C) (the set o arrows o C) with N(C) 1 (the set o edges o N(C)) i C is a category. Usually, we will not distinguish between N(C op ) and N(C) op or C a category, a (2, 1)-category or a simplicial category. We denote the homotopy category [28, , ] o an -category C by hc and we view it as an ordinary category. In other words, we ignore the H-enrichment o hc. Let C be an -category and let c : N( ) C (resp. c : N( op ) C) be a cosimplicial (resp. simplicial) object o C. Then the limit [28, ] lim(c ) (resp. colimit or geometric realization lim(c )), i it exists, is denoted by lim c n (resp. lim c n n op n). It is viewed as an object (up to equivalences parameterized by a contractible Kan complex) o C. Let C be an ( -)category, and let C C be a ull subcategory. We say a morphism : y x in C is representable in C i or every Cartesian diagram [28, 4.4.2] w z y x such that z is an object o C, w is equivalent to an object o C. We reer the reader to the beginning o [28, 2.3.3] or the terminology homotopic relative to A over S. We say and are homotopic over S (resp. homotopic relative to A) i A = (resp. S = ). Recall that Cat is the -category o V-small -categories. In [28, ], the subcategories Pr L, Pr R Cat are deined 4. We deine subcategories Pr L st, Pr R st Cat as ollows: 4 Under our convention, the objects o Pr L and Pr R are the U-presentable -categories in V.

11 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 11 The objects o both Prst L and Prst R are the U-presentable stable -categories in V [28, ], [29, ]. A unctor F : C D o presentable stable -categories is a morphism o Prst L i and only i F preserves small colimits, or, equivalently, F is a let adjoint unctor [28, , (1)]. A unctor G: C D o presentable stable -categories is a morphism o Prst R i and only i G is accessible and preserves small limits, or, equivalently, G is a right adjoint unctor [28, (2)]. We adopt the notation o [28, ]: or -categories C and D, we denote by Fun L (C, D) (resp. Fun R (C, D)) the ull subcategory o Fun(C, D) [28, ] spanned by let (resp. right) adjoint unctors. Small limits exist in Cat, Pr L, Pr R, Prst L and Prst. R Such limits are preserved by the natural inclusions Prst L Pr L Cat and Prst R Pr R Cat by [28, , ] and [29, ]. For the simplicial model category Set + o marked simplicial sets in V [28, ] with respect to the Cartesian model structure [28, , ], we ix a ibrant replacement simplicial unctor Fibr: Set + (Set+ ) via the Small Object Argument [28, A.1.2.5, A.1.2.6]. By construction, it commutes with inite products. I C is a V-small simplicial category [28, ], we let Fibr C : (Set + )C ((Set + ) ) C (Set + )C be the induced ibrant replacement simplicial unctor with respect to the projective model structure [28, A.3.3.1] Acknowledgments. We thank Oer Gabber, Luc Illusie, Aise Johan de Jong, Joël Riou, Shenghao Sun, and Xinwen Zhu or useul conversations. Part o this work was done during a visit o the irst author to the Morningside Center o Mathematics, Chinese Academy o Sciences, in Beijing. He thanks the Center or its hospitality. The irst author was partially supported by NSF grant DMS The second author was partially supported by China s Recruitment Program o Global Experts; National Natural Science Foundation o China Grant ; National Center or Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Key Laboratory o Mathematics, Chinese Academy o Sciences. 1. Preliminaries on -categories This chapter is a collection o preliminaries on -categories. In 1.1, we record some basic lemmas. In 1.2, we recall a key lemma and its variant established in [26], which will be subsequently used in this article. In 1.3, we recall the deinitions o multisimplicial sets and multi-marked simplicial sets rom [26]. In 1.4, we develop a method o taking partial adjoints, namely, taking adjoint unctors along given directions. This will be used to construct the initial enhanced operation map or schemes. In 1.5, we collect some general acts about symmetric monoidal -categories, including a closure property o closed symmetric monoidal presentable -categories. We also introduce an -operad P to coherently encode the projection ormula in the construction o enhanced operation maps in latter chapters Elementary lemmas. Let us start with the ollowing lemma, which shows up as [30, 2.4.6]. We include a proo or the convenience o the reader. Lemma Let C be a nonempty -category that admits product o two objects. Then the geometric realization C is contractible. Proo. Fix an object X o C and a unctor C C sending Y to X Y. The projections X Y X and X Y Y deine unctors h, h : 1 C C such that h {0} C = h {0} C; h {1} C is the constant unctor o value X; h {1} C = id C.

12 12 YIFENG LIU AND WEIZHE ZHENG Then h and h provide a homotopy between id C and the constant map o value X. The ollowing is a variant o the Adjoint Functor Theorem [28, ]. Lemma Let F : C D be a unctor between presentable -categories. Let hf : hc hd be the unctor o (unenriched) homotopy categories. (1) The unctor F has a right adjoint i and only i it preserves pushouts and hf has a right adjoint. (2) The unctor F has a let adjoint i and only i it is accessible and preserves pullbacks and hf has a let adjoint. Proo. The necessity ollows rom [28, ]. The suiciency in (1) ollows rom the act that small colimits can be constructed out o pushouts and small coproducts [28, ] and preservation o small coproducts can be tested on hf. The suiciency in (2) ollows rom dual statements. We will apply the above lemma in the ollowing orm. Lemma Let F : C D be a unctor between presentable stable -categories. Let hf : hc hd be the unctor o (unenriched) homotopy categories. Then (1) The unctor F admits a right adjoint i and only i hf is a triangulated unctor and admits a right adjoint. (2) The unctor F admits a let adjoint i F admits a right adjoint and hf admits a let adjoint. Proo. By [29, ], a unctor G between stable -categories is exact i and only i hg is triangulated. The lemma then ollows rom Lemma and [29, ] Constructing unctors via the category o simplices. In this section we recall the technique in [26, 2] or constructing unctors to -categories. It is crucial or many constructions in both articles. Let K be a simplicial set. Recall that the category o simplices o K [28, ], denoted by /K, is the strict iber product Set (Set ) /K. An object o /K is a pair (J, σ), where J and σ Hom Set ( J, K). A morphism (J, σ) (J, σ ) is a map d: J J such that σ = σ d. Notation For a marked simplicial set M, we deine an object Map[K, M] o the diagram category (Set ) ( /K) op by Map[K, M](J, σ) = Map (( J ), M), or every (J, σ) /K. For an -category C, we let Map[K, C] = Map[K, C ]. We have Γ(Map[K, M]) Map (K, M). Note that Map (K, C ) is the largest Kan complex [28, ] contained in Fun(K, C). The right adjoint o the diagonal unctor Set (Set ) ( /K) op is the global section unctor Γ: (Set ) ( /K) op Set, Γ(N) q = Hom (Set ) ( /K )op ( q K, N), where q K : ( /K) op Set is the constant unctor o value q. We have Γ(Map[K, C]) = Map (K, C ). I g : K K is a map, composition with the unctor /K /K induced by g deines a unctor g : (Set ) ( /K) op (Set ) ( /K )op. We have g Map[K, M] = Map[K, M]. Let Φ: N R be a morphism o (Set ) ( /K) op. We denote by Γ Φ (R) Γ(R) the simplicial subset, union o the images o Γ(Ψ): Γ(M) Γ(R) or all decompositions N M Ψ R

13 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 13 o Φ such that N(σ) M(σ) is anodyne [28, ] or all objects σ o /K. The map Γ(Φ): Γ(N) Γ(R) actorizes through Γ Φ (R). For every map g : K K, the canonical map Γ(R) Γ(g R) carries Γ Φ (R) into Γ g Φ(g R). Proposition ([26, 2.2, 2.4]). Let : Z T be a ibration in Set + with respect to the Cartesian model structure, let K be a simplicial set, let a: K T be a map, and let N (Set ) ( /K) op be such that N(σ) is weakly contractible or all σ /K. We let Map[K, ] a denote the iber o Map[K, ]: Map[K, Z] Map[K, T ] at the section 0 K Map[K, T ] corresponding to a. (1) For morphism Φ: N Map[K, ] a, Γ Φ (Map[K, ] a ) is a weakly contractible Kan complex. (2) For homotopic Φ, Φ : N Map[K, ] a, Γ Φ (Map[K, ] a ) and Γ Φ (Map[K, ] a ) lie in the same connected component o Γ(Map[K, ] a ). The condition in (2) means that there exists a morphism H : 1 K N Map[K, ] a in (Set ) ( /K) op such that H {0} K N = Φ, H {1} K N = Φ. Corollary ([26, 2.7]). Let K be a simplicial set, let C be an -category, and let i: A B be a monomorphism o simplicial sets. Let : Fun(B, C) Fun(A, C) be the morphism induced by i. Let N be an object o (Set ) ( /K) op such that N(σ) is weakly contractible or all σ, and let Φ: N Map[K, Fun(B, C)] be a morphism such that Map[K, ] Φ: N Map[K, Fun(A, C)] actorizes through 0 K to give a unctor a: K Fun(A, C). Then there exists b: K Fun(B, C) liting a, such that or every map g : K K and every global section ν o g N, b g and g Φ ν : K Fun(B, C) are homotopic over Fun(A, C) Multisimplicial sets. We recall the deinitions o multisimplicial sets and multi-marked simplicial sets rom [26, 3]. Let I, J be V-small sets. Deinition (Multisimplicial set). We deine the category o I-simplicial sets to be Set I := Fun(( I ) op, Set), where I := Fun(I, ). For an integer k 0, we deine the category o k- simplicial sets to be Set k := Set I, where I = {1,..., k}. We denote by ni i I the I-simplicial set represented by the object ([n i ]) i I o I. For an I- simplicial set S, we denote by S ni i I the value o S at the object ([n i ]) i I o I. An (n i ) i I -simplex o an I-simplicial set S is an element o S ni i I. By Yoneda s lemma, there is a canonical bijection o the set S ni i I and the set o maps rom ni i I to S. Let J I. Composition with the partial opposite unctor I I sending (..., S j,..., S j,... ) to (..., S j,..., S op j,... ) (taking op or S j when j J) deines a unctor op I J : Set I Set I. We deine n i i I J = op I J ni i I. Although n i i I J is isomorphic to ni i I, it will be useul in speciying the variance o many constructions. When I = {1,..., k}, we use the notation op k J and n 1,...,n k J. Notation Let : J I be a map o sets. Composition with deines a unctor : I J. Composition with induces a unctor ( ) : Set J Set I, which has a right adjoint ( ) : Set I Set J. We will now look at two special cases. Let : J I is an injective map o sets. Then has a right adjoint c : J I given by c (F ) i = F j i (j) = i and c (F ) i = [0] i i is not in the image o. We have c = id J. In this case, we write ( ) can be identiied with the unctor ɛ induced by composition with c. We have ɛ ( ) = id SetJ so that the adjunction map ( ) ɛ id SetI is a split monomorphism. I J = {1,..., k }, we write ɛ I (1) (k ) or ɛ. Let : I {1}. Then δ I := : I is the diagonal map. Composition with δ I induces a unctor δi = ( ) : Set I Set. For J I, we deine [n i] i I J := δi n i i I J = n i j J( n j ) op. i I J

14 14 YIFENG LIU AND WEIZHE ZHENG When J =, we simply write [n i] i I or [n i] i I = i I n i. We deine the multisimplicial nerve unctor to be the right adjoint δ I : Set Set I o δi. An (n i) i I -simplex o δ X I is given by a map [n i] i I X. For J I, we deine the twisted diagonal unctor δi,j = δ I opi J : Set I Set. When J =, op I J is the identity unctor so that δ I, = δ I. When I = {1,..., k}, we write k instead o I in the previous notation. In particular, we have δk : Set k Set so that (δk X) n = X n,...,n. Moreover, (ɛ k j K) n = K 0,...,n,...,0, where n is at the j-th position and all other indices are 0. We deine a biunctor : Set I Set J Set (I J) by the ormula S S = ( ι I ) S ( ι J ) S, where ι I : I I J, ι J : J I J are the inclusions. In particular, when I = {1,..., k}, J = {1,..., k }, we have : Set k Set k Set (k+k ), S S = ( ι ) S ( ι ) S, where ι: {1,..., k} {1,..., k + k } is the identity and ι : {1,..., k } {1,..., k + k } sends j to j + k. In other words, (S S ) n1,...,n k+k = S n1,...,n k S n k+1,...,n k+k. We have n 1 n k = n 1,...,n k. For a map : J I, an (n j ) j J -simplex o ( ) X is given by i I [n j] j 1 (i) X. Deinition (Multi-marked simplicial set). An I-marked simplicial set (resp. I-marked - category) is the data (X, E = {E i } i I ), where X is a simplicial set (resp. an -category) and, or all i I, E i is a set o edges o X that contains every degenerate edge. A morphism : (X, {E i } i I ) (X, {E i } i I) o I-marked simplicial sets is a map : X X having the property that (E i ) E i or all i I. We denote the category o I-marked simplicial sets by Set I+. It is the strict iber product o I copies o Set + above Set. Notation For an I-marked -category (C, E), we denote by C cart E δ C I the Cartesian I- simplicial nerve o (C, E) ([26, 3.7]). Roughly speaking, its (n i ) i I -simplices are unctors [n i] i I C such that the image o a morphism in the i-th direction is in E i or i I, and the image o every unit square is a Cartesian square. For a marked -category (C, E), we write C E or Map (( 0 ), (C, E)). C cart E 1.4. Partial adjoints. Deinition Consider diagrams o -categories C F D C G D U σ V U τ V F C G D C D which commute up to speciied equivalences α: F V U F and β : V G G U. We say that σ is a let adjoint to τ and τ is a right adjoint to σ, i F is a let adjoint o G, F is a let adjoint o G, and α is equivalent to the composite transormation F V F V G F β F G U F U F. The diagram τ has a let adjoint i and only i τ is let adjointable in the sense o [28, ] and [29, ]. I G and G are equivalences, then τ is let adjointable. We have analogous notions or ordinary categories. A square τ o -categories is let adjointable i and only i G and G admit let adjoints and the square hτ o homotopy categories is let adjointable. When visualizing a square 1 1 C, we adopt the convention that the irst actor o 1 1 is vertical and the second actor is horizontal.

15 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 15 Lemma Consider a diagram o right Quillen unctors C G D U V C G D o model categories, which commutes up to a natural equivalence β : V G G U and is endowed with Quillen equivalences (F, G) and (F, G ). Assume that U preserves weak equivalences and all objects o D are coibrant. Let α be the composite transormation F V F V G F β F G U F U F. Then or every ibrant-coibrant object Y o D, the morphism α(y ): (F V )(Y ) (U F )(Y ) is a weak equivalence. Proo. The square Rβ hc RG hd RU RV hc RG hd o homotopy categories is let adjointable. Let σ : LF RV RU LF be its let adjoint. For ibrant-coibrant Y, α(y ) computes σ(y ). We apply Lemma to the straightening unctor [28, 3.2.1]. Let p: S S be a morphism o simplicial sets and let π : C C be a unctor o simplicial categories itting into a diagram C[S ] C[p] C[S] φ φ C op C op π op which is commutative up to a simplicial natural equivalence. By [28, ], we have a diagram (Set + Un+φ )C (Set + ) /S π p (Set + )C Un + φ (Set + ) /S, which satisies the assumptions o Lemma i φ and φ are equivalences o simplicial categories. In this case, or every ibrant object : X S o (Set + ) /S, endowed with the Cartesian model structure, the morphism (St + φ p )X (π St + φ )X is a pointwise Cartesian equivalence. Similarly, i g : C D is a unctor o (V-small) categories, then [28, ] provides a diagram (Set + )D N+ (D) (Set + ) /N(D) g N(g) (Set + )C N+ (C) (Set + ) /N(C)

16 16 YIFENG LIU AND WEIZHE ZHENG satisying the assumptions o Lemma Thus or every ibrant object Y o (Set + ) /N(D), endowed with the cocartesian model structure, the morphism is a pointwise cocartesian equivalence. F + N(g) Y (C) g F + Y (D) Proposition Consider quadruples (I, J, R, ) where I is a set, J I, R is an I-simplicial set and : δ I R Cat is a unctor, satisying the ollowing conditions: (1) For every j J and every edge e o ɛ I j R, the unctor (e) has a let adjoint. (2) For all i J c = I\J, j J, τ (ɛ I i,j R) 11, the square (τ): 1 1 Cat is let adjointable. There exists a way to associate, to every such quadruple, a unctor J : δ I,J R Cat, satisying the ollowing conclusions: (1) J δ J c( ι ) R = δ J c( ι ) R, where ι: J c J is the inclusion. (2) For every j J and every edge e o ɛ I j R, the unctor J(e) is a let adjoint o (e). (3) For all i J c, j J, τ (ɛ I i,j R) 11, J (τ) is a let adjoint o (τ). (4) For two quadruples (I, J, R, ), (I, J, R, ) and maps µ: I I, u: ( µ ) R R such that J = µ 1 (J) and = δ I u, the unctor J is equivalent to J δ I,J u. When visualizing (1, 1)-simplices o ɛ I i,jr, we adopt the convention that direction i is vertical and direction j is horizontal. I J c is nonempty, then assumption (2) implies assumption (1), and conclusion (3) implies conclusion (2). Proo. Recall that we have ixed a ibrant replacement unctor Fibr: Set + Set+. Let σ (δ I,J R) n be an object o /δ I,J R, corresponding to n i i I J R, where n i = n. It induces a unctor (σ): N(D) [n i] i I J Cat, where D is the partially ordered set S T op, S = [n] J c, T = [n] J. This corresponds to a projectively ibrant simplicial unctor F : C[N(D)] Set +. Let φ D : C[N(D)] D be the canonical equivalence o simplicial categories and let F = (Fibr D St + φ op D Un + N(D) )F : D Set + op. We have weak equivalences F (St + N(D) op Un + N(D) op )F (φ D φ D! St + N(D) op Un + N(D) op )F (φ D St + φ op D Un + N(D) op )F φ D(F ). Thus, or every τ (ɛ I i,j N(D)) 1,1, F (τ) is equivalent to (τ). Let F be the composition S (Set + op Un+ )T φ T (Set + ) /N(T ), where the irst unctor is induced by F. For every s S, F (s): X N(T ) is ibrant or the Cartesian model structure. In other words, there exists a Cartesian ibration p: Y N(T ) and an isomorphism X Y. By assumption (1), or every morphism t t o T, the induced unctor Y t Y t has a let adjoint. By [28, ], p is also a cocartesian ibration. We consider the object (p, E) o (Set + ) /N(T ), where E is the set o p-cocartesian edges. By assumption (2), this construction is unctorial in s, giving rise to a unctor G : S (Set + ) /N(T ). The composition S G (Set + ) /N(T ) induces a projectively ibrant diagram F + (T ) (Set + )T G: S T Set +. Fibr T (Set + )T

17 ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM 17 We denote by G σ : [n] Set + the composition [n] S T Set +, where the irst unctor is the diagonal unctor. The construction o G σ is not unctorial in σ because the straightening unctors do not commute with pullbacks, even up to natural equivalences. Nevertheless, or every morphism d: σ σ in /δ I,J R, we have a canonical morphism G σ d G σ in (Set + )[n], which is a weak equivalence by Lemma The unctor sending d: σ σ to d G σ induces a map ( /δ I,J R) σ/ (Set + )[n] N(σ) := N(( /δ I,J R) σ/ ) Map (( n ), (Cat ) ), which we denote by Φ(σ). Since the category ( /δ I,J R) σ/ has an initial object, N(σ) is weakly contractible. This construction is unctorial in σ so that Φ: N Map[δ I,J R, Cat ] is a morphism o (Set ) ( /δ I,J R)op. Applying Lemma (1), we obtain a unctor J : δ I,J R Cat satisying (2), (3) up to homotopy. Under the assumptions o (4), δ I,J u: δ I,J R δ I,J R induces ϕ: N (δ I,J u) N. By construction, there exists a homotopy between Φ and ((δ I,J u) Φ) ϕ. By Lemma (2), this implies that J and J δi,j u are homotopic. By construction, there exists a homotopy between r Φ and the composite map r N 0 Q Map[Q, Cat ], where Q = δj c( ι ) R and r : Q δi,j R is the inclusion. By Lemma (2), this implies that J Q and Q are homotopic. Since the inclusion Q ( 1 ) Q ( {0} ) (δ I,JR) ( {0} ) (δ I,JR) ( 1 ) is marked anodyne, there exists J : δ I,J R Cat homotopic to J such that J Q = Q. Remark (1) There is an obvious dual version o Proposition or right adjoints. (2) Proposition holds without the assumption that R is V-small. To see this, it suices to apply the proposition to the composite map δi R Cat Cat W, where W V is a universe containing R and Cat W is the -category o -categories in W. (3) Applying Proposition (and Remark (2)) to the quadruple (2, {1}, δ 2 Cat, ), where : δ2 δ 2 Cat Cat is the counit map, we get a universal morphism δ2,{1} δ 2 Cat Cat. In act, or any quadruple (I, J, R, ), i we denote by µ: I {1, 2} the map given by µ 1 (1) = J, then : δ2 ( µ ) R Cat uniquely determines a map u: ( µ ) R δ 2 Cat by adjunction and J can be taken to be the composite map δ I,JR δ 2,{1} ( µ ) R δ 2,{1} u δ 2,{1} δ 2 Cat Cat. Q (4) For the quadruple (1, {1}, Pr R, ) where : Pr R Cat is the natural inclusion, the map J constructed in Proposition induces an equivalence Pr : (Pr R ) op Pr L. This gives another proo o the second assertion o [28, ]. By restriction, this equivalence induces an equivalence Prst : Pr L st (Pr R st) op. (5) For the quadruple (2, {1}, S op Fun LAd (S op, Cat ), ) where : S op Fun LAd (S op, Cat ) Cat