DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS

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1 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN arxiv: v1 [math.ct] 7 Sep 2014 Contents 1. Overview 2 2. Twisted arrow -cateories 4 3. The deinition o the dal 5 4. The doble dal 8 5. The dalit pairin 14 Appendi A. Cartesian and cocartesian ibrations 18 Reerences 19 Anone who has worked seriosl with qasicateories has had to spend some qalit time with cartesian and cocartesian ibrations. (For a crash corse in the basic deinitions and constrctions, see Appendi A; or an in-depth std, see [4, 2.4.2].) The prpose o (co)cartesian ibrations is to inesse the varios homotop coherence isses that natrall arise when one wishes to speak o nctors valed in the qasicateor Cat o qasicateories. A cartesian ibration p : X S is essentiall the same thin as a nctor X : S op Cat, and a cocartesian ibration q : Y T is essentiall the same thin as a nctor Y : T Cat. We sa that the (co)cartesian ibration p or q is classiied b X or Y (A.4). It has thereore been a continal sorce o irritation to man o s who work with qasicateories that, iven a cartesian ibration p : X S, there has been no eplicit wa to constrct a cocartesian ibration p : X S op that is classiied b the same nctor S op Cat. Man constrctions reqire as inpt eactl one o these two, and i one has become sidled with the wron one, then in the absence o an eplicit constrction, one is orced to etrde the desired ibration throh tortos epressions sch as the cocartesian ibration p classiied b the nctor b which the cartesian ibration p is classiied. We know o corse that sch a thin eists, bt we have little hope o sin it i we don t have access to a model that lets s precisel identi an n-simple o X in terms o p. In this technical note, we pt an end to this maddenin state o aairs: we proer a ver eplicit constrction o the dal cocartesian ibration p o a cartesian ibration p, and we show the are classiied b the same nctor to Cat. Amsinl, the constrction o the dal itsel is qite simple; however, provin that it works as advertised (and or that matter, even provin that p is a cocartesian ibration) is a nontrivial matter. The main technical tool we se is the technolo o eective Brnside -cateories and the nrlin constrction o the irst athor [1]. In the irst section, we will ive an inormal bt ver concrete description o the dal, and we will state the main theorem, Th Some sers o this technolo 1

2 2 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN will be happ to stop readin riht there. For those who press on, in 2, we briel recall the deinition o the twisted arrow cateor, which plas a siniicant role in the constrction. In 3, we ive a precise deinition o the dal o a cartesian ibration, and we prove that it is a cocartesian ibration. In particlar, we can sa eactl what the n-simplices o X are (3.8). In 4, we prove Pr. 4.1, which asserts that the doble dal is homotopic to the identit, and we se this to prove the main theorem, Th Finall, in 5, we constrct a relative version o the twisted arrow -cateor or a cocartesian ibration and its dal. provides another wa to witness the eqivalence between the nctor classiin p and the nctor classiin p. 1. Overview 1.1. Beore we describe the constrction, let s pase to note that simpl takin opposites will not address the isse o the da: i p : X S is a cartesian ibration, then it is tre that p op : X op S op is a cocartesian ibration, bt the nctor S op Cat that classiies p op is the composite o the nctor X : S op Cat that classiies p with the involtion op : Cat Cat that carries a qasicateor to its opposite. This discssion does, however, permit s to rephrase the problem in an enlihtenin wa: the morphism (p ) op : (X ) op S mst be another cartesian ibration that is classiied b the composite o the nctor that classiies p with the involtion op. The dal cocartesian ibration to (p ) op shold be eqivalent to p op, so that we have a dalit ormla ((p ) op ) p op. In particlar, it will be sensible to deine the dal q o a cocartesian ibration q : Y T as ((q op ) ) op, so that p p. We ths smmarize: The cartesian ibration and the cocartesian ibration are each classiied b p : X S p : X S op X : S op Cat ; (p ) op : (X ) op S p op : X op S op op X : S op Cat ; q : Y T op q : Y T Y : T Cat ; q op : Y op T op (q ) op : (Y ) op T op Y : T Cat We can describe or constrction ver eicientl i we ive orselves the lr o temporaril skippin some details. For an qasicateor S and an cartesian ibration p : X S, we will deine X as a qasicateor whose objects are those o X and whose morphisms are diarams (1.2.1)

3 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 3 o X in which is a p-cartesian ede, and p() is a deenerate ede o S. Composition o morphisms in X will be iven b ormin a pllback: w v z The n-simplices or n 3 are described completel in 3.8. One now has to eplain wh this deines a qasicateor, bt it does indeed (D. 3.5), and it admits a natral nctor to S op that carries an object to p() and a morphism as in (1.2.1) to the ede p() : p() p() = p() in S op. This is or nctor p : X S op, and we have ood news Proposition. I p : X S is a cartesian ibration, then p : X S op is a cocartesian ibration, and a morphism as in (1.2.1) is p -cocartesian jst in case is an eqivalence. This mch will actall ollow triviall rom the ndamental nrlin lemmas o the irst athor [1, Lm and Lm. 11.5], bt the dalit statement we re ater is more than jst the constrction o this cocartesian ibration. I one inspects the iber o p over a verte s S op, one inds that it is the qasicateor whose objects are objects o X s := p 1 (s), and whose morphisms are diarams (1.2.1) o X s in which is an eqivalence. This is visibl eqivalent to X s itsel. Frthemore, we will prove that this eqivalence is nctorial: 1.4. Proposition. The nctor S op Cat that classiies a cartesian ibration p is eqivalent to the nctor S op Cat that classiies its dal p. Eqivalentl, we have the ollowin Proposition. I X : S op Cat classiies p, then op X : S op Cat classiies (p ) op We will deine the dal o a cocartesian ibration q : Y T over a qasicateor T as sested above: q := ((q op ) ) op. In other words, Y will be the qasicateor whose objects are those o Y and whose morphisms are diarams o Y in which q() is a deenerate ede o T, and is q-cocartesian. Composition o morphisms in Y will be iven b ormin a pshot: w v z

4 4 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN The three propositions above will immediatel dalize. In smmar, the objects o X and (X ) op = (X op ) are simpl the objects o X, and the objects o Y and (Y ) op = (Y op ) are simpl the objects o Y. A morphism η : in each o these -cateories is then as ollows: In η is a diaram o in which and X X is p-cartesian, lies over an identit; (X ) op X lies over an identit, is p-cartesian; Y Y lies over an identit, is q-cocartesian; (Y ) op Y is q-cocartesian, lies over an identit. The propositions above are all sbsmed in the ollowin statement o or main theorem, which emplos some o the notation o A Theorem. The assinments p p and q q deine homotop inverse eqivalences o -cateories ( ) : Cat cart /S Cat cocart /Sop : ( ) o cartesian ibrations over S and cocartesian ibrations over S op. These eqivalences are compatible with the straihtenin/nstraihtenin eqivalences s in the sense that the diaram o eqivalences Cat cart /S s ( ) Fn(S op, Cat ) s Cat cocart /S op op Cat cocart /S op op Fn(S op, Cat ) s s ( ) op Catcart /S commtes p to a (canonical) homotop. 2. Twisted arrow -cateories 2.1. Deinition. I X is an -cateor (i.e., a qasicateor), then the twisted arrow -cateor Õ(X) is the simplicial set iven b the ormla Õ(X) n = Mor( n,op n, X) = X 2n+1.

5 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 5 The two inclsions n,op n,op n and n n,op n ive rise to a map o simplicial sets Õ(X) X op X The vertices o Õ(X) are edes o X; an ede o Õ(X) rom v to can be viewed as a commtative diaram (p to chosen homotop) v When X is the nerve o an ordinar cateor C, Õ(X) is isomorphic to the nerve o the twisted arrow cateor o C in the sense o [3]. When X is an -cateor, or terminolo is jstiied b the ollowin Proposition (Lrie, [6, Pr ]). I X is an -cateor, then the nctor Õ(X) X op X is a let ibration; in particlar, Õ(X) is an -cateor Eample. To illstrate, or an object p, the -cateor Õ( p ) is the nerve o the cateor (Here we write n or p n.) In [6, 4.2], Lrie oes a step rther and ives a characterization the let ibrations that (p to eqivalence) are o the orm Õ(X) Xop X. We ll discss (and se!) this reslt in more detail in The deinition o the dal We now ive a precise deinition o the dal o a cartesian ibration and, conversel, the dal o a cocartesian ibration. The deinitions themselves will not depend on previos work, bt the proos that the constrctions have the desired properties ollow triviall rom eneral acts abot the nrlin constrction o the irst athor [1, Lm and 11.5].

6 6 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN 3.1. Notation. Throhot this section, sppose S and T two -cateories, p : X S a cartesian ibration, and q : Y T a cocartesian ibration. As in Nt. A.5, denote b ιs S the sbcateor that contains all the objects and whose morphisms are eqivalences. Denote b ι S X X the sbcateor that contains all the objects, whose morphisms are p-cartesian edes. Similarl, denote b ιt T the sbcateor that contains all the objects, whose morphisms are eqivalences. Denote b ι T Y Y the sbcateor that contains all the objects and whose morphisms are q-cocartesian edes It is eas to see that (S, ιs, S) and (X, X S ιs, ι S X) are adeqate triples o -cateories in the sense o [1, D. 5.2]. Dall, (T op, ιt op, T op ) and (Y op, Y op T op ιt op, (ι T Y ) op ) are adeqate triples o -cateories. Frthermore, the cartesian ibrations p : X S and q : Y op T op are adeqate inner ibrations over (S, ιs, S) and (T op, ιt op, T op ) (respectivel) in the sense o [1, D. 10.3] Deinition. For an -cateor C and an two sbcateories C C and C C that each contain all the eqivalences, we deine A e (C, C, C ) as the simplicial set whose n-simplices are those nctors : Õ( n ) op sch that or an inteers 0 i k l j n, the sqare C ij kj il kl is a pllback in which the morphisms ij kj and il kl lie in C and the morphisms ij il and kj kl lie in C. When A e (C, C, C ) is an -cateor (which is the case, or eample, when (C, C, C ) is an adeqate triple o -cateories in the sense o [1, D. 5.2]), we call it the eective Brnside -cateor o (C, C, C ). Note that the projections Õ( n ) op n and Õ( n ) op ( n ) op indce inclsions and C A e (C, C, C ) and (C ) op A e (C, C, C ). Now it is eas to see that p and q indce morphisms o simplicial sets p : A e (X, X S ιs, ι S X) A e (S, ιs, S) q : A e (Y op, Y op T op ιt op, (ι T Y ) op ) op A e (T op, ιt op, T op ) op, respectivel. We wish to see that p is a cocartesian ibration and that q is a cartesian ibration, bt it s not even clear that the are inner ibrations.

7 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 7 Lckil, the ndamental nrlin lemmas [1, Lm and Lm. 11.5] o the irst athor address eactl this point. The basic observation is that the nrlin Υ(X/(S, ιs, S)) (respectivel, Υ(Y op /(T op, ιt op, T op )) ) o the adeqate inner ibration p (resp., q op ) [1, D. 11.3] is then the eective Brnside -cateor A e (X, X S ιs, ι S X) (resp., A e (Y op, Y op T op ιt op, (ι T Y ) op ) ), and the nctor Υ(p) (resp., the nctor Υ(q op ) op ) is the nctor p (resp., the nctor q) described above. The ndamental lemmas [1, Lm and Lm. 11.5] now immediatel impl the ollowin Proposition. The simplicial set A e (S, ιs, S) is an -cateor, and the nctor p is a cocartesian ibration. Frthermore, a morphism o A e (X, X S ιs, ι S X) o the orm is p-cocartesian jst in case is an eqivalence. Dall, the simplicial set A e (T, T, ιt ) is an -cateor, and the nctor q is a cartesian ibration. Frthermore, a morphism o A e (Y op, Y op T opιt op, (ι T Y ) op ) op o the orm is q-cocartesian jst in case is an eqivalence Deinition. The dal o p is the projection p : X := A e (X, X S ιs, ι S X) A e (S,ιS,S) S op S op, which is a cocartesian ibration. Dall, the dal o q is the projection q : Y := A e (Y op, Y op T op ιt op, (ι T Y ) op ) op A e (T op,ιt op,t op ) op T T, which is a cartesian ibration The ormation o the dal and the ormation o the opposite are b constrction dal operations with respect to each other; that is, one has b deinition 3.7. Observe that the inclsions (p op ) = (p ) op and (q op ) = (q ) op. S op A e (S, ιs, S) and T A e (T op, ιt op, T op ) op are each eqivalences. Conseqentl, the projections X A e (X, X S ιs, ι S X) and Y A e (Y op, Y op T op ιt op, (ι T Y ) op ) op are eqivalences as well.

8 8 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN 3.8. Note also that the description o X and Y iven in the introdction coincides with the one iven here: an n-simple o X, or instance, is a diaram in which an j-simple o the orm 0j 1j jj covers a totall deenerate simple o S (i.e., a j-simple in the imae o S 0 S j ), and all the morphisms ij il are p-cartesian. In particlar, note that the ibers (X ) s are eqivalent to the ibers X s, and the ibers (Y ) t are eqivalent to the ibers Y t. 4. The doble dal 4.1. Proposition. Sppose S and T two -cateories, p : X S a cartesian ibration, and q : Y T a cocartesian ibration. There are natral eqivalences p p and q q o cartesian ibrations X S and cocartesian ibrations Y T, respectivel. We postpone the proo (which is qite a chore) till the end o this section. In the meantime, let s reap the rewards o or deerred labor: in the notation o A.4, we obtain the ollowin Corollar. The ormation o the dal deines an eqivalence o -cateories ( ) : Cat cart /S Cat cocart /Sop : ( ) Proo. The onl thin let to observe that ( ) is a nctor rom the ordinar cateor o cartesian (respectivel, cocartesian) ibrations to the ordinar cateor o cocartesian (resp., cartesian) ibrations, and this nctor preserves weak eqivalences (since the are deined iberwise), whence it descends to a nctor o -cateories Cat cart /S Cat cocart /S op (resp., Catcocart /S op Catcart /S ). Let s now prove the main theorem, Th To do so, we mst enae with some size isses Notation. We recall the set-theoretic technicialities and notation sed in [4, , Rk ]. Let s choose two stronl inaccessible ncontable cardinals κ < λ. Denote b Cat (repsectivel, Top) -cateor o κ-small -cateories (resp., o κ-small Kan complees). Similarl, denote b Ĉat (resp., Top) the -cateor o λ-small -cateories (resp., o λ-small Kan complees).

9 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 9 Note that Cat and Top are essentiall λ-small and locall κ-small, whereas Ĉat and Top are onl locall λ-small. Proo o Th For an -cateor S, consider the composite eqivalence Fn(S op, Cat ) Cat cart /S Cat cocart /S op Fn(S op, Cat ), where the irst eqivalence is iven b nstraihtenin, the second is iven b the ormation o the dal, and the last is iven b straihtenin. It is eas to see that all o these eqivalences are natral in S [4, Pr (3)], so we obtain an atoeqivalence η o the nctor Fn(( ) op, Cat ) : Cat op Ĉat, and ths o the nctor Map(( ) op, Cat ) : Cat op Top. Now the let Kan etension o this nctor alon the inclsion Cat op Ĉat op is the nctor h : Ĉat op Top represented b Cat. The atoeqivalence η thereore also etends to an atoeqivalence η o h. The Yoneda lemma now implies that η is indced b an atoeqivalence o Ĉat itsel. B the Unicit Theorem o Toën [7], Lrie [5, Th ], and the irst athor and Chris Schommer-Pries [2], we dedce that η is canonicall eqivalent either to id or to op, and considerin the case S = 0 shows that it s the ormer option. This proves the commtativit o the trianle o eqivalences Cat cart /S ( ) Cat cocart /S op s s Fn(S op, Cat ), and the commtativit o the remainder o the diaram in Th. 1.7 ollows rom dalit. We ve delaed the inevitable lon enoh. Proo o Lm We prove the irst assertion; the second is dal. To bein, let s nwind the deinitions o the dals to describe X eplicitl. First, or an -cateor C, denote b Õ(2) (C) the simplicial set iven b the ormla Õ (2) (C) k = Mor(( k ) op k ( k ) op k, C) = C 4k+3. (This is a two-old edewise sbdivision o C. It can eqall well be described as a twisted 3-simple -cateor o C. ) Now the n simplices o X are those nctors : Õ(2) ( n ) op X sch that an r-simple o the orm (ab 1 c 1 d 1 ) (ab 2 c 2 d 2 ) (ab r c r d r ) covers a totall deenerate r-simple o S, and, or an inteers 0 a a b b c c d d n (which toether represent an ede abcd a b c d o Õ(2) (C)) we have (4.1.1) the morphism (a bcd) (abcd) is p-cartesian; (4.1.2) the morphism (ab cd) (abcd) is an eqivalence;

10 10 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN (4.1.3) the morphism (abcd ) (abcd) is an eqivalence. In other words, an object o X is an object o X, and a morphism o X is a diaram z v φ ψ in X sch that φ,, and ψ all cover deenerate edes o S, and (4.1.1-bis) the morphism is p-cartesian; (4.1.2-bis) the morphism ψ is an eqivalence; (4.1.3-bis) the morphism φ is an eqivalence. We will now constrct a cartesian ibration p : X S, a trivial ibration α : X X over S and a iberwise eqivalence β : X X over S. These eqivalences will all be the identit on objects. We will identi X with the sbcateor o X whose morphisms are as above with ψ and φ are deenerate; the inclsion will be the iberwise eqivalence β. The eqivalence α : X X will then in eect be obtained b composin and. To constrct p, we write, or an -cateor C, O(C) := Fn( 1, C). Note that the nctor s : O(C) C iven b evalation at 0 is a cartesian ibration (E. A.3). We now deine X as the simplicial set whose n-simplices are those commtative sqares O( n ) s X n S, sch that carries s-cartesian edes to p-cartesian edes. We deine p : X S to be the map that carries an n-simple as above to σ S n. We now constrct the desired eqivalences. The basic observation is that or an inteer k 0, we have nctors k k 1 k k k ( k ) op k ( k ) op : on the let we have the projection onto the irst actor; in the middle we have the nctor correspondin to the niqe natral transormation between the two inclsions k k k ; on the riht we have the obvios inclsion. These nctors indce, or an n 0, nctors These in trn indce a ziza o nctors n O( n ) Õ (2) ( n ) op. σ X α X β X over S, which are each the identit on objects. On morphisms, α carries a morphism iven b z to the composite z, and β carries a morphism iven p

11 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 11 b z to the morphism o X iven b the diaram z. We now have the ollowin, whose proo we postpone or a moment Lemma. The morphism X X constrcted above is a trivial Kan ibration. Ths p is the composite o two cartesian ibrations, and thereore a cartesian ibration. Now to complete the proo o Pr. 4.1, it sices to remark that X maniestl a iberwise eqivalence. X is Let s now set abot provin that X X is indeed a trivial ibration. For this, we will need to make sstematic se o the cartesian model cateories o marked simplicial sets as presented in [4, 3.1]. Proo o Lm We make O( n ) into a marked simplicial set O( n ) b markin those edes that map to deenerate edes nder the taret map t : O( n ) n. Frthermore, or an simplicial sbset K O( n ), let s write K or the marked simplicial set (K, E) in which E K 1 is the set o edes that are marked as edes o O( n ). Now write n O( n ) := O( {0,...,î,...,n} ) O( n ), i=0 which is a proper simplicial sbset o Fn( 1, n ) when n > 2. Observe that O( n ) has the propert that there is a bijection Map( O( n ), X) = Map( n, X ). Recastin the statement the Lemma in terms o litin properties, we see that it will ollow rom the claim that or an n 0 and an morphism O( n ) S o marked simplicial sets, the natral inclsion ι n : O( n ) ( n ) ( n ) O( n ) is a trivial coibration in the cartesian model strctre or marked simplicial sets over S, where the n in O( n ) is the bondar o the lon n-simple whose vertices are the identit edes in n. In act, we will prove slihtl more. Let I denote the smallest class o monomorphisms o marked simplicial sets that contains the marked anodne morphisms and satisies the two-ot-o-three aiom. We call these morphisms eectivel anodne maps o marked simplicial sets. Clearl, or an morphism Y S, an eectivel anodne morphism X Y is a trivial coibration in the cartesian model strctre on marked simplicial sets over S. It s clear that ι 1 is marked anodne, becase it s isomorphic to the inclsion ( {0,2} ) ( 2 ) ( {1,2} ) ( {1,2} ). Or claim or n > 1 will in trn ollow rom the ollowin sblemma.

12 12 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN 4.5. Lemma. The inclsion ( n ) O( n ) o the lon n-simple is eectivel anodne. Let s assme the veracit o this lemma or now, and let s complete the proo o Lm It s enoh to show that the inclsion ( n ) O( n ) ( n ) ( n ) is eectivel anodne, or then ι n will be a eectivel anodne b the two-ot-o three propert. We ll deplo indction and assme that Lemma 4.4 has been proven or each l < n. Now or each l, let so that B Lemma 4.4 or ι l, we have that sk l O( n ) := colim I n, I l O( I ) sk l 1 O( n ) (sk l 1 n ) ( n ) sk n 1 O( n ) = O( n ). skl O( n ) (sk l n ) ( n ) is a trivial coibration, becase it s a composition o pshots alon maps isomorphic to ι l. Since sk 0 O( n ) (sk0 n ) ( n ) = ( n ), iteratin this p to l = n 1 ives the reslt. Proo o Lm Write S or the set o nondeenerate (2n)-simplices = [00 = i 0 j 0 i 1 j 1 i 2n j 2n = nn] o O( n ). For S as above, deine ( A() = 1 2 n + 2n r=0 (j r i r ) Drawin O( n ) as a staircase-like diaram and as a path therein, it s easil checked that A() is the nmber o sqares enclosed between and the stairs iven b the simple 0 = [ (n 1)n nn]. We ll ill in the simplices o S b indction on A(). For k 0, let and S k = { S A() = k} and T k = { S A() k} We make the convention that O k ( n ) := T k O( n ). O 1 ( n ) := n. We mst now show that or all k with 0 k 1 2n(n 1), the inclsion O k 1 ( n ) O k ( n ) is marked anodne, and or each k it will be a matter o determinin O k 1 ( n ) or each S k and showin that the inclsion is eectivel anodne. O k 1 ( n ) ).

13 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 13 The case k = 0 is eceptional, so let s do it irst. The set S 0 has onl one element, the simple 0 = [ (n 1)n nn]. We claim that the inclsion o O 1 ( n ) the marked 2-simplices o the orm 0 [ii i(i + 1) (i + 1)(i + 1)] is eectivel anodne. Stickin all onto O 1 ( n ) is a marked anodne operation, so let s do that and call the reslt. Clearl the spine o 0 is inner anodne in, so the inclsion 0 is a trivial coibration. This proves the claim. Now we sppose k > 0, and sppose = [00 = i 0 j 0 i 1 j 1 i 2n j 2n = nn] S k. We call a verte v = (i r j r ) o a lipverte i it satisies the ollowin conditions: 0 < r < 2n; j r > i r ; i r 1 = i r (and hence j r 1 = j r 1); j r+1 = j r (and hence i r+1 = i r + 1). Observe that mst contain some lipvertices, and it is niqel determined b them. Note also that i is an arbitar simple o O( n ) containin all the lipvertices o, and i z S contains as a sbsimple, then A(z) A(), with eqalit i and onl i z =. We deine the lip o at v Φ(, v) as the modiication o in which the seqence i r (j r 1) i r j r (i r + 1)j r has been replaced b the seqence i r (j r 1) (i r + 1)(j r 1) (i r + 1)j r. Then Φ(, v) S k 1, so we have Φ(, v) O k 1 ( n ). We have thereore established that O k 1 ( n ) is the nion o the aces v = Φ(, v) as v ranes over lipvertices o. Eqivalentl, i {v 1,, v m } is the set o lipvertices o, then O k 1 ( n ) is the eneralized horn O k 1 ( n ) = Λ 2n {0,,2n}\{v 1,,v m} 2n = in the sense o [1, Nt. 12.6]. I m > 1, since lipvertices cannot be adjacent, it ollows that the set {0,, 2n} \ {v 1,, v m } satisies the hpothesis o [1, Lm ], and so the inclsion O k 1 ( n ) is inner anodne, whence O k 1 ( n ) is eectivel anodne. On the other hand, i m = 1, then O k 1 ( n ) is a ace: O k 1 ( n ) = v = {0,...,î+j,...,2n} 2n =, where v = (ij) is the niqe lipverte o. We mst show that the inclsion O k 1 ( n )

14 14 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN is eectivel anodne. We denote b the nion o v with the 2-simple [i(j 1) ij (i + 1)j]. The inclsion v is marked anodne; we claim that the inclsion is inner anodne. Indeed, somethin more eneral is tre: sppose s is an inner verte o m and F is a sbset o [m] which has s as an inner verte and is contios, meanin that i t 1, t 2 F and t 1 < < t 2 then F. Then the inclsion s m F m is inner anodne. We prove this b indction on m F. I F = m, then F = m and the claim is vacos. Otherwise, let F be a contios sbset o [n] containin F with F = F + 1. Then F ( F s m ) = F s F. Bt F s F is the eneralized horn Λ F F \{s}, and F \{s} satisies the hpothesis o [1, Lm ] as a sbset o F since s was alread an inner verte o F. Ths s n F s n F is inner anodne, and b the indction hpothesis, we are done. 5. The dalit pairin In this section we ive constrct a pairin that concretel ehibits the eqivalence between the nctor Y : T Cat that classiies a cocartesian ibration q : Y T and the opposite o the nctor that classiies the cocartesian ibration (q ) op. The wa we ll o abot this is the ollowin: we will constrct a let ibration M : Õ(Y/T ) (Y ) op T Y sch that or an object t T, the plled back ibration Õ(Y/T ) t ((Y ) op ) t Y t Y op t Y t is a perect pairin; i.e., it satisies the conditions o the ollowin reslt o Lrie Proposition ([6, Cor ]). Sppose σ : X A and τ : X B two nctors that toether deine a let ibration λ : X A B. Then λ is eqivalent to a ibration o the orm Õ(C) Cop C (and in particlar A B op ) jst in case the ollowin conditions are satisied. (5.1.1) For an object a A, there eists an initial object in the -cateor X a := σ 1 ({a}). (5.1.2) For an object b B, there eists an initial object in the -cateor X b := τ 1 ({b}). (5.1.3) An object X is initial in X σ() jst in case it is initial in X τ(). In or case, the nctor that classiies M will be the iberwise mappin space nctor Map Y/T : (Y ) op T Y Top. This nctor carries an object (, ) (Y ) op T Y to the space Map Y(t) (, ), where t = q() = q(). I φ : s t is a morphism o S, then a morphism (, ) : (, v) (, )

15 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 15 o (Y ) op T Y coverin φ is iven, in eect, b morphisms : Y(φ)() and : Y(φ)(v) o Y(s). The nctor Map Y/T will then carr (, ) to the morphism Map Y(s) (, v) Y(φ) Map Y(t) (Y(φ)(), Y(φ)(v)) Map Y(t) (, ) Beore we proceed headlon into the details o the constrction, let s irst ive an inormal bt ver concrete description o both Õ(Y/T ) and M. The objects o Õ(Y/T ) will be morphisms : v o Y sch that q() is an identit morphism in T. Now a morphism rom an arrow : v to an arrow : is a commtative diaram v φ w ξ in which φ is q-cocartesian, q(ψ) is an identit morphism. Composition is perormed b ormin sitable pshots on the sorce side and simple composition on the taret side. We will establish below that there is indeed an -cateor that admits this description. The nctor M will carr an object Õ(Y/T ) as above to the pair o objects (, v) (Y ) op Y, and it will carr a morphism as above to the pair o morphisms w φ ψ, v ξ (Y ) op Y. We call M the dalit pairin or q. We will prove below that it is let ibration, whence it ollows readil rom this description that the nctor that classiies it is indeed be the iberwise mappin space nctor deined above. ψ Map Y/T : (Y ) op T Y Top 5.3. Proposition. Both an -cateor Õ(Y/T ) and a let ibration M as described above eist. We postpone the precise constrction o Õ(Y/T ) and M till the end o this section (Constr. 5.5). Or concrete description sices to dedce the main reslt o this section Theorem. For an object t T, the let ibration Õ(Y/T ) t ((Y ) op ) t Y t plled back rom the dalit pairin M is a perect pairin; i.e., it satisies the conditions o Pr Proo. Sppose ((Y ) op ) t and Y t. Then it is eas to see that the identit map id is the initial object o the iber Õ(Y/T ) : or an morphism :

16 16 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN sch that q() is a deenerate ede, the essentiall niqe morphism id Õ(Y/T ) is iven b the diaram o Dall, the identit map id is the initial object o the iber Õ(Y/T ) : the essentiall niqe morphism id o Õ(Y/T ) is iven b the diaram The reslt now ollows immediatel. In liht o Pr. 5.1, we dedce an identiication that is nctorial in t, as desired. ((Y ) op ) t Y op t 5.5. Constrction. We now set abot ivin a precise constrction o the - cateor Õ(Y/T ) and the let ibration M described in 5.2. We se ver heavil the technolo o eective Brnside -cateories rom [1]. We bein b identiin two sbcateories o the arrow -cateor O(Y ), each o which contains all the objects. Sppose : v and : morphisms o Y. A morphism η : o O(Y ) iven b a sqare s(η) v t(η) lies in O(Y ) jst in case q(s(η)) is an eqivalence o T and t(η) is an eqivalence o Y ; the morphism η lies in O(Y ) jst in case s(η) is q-cocartesian. Now orm the eective Brnside -cateories Õ (Y ) := A e (O(Y ) op, (O(Y ) ) op, (O(Y ) ) op ), Ô(T ) := A e (O(T ) op, ιo(t ) op, O(T ) op ), (Y ) op := A e (Y op, Y op T op ιt op, (ι T Y ) op ), Ŷ := A e (Y op, ιy op, Y op ), T := A e (T op, ιt op, T op ).

17 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 17 The objects o Õ (Y ) are ths morphisms : v o Y, and a morphism rom an arrow : v to an arrow : is a commtative diaram φ v ξ in which: φ is q-cocartesian, q(ψ) is an eqivalence, and η is an eqivalence. The sorce and taret nctors O(Y ) op Y op alon with the cocartesian ibration q toether indce a diaram o nctors ψ Õ (Y ) Ô(T ) η (Y ) op Ŷ T T Observe that the omnibs theorem o the irst athor [1, Th. 12.2] implies that all o the nctors that appear in this qadrilateral are inner ibrations. Frthermore, since the ormation o the eective Brnside -cateor respects iber prodcts, one ma emplo [1, Th. 12.2] to show not onl that the natral map M : Õ (Y ) ( (Y ) op Ŷ ) Ô(T ) T T is an inner ibration, bt also that ever morphism o Õ (Y ) is M -cocartesian. It is clear that M admits the riht litin propert with respect to the inclsion {0} 1, one dedces that M is a let ibration. As we see, the -cateor Õ (Y ) is mch too lare, bt we now proceed to ct both it and the let ibration M down to size via pllbacks: (5.5.1) The irst pllback in eect reqires all eqivalences in the description o the morphisms o Õ (Y ) above to be identities. We pll back M alon the inclsion ( ) (Y ) op Ŷ ((Y ) op Y ) O(T ) T T Ô(T ) T T (which is o corse an eqivalence) to obtain a let ibration M : Õ (Y ) ((Y ) op Y ) O(T ). T T (5.5.2) Second, we pll back the composite Õ (Y ) M ((Y ) op Y ) O(T ) O(T ) T T alon the inclsion T desired let ibration M : Õ(Y/T ) O(T ) o the deenerate arrows to obtain the (Y ) op T Y It is now plain to see that Õ(Y/T ) is the -cateor described in 5.2, and M is the let ibration described there.

18 18 CLARK BARWICK, SAUL GLASMAN, AND DENIS NARDIN Appendi A. Cartesian and cocartesian ibrations A.1. Deinition. Sppose p: X S an inner ibration o simplicial sets. Recall [4, Rk ] that an ede : 1 X is p-cartesian jst in case, or each inteer n 2, an etension {n 1,n} X, Λ n n F and an solid arrow commtative diaram Λ n F n F X n S, p the dotted arrow F eists, renderin the diaram commtative. We sa that p is a cartesian ibration [4, D ] i, or an ede η : s t o S and or ever verte X 0 sch that p() = s, there eists a p-cartesian ede : sch that η = p(). Cocartesian edes and cocartesian ibrations are deined dall, so that an ede o X is p-cocartesian jst in case the correspondin ede o X op is p op -cartesian, and p is a cocartesian ibration jst in case p op is a cartesian ibration. A.2. Eample. A nctor p: D C between ordinar cateories is a Grothendieck ibration i and onl i the indced nctor N(p): ND NC on nerves is a cartesian ibration [4, Rk ]. A.3. Eample. For an -cateor C, write O(C) := Fn( 1, C). B [4, Cor ], evalation at 0 deines a cartesian ibration s: O(C) C, and evalation at 1 deines a cocartesian ibration t: O(C) C. One can ask whether the nctor s: O(C) C is also a cocartesian ibration. One ma observe [4, Lm ] that an ede 1 O(C) is s-cocartesian jst in case the correspondin diaram is a pshot sqare. (Λ 2 0) = 1 1 C A.4. Sppose S a simplicial set. Then the collection o cartesian ibrations to S with small ibers is natrall oranized into an -cateor Cat cart /S. To deine it, let Cat cart be the ollowin sbcateor o O(Cat ): an object X U o O(Cat ) lies in Cat cart i and onl i it is a cartesian ibration, and a morphism p q in O(Cat ) between cocartesian ibrations represented as a sqare p X U Y V q

19 DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 19 lies in in Cat cart i and onl i carries p-cartesian edes to q-cartesian edes. We now deine Cat cocart /S as the iber over S o the taret nctor t: Cat cart O(Cat ) Cat. Eqivalentl [4, Pr ], one ma describe Cat cart /S as the simplicial nerve o the (ibrant) simplicial cateor o marked simplicial sets [4, D ] over S that are ibrant or the cartesian model strctre i.e., o the orm X S or X S a cartesian ibration [4, D ]. The straihtenin/nstraihtenin Qillen eqivalence o [4, Th ] now ields an eqivalence o -cateories Cat cart /S Fn(Sop, Cat ). So we obtain a dictionar between cartesian ibrations p: X S with small ibers and nctors X: S op Cat. For an verte s S 0, the vale X(s) is eqivalent to the iber X s, and or an ede η : s t, the nctor X(t) X(s) assins to an object X t an object X s with the propert that there is a cocartesian ede that covers η. We sa that X classiies p [4, D ]. Dall, the collection o cocartesian ibrations to S with small ibers is natrall oranized into an -cateor Cat cocart /S, and the straihtenin/nstraihtenin Qillen eqivalence ields an eqivalence o -cateories Cat cocart /S Fn(S, Cat ). A.5. Notation. A cartesian (respectivel, cocartesian) ibration with the propert that each iber is a Kan comple or eqivalentl, with the propert that the nctor that classiies it actors throh the ll sbcateor Top Cat o Kan complees is called a riht (resp., let) ibration. These are more eicientl described as maps that satis the riht litin propert with respect to horn inclsions Λ n k n sch that 1 k n (resp., 0 k n 1) [4, Pr ]. For an cartesian (resp., cocartesian) ibration p: X S, one ma consider the smallest simplicial sbset ι S X X that contains the p-cartesian (resp., p- cocartesian) edes. The restriction ι S (p): ι S X S o p to ι S X is a riht (resp., let) ibration. The nctor S op Top (resp., S Top) that classiies ι S p is then the nctor iven b the composition ι X, where X is the nctor that classiies p, and ι is the nctor Cat Top that etracts the maimal Kan comple contained in an -cateor. Reerences 1. C. Barwick, Spectral Macke nctors and eqivariant alebraic K-theor (I), Preprint arxiv: , April C. Barwick and C. Schommer-Pries, On the nicit o the homotop theor o hiher cateories, Preprint arxiv: , December W. G. Dwer and D. M. Kan, Fnction complees or diarams o simplicial sets, Nederl. Akad. Wetensch. Inda. Math. 45 (1983), no. 2, MR (85e:55038) 4. J. Lrie, Hiher topos theor, Annals o Mathematics Stdies, vol. 170, Princeton Universit Press, Princeton, NJ, MR (2010j:18001) 5., (, 2)-cateories and the Goodwillie calcls I, Preprint rom the web pae o the athor, October , Derived alebraic eometr X. Formal modli problems, Preprint rom the web pae o the athor, September B. Toën, Vers ne aiomatisation de la théorie des catéories spérieres, K-Theor 34 (2005), no. 3, MR (2006m:55041)