aa r X i v : . [ m a t h . C T ] J un Enlargement of (fibered) derivators
Fritz H¨ormannMathematisches Institut, Albert-Ludwigs-Universit¨at FreiburgJune 29, 2017
Keywords: fibered multiderivators, monoidal derivators, derivators
Abstract
We show that the theory of derivators (or, more generally, of fibered multiderivators) on all smallcategories is equivalent to this theory on partially ordered sets , in the following sense: Every deriva-tor (more generally, every fibered multiderivator) defined on partially ordered sets has an enlarge-ment to all small categories that is unique up to equivalence of derivators. Furthermore, extendinga theorem of Cisinski, we show that every bifibration of multi-model categories (basically a collec-tion of model categories, and Quillen adjunctions in several variables between them) gives rise to aleft and right fibered multiderivator on all small categories.
Let M be a model category. Cisinski has shown in [2] that the pre-derivator associated with M ,defined on all small categories , is a left and right derivator. This does not use any additionalproperties of M such as being combinatorial, or left or right proper. In this article we showthat the analogous statement holds true also for fibered multiderivators (in particular for fiberedderivators and monoidal derivators). More precisely: Recall [3, Definition 4.1.3] that a bifibrationof multi-model categories is a bifibration D → S of multicategories together with the choice ofmodel category structures on the fibers such that the push-forward and pull-back functors alongany multimorphism in S form a Quillen adjunction in n variables (and an additional conditionconcerning “units”, i.e. 0-ary push-forwards). Theorem 1.1 (Theorem 4.6) . Let
D → S be a bifibration of multi-model categories, and let S be therepresented pre-multiderivator of S . For each small category I denote by W I the class of morphismsin Fun ( I, D) which point-wise are weak equivalences in some fiber D S (cf. [3, Definition 4.1.2]).The association I ↦ D ( I ) ∶ = Fun ( I, D )[ W − I ] defines a left and right fibered multiderivator over S with domain Cat . Furthermore the categories D ( I ) are locally small. Small categories intrinsically enlarged to a larger diagram category, whenever some nerve-like construction is available, relating the two diagram categories. This is more of theoreticalinterest because most derivators occurring in nature come from model categories. One concreteresult of the construction is the following:
Corollary 1.2.
Let D → S be a left (resp. right) fibered multiderivator with domain Invpos (resp. Dirpos ) such that S is defined on Cat and such that also (FDer0 right) (resp. (FDer0 left)) holds.Then there exists an enlargement of D to a left (resp. right) fibered multiderivator E → S withdomain Cat , such that its restriction to
Invpos (resp.
Dirpos ) is equivalent to D . Any other suchenlargement is equivalent to E . Note that this holds, in particular, for usual derivators (take for S the final pre-derivator) andclosed monoidal derivators (take for S the final pre-multiderivator). Proof.
In the left case, apply the machine of Theorem 4.1 twice using the functors N constructedin Proposition 2.5, firstly for the pair ( Invpos ⊂ Cat ○ ) , and secondly for the pair ( Inv ⊂ Cat ) .Similarly for the right case.If we start with a fibered multiderivator on all of Pos, however, we show that the two extensionsto Cat agree. Therefore we arrive at the following
Corollary 1.3.
Let D → S be a left and right fibered multiderivator with domain Pos such that S is defined on Cat . Then there exists a canonical enlargement of D to a left and right fiberedmultiderivator E → S with domain Cat , such that its restriction to
Pos is equivalent to D . Theenlargement is unique up to equivalence of fibered multiderivators over S . Actually, here Pos can be even replaced by the smallest diagram category containing both Invposand Dirpos.
Proof.
We consider this time the pairs ( Pos ⊂ Cat ○ ) and ( Cat ○ ⊂ Cat ) . For each of these pairs wedispose of functors N as in 2.3 for the left and the right case simultaneously by Proposition 2.5 (byenlarging Dia ′ we only weaken the axioms). Hence we may conclude by applying Proposition 4.10twice.For the reader mainly interested in plain (left and right) derivators, we state explicitly: Corollary 1.4.
Let D be a derivator with domain Pos . Then there exists a canonical enlargementof D to a derivator E with domain Cat , such that its restriction to
Pos is equivalent to D . Theenlargement is unique up to equivalence of derivators. Thanks to Falk Beckert for pointing out that a weaker statement in the direction of Corollary 1.4 hasbeen proven by Jan Willing [5] in a diploma thesis under the supervision of Jens Franke. There, onlystable derivators were considered under the name “verfeinerte triangulierte Diagrammkategorien”. Inverse posets Directed posets Relating different diagram categories via nerve-like construc-tions
Let Dia be a diagram category, cf. [3, Definition 1.1.1]. In contrast to Axiom (Dia3) of [loc. cit.], inthis article we require that Dia permits the construction of comma categories I × / J K for arbitraryfunctors α ∶ I → J and β ∶ K → J in Dia. We assume that the reader is, to some extent, familiarwith the definition of fibered multiderivator [3, Section 1.2–3]. The reader mainly interested inusual derivators or monoidal derivators can let S be the final pre-derivator (resp. the final pre-multiderivator) and then a “left (resp. right) fibered multiderivator over S ” is just a left (resp.right) derivator (resp. a monoidal left (resp. closed right) derivator).Recall the following [3, Definition 2.4.1]: Definition 2.1.
Let D → S be a (left or right) fibered derivator with domain Dia . Let
I, E ∈ Cat bediagrams with I ∈ Dia and let π ∶ I → E be a functor. We say that an object X ∈ D ( I ) is π - (co)Cartesian , if for any morphism µ ∶ i → j in I mapping to an identity under π , thecorresponding morphism D ( µ ) ∶ i ∗ X → j ∗ X is (co-)Cartesian.If E is the trivial category, we omit π from the notation, and talk about (absolutely) (co-)Cartesianobjects. Note: If S is trivial or if X lies over an object of the form π ∗ S for S ∈ S ( E ) the notions π -coCartesianand π -Cartesian coincide. Definition 2.2 (cf. [3, Definition 3.3.1]) . Let D → S and π ∶ I → E be as in the previous Definitionand let S ∈ S ( I ) . If the fully-faithful inclusion D ( I ) π − cart S ↪ D ( I ) S D ( I ) π − cocart S ↪ D ( I ) S has a left (resp. right) adjoint, we call that adjoint a left (resp. right) (co)Cartesian projector ,denoted ◻ π ! (resp. ◻ π ∗ ). In this article it is always clear from the context, whether Cartesian orcoCartesian objects are considered hence we will not use the notation ∎ ! , ∎ ∗ from [loc. cit.]. We want to extend (left or right) fibered multiderivators D → S from a diagram category Dia ′ to alarger diagram category Dia. Here S can be any pre-multiderivator (or even a 2-pre-multiderivator)satisfying (Der1) and (Der2). We assume that S is already defined on the larger diagram categoryDia and that D is a left (resp. right) fibered multiderivator over S such that also (FDer0 right) resp. (FDer0 left) hold true. Let Dia ′ ⊂ Dia be diagram categories. We suppose given a functor N ∶ Dia → Dia ′ in which, forgetting 2-morphisms, Dia and Dia ′ are considered to be usual 1-categories, togetherwith a natural transformation π ∶ N ⇒ idwith the following properties: at least when neglecting the multi-aspect I ∈ Dia, the comma category N ( I ) × / I N ( I ) (formed w.r.t. the functors π I ) is in Dia ′ as well.(N2) For all I, J ∈ Dia, we have N ( I ∐ J ) = N ( I ) ∐ N ( J ) . Furthermore N ( ∅ ) = ∅ .(N3) For all I ∈ Dia, π I is surjective on objects and morphisms, and has connected fibers.(N4 left) For any functor α ∶ I → J in Dia and for any object j ∈ J the diagram N ( I × / J j ) ⇙ / / (cid:15) (cid:15) N ( I ) α ○ π I (cid:15) (cid:15) j / / J is 2-Cartesian (i.e. identifies the top left category with the corresponding comma category).(N5 left) For any pre-derivator D satisfying (Der1) and (Der2) and for all I ∈ Dia with final object i the functors D ( N ( I )) π I − cart n ∗ / / D ( ⋅ ) p ∗ o o form an adjunction, with n ∗ left adjoint, where n is some object with π I ( n ) = i and p ∶ N ( I ) → ⋅ is the projection. Furthermore the counit of the adjunction is the natural isomorphism andthe unit is an isomorphism on (absolutely) Cartesian objects.Some immediate consequences of the axioms are listed in the following: Lemma 2.4 (left) .
1. Property (N5 left) is true w.r.t. any object n with π I ( n ) = i .2. Let D → S be a left fibered derivator satisfying also (FDer0 right) . Let J ∈ Dia ′ , let I ∈ Dia with final object i , and let S ∈ S ( I × J ) be an object. Let n ∈ N ( I ) with π I ( n ) = i . We denote π I,J ∶ = ( π I × id J ) ∶ N ( I ) × J → I × Ji J ∶ = ( i × id J ) ∶ J → I × Jp J ∶ = ( p × id J ) ∶ N ( I ) × J → Jn J ∶ = ( n × id J ) ∶ J → N ( I ) × Jf ∶ = S ( ν )( S ) ∶ S → p ∗ J i ∗ J S where ν ∶ id I × J ⇒ i J p J is the obvious natural transformation. The functors D ( N ( I ) × J ) π I,J − cart π ∗ I,J
S n ∗ J / / D ( J ) ( i J ) ∗ S ( π ∗ I,J f ) ● p ∗ J o o form an adjunction, with n ∗ J left adjoint. The counit is the natural isomorphism and the unitis an isomorphism on pr -Cartesian objects.3. The natural morphism n ∗ J → p J, ! ( π ∗ I,J f ) ● is an isomorphism on the subcategory D ( N ( I ) × J ) π I,J − cart π ∗ I,J S . at least when neglecting the multi-aspect . The functor ( π ∗ I,J f ) ● p ∗ J n ∗ J defines a left p J -Cartesian projector (i.e. a left adjoint to theinclusion) D ( N ( I ) × J ) π I,J − cart π ∗ I,J S → D ( N ( I ) × J ) p J − cart π ∗ I,J S . Proof.
1. The fiber over i in N ( I ) is connected (N3). Hence on the subcategory D ( N ( I )) π I − cart allfunctors n ∗ for n ∈ N ( I ) with π I ( n ) = i are isomorphic. Any of them can be thus taken as adjoint.2. The adjunction in question is the composition of the adjunctions D ( N ( I ) × J ) π I,J − cart π ∗ I,J S ( π ∗ I,J f ) ● / / D ( N ( I ) × J ) π I,J − cart p ∗ J i ∗ J S ( π ∗ I,J f ) ● o o n ∗ J / / D ( J ) i ∗ J S . π ∗ I,J o o For the second apply (N5 left) to the pre-derivator (fiber) D ( J,i ∗ J S ) ∶ K ↦ D ( K × J ) pr ∗ i ∗ J S .3. Essentially uniqueness of adjoints.4. Follows from Lemma 3.2 applied to the monad ( π ∗ I,J f J ) ● p ∗ J n ∗ J associated with the adjunctionof 2. The assumptions are true because this monad has obviously values in absolutely Cartesianobjects and by (N5 left) the unit is an isomorphism on absolutely Cartesian objects.There are corresponding dual axioms (with a corresponding dual version of the Lemma which weleave to the reader to state):(N4 right) For any functor α ∶ I → J in Dia and for any object j ∈ J the diagram N ( j × / J I ) ⇗ / / (cid:15) (cid:15) N ( I ) (cid:15) (cid:15) j / / J is 2-Cartesian (i.e. identifies the top left category with the corresponding comma category).(N5 right) For any pre-derivator D satisfying (Der1) and (Der2) with domain Dia ′ and for all I ∈ Diawith initial object i the functors D ( N ( I )) π I − cart n ∗ / / D ( ⋅ ) p ∗ o o form an adjunction, with n ∗ right adjoint, where n is some object with π I ( n ) = i and p ∶ N ( I ) → ⋅ is the projection. Furthermore the unit of the adjunction is the natural isomorphismand the counit is an isomorphism on (absolutely) Cartesian objects. Proposition 2.5.
A strict functor N as in 2.3 exists in the following cases and satisfies axioms(N1–3), (N4–5 left) (resp. (N4–5 right)): Dia ′ DiaInv CatInvpos Cat ○ Dir CatDirpos Cat ○ Here Cat ○ is the category of those small categories in which identities do not factor nontrivially.Observe that Cat ○ is self-dual, and that Dir ⊂ Cat ○ and Inv ⊂ Cat ○ .5 roof. The functors N are the following: For the pair ( Dir ⊂ Cat ) denote by N ○ ( I ) the semi-simplicial nerve of I . By applying the Grothendieck construction to the semi-simplicial set N ○ ( I ) we obtain a directed diagram which is an opfibration with discrete fibers over ( ∆ ○ ) op : N ( I ) ∶ = ∫ N ○ ( I ) → ( ∆ ○ ) op . It comes equipped with a natural functor π I ∶ N ( I ) → I mapping ( ∆ n , i → ⋅ ⋅ ⋅ → i n ) to i .For the pair ( Dirpos ⊂ Cat ○ ) denote by N ○ ( I ) ′ the subobject of the semi-simplicial nerve of I givenby simplices ∆ n → I in which no non-identity morphism is mapped to an identity. N and π aredefined similarly and it is clear that N ( I ) is a directed poset.For the pair ( Inv ⊂ Cat ) , by taking the opposite of the functor N constructed for the pair ( Dir ⊂ Cat ) , we get an inverse diagram with a fibration to ∆ ○ : N ( I ) ∶ = ( ∫ N ○ ( I )) op → ∆ ○ . It comes equipped with a natural functor π I ∶ N ( I ) → I mapping ( ∆ n , i → ⋅ ⋅ ⋅ → i n ) to i n .For the pair ( Invpos ⊂ Cat ○ ) we have the obvious fourth construction.We have to check the axioms in each case, but will concentrate on the pairs ( Dir ⊂ Cat ) (in thefollowing called case A) and ( Dirpos ⊂ Cat ○ ) (in the following called case B), the others being dual.(N1–3) and (N4 right) are obvious.(N5 right) Let I ∈ Dia be a diagram with initial object. We let n (in both cases) be the object ( ∆ , i ) of N ( I ) . The unit of the adjunction is the natural isomorphism u ∶ id ⇒ n ∗ p ∗ given by the equation p ○ n = id.Recall from [3, Lemma 2.3.3] (cf. also [2, Proposition 6.6]) the definition of the functor ξ ∶ N ( I ) → N ( I ) which in case A is defined by ( ∆ n , i → ⋯ → i n ) ↦ ( ∆ n + , i → i → ⋯ → i n ) , and in case B by ( ∆ n , i → ⋯ → i n ) ↦ ⎧⎪⎪⎨⎪⎪⎩( ∆ n + , i → i → ⋯ → i n ) i / = i, ( ∆ n , i → ⋯ → i n ) i = i. There are (in both cases) natural transformationsid N ( I ) ξ k s + n ○ p. (1)The counit of the adjuntion c ∶ p ∗ n ∗ ⇒ idis given as follows: Applying D to the sequence (1) we getid ξ ∗ k s + ( n ○ p ) ∗ where the morphism to the right is obviously an isomorphism on π I -Cartesian objects. We let c bethe composition going from right to left. 6e will now check the counit/unit equations.1. We have to show that the composition n ∗ un ∗ / / n ∗ p ∗ n ∗ n ∗ c / / n ∗ (2)is the identity. Inserting the definitions, we get that (2) is D applied to the following sequence offunctors and natural transformations: n ξn e o o e / / npn n where ξn is the inclusion of ( ∆ , id i ) in case A and is n in case B. The morphisms e , are the(opposite of the) two inclusions ∆ → ∆ in case A and the identity in case B. Hence in case B itis obvious that the composition (2) is the identity while in case A it follows from Lemma 2.6 (afterapplying D ).2. We have to show that the composition p ∗ p ∗ u / / p ∗ n ∗ p ∗ cn ∗ / / p ∗ (3)is the identity. Inserting the definitions, we get that (3) is D applied to the following sequence offunctors and natural transformations: p pξ pnp p which consists only of identities. Hence the composition (3) was the identity as well. Lemma 2.6 (right) . Let D be a pre-derivator with domain Dia ′ , let I be a diagram in Dia withinitial object i , and let E ∈ D ( ∫ N ○ ( I )) π I − cart . Then the two isomorphisms ( ∆ , i ) ∗ E o o D ( e ) o o D ( e ) ( ∆ , id i ) ∗ E in D ( ⋅ ) are equal.Proof. The underlying diagram of ι ∗ E , where ι ∶ ( ∆ ○ ) op = ∫ N ○ ( i ) → ∫ N ○ ( I ) is the inclusion, is afunctor ( ∆ ○ ) op → D ( ⋅ ) which maps all morphisms to isomorphisms. Since π (( ∆ ○ ) op ) =
1, necessarily all parallel mor-phisms are mapped to the same isomorphism. π I -Cartesian projectors Let Dia ′ ⊂ Dia be diagram categories and let N be a functor as in 2.3. We also use the notation ofLemma 2.4. Proposition 3.1 (left) . Let D → S be a left fibered derivator satisfying also (FDer0 right) withdomain Dia ′ . For all I ∈ Dia , J ∈ Dia ′ and S ∈ S ( I × J ) there exists a left π I,J -Cartesian projector(cf. Definition 2.2) ◻ π I,J ! ∶ D ( N ( I ) × J ) π ∗ I,J S → D ( N ( I ) × J ) π I,J − cart π ∗ I,J S . at least when neglecting the multi-aspect Proof.
Recall from [4, 6.8] that a left fibered derivator with domain Dia ′ gives rise to a pseudo-functor of 2-categories (the multi-aspect is not needed here):Ψ ∶ ( Dia ′ ) cor ( S ) → CAT ( I, S ) ↦ D ( I ) S . For each triple ( I, J, S ) as in the statement, we define the following monad T in ( Dia ′ ) cor ( S ) . Ithas the properties that Ψ ( T ) has values in D ( N ( I ) × J ) π I,J − cart π ∗ I,J S , and that the unit id ⇒ Ψ ( T ) is anisomorphism on D ( N ( I ) × J ) π I,J − cart π ∗ I,J S . By Lemma 3.2 it follows that Ψ ( T ) is a left π I,J -Cartesianprojector.We have the 1-morphism [ π ( S ) I,J ] in Dia cor ( S ) and its left adjoint [ π ( S ) I,J ] ′ , cf. [4, 6.1–3 and Lemma 6.7].This adjunction defines a monad T ∶ = [ π ( S ) I,J ] ○ [ π ( S ) I,J ] ′ on ( N ( I ) × J, π ∗ I,J S ) . Actually T lies in ( Dia ′ ) cor ( S ) because of axiom (N1). Let us explicitly write down (a correspondence isomorphic to) T as well as the unit: ( N ( I ) × / I N ( I )) × J ( pr , id J ) v v ❧❧❧❧❧❧❧❧❧❧❧❧❧ ( pr , id J ) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ N ( I ) × J N ( I ) × JN ( I ) × J ( ∆ , id J ) O O h h ❘❘❘❘❘❘❘❘❘❘❘❘❘ ❧❧❧❧❧❧❧❧❧❧❧❧❧ Here the topmost correspondence is equipped with the morphism f ∶ pr ∗ π ∗ I,J S ⇒ pr ∗ π ∗ I,J S inducedby the natural transformation associated with the comma category.Point-wise for ( n, j ) ∈ N ( I ) × J and E ∈ D ( N ( I ) × J ) pr ∗ J S we thus have for i ∶ = π I ( n ) : ( n, j ) ∗ ( Ψ ( T ) E ) = p N ( i × / I I ) , ! (( n, id N ( I ) , j ) ∗ f ) ● ( pr , j ) ∗ E . Obviously the right hand side depends only on ( i, j ) . Therefore the object Ψ ( T ) E is π I,J -Cartesian.Note that by (N4 left) we have i × / I N ( I ) = N ( i × / I I ) .The unit is given point-wise by the natural morphism ( n, n, j ) ∗ ( N ( pr ) , j ) ∗ E / / p N ( i × / I I ) , ! (( n, id N ( I ) , j ) ∗ f ) ● ( N ( pr ) , j ) ∗ E . If E is π I,J -Cartesian ( N ( pr ) , j ) ∗ E is π I,J -Cartesian as well, and this map is an isomorphism byLemma 2.4, 3.The following Lemma is well-known but due to lack of reference in this precise formulation weinclude its proof.
Lemma 3.2 (left) . Let ( C , T, u, µ ) with T ∶ C → C , u ∶ id ⇒ T , and µ ∶ T ⇒ T be a monad in CAT and let D ⊂ C be a full subcategory such that1. T takes values in D ,2. the unit u ∶ id ⇒ T is an isomorphism on objects of D . hen T , considered as functor C → D , is left adjoint to the inclusion
D ↪ C . There is a corresponding right version in which a comonad gives rise to a right adjoint to theinclusion.
Proof.
Consider T as functor C → D , which is possible by assumption 1., and denote ι the inclusion D ↪ C . We define the unit id → ιT of the adjunction to be the unit u of the monad. The counit T ι → id is its inverse which exists by assumption 2.To show that this defines indeed an adjunction, we have to show the equation uT = T u as naturaltransformations T ⇒ T (which is to say that the monad is an idempotent monad).By the definition of monad, we have the diagram T uT / / T u / / T µ / / T in which both compositions are the identity. Hence to show that uT = T u we have to show thatone of them is an isomorphism, for then µ is an isomorphism as well, and hence after canceling µ we have uT = T u . However uT is an isomorphism by the assumptions. Proposition 3.3 (left) . Let D → S be a left fibered derivator with domain Dia ′ satisfying also(FDer0 right) . For each I ∈ Dia , J ∈ Dia ′ , and S ∈ S ( I × J ) the functor ( N ( pr ) , pr ○ π I,J ) ∗ ∶ D ( N ( I ) × J ) π I,J − cart π ∗ I,J S → D ( N ( I × J )) π I × J − cart π ∗ I × J S is an equivalence of categories. Its inverse is given by ( N ( pr ) , pr ○ π I,J ) ! followed by the left π I,J -Cartesian projector of Proposition 3.1.Proof.
Set π ∶ = ( N ( pr ) , π J N ( pr )) and π ∶ = π I,J . Consider the composition: N ( I × J ) π I × J π / / N ( I ) × J π / / I × J With the following notation L ∶ = [ π ( π ∗ S ) ] ′ R ∶ = [ π ( π ∗ S ) ] L ∶ = [ π ( S ) ] ′ R ∶ = [ π ( S ) ] the two monads in ( Dia ′ ) cor ( S ) associated with π I × J and π I,J are respectively: T I,J ∶ = R ○ L ,T I × J ∶ = R ○ R ○ L ○ L . Consider the following diagram in which the objects are 1-morphisms in ( Dia ′ ) cor ( S ) and in whichthe 2-morphisms are given by the obvious units and counits: R ○ R ○ L ○ L ○ R / / R ○ R ○ L R R ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ / / R ○ L ○ R O O / / R O O at least when neglecting the multi-aspect ( Dia ′ ) cor ( S ) although L and R do only lie in Dia cor ( S ) .Hence after applying the functor Ψ ∶ ( Dia ′ ) cor ( S ) → CAT and evaluating everything at a π I × J -Cartesian object we obtain a diagram in which 1 is mapped to an isomorphism because T I × J = L ○ L ○ R ○ R is mapped to a left π I × J -Cartesian projector. Also 3 is mapped to an isomorphismbecause T I,J = R ○ L is mapped to a left π I,J -Cartesian projector. Hence 2 is mapped to anisomorphism. However the image of 2 is Ψ ( R ) = π ∗ applied to the unitΨ ( T I,J ) π , ! π ∗ ← idThat is hence an isomorphism.The morphism L ○ L ○ R ○ R → idis mapped to an isomorphism on π I × J -Cartesian objects, hence the counit π ∗ Ψ ( T I,J ) π , ! → idis an isomorphism on π I × J -Cartesian objects. Therefore π ∗ and Ψ ( T I,J ) π , ! constitute an equiva-lence as claimed. Theorem 4.1.
Let
Dia ′ ⊂ Dia be two diagram categories. Let D → S be a left (resp. right) fiberedmultiderivator with domain Dia ′ satisfying also (FDer0 right) (resp. (FDer0 left)) such that S isdefined on all of Dia . Let N ∶ Dia → Dia ′ be a functor as in 2.3 satisfying axioms (N1–3) and (N4–5left) (resp. (N4–5 right)). Then E ( I ) S ∶ = D ( N ( I )) π I − cart π ∗ I S defines a left (resp. right) fibered multiderivator satisfying also (FDer0 right) (resp. (FDer0 left))with domain Dia . The restriction of E to Dia ′ is canonically equivalent to D . Any other suchenlargement of D to Dia is equivalent to E .If Posf ⊂ Dia ′ and the fibers of D are in addition right (resp. left) derivators with domain Posf , soare the fibers of E .Proof. We begin by explaining the precise construction of E → S . The category E ( I ) as a bifibrationover S ( I ) is defined as the pull-back (cf. [4, 2.23]) E ( I ) / / (cid:15) (cid:15) D ( N ( I )) π I − cart (cid:15) (cid:15) S ( I ) π ∗ I / / S ( N ( I )) . Note that D ( N ( I )) π I − cart over S ( N ( I )) is not necessarily bifibered (the pull-back, resp. push-forward functors will not preserve the D ( N ( I )) π I − cart subcategories, whereas the pull-back E ( I ) isbifibered over S ( I ) by the following argument. CoCartesian morphisms exist because for morphisms at least when neglecting the multi-aspect neglecting the multi-aspect
10n the image of π ∗ I ∶ S ( I ) → S ( N ( I )) the push-forward preserves the condition of being π I -Cartesian.In the same way, 1-ary Cartesian morphisms exist because for morphisms in the image of π ∗ I ∶ S ( I ) → S ( N ( I )) the pull-back will preserve the condition of being π I -Cartesian. For n -ary morphsims, n ≥ n -ary pull-backs are not necessarily “computed point-wise”). However,let f ∈ Hom S ( I ) ( S , . . . , S n ; T ) be a multimorphism. An adjoint of the push-forward ( π ∗ I f ) ● ∶ D ( N ( I )) π I − cart π ∗ I S × ⋅ ⋅ ⋅ × D ( N ( I )) π I − cart π ∗ I S n → D ( N ( I )) π I − cart π ∗ I T w.r.t. the i -th slot always exists, and is given by the usual pull-back ( π ∗ I f ) ● ,i followed by the right π I -Cartesian projector of Proposition 3.1 (right). Since the right π I -Cartesian projector exists onlywhen D → S is right fibered we can only show (FDer0 right) neglecting the multi-aspect if D → S is not assumed to be right fibered as well. This shows that the pull-back E ( I ) → S ( I ) is bifibered(with the mentioned restriction).A functor α ∶ I → J induces the following commutative diagram D ( N ( J )) π J − cart α ∗ / / (cid:15) (cid:15) D ( N ( I )) π I − cart (cid:15) (cid:15) S ( N ( J )) α ∗ / / S ( N ( I )) Hence via pullback we get a diagram E ( J ) α ∗ / / (cid:15) (cid:15) E ( I ) (cid:15) (cid:15) S ( J ) α ∗ / / S ( I ) and the upper horizontal functor maps coCartesian morphism to coCartesian morphisms and Carte-sian 1-ary morphisms to Cartesian 1-ary morphisms. This shows (FDer0 left) and the first part of(FDer0 right). The remaining part of (FDer0 right) will be shown in the end.We now construct the 2-functoriality of E and concentrate on the left case, the other being dual.A natural transformation µ ∶ α ⇒ β where α, β ∶ I → J are functors can be encoded by a functor µ ∶ I × ∆ → J such that µ (restriction to I = I × e ) is α and µ (restriction to I = I × e ) is β . We use theequivalence ( N ( pr ) , pr ○ π N ( I )× ∆ ) ∗ ∶ D ( N ( I ) × ∆ ) π I, ∆1 − cart π ∗ I, ∆1 S ∼ Ð→ D ( N ( I × ∆ )) π I × ∆1 − cart π ∗ I × ∆1 S (cf. Proposition 3.3). From an object E ∈ E ( J ) over S ∈ S ( J ) we get an object ◻ π I, ∆1 ! ( N ( pr ) , pr ○ π N ( I )× ∆ ) ! N ( µ ) ∗ E which defines a morphism f ● α ∗ E → β ∗ E f is the composition N ( α ) ∗ π ∗ J S ∼ (cid:15) (cid:15) N ( β ) ∗ π ∗ J S ∼ (cid:15) (cid:15) π ∗ J α ∗ S ( π ∗ J ∗ S ( µ ))( S ) / / π ∗ J β ∗ S This defines the 2-functoriality.The axioms (Der1–2) for E are clear (use axiom (N2) for (Der1)).For the axioms (FDer3–4) we concentrate on the left case again, the other is dual.(FDer3 left) Let α ∶ I → J be a functor in Dia. By assumption relative left Kan extensions existfor D , i.e. the functor N ( α ) ∗ ∶ D ( N ( J )) π ∗ J S → D ( N ( I )) π ∗ I α ∗ S has a left adjoint N ( α ) ! . Since by Proposition 3.1 a left π J -Cartesian projector ◻ π J ! exist on D ( N ( J )) π ∗ J S , we obtain also a left adjoint to N ( α ) ∗ restricted to the respective subcategories,namely ◻ π J ! N ( α ) ! ∶ D ( N ( I )) π I − cart π ∗ I α ∗ S → D ( N ( J )) π J − cart π ∗ J S . (FDer4 left) Consider a diagram as in the axiom: I × / J j ι / / p (cid:15) (cid:15) ⇙ I α (cid:15) (cid:15) j (cid:31) (cid:127) / / J and the following induced diagram: N ( I × / J j ) = N ( I ) × / J j N ( ι ) / / (cid:15) (cid:15) N ( I ) N ( α ) (cid:15) (cid:15) N ( J ) × / J j ι / / p N ( J ×/ J j ) (cid:15) (cid:15) ⇙ µ N ( J ) (cid:15) (cid:15) j (cid:31) (cid:127) / / J By definition of the left π J -Cartesian projector we have that n ∗ ◻ π J ! ≅ p N ( J × / J j ) , ! ( S ( µ )) ● ι ∗ where n is any element of N ( J ) mapping to j . Therefore n ∗ ◻ π J ! N ( α ) ! ≅ p N ( I × / J j ) , ! ( N ( α ) ∗ ( S ( µ ))) ● N ( ι ) ∗ . Finally note that the composition D ( N ( I × / J j )) π N ( I ×/ J j ) − cart p ∗ j ∗ S ◻ π ! N ( p ) ! / / D ( N ( j )) π j − cart π ∗ j j ∗ S π j, ! / / D ( j ) j ∗ S
12s isomorphic to p N ( I × / J j ) , ! because left Cartesian projectors commute with relative left Kan exten-sions. By Lemma 2.4 the second functor is an equivalence, and we deduce the isomorphism N ( j ) ∗ ◻ π J ! N ( α ) ! ≅ ◻ π j ! N ( p ) ! ( N ( α ) ∗ ( S ( µ ))) ● N ( ι ) ∗ . A tedious check shows that this isomorphism can be identified with the base change morphism of(FDer4 left).(FDer5 right) In the right case of the Theorem (FDer0 left) has been established already. (FDer5right) is the adjoint statement of the second part of (FDer0 left) [3, Lemma 1.3.8].(FDer5 left) By Lemma 4.4, it suffices to prove (FDer5 left) for p ∶ I → ⋅ , which means that thepush-forward commutes with (homotopy) colimits in each variable. That is, we have to see thatthe natural morphism ◻ ! N ( p ) ! ( N ( p ) ∗ π ∗ f ) ● ( N ( p ) ∗ − , . . . , − , . . . , N ( p ) ∗ − ) → ( π ∗ f ) ● ( − , . . . , ◻ ! N ( p ) ! , . . . , − ) is an isomorphism. Now ◻ ! on D ( N ( ⋅ )) is given by π ! π ∗ for the projection π ∶ N ( ⋅ ) → ⋅ (cf.Lemma 2.4, 3–4.). Therefore we may rewrite the morphism as π ∗ π ! N ( p ) ! ( N ( p ) ∗ π ∗ f ) ● ( N ( p ) ∗ − , . . . , − , . . . , N ( p ) ∗ − ) → ( π ∗ f ) ● ( − , . . . , π ∗ π ! N ( p ) ! , . . . , − ) . Since all arguments, except the i -th one, are on N ( ⋅ ) and supposed to be π -Cartesian, they are inthe essential image of π ∗ as well ( π ! π ∗ is isomorphic to the identity on them, as just explained).Therefore it suffices to show (using FDer0 left) that π ∗ π ! N ( p ) ! ( N ( p ) ∗ π ∗ f ) ● ( N ( p ) ∗ π ∗ , . . . , − , . . . , N ( p ) ∗ π ∗ − ) → π ∗ ( f ● ( − , . . . , π ! N ( p ) ! , . . . , − )) is an isomorphism. This follows from (FDer5 left) for the original left fibered multiderivator D → S .The remaining part of (FDer0 right): In the left case of the Theorem (FDer5 left) has been estab-lished already. The remaining part of (FDer0 right) is just the adjoint statement of (FDer5 left) [3,Lemma 1.3.8], hence it is satisfied automatically. In the right case of the Theorem, by Lemma 4.3,it suffices to show (FDer0 right) for opfibrations of the form α ∶ i × / J I → I . Axiom (N4 right)implies that also N ( α ) ∶ N ( i × / J I ) → N ( I ) is an opfibration. By Lemma 4.5, N ( α ) ∗ commuteswith the right Cartesian projectors as well. Therefore the statement is clear for opfibrations of thisform.That E enlarges D and that any other enlargement F is equivalent to E is shown as follows. FromProposition 3.3 applied for I = ⋅ and for J ∈ Dia ′ we get an equivalence D ( N ( ⋅ ) × J ) π ⋅ ,J − cart π ∗⋅ ,J S ≅ D ( N ( J )) π J − cart π ∗ J S Def. = E ( J ) S . By (N5 left) applied to I = ⋅ , and the derivator I ↦ D ( I × J ) pr ∗ S we get D ( N ( ⋅ ) × J ) π ⋅ ,J − cart π ∗⋅ ,J S ≅ D ( J ) S . All equivalences are compatible with pull-backs α ∗ and push-forwards, resp. pull-backs along mor-phisms in S .With the same reasoning, setting Dia ′ = Dia, and D = F , we have for all J ∈ Dia F ( J ) S ≅ F ( N ( J )) π J − cart π ∗ J S . F is equivalent to D on Dia ′ we have F ( N ( J )) π J − cart π ∗ J S ≅ D ( N ( J )) π J − cart π ∗ J S Def. = E ( J ) S . Finally, if the fibers of D are right derivators with domain Posf (in the left case of the Theorem)then for all S ∈ S ( K ) , with K ∈ Dia, and for all functors α ∶ I → J in Posf the pull-back ( id × α ) ∗ ∶ D ( N ( K ) × J ) π ∗ K,J pr ∗ S → D ( N ( K ) × I ) π ∗ K,I pr ∗ S has a right adjoint ( id × α ) ∗ such that Kan’s formula holds true for it. It is easy to see that both ( id × α ) ∗ and ( id × α ) ∗ respect the subcategories of π K,I -Cartesian, resp. π K,J -Cartesian objects.Therefore the fibers of E are left derivators with domain Posf again, because by Proposition 3.3 wehave an equivalence D ( N ( K ) × J ) π K,J − cart π ∗ K,J pr ∗ S → D ( N ( K × J )) π K × J − cart π ∗ K × J pr ∗ S Def. = E ( K × J ) pr ∗ S (via the pull-back). Remark 4.2.
The additional statement shows that if D → S has stable, hence triangulated fibers(for this it is sufficient that the fibers are stable left and right derivators with domain Posf ) then also E → S has stable, hence triangulated fibers. This allows to establish, under additional conditions,that a left fibered multiderivator is automatically a right fibered multiderivator as well and vice versa(see [3, § In the proof of Theorem 4.1 we used the following Lemmas:
Lemma 4.3.
The axiom (FDer0 right) in the definition of a right fibered multiderivator can bereplaced by the following weaker axiom:(FDer0 right’) For each I in Dia the morphism p ∶ D → S specializes to a fibered (multi)category and anyfunctor of the form α ∶ i × / I I → I (note that this is an opfibration) in Dia induces a diagram D ( J ) α ∗ / / (cid:15) (cid:15) D ( I ) (cid:15) (cid:15) S ( J ) α ∗ / / S ( I ) of fibered (multi)categories, i.e. the top horizontal functor maps Cartesian morphisms w.r.t.the i -th slot to Cartesian morphisms w.r.t. the i -th slot.Proof. Let α now be an arbitrary opfibration. By axiom (Der2) it suffices to see that the naturalmorphism α ∗ f ● ( − , . . . , − ; − ) → ( α ∗ f ) ● ( α ∗ − , . . . , α ∗ − ; α ∗ − ) is an isomorphism point-wise.Using the homotopy Cartesian squares i × / I I ⇗ µi ι I / / p i (cid:15) (cid:15) Ii / / I j × / J J ⇗ µj ι J / / p j (cid:15) (cid:15) Jj / / J j = α ( i ) , we have that i ∗ ≅ p i, ∗ S ( µ i ) ● ι ∗ I and j ∗ ≅ p j, ∗ S ( µ j ) ● ι ∗ J . Using the assumption, i.e.(FDer0 right’), the second statement of (FDer0 right) holds for ι I , and ι J , respectively. Thus, weare left to show that p j, ∗ S ( µ j ) ● ( ι ∗ J f ) ● ( ι ∗ J − , . . . , ι ∗ J − ; ι ∗ J − ) → p i, ∗ S ( µ i ) ● ( ι ∗ I f ) ● ( ι ∗ I α ∗ − , . . . , ι ∗ I α ∗ − ; ι ∗ I α ∗ − ) is an isomorphism. We have p I = p J ○ ρ , where ρ ∶ i × / I I → j × / J J is the functor induced by α .Using (FDer5 right) we arrive at the morphism p j, ∗ S ( µ j ) ● ( ι ∗ J f ) ● ( ι ∗ J − , . . . , ι ∗ J − ; ι ∗ J − ) → p j, ∗ S ( µ j ) ● ( ι ∗ J f ) ● ( ι ∗ J − , . . . , ι ∗ J − ; ρ ∗ ρ ∗ ι ∗ J − ) induced by the unit id → ρ ∗ ρ ∗ . Since α is an opfibration, ρ has a left adjoint ρ ′ given as follows: Itmaps an object j → j ′ in j × / J J to some (chosen for each such morphism) corresponding coCartesianmorphism i → i ′ . Hence ρ ∗ = ( ρ ′ ) ∗ . Since the unitid = ρ ○ ρ ′ is an equality the statement follows. Lemma 4.4.
The axiom (FDer5 left) in the definition of a left fibered multiderivator can be replacedby the following weaker axiom.(FDer5 left’) For any diagram I ∈ Dia , and for any morphism f ∈ Hom ( S , . . . , S n ; T ) in S ( ⋅ ) for some n ≥ , the natural transformation of functors p ! ( p ∗ f ) ● ( p ∗ − , ⋯ , p ∗ − , − , p ∗ − , ⋯ , p ∗ − ) → f ● ( − , ⋯ , − , p ! − , − , ⋯ , − ) , where p ∶ I → ⋅ is the projection, is an isomorphism.Proof. Let α ∶ I → J be an arbitrary opfibration. We have to show that the natural morphism α ! ( α ∗ f ) ● ( α ∗ − , . . . , − , . . . , α ∗ − ) → f ● ( − , . . . , α ! − , . . . , − ) is an isomorphism. This can be proven point-wise by (Der2). Applying j ∗ for an object j ∈ J , wearrive at p I j , ! ( p ∗ I j j ∗ f ) ● ( p ∗ I j − , . . . , − , . . . , p ∗ I j − ) → ( j ∗ f ) ● ( − , . . . , p I j , ! − , . . . , − ) using (FDer0 left) and that α ! is computed fiber-wise for opfibrations. This is the statement of(FDer5 left’). Lemma 4.5.
For an opfibration of the form α ∶ i × / I I → I the pullback N ( α ) ∗ commutes with theright π -Cartesian projector, i.e. the natural (exchange) morphism N ( α ) ∗ ◻ π I ∗ → ◻ π i ×/ I I ∗ N ( α ) ∗ is an isomorphism. roof. Consider the following cube (where the objects in the rear face have been changed using(N4 right)) i × / I N ( I ) × / I N ( I ) pr i ×/ I I / / ⇗ µi × I I N ( α )× / I N ( α ) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ pr i ×/ I I (cid:15) (cid:15) i × / I N ( I ) (cid:15) (cid:15) N ( α ) y y sssssssss N ( I ) × / I N ( I ) ⇗ µI pr I / / pr I (cid:15) (cid:15) N ( I ) (cid:15) (cid:15) i × / I N ( I ) N ( α ) / / N ( α ) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ i × / I I α y y sssssssssss N ( I ) / / I We have by definition and (N4 right) ◻ π i ×/ I I ∗ = pr i × / I I , ∗ S ( µ i × / I I ) ● ( pr i × / I I ) ∗ , and ◻ π I ∗ = pr I , ∗ S ( µ I ) ● ( pr I ) ∗ . Note that, in the cube, the rear face is just a pull-back of the front face. Since N ( α ) is an opfibrationby (N4 right) we have therefore N ( α ) ∗ pr I , ∗ S ( µ I ) ● ( pr I ) ∗ ≅ pr i × / I I , ∗ ( N ( α ) × / I N ( α )) ∗ S ( µ I ) ● ( pr I ) ∗ ≅ pr i × / I I , ∗ S ( µ i × / I i ) ● ( N ( α ) × / I N ( α )) ∗ ( pr I ) ∗ ≅ pr i × / I I , ∗ S ( µ i × / I i ) ● ( pr i × / I I ) ∗ N ( α ) ∗ . Theorem 4.6.
Let
D → S be a bifibration of multi-model categories, and let S be the represented pre-multiderivator of S . For each small category I denote by W I the class of morphisms in Fun ( I, D ) which point-wise are weak equivalences in some fiber D S (cf. [3, Definition 4.1.2]). The association I ↦ D ( I ) ∶ = Fun ( I, D )[ W − I ] defines a left and right fibered multiderivator over S with domain Cat . Furthermore the categories D ( I ) are locally small.Proof. In view of Theorem 4.1 it suffices to establish that we have equivalences of pre-multiderivators(compatible with the morphism to S ) D ≅ E left D ≅ E right where E left (resp. E right ) is the left (resp. right) fibered multiderivator — the enlargement of D —constructed there w.r.t. the N given for the pair ( Inv ⊂ Cat ) (resp. ( Dir ⊂ Cat ) ).16e sketch the left case, the other can be proven similarly. The idea goes back to Deligne in [1,Expos´e XVII, § S , the localization of the fiber Fun ( I, D ) S is isomorphic to the fiber E ( I ) S (Proposi-tion 4.7 below) and that the push-forward and pull-back functors can be derived using left or rightreplacement functors, which exist by Lemma 4.8.Let I ∈ Cat be a small category. We fix push-forward functors f D● , and f E ● , for the bifibrationsFun ( I, D ) → S ( I ) , and E ( I ) → S ( I ) , respectively. Note that it is not yet established that D ( I ) = Fun ( I, D )[ W − I ] → S ( I ) is an opfibration.We have a natural functor Fun ( I, D )[ W − I ] → E ( I ) induced by π ∗ I and we will construct a functor going in the other direction E ( I ) → Fun ( I, D )[ W − ] . By Proposition 4.7 every object in E ( I ) lies in the essential image of π ∗ I , hence any morphism in E ( I ) is of the form: ξ ∶ π ∗ I E , . . . , π ∗ I E n → π ∗ I F lying over π ∗ I f for some f ∈ Hom S ( I ) ( S , . . . , S n ; T ) — or equivalently — ( π ∗ f ) E ● ( π ∗ E , . . . , π ∗ E n ) → π ∗ F in the fiber E ( I ) T . It can be represented by a morphism in Fun ( N ( I ) , D ) π ∗ T of the form π ∗ I ( f D● ( ̃ Q E , . . . , ̃ Q E n )) → π ∗ I F (in which all functors are underived functors). Since the underived π ∗ I is fully-faithful by (N3) ,this is the image under π ∗ I of a morphism ξ ′′ ∶ f D● ( ̃ Q E , . . . , ̃ Q E n ) → F . Proposition 4.7 shows that ξ ′′ is well-defined in Fun ( I, D ) π ∗ T [ W − ( I,T ) ] hence also well-defined inFun ( I, D )[ W − I ] because W ( I,T ) ⊂ W I .Equivalently ξ ′′ gives rise to a morphism ξ ′ ∶ ̃ Q E , . . . , ̃ Q E n → F over f , which composed with the formal inverses of the morphisms ̃ Q E i → E i , we define to be the image of ξ . A small check shows that this defines indeed a functor which isinverse to the one induced by π ∗ I .Let D → S be a bifibration of multi-model categories and let D → S be the morphism of pre-multiderivators defined as in Theorem 4.6 (cf. also [3, Definition 4.1.2]), however, with domain Inv.It is a left fibered multiderivator, satisfying also (FDer0 right), by [3, Theorem 4.1.5]. This holds true for α ∗ ∶ Fun ( J, D) →
Fun ( I, D) for any category D and any functor α ∶ I → J which is surjectiveon objects and morphisms, and with connected fibers. roposition 4.7 (left) . Let I ∈ Cat be a small category. Then π ∗ I induces an equivalence Fun ( I, D ) S [ W − ( I,S ) ] ≅ D ( N ( I )) π I − cart π ∗ I S where W − ( I,S ) is the class of morphisms in Fun ( I, D ) S which are point-wise in the corresponding W S i (weak equivalences in the model structure on the fiber D S i ). There is an obvious right variant of the Proposition which we leave to the reader to state. Note,however, that for given I , the left hand side category is the same in both cases! Note that D ( N ( I )) π ∗ I S = Fun ( N ( I ) , D ) π ∗ I S [ W − ( N ( I ) ,π ∗ I S ) ] by [3, Proposition 4.1.29]. Proof.
We have the (underived) adjunctionFun ( I, D ) S π ∗ I % % Fun ( N ( I ) , D ) π ∗ I Sπ ( S ) I, ! e e with π ( S ) I, ! left adjoint. Both sides are equipped with classes classes W ( I,S ) , and W ( N ( I ) ,π ∗ I S ) , re-spectively, of weak equivalences, and the right hand side is equipped even with the Reedy modelcategory structure defined in [3, 4.1.18]. For the functors the following holds true:1. π ∗ I is exact (i.e. respects the classes W ( I,S ) and W ( N ( I ) ,π ∗ I S ) ).2. π ∗ I , when restricted to the localizations, has still a left adjoint defined by π I, ! Q , where Q isthe cofibrant resolution. Proof: It suffices to show that π I, ! Q defines a absolute left derivedfunctor. For this it suffices to see that π I, ! maps weak equivalences between cofibrant objectsto weak equivalences. This can be checked point-wise. Consider the 2-Cartesian diagram N ( I ) × / I i ι / / p (cid:15) (cid:15) ⇙ µ N ( I ) π I (cid:15) (cid:15) i / / I (4)We have the following isomorphism between underived functors i ∗ π I, ! ≅ p ! S ( µ ) ● ι ∗ . Now, ι ∗ preserves cofibrant objects (w.r.t. the model structure considered in [3, 4.1.18]) by [3,Lemma 4.1.27] and we know that S ( µ ) ● and p ! both map weak equivalences between cofibrantobjects to weak equivalences (both are left Quillen). Therefore π I, ! maps weak equivalencesbetween cofibrant objects to weak equivalences as well.We have to show that the unit (between the derived functors)id → π ∗ I π I, ! is an isomorphism on π I -Cartesian objects. This can be shown after applying n ∗ for any n ∈ N ( I ) .Set i ∶ = π I ( n ) . We get: n ∗ → i ∗ π I, ! ( n ′ ) ∗ ι ∗ → p ! S ( µ ) ● ι ∗ where n ′ = ( n, id i ) ∈ N ( I ) × / I i . However this is an isomorphism by Lemma 2.4, 3.We have to show that the counit (between the derived functors) π I, ! π ∗ I → idis an isomorphism. Again, it suffices to see this point-wise. After applying i ∗ and using the abovewe arrive at the morphism p ! S ( µ ) ● ι ∗ π ∗ I → i ∗ for p and ι as in 2.On π I -Cartesian objects, p N ( I )× / I i, ! is equal to the evaluation at any n mapping to i by Lemma 2.4,3.Recall that for a functor F ∶ C × ⋯ × C n → D , and for given classes of “weak equivalences” W C , , . . . , W C ,n , W D we call a left replacement functor adapted to F a collection of endofunctors Q i ∶ C i → C i with natural transformations Q i ⇒ id C i consisting point-wise of weak equivalences suchthat F ○ ( Q , . . . , Q n ) maps weak equivalences to weak equivalences. It follows that F ○ ( Q , . . . , Q n ) is an absolute left derived functor of F . Similarly for the right case. Lemma 4.8.
Let ( N, π ) be the functor and natural transformation constructed in Proposition 2.5for the pair ( Inv ⊂ Cat ) (a left case) and ( ̃ N , ̃ π ) the ones for the pair ( Dir ⊂ Cat ) (a rightcase). Let I ∈ Cat be a small category and let f ∈ Hom S ( I ) ( S , . . . , S n ; T ) be a multimorphism.On ∏ i Fun ( I, D ) S i the functor ̃ Q ∶ = π ( S i ) I, ! Qπ ∗ I , where Q is the cofibrant replacement in the Reedymodel category Fun ( N ( I ) , D ) π ∗ I S i [3, 4.1.18], is a left replacement functor adapted to f ● by virtueof the composition π ( S i ) I, ! Qπ ∗ I → π ( S i ) I, ! π ∗ I → id . In particular f ● has a total left derived functor. Similarly the functor ̃ R ∶ = ̃ π ( T ) I, ∗ R ̃ π ∗ I , where R is thefibrant replacement in the Reedy model category Fun ( ̃ N ( I ) , D ) ̃ π ∗ I T (the opposite of [3, 4.1.18]), is aright replacement functor adapted to f ● ,i (in the covariant argument) by virtue of the composition ̃ π ( T ) I, ∗ R ̃ π ∗ I ← ̃ π ( T ) I, ∗ ̃ π ∗ I ← id . More precisely, the functor f ● ,i ( ̃ Q op , ̂ i . . ., ̃ Q op ; ̃ R ) maps weak equivalences to weak equivalences. In particular f ● ,i as a functor Fun ( I, D ) op S × ̂ i ⋯ × Fun ( I, D ) op S n × Fun ( I, D ) T → Fun ( I, D ) S i has a total right derived functor. The so constructed derived functors form an adjunction in n variables again.Proof. As usual, we omit the bases from the relative Kan extension functors, they are clear fromthe context. Let i ∈ I be an object. Note that i ∗ π I, ! Qπ ∗ I ≅ p ∗ S ( µ ) ● ι ∗ Qπ ∗ I p ∗ , S ( µ ) ● , and ι ∗ preservecofibrations and weak equivalences between cofibrations. Therefore π I, ! Qπ ∗ I has image in point-wisecofibrant objects. f ● maps point-wise weak equivalences between point-wise cofibrant objects topoint-wise weak equivalences.We have f ● ,j ( π I, ! Qπ ∗ I , . . . , π I, ! Qπ ∗ I ; ̃ π I, ∗ R ̃ π ∗ I ) ≅ ̃ π I, ∗ (̃ π ∗ I f ) ● ,j (̃ π ∗ I π I, ! Qπ I ∗ , . . . , ̃ π ∗ I π I, ! Qπ ∗ I ; R ̃ π ∗ I ) and (̃ π ∗ I f ) ● ,j (̃ π ∗ I π I, ! Qπ ∗ I , . . . , ̃ π ∗ I π I, ! Qπ ∗ I ; R ̃ π ∗ I ) maps weak equivalences to weak equivalences andmaps to fibrant objects in the Reedy model category structure (opposite to [3, 4.1.18]) because (̃ π ∗ I f ) ● ,j is part of a Quillen adjunction in n variables and cofibrations are the point-wise ones in thatmodel-category structure. Therefore f ● ,j ( π I, ! Qπ ∗ I , . . . , π I, ! Qπ ∗ I ; ̃ π I, ∗ R ̃ π ∗ I ) maps weak equivalences toweak equivalences. Remark 4.9.
In the non-fibered case, Cisinski [2, Th´eor`eme 6.17] shows that for a right propermodel category D where the cofibrations are the monomorphisms, a similar construction like inthe Lemma may even be used to construct a model category structure on Fun ( I, D ) itself in whichthe weak equivalences are the point-wise ones. Probably a similar statement is true in the fiberedsituation, but we have not checked this. Proposition 4.10.
Let
Dia ′ ⊂ Dia be two diagram categories and N , and ̃ N , be two functors asin 2.3 satisfying (N1–3) and (N4–5 left), (resp. (N4–5 right)). And suppose, in addition, that foreach I ∈ Dia the diagram N ( I ) × / I ̃ N ( I ) is in Dia ′ as well. Let D → S be a left and right fiberedmultiderivator such that S extends to all of Dia . Then we have an equivalence of categories D ( N ( I )) π I − cart π ∗ I S ≅ D ( ̃ N ( I )) ̃ π I − cart ̃ π ∗ I S compatible with pull-back along functors α ∶ I → J and, for all morphisms f in S ( I ) , intertwiningpush-forward along π ∗ I f with that along ̃ π ∗ I f .Proof. Consider the adjunction D ( N ( I )) π ∗ I S pr , ! S ( µ ) ● pr ∗ ( ( D ( ̃ N ( I )) ̃ π ∗ I S pr , ∗ S ( µ ) ● pr ∗ h h (5)induced by the following 2-commutative diagram: N ( I ) × / I ̃ N ( I ) ⇗ µ pr / / pr (cid:15) (cid:15) ̃ N ( I ) ̃ π I (cid:15) (cid:15) N ( I ) π I / / I It suffices to show that the unit (resp. the counit) of the adjunction are isomorphisms on π I -Cartesian (resp. ̃ π I -Cartesian) objects. We concentrate on the counit (the unit case is analogous)and show that c ∶ pr , ! S ( µ ) ● pr ∗ pr , ∗ S ( µ ) ● pr ∗ → id20s an isomorphism on ̃ π I -Cartesian objects. It suffices to see this after pull-back to any ̃ n . Hencewe have to show that ̃ n ∗ c ∶ ̃ n ∗ pr , ! S ( µ ) ● pr ∗ pr , ∗ S ( µ ) ● pr ∗ → ̃ n ∗ is an isomorphism. Consider also an object n ∈ N ( I ) mapping to the same i ∈ I as ̃ n . They giverise to an element κ = ( n, ̃ n, id i ) ∈ N ( I ) × / I ̃ N ( I ) and we may rewrite the morphism as κ ∗ pr ∗ pr , ! S ( µ ) ● pr ∗ pr , ∗ S ( µ ) ● pr ∗ → κ ∗ pr ∗ . Consider the following commutative diagram of functors and natural transformations (all given byunits and counits of the obvious adjunctions): κ ∗ pr ∗ κ ∗ pr ∗ pr , ! S ( µ ) ● pr ∗ pr , ∗ S ( µ ) ● pr ∗ / / κ ∗ pr ∗ c ❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝ κ ∗ pr ∗ pr , ! S ( µ ) ● S ( µ ) ● pr ∗ / / κ ∗ pr ∗ pr , ! pr ∗ O O κ ∗ S ( µ ) ● pr ∗ pr , ∗ S ( µ ) ● pr ∗ O O / / κ ∗ S ( µ ) ● S ( µ ) ● pr ∗ O O / / κ ∗ pr ∗ g g O O It suffices to verify that 1 and 2 are both isomorphisms on ̃ π I -Cartesian objects.1 : It is clear that both functors in the adjunction (5) have image in π I -Cartesian (resp. ̃ π I -Cartesian) objects. Hence is suffices to see that κ ∗ S ( µ ) ● pr ∗ → κ ∗ pr ∗ pr , ! S ( µ ) ● pr ∗ is an isomorphism on ̃ π I -Cartesian objects. This is the same as the natural morphism n ∗ → p ! S ( µ ) ● ι ∗ induced by the 2-commutative diagram N ( I × / I i ) ⇗ µ p / / ι (cid:15) (cid:15) i (cid:15) (cid:15) N ( I ) π I / / I and this is an isomorphism by Lemma 2.4, 3 (left case).2 : We may rewrite 2 as: n ∗ pr , ∗ S ( µ ) ● pr ∗ → ̃ n ∗ and have to show that it is an isomorphism on ̃ π I -Cartesian objects. This is the same as the naturalmorphism p ∗ S ( µ ) ● ι ∗ → ̃ n ∗ for the 2-commutative diagram ̃ N ( i × / I I ) ⇗ µ ι / / p (cid:15) (cid:15) ̃ N ( I ) ̃ π I (cid:15) (cid:15) i / / I and this is an isomorphism by Lemma 2.4, 3 (right case).21 eferences [1] Th´eorie des topos et cohomologie ´etale des sch´emas. Tome 3 . Lecture Notes in Mathematics,Vol. 305. Springer-Verlag, Berlin-New York, 1973. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1963–1964 (SGA 4), Dirig´e par M. Artin, A. Grothendieck et J. L. Verdier. Avec lacollaboration de P. Deligne et B. Saint-Donat.[2] D. C. Cisinski. Images directes cohomologiques dans les cat´egories de mod`eles.