Filtered Derived Categories and Their Applications

Filtered derived categories and their applications

D. Kaledin∗

Contents

1 f-categories. 1 1.1 Definitions and examples...... 1 1.2 Reflecting the filtration...... 3 1.3 The comparison functor...... 5 1.4 Further adjunctions...... 7 1.5 Morphisms...... 9

2 End(N)-action. 12 2.1 Prototype: End(N)-action on functor categories...... 12 2.2 N-categories...... 15 2.3 Uniqueness...... 16 2.4 Full embeddings...... 19 2.5 Biﬁbrations...... 22

3 Λ∞-categories. 28 3.1 The category Λ∞...... 28 3.2 Unbounded maps...... 30 3.3 The biﬁbration ΛD: preliminaries...... 32 3.4 The biﬁbration ΛD: the construction...... 35 3.5 Functoriality...... 37

Introduction.

1 f-categories.

1.1 Deﬁnitions and examples. Out starting point is a version of a deﬁnition given by Beilinson in [B].

∗Partially supported by grant NSh-1987.2008.1

1 Preliminary version – please do not distribute, use at your own risk 2

Deﬁnition 1.1. An f-category is a triple hD, F, si of a triangulated category D, a fully faithful left-admissible triangulated embedding F : D → D, called the twist functor, and a morphism s : F → Id from F to the identity functor Id : D → D, such that

2 (i) for any M ∈ D, we have sF (M) = F (sM ): F (M) → F (M), and

⊥ (ii) for any M ∈ F (D), the cone of the map sM : F (M) → M lies in F (D)⊥.

The quotient category D/F (D) is called the core of the f-category hD, F, si and denoted by C(D,F ), or simply C(D) when there is no danger of confu- sion. We note that the existence of the quotient C(D) is not a problem: it is canonically identified with ⊥F (D) ⊂ D, hence Hom-sets are small. Given an f-category hD, F, si, denote by L : D → D the left-adjoint functor to F . It immediately follows from the definition that for any integer n ≥ 2, the composition F n : D → D is also a left-admissible full triangulated embedding, with the left-adjoint functor Ln : D → D. We will denote the n n image of F by D≥n = F (D) ⊂ D, n ≥ 1, and we will set D≥0 = D. For n n any n ≥ 0, we will denote the composition F ◦ L : D → D≥n ⊂ D by σ≥n. By definition, for any M ∈ D, we have the adjunction map M → σ≥nM; the ⊥ cone of this map lies in the orthogonal D≥n, which we denote by D≤n−1. For any n ≥ 0, D≤n ⊂ D is a right-admissible full triangulated subcategory; we denote by σ≤n : D → D the composition of the embedding D≤n ⊂ D and its right-adjoint functor D → D≤n. For any M ∈ D, we have a functorial distiguished triangle

(1.1) σ≤nM −−−−→ M −−−−→ σ≥n+1M −−−−→ .

Deﬁnition 1.2. An f-category D is called non-degenerate if [ D = D≤n. n≥0

An f-structure on a triangulated category D is given by a non-degenerate f-category hDF, F, si and a triangulated equivalence D =∼ C(DF) between D and the core C(DF) of the f-category DF. To give a basic motivating example of an f-structure, we introduce some notation. Consider the ordered set N of natural numbers as a small category Preliminary version – please do not distribute, use at your own risk 3 in the usual way – objects are natural numbers, that is, non-negative integers, the set N(n, m) is empty if m < n, and consists of exactly one element ιn,m otherwise. For any category A, Fun(N, A) is the category of diagrams E(0) → E(1) → · · · → E(n) → ... in A. Let Funf (N, A) ⊂ Fun(N, A) be the full subcategory spanned by all such diagrams that stabilize at some n – that is, there exists a number n ∈ N such that the map E(i) → E(i + 1) is an isomorphism for i ≥ n.

Example 1.3. Let A be an abelian category, and consider its derived category D(A). Then the derived category DF(A) = D(Funf (N, A)) gives an f-structure on D(A). The twist functor F is induced by the functor Funf (N, A) → Funf (N, A) given by ( E(n − 1), n ≥ 1, F (E)(n) = 0, n = 0, and the morphism s : F (E) → E is induced by the natural maps E(n) → F (E)(n + 1) = E(n) → E(n + 1), n ≥ 0. It is well-known (and easy to check) that in the definition of the derived category D(Funf (N, A)), it suffices to consider such complexes E of functors from N to A that all the maps E (n) → E (n + 1), n ≥ 0, areq injective. Such a complex is nothing but a filteredq complexq in A, and DF(A) is usually called the filtered derived category of the abelian category A. In these terms, the twist functor F is given by the shift in the filtration.

1.2 Reﬂecting the ﬁltration. Assume given a non-degenerate f-category hD, F, si.

Lemma 1.4. For any n ≥ 0, the left-admissible embedding D≥n ⊂ D is also ⊥ ⊥ right-admissible, and the projection D≥n+1 → D → D≥n+1 = D≤n is an equivalence of categories.

Proof. It is a standard fact that for any left-admissible embedding D ⊂ D0 of triangulated categories, the composition functor D⊥ → D0 → ⊥D is fully faithful, and it is an equivalence if and only if D ⊂ D0 is right- admissible. Thus the second claim is equivalent to the ﬁrst. Moreover, since the composition of right-admissible embeddings is right-admissible, it suﬃces to consider the case n = 1. Thus all we have to prove is that the ⊥ projection D≥n+1 → D → D≤n is essentially surjective for n = 0. This is Preliminary version – please do not distribute, use at your own risk 4

0 Deﬁnition 1.1 (ii): for any M ∈ D≤0, any cone M of the map s : F (M) → M ⊥ 0 ∼ lies in D≥1, and we obviously have σ≤0(M ) = M.

In particular, denote by π : D → D/D≥1 = C(D) the projection functor, ∼ ⊥ and let λ : C(D) = D≤0 = D≥1 ⊂ D be its left-adjoint; then we also have ∼ ⊥ the right-adjoint ρ : C(D) = D≥1 ⊂ D, and the map s : F → Id has a functorial cone: for every M ∈ D, we have a functorial distiguished triangle

F (M) −−−−→s M −−−−→t ρ(π(M)) −−−−→ , where t : M → ρ(π(M)) is the adjunction map. Applying this to M = Λ(M 0), M 0 ∈ C(D), we obtain a functorial distiguished triangle

(1.2) F (λ(M 0)) −−−−→s λ(M 0) −−−−→t ρ(M 0) −−−−→ .

Denote by R : D → D the right-adjoint functor to F : D → D; for any n ≥ 2, Rn is then right-adjoint to F n. Denote by s0 : Id → R the map ≥n adjoint to s : F → Id. Denote by σ : D → D≥n ⊂ D the composition n n ≤n ⊥ F ◦ R , and denote D = D≥n+1 ⊂ D. This is a category equivalent to D≤n, and both are equivalent to the quotient category D/D≥n+1; the ∼ ∼ ≤n embeddings D/D≥n+1 = D≤n ⊂ D and D/D≥n+1 = D ⊂ D are left and right-adjoint to the projection D → D/D≥n+1. We will denote the ∼ ≥n ≤n composition D → D/D≥n+1 = D ⊂ D by σ . For any M ∈ D and any n ≥ 0, we have a functorial distinguished triangle

(1.3) σ≥n+1M −−−−→ M −−−−→ σ≤nM −−−−→ , similar to (1.1). Loosely speaking, the collections σ≥nM gives a (generalized) descreasing filtration on any object M ∈ D, a “reflection” of the (generalized) increasing filtration σ≤nM. It is also possible to do a second reflection.

0 Lemma 1.5. For any object M ∈ D≤0 ⊂ D, the map s : M → R(M) is an isomorphism.

Proof. By deﬁnition, s0 is the composition of the adjunction map M → R(F (M)) and the map R(s): F (M) → M. The adjunction map is an isomorphism. Since R is a triangulated functor, the cone M 0 of the map s0 : M =∼ R(F (M)) → R(M) is then isomorphic to R(M 00), where M 00 is the cone of the map s : F (M) → M. By Deﬁnition 1.1 M 00 lies in ≤0 0 00 D = Ker R ⊂ D, so that M = R(M ) = 0. Preliminary version – please do not distribute, use at your own risk 5

Corollary 1.6. The left-admissible subcategory D≤0 ⊂ D is right-admissible.

Proof. As follows from the proof of Lemma 1.4, every object in D≤0 is a cone of the map s : F (M) → M for some M ∈ D≤0. In particular, we have ≤0 D ⊂ D≤1. Since D≤n ⊂ D is right-admissible for any n ≥ 0, it suﬃces to ≤0 ⊥ ⊥ prove that D is right-admissible in D≤1. Let D , D ⊂ D≤1 be its right and left orthogonal. As in the proof of Lemma 1.4, we have to prove that the projection p : D⊥ → ⊥D is essentially surjective. ⊥ We have D = D≥1 ∩ D≤1 = F (D≤0). Moreover, for any M ∈ D≤0, the ≤0 ∼ cone of the map sM : F (M) → M lies in D , so that p(M) = p(F (M)) = ⊥ F (M). Thus it suﬃces to prove that every object M ∈ D≤0 lies in D . This is obvious: Lemma 1.5 implies by adjunction that for any M,N ∈ D≤0, the cone of the map s : F (N) → N is left-orthogonal to M. By this Corollary, the functor R : D → D has a right-adjoint T : D → D. Then for any n ≥ 2, T n is right-adjoint to Rn, so that D≤n−1 = Ker(Rn) ⊂ D is also right-admissible. We denote its right orthogonal by D≥n ⊂ D. The functor T n provides an equivalence D =∼ D≥n. To sum up, for any n ≥ 0 we have the following natural full subcategories in D:

≤n ≥n+1 (1.4) D≤n ⊥ D≥n+1 ⊥ D ⊥ D , where ⊥ indicates the orthogonality relations.

1.3 The comparison functor. For any n ≥ 0, denote by Rn : D → C(D) n n the composition Rn = π◦R . This functor has a left-adjoint F ◦λ : C(D) → D and a right-adjoint T n ◦ ρ : C(D) → D; both are full embeddings, with images given by

n F (λ(C(D))) = D≥n ∩ D≤n ⊂ D, T n(ρ(C(D))) = D≥n ∩ D≤n ⊂ D

By virtue of Deﬁnition 1.1 (i), the map s induces a unique natural map n+m m sn,m : F → F for any n, m ≥ 0 such that, in particular, sn,m◦sn0,n+m = 0 0 sn+n0,m for any n, n , n ≥ 0. By adjunction, we obtain natural maps sn,m : m n+m 0 R → R , and applying the projection π, we obtain natural maps sn,m : Rm → Rn+m. Sending M ∈ D to the collection of objects Rn(M) with the 0 maps sn,m between them gives a comparison functor

(1.5) R : D → Fun(N, C(D)). Preliminary version – please do not distribute, use at your own risk 6

In the case D = DF(A) of Example 1.3, this is just the forgetful functor which sends an object in D represented by a complex E ∈ Fun(N, A) to the collection of complexes E (n), n ≥ 0, with the naturalq maps between them. The comparison functorq R is of course not an equivalence; however, R is useful when one wants to describe various full subcategories in D. Here is the description of the middle two of the categories (1.4).

Lemma 1.7. For any non-degenerate f-category D, the comparison functor R of (1.5) is conservative, and R(M) = 0 implies M = 0. Moreover,

•D≥n ⊂ D is the full subcategory spanned by objects M ∈ D such that Rm(M) = 0 for m < n, and •D≤n ⊂ D is the full subcategory spanned by objects M ∈ D such that Rm(M) = 0 for m > n, ∼ Proof. By deﬁnition, for any M ∈ D and n ≥ 0 we have Rn(R(M)) = Rn+1(M). Also by deﬁnition, R0(M) = π(M) = 0 if and only if M ∈ D lies in D≥1 = F (D) ⊂ D. If this happens, the adjunction map F (R(M)) → M is an isomorphism. Therefore for any n ≥ 1, R(M) ∈ D≥n ⊂ D≥1 implies that ∼ M = F (R(M)) lies in D≥n+1 = F (D≥n). By induction, if Rm(M) = 0 for ∼ all m < n, then M lies in D≥n ⊂ D. Conversely, since R ◦ F = Id, we have Rn+1(F (M)) = Rn(M) for any M ∈ D and any n ≥ 0, so that by induction, M ∈ D≥n implies Rm(M) = 0 for all m < n. Since D is non-degenerate, we have

 ⊥ \ [ ⊥ D≥n ⊂  D≤n = D = 0, n≥0 n≥0 so that R(M) = 0 implies M = 0. Thus in particular, Rm(M) = 0 for m > n if and only if Rn+1(M) = 0, which by adjunction is equivalent to ≤n ⊥ n+1 ⊥ M ⊂ D = D≥n+1 = F (D) . Moreover, R is conservative – that is, if we have a map f : M → M 0 in the category D such that R(f) is an isomorphism, then f itself is an 00 isomorphism. Indeed, let M be the cone of such a map; since Rn is a 00 00 triangulated functor for any n ≥ 0, Rn(M ) = 0, so that R(M ) = 0 and 00 M = 0. Preliminary version – please do not distribute, use at your own risk 7

1.4 Further adjunctions. To proceed further, we need to introduce two n more families D , Dn ⊂ D, n ≥ 0 of full subcategories in D. These are: •Dn ⊂ D is the full subcategory spanned by objects M ∈ D such that Rm(M) = 0 for m 6= n.

•Dn ⊂ D is the full subcategory spanned by objects M ∈ D such that the map Rn(M) → Rn+1(M) is an isomorphism.

n ≤n By Lemma 1.7, we have D = D≥n ∩ D ⊂ D. Since both D≥n ⊂ D and D≤n ⊂ D are right and left-admissible, Dn ⊂ D is also right and left- admissible. It is obviously equivalent to C(D), with equivalence given by F n ◦ ρ : C(D) =∼ Dn ⊂ D.

n⊥ ⊥ n+1 Lemma 1.8. For any n ≥ 0, we have D = Dn = D ⊂ D.

Proof. For any m ≥ 0, let lm, rm : D → C(D) be the left and right-adjoint functors to the embedding F m ◦ ρ : C(D) =∼ Dm ⊂ D of the admissible category Dm ⊂ D. The natural map t : λ → ρ of (1.2) induces a map n n F ◦ λ → F ◦ ρ which gives by adjunction a natural map t : rn → Rn; moreover, since (1.2) is a distinguished triangle, for any M ∈ D we have a distinguished triangle

t s0 rn(M) −−−−→ Rn(M) −−−−→ Rn+1(M) −−−−→ .

n⊥ Therefore the orthogonal D = Ker(rn) is exactly the subcategory Dn ⊂ D. By Lemma 1.5, the map s0 : M → R(M) is an isomorphism for any M ∈ D≤0 = λ(C(D)) ⊂ D. By adjunction, for any M ∈ D the adjoint map s00 : T (M) → M becomes an isomorphism after applying the projection π : D → C(D). Analogously to (1.2), for any M ∈ D we have a functorial distinguished triangle

s F (M) =∼ F (R(T (M))) −−−−→ T (M) −−−−→ ρ(π(M)) −−−−→ , where s is the composition of the adjunction map T (M) → ρ(π(T (M))) and the isomorphism ρ(π(s00)) : ρ(π(T (M))) → ρ(π(M)). In particular, if M lies ∼ in D≥1 = F (D) = Ker π, we have F (M) = T (M), so that for any M ∈ D and any n ≥ 0, F n+1 =∼ T n ◦ F . On the other hand, if M lies in D≤0 ⊂ D, we have M =∼ ρ(π(M)); applying T n, we obtain a distinguished triangle

t s00 F n+1(M) =∼ T n(F (M)) −−−−→ T n+1(M) −−−−→ T n(M) −−−−→ Preliminary version – please do not distribute, use at your own risk 8

for any n ≥ 0. Then by adjunction, we have a natural map t : Rn+1 → ln+1, and for any M ∈ D, we have an distinguished triangle

s0 Rn(M) −−−−→ Rn+1(M) −−−−→ ln+1(M) −−−−→ .

⊥ n+1 Therefore D = Ker(ln+1) is again exactly the subcategory Dn ⊂ D. This allows to describe the remaining two subcategories of (1.4) in terms of the comparison functor (1.5).

•D≥n ⊂ D is the full subcategory spanned by objects M ∈ D such that the map Rm(M) → Rm+1(M) is invertible for m < n.

•D≤n ⊂ D is the full subcategory spanned by objects M ∈ D such that the map Rm(M) → Rm+1(M) is invertible for m ≥ n.

m Indeed, on one hand, we have D ⊂ D≥n for any m ≥ n by Lemma 1.7, so that

⊥ \ ⊥ m+1 \ (1.6) D≤n = D≥n+1 ⊂ D = Dm, m≥n m≥n T and on the other hand, D≥n+1 ∩ m≥n Dm = 0 by the same Lemma 1.7, so that the inclusion is an equality; and similarly for D≥n.

Corollary 1.9. There exists a series of functors Tn : D → D, n ≥ 0 such that T0 = T , Tn+1 is right-adjoint to Tn for any n ≥ 0, and Tn : D → D sends T (D) into itself for any n ≥ 2.

Proof. By deﬁnition, T = T0 : D → D is a fully faithful embedding onto 0⊥ T (D) = D0 = D ⊂ D. Assume by induction that we have constructed the functors Tl for all l ≤ 2n, and that T2n : D → D factors through an ∼ ⊥ n+1 equivalence D = Dn ⊂ D. Then since Dn = D is right-admissible, there exists a right-adjoint functor T2n+1 : D → D, and it factors through n+1 ∼ n+1 an equivalence D/D = D. Since D is also right-admissible, T2n+1 has a right-adjoint T2n+2 : D → D which factors through an equivalence ∼ n+1⊥ D = Dn+1 = D ⊂ D. To analyze what the functors Tn do to T (D) ⊂ D, consider the full subcategory D≤0 ⊂ D. For any M ∈ D≤0, the map s : M → R(M) is an isomorphism by Lemma 1.5. Moreover, since D≤0 ⊂ Dn for any n ≥ 0, the adjunction map M → T (R(M)) is an isomorphism, and so is the adjunction map M → Tn+1(Tn(M)) for any n ≥ 0. Therefore we have isomorphisms ∼ M = Tn(M) for any n ≥ 0, so that the functor R and all the functors Tn, Preliminary version – please do not distribute, use at your own risk 9

⊥ n ≥ 0 send D≤0 ⊂ D into itself. By adjunction, Tn sends D≥1 = D≤0 ⊂ D ≤0 ⊥ into itself for n ≥ 0, then by adjunction it sends D = D≥1 ⊂ D into itself for n ≥ 1, and ﬁnally it sends T (D) = D≥1 = D≥0⊥ ⊂ D into itself for any n ≥ 2.

Corollary 1.10. For any non-degenerate f-category hD, F, si, the limit

lim Rn : D → C(D) n → ∼ exists, and is left-adjoint to the embedding λ : C(D) = D≤0 ⊂ D.

Proof. The existence immediately follows from non-degeneracy: the inductive system stabilizes at some ﬁnite step for any argument M ∈ D. Moreover, (1.6) together with Lemma 1.8 shows that D≤0 ⊂ D is right orthogonal to Dn ⊂ D for any n ≥ 0, hence also to D≤n ⊂ D for any n ≥ 0. Since for n n n any M ∈ D, R (M) lies in D≤0 for n 0, so does T (R (M)), while the cone of the adjunction map M → T n(Rn(M)) lies in D≤n. Therefore the subcategory D≤0 ⊂ D is left-admissible, with orthogonal given by

⊥ [ ≤n D≤0 = D . n

n Thus λ does admit a left-adjoint. To identify it with lim→ Rn, it suﬃces to notice that this limit vanishes on D≤n for any n ≥ 0, and the map ∼ M = R0(λ(M)) → lim Rn(λ(M)) n → is an isomorphism for any M ∈ C(D).

1.5 Morphisms. Assume given two non-degeneratef-categories hD, F, si, hDe, F,e sei. Let R : D → D, Re : De → De be the right adjoint functors to F , resp. Fe, and let R, Re be the comparison functors (1.5) for D, resp. De. By a morphism from hD, F, si to hDe, F,e sei we will understand a pair hE, εi of a triangulated functor E : D → De and an isomorphism ε : E ◦ F =∼ Fe ◦ E such that se◦ ε = E(s). Lemma 1.11. The following conditions on a morphism hE, εi are equivalent:

(i) E(D≤0) ⊂ De≤0. Preliminary version – please do not distribute, use at your own risk 10

(ii) E(D≤0) ⊂ De≤0.

(iii) The base change map ε0 : E ◦ R → Re ◦ E associated to ε is an isomorphism.

Proof. The implication (i)⇒(ii) is obvious: since se◦ε = E(s), E sends a cone of the map s : F (M) → M, M ∈ D, to a cone of the map se : F (E(M)) → 0 E(M). To prove (ii)⇒(iii), note that for any M = F (M ) ∈ F (D) = D≥1, the map

ε0(M): E(R(M)) =∼ E(R(F (M 0))) =∼ E(M 0) =∼ Re(Fe(E(M 0))) → → Re(E(F (M 0))) =∼ Re(E(M)) is equal to Re(ε(M 0)), thus an isomorphism. Thus by (1.3), it suﬃces to check that ε0(M) is an isomorphism for M ∈ D≤0. But for such an object M, E(R(M)) = 0, so that ε0(M) is an isomorphism if an only if E(M) lies in Ker Re = De≤0 ⊂ De. Finally, (iii) together with Lemma 1.5 imply that 0 for any M ∈ D≤0 ⊂ D, the natural map s : E(M) → Re(E(M)) is an isomorphism. Therefore all the transition maps Ren(E(M)) → Ren+1(E(M)) are isomorphisms, so that E(M) lies in De≤0 ⊂ De.

Deﬁnition 1.12. A morphism hE, εi is admissible if it satisﬁes the equivalent conditions of Lemma 1.11.

Remark 1.13. This condition is not automatic: for example, hF, idi is a non-admissible morphism from hD, F, si to itself.

Assume given an admissible morphism hE, εi from hD, F, si to hDe, F,e sei. Then in particular, E : D → De sends F (D) ⊂ D into F (De) ⊂ De, thus induces a functor C(E): C(D) = D/F (D) → C(D0) = De/ C(De). Extend C(E) to a functor

C(E) : Fun(N, C(D)) → Fun(N, C(De)) by making it act pointwise; then by Lemma 1.11 (iii), the base change map ε0 induces an isomorphism C(E)◦R =∼ Re◦E. This immediately shows, in particular, that E sends all the subcategories (1.4) in D into the corresponding subcategories in De. In general, an admissible morphism hE, εi contains strictly more infor- mation than the induced functor C(E): D → De: if we are given two triangulated categories with f-structures, then there is no reason why any Preliminary version – please do not distribute, use at your own risk 11 triangulated functor between them should lift to an admissible morphism, nor why such a lifting should be unique. Let us show that at least, if we are given an adjoint pair of functors, then it suﬃces to lift one of them.

Lemma 1.14. Assume given an admissible morphism hE, εi from a non- degenerate f-category hDe, F,e sei to a non-degenerate f-category hD, F, si. Then the functor E : De → D admits a left-adjoint E] : D → De if and only if the functor C(E): C(De) → C(D) admits a left-adjoint C(E)] : C(D) → C(De), and if this happens, then the base change map ε] : E] ◦ F → Fe ◦ E] is ∼ an isomorphism, hE], ε]i is an admissible morphism, and C(E]) = C(E)]. Analogously, the functor E : De → D admits a right-adjoint E] : D → De if and only if the functor C(E): C(De) → C(D) admits a right-adjoint C(E)] : C(D) → C(De), and if this happens, then we have the base change isomorphims ε] : E] ◦ F =∼ Fe ◦ E], hE], ε]i is an admissible morphism, and C(E]) =∼ C(E)].

Proof. By definition, E : De → D admits a left-adjoint if and only if Hom(M,E(−)) is representable for any M ∈ D. By (1.1), it suffices to check n 0 0 n ∼ n it when M = F (M ) for some n ≥ 0, M ∈ D≤0. Since R ◦ E = E ◦ Re , we may assume n = 0, M = M 0 = λ(M 00) for some M 00 ∈ C(D). Then ∼ 00 Hom(M,E(−)) = Hom(M , C(E)(πe(−))), and it is representable for any M 00 ∈ C(D) if and only if C(E) admits a left-adjoint. Analogously, for the existence of a right-adjoint, we have to check representability of Hom(E(−),M); by induction, it suffices to check it for n 0 0 n+1 ∼ n M = F (M ), M ∈ D≤0, and since F = T ◦ F , it suffices to consider the cases n = 0 and n = 1. Equivalently, we may assume M ∈ D≤0 or M ∈ D≤0. If M ∈ D≤0, then again the required representability is equivalent to the one needed for the existence of C(E)]. By Corollary 1.10, this also implies the representability for M ∈ D≤0. Assume now existence of either the left-adjoint E] or the right-adjoint E]. Then by adjunction, (i) resp. (ii) of Lemma 1.11 imply that E], resp. E] sends F (D) ⊂ D into F (De) ⊂ De. The induced functor C(D) → C(De) ] on the quotient categories is obviously isomorphic to C(E) , resp. C(E)]. ] ⊥ Moreover, by adjunction the cone of the map ε resp. ε] lies in F (De) resp. F (De)⊥, hence is equal to 0. And since E sends F (De) ⊂ De into F (D) ⊂ D, ] E resp. E] satisfy by adjunction (ii) resp. (i) of Lemma 1.11. Using Definition 1.12, we can iterate the definition of an f-category to axiomatize the notion of a “filtered category with several filtrations”. This deserves a separate detailed study; pending such a study, we would prefer Preliminary version – please do not distribute, use at your own risk 12 not to go into the subject at all in the present paper. But unfortunately, the existence of a “bifiltered” derived category is required for some of the the- orems we are going to prove. Thus we introduce the following provisionary notion.

Deﬁnition 1.15. A non-degenerate f 2-category is a triangulated category D equipped with two functors F1,F2 : D → D, two maps s1 : F1 → Id, d2 : F2 → Id, and an isomorphism ε : F1 ◦ F2 → F2 ◦ F1 such that

(i) hD,F1, s1i and hD,F2, s2i are non-degenerate f-categories in the sense of Deﬁnition 1.1, Deﬁnition 1.2, and

(ii) the pair hF2, εi is a morphism from hD,F1, s1i to itself in the sense of Deﬁnition 1.12. An obvious example of a non-degenerate f 2-category is the derived category D(Funf (N, Funf (N, A))) ⊂ D(Fun(N × N, A)) with abelian A, as in Example 1.3.

2 End(N)-action.

2.1 Prototype: End(N)-action on functor categories. Assume given an arbitrary category C, and consider the functor category Fun(N, C). Any order-preserving map f : N → N induces by restriction the pullback functor ∗ f : Fun(N, C) → Fun(N, C).

This functor sends Funf (N, C) ⊂ Fun(N, C) into itself. For any two such maps f1, f2 : N → N we have a natural isomorphism ∗ ∼ ∗ ∗ (2.1) (f2 ◦ f1) = f1 ◦ f2 .

Moreover, equip the set of all order-preserving maps f : N → N with point- 0 0 wise order – that is, f ≤ f iﬀ f(n) ≤ f (n) for any n ∈ N. Then whenever 0 f ≤ f , we have map of corresponding functors from N to itself, and the induced natural map

0 (2.2) f ∗ → f ∗.

Say that a map f : S0 → S between two sets S, S0 is proper if f −1(s) ⊂ S0 is ﬁnite for any s ∈ S. For any proper order-preserving map f : N → N, setting

0 0 (2.3) f](n) = min{n ∈ N | f(n ) ≥ n} Preliminary version – please do not distribute, use at your own risk 13

gives a well-deﬁned order-preserving map f] : N → N. In the language of functors, f] : N → N is left-adjoint to f. Therefore the pullback functor ∗ ∗ f is left-adjoint to the pullback functor f] : Fun(N, C) → Fun(N, C). By adjunction, (2.2) induces a map

0∗ ∗ (2.4) f] → f]

0 0 whenever f ≤ f (in fact in this case f] ≤ f], and this adjoint map is also 0 the map (2.2) for f] and f]). Any injective map i : N → N is obviously proper. A proper order- preserving map i : N → N is injective if and only if the adjoint map i] : N → ∗ N is surjective; in this case, we have i]◦i = id, and i] : Fun(N, C) → Fun(N, C) is a fully faithful embedding. More generally, for any commutative square

f 0 N −−−−→ N   (2.5) 0 g g y y f N −−−−→ N 0 0 of proper order-preserving maps, we have f ◦ f] ≥ id by adjunction, so that

0 0 0 0 g ≤ g ◦ f ◦ f] = f ◦ g ◦ f],

0 0 and by adjunction, f] ◦ g ≤ g ◦ f] (this is of course the usual construction of the base-change map). Say that a square (2.5) is perfect if we actually have 0 0 f] ◦ g = g ◦ f]. Then for a perfect square, the isomorphism

∗ ∗ ∼ ∗ 0 0 ∗ ∼ 0∗ 0∗ (2.6) g ◦ f] = (f] ◦ g) = (g ◦ f]) = f] ◦ g induced by the isomorphisms (2.1) is the base-change map for the commutative square of pullback functors f ∗, g∗, f 0∗, g0∗ – that is, (2.6) is adjoint to the map

0∗ 0 ι 0 ◦f ∗ (2.7) 0∗ ∗ ∗ f ,g ] 0∗ ∗ ∗ g ◦af 0∗ f ◦ g ◦ f] −−−−−→ g ◦ f ◦ f] −−−−→ g ,

∗ ∗ 0∗ ∗ ∼ 0∗ ∗ where af : id → f ◦ f] is the adjunction map, and ιf 0,g : f ◦ g = g ◦ f is the isomorphism induced by (2.1). The standard way to package the functors f ∗ and the isomorphisms (2.1) between them is by using the Grothendieck construction. Namely, let Preliminary version – please do not distribute, use at your own risk 14

End(N) be the category with one object N, and all proper order-preserving maps f : N → N as morphisms. Then we can deﬁne a category

opp Tot(End(N) , Fun(N, C)) as follows: objects are the same as objects in Fun(N, C), and a morphism from E to E0 is given by a pair hf, ei of a proper order-preserving map ∗ 0 f : N → N and a map e : f E → E . We have a natural forgetful functor

opp opp (2.8) Tot(End(N) , Fun(N, C)) → End(N) which sends any object to the unique object N ∈ End(N), and sends a morphism hf, ei to f opp – that is, the map f considered as a morphism in opp the opposite category End(N) . This forgetfull functor is a biﬁbration: for opp opp any morphism f in the category End(N) , the transition functors are opp ∗ opp∗ ∗ given by f∗ = f , f = f] . The base-change description of the isomorphism (2.6) can be formalized in the language of the Grothendieck construction as follows.

opp opp Deﬁnition 2.1. A category C/End(N) biﬁbered over End(N) has the adjunction property if for any perfect commutative square (2.5), the base change map

opp opp∗ 0opp∗ 0opp (2.9) g∗ ◦ f → f ◦ g∗ is an isomorphism.

opp Lemma 2.2. Assume given a biﬁbered category C/End(N) which has the adjunction property. Then for any proper order-preserving map f : N → N, opp∗ opp the transition functor f is isomorphic to the transition functor f]∗ (and opp∗ opp∗ therefore right-adjoint to the transition functor f] ). If f is injective, f opp ∼ is a fully faithful embedding. Moreover, under these identiﬁcations f∗ = opp∗ 0opp ∼ 0opp∗ f] , f∗ = f] , the isomorphism (2.6) for any perfect commutative square (2.5) is adjoint to the isomorphism (2.7).

Proof. For any injective order-preserving map i : N → N, the square (2.5) 0 0 with g = i], f = i, f = g = id is commutative and perfect. Then the opp∗ ∼ opp opp opp∗ ∼ adjunction property provides an isomorphism i = i]∗ . Thus i∗ ◦i = opp opp ∼ opp∗ i∗ ◦ i]∗ = id, so that i is fully faithful. Moreover, for any surjective 0 0 order-preserving map p : N → N, setting f = p, g = p], g = f = id also gives a perfect commutative square (2.5), and the adjunction property Preliminary version – please do not distribute, use at your own risk 15

opp∗ ∼ opp provides an isomorphism p = p]∗ . Since every map f : N → N factors uniquely as f = i ◦ p with surjective p and injective i, we obtain a canonical opp∗ ∼ opp isomorphism f = f]∗ . The last claim is left to the reader. The isomorphisms (2.4) do not ﬁt naturally into the framework of the Grothendieck construction, but in principle, some of them can be recovered using the adjunction property. In particular, assume given a commutative square (2.5) which is not perfect. Then we still have the adjunction map opp ∼ opp∗ 0opp ∼ 0opp∗ (2.9), and if we identify g∗ = g] , g∗ = g] using the adjunction opp∗ 0 0 opp∗ property, we obtain a natural map (g] ◦ f) → (f ◦ g]) .

2.2 N-categories. If the category C = A is abelian, then all the structures on the functor category Fun(N, A) described in Subsection 2.1 descend to the derived category DF(A) = D(Funf (N, A)) of Example 1.3. In particular, we have a bifibration opp opp Tot(End(N) , DF(A))/End(N) analogous to (2.8). It is natural to try to recover this bifibration using only the f-category structure on DF(A). As it happens, even less is needed – there is a unified construction which applies both to the functor category Fun(N, C) with an arbitrary C, and to an arbitrary f-category D.

Deﬁnition 2.3. An N-category is a category D equipped with a functor R : D → D. An N-category D is admissible if there exists a sequence of functors Tn : D → D, n ≥ 0, such that T0 is right-adjoint to R, Tn+1 is right-adjoint to Tn for all n ≥ 0, and (i) T : D → D is fully faithful,

(ii) for any n ≥ 2, Tn sends the essential image T (D) ⊂ D into itself.

Example 2.4. Any f-category D with the functor R of Subsection 1.2 is an N-category, and it is admissible by Corollary 1.9.

∗ Example 2.5. For any category C, Fun(N, C) is an N-category, with T = t being the pullback with respect to the shift map t : N → N, t(n) = n + 1. The N-category of Example 2.5 is also admissible; this can be checked by an easy direct computation. However, it also follows from the adjunction property of the bifibration (2.8). Namely, assume given an arbitrary opp bifibration C/End(N) . Let CN be the fiber of this bifibration, and let opp R = t∗ : CN → CN be the transition functor associated to the shift map t of Example 2.5. Preliminary version – please do not distribute, use at your own risk 16

opp Lemma 2.6. Assume that the biﬁbration C/End(N) has the adjunction opp∗ property in the sense of Deﬁnition 2.1. Then the pair hCN, t i is an admissible N-category.

Proof. Let t0 = t, and deﬁne inductively tn : N → N, n ≥ 1, by setting tn = (tn−1)]. Explicitly, we have ( ( n, n ≤ k, n, n ≤ k, (2.10) t2k(n) = t2k+1(n) = n + 1, n > k, n − 1, n > k.

opp∗ opp∗ opp Let Tn = tn . By deﬁnition, T0 = t is right-adjoint to R = t0 . By Lemma 2.2, Tn+1 is right-adjoint to Tn for any n ≥ 0, as required, and since t is injective, T = topp∗ is fully faithful, which is (i) of Deﬁnition 2.3. Finally, it immediately follows from (2.10) that

(2.11) tn+1 ◦ t = t ◦ tn ∼ for n ≥ 1, so that Tn+1 ◦ T = T ◦ Tn. This yields (ii). Now, it turns out that Lemma 2.6 has a partial converse. To formulate it, we need to impose a restriction on order-preserving maps N → N.

Deﬁnition 2.7. An order-preserving map f : N → N is bounded if f(n + 1) = f(n) + 1 for n 0. b Let End (N) ⊂ End(N) be the subcategory spanned by bounded maps. If f : N → N is bounded, then so is the adjoint f], so that the adjunction b opp property make sense for biﬁbrations over End (N) .

Theorem 2.8. Assume given an admissible N-category hD,Ri. Then there exists a biﬁbration opp b opp (2.12) Tot(End(N) , D)/End (N) opp ∼ opp ∼ with the adjunction property such that Tot(End(N) , D)N = D and t∗ = R. Moreover, such a biﬁbration is unique up to an equivalence.

2.3 Uniqueness. The uniqueness part of Theorem 2.8 can be proved right away. For any admissible N-category hD,Ri, both adjunction maps R◦T0 → id → T1 ◦ T0 are invertible, so that we have a commutative square D −−−−→T0 D   (2.13)   T0y yR D −−−−→DT1 . Preliminary version – please do not distribute, use at your own risk 17

Lemma 2.9. Assume given an admissible N-category hD,T i. Then there exists a unique series of isomorphisms ∼ γm,n : Tm+n+1 ◦ Tm = Tm ◦ Tm+n−1, m > n ≥ 0, such that γ1,0 is the base change map associated to (2.13), γm+1,n+1 is adjoint to γm,n, and γm,n+1 is the base-change map associated to γm,n.

Proof. Uniqueness is obvious, it is existence that is the issue: in order to define γm,n+1, we have to prove that γm,n is an isomorphism. Use induction on n−m. Since the adjoint map to an isomorphism is itself an isomorphism, we may assume m = 0. Since by Definition 2.3 (ii), Tn+1 ◦ T sends D into T (D) ⊂ D, it suffices to prove that T1 ◦ γ0,n+1 : T1 ◦ Tn+1 ◦ T → T1 ◦ T ◦ Tn−1 is an isomorphism. By definition, this map factors as

−1 γ1,n+1◦T T1 ◦ Tn+1 ◦ T −−−−−−→ Tn ◦ T1 ◦ T −−−−→ Tn, which ﬁnishes the proof.

Deﬁnition 2.10. A morphism from an N-category hD,Ri to an N-category hD0,R0i is a pair hE, εi of a functor E : D → D and an isomorphism ε : E ◦ R =∼ R0 ◦E. If both D and D0 are admissible, then a map hE, εi is admissible ∼ 0 if and only if there exists a sequence of isomorphisms εn : E ◦ Tn = Tn ◦ E, n ≥ 0, such that ε0 is the base change map for the isomorphism ε, and for any n ≥ 0, εn+1 is the base change map for the isomorphism εn.

As in Lemma 2.9, the isomorphisms εn, n ≥ 0 are obviously unique, so that admissibility is a condition on a morphism, not an extra structure. If E is an equivalence, this condition is obviously automatically satisﬁed.

Proposition 2.11. Assume given two categories C, C0 biﬁbered over the b opp category End (N) , and assume that both biﬁbrations have the adjunction property. Let hC ,Ri, hC0 ,R0i be the corresponding admissible -categories. N N N Then any morphism hE, εi from hC ,Ri to hC0 ,R0i extends uniquely to a N N bicartesian functor Ee : C =∼ C0.

Proof. By deﬁnition, to specify the functor Ee, we have to specify its action on morphisms – that is, we need to specify a map

0 0 Ef : HomC,f opp (M,M ) → HomC0,f opp (E(M),E(M )) Preliminary version – please do not distribute, use at your own risk 18

0 for any objects M,M ∈ CN and any bounded order-preserving map f : N → N, and we have to check that for two such maps f, g, Ef , Eg and Ef◦g are compatible with the compositions. An injective order-preserving map i : N → N is bounded if and only if the complement N\i(N) is ﬁnite. Any such map admits a unique decomposition of the form

i = t2n1 ◦ · · · ◦ t2nk with n1 ≤ · · · ≤ nk — indeed, it is easy to see that it is necessary and 0 0 0 0 suﬃcient to take ni = ni − i, where N = f(N) ∪ {n1, . . . , nk}, and n1 < ··· < 0 nk. By adjunction, a surjective bounded order-preserving map p : N → N has a unique decomposition

p = t2m1+1 ◦ · · · ◦ t2ml+1, m1 ≥ · · · ≥ ml.

An arbitrary bounded order-preserving map f : N → N decomposes uniquely as f = i ◦ p with surjective bounded p and injective bounded i; thus we have a unique decomposition

(2.14) f = t2n1 ◦ · · · ◦ t2nk ◦ t2m1+1 ◦ · · · ◦ t2ml+1 with m1 ≥ · · · ≥ ml and n1 ≤ · · · ≤ nk. In other words, the monoid of bounded order-preserving maps N → N is generated by tn, n ≥ 0, modulo the relations tm+n+1 ◦ tm = tm ◦ tm+n−1, m > n ≥ 0. Therefore it suﬃces to construct the maps Ef for f = tn, n ≥ 0, and moreover, it suﬃces to check the compatibility with compositions for g = tm, f = tm+n+1, m > n ≥ 0. But by the adjunction property, the isomorphisms

Etn must be induced by the isomorphism εn of Deﬁnition 2.10. Moreover, by the same adjunction property these isomorphisms are compatible with the canonical isomorphisms γm,n, m > n ≥ 0 of Lemma 2.9, which exactly means the required compatibility with compositions. To prove the existence part of Theorem 2.8, we can imitate the proof of Lemma 2.11: for any bounded order-preserving f : N → N with decomposition (2.14), we deﬁne a transition functor f opp∗ : D → D by

opp∗ (2.15) f = T2n1 ◦ · · · ◦ T2nk ◦ T2m1+1 ◦ · · · ◦ T2ml+1, and we can use the maps γm,n of Lemma 2.9 to define the isomorphisms opp∗ ∼ opp∗ opp∗ τf,g :(f ◦ g) = f ◦ g . However, in order to define a bifibration, we need to check a compatibility condition for the maps τ , for every triple of b maps in End (N), and this would require much more combinatoricsq q than a Preliminary version – please do not distribute, use at your own risk 19 human being ought to be expected to handle. To control the combinatorics, we first construct natural maps

opp∗ opp∗ (2.16) i1 → i2 for all pairs of bounded injective maps i1, i2 : N → N such that i1 ≥ i2 with respect to pointwise order, and the complements N \ i1(N), N \ i2(N) have the same cardinality (in the situation of Example 2.5, these are the maps (2.4)). This done in Subsection 2.4. Then in Subsection 2.5, we use these maps to construct a biﬁbration (2.12) and ﬁnish the proof of Theorem 2.8.

2.4 Full embeddings. Fix an N-category hD,Ri. Assume that it is admissible in the sense of Deﬁnition 2.3, so that we have the sequence of adjoint functors Tn : D → D, n ≥ 0. The functor T0 is fully faithful by deﬁnition; by adjunction, so are the functors T2n, n ≥ 0. For any n ≥ 0, the adjunction map T2n+1 ◦ T2n+2 → id is an isomorphism, and the inverse isomorphism n+1 gives by adjunction a map βn : T2n → T2n+1; composing these maps, we obtain canonical maps

m m n+1 βn = βm−1 ◦ · · · ◦ βn : T2n → T2m n for any m > n ≥ 0. Let βn = id, n ≥ 0. Let D = D × N (where N is considered as a small category as in Subsection 1.1).q Then the functors T2n m and the maps βn together deﬁne a functor T : D → D by the following rule: q q

T (M × n) = T2n(M),M ∈ D, n ∈ N, (2.17) m T (fq × ιm,n) = βn ◦ T2n(f), m ≥ n. q For any n ≥ 0, let Dn = T2n(D) ⊂ D be the essential image of the functor T2n. More generally, for any injective bounded order-preserving map f : N → N with canonical decomposition (2.14), let Df ⊂ D be the essential image of the fully faithful functor f opp∗ : D → D given by (2.15) (since f is injective, (2.14) has no terms of the form t2m+1). It immediately follows from 0 Lemma 2.9 that for any injective bounded order-preserving f, f : N → N, an opp∗ object M ∈ D lies in Df 0 ⊂ D if and only if f (M) ∈ D lies in Df◦f 0 ⊂ D. Applying this to the decomposition (2.14) of the map f, we see that \ (2.18) Df = Dn ⊂ D. n∈N\f(N) Preliminary version – please do not distribute, use at your own risk 20

opp∗ ∼ opp∗ The embedding f : D = Df ⊂ D has a right-adjoint functor f] : D → D, again given by (2.15) for the canonical decomposition of the map f]. For any bounded order-preserving map f : N → N, there exists a unique integer l such that f(n) = n + l for n 0; denote this integer by l(f). We obviously have l(f]) = −l(f). If f is injective, l(f) = |N \ f(N)| is the cardinality of the complement N \ f(N) (in particular, it is non-negative). Let I be the set of all bounded injective order-preserving maps f : N ⊂ N. For any l ≥ 0, let Il ⊂ I be the subset of maps f with l(f) = l, and equip 0 0 Il with pointwise partial order: f ≤ f iﬀ f(m) ≤ f (m) for any m ∈ N.

0 0 Definition 2.12. Two maps f, f ∈ Il are adjacent if f(m) = f (m) for all m ∈ N except for a single value i ∈ N. They are adjacent at n ∈ N if in addition, f(i) = n + 1 and f 0(i) = n. For any f ∈ I, let t(f) ∈ I be given by t(f)(m) = f(m) + 1 (or equivalently, t(f) = t ◦ f). The map t : I → I sends Il ⊂ I into Il+1 ⊂ I. 0 It preserves adjacency; moreover, if f, f ∈ Il are adjacent at n, then 0 t(f), t(f ) ∈ Il+1 are adjacent at n + 1. Define a partial order on I as follows: f ≤ f 0 iff l(f) ≥ l(f 0) and tl(f)−l(f 0)(f 0) ≤ f. By definition, the opp projection l : I → N is order-preserving. If we treat partially ordered sets as small categories, then l is a cofibra- opp l0−l opp opp tion, with fiber Il at l ∈ N, and with transition functor t : Il → Il0 corresponding to a morphism ιl,l0 in N. T T Let D be another cofibration over N, with fibers Dl = D, l ∈ N, and l0−l with transition functor T0 : D → D corresponding to a morphism ιl,l0 in N. T T Let Cart(I, D ) be the category of all Cartesian functors I → D over N. For T opp any l, a functor E ∈ Cart(I, D ) gives by evaluation a functor El : Il → D; T opp explicitly, E ∈ Cart(I, D ) is defined by the collection El ∈ Fun(Il , D) and transition isomorphisms ∗ ∼ t El+1 = T0(El) for any l ≥ 0. Since I0 consists of the single element, namely, the identity map id : N → N, E0 : pt → D is just an object in D.

opp Deﬁnition 2.13. For any l ≥ 0, the subcategory De(l) ⊂ Fun(Il , D) is the full subcategory consisting of such functors E that

(i) for any f ∈ Il and n ∈ N \ f(N), E(f) ∈ D lies in Dn ⊂ D, and 0 (ii) for any f, f ∈ Il which are adjacent at some n ∈ N, the map

0 El(f) → El(f ) Preliminary version – please do not distribute, use at your own risk 21

becomes an isomorphism after applying the functor T2n+1.

The category De is the full subcategory De ⊂ Cart(Iopp, DT ) consisting of such opp T opp E ∈ Cart(I , D ) that El ∈ De(l) ⊂ Fun(Il , D) for any l ≥ 0.

Proposition 2.14. Sending E ∈ De to E0 ∈ D gives an equivalence of categories ϕ : De =∼ D. S Proof. Let I

We have to check the conditions (i), (ii) of Definition 2.13 for the functor opp E = T ◦ Ee : Il → D. Both conditions are automatic for n = a(f), so that weq may assume that n > a(f). Then both condition only depend on a(f) the restriction of Ee to the fiber Il ⊂ Il. This restriction is isomorphic to ∗ 0 a(f) 0 α E , where α : Il → Il is the transition functor, and it suffices to notice that α does not change the complement N \ (f(N) ∪ {a(f)}).

2.5 Biﬁbrations. In order to apply Proposition 2.14, we need to change slightly Deﬁnition 2.13.

Lemma 2.15. The condition (ii) of Deﬁnition 2.13 is equivalent to the following stronger condition: 0 0 (ii’) for any adjacent f, f ∈ Il with f ≤ f , the map 0 El(f) → El(f ) opp becomes an isomorphism after applying the functor f∗ of (2.15). Proof. Note that for any two adjacent f, f 0 ∈ I with f ≤ f 0, there exists a 0 unique sequence f0, . . . , fm ∈ I such that f0 = f, fm = f , and fn is adjacent to fn+1 at (j+n), for any n, 0 ≤ n < m, and some ﬁxed j ∈ N. Moreover, the interval H = [j +1, j +m−1] ⊂ N lies both in the complement S = N\f(N) 0 0 and in the complement S = N \ f (N). For any n, j ≤ n < j + m, the full subcategory \ DH = Dl ⊂ D l∈H lies in the intersection Dn ∩ Dn+1, so for any M ∈ DH , the adjunction map T2n ◦ T2n+1(M) → M is an isomporphism. Since DH contains both Df and Df 0 , this proves the claim. b Let Mono(N) ⊂ End (N) be the subcategory spanned by bounded injective order-preserving maps. Any such map f : N → N induces a map c(f): I → I, c(f 0) = f 0 ◦ f. If l(f) = l, then for any n ≥ 0, c(f) sends In into In+l. All the maps c(f): In → In+l are obviously order-preserving and preserve adjacency (although not the adjacency at n). Moreover, c(f) commutes with t, and we have Cartesian squares

c(f) Iopp −−−−→ Iopp DT −−−−→DT         y y y y tl(f) tl(f) N −−−−→ N, N −−−−→ N. Preliminary version – please do not distribute, use at your own risk 23

Therefore we have a natural functor c(f)∗ : Cart(Iopp, DT ) → Cart(Iopp, DT ). If we have two maps f, f 0 ∈ I, then c(f ◦f 0) = c(f 0)◦c(f), we have a natural isomorphism c(f)∗ ◦ c(f 0)∗ =∼ c(f ◦ f 0)∗, and these data glue together to give opp T opp a category Tot(Mono(N) , Cart(I, D )) ﬁbered over Mono(N) , with ﬁber Cart(I, DT ) and transition functors f opp∗ = c(f)∗. Let

opp opp T Tot(Mono(N) , De) ⊂ Tot(Mono(N) , Cart(I, D )) be the full subcategory consisting of those objects which lie in the subcategory De ⊂ Cart(I, DT ).

Proposition 2.16. The natural projection

opp opp (2.19) Tot(Mono(N) , De) → Mono(N)

opp T opp induced by the projection Tot(Mono(N) , Cart(I, D )) → Mono(N) is a biﬁbration. Moreover, for any map f : N → N in Mono(N), the transition functor f opp∗ : De → De is isomorphic to ϕ−1 ◦ f opp∗ ◦ ϕ, where ϕ : De → D is the equivalence of Proposition 2.14, and f opp∗ : D → D is given by (2.15).

opp opp Proof. To prove that Tot(Mono(N) , De) → Mono(N) is a fibration, it suffices to prove that for any f ∈ I, the corresponding transition functor c(f)∗ : Cart(I, D) → Cart(I, D) sends D ⊂ Cart(I, D) into itself. In other words, we have to prove that c(f) preserves the condition (i), (ii) of Defini- tion 2.13. This is obvious for (i). Since c(f) preserves the adjacency, this is also obvious for (ii’); by Lemma 2.15, this is sufficient. Since every bounded order-preserving f : N → N admits the canonical decomposition (2.14), it suffices to construct isomorphisms f opp∗ =∼ ϕ−1 ◦ opp∗ f ◦ ϕ for f = t2n, n ≥ 0. Moreover, by adjunction, it suffices to do it for n = 0, that is, for f = t. In this case, it immediately follows from Definition 2.13. Finally, to check that (2.19) is a bifibration, we have to check that all the transition functors f opp∗ have left-adjoint functors. Since f opp∗ =∼ ϕ−1 ◦ opp∗ f ◦ ϕ, it suffices to construct left-adjoints for all the functors Tk that appear in (2.15). This is part of Definition 2.3: Tk−1 is left-adjoint to Tk when k ≥ 1, and R : D → D is left-adjoint to T0. By virtue of Proposition 2.16, for any bounded injective order-preserving opp∗ map f : N → N, we have a transition functor f : De → De, for any two such maps f1, f2, we have an isomorphism opp∗ opp∗ ∼ opp∗ τf1,f2 : f1 ◦ f2 = (f1 ◦ f2) , Preliminary version – please do not distribute, use at your own risk 24

and for any three maps f1, f2, f3, we have

opp∗ opp∗ (2.20) τf1◦f2,f3 ◦ (τf1,f2 ◦ f3 ) = τf1,f2◦f3 ◦ (f1 ◦ τf2,f3 ). opp We also have adjoint functors f∗ and adjoint isomorphisms

τ ] : f opp ∗ ◦f opp∗ =∼ (f ◦ f )opp∗ f1,f2 1 2 2 1 which satisfy an identity adjoint to (2.20). However, Proposition 2.14 gives even more. Namely, if we have two bounded injective order-preserving maps opp opp f1, f2 ∈ Il such that f1 ≥ f2, then c(f1) ≥ c(f2) on every In ⊂ I with respect to the pointwise partial order. Therefore we have a natural map ∗ ∗ c(f1) → c(f2) which gives by restriction a natural map

opp∗ opp∗ ιf1,f2 : f1 → f2 between the transition functors of the biﬁbration (2.19) (under the equivalence ϕ : De =∼ D, these are the maps (2.16)). If we are given a third injective bounded order-preserving map f3 : N → N, then opp∗ opp∗ (2.21) f3 ◦ ιf1,f2 = ιf3◦f1,f3◦f2 , ιf1,f2 ◦ f3 = ιf1◦f3,f2◦f3 , and if f2 ≥ f3, l(f3) = l, then

(2.22) ιf1,f3 = ιf2,f3 ◦ ιf1,f3 .

In particular, assume given bounded order-preserving maps i, p : N → N, i injective, p surjective, and consider the commutative square

p0 N −−−−→ N   (2.23)   0 iy yi p N −−−−→ N, 0 0 where i ◦ p = p ◦ i is the canonical factorization of the map p ◦ i : N → N 0 0 0 0 with injective i and surjective p . Then by adjunction, p] ◦ i ≤ i ◦ p], and

0 0 0 0 0 0 l(p] ◦ i ) − l(i ◦ p]) = l(i ) + l(p]) − l(i) − l(p]) = l(i ) + l(p ) − l(p) − l(i) = l(i0 ◦ p0) − l(p ◦ i) = 0, so that we have the map

0 opp∗ 0 opp∗ (2.24) ι 0 0 :(i ◦ p ) → (p ◦ i ) . i◦p],p]◦i ] ] Preliminary version – please do not distribute, use at your own risk 25

Let opp opp∗ 0opp∗ 0opp τp,i : p]∗ ◦ i → i ◦ p]∗ be the corresponding base change map – that is, the composition

0 popp◦iopp∗◦a ι]◦p opp opp opp∗ ]∗ opp opp∗ 0opp∗ 0opp ]∗ 0opp∗ 0opp p]∗ ◦ i −−−−−−−−→ p]∗ ◦ i ◦ p] ◦ p]∗ −−−−−→ i ◦ p]∗ ,

0opp∗ 0opp ] opp opp∗ 0opp∗ where a : id → p] ◦p]∗ is the adjunction map, and ι : p]∗ ◦i ◦p] → i0opp∗ is adjoint to (2.24).

Lemma 2.17. For every commutative square (2.23), the map τp,i is an isomorphism. For any bounded order-preserving maps i1, i2, p1, p2 : N → N, i1, i2 injective, p1, p2 surjective, we have (2.25) opp 00opp opp∗ 0opp∗ τ ◦ (p ◦ τ ) = (τ 0 0 ◦ p ) ◦ (τ ◦ i ) ◦ (i ◦ τ 0 ), p1,i2◦i1 1]∗ i1,i2 i1,i2 1]∗ p1,i2 1 2 p1,i1 ] opp∗ opp 0opp 00opp∗ ] 0 τp2◦p1,i2 ◦ (τp ,p ◦ i2 ) = (p ◦ τp1,i2 ) ◦ (τp2,i ◦ p ) ◦ (i2 ◦ τ 0 0 ), 1] 2] 2]∗ 2 1]∗ p1],p2]

0 0 00 0 0 00 0 0 0 0 where i2 ◦ p1 = p1 ◦ i2, p1 ◦ i1 = p1 ◦ i1, i2 ◦ p2 = p2 ◦ i2 with injective i1, i2, 00 0 00 0 i2 and surjective p1, p1, p2.

Proof. The identities (2.25) follow from (2.20), its adjoint identity, (2.21), and (2.22) by straightforward diagram chasing liberally spiced with adjunction; here is the diagram

i1 i2 N −−−−→ N −−−−→ N    00 0 p  p  p1 1 y 1y y 0 0 i1 i2 N −−−−→ N −−−−→ N   0 p  p2 2y y 00 i2 N −−−−→ N. For example, for the ﬁrst of the identities (2.25), one easily shows that the two possible ways to cook up a map

opp∗ opp∗ 00opp∗ opp∗ 0 0 opp∗ ι : i2 ◦ i1 ◦ p1] → p1] ◦ (i2 ◦ i1) coincide, hence the corresponding base change maps also coincide; and similarly for the second identity. By virtue of these identities and the canonical Preliminary version – please do not distribute, use at your own risk 26

decomposition (2.14), it suﬃces to prove that τp,i is an isomorphism for p = t2m+1, i = 2n, m, n ≥ 0. This immediately follows from Lemma 2.9.

Proof of Theorem 2.8. For any bounded order-preserving map f : N → N with canonical factorization f = i◦p, p surjective, i injective, deﬁne a functor f opp∗ : De → De by opp∗ opp∗ opp f = i ◦ p]∗ .

For any two such maps f1 = i1 ◦ p1, f2 = i2 ◦ p2, deﬁne an isomorphism opp∗ opp∗ ∼ opp∗ τf1,f2 : f1 ◦ f2 = (f1 ◦ f2) by

opp∗ opp ] 0 τf1,f2 = (i1 ◦ τp1,i2 ◦ p ) ◦ (τi1,i ◦ τ 0 ), 2]∗ 2 p1],p2]

0 0 0 0 where p1 ◦ i2 = i2 ◦ p1 with injective i2 and surjective p1. We note that this is consistent with our earlier maps τf1,f2 for injective f1, f2, and τp,i with surjective p and injective i. To show that the functors f opp∗ with the isomorphisms τf1,f2 deﬁne a ﬁbration

b opp b opp (2.26) Tot(End (N) , De)/End (N) with ﬁber De, we have to check the compatibility condition (2.20) for all bounded f1, f2, f3. This is a tedious but straightfoward computation using (2.20) for injective f1, f2, f3, its adjoint identity, and (2.25): let fl = il ◦ pl with surjective pl and injective il, l = 1, 2, 3, extend these factorizations to a canonical commutative diagram

N  p1 y i1 N −−−−→ N   0 p  p2 2y y 0 i1 i2 N −−−−→ N −−−−→ N    00 0 p  p p3 3 y y 3 y 00 0 i1 i2 i3 N −−−−→ N −−−−→ N −−−−→ N 0 0 00 0 00 0 with surjective p2, p3, p3 and injective i1, i1, i2, and use (2.25), (2.20) and its adjoint identity to show that all the possible ways to obtain a map

opp∗ opp∗ opp∗ opp∗ opp∗ opp∗ 0 00 opp∗ 00 0 opp∗ τ : i3 ◦ p3 ◦ i2 ◦ p2 ◦ i1 ◦ p1 → (i3 ◦ i2 ◦ i1) ◦ (p3 ◦ p2 ◦ p1) Preliminary version – please do not distribute, use at your own risk 27

from the canonical maps τ−,− give the same result. We leave the details to the reader. Every surjective bounded order-preserving map p : N → N is of the ] ] form (p )] for a unique injective bounded order-preserving map p : N → N. opp∗ opp∗ opp opp opp ]opp∗ Therefore f = i ◦ p]∗ has an adjoint f∗ = i∗ ◦ p , so that (2.26) is a biﬁbration. It remains to check that (2.26) has the adjunction property. We already know it for perfect squares (2.5) with f, f 0, g, g0 injective. For squares with f, f 0 injective, g, g0 injective, it holds by Lemma 2.17. By adjunction, we also obtain the adjunction properties for f, f 0 injective, g, g0 surjective, and for 0 0 0 0 0 f, f , g, g surjective. In the general case, let f = i1◦p1, f = i1◦p1, g = i2◦p2, 0 0 0 0 0 0 0 g = i2 ◦ p2, with surjective p1, p1, p1, p2 and injective i1, i1, i2, i2. Then since b the epi-mono factorization in End (N) is functorial, for any commutative square (2.5) we have a commutative diagram

0 0 p1 i1 N −−−−→ N −−−−→ N    0 00 p  p p2 2y y 2 y 00 00 (2.27) p1 i1 N −−−−→ N −−−−→ N    i0  i00  2y y 2 yi2 p1 i1 N −−−−→ N −−−−→ N 00 00 00 00 with surjective p1, p2 and injective i1, i2. We have 00 00 0 00 00 f] ◦ g = p1] ◦ i1] ◦ ◦i2 ◦ p2 ≤ p1] ◦ i2 ◦ i1] ◦ p2 ≤ i2 ◦ p1] ◦ i1] ◦ p2 0 00 00 0 0 0 0 0 0 0 ≤ i2 ◦ p1] ◦ p2 ◦ i1] ≤ i2 ◦ p2 ◦ p1] ◦ i1] = g ◦ f], and if the square is perfect, all these inequalities become equalities. There- fore all the four small squares in (2.27) are also perfect. Since we already know that all four have the adjunction property, so does the original perfect commutative square (2.5). We finish the section by observing one easy corollary of Theorem 2.8. Let ∆+ be the category of non-empty finite totally ordered sets and maps between them that send the maximal element to the maximal element. We b have a functor δ : ∆+ → End (N) which identifies a set [n] of cardinality n with the initial interval [0, n − 1] ⊂ N, and sends a map f :[n] → [m] to the map fe : N → N given by the following rule: ( f(l), 0 ≤ l < n, fe(l) = l + m − n, l ≥ n. Preliminary version – please do not distribute, use at your own risk 28

By restriction, the biﬁbration (2.8) induces a biﬁbration

∗ b b opp δ Tot(End ( ), D) = Tot(End ( ), D) × b opp ∆ N N End (N) + opp over the category ∆+ . For any n ≥ 0, let \ D≤n = Dl ⊂ D. l≥n where Dn = T2n(D) ⊂ D is the essential image of the fully faithful embedding T2n : D → D, as in Subsection 2.4. Let

∗ b opp P D ⊂ δ Tot(End (N), D)/∆+ be the full subcategory spanned by those pairs hM, [n]i for which M ∈ opp D≤n−1 ⊂ D. Then P D/∆ is also a bifibration. Indeed, for any map f :[n] → [m] and any l ≥ m − 1, we have t2l+1 ◦ fe = fe◦ t2(l+n−m)+1 ◦ fe, opp opp so that the transition functor fe∗ sends Dl = T2l(D) = t(2l+1)∗(D) ⊂ D opp∗ into Dl+n−m ⊂ D, and analogously for fe . In the case D = Fun(N, C) of opp Example 2.5, P D/∆+ is the standard bifibration with fibers

P D[n] = Fun([n], C), where the totally ordered set [n] in the right-hand side is considered as a small category in the usual way.

3 Λ∞-categories.

Assume given a non-degenerate f-category hD, F, si. Then D is an admissible N-category as in Example 2.4, so that by Theorem 2.8, we have the biﬁ- opp bration (2.8) and the biﬁbration P D/∆+ . However, both these structures just imitate the prototype situation D = Fun(N, C) of Subsection 2.1. As it happens, the triangulated structure on D leads to additional symmetries which do not exists for Fun(N, C) (even when C is abelian or triangulated).

3.1 The category Λ∞. The small category Λ∞ has been introduced by B. Feigin and B. Tsygan in [FT] in relation to A. Connes’ cyclic category Λ. For the convenience of the reader, let us briefly recall the definition. Consider the set Z of all integers as a partially ordered set with the usual order. As for the set N, say that an order-preserving map f : Z → Z −1 is proper if f (n) is finite for any n ∈ Z. For any such map f : Z → Z, Preliminary version – please do not distribute, use at your own risk 29

(2.3) with N replaced by Z deﬁnes an adjoint map f] : Z → Z. For Z, unlike for N, the correspondence f 7→ f] is a bijection, with the inverse bijection ] sending f to the map f : Z → Z given by

] 0 0 (3.1) f (n) = max{n ∈ Z | f(n ) ≤ n}.

The category Λ∞ is the category of pairs hS, σi of a partially ordered set ∼ S and an order-preserving automorphism σ : S → S such that S = Z as a partially ordered set, and σ > id. Morphisms in Λ∞ are order-preserving maps which commute with σ. For any positive integer n, let [n] ∈ Λ∞ be Z with the automorphism σ given by σ(l) = l + n. Then any hS, σi ∈ Λ∞ is isomorphic to [n] for some n ≥ 1, so that, although it helps to have an invariant deﬁnition, eﬀectively objects of Λ∞ are just positive integers. Explicitly, for any n, m ≥ 1, the set Λ∞([n], [m]) of maps from [n] to [m] is the set of all order-preserving maps f : Z → Z such that

(3.2) f(l + n) = f(l) + m for any l ∈ Z. Any such map f is automatically proper. Both adjoint ] maps f], f are maps from [m] to [n], so that we have two mutually inverse equivalences

∼ opp (3.3) Λ∞ = Λ∞ .

Another description of the object [n] ∈ Λ∞ is as follows: take the finite totally ordered set [n] of cardinality [n], consider the product Z × [n] with lexicographic order, and let σ : Z×[n] → Z×[n] act as σ(l ×s) = (l +1)×s, l ∈ Z, s ∈ [n]. This defines a canonical functor κ : ∆ → Λ∞ (where ∆, as usual, is the category of non-empty finite totally ordered sets). Applying the adjunction equivalence (3.3), we obtain a faithful essentially surjective functor ] opp j = κ : ∆ → Λ∞ . It is easy to characterize its image: for any [m], [n] ∈ ∆, the subset

∆([n], [m]) ⊂ Λ∞([m], [n]) consists of such maps f : Z → Z that f(0) = 0 in addition to (3.2). Consider the functor ∆+ → ∆ which adds a new ﬁrst element to a totally ordered set [n] ∈ ∆+, and let

ρ : ∆+ → Λ∞ Preliminary version – please do not distribute, use at your own risk 30 be its composition with the canonical embedding κ : ∆ → Λ. In this section, opp we are going to shows that the biﬁbration P D/∆+ extends to the category opp Λ∞ . More precisely, we will construct a biﬁbration

opp ΛD/Λ∞ such that ρ∗ΛD =∼ P D. We will also show that the construction is functorial with respect to admissible morphisms of f-categories.

Remark 3.1. In fact, the adjunction property of Definition 2.1 extends literally to bifibrations over Λopp, and our bifibration ΛD will have it; moreover, it is not too difficult to show that this characterizes ΛD/Λopp uniquely, similarly to Proposition 2.11. We will not do it since we will never need it.

3.2 Unbounded maps. Unfortunately, the maps f : Z → Z used in the deﬁnition of the category Λ∞ are usually not bounded in the sense of Deﬁnition 2.7 (f(l + 1) = l + 1 for l 0 together with f(l + n) = b f(l) + m implies m = n). Thus we need to extend the End (N)-action on D constructed in Section 2 to an action of unbounded maps. We do it using the non-degeneracy assumption of the f-category D. For any n ≥ 0, let D≤n ⊂ D be as in Subsection 1.1. Then by assumption, we have [ (3.4) D = D≤n. n

Recall that for any bounded order-preserving f : N → N, we have deﬁned t(f): N → N by t(f)(0) = 0, t(f)(l) = f(l − 1) + 1 for l ≥ 1. Then for any n such f : N → N and any n ≥ 0, t (f)(l) = l for l = 0, . . . , n − 1, and in Example 1.3, it is obvious that the transition functor tn(f)opp∗ is identical on D≤n ⊂ D. The same is true in the general case; here is the precise formulation.

Lemma 3.2. For any n ≥ 0, the embedding D≤n ⊂ D extends to a commutative square

b opp σn b D≤n × End (N) −−−−→ Tot(End (N), D)     y y b tn b End (N) −−−−→ End (N) Preliminary version – please do not distribute, use at your own risk 31

with Cartesian functor σn, and we can choose these functors σn in such a way that for any m > n, we have a commutative square

m−n b opp Id ×t b opp D≤n × End (N) −−−−−−→D≤n × End (N)   (3.5)  σ y y n b opp σm b D≤m × End (N) −−−−→ Tot(End (N), D).

Proof. The map s : F → Id induces by adjunction a map s0 : Id → R and opp maps s2n+1 : Id → T2n+1 = t2n∗, s2n : T2n → Id for any n ≥ 0. We claim that for any n ≥ 0 and m ≥ 2n, the functor Tm sends D≤n ⊂ D into itself, and for any M ∈ D≤n and m ≥ n, the maps s2m : T2m(M) → M and s2m+1 : M → T2m+1(M) are isomorphisms. Indeed, for n = 0 this has been shown in the proof of Corollary 1.9; if n ≥ 1, then M ∈ D lies in D≤n ⊂ D if n and only if R (M) lies in D≤0, so that it suﬃces to notice that by Lemma 2.9 n ∼ n and adjunction, we have R ◦ Tm = Tm−2n ◦ R for any m ≥ 2n, and these isomorphisms are compatible with the maps s . n b b Now, for any n ≥ 1, the map t : End (N) →q End (N) obviously satisﬁes n n t (f]) = t (f)], thus sends perfect commutative squares (2.5) to perfect commutative square. Therefore the pullback

n∗ b b opp t Tot(Endn(N), D)/End (N) of the cofibration (2.8) also has the adjunction property of Definition 2.1. The corresponding admissible N-category is hD,T2n−1i. The natural embedding D≤n ⊂ D together with the map s2n−1 then define a morphism

σn : hD≤n, Idi → hD,T2n−1i of N-categories, where D≤n is considered as an N-category by equipping it with the identity functor, and since sm is an isomorphism on D≤n for m ≥ 2n, this morphism is admissible in the sense of Definition 2.10. But the bifibration (2.8) of Theorem 2.8 for hD≤n, Idi is obviously trivial. Thus to obtain the desired embedding σn, it suffices to invoke Proposition 2.11. Moreover, the restriction of the isomorphism s2m−1 to D≤n ⊂ D≤m is exactly n opp the isomorphism Id → t (t2(m−n))∗ provided by Proposition 2.11, so that we have the compatibility condition (3.5). Assume now given any proper order-preserving map f : N → N, and for any n ≥ 0, let σn(f): N → N be given by σn(f)(l) = f(l) for l ≤ n, Preliminary version – please do not distribute, use at your own risk 32

and σn(f)(l) = l − n + f(n) for l > n. Then σn(f) is bounded, we have σn(σm(f)) = σn(f) for m ≥ n, and we have a canonical decomposition

n n n f = t (t] ◦ f ◦ t ) ◦ σn(f). where t : N → N is the shift map of Example 2.5, and t] = t1 is its adjoint map. If f itself is bounded, Lemma 3.2 provides a canonical isomorphism

opp∗ ∼ opp∗ f = σn(f) on D≤n ⊂ D. If f is not bounded, we at least have a canonical isomorphism opp∗ ∼ opp∗ σm(f) = σn(f) on D≤n for any m ≥ n. Thus we can set

opp∗ opp∗ f = lim σn(f) on D≤n ⊂ D, ←m>n

opp∗ and this is compatible with the embeddings D≤n ⊂ D≤m. By (3.4), f is then defined on the whole D. For the same reason, we have natural isomorphisms (f ◦ g)opp∗ =∼ f opp∗ ◦ gopp∗ for any two proper order-preserving maps f, g : N → N, they satisfy the compatibiliy condition for triples of such opp maps, and we also have adjoint functors f∗ . In other words, the bifibration (2.8) constructed in Theorem 2.8 extends to a bifibration

opp (3.6) Tot(End(N), D)/End(N) .

3.3 The biﬁbration ΛD: preliminaries. For any n ≥ 0, let D≤n, D≥n ⊂ D be as in (1.4). The functors R,F : D → D send D≤n into D≤n−1, resp. ≤n+1 D . Moreover, equip the functor category Fun(N, C(D)) with an N- category structure as in Example 2.5. Then we have the following general fact.

Lemma 3.3. The comparison functor R : D → Fun(N, C(D)) of (1.5) extends to an admissible morphism of N-categories, and for any proper order- preserving f : N → N, any l ∈ N and any M ∈ D, we have a natural isomorphism

opp∗ ∼ (3.7) Rlf (M) = Rf(l)(M). ∼ Proof. By deﬁnition, we have Rl ◦ R = Rl+1 for any l ≥ 0, and these ∼ opp isomorphisms glue together to a functorial isomorphism ε : R ◦ R = t0∗ , so that hR, εi is a morphism of N-categories in the sense of Deﬁnition 2.10. ∼ ∼ Moreover, since R ◦ T = Id, we have Rl+1 ◦ T = Rl for l ≥ 0, and together with the isomorphism π(s): π(T (M)) =∼ π(M) these isomorphisms give an Preliminary version – please do not distribute, use at your own risk 33

∼ opp isomorphism ε0 : R ◦ T = t1∗ ◦ R adjoint to ε. Assume by induction that for opp some n ≥ 1, we have maps εl : R◦Tl → tl∗ ◦R for l ≤ n, εl is an isomorphism for any l < n, and εl+1 is the base change map associated to εl. To prove that opp hR, εi is admissible, we have to show that εn(M): R(Tn(M)) → tn∗ (R(M)) is an isomorphism for any M ∈ D. If M = T (M 0) for some M 0 ∈ D, this follows from the inductive assumption and Lemma 2.9, so we may assume ≤0 ≤0 M ∈ D . Since D ⊂ D≤1, the claim then follows from Lemma 3.2. We conclude that the morphism hR, εi is indeed admissible. By Propo- sition 2.11, this yields (3.7) for bounded f; since Rn(M) only depends on ≤n ≤n the truncation σ (M) ⊂ D ⊂ D≤n+1, (3.7) extends to the case of an arbitrary proper order-preserving f. Coupled with Lemma 1.7, (3.7) immediately shows that for any proper order-preserving f : N → N and any n ≥ 0, we have

opp ≤f(n) ≤n opp ≥f(n+1) ≥n+1 (3.8) f∗ (D ) ⊂ D , f∗ (D ) ⊂ D .

For any n ≥ 1, the full triangulated subcategory T n(C(D)) = D≥n ∩ D≤n ⊂ D≤n is left-admissible, with orthogonal ⊥(D≥n∩D≤n) = D≤n−1. The natural projection

≤n (3.9) Pn : D → D≤n

n ≤n is an equivalence, and it sends T (C(D)) ⊂ D into D≤0 ⊂ D≤n. Therefore T n(C(D)) ⊂ D≤n is also left-admissible, with orthogonal (D≥n ∩ D≤n)⊥ = −1 ≥n−1 pn (F (D≤n−1)) = F (D ). The projection

≤n−1 ≤n−1 (3.10) Qn : D → F (D ) is then an equivalence of categories. S Consider the ordered set Z as a small category, and let D /Z be cofibration with fibers D, and with transition functor corresponding to a map ιl,l0 , l ≤ l0 given by Rl0−l : D → D. Let Sec(DS) be the category of sections of this cofibration.

Deﬁnition 3.4. The full subcategory

De≤n ⊂ Sec(DS)

S is spanned by such sections E : Z → D that for any l ∈ Z, (i) E(l) ∈ D lies in F (D≤n−1) ⊂ D, and Preliminary version – please do not distribute, use at your own risk 34

(ii) the cone of the transition map

ιl(E): RE(l) → E(l + 1) lies in D≥n ⊂ D.

Lemma 3.5. For any l ∈ Z, evaluation at l gives an equivalence De≤n =∼ F (D≤n−1). Proof. The functor R : D → D induces an equivalence of categories R : ≤n−1 ≤n−1 F (D ) → D ; composing it with the equivalence Qn of (3.10), we obtain an equivalence ≤n−1 ≤n−1 Sn : F (D ) → F (D ). ≤n−1 We also have the adjunction map ψ : R◦i → i◦Sn, where i : F (D ) → D l is the embedding functor. The functors i ◦ Sn, l ∈ Z together with the maps l ψ ◦ Sn glue together to a functor ≤n−1 S Sn : F (D ) × Z → D , q l ≤n−1 ≤n−1 with S = i ◦ Sn on F (D ) × l ⊂ F (D ) × Z. Then the condition (i) of Definitionq 3.4 exactly means that the section E is of the form S ◦ Ee ≤n−1 for some section Ee : Z → F (D ) × Z of the tautological cofibrationq ≤n−1 F (D ) × Z → Z, and the condition (ii) means that this section Ee must be Cartesian, hence constant. For proper order-preserving map f : Z → Z and any l ∈ Z, let τl(f): N → N be the map given by τl(f)(a) = f(l + a) − f(l). This is also a proper order-preserving map. Moreover, for any l0 ≥ l, we obviously have l0−l f(l0)−f(l) τl(f) ◦ t = t ◦ τl0 (f). Therefore we have a compatible system of isomorphisms l0−l opp ∼ opp f(l0)−f(l) R ◦ τl(f)∗ = τl0 (f)∗ ◦ R , opp opp where τl(f)∗ , τl0 (f)∗ : D → D are the transition functors of the bifibration opp∗ (3.6). The adjoint functors τl(f) , l ∈ Z together with the corresponding base change isomorphisms define a functor f opp∗ : DS → DS which fits into a commutative diagram f opp∗ DS −−−−→DS     y y f Z −−−−→ Z. Preliminary version – please do not distribute, use at your own risk 35

On the level of sections, we have a natural functor

opp S S f∗ : Sec(D ) → Sec(D )

S such that for any l ∈ Z and E ∈ Sec(D ), we have a natural isomorphism opp ∼ opp f∗ (E)(l) = τl(f)∗ E(f(l)). These functors are obviously compatible with compositions, so that we have a coﬁbration

S opp (3.11) Tot(Λ∞, Sec(D ))/Λ∞

S opp with ﬁbers Sec(D ) and transtion functors f∗ .

Lemma 3.6. Assume given two integers n, m ≥ 0, and assume that a bounded ordered preserving f : Z → Z is such that f(l+n) = f(l)+m for any opp ≤m S ≤n S l ∈ Z. Then the functor f∗ sends De ⊂ Sec(D ) into De ⊂ Sec(D ).

m opp Proof. We have to show that for any E ∈ De , the pullback f∗ E satisfies the conditions (i), (ii) of Definition 3.4. The first condition immediately follows from the first embedding of (3.8). To check the second condition, take an arbitrary l ∈ Z, let k = f(l + 1) − f(l), and note that the map opp opp ∼ k opp ιl(f∗ E): R ◦ τl(f)∗ E(l) = R ◦ τl+1(f)∗ E(f(l)) → opp → τl+1(f)∗ E(f(l + 1)) factors as opp ιl(f∗ E) = ιk ◦ ιk−1 ◦ · · · ◦ ι2 ◦ ι1, where we denote

k−j opp k−j+1 opp ιj = R τl+1(f)∗ ιf(l)+j−1(E):R τl+1(f)∗ E(f(l) + j − 1) → k−j opp → R τl+1(f)∗ E(f(l) + j) for j = 1, . . . , k. Thus it suﬃces to show that the cone of each of the maps ιj, 1 ≤ j ≤ k, lies in D≥n−1 ⊂ D. This immediately follows from the condition (ii) for E and the second embedding of (3.8).

3.4 The bifibration ΛD: the construction. We can now define our opp cofibration ΛD/Λ∞ . Let

S ΛD ⊂ Tot(Λ∞, Sec(D )) be the full subcategory in the category (3.11) spanned by such objects S ≤n hM, [n]i in Tot(Λ∞, Sec(D )) that M lies in De ⊂ Sec(D). Preliminary version – please do not distribute, use at your own risk 36

opp Lemma 3.7. The projection ΛD → Λ∞ is a biﬁbration.

opp Proof. By Lemma 3.6, the transition functor f∗ of the cofibration (3.11) ≤n S associated to a map f :[n] → [m] sends ΛD[m] = De ⊂ Sec(D ) into ΛD[n]; opp therefore ΛD/Λ∞ is a cofibration. Explicitly, choose an arbitrary l ∈ Z, and apply Lemma 3.5 at l and f(l) to identify the fibers ΛD[n],ΛD[m] with F (D≤n−1) ⊂ D≤n, resp. F (D≤m−1) ⊂ D≤m. Then the transition functor opp f∗ is induced by the transition functor

opp τl(f)∗ : D≤n → D≤m of the biﬁbration (3.6). Since τlf : N → N sends 0 ∈ N to 0 ∈ N, so does the adjoint map τlf]; moreover, since τl(f)(i + n) = τl(f)(i) + m for any i ≥ 0, we have τl(f)](i + m) = τl(f)](i) + n. Therefore the right-adjoint functor opp∗ ∼ opp opp τlf = τlf]∗ sends F (D≤m−1) ⊂ D into F (D≤n−1), so that ΛD/Λ∞ is a biﬁbration.

Lemma 3.8. There exists a Cartesian diagram

ρ0 P D −−−−→ ΛD     y y ρ opp ∆+ −−−−→ Λ∞ with Cartesian functor ρ0.

Proof. By deﬁnition, for any map f :[n] → [m] in the category ∆+, the map ρ(f): Z × [n + 1] → Z × [m + 1] sends 0 = 0 × 0 ∈ Z × [n + 1] to 0 ∈ Z × [m + 1]. Therefore the equivalences 0 ∗ ≤n+1 ∼ ≤n (3.12) δ[n] :(ρ ΛD)[n] = ΛD[n+1] = De = F (D ) ⊂ D

opp of Lemma 3.5 at l = 0 are compatible with the transition functors ρ(f)∗ , so that we have a commutative diagram

0 δ ∗ ρ Tot(End(N), D) ←−−−− ρ ΛD −−−−→e ΛD       y y y opp t◦δ opp ρ opp End(N) ←−−−− ∆+ −−−−→ Λ∞ Preliminary version – please do not distribute, use at your own risk 37

0 0 with Cartesian δ , ρe, where δ is given by (3.12) on the ﬁber over [n] ∈ ∆+. For any [n] ∈ ∆+, the equivalence Pn of (3.9) induces an equivalence

≤n Pn ◦ R : F (D ) → D≤n.

Since t ◦ δ(f) = (t ◦ δ)(f) ◦ t, the inverse equivalences glue together to give 0 ∗ 0 0 a Cartesian functor ρe : P D → ρ ΛD. We set ρ = ρe◦ ρe : P D → ΛD.

3.5 Functoriality. Assume now given two non-degenerate f-categories hD, F, si, hDe, F,e sei, and an admissible morphism hE, εi, E : D → De between them in the sense of Definition 1.12. Let R, Re be the right-adjoint functors to F resp. Fe. Then in particular, D and De are admissible N-categories in the sense of Definition 2.3, as in Example 2.4, and Lemma 1.11 (iii) gives an 0 ∼ 0 isomorphism ε : Re ◦E = E ◦R, so that hE, ε i is a morphism of N-categories in the sense of Definition 2.10.

Lemma 3.9. The morphism hR, ε0i is admissible in the sense of Deﬁni- tion 2.10.

Proof. We have to prove by induction that the maps εn : E ◦ Tn → Ten ◦ E adjoint to ε0 are isomorphism. Since hE, εi is an admissible morphism of f-categories, it commutes with the comparison functors R, Re of (1.5). 0 The pair hC(E), Re(ε )i is a morphism of N-categories from Fun(N, C(D)) to Fun(N, C(De)), and this morphism is obviously admissible. Therefore Re(εn) is an isomorphism. But since Re is conservative by Lemma 1.7, εn itself must be an isomorphism.

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Steklov Math Institute Moscow, USSR

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