A categorified Dold-Kan correspondence

In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. The result may be regarded as a contribution to the foundations of an emerging subject that could be termed categorified homological algebra.


Introduction
A central tool in classical homological algebra is the construction of a chain complex from a simplicial abelian group via the formula The fact that a large number of interesting complexes arise via this procedure is not a coincidence -the classical Dold-Kan correspondence [Dol58,Kan58] states that the passage to normalized chains establishes an equivalence of categories: We refer the reader to §3.1 and §3.2 for an explication of the terminology and a precise statement of the theorem. A key ingredient of the proof is an explicit construction of the categorified Dold-Kan nerve N. Its classical counterpart N associates to a chain complex B • of abelian groups the simplicial abelian group N(B • ) which can be described as follows: the group of n-simplices is given by collections {x σ }, parametrized by monotone maps σ : [k] ֒→ [n], of elements x σ ∈ B k subject to the equations
Note that the data of a 2-simplex in N(B • ) defines, upon passage to classes in the respective Grothendieck groups, a 2-simplex in N(K 0 (B • )). This observation generalizes to simplices in all dimensions and is a justification for the use of the term categorification.
The categorified Dold-Kan nerve N unifies various known constructions from algebraic K-theory:  ) is an ∞-categorical version of Waldhausen's S • -construction (cf. [Wal85]). Waldhausen's S • -construction is usually considered as a simplicial object in the category Cat (or Cat ∞ ). While the additional 2-functoriality present in our treatment does not seem to appear explicitly in the literature, it does feature implicitly, yet crucially, in Waldhausen's proof of the additivity theorem [Wal85] and its modification for stable ∞-categories presented in [Lur14]. This observation will be explored in detail elsewhere.
(II) Let f : B 1 → B 0 be an exact functor of stable ∞-categories. Then applying N to the complex B 0 concentrated in degrees {0, 1} yields an ∞-categorical version of Waldhausen's relative S • -construction. Besides its appearance in Waldhausen's own work, this construction also features prominently in [DKSS17] where it provides a local description of perverse schobers (cf. [KS14]).
(III) Let B be a stable ∞-category and let B[2] denote the chain complex ) is an ∞-categorical version of Hesselholt-Madsen's S 2,1 • -construction. As explained in [HM13], a duality on the category B furnishes N(B[2]) with the structure of a real object which can then be utilized to upgrade the K-theory spectrum of B to a genuine Z/(2)-equivariant spectrum.
(IV) Let B be a stable ∞-category and let B[k] denote the chain complex with B concentrated in degree k.
• -construction introduced for abelian categories in [Pog17]. These higher-dimensional Waldhausen constructions have an interesting interpretation in the context of higher algebraic K-theory: Let us denote by A ≃ the Kan complex obtained from an ∞-category A by discarding noninvertible morphisms. Then, for every k ≥ 1, there is a canonical weak equivalence of spaces The kth cohomology group of a topological space X with coefficients in an abelian group B can be described as homotopy classes of maps from X to K(B, k). The interpretation of the 2-simplicial stable ∞-category N(B[k]) as a categorified Eilenberg-MacLane space predicts the existence of a categorified notion of cohomology. This circle of ideas will be explored in future work with a view towards applications to topological Fukaya categories. For k = 1, the results of [DK17,Dyc17b] can be interpreted as describing the topological Fukaya category of a marked Riemann surface as categorified a relative 1st cohomology group.
Acknowledgements. It is a pleasure to thank Rune Haugseng, Gustavo Jasso, Dima Kaledin, Mikhail Kapranov, Jacob Lurie, Thomas Nikolaus, Thomas Poguntke, Vadim Schechtman, Ed Segal, and Nicolo Sibilla for inspiring conversations that helped shape my perspective on the material presented in this work. Specifically, I would like to thank Mikhail Kapranov and Vadim Schechtman for many discussions on perverse schobers, which are one of the sources of inspiration for this work, and I am indebted to Thomas Nikolaus for conversations which convinced me to focus on 2-categorical methods.

The classical Dold-Kan correspondence
Let Ab denote the category of abelian groups. The Dold-Kan correspondence establishes an equivalence between the category Ab ∆ of simplicial abelian groups and the category Ch ≥0 (Ab) of connective chain complexes of abelian groups. We present a particular proof of this result which is designed so that the proof of our main result, provided in §3, can be regarded as a step-by-step categorification of the involved arguments.
Let A • be a simplicial abelian group. The associated chain complex ( and D n ⊂ A n is the subgroup generated by the degenerate n-simplices. For each element j = (j 1 , . . . , j n ) of the cube {0, 1} n , we consider setting j 0 = 1, and introduce the map Proposition 2.2. Let n ≥ 0, and let D n ⊂ A n be the subgroup generated by the degenerate simplices. Then the map π : A n → A n is a retraction onto A n with kernel D n . In particular, it induces an isomorphism Proof. To show that im(π) ⊂ A n , we observe that, for 0 < i ≤ n, the face map d i maps the two faces of the cube {f * j a} that are orthogonal to the ith coordinate direction to the same (n − 1)-dimensional cubes in A n−1 . Since the contributions of these faces in the formula for π appear with opposite signs, we obtain, for every a ∈ A n , d i π(a) = 0. A similar argument shows that D n ⊂ ker(π): for a = s i a ′ , we have π(a) = 0 since the opposing faces orthogonal to the ith coordinate direction of the cube {f * j a} cancel in formula (2.1). We show that π is a retraction. Let a ∈ A n . For j = (1, . . . , 1), the map f j factors through some face map ∂ i : [n − 1] → [n], i > 0, so that f * j a = 0. Since f (1,...,1) = id, we have π(a) = a so that π is a retraction. Finally, formula (2.1) implies that A n = A n + D n which in combination with the statements established above yields A n = A n ⊕ D n so that D n = ker(π).
Corollary 2.3. We have an isomorphism of complexes where the projection onto the first summand is given by π.
We consider the functor Theorem 2.4. The adjunction is a pair of inverse equivalences.
Proof. We analyze the counit of the adjunction C(N(B • )) → B • . An n-simplex in N(B • ) is given by a collection {b σ } and the counit maps this collection to b id ∈ B n . The condition that {b σ } be a normalized chain translates to the requirement that b σ = 0 for all σ that factor through one of the face maps ∂ i , i > 0. But this implies that the only possibly nonzero elements are b id and b ∂ 0 . Further, the element b ∂ 0 is determined as the image of b id under d. Therefore, the counit is an isomorphism. The unit u : A • → N(C(A • )) is given by associating to a in A n the n-simplex in N((A • , d)) given by the collection {a σ } with a σ = σ * a and then postcomposing with the map N(π) : By an argument similar as for the counit, it is immediate that C(u) is an isomorphism. We conclude the proof in virtue of Proposition 2.5 below.
Proposition 2.5. The normalized chains functor C is conservative: a morphism f : We show by induction on n ≥ 0 that, for every map f : A • → A ′ • such that C(f ) is an isomorphism of chain complexes, the map f n : A n → A ′ n is an isomorphism. For n = 0, the claim is apparent. Assume the induction hypothesis holds for a fixed n ≥ 0 and all maps of simplicial abelian groups. For a given map f : • , consider the diagram simplicial abelian groups where the horizontal sequences are short exact. Then the induction hypothesis implies that f n and f n are isomorphisms so that P(f ) n = f n+1 is an isomorphism as well.

The categorified Dold-Kan correspondence
In this section, we prove the main result of this work: a categorification of the Dold-Kan correspondence relating simplicial objects and connective chain complexes with values in the category of stable ∞-categories. We begin by defining these notions in detail.

Basic definitions
3.1.1 Model for (∞, 2)-categories Let Set ∆ denote the category of simplicial sets. Following [Lur09a], we define an ∞category to be a simplicial set satisfying the inner horn filling conditions. We define Cat ∞ to be the full subcategory of Set ∆ spanned by the ∞-categories. ∞-categories are the fibrant objects of a model structure on the category of marked simplicial sets Set + ∆ with marked edges given by the equivalences. As explained in [Lur09b], the category of Set + ∆ -enriched categories carries a model structure which can be regarded as a model for the theory of (∞, 2)-categories. The (∞, 2)-categorical structures that appear in this work will be organized within this model and related to other models via the theory developed in [Lur09b]. The fibrant objects within this model structure can be identified with the Cat ∞ -enriched categories.

Stable ∞-categories
The simplicial set of functors between a pair of ∞-categories forms another ∞-category so that Cat ∞ becomes a Cat ∞ -enriched category with respect to the Cartesian monoidal structure. Marking equivalences in the various functor ∞-categories, Cat ∞ becomes a fibrant object in the model category of Set + ∆ -enriched categories from 3.1.1. We may therefore interpret Cat ∞ as a specific model for the (∞, 2)-category of ∞-categories. We further denote by St ⊂ Cat ∞ the Cat ∞ -enriched subcategory with stable ∞-categories as objects and functor ∞-categories spanned by the exact functors in the sense of [Lur11].

The simplex 2-category
By a 2-category, we mean a category enriched in categories. A 2-category defines a Set + ∆ -enriched category by passing to nerves of the enriched mapping categories and marking equivalences. We will typically leave this passage implicit so that, referring to 3.1.1, we may consider any 2-category as an (∞, 2)-category. We denote by Cat the 2-category of small categories and by ∆ ⊂ Cat the full 2-subcategory spanned by the standard ordinals {[n]}, considered as categories.

Localization
We will construct ∞-categories via localization: Given a small category C and a set of morphisms W , there is an associated ∞-category L W C, equipped with a functor N(C) → L W C universal among all functors that send W to equivalences (cf. [Lur11, 1.3.4]).

2-simplicial stable ∞-categories
We denote by St ∆ the category of Cat ∞ -enriched functors from the opposite 2-simplex category ∆ (op,−) to St. This category comes equipped with a collection of weak equivalences given by those Cat ∞ -enriched natural transformations that are levelwise equivalences of stable ∞-categories. Via localization, we obtain a corresponding ∞-category L(St ∆ ) of 2-simplicial stable ∞-categories.
The additional data captured by the 2-functoriality of A • contains unit and counit transformations which exhibit a sequence of adjunctions

Connective chain complexes of stable ∞-categories
We denote by N the poset of nonnegative integers, considered as a category. We denote by Fun(N op , St) the category of (strict) functors from the opposite category of N to the category St of stable ∞-categories. We introduce the full subcategory given by those diagrams of stable ∞-categories that satisfy the following condition: for every i ≥ 0, the functor comes equipped with a class of weak equivalences given by those natural transformations that are levelwise equivalences. We refer to the corresponding localization L Ch ≥0 (St) as the ∞-category of connective chain complexes of stable ∞-categories.
Remark 3.2. At first sight, our definition of a complex of stable ∞-categories may seem too naive. For example, the analogous definition of a connective complex of objects in a stable ∞-category A as a functor satisfying the condition d 2 ≃ 0 really is too naive. The reason is that, for every i ≥ 0, there is a potentially nontrivial space of paths in the Kan complex Map A (X i+2 , X i ) from d 2 to 0. Following general principles, the condition d 2 ≃ 0 should be replaced by the choice of a path in Map A (X i+2 , X i ) between d 2 and 0. Further, these choices are supposed to be part of a coherent system of trivializations d n ≃ 0, n ≥ 2, corresponding to trivializations of higher Massey products. This coherence data is important: it is, for example, needed to form the totalization of a complex. One way to codify all this data is to remember the complex in terms of the filtered object formed by the totalizations of its various truncations. This is the point of view taken in [Lur11] where a connective complex of objects in a stable ∞-category corresponds to a functor without any further conditions. The actual terms of the complex captured by such a datum are then given as shifts of the cofibers of the maps Y i → Y i+1 . One concrete justification for this being a reasonable notion of a complex is provided by a Dold-Kan correspondence relating simplicial objects and connective chain complexes with values in a given stable ∞-category ([Lur11, 1.2.4]).
In contrast, given a chain complex of stable ∞-categories in our sense, the space of identifications d 2 ≃ 0 is contractible, since it is the space of equivalences between d 2 and the zero object 0 in the ∞-category Fun(B i+2 , B i ). Therefore, in this context, there is no analog of the coherent system of trivializations captured by (3.3).

Statement of the theorem
Using the terminology introduced in §3.1, we may formulate the main theorem.
Theorem 3.4. There exist functors and which induce a pair of inverse equivalences Remark 3.7. The classical Dold-Kan correspondence generalizes to categories that are additive and idempotent complete. While the (∞, 2)-category St does not have direct analogs of these two properties, it does admit certain categorified variants: (i) Given two functors f and g between stable ∞-categories A and A ′ , equipped with a natural transformation η : f ⇒ g, we may form the cone of η as a replacement for the difference of two maps.
(ii) Given an idempotent e : A → A that arises from a fully faithful embedding i : A ′ ⊂ A with right adjoint q : A → A ′ as e = i•q, then there is a canonical natural transformation e ⇒ id A whose cone will be a projector onto the subcategory ker(q) ⊂ A. Together, the subcategories ker(q) and A ′ form a semiorthogonal decomposition of A (cf. [BK89]).
Our proof of Theorem 3.4 relies on a systematic utilization of these features of St.
Abelian categories form a pleasant class of categories for which the Dold-Kan correspondence holds. It is an interesting task to introduce a suitable categorified axiomatic framework of "2-abelian" (∞, 2)-categories. One basic requirement would be that the proof of the categorified Dold-Kan correspondence generalizes to this context.
Before we construct the functors in Theorem 3.4 and provide its proof, we need some preliminaries on Grothendieck constructions.

Grothendieck constructions
Let C be a category and let Grp denote the category of small categories. Recall that, given a functor The construction provides a functor into the category Fib /C of categories fibered in groupoids over C. The functor χ has a left adjoint Ξ given by and a right adjoint Γ provided by Upon localizing the categories Fun(C op , Grp) and Fib /C along suitable classes of weak equivalences, the functor χ becomes an equivalence with inverse provided by both Ξ and Γ. As a central tool in his approach to higher category theory, Lurie has introduced various generalizations of the adjunction in the context of Quillen's theory of model categories. In this work, we will use various explicit versions of these Grothendieck constructions and a description of their inverses in terms of generalizations of the right adjoint functor Γ. In this section, we survey the theory in the form it will be applied below.

The (∞, 1)-categorical Grothendieck construction
Let C be an ordinary category and let be a functor into the category Cat ∞ of ∞-categories, considered as a full subcategory of the category Set ∆ of simplicial sets. The simplicial set χ(F ) comes equipped with an apparent forgetful map χ(F ) → N(C). We consider χ(F ) as an object of Set + ∆/ N(C) marking the Cartesian edges, and refer to it as the Grothendieck construction of F .
As a consequence of results in [Lur09a], the Grothendieck construction induces an equivalence of ∞-categories obtained by localizing along levelwise and fiberwise categorical equivalences. The righthand symbol • signifies that we restrict to the full subcategory of Cartesian fibrations with Cartesian edges marked. We will use an explicit model for an inverse equivalence to χ which we now construct. For an object c ∈ C, we denote by C /c the overcategory of c. Keeping track of the forgetful functor C /c → C, the various overcategories organize to define a functor Given an object X ∈ Set + ∆/ N(C) , we thus obtain a functor Proposition 3.9. There is a natural transformation which is a levelwise weak equivalence. In particular, upon localization, the functor Γ defines an inverse to χ.
Proof. For a functor F : C op −→ Cat ∞ and c ∈ C, we provide a map of simplicial sets which, by adjunction, can be identified with a map of simplicial sets over N(C). We define this map by associating to an n-simplex (x, c 0 → c 1 → · · · → c n → c) of F (c) × N(C /c ) the n-simplex of χ(F ) that is given by (1) the n-simplex σ = c 0 → · · · → c n in N(C) (2) for I ⊂ [n], the I-simplex of F (σ(min(I))) obtained as the composite This association is natural in c and provides the value of the transformation η at F . To show that η is a weak equivalence, we need to show that, for every F : C op → Cat ∞ and for every c ∈ C, the map η F (c) from (3.11) is an equivalence of ∞-categories.
To this end, we note that there is a natural evaluation map

2-categorical terminology
Let C be a 2-category. We denote the category of morphisms between objects x and y by C(x, y). Example 3.12. The 2-category Cat of small categories admits a self-duality We introduce 2-categorical versions of undercategories and overcategories. Since there is potential confusion with the orientation of the 2-morphisms, we provide explicit descriptions of both. For an object c in a 2-category C, we define the lax undercategory C c/ with • objects given by 1-morphisms c → x, • a 1-morphism from ϕ : c → x to ψ : c → y consists of a 2-commutative triangle x y • a 2-morphism γ from f to g is given by a 2-commutative diagram For every object c in C, we define the lax overcategory C /c with • objects given by 1-morphisms x → c, • a 1-morphism from ϕ : x → c to ψ : y → c consists of a 2-commutative triangle x y c f ϕ ψ where f is a 1-morphism in C, • a 2-morphism γ from f to g is given by a 2-commutative diagram x y c. In this section, we introduce a combinatorial variant of the unstraightening functor for (∞, 2)-categories introduced in [Lur09b]. It can be applied to any strict 2-functor where C is a 2-category. It is a lax analog of the relative nerve construction, provided in [Lur09a] as a combinatorial alternative to the straightening construction applicable to strict functors C → Cat ∞ where C is an ordinary category. For every nonempty finite linearly ordered set I, we define a 2-category O I as follows: • The set of objects of O I is I, • The category O I (i, j) of morphisms between objects i and j is the poset consisting of those subsets S ⊂ I satisfying min S = i and max S = j.
• The composition law is given by the formula The various 2-categories O [n] , n ≥ 0, assemble to a functor ∆ −→ Cat 2 (3.13) into the category Cat 2 of 2-categories.
Definition 3.14. Let C be a 2-category. We define the nerve N sc (C) of C to be the scaled simplicial set with N sc (C) n = Fun(O [n] , C) with functoriality in n provided by (3.13). The thin 2-simplices are the ones that correspond to invertible natural transformations. N(C(σ(min(I)), σ(min(J))) op ) × F (σ(min(J))) F (σ(min(I))) commutes where the top row is obtained from (3.15) by passing to nerves.
By construction, the lax Grothendieck construction comes equipped with a forgetful functor π : χ(F ) → N sc (C). We further introduce a marking on χ(F ) consisting of the π-Cartesian edges so that we have χ(F ) ∈ Set + ∆/ N sc (C) .
We summarize some basic properties whose, in parts somewhat technical, proofs are deferred to [Dyc17a]. (2) For every Cat-enriched functor D → C, we have (3) Suppose that C is a 2-category with discrete morphism categories which we may therefore identify with a 1-category. Then restriction along the functor between the ordinary and lax Grothendieck constructions which is a fiberwise equivalence of Cartesian fibrations.
Let C be a 2-category and c an object of C. Consider the lax overcategory C /c as defined in §3.3.2. The forgetful functor C /c → C induces a map N sc (C /c ) → N sc (C) of simplicial sets. We further introduce a marking on N sc (C /c ) given by those edges where the corresponding 2-morphism is an isomorphism. Thus, we have N sc (C /c ) ∈ Set + ∆/ N sc (C) . The functoriality of this construction in c is captured by a Cat ∞ -enriched functor We now define for X ∈ Set + ∆/ N sc (C) and c ∈ C, the marked simplicial set Pulling back (3.19), this construction is functorial in c and defines a functor Γ(X) : C (op,op) −→ Set + ∆ . The additional functoriality in X provides

The categorified normalized chains functor C
We provide the definition of the functor C from Theorem 3.4 whose construction is a mutatis mutandis modification of the normalized chains functor appearing in the classical Dold-Kan correspondence.
Definition 3.21. Given a 2-simplicial stable ∞-category A • , we define, for n ≥ 0, the stable ∞-category A n ⊂ A n as the full subcategory spanned by those vertices a of A n such that, for every 1 ≤ i ≤ n, the object d i (a) is a zero object in A n−1 . As a result of the relations among the face maps in ∆, the various functors d 0 : A n → A n−1 , n > 0, restrict to define a chain complex of stable ∞-categories. We denote this chain complex by C(A • ) and refer to it as the categorified normalized chain complex associated to A • .
The construction A • → C(A • ) on objects extends to define a functor which preserves weak equivalences.

The categorified Dold-Kan nerve N
We provide a definition of the functor N from Theorem 3.4. It can be regarded as a categorification of the classical Dold-Kan nerve appearing in §2. (1) f : where we set j 0 = 1, We define, for every n ≥ 0, the ∞-category as the full subcategory spanned by the diagrams that satisfy the following conditions: (N1) For every k ≥ 1 and every degenerate k-simplex τ : [k] → [n] of ∆ n , the object A τ is a zero object in the ∞-category B k .
(N2) For every k ≥ 1 and every nondegenerate k-simplex σ : obtained by restricting A to the pullback of σ along the canonical cube from Definition 3.24, is a π-limit diagram with limit vertex (0, 0, . . . , 0). To gain familiarity with the categorified Dold-Kan nerve, we provide an explicit description of the low-dimensional simplices of N(A • ) for a given chain complex B • of stable ∞-categories: (1) The ∞-category of 1-simplices of N(B • ) is equivalent to the ∞-category of diagrams of the form in B 0 induced by A is biCartesian so that it exhibits the object d(A 01 ) as the cofiber of the map A 0 → A 1 .
(2) The ∞-category N(B • ) 2 is equivalent to the ∞-category of diagrams in (b) A 00 and A 11 are zero objects in B 1 , A 001 , A 011 , and A 002 are zero objects in B 2 , (c) the diagram A exhibits the objects d(A 01 ), d(A 02 ), and d(A 12 ) as cofibers of the morphisms A 0 → A 1 , A 0 → A 2 , and A 1 → A 2 , respectively, as detailed in (1). In particular, by the octahedral lemma, the square is biCartesian.
in B 1 induced by the diagram A is biCartesian so that it exhibits the object d(A 012 ) as a totalization of the 3-term complex A 01 → A 02 → A 12 .
(n) Similarly, the higher-dimensional simplices of N(B • ) consist of collections of diagrams in B k parametrized by the various posets of k-simplices of ∆ n , together with additional compatibility data that, for every nondegenerate k-simplex σ in ∆ n , exhibits the object d(A σ ) as the totalization of a (k + 1)-term complex with underlying sequence of maps Our next goal is to study special cases of the categorified Dold-Kan nerve and exhibit how they relate to previously studied constructions.
• the objects {A ii } are zero objects in B 1 , • for every 0 ≤ i < j ≤ n, the diagram • for every 0 ≤ i < j < k ≤ n, the diagram In this special case, the categorified Dold-Kan nerve N(B • ) can thus be regarded as an ∞-categorical variant of Waldhausen's relative S • -construction associated to the functor d : B 1 → B 0 of stable ∞-categories. In particular, for B 0 = 0, we recover Waldhausen's S • -construction of the stable ∞-catgeory B 1 . (1) For every degenerate k-simplex τ : [k] → [n], the object A τ is a zero object in B.
As where N /n denotes the overcategory 0 → 1 → · · · → n of n in the poset N, induces an equivalence C(N(B • )) n ≃ Γ(χ(B • )) n . Since this map is functorial in n and B • , we obtain a natural weak equivalence Proof. Immediate from Proposition 4.1 and Proposition 3.9.

The equivalence N • C ≃ id
We proceed by showing N • C ≃ id. To this end, we produce a zigzag diagram Proposition 4.4. There is a weak equivalence Proof. This is an immediate consequence Proposition 3.17(3).

The functor F
To simplify notation, we introduce ∆ ′ = ∆ (−,op) . For n ≥ 0, we introduce the pushout is a biCartesian cube in the fiber π −1 ([k]) where f denotes the cube from Definition 3.24.
Proof. This follows immediately from the fact that there exists an exact triangle Collecting all results of this section, we obtain the following main result: Theorem 4.8. There is a natural equivalence Proof. The various results of this section imply the existence of a diagram of natural weak equivalences which leads to the desired conclusion after localization.

The functor C is conservative
Proposition 4.9. The categorified normalized chains functor is conservative: a morphism f in St ∆ is a weak equivalence if and only if C(f ) is a weak equivalence.
Proof. The proof is a step-by-step categorification of the proof of Proposition 2.5. Given a 2-simplicial stable ∞-category A • , we introduce its path object P(A • ), which is the 2-simplicial object obtained by pullback along the 2-functor The values of the path object are given by P(A • ) n = A n+1 . The various omitted face maps d n : A n → A n−1 define a natural map of 2-simplicial stable ∞-categories d : P(A • ) → A • . For every n ≥ 0, we denote by Ω(A • ) n the full subcategory of A n+1 spanned by the objects X such that d n+1 (X) is a zero object in A n . We obtain a 2-simplicial stable ∞-category Ω(A • ) which is part of a sequence We now proceed by showing the following statement by induction on n: (I) Let n ≥ 0. Then, for every map f : A • → A ′ • of 2-simplicial stable ∞-categories, such that C(f ) is a weak equivalence, the map f n : A n → A ′ n is an equivalence of stable ∞-categories.
The statement is obvious for n = 0, since C(f ) 0 = f 0 . Assume that (I) holds for a fixed n ≥ 0. Given a map A • → A ′ • , we consider the commutative diagram Evaluating the diagram at [n] ∈ ∆, we obtain the diagram n Ω(f )n d n+1 f n+1 fn d n+1 (4.11) of stable ∞-categories. By induction hypothesis, the functor f n is an equivalence. Further, by (4.10), we have that C(Ω(f )) = Ω(C(f )) is a weak equivalence so that, again by induction hypothesis, the functor Ω(f ) n is an equivalences. Note that the right square in (4.11) can be completed to a commutative diagram where d n and s n denote the respective face and degeneracy maps and we leave the 2categorical data implicit. We deduce that f n+1 is an equivalence by Lemma 4.13. This concludes the proof of (I) and the lemma. of stable ∞-categories with sp = sq = id A , s ′ p ′ = s ′ q ′ = id A ′ so that these identities are counits and units, respectively, of adjunctions p ⊣ s ⊣ q and p ′ ⊣ s ′ ⊣ q ′ .
Set B = ker(q) and B ′ = ker(q ′ ). Suppose that the induced functors f : A → A ′ and g : B → B ′ are equivalences. Then the functor g is an equivalence.
Proof. Consider the relative nerve π : N s (∆ 1 ) −→ ∆ 1 of the functor ∆ 1 → Cat ∞ determined by s : A → X (cf. [Lur09a, 3.2.5.12]). Since s has the right adjoint q, the coCartesian fibration π is Cartesian as well so that we have an equivalence X ≃ Map # ∆ 1 (∆ 1 , N s (∆ 1 )) with the ∞-category of Cartesian sections of N s (∆ 1 ). The latter ∞-category can be identified with the full subcategory of Fun(∆ 1 , X) spanned by the counit edges, i.e., edges equivalent to s(q(X)) → X. Here, an edge e is a counit edge if and only if q(e) is an equivalence in A. But this is in turn equivalent to the statement that the cofiber of e lies in B = ker(q). Consider the ∞-category X(B, A) of diagrams A X 0 0 B A ′ in X where A ∈ A, B ∈ B, and both squares are biCartesian (which implies A ′ ∈ A). The above discussion implies that the evaluation map at X establishes an equivalence of ∞-categories X(B, A) ≃ X. Let Map X (B, A) ⊂ Fun(∆ 1 , X) denote the full subcategory spanned by those edges e in X so that d 1 (e) is a vertex in B and d 0 (e) is a vertex in A.
Note that Map X (B, A) can be identified with the ∞-category of sections of θ. The assumption that p is a left adjoint to s implies that the map θ is a coCartesian fibration where a section e is a coCartesian edge if and only if p(e) is an equivalence in A.
We have thus produced a diagram of equivalences of ∞-categories The diagram (4.14) induces a map t : Y → Y ′ that preserves coCartesian edges and is a fiberwise equivalence. By [Lur09a, 3.3.1.5], it follows that t itself is an equivalence. We conclude by noting the commutative diagram where all horizontal arrows are equivalences and, since t is an equivalence, the rightmost vertical map is an equivalence. By the two-out-of-three property the leftmost arrow f is an equivalence as well.
Remark 4.15. The equivalence appearing in the proof of Lemma 4.13 admits the following interpretation: the stable ∞category X comes equipped with a semiorthogonal decomposition X = B, A satisfying a certain admissibility condition. In this situation, the equivalence (4.16) shows that the ∞-category X can be recovered from the two components B and A of the decomposition together with the gluing functor p : B → A, using the terminology of [BK89].