K-theory and polynomial functors
aa r X i v : . [ m a t h . K T ] F e b K -THEORY AND POLYNOMIAL FUNCTORS CLARK BARWICK, SAUL GLASMAN, AKHIL MATHEW, AND THOMAS NIKOLAUS
Abstract.
We show that the algebraic K -theory space of stable ∞ -categoriesis canonically functorial in polynomial functors. Introduction
The purpose of this note is to provide an additional structure on the higheralgebraic K -theory of stable ∞ -categories, arising from polynomial rather thanexact functors.In the case of the Grothendieck group K , the construction is due to Dold [Dol72]and Joukhovitski [Jou00]. Let A be an additive category. The group K ( A ) isdefined to be the group completion of the additive monoid of isomorphism classesof objects of A . By construction, an additive functor F : A → B induces a map ofabelian groups K ( A ) → K ( B ).The results of loc. cit. provide additional functoriality on the construction K ,and show that if F : A → B is merely a polynomial functor in the sense of [EML54],then F nevertheless induces a canonical map of sets F ∗ : K ( A ) → K ( B ), such that F ∗ carries the class of an object x ∈ A to the class of F ( x ) ∈ B . This polynomialfunctoriality yields, for example, the λ -operations on K ( R ) for a commutative ring R , which arise from the exterior power operations on R -modules: the i th exteriorpower functor V i induces a polynomial endofunctor on finitely generated projective R -modules, and hence a map of sets λ i : K ( R ) → K ( R ). Here we will extend thispolynomial functoriality to higher algebraic K -theory. To do this, it is convenientto use the setup of the K -theory of stable ∞ -categories.Let C be a stable ∞ -category. As in [BGT13, Bar16], one constructs an algebraic K -theory space K ( C ) via the Waldhausen S • -construction applied to C ; an exactfunctor C → D of stable ∞ -categories induces a map of spaces K ( C ) → K ( D ).For example, when C = Perf( X ) is the stable ∞ -category of perfect complexesover a quasi-compact and quasi-separated scheme X , this is the K -theory space of X (introduced in [TT90], or [Qui72] if X is affine). Moreover, one characterizes[BGT13, Bar16] the construction C 7→ K ( C ), when considered as an invariant of allstable ∞ -categories and exact functors between them, via a universal property.In this paper, we provide additional structure on the construction of algebraic K -theory in analogy with the results on K from [Dol72, Jou00], and characterizeit by the same universal property.To formulate the result, let Cat perf ∞ denote the ∞ -category of small, idempotent-complete stable ∞ -categories and exact functors between them, and let S be the Date : February 2, 2021. ∞ -category of spaces. Algebraic K -theory defines a functor K : Cat perf ∞ → S . It receives a natural transformation from the functor ι : Cat perf ∞ → S which carries C ∈
Cat perf ∞ to the space of objects in C , i.e., we have a map ι → K of functorsCat perf ∞ → S . The universal property of K -theory [BGT13, Bar16] states that K isthe initial functor Cat perf ∞ → S receiving a map from ι such that K preserves finiteproducts, splits semiorthogonal decompositions, and is grouplike.Let C , D be small, stable idempotent-complete ∞ -categories. A functor f : C → D is said to be polynomial if it is n -excisive for some n in the sense of[Goo92]. Let Cat poly ∞ denote the ∞ -category of small, idempotent-complete stable ∞ -categories and polynomial functors between them. Thus, we have an inclusionCat perf ∞ → Cat poly ∞ ; note that Cat perf ∞ , Cat poly ∞ have the same objects, but Cat poly ∞ has many more morphisms. Our main result states that K -theory can be definedon Cat poly ∞ . Theorem 1.1.
There is a canonical extension of the functor K : Cat perf ∞ → S to afunctor e K : Cat poly ∞ → S . In fact, the construction e K is characterized by a similar universal property.Namely, one has a canonical extension of the functor ι : Cat perf ∞ → S to a functor ι : Cat poly ∞ → S since ι can be defined on all ∞ -categories and functors betweenthem. One defines the functor e K (with a natural map ι → e K ) by enforcing thesame universal property on Cat poly ∞ . The main computation one then carries out isthat e K restricts to K on Cat perf ∞ , i.e., one recovers the original K -theory functor. Remark 1.2.
Theorem 1.1, together with the theory of the Bousfield–Kuhn functor[Kuh89, Bou01], implies that for n ≥
1, the (telescopic) T ( n )-localization of thealgebraic K -theory spectrum of a stable ∞ -category is functorial in polynomialfunctors, i.e., extends to Cat poly ∞ . Motivation and related work.
Many previous authors have considered vari-ous types of non-additive operations on algebraic K -theory spaces, which providedsubstantial motivation for this work.An important example is given by operations in the K -theory space (and onthe K -groups) of a ring R arising from exterior and symmetric power functorson R -modules. Constructions of such maps appear in many sources, including[Hil81, Kra80, Sou85, Gra89, Nen91, Lev97, HKT17]. We expect, but have notchecked, that these maps agree with the maps provided by Theorem 1.1 using thederived exterior and symmetric power functors on Perf( R ). Another example in thisvein is given by the multiplicative norm maps along finite ´etale maps constructedin [BH17].A different instance of non-additive operations in K -theory arises in Segal’s ap-proach to the Kahn–Priddy theorem [Seg74]. These maps arise from the K -theoryof non-additive categories (such as the category of finite sets) and cannot be ob-tained from Theorem 1.1. Notation and conventions.
We freely use the language of ∞ -categories andhigher algebra as in [Lur09, Lur14]. Throughout, we let S denote the ∞ -categoryof spaces, and Sp the ∞ -category of spectra. -THEORY AND POLYNOMIAL FUNCTORS 3 Acknowledgments.
We are very grateful to Bhargav Bhatt, Lukas Brantner,Dustin Clausen, Rosona Eldred, Matthias Flach, Marc Hoyois, Jacob Lurie andPeter Scholze for helpful discussions.The third author was supported by the NSF Graduate Fellowship under grantDGE-114415 as this work was begun and was a Clay Research Fellow when thiswork was completed. The third author also thanks the University of Minnesotaand Hausdorff Institute of Mathematics (during the fall 2016 Junior Trimesterprogram “Topology”) for their hospitality. The fourth author was funded bythe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un-der Germany’s Excellence Strategy EXC 2044 390685587, Mathematics M¨unster:Dynamics-Geometry-Structure.2.
Polynomial functors
Simplicial and filtered objects.
In this subsection, we review basic factsabout simplicial objects in a stable ∞ -category. In particular, we review the stableversion of the Dold-Kan correspondence, following Lurie [Lur14], which connectssimplicial and filtered objects.To begin with, we review the classical Dold-Kan correspondence. A generalreference for this is [GJ99, III.2] for the category of abelian groups or [Wei94, 8.4]for an abelian category. We refer to [Lur14, 1.2.3] for a treatment for arbitraryadditive categories. Theorem 2.1 (Dold-Kan correspondence) . Let A be an additive category which isidempotent-complete. Then we have an equivalence of categories Fun(∆ op , A ) ≃ Ch ≥ ( A ) , between the category Fun(∆ op , A ) of simplicial objects in A and the category Ch ≥ ( A ) of nonnegatively graded chain complexes in A . The Dold-Kan equivalence arises as follows. Given a simplicial object X • ∈ Fun(∆ op , A ), we form an associated chain complex C ∗ such that:(1) C n is a direct summand of X n , and is given by the intersection of thekernels T i ≥ ker( d i ) where the d i ’s give the face maps X n → X n − . If A isonly assumed additive, the existence of this kernel is not a priori obvious(cf. [Lur14, Rmk. 1.2.3.15]). However, we emphasize that the object C n depends only on the face maps d i , i ≥ C n → C n − comes from the face map d in the simplicialstructure.The Dold-Kan correspondence has an analog for stable ∞ -categories, formulatedin [Lur14, Sec. 1.2.4], yielding a correspondence between simplicial and filtered objects. Theorem 2.2 (Lurie [Lur14, Th. 1.2.4.1]) . Let C be a stable ∞ -category. Then wehave an equivalence of stable ∞ -categories Fun(∆ op , C ) ≃ Fun( N Z ≥ , C ) , which sends a simplicial object X • ∈ Fun(∆ op , C ) to the filtered object | sk X • | →| sk X • | → | sk X • | → . . . . CLARK BARWICK, SAUL GLASMAN, AKHIL MATHEW, AND THOMAS NIKOLAUS
Remark 2.3 (Making Dold-Kan explicit) . We will need to unwind the correspon-dence as follows. Given a filtered object Y → Y → Y → . . . in C , we formthe sequence of cofibers Y , Y /Y , Y /Y , . . . , i.e., the associated graded. We haveboundary maps Y /Y → Σ Y , Y /Y → Σ Y /Y , . . . , and the sequence(1) · · · → Σ − Y /Y → Σ − Y /Y → Y forms a chain complex in the homotopy category Ho( C ): the composite of anytwo successive maps is nullhomotopic. If C is an idempotent-complete stable ∞ -category, then Ho( C ) is an idempotent-complete additive category. Given a sim-plicial object X • ∈ Fun(∆ op , C ), we can also extract a simplicial object in Ho( C )and thus a chain complex in Ho( C ) by the classical Dold-Kan correspondence. Abasic compatibility states that this produces the sequence (1), i.e., the additive andstable Dold-Kan correspondences are compatible [Lur14, Rem. 1.2.4.3].2.2. Polynomial functors of stable ∞ -categories. In this subsection, we re-view the notion of polynomial functor between stable ∞ -categories. We first discussthe analogous notion for additive ∞ -categories. Definition 2.4 (Eilenberg-MacLane [EML54]) . Let F : A → B be a functor be-tween additive ∞ -categories and assume first that B is idempotent complete. Then: • F is called polynomial of degree ≤ − • F is called polynomial of degree ≤ • Inductively, F is called polynomial of degree ≤ n if for each Y ∈ A thefunctor ( D Y F )( X ) := ker (cid:16) F ( Y ⊕ X ) F (pr X ) −−−−−→ F ( X ) (cid:17) is polynomial of degree at most n −
1. Note that this kernel exists as wehave assumed B to be idempotent complete and it is the complementarysummand to F ( X ).We let Fun ≤ n ( A , B ) ⊂ Fun( A , B ) be the subcategory spanned by functors of degree ≤ n . Finally, a functor F : A → B is polynomial if it is polynomial of degree ≤ n for some n .The notion of a polynomial functor behaves in a very intuitive fashion. Forexample, the composite of a functor of degree ≤ m with one of degree ≤ n is ofdegree ≤ mn . As another example, we have: Example 2.5.
Let R be a commutative ring. The symmetric power functorsSym i and exterior power functors V i on the category Proj ωR of finitely generatedprojective R -modules are polynomial of degree ≤ i .Let C be a stable ∞ -category. Definition 2.6 ( n -skeletal simplicial objects) . We say that a simplicial object X • ∈ Fun(∆ op , C ) is n -skeletal if it is left Kan extended from its restriction to∆ ≤ n ⊂ ∆. This notion easily generalizes when B is not idempotent complete. If B is not idempotentcomplete then F : A → B is called polynomial of degree ≤ n if the composition A → B → B ispolynomial of degree ≤ n , where B → B is an idempotent completion. -THEORY AND POLYNOMIAL FUNCTORS 5
Remark 2.7 ( n -skeletal geometric realizations exist) . Note that if X • is n -skeletalfor some n , then the geometric realization | X • | exists in C . Remark 2.8 ( n -skeletal objects via Dold-Kan) . The condition that X • should be n -skeletal depends only on the underlying homotopy category of C , considered asan additive category. Namely, using the Dold-Kan correspondence, we can form achain complex C ∗ in the homotopy category of C from X • , and then we claim that X • is n -skeletal if and only if C ∗ vanishes for ∗ > n .Indeed, if X • is n -skeletal, then clearly the maps | sk i X • | → | sk i +1 X • | are equiv-alences for i ≥ n , which implies that C ∗ = 0 for ∗ > n . Conversely, if C ∗ = 0 for ∗ > n , then the map of simplicial objects sk n X • → X • has the property that itinduces an equivalence on i -truncated geometric realizations for all i ∈ Z ≥ , whichimplies by Theorem 2.2 that it is an equivalence of simplicial objects. Definition 2.9.
We say that a functor f : C → D of stable ∞ -categories preserves finite geometric realizations if for every simplicial object X • in C which is n -skeletalfor some n , the colimit | F ( X • ) | exists and the natural map | F ( X • ) | → F ( | X • | ) isan equivalence in C . Proposition 2.10.
Let C , D be stable ∞ -categories. Let F : C → D be a functorsuch that the underlying functor on additive categories
Ho( C ) → Ho( D ) is polyno-mial of degree ≤ d . Then if X • is an n -skeletal simplicial object in C , F ( X • ) is an nd -skeletal simplicial object in D .Proof. As above in Remark 2.8, this is a statement purely at the level of homotopycategories. That is, the simplicial object X • defines a simplicial object of theadditive category Ho( C ), and thus a chain complex C ∗ in Ho( C ). Similarly F ( X • )defines a simplicial object of Ho( D ) and thus a chain complex D ∗ of Ho( D ). Theclaim is that if C ∗ = 0 for ∗ > n then D ∗ = 0 for ∗ > nd . This is purely a statementabout additive categories. For proofs, see [GS87, Lem. 3.3] or [DP61, 4.23]. (cid:3) Definition 2.11 (Polynomial functors) . Let C , D be stable ∞ -categories. We saythat a functor F : C → D is polynomial of degree ≤ d if the underlying functorHo( F ) : Ho( C ) → Ho( D ) of additive categories is polynomial of degree ≤ d and if F preserves finite geometric realizations. We let Fun ≤ d ( C , D ) ⊂ Fun( C , D ) denotethe full subcategory spanned by functors of degree ≤ d .This notion of a polynomial functor will be fundamental to this paper. Thedefinition is equivalent to the more classical definition of a polynomial functor,due to Goodwillie [Goo92], via n -excisivity. Of course, the primary applications ofthe theory treat functors where either the domain or codomain is not stable. Forconvenience, we describe the comparison below. Definition 2.12.
Let [ n ] = { , , . . . , n } , and let P ([ n ]) denote the nerve of theposet of subsets of [ n ], so that P ([ n ]) ≃ (∆ ) n +1 . An n -cube in C is a functor f : P ([ n ]) → C . The n -cube is said to be strongly coCartesian if it is left Kanextended from the subset P ≤ ([ n ]) ⊂ P ([ n ]) spanned by subsets of cardinality ≤ coCartesian if it is a colimit diagram. Note that a diagram is coCartesian ifand only if it is a limit or Cartesian diagram by [Lur14, Prop. 1.2.4.13].
Definition 2.13 (Goodwillie [Goo92, Def. 3.1]) . Let C , D be small stable ∞ -categories and let F : C → D be a functor. For n ≥
0, we say that F is n -excisive if F carries strongly coCartesian n -cubes to coCartesian n -cubes. CLARK BARWICK, SAUL GLASMAN, AKHIL MATHEW, AND THOMAS NIKOLAUS
Example 2.14.
For n = 0, the condition is that F should be a constant functor.For n = −
1, we say that F is n -excisive if F is identically zero. For n = 1, F is1-excisive (or simply excisive) if it carries pushouts to pullbacks. Proposition 2.15.
The functor F : C → D is polynomial of degree ≤ n (in thesense of Definition 2.11) if and only if it is n -excisive.Proof. Suppose first that F is polynomial of degree ≤ n , and fix a strongly co-Cartesian ( n + 1)-cube in C , which is determined by a collection of maps X → Y i , i = 0 , , . . . , n , in C . We would like to show that F carries this cube to a coCartesianone. Since every object in Fun(∆ , C ) is a finite geometric realization of arrows ofthe form A → A ⊕ B , and F preserves finite geometric realizations, we may assumethat we have Y i ≃ X ⊕ Z i for objects Z i , i = 0 , , . . . , n .Denote the above strongly coCartesian cube by f : P ([ n + 1]) → C . Thecofiber lim −→ P ′ ([ n +1]) F ◦ f → F ( f ([ n ])) is given precisely by the iterated derivative D Z D Z . . . D Z n F ( X ), as follows easily by induction on n . Thus, if F is polynomialof degree ≤ n , we see that F is n -excisive.Conversely, if F is n -excisive, the previous paragraph shows that Ho( F ) is poly-nomial of degree ≤ n . It remains to show that F preserves finite geometric realiza-tions. This follows from the general theory and classification of n -excisive functors,cf. also [BM19, Prop. 3.36] for an account. We can form the embedding D ֒ → Ind( D )to replace D by a presentable stable ∞ -category; this inclusion preserves finite geo-metric realizations. In this case, the general theory shows that F can be built upvia a finite filtration from its homogeneous layers , and each homogeneous layer ofdegree i is of the form X B ( X, X, . . . , X ) h Σ i for B : C i → Ind( D ) a functorwhich is exact in each variable and symmetric in its variables (see [Lur14, Sec. 6.1]for a reference in this setting). Therefore, it suffices to show that if B : C i → Ind( D )is a functor which is exact in each variable, then X B ( X, X, . . . , X ) preservesfinite geometric realizations. This now follows from the cofinality of the diagonal∆ op → (∆ op ) i and the fact that B preserves finite geometric realizations in eachvariable separately, since it is exact. (cid:3) Construction of polynomial functors.
We now discuss some examplesof polynomial functors on stable ∞ -categories; these will arise from the derivedfunctors of polynomial functors of additive ∞ -categories. We first review the rela-tionship between additive, prestable, and stable ∞ -categories. Compare [Lur, Sec.C.1.5]. Construction 2.16 (The stable envelope) . Given any small additive ∞ -category A , there is a universal stable ∞ -category Stab( A ) equipped with an additive, fullyfaithful functor A →
Stab( A ). Given any small stable ∞ -category B , any additivefunctor A → B canonically extends to an exact functor Stab( A ) → B . In otherwords, Stab is a left adjoint from the natural forgetful functor from the ∞ -categoryof small stable ∞ -categories to the ∞ -category of small additive ∞ -categories. Werefer to Stab( A ) as the stable envelope of A .Explicitly, Stab( A ) is the stable subcategory of the ∞ -category Fun × ( A op , Sp)of finitely product-preserving presheaves of spectra on A generated by the imageof the Yoneda embedding. Construction 2.17 (The prestable envelope Stab( A ) ≥ ) . Let A be a small additive ∞ -category as above, and let Stab( A ) be its stable envelope. We let Stab( A ) ≥ -THEORY AND POLYNOMIAL FUNCTORS 7 denote the subcategory of the nonabelian derived ∞ -category [Lur09, Sec. 5.5.8] P Σ ( A ) generated under finite colimits by A . Then Stab( A ) ≥ is a prestable ∞ -category [Lur, Appendix C] and is the universal prestable ∞ -category receiving anadditive functor A → P Σ ( A ), which we call the prestable envelope of A . There arefully faithful embeddings A ⊂
Stab( A ) ≥ ⊂ Stab( A ) , and Stab( A ) is also the stabilization of the prestable ∞ -category Stab( A ) ≥ . Example 2.18.
Let R be a ring, or more generally a connective E -ring spectrum.Then one has a natural additive ∞ -category Proj ωR of finitely generated, projective R -modules. The stable envelope is given by the ∞ -category Perf( R ) of perfect R -modules, and the prestable envelope is given by the ∞ -category Perf( R ) ≥ ofconnective perfect R -modules.Our main result is the following extension principle, which allows one to extendpolynomial functors from an additive ∞ -category to the stable envelope. Theorem 2.19 (Extension of polynomial functors) . Let D be a stable, idempotentcomplete ∞ -category and A be an additive ∞ -category. Pullback along the functor A →
Stab( A ) induces an equivalence of ∞ -categories Fun ≤ n ( A , D ) ≃ Fun ≤ n (Stab( A ) , D ) , between degree ≤ n functors A → D (in the sense of additive ∞ -categories) anddegree ≤ n functors (in the sense of stable ∞ -categories) Stab( A ) → D . We use the following crucial observation due to Brantner.
Lemma 2.20 (Extending from the connective objects, cf. [BM19, Th. 3.35]) . Let D be a presentable, stable ∞ -category. Restriction induces an equivalence of ∞ -categories Fun ≤ n (Stab( A ) ≥ , D ) ≃ Fun ≤ n (Stab( A ) , D ) , between n -excisive functors Stab( A ) ≥ → D and n -excisive functors Stab( A ) → D . Lemma 2.21.
Let C , D be stable ∞ -categories. Let A ⊂ C be an additive subcat-egory which generates C as a stable subcategory. Let F : C → D be a degree ≤ n functor. Suppose F | A has image contained in a stable subcategory D ′ ⊂ D . Then F has image contained in D ′ .Proof. We use essentially the notions of levels in C , cf. [ABIM10, Sec. 2], althoughwe are not assuming idempotent completeness.We define an increasing and exhaustive filtration of subcategories C ≤ ⊂ C ≤ ⊂· · · ⊂ C as follows. The subcategory C ≤ is the additive closure of A under shifts,so any object of C ≤ can be written in the form Σ i A ⊕ · · · ⊕ Σ i n A n for some A , . . . , A n ∈ A and i , . . . , i n ∈ Z . Inductively, we let C ≤ n denote the subcategoryof objects that are extensions of objects in C ≤ a and C ≤ b for 0 < a, b < n with a + b ≤ n . Each C ≤ n is closed under translates, and it is easy to see that S n C ≤ n isstable and contains A and hence equals C .We claim first that F ( C ≤ ) ⊂ D . That is, we need to show that for A , . . . , A n ∈A and i , . . . , i n ∈ Z , we have F (Σ i A ⊕ · · · ⊕ Σ i n A n ) ∈ D . If i , . . . , i n ≥
0, thenwe can choose a simplicial object in A which is d -truncated for some d and whosegeometric realization is Σ i A ⊕ · · · ⊕ Σ i n A n . Since F preserves finite geometricrealizations, the claim follows. In general, for any object X ∈ C , we can recover CLARK BARWICK, SAUL GLASMAN, AKHIL MATHEW, AND THOMAS NIKOLAUS F ( X ) as a finite homotopy limit of F (0) , F (Σ X ) , F (Σ X ⊕ Σ X ) , . . . via the T n -construction, since F is n -excisive. Using this, we can remove the hypotheses that i , . . . , i n ≥ F ( C ≤ ) ⊂ D . Given an object of C ≤ n with n >
1, it is an extension of objects in C ≤ a and C ≤ b forsome a, b < n . In view of Construction 3.8 below, it follows that any such object canbe written as a finite geometric realization of objects of level < n . Since F preservesfinite geometric realizations, it follows by induction that F ( C ≤ n ) ⊂ D . (cid:3) Proof of Theorem 2.19.
Embedding D inside Ind( D ), we may assume that D isactually presentable. By Lemma 2.21, we do not lose any generality by doing so.By the universal property of P Σ , we have an equivalence(2) Fun ≤ n ( A , Ind( D )) ≃ Fun Σ ≤ n ( P Σ ( A ) , Ind( D )) , where Σ denotes functors which preserve sifted colimits; the universal propertygives this without the ≤ n condition, which we can then impose. Now the inclusionStab( A ) ≥ ⊂ P Σ ( A ) exhibits the target as the Ind-completion of the source, whichyields an equivalence(3) Fun ≤ n (Stab( A ) ≥ , Ind( D )) ≃ Fun ω ≤ n ( P Σ ( A ) , Ind( D )) , where ω denotes functors which preserve filtered colimits. By definition, any func-tor in Fun ω ≤ n ( P Σ ( A ) , Ind( D ) preserves finite geometric realizations, and hence allgeometric realizations; thus it also preserves all sifted colimits. This shows that thecategories in (2) and (3) are identified. Now the result follows by combining thisidentification with Lemma 2.20. (cid:3) Corollary 2.22.
Let
A → B be additive ∞ -categories. Then a degree ≤ n func-tor A → B canonically prolongs to a degree ≤ n functor of stable ∞ -categories, Stab( A ) → Stab( B ) . (cid:3) Example 2.23.
Let R be a commutative ring. Then we have a functor Sym i :Perf( R ) → Perf( R ) which is i -excisive and which extends the usual symmetricpowers of finitely generated projective R -modules. We can regard this as a derivedfunctor of the usual symmetric power, although we are allowing nonconnectiveobjects as well.The above construction of extending functors, for Stab( A ) ≥ , is the classical oneof Dold-Puppe [DP61] of “nonabelian derived functors” constructed using simplicialresolutions. Compare also [JM99] for the connection between polynomial functorson additive categories and n -excisive functors. The extension to Stab( A ), at least incertain cases, goes back to Illusie [Ill71, Sec. I-4] in work on the cotangent complex,using simplicial cosimplicial objects.3. K and polynomial functors Additive ∞ -categories. In this section, we review the result of [Dol72,Jou00] that K of additive ∞ -categories is naturally functorial in polynomial func-tors; this special case of Theorem 1.1 will play an essential role in its proof. Definition 3.1 (Passi [Pas74]) . Let M be an abelian monoid and A be an abeliangroup. We will define inductively when a map f : M → A (of sets) is calledpolynomial of degree ≤ n . • A map f is called polynomial of degree ≤ − -THEORY AND POLYNOMIAL FUNCTORS 9 • A map f if called polynomial of degree ≤ n if for each y ∈ M the map D y f : M → A defined by( D y f )( x ) := f ( x + y ) − f ( x )is polynomial of degree ≤ n − f is polynomial if it is polynomial of degree n for some n .We denote the set of polynomial maps M → A of degree ≤ n by Hom ≤ n ( M, A ).It is straightforward to check that composing polynomial maps whenever this isdefined is again polynomial and changes the degree in the obvious way.
Example 3.2.
A map f : Z → Z is polynomial of degree ≤ n precisely if it can berepresented by a polynomial of degree n with rational coefficients. In this case ithas to be of the form f ( x ) = n X i =0 α i (cid:18) xi (cid:19) with α i ∈ Z .Now let i : M → M + be the group completion of the abelian monoid M . Thenthe following result states that we can always extend polynomial maps uniquelyover the group completion. This is surprising if one thinks about how to extend toa formal difference. Theorem 3.3 ([Jou00, Prop. 1.6]) . For any abelian monoid M and abelian group A , the map i ∗ : Hom ≤ n ( M + , A ) → Hom ≤ n ( M, A ) is a bijection. The proof in loc. cit. gives an explicit argument. For the convenience of thereader, we include an abstract argument via monoid and group rings.
Proof.
The first step is to reformulate the condition for a map f : M → A to bepolynomial of degree ≤ n . To do this we will temporarily for this proof write M multiplicatively, in particular 1 ∈ M is the neutral element.A map of sets f : M → A is polynomial of degree at most n precisely if theinduced map f : Z [ M ] → A defined by X m ∈ M α m · m X m ∈ M α m f ( m )(with α m ∈ Z ) factors over the ( n + 1)’st power I n +1 of the augmentation ideal I ⊆ Z [ M ]. In other words, there is a canonical bijection(4) Hom ≤ n ( M, A ) ≃ −→ Hom Ab ( Z [ M ] /I n +1 , A ) . This fact is proven in [Pas74], but let us give an argument. The augmentation ideal I is generated additively by elements of the form ( m −
1) with m ∈ M . Therefore I n +1 is generated additively by elements of the form( m − · . . . · ( m n − m i ∈ M . A slightly bigger additive generating set for I n +1 is then given by x · ( m − · . . . · ( m n − with m i , x ∈ M . Using this fact we have to show that f : M → A is polynomialof degree at most n precisely if f : Z [ M ] → A vanishes on these products for all x, m i ∈ M . This follows from the following pair of observations: • A map f : M → A is polynomial of degree at most n precisely if for eachsequence m , ..., m n , x of elements in M we have( D m D m ...D m n f )( x ) = 0 . • There is an equality( D m D m ...D m n f )( x ) = f (cid:16) x · ( m − · ( m − · . . . · ( m n − (cid:17) . The first of these observations is the definition. The second observation followsinductively from the case n = 0 which is obvious.Now we can proceed to the proof of the theorem. By virtue of the naturalbijection (4) it suffices to show that the map Z [ M ] /I n +1 → Z [ M + ] /I n +1 is an isomorphism of abelian groups. Both sides are actually rings and the map inquestion is a map of rings. In order to construct an inverse ring map, it sufficesto check that all elements m ∈ M represent multiplicative units in Z [ M ] /I n +1 ;however, this follows because m − (cid:3) The last result shows that group completion is universal with respect to polyno-mial maps and not only for additive maps. From this, the extended functorialityof K readily follows, as in [Jou00]; we review the details below. Definition 3.4 ( K of additive ∞ -categories) . For an additive ∞ -category A , thegroup K ( A ) is the group completion of the abelian monoid π ( A ) of isomorphismclasses of objects with ⊕ as addition. Concretely K ( A ) is the abelian group gener-ated from isomorphism classes of objects subject to the relation [ A ]+ [ B ] = [ A ⊕ B ].Let Cat add be the ∞ -category of additive ∞ -categories and additive functors.Let Ab denote the ordinary category of abelian groups. Then K defines a functor K : Cat add → AbLet Cat add , poly be the ∞ -category of additive ∞ -categories and polynomial func-tors between them. The next result appears in the present form in [Jou00] and (formodules over a ring) in [Dol72]. Proposition 3.5 ([Jou00, Prop. 1.8]) . There is a functor e K : Cat addpoly → Set witha transformation π → e K such that the diagram Cat add (cid:15) (cid:15) K / / Ab (cid:15) (cid:15) Cat addpoly e K / / Set commutes up to natural isomorphism.Proof.
For a polynomial functor F : A → B the map π ( A ) → π ( B ) → K ( B )is polynomial. Thus by Theorem 3.3 it can be uniquely extended to a polynomialmap K ( A ) → K ( B ). This gives the desired maps, and it is easy to see that theydefine a functor e K : Cat add , poly → Set. (cid:3) -THEORY AND POLYNOMIAL FUNCTORS 11
Stable ∞ -categories. In this section, we extend the results of the previ-ous section to show that K is functorial in polynomial functors of stable ∞ -categories; the strategy of proof is similar to that of [Dol72]. Recall that for stable ∞ -categories, one has the following definition of K , which only depends on theunderlying triangulated homotopy category. Definition 3.6.
Given a stable ∞ -category C , we define the group K ( C ) as thequotient of K add0 ( C ) (i.e., K of the underlying additive ∞ -category) by the relations[ X ] + [ Z ] = [ Y ] for cofiber sequences X → Y → Z in C . Proposition 3.7.
Let F : C → D be a polynomial functor between the stable ∞ -categories C , D . Then there is a unique polynomial map F ∗ : K ( C ) → K ( D ) suchthat F ∗ ([ X ]) = [ F ( X )] for X ∈ C . We have already seen an analog of this result for K add0 , the K of the underlyingadditive ∞ -category (Proposition 3.5). The obstruction is to understand the in-teraction with cofiber sequences. For this, we will need the following construction,and a general lemma about simplicial resolutions. Construction 3.8.
Let C be a stable ∞ -category. Suppose given a cofiber sequence X ′ → X → X ′′ in C . Then we form the ˇCech nerve of the map X → X ′′ . Thisconstructs a 1-skeletal simplicial object A • in C of the form . . . / / / / / / / / X ′ ⊕ X ′ ⊕ X / / / / / / X ′ ⊕ X / / / / X .
Alternatively, we can consider this simplicial object as the two-sided bar construc-tion of the abelian group object X ′ ∈ C acting on X (via X ′ → X ). We observethat each of the terms in the simplicial object, and each of the face maps d i , i ≥ X ′ , X (and not on the map X ′ → X ). Also, A • isaugmented over X ′′ and is a resolution of X ′′ . Lemma 3.9.
Suppose C is a stable ∞ -category and X • , X • ∈ Fun(∆ op , C ) are twosimplicial objects such that: (1) Both X • , X • are d -skeletal for some d . (2) We have an identification X n ≃ X n for each n . (3) Under the above identification, the face maps d i , i ≥ for both simplicialobjects are homotopic.Then | X • | , | X • | define the same class in K ( C ) .Proof. This follows from the fact that X • , X • have finite filtrations whose associatedgradeds are identified in view of the stable Dold-Kan correspondence. (cid:3) Proposition 3.10.
Let f : A → A ′ be a polynomial map between abelian groups.Let M ⊂ A be an abelian submonoid. Suppose that for m ∈ M and x ∈ A , we have f ( x + m ) = f ( x ) . Then for any m ′ belonging to the subgroup M ′ generated by M and x ∈ A , we have f ( x + m ′ ) = f ( x ) and f factors over A/M ′ .Proof. Fix x ∈ A . Consider the polynomial map A → A ′ sending y f ( x + y ) − f ( x ). Since this vanishes for y ∈ M , it vanishes on the image of M + → A and theresult follows. (cid:3) Proposition 3.11.
Let f : M → A be a polynomial map from an abelian monoid M to an abelian group A . Let N ⊂ M × M be a submonoid which contains thediagonal. Suppose that for each ( m , m ) ∈ N , we have f ( m ) = f ( m ) . Then the unique polynomial extension f + : M + → A factors over the quotient of M + by thesubgroup generated by { m − m } ( m ,m ) ∈ N .Proof. Note first that the collection C = { m − m } ( m ,m ) ∈ N ⊂ M + is a sub-monoid. We claim that for any x ∈ M + and c ∈ C , we have f + ( x ) = f + ( x + c ).Equivalently, for any y ∈ M + , f + ( y + m ) = f + ( y + m ). Since both are poly-nomial maps, it suffices to check this for y ∈ M , in which case it follows fromour assumptions. Thus the function f + : M + → A is invariant under translationsby elements of C . Since C is a monoid, it follows by Proposition 3.10 that f + isinvariant under translations by elements of the subgroup generated by C . (cid:3) Proof of Proposition 3.7.
By Theorem 3.5, we have a natural polynomial map onadditive K -theory K add0 ( C ) F add ∗ −−−→ K add0 ( D ) , such that F add ∗ ([ X ]) = [ F ( X )] for X ∈ C . It suffices to show the composite K add0 ( C ) F add ∗ −−−→ K add0 ( D ) ։ K ( D ) factors through K ( C ). To see this, recall that K ( C ) is the quotient of K add0 ( C ) (in turn the group completion of π ( C )) by therelations [ X ] = [ X ′ ⊕ X ′′ ] for each cofiber sequence(5) X ′ → X → X ′′ . The collection of such defines a submonoid of π ( C ) × π ( C ) containing the diagonal.To prove the assertion, we need to show that if (5) is a cofiber sequence in C , then[ F ( X )] = [ F ( X ′ ⊕ X ′′ )] . To see this, we construct two simplicial objects C • and C • as in Construction 3.8such that:(1) C • , C • are identified in each degree n with X ′ ⊕ X ′′ [ − ⊕ n and the facemaps d i , i ≥ | C • | ≃ X ′ ⊕ X ′′ and | C • | ≃ X .(3) Both C • , C • are 1-skeletal.Namely, C • is the ˇCech nerve of X ′ (id , −−−→ X ′ ⊕ X ′′ while C • is the ˇCech nerve of X ′ → X . We then find that the simplicial objects F ( C • ) , F ( C • ) are n -skeletal (if F has degree ≤ n ) and the geometric realizations are given by F ( X ′ ⊕ X ′′ ) , F ( X )respectively. Moreover, F ( C • ) , F ( C • ) agree in each degree and the face maps d i , i ≥ K , as desired. (cid:3) The main result
The universal property of higher K -theory. Our first goal is to reviewthe axiomatic approach to higher K -theory, and its characterization. We will usethe K -theory of stable ∞ -categories, as developed by [BGT13] and [Bar16], follow-ing ideas that go back to Waldhausen [Wal85] and ultimately Quillen [Qui73].Throughout, we fix (for set-theoretic reasons) a regular cardinal κ . Recall thatCat perf ∞ is compactly generated [BGT13, Cor. 4.25]. Let Cat perf ∞ ,κ denote the subcat-egory of κ -compact objects. Compare also [Jou00, Th. A] for a related type of statement. -THEORY AND POLYNOMIAL FUNCTORS 13
Definition 4.1 (Additive invariants) . (1) Let Fun π (Cat perf ∞ ,κ , S ) denote the ∞ -category of finitely product-preserving functors Cat perf ∞ ,κ → S .(2) We say that f ∈ Fun π (Cat perf ∞ ,κ , S ) is additive if f is grouplike and f carriessemiorthogonal decompositions in Cat perf ∞ ,κ to products.We let Fun π add (Cat perf ∞ ,κ , S ) ⊂ Fun π (Cat perf ∞ ,κ , S ) be the subcategory of additiveinvariants. This inclusion admits a left adjoint ( − ) add , called additivization .The construction ι which carries C ∈
Cat perf ∞ ,κ to its underlying space (i.e., thenerve of the maximal sub ∞ -groupoid) yields an object of Fun π (Cat perf ∞ ,κ , S ). Theconstruction of the algebraic K -theory space K ( − ) yields an additive invariant, byWaldhausen’s additivity theorem. Theorem 4.2 (Compare [BGT13, Bar16]) . The K -theory functor K : Cat perf ∞ ,κ → S is the additivization of ι ∈ Fun π (Cat perf ∞ ,κ , S ) . Remark 4.3.
As the results in loc. cit. are stated slightly differently (in partic-ular, κ = ℵ is assumed), we briefly indicate how to deduce the present form ofTheorem 4.2.To begin with, we reduce to the case κ = ℵ . Let F = ( ι ) add denote theadditivization of ι considered as an object of Fun π (Cat perf ∞ ,κ , S ). We can also considerthe additivization of ι considered as an object of Fun π (Cat perf ∞ ,ω , S ) and left Kanextend from Cat perf ∞ ,ω to Cat perf ∞ ,κ ; we denote this by F ′ : Cat perf ∞ ,κ → S . By left Kanextension, we also have a map ι → F ′ in Fun π (Cat perf ∞ ,κ , S ).Now F ′ is also an additive invariant, thanks to [HSS17, Prop. 5.5]. It followsthat we have maps in Fun π (Cat perf ∞ ,κ , S ) under ι from F ′ → F and F → F ′ , us-ing the universal properties. It follows easily (from the universal properties inFun π (Cat perf ∞ ,κ , S ) and Fun π (Cat perf ∞ ,ω , S )) that the composites in both directions arethe identity, whence F ≃ F ′ .Thus, we may assume κ = ℵ for the statement of Theorem 4.2. For κ = ℵ , wehave that Fun π add (Cat perf ∞ ,ω , S ) is the ∞ -category of Sp ≥ -valued additive invariantsin the sense of [BGT13], whence the result.We will also need a slight reformulation of the universal property, using a variantof the definition of an additive invariant, which turns out to be equivalent. In thefollowing, we write Fun ex ( − , − ) denote the ∞ -category of exact functors betweentwo stable ∞ -categories. Definition 4.4 (Universal K -equivalences) . A functor F : C → D in Cat perf ∞ is saidto be a universal K -equivalence if there exists a functor G : D → C such that(6) [ G ◦ F ] = [id C ] ∈ K (Fun ex ( C , C )) , [ F ◦ G ] = [id D ] ∈ K (Fun ex ( D , D )) . Equivalently, this holds if and only if for every
E ∈
Cat perf ∞ , the natural mapFun ex ( D , E ) → Fun ex ( C , E ) induces an isomorphism on K . Example 4.5.
The shear map
C ×C → C ×C , i.e., the functor (
X, Y ) ( X ⊕ Y, Y ),is a universal K -equivalence. If C admits a semiorthogonal decomposition intosubcategories C , C , then the projection C → C × C is a universal K -equivalence. Proposition 4.6.
A functor in
Fun π (Cat perf ∞ ,κ , S ) is additive if and only if it carriesuniversal K -equivalences to equivalences. Proof.
By the above examples, any object in Fun π (Cat perf ∞ ,κ , S ) which preserves uni-versal K -equivalences is necessarily additive. Thus, it remains only to show thatan additive invariant carries universal K -equivalences to equivalences. Note thatan additive invariant f : Cat perf ∞ ,κ → S naturally lifts to Sp ≥ , since it is grouplike.Moreover, given C , D ∈
Cat perf ∞ ,κ , the map obtained by applying f , π (Fun ex ( C , D ) ≃ ) f −→ π Hom Sp ≥ ( f ( C ) , f ( D ))has the property that it factors through K (Fun ex ( C , D )): indeed, this follows usingadditivity for D ∆ . This easily shows that f sends universal K -equivalences toequivalences. (cid:3) We thus obtain the following result, showing that additivization is the Bousfieldlocalization at the universal K -equivalences. Corollary 4.7.
Fun π add (Cat perf ∞ ,κ , S ) is the Bousfield localization of Fun π (Cat perf ∞ ,κ , S ) at the class of maps in Cat perf ∞ ,κ (via the Yoneda embedding) which are universal K -equivalences. (cid:3) The universal property with polynomial functors.
In this subsection,we formulate the main technical result (Theorem 4.9) of the paper, which controlsthe additivization of a theory functorial in polynomial functors. Throughout, wefix a regular uncountable cardinal κ . Definition 4.8.
We let Cat poly ∞ ,κ denote the ∞ -category whose objects are κ -compactidempotent-complete, stable ∞ -categories and whose morphisms are polynomialfunctors between them.We consider the ∞ -category Fun π (Cat poly ∞ ,κ , S ) of functors Cat poly ∞ ,κ → S whichpreserve finite products. We say that an object T ∈ Fun π (Cat poly ∞ ,κ , S ) is additive ifits restriction to Cat perf ∞ ,κ is additive. We let Fun π add (Cat poly ∞ ,κ , S ) ⊂ Fun π (Cat poly ∞ ,κ , S )denote the subcategory of additive objects. This inclusion admits a left adjoint( − ) addp , called polynomial additivization. As an example, the underlying ∞ -groupoid functor still defines a functor ι :Cat poly ∞ ,κ → S which preserves finite products, and hence an object of Fun π (Cat poly ∞ ,κ , S ).We now state the main technical result, which states that the polynomial additiviza-tion recovers the additivization when restricted to Cat perf ∞ ,κ . This will be provedbelow in section 4.4. Theorem 4.9.
Let T ∈ Fun π (Cat poly ∞ ,κ , S ) and let T addp denote its polynomial ad-ditivization. Then the map T → T addp , when restricted to Cat perf ∞ ,κ , exhibits therestriction T addp | Cat perf ∞ ,κ as the additivization of T | Cat perf ∞ ,κ . A direct consequence of the theorem is a sort of converse: given any map T → T ′ such that the restricted transformation exhibits T ′ | Cat perf ∞ ,κ as the additivization of T | Cat perf ∞ ,κ , then T ′ is already the polynomial additivization. To see this simply notethat T ′ is additive since this is only a condition on the restricted functor. Thus weget a map T addp → T ′ which is, by the theorem, an equivalence when restricted toCat perf ∞ ,κ and therefore an equivalence.Taking T = ι and using the universal property of K -theory (as in Theorem 4.2),we obtain the polynomial functoriality of K -theory (Theorem 1.1 from the intro-duction). -THEORY AND POLYNOMIAL FUNCTORS 15 Corollary 4.10.
There is a (unique) functor e K : Cat poly ∞ ,κ → S together with amap ι → e K in Fun π (Cat poly ∞ ,κ , S ) , such that the underlying map ι | Cat perf ∞ ,κ e K | Cat perf ∞ ,κ identifies e K | Cat perf ∞ ,κ with K . Moreover, e K is the polynomial additivization of ι . (cid:3) Remark 4.11.
Proving such a result directly (e.g., by examining the S • -construction)seems to be difficult. In fact, since the maps on K -theory spaces induced by poly-nomial functors are in general not loop maps they cannot be induced by maps ofthe respective S • -constructions.4.3. Generalities on Bousfield localizations.
We need some preliminaries aboutstrongly saturated collections, cf. [Lur09, Sec. 5.5.4].
Definition 4.12.
Let E be a presentable ∞ -category. A strongly saturated classof maps is a full subcategory of Fun(∆ , E ) which is closed under colimits, base-changes, and compositions. Construction 4.13 (Strongly saturated classes correspond to Bousfield localiza-tions) . Given a set of maps in E , they generate a smallest strongly saturated class.A strongly saturated class arising in this way is said to be of small generation. The class of maps in E that map to equivalences under a Bousfield localization E → E ′ of presentable ∞ -categories is strongly saturated and of small generation,and this in fact establishes a correspondence between accessible localizations andstrongly saturated classes of small generation [Lur09, Props. 5.5.4.15-16]. Specifi-cally, given a set S of maps, the Bousfield localization corresponding to the stronglysaturated class generating is the Bousfield localization whose image consists of the S -local objects. To summarize, given a presentable ∞ -category E , we have a cor-respondence between the following collections: • Presentable ∞ -categories E ′ , equipped with fully faithful right adjoints E ′ → E (so the left adjoint is a localization functor). • Strongly saturated classes of maps in E which are of small generation. • Accessible localization functors L : E → E . Proposition 4.14.
Let C be a presentable ∞ -category which is given as the non-abelian derived ∞ -category of a subcategory C ⊂ C closed under finite coprod-ucts. Let S be the strongly saturated collection of maps in C generated by a subset S ⊂ Fun(∆ , C ) . Suppose that S is closed under finite coproducts and containsthe identity maps. Let F : C → D be a functor which preserves sifted colimits andlet V be a strongly saturated class in D . If F ( S ) ⊂ V , then F ( S ) ⊂ V .Proof. Consider the collection M of maps x → y in C such that for every map x → x ′ , the map F ( x ′ → y ∪ x x ′ )) ∈ V . This collection M (in Fun(∆ , C )) isclearly closed under base-change, composition, and sifted colimits. Therefore, M is closed under all colimits and is in particular a strongly saturated class.We claim that this collection M contains all of S ; it suffices to see that S ⊂ M .To see this, let x → y be a map in S . We need to see that the base-changeof this map along a map x → x ′ is carried by F into V . Any map x → x ′ can be written as a sifted colimit of maps x → x ⊔ z for z ∈ C , so one reducesto this case. Writing z as a sifted colimit of objects in C , we reduce to the casewhere z = z ∈ C . Then the assertion is part of the hypotheses, so we obtain( x → y ) ∈ M as desired. (cid:3) Corollary 4.15.
Let A , B be ∞ -categories admitting finite coproducts. Let F : A → B be a functor preserving finite coproducts, inducing a cocontinuous functor F : P Σ ( A ) → P Σ ( B ) with a right adjoint G : P Σ ( B ) → P Σ ( A ) which preservessifted colimits.Let S be a class of maps in A and let T = F ( S ) denote the induced class ofmaps in B ; let S, T be the induced strongly saturated classes of maps in P Σ ( A ) →P Σ ( B ) , and let L S , L T be the associated Bousfield localization functors. Supposethat the class of maps G ( T ) = GF ( S ) in P Σ ( A ) belongs to the strongly saturatedclass generated by S .Then the functor G : P Σ ( B ) → P Σ ( A ) commutes with the respective localizationfunctors. More precisely: (1) For any Y ∈ P Σ ( B ) which is T -local, G ( Y ) is S -local. (2) For any Y ′ ∈ P Σ ( B ) , the natural map Y ′ → L T ( Y ′ ) induces (by the property(1)) a map L S G ( Y ′ ) → G ( L T ( Y ′ )) ; this map is an equivalence. (3) G induces a functor L T P Σ ( B ) → L S P Σ ( A ) which commutes with limitsand sifted colimits, which is right adjoint to the functor L T F : L S P Σ ( A ) → L T P Σ ( B ) .Proof. Part (1) follows because F (which preserves colimits) carries S into T , sothe right adjoint necessarily carries T -local objects into S -local objects.By Proposition 4.14, G carries the strongly saturated class T in P Σ ( B ) into thestrongly saturated class S in P Σ ( A ). Now in (2), the map Y ′ → L T ( Y ′ ) belongsto the strongly saturated class T , whence G ( Y ′ ) → G ( L T ( Y ′ )) belongs to thestrongly saturated class S . Since the target of this map is S -local, it follows that L S G ( Y ′ ) ∼ −→ G ( L T ( Y ′ )). This proves part (2).For (3), we already saw in (1) that G induces a functor L T P Σ ( B ) → L S P Σ ( A ),and clearly this commutes with limits. It also commutes with sifted colimits sincethe functor G : P Σ ( B ) → P Σ ( A ) commutes with sifted colimits and since G carriesthe L T -localization into the L S -localization. From this (3) follows. (cid:3) Proof of Theorem 4.9.
The proof of Theorem 4.9 will require some morepreliminaries. To begin with, we will need the construction of a universal target fora degree ≤ n functor. Construction 4.16.
Given
C ∈
Cat perf ∞ , we define the object Γ n C ∈
Cat perf ∞ suchthat we have a natural equivalence for any D ∈
Cat perf ∞ ,Fun ex (Γ n C , D ) ≃ Fun ≤ n ( C , D ) . In particular, Γ n receives a degree ≤ n functor C → Γ n C and Γ n C is universal forthis structure. Explicitly, Γ n C is obtained by starting with the free idempotent-complete stable ∞ -category on C , i.e., compact objects in Sp-valued presheaves on C , and then forming the minimal exact localization such that the Yoneda functorbecomes n -excisive. Remark 4.17 (Some cardinality estimation) . Recall again that κ is assumed tobe uncountable. If C ∈
Cat perf ∞ ,κ , then we claim that Γ n C ∈
Cat perf ∞ ,κ for all n ≥ C ∈
Cat perf ∞ ,κ if and only if C is κ -compact as an object ofCat ∞ ; moreover, this holds if and only if C has < κ isomorphism classes of objectsand the mapping spaces in C are κ -small. -THEORY AND POLYNOMIAL FUNCTORS 17 The crucial observation, for our purposes, is simply that Γ n behaves relativelywell with respect to semiorthogonal decompositions: it transforms them into some-thing that, while slightly more complicated, is very controllable on K -theory. Proposition 4.18.
Let F : C → C ′ be a universal K -equivalence. Then the map Γ n C → Γ n C ′ is a universal K -equivalence. That is, for every D ∈
Cat perf ∞ , thefunctor F ∗ : Fun ≤ n ( C ′ , D ) → Fun ≤ n ( C , D ) , induces an isomorphism on K .Proof. Let G : C ′ → C be a functor such that F ◦ G, G ◦ F satisfy (6). It sufficesto show that the composite Fun ≤ n ( C , D ) G ∗ → Fun ≤ n ( C ′ , D ) F ∗ → Fun ≤ n ( C , D ) is theidentity on K (and the converse direction follows by symmetry).To see this, fix a functor f ∈ Fun ≤ n ( C , D ). We then have a degree n functor f ◦ · : Fun ex ( C , C ) → Fun ≤ n ( C , D ) , φ f ◦ φ. By Proposition 3.7, this induces a unique map on K . It follows that since G ◦ F, iddefine the same class in K (Fun ex ( C ′ , C )), the functors f ◦ G ◦ F, f define the sameclass in K (Fun ≤ n ( C , D )). This shows precisely that F ∗ ◦ G ∗ induces the identityon K . Similarly, G ∗ ◦ F ∗ induces the identity on K . This completes the proof. (cid:3) Proof of Theorem 4.9.
Consider the commutative diagramFun π add (Cat poly ∞ ,κ , S ) Res (cid:15) (cid:15) / / Fun π (Cat poly ∞ ,κ , S ) Res (cid:15) (cid:15)
Fun π add (Cat perf ∞ ,κ , S ) / / Fun π (Cat perf ∞ ,κ , S ) , where the horizontal rows are the inclusions and the vertical arrows are given byrestriction along Cat perf ∞ ,κ ⊂ Cat poly ∞ ,κ . Our goal is to show that when we reversethe horizontal arrows by replacing the inclusion functors by additivizations, thediagram still commutes.This statement fits into the setup of Corollary 4.15. Here we take A = (Cat perf ∞ ,κ ) op and B = (Cat poly ∞ ,κ ) op , and F to be the opposite of the inclusion Cat perf ∞ ,κ → Cat poly ∞ ,κ .Moreover, S can be taken to be the class of universal K -equivalences in A =(Cat perf ∞ ,κ ) op . The local objects then correspond to the additive invariants (Corol-lary 4.7).Unwinding the definitions, we find that in order to apply Corollary 4.15, we nowneed to verify that if C → D is a universal K -equivalence in Cat perf ∞ ,κ , then the mapin Fun π (Cat perf ∞ ,κ , S ) given byHom Cat poly ∞ ,κ ( D , − ) → Hom
Cat poly ∞ ,κ ( C , − )induces an equivalence upon additivizations. Now by definition we haveHom Cat poly ∞ ,κ ( D , − ) = lim −→ n Hom
Cat perf ∞ (Γ n D , − ) , and similarly for Hom Cat poly ∞ ,κ ( C , − ). It therefore suffices to show thatHom Cat perf ∞ (Γ n D , − ) → Hom
Cat perf ∞ (Γ n C , − ) (as a map in Fun π (Cat perf ∞ ,κ , S )) induces an equivalence on additivizations. Thisfollows from Proposition 4.18 and Corollary 4.7, noting that Γ n C , Γ n D belong toCat perf ∞ ,κ since C , D do and κ is uncountable. (cid:3) References [ABIM10] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Srikanth B. Iyengar, and Claudia Miller,
Homology of perfect complexes , Adv. Math. (2010), no. 5, 1731–1781. MR 2592508[Bar16] Clark Barwick,
On the algebraic K -theory of higher categories , J. Topol. (2016),no. 1, 245–347. MR 3465850[BGT13] Andrew J. Blumberg, David Gepner, and Gon¸calo Tabuada, A universal characteriza-tion of higher algebraic K -theory , Geom. Topol. (2013), no. 2, 733–838. MR 3070515[BH17] Tom Bachmann and Marc Hoyois, Norms in motivic homotopy theory , arXiv preprintarXiv:1711.03061 (2017).[BM19] Lukas Brantner and Akhil Mathew,
Deformation theory and partition Lie algebras ,arXiv preprint arXiv:1904.07352 (2019).[Bou01] A. K. Bousfield,
On the telescopic homotopy theory of spaces , Trans. Amer. Math. Soc. (2001), no. 6, 2391–2426 (electronic). MR MR1814075 (2001k:55030)[Dol72] Albrecht Dold, K -theory of non-additive functors of finite degree , Math. Ann. (1972), 177–197. MR 301078[DP61] Albrecht Dold and Dieter Puppe, Homologie nicht-additiver Funktoren. Anwendungen ,Ann. Inst. Fourier Grenoble (1961), 201–312. MR 0150183[EML54] Samuel Eilenberg and Saunders Mac Lane, On the groups H (Π , n ) . II. Methods ofcomputation , Ann. of Math. (2) (1954), 49–139. MR 65162[GJ99] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory , Progress in Mathe-matics, vol. 174, Birkh¨auser Verlag, Basel, 1999. MR MR1711612 (2001d:55012)[Goo92] Thomas G. Goodwillie,
Calculus. II. Analytic functors , K -Theory (1991/92), no. 4,295–332. MR 1162445[Gra89] Daniel R. Grayson, Exterior power operations on higher K -theory , K -Theory (1989),no. 3, 247–260. MR 1040401[GS87] H. Gillet and C. Soul´e, Intersection theory using Adams operations , Invent. Math. (1987), no. 2, 243–277. MR 910201[Hil81] Howard L. Hiller, λ -rings and algebraic K -theory , J. Pure Appl. Algebra (1981),no. 3, 241–266. MR 604319[HKT17] Tom Harris, Bernhard K¨ock, and Lenny Taelman, Exterior power operations on higher K -groups via binary complexes , Ann. K-Theory (2017), no. 3, 409–449. MR 3658990[HSS17] Marc Hoyois, Sarah Scherotzke, and Nicol`o Sibilla, Higher traces, noncommuta-tive motives, and the categorified Chern character , Adv. Math. (2017), 97–154.MR 3607274[Ill71] Luc Illusie,
Complexe cotangent et d´eformations. I , Lecture Notes in Mathematics,Vol. 239, Springer-Verlag, Berlin-New York, 1971. MR 0491680[JM99] Brenda Johnson and Randy McCarthy,
Taylor towers for functors of additive cate-gories , J. Pure Appl. Algebra (1999), no. 3, 253–284. MR 1685140[Jou00] Seva Joukhovitski, K -theory of the Weil transfer functor , vol. 20, 2000, Special issuesdedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part I, pp. 1–21.MR 1798429[Kra80] Ch. Kratzer, λ -structure en K -th´eorie alg´ebrique , Comment. Math. Helv. (1980),no. 2, 233–254. MR 576604[Kuh89] Nicholas J. Kuhn, Morava K -theories and infinite loop spaces , Algebraic topology(Arcata, CA, 1986), Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 243–257. MR MR1000381 (90d:55014)[Lev97] Marc Levine, Lambda-operations, K -theory and motivic cohomology , Algebraic K -theory (Toronto, ON, 1996), Fields Inst. Commun., vol. 16, Amer. Math. Soc., Provi-dence, RI, 1997, pp. 131–184. MR 1466974[Lur] Jacob Lurie, Spectral algebraic geometry .[Lur09] ,
Higher topos theory , Annals of Mathematics Studies, vol. 170, Princeton Uni-versity Press, Princeton, NJ, 2009. MR MR2522659 -THEORY AND POLYNOMIAL FUNCTORS 19 [Lur14] J. Lurie,
Higher algebra , Available at ,2014.[Nen91] A. Nenashev,
Simplicial definition of λ -operations in higher K -theory , Algebraic K -theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 9–20.MR 1124623[Pas74] I. B. S. Passi, Polynomial maps , 550–561. Lecture Notes in Math., Vol. 372.MR 0369416[Qui72] Daniel Quillen,
On the cohomology and K -theory of the general linear groups over afinite field , Ann. of Math. (2) (1972), 552–586. MR MR0315016 (47 Higher algebraic K -theory. I , Algebraic K -theory, I: Higher K -theories (Proc.Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR MR0338129 (49 Operations in stable homotopy theory , New developments in topology(Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London,1974, pp. 105–110. London Math Soc. Lecture Note Ser., No. 11. MR 0339154 (49