Higher semiadditive algebraic K-theory and redshift

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$- and $\mathrm {T}(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$.

Algebraic K-theory K : Cat st → Sp is a rich invariant of stable ∞-categories and thus of rings and ring spectra. Ausoni-Rognes [AR02, AR08] suggested a fascinating program concerning the interaction between algebraic K-theory and the chromatic filtration on spectra, now known as the redshift philosophy. Namely, that algebraic K-theory increases the chromatic height of ring spectra by 1. They demonstrated this phenomenon at height 1, and conjectured that it persists to arbitrary heights. Another interesting aspect of algebraic K-theory is its descent properties. For example, it is known by [TT90] that it satisfies Nisnevich descent for ordinary rings, while it fails to satisfyétale descent due to its failure to satisfy Galois descent. The recent breakthroughs of [CMNN20,LMMT20] have shown that chromatically localized K-theory does satisfy Galois descent under certain hypotheses, which was used to prove the following part of the redshift conjecture. In addition, Hahn-Wilson [HW22] and Yuan [Yua21] give the first examples of non-vanishing of T(n + 1)-localized K-theory for ring spectra of chromatic height n, at arbitrary heights n ≥ 0. Building on this, Burklund-S.-Yuan [BSY22] have recently proved the non-vanishing of T(n + 1)localized K-theory for all commutative ring spectra of chromatic height n.

Higher Semiadditivity
Hopkins-Lurie [HL13, Theorem 5.2.1] and Carmeli-S.-Yanovski [CSY22,Theorem A] proved that the chromatically localized ∞-categories Sp K(n) and Sp T(n) (respectively) are ∞-semiadditive. Namely, that there is a canonical natural equivalence between limits and colimits indexed by π-finite spaces (i.e. spaces with finitely many connected components and finitely many non-zero homotopy groups all of which are finite). In this paper we will only make use of p-typical higher semiadditivity, that is, relaxing the condition to π-finite p-spaces (i.e. π-finite spaces whose homotopy groups are all p-groups), which we thus simply call higher semiadditivity.
Harpaz [Har20] studied the connection between ∞-semiadditivity and ∞-commutative monoids. Recall that a (0-)commutative monoid is, roughly speaking, the structure of summation of finite families of elements (in a coherently associative and commutative way). Similarly, a (p-typical) ∞-commutative monoid is, roughly speaking, the structure of "integration" of families of elements indexed by a π-finite p-space (in a coherently associative and commutative way). More precisely, given an ∞-category C, the ∞-category of (p-typical) ∞-commutative monoids in C is defined to be CMon (p) ∞ (C) = Fun seg (Span(S (p) π-fin ) op , C), the full subcategory of those functors from spans of π-finite p-spaces that satisfy the ∞-Segal condition. In [Har20,Corollary 5.19] and [CSY21a, Proposition 5.3.1] it is shown that the property of being a (p-typically) ∞-semiadditive presentable ∞-category is classified by the mode CMon (p) ∞ (S) of ∞-commutative monoids in spaces. 1 That is, a presentable ∞-category C is ∞-semiadditive if and only if it admits a (necessarily unique) module structure over CMon (p) ∞ (S) in Pr L . Furthermore, any object X ∈ C in an ∞-semiadditive presentable ∞category C is canonically endowed with the structure of an ∞-commutative monoid, that is, there is an equivalence C ∼ − → CMon (p) ∞ (C). Using this ∞-commutative monoid structure, [CSY21a, Definition 3.1.6] introduces the semiadditive height of an object X ∈ C, denoted by ht(X). The notion of semiadditive height, which is defined in arbitrary ∞-semiadditive ∞-categories, is related to the chromatic height. For example, all objects of Sp K(n) and Sp T(n) are of semiadditive height n by [CSY21a,Theorem 4.4.5].
A particularly interesting example of an ∞-semiadditive presentable ∞-category, which is studied in [CSY21a], is the mode classifying the property of being a p-local stable ∞-semiadditive presentable ∞-category, ‫צ‬ = CMon (p) ∞ (Sp (p) ), consisting of (p-typical) ∞-commutative monoids in p-local spectra (see Definition 4.11). By construction, there is a canonical map of modes (−) gpc : CMon (p) ∞ (S) → ‫,צ‬ which we call the group-completion. Additionally, there is a canonical map of modes L ‫צ‬ T(n) : ‫צ‬ → Sp T(n) , which by [CSY21a,Corollary 5.5.14] is a smashing localization, and in particular has a fully faithful right adjoint.
Another important example of an ∞-semiadditive presentable ∞-category is Cat π-fin , consisting of ∞-categories admitting colimits over all π-finite p-spaces (see [Har20,Theorem 5.23] and [CSY21a, Proposition 2.2.7]). As an ∞-semiadditive ∞-category, its objects, which are themselves ∞-categories, can have a semiadditive height. Additionally, there is an interplay between the semiadditive height of objects in an ∞-semiadditive ∞-category and the semiadditive height of the ∞-category itself as an object of Cat π-fin , which we view as the crucial step at which redshift happens.
Theorem 1.2 (Semiadditive Redshift [CSY21a,Theorem B]). Let C be an ∞-semiadditive ∞category, then ht(X) ≤ n for all X ∈ C if and only if ht(C) ≤ n + 1 as an object of Cat π-fin .

Higher Semiadditive Algebraic K-Theory of Categories
In this paper we study the confluence of the above ideas. As Cat π-fin is itself ∞-semiadditive, the ∞categories therein admit a canonical structure of ∞-commutative monoids, called the ∞-cocartesian structure, via the equivalence Cat st π-fin ∞ (Cat st π-fin ), where the integration of a family of objects is given by their colimit. We observe that the S • -construction preserves limits, thus it preserves ∞-commutative monoid structure. This observation along with the group-completion functor described above lead us to the main definition of the present paper. By analogy with the definition of ordinary algebraic K-theory, in Definition 6.5 we define ∞-semiadditive algebraic K-theory K [∞] : Cat st π-fin → ‫צ‬ as the composition Theorem A (Theorem 7.12). Let R ∈ Alg(‫)צ‬ have ht(R) ≤ n and let m > n, then ht(K [m] (R)) ≤ n + 1.
To give a lower bound on the height, we make use of the higher height analogues of cyclotomic extensions defined in [CSY21b,Definition 4.7]. Recall that for R ∈ Alg(‫)צ‬ of ht(R) = n, there is a ((Z/p) × -equivariant) splitting of algebras R[B n C p ] ∼ = R × R[ω (n) p ], where R[ω (n) p ] is called the (height n) p-cyclotomic extension of R, which generalizes ordinary cyclotomic extensions at height 0 (i.e. for algebras over the rationals). We say that R has (height n) p-th roots of unity if the cyclotomic extension splits as a product R[ω (n) p ] ∼ = (Z/p) × R (see Definition 7.17). For example, by [CSY21b,Proposition 5.1], the Lubin-Tate spectrum E n has (height n) p-th roots of unity. For such R, we get an equivalence of R-modules R[B n C p ] ∼ = R p , from which we immediately deduce the following strengthening of Theorem A.

A natural question left open is the following:
Question 1.3. Can the assumption of having (height n) p-th roots of unity be dropped? Namely, is it true that if R ∈ Alg(‫)צ‬ is of height n, then K [m] (R) is of height exactly n + 1?

Relationship to Chromatically Localized K-Theory
As we have seen in Theorem A and Theorem B, K [m] satisfies a form of the redshift conjecture with respect to semiadditive height. A natural next direction is connecting these results to ordinary algebraic K-theory and the chromatic height. Let R ∈ Alg(Sp T(n) ). The inclusion Sp T(n+1) ⊂ Sp admits a left adjoint L T(n+1) : Sp → Sp T(n+1) . Since K(R) ∈ Sp, we can consider L T(n+1) K(R) ∈ Sp T(n+1) .
(2) Is the comparison map an equivalence?
A positive answer to both questions will imply that K [m] (R) ∼ = L T(n+1) K(R) (see Conjecture 1.4 below). The first question is closely related to the Quillen-Lichtenbaum conjecture for R, in the guise of having a non-zero finite spectrum X such that K(R) ⊗ X is bounded above, as we show in Proposition 8.4. The second question is equivalent to L T(n+1) K [m] (R) satisfying the m-Segal condition. More informally, having descent properties for chromatically localized K-theory.
Using the Galois descent results for T(n + 1)-localized K-theory of [CMNN20], this second question is answered in the affirmative for m = 1 in Proposition 8.6. In work in progress with Carmeli and Yanovski [BMCSY] we show that the descent result for chromatically localized Ktheory generalizes from finite p-groups to arbitrary π-finite p-spaces. This would give a positive answer to the second question for every m ≥ 1.
Next, we focus on the case of height 0, answering both question in the affirmative in complete generality. Using the Quillen-Lichtenbaum property of S[p −1 ] together with Galois descent we obtain the following: Theorem C (Theorem 8.10). Let R ∈ Alg(Sp[p −1 ]) and let m ≥ 1, then In particular, K [m] (Q) ∼ = KU p .
Finally, we study the completed Johnson-Wilson spectrum E(n) at height n ≥ 1. In [HW22], Hahn-Wilson produced an E 3 -algebra structure on BP⟨n⟩, for which they have proven a version of the Quillen-Lichtenbaum conjecture. This structure also endows E(n) with an E 3 -algebra structure. Using their Quillen-Lichtenbaum result, along with a comparison of two direct computations of the higher commutative monoid structure on K [m] ( E(n)), we obtain the following strengthening of Theorem B for E(n)-algebras. We would like to thank the anonymous referee for suggesting crucial parts of the proof of this result.
As mentioned above, our upcoming work with Carmeli and Yanovski [BMCSY] implies that Theorem E generalizes to m-semiadditive K-theory for any m ≥ 1. This generalization, along with Theorem C answering the case of height 0, lead us to conjecture the following: Conjecture 1.4. For any R ∈ Alg(Sp T(n) ) and m ≥ 1 we have We would like to highlight two interesting phenomena exemplified by Theorem C and Theorem E. First, higher semiadditive algebraic K-theory lands in the highest non-zero height predicted by the redshift conjecture, without forcing it be in a pure height from the outside. Second, algebraic K-theory can be modified to have a higher commutative monoid structure in two ways -either by chromatically localizing it from the outside, or by internally remembering the higher commutative monoid structure on the input ∞-category. These results show that these two a priori distinct objects coincide, at least in some cases. This identification gives different approaches to study the higher commutative monoid structure, similarly to the proof of Theorem D itself.

Atomic Objects and a Monoidal Natural Yoneda
Recall that in the construction of the higher semiadditive algebraic K-theory of R ∈ Alg(‫)צ‬ described above, we passed to the left dualizable objects. In order to study the functoriality of this construction in R, as well as to generalize the construction to stable ∞-semiadditive presentable ∞-categories other than ∞-categories of modules, we define and study M-atomic objects for any mode M (see Definition 2.3). One of our main results is that M-atomic objects indeed coincide with left dualizable objects in left modules, i.e. LMod at R = LMod ldbl R for any R ∈ Alg(M) (see Proposition 2.54). Another direction of generalization is the case M = Sp, where Sp-atomic objects coincide with compact objects. We also show that for any absolute limit of M, the M-atomic objects are closed under I op -shaped colimits (see Proposition 2.24). These two results are then applied in Proposition 4.15 to show that for R ∈ Alg(‫,)צ‬ we have LMod ldbl R ∈ Cat st m-fin , so that it can be used as an input to higher semiadditive algebraic K-theory.
Another key result is the strong connection between the functor P M taking M-valued presheaves and the functor taking M-atomic objects. Let Mod iL M denote the subcategory of Pr L consisting of ∞-categories in the mode M and internally left adjoint functors (that is, left adjoint functors whose right adjoint admits a further right adjoint), which inherits a symmetric monoidal structure from Pr L . We then have the following: Theorem F (Theorem 2.46). There is a symmetric monoidal adjunction i.e. P M is symmetric monoidal with a lax symmetric monoidal right adjoint (−) M-at .
Building on the work of Glasman [Gla16] and Haugseng-Hebestreit-Linskens-Nuiten [HHLN20b, Theorem 8.1] on the Yoneda embedding, the adjunction is constructed such that the unit is (the factorization through the M-atomic objects of) the Yoneda map よ M : C 0 → P M (C 0 ), reproducing the ordinary Yoneda embedding for M = S. As an immediate consequence, we obtain a monoidal and natural version of the Yoneda map for any operad O, which may be of independent interest.

Organization
In Section 2, we develop the notion of M-atomic objects in a presentable ∞-category in the mode M. We study the connection between M-atomic objects and M-valued presheaves, and leverage this connection to endow the functor taking the M-atomic objects with a lax symmetric monoidal structure. As a byproduct, we obtain a monoidal natural version of the Yoneda map.
In Section 3, we recall the universal property of the Day convolution, and study its functoriality in the source and the target.
In Section 4, we recall some facts about (p-typical) (pre-)m-commutative monoids, and study their multiplicative structure. We observe that the ∞-category of m-commutative monoids can naturally be endowed with two symmetric monoidal structures, and we show that these two structures coincide.
In Section 5, we recall the definition of the higher cocartesian structure, and show that it satisfies certain expected properties. In particular, we show that tensoring a family of objects is indeed given by their colimit.
In Section 6, we define m-semiadditive algebraic K-theory using the tools developed in the previous sections, and study its properties. We construct it in two different ways, first using the S •construction, and second by exhibiting it as the universal way to make ordinary algebraic K-theory into an m-semiadditive functor. We leverage the second definition of m-semiadditive algebraic K-theory to endow it with a lax symmetric monoidal structure.
In Section 7, we study the interplay between m-semiadditive algebraic K-theory and semiadditive height. In particular, we show that it can increase the height of rings at most by one. Furthermore, we show that if the ring has (height n) p-th roots of unity, then the height of its m-semiadditive algebraic K-theory is exactly n + 1.
In Section 8, we study the connection between higher semiadditive algebraic K-theory and chromatically localized K-theory. We apply the Quillen-Lichtenbaum conjecture and the Galois descent result for chromatically localized K-theory, to show that the higher semiadditive algebraic K-theory of p-invertible algebras coincides with their T(1)-localized algebraic K-theory. Finally, we use the Quillen-Lichtenbaum result for BP⟨n⟩ to show that the higher semiadditive algebraic K-theory of E(n)-algebras lands in T(n + 1)-local spectra, and that specifically their 1-semiadditive algebraic K-theory coincides with their T(1)-localized algebraic K-theory.

Conventions
Throughout the paper, we work in the framework of ∞-categories, mostly following the notations of [Lur09,Lur17]. For brevity, we use the word category to mean an ∞-category. We also generally follow the notation and terminology of [CSY21a] related to higher semiadditivity, but we diverge by working exclusively in the p-typical case.
(1) We denote the space of morphisms between two objects X, Y ∈ C by hom C (X, Y ) and omit C when it is clear from the context. If C is D-enriched (e.g. in a mode D = M, or closed symmetric monoidal D = C), we denote by hom D C (X, Y ) the D-object of morphisms and omit C when it is clear from the context.
(2) We say that a space A ∈ S is (a) a p-space, if all the homotopy groups of A are p-groups.
(b) m-finite for m ≥ −2, if m = −2 and A is contractible, or m ≥ −1, the set π 0 A is finite and all the fibers of the diagonal map ∆ : (3) For −2 ≤ m ≤ ∞, we denote by S (p) m ⊂ S the full subcategory spanned by all m-finite p-spaces.
(4) We say that a category C is (p-typically) m-semiadditive if all m-finite p-spaces A ∈ S (p) m are C-ambidextrous.
(5) We denote by Cat st ⊂ Cat the subcategory spanned by all stable categories and exact functors.
(6) For a collection K of indexing categories, we let Cat K ⊂ Cat be the subcategory spanned by all categories admitting all colimits indexed by I ∈ K and functors preserving them.
m , and we let Cat st m-fin ⊂ Cat m-fin be the subcategory of those categories which are additionally stable and functors which are additionally exact.

Acknowledgements
We are grateful to the anonymous referees for valuable suggestions and corrections, particularly for suggesting the strategy for Theorem D. We would like to thank the entire Seminarak group, especially Shaul Barkan, Shachar Carmeli, Shaul Ragimov and Lior Yanovski, for useful discussions on different aspects of this project, and for their valuable comments on the paper's first draft. In particular, we would like to thank Lior Yanovski for sharing many ideas regarding atomic objects appearing in Section 2, and for suggesting the proof of Proposition 3.6. We would also like to thank Bastiaan Cnossen and Maxime Ramzi for helpful comments on the paper's first version. The second author is supported by ISF1588/18 and BSF 2018389.

Atomic Objects
Let M be a mode, that is, an idempotent algebra in Pr L (see [CSY21a, Section 5] for generalities on modes). In this section we study M-atomic objects (see Definition 2.3), a finiteness property of objects in categories C ∈ Mod M in the mode M, which generalizes both compactness (for the case M = Sp, see Proposition 2.8) and dualizability of modules (for the case C = LMod R (M), see Proposition 2.54). The results of this section are subsequently used in Definition 6.20 to define the higher semiadditive algebraic K-theory of algebras in ‫צ‬ [m] (see Definition 4.11), and in particular for algebras in Sp T(n) , including its lax symmetric monoidal structure.
In Subsection 2.1 we give the definition of atomic objects (see Definition 2.3) and study their basic properties. We show that taking atomic objects is functorial in internally left adjoint functors (see Definition 2.12). Analogously to the condition of being compactly generated, we study the condition of being generated under colimits and the action of M from the M-atomic objects, which we call being M-molecular (see Definition 2.10), and we explain its relationship to internally left adjoint functors. Lastly, in Proposition 2.24 we show that for any absolute limit I of M (see Definition 2.18), the atomic objects are closed under I op -shaped colimits. This yields a functor (−) at : Mod iL M → Cat K where K is any small collection of opposites of absolute limits of M. In Subsection 2.2 and Subsection 2.3 we study the connection between M-atomic objects and M-valued presheaves (see Definition 2.27), and the multiplicative structure of both functors. The main result of this section is Theorem 2.46, exhibiting a symmetric monoidal adjunction Moreover, the unit of this adjunction is the Yoneda map, and as an immediate consequence, Corollary 2.47 shows that よ Lastly, in Subsection 2.4 we study atomic objects in categories of left modules. In Proposition 2.54 we show that atomic objects and left dualizable left modules coincide, i.e. LMod at R = LMod ldbl R . Remark 2.1. Many of the results of this section can be generalized to modules in Pr L over any presentably monoidal category V ∈ Alg(Pr L ) and V-linear functors. Parts of these generalizations were carried out by the first author in [BM23], building on the works of [GH15,Hin20,Hin21,Hei23]. The main feature of modes is that being an M-module is a property rather than extra structure, and that any left adjoint functor is automatically M-linear. As such, working over a mode simplifies the definitions and proofs, and avoids using enriched category theory. Since this suffices for our applications in the rest of the paper, we have restricted to this case.

Atomics and Internally Left Adjoints
Proof. We prove this using the Yoneda lemma. Let m ∈ M. Recall that F : C → D is a map in Mod M , so that it commutes with m ⊗ −, so we conclude that The fact that it is a lift of the S-enriched hom is the case m = 1 M .
Definition 2.3. Let C ∈ Mod M . An object X ∈ C is called M-atomic, if hom M (X, −) : C → M commutes with colimits. We denote by C M-at ⊆ C the full subcategory of the M-atomic objects. When the mode is clear from the context, it is dropped from the notation.
Remark 2.4. The definition of atomic objects will be made functorial in Definition 2.17. Proposition 2.7. The only S-atomic object in S is the point * .
Proof. Let X ∈ S at be atomic, then for any Y ∈ S we have hom(X, Y ) ∼ = hom(X, colim Thus X corepresents the identity functor id : S → S, namely X = * .
Proposition 2.8. Let C ∈ Mod Sp be a presentable stable category, then the Sp-atomics are the compact objects, i.e. C Sp-at = C ω .
Proof. Let X ∈ C. First assume that X is atomic. Recall that Ω ∞ : Sp → S commutes with filtered colimits, so that hom(X, −) ∼ = Ω ∞ hom Sp (X, −) commutes with filtered colimits, i.e. it is compact. Now assume that X is compact. Recall that for any n ∈ Z, the functor Σ n : Sp → Sp commutes with all limits and colimits and in particular with filtered colimits, thus hom(X, Σ n −) ∼ = Ω ∞ Σ n hom Sp (X, −) also commutes with filtered colimits. Additionally, the functors Ω ∞ Σ n : Sp → S are jointly conservative, implying that hom Sp (X, −) commutes with filtered colimits. Furthermore, it commutes with finite limits, thus by stability also with all finite colimits, which together with filtered colimits generate all colimits.
Proposition 2.9. Let C ∈ Mod M , then C at ∈ Cat is a small category.
Proof. Let κ be a regular cardinal such that the unit 1 M ∈ M is κ-compact. We show that C at ⊆ C κ , that is the atomics are κ-compact. Let X ∈ C be an atomic object, so in particular hom M (X, −) commutes with κ-filtered colimits. Since 1 M is κ-compact, hom(1 M , −) commutes with κ-filtered colimits, implying that the composition hom(X, −) ∼ = hom(1 M , hom M (X, −)) commutes with κfiltered colimits.
Definition 2.10. Let C ∈ Mod M . We say that a collection of atomic objects B ⊆ C at are M-atomic generators, if C is generated from B under colimits and the action of M. 4 If such B exists, we say that C is M-molecular . 5 If the mode is clear from the context, we call C molecular and say that B are atomic generators.
Example 2.11. Every mode M is itself M-molecular, because the unit 1 M is atomic and any object m can be written as m ⊗ 1 M . 6 Definition 2.12. Let C, D ∈ Pr L . We say that a functor F : C → D is internally left adjoint if it is left adjoint in Pr L , namely if it is a left adjoint functor and its right adjoint G : D → C is itself a left adjoint. We denote by Fun iL (C, D) ⊆ Fun L (C, D) the full subcategory of internally left adjoint functors. We let Mod iL M be the wide subcategory of Mod M with the same objects, and morphisms the internally left adjoint functors.
Proposition 2.13. Let C, D ∈ Mod M , and let F : C → D be an internally left adjoint functor, then it sends atomic objects to atomic objects.
Proof. By assumption the right adjoint G : D → C is itself a left adjoint, thus preserves colimits. Let X ∈ C at be an atomic object, then using Lemma 2.2 hom M (F X, −) ∼ = hom M (X, G−), which is the composition of G and hom M (X, −), both of which preserve colimits, so that F X is atomic.
Proposition 2.14. Let C, D ∈ Mod M , and let F : C → D be a left adjoint functor. If C is molecular and F sends a collection of atomic generators B ⊂ C to atomic objects in D, then F is internally left adjoint.
Proof. We wish to show that G, the right adjoint of F , is itself a left adjoint, namely that it preserves colimits. Let Y i : I → D be a diagram, and we wish to show that G(colim Y i ) ∼ = colim GY i . By the Yoneda lemma, this is equivalent to checking that for every X ∈ C we have hom(X, G(colim Y i )) ∼ = hom(X, colim GY i ).
Since hom(−, −) ∼ = hom(1 M , hom M (−, −)), it suffices to check that for every X ∈ C we have Let A denote the collection of X ∈ C for which this condition holds, and we shall show that A = C.
First, for every X ∈ B, we know that where the first and third steps follow from Lemma 2.2, the second step follows from the assumption that F X is atomic since X ∈ B and F sends B to atomic objects, and the fourth step follows from the fact X is atomic. Therefore, B ⊆ A. Second, for every X ∈ A and m ∈ M, we know that hom M (m⊗X, −) ∼ = hom M (m, hom M (X, −)) so that m ⊗ X ∈ A, i.e. A is closed under the action of M.
Third, for every diagram X j : J → C landing in A, we know that hom M (colim J X j , −) ∼ = lim J op hom M (X j , −) so that colim J X j ∈ A, i.e. A is closed under colimits.
We have shown that B ⊆ A and that A is closed under the action of M and colimits, and by assumption B are atomic generators, thus A = C as needed.
Recall that for C ∈ Mod M we have an equivalence Fun L (M, C) ∼ − → C given by evaluation at 1 M . Its inverse sends X ∈ C to the functor − ⊗ X : M → C (part of the data admitting C as an M-module). Furthermore, the right adjoint of − ⊗ X : Proof. First, by Proposition 2.13 and the fact that the 1 M is atomic, the functor indeed lands in the full subcategory C at . In particular, it is also fully faithful as the restriction of an equivalence to two full subcategories. We need to show that it is essentially surjective, i.e. that if X ∈ C at then − ⊗ X : M → C is internally left adjoint. This holds since its right adjoint is hom M (X, −) : C → M, which by assumption preserves colimits. For the last part, as − ⊗ X : M → C is internally left adjoint, Proposition 2.13 implies that it sends atomic objects to atomic objects.
Remark 2.16. In Corollary 2.48 we extend the last part of the proposition to show that in fact C at is a module over M at .
In light of this proposition, we construct the functor of taking atomics functorially. We also recall from Proposition 2.9 that C at is a small category. Definition 2.18. Let I be an indexing category. We say that I is an absolute limit of M if for any C ∈ Mod M , I-shaped limits in C commute with colimits.
Remark 2.19. The term absolute limit is usually used in the context of enriched categories, saying that I is an absolute limit of V ∈ Mon(Cat) if any V-enriched functor commutes with I-shaped limits. We will not use this condition in this paper, but for the convenience of the reader we remark on the connection between this condition and the one appearing in Definition 2.18 when V is a mode. Assume that I is an absolute limit in the ordinary sense, namely that V-enriched functors commute with I-shaped limits. Let C ∈ Mod V . For any indexing category J consider colim J : C J → C. This functor commutes with colimits, and since V is a mode, it is a morphism in Mod V , so, as referred to in Remark 2.1, it is canonically V-enriched, and therefore commutes with I-shaped limits. This holds for any J, meaning that I-shaped limits in C commute with colimits, reproducing Definition 2.18.
The implication in the other direction should follow from a working theory of enriched left Kan extensions and their compatibility with the enriched Yoneda embedding, which we are unaware of a reference for.
Lemma 2.20. If I is an absolute limit of M and M → N is map of modes, then I is an absolute limit of N as well.
Proof. This is immediate from the fact that Mod N ⊆ Mod M .
Lemma 2.21. Let I be an absolute limit of M, and C ∈ Mod M . Then m ⊗ − : C → C commutes with I-shaped limits, for any m ∈ M.
Proof. By assumption, lim I : C I → C commutes with colimits. Therefore, it is a map in Mod M , so that it also commutes m ⊗ − : C → C for any m ∈ M.
Proposition 2.22. Let I be an absolute limit of M, then for any C ∈ Mod M , the atomics C at ⊂ C are closed under I op -shaped colimits.
Proof. Let X i : I op → C be a diagram landing in the atomics. Recall that hom M (−, −) : C op × C → M commutes with limits in the first coordinate, thus hom M (colim I op X i , −) is equivalent to ∆ commutes with colimits since colimits in functor categories are computed level-wise. Since each X i is atomic, each hom M (X i , −) commutes with colimits, and as colimits in functor categories are computed level-wise, we get that (hom M (X i , −)) I commutes with colimits. By assumption, I is an absolute limit of M, thus lim I commutes with colimits. This shows that hom M (colim I op X i , −) commutes with colimits, i.e. that colim I op X i is indeed atomic.
Remark 2.23. Let F : C → D be an internally left adjoint functor, and let I be an absolute limit of M. Then F preserves colimits, and the atomics are closed under I op -shaped colimits, so that the induced functor between the atomics preserves I op -shaped colimits.
The following claim immediately follows. Proof. Recall that C itself is stable, so the first part follows from the commutativity of finite limits and colimits in stable categories. For the second part, first note that the zero object is obviously atomic. As finite limits are absolute, the atomics are closed under finite colimits, so it suffices to show that the atomics are also closed under desuspensions. Let

Atomics and Presheaves
Throughout this subsection, let K ⊂ {I op | I absolute limit of M} be some small collection of opposites of absolute limits of M (not necessarily all of them, for instance, K is allowed to be empty). We also let K op = {I | I op ∈ K} be the collection of all of the opposite categories.
Remark 2.28. The definition will be made functorial in Definition 2.34.
For the case M = S, [GHN17, Lemma 10.6] shows that P K (C 0 ) is presentable. From this we deduce the following: Proposition 2.29. There is an equivalence P M K (C 0 ) ∼ = P K (C 0 ) ⊗ M, and in particular it is presentable and in the mode M.
Proof. Indeed, we have an equivalence where the first equality is [Lur17, Proposition 4.8.1.17], the second is passing to the opposite, the third is the universal property of P K given in [Lur09, Corollary 5.3.6.10], the fourth is by passing to the opposite, and the last is by definition.
is closed under limits and colimits, thus the inclusion has both adjoints.
. We need to show that colim J F j and lim J F j are again in P M K (C 0 ), i.e. that they commute with all limits indexed by I ∈ K op . Let X i : I op → C 0 be a diagram. Using the fact that colimits and limits in functor categories are computed level-wise, and that I is an absolute limit, we get: Similarly lim J F j ∈ P M K (C 0 ), since limits commute with limits.
Definition 2.31. Let C 0 ∈ Cat K . We denote by L K : . Proof. f * is given by pre-composition with f op : C op 0 → D op 0 , which preserves limits indexed by I ∈ K op , as the opposite of a morphism in Cat K .
preserves all limits and colimits and thus has a right adjoint f * and a left adjoint f ! .
Proof. By Lemma 2.30, P M K (C 0 ) is closed under limits and colimits in P M (C 0 ), which are thus computed level-wise, and similarly for D 0 . Therefore, we get showing that f * commutes with colimits, and similarly for limits.
Lemma 2.32 shows that the functor Fun and f to f * . By Proposition 2.29, the categories P M K (C 0 ) are in the mode M, and by Lemma 2.33, the morphism f * is a right adjoint, so that the functor factors as Definition 2.34. We define the functor P M K : Cat K → Mod M by passing to the left adjoints in Proposition 2.36. There is a natural transformation L K : P M ⇒ P M K of functors Cat K → Mod M , making the construction of Definition 2.31 natural. Definition 2.37. We define the Yoneda map よ where よ is the ordinary Yoneda embedding, and the second map is given by tensoring with the unit map S → M.
Remark 2.38. Generally, the Yoneda map よ Proposition 2.39. The Yoneda map can be upgraded to a natural transformation よ Proof. The natural transformation is obtained by the following diagram: Here よ : ι ⇒ P is the ordinary Yoneda natural transformation constructed in [HHLN20b, Theorem 8.1], the natural transformation u : id ⇒ − ⊗ M is the unit map of the free-forgetful adjunction − ⊗ M : Pr L ⇄ Mod M : (−) given by tensoring with S → M, and L K : P M ⇒ P M K is the natural transformation of Proposition 2.36.
Proof. We first reduce to the case where K = ∅. Since L K : P M (C 0 ) → P M K (C 0 ) is the left adjoint of the inclusion, using Lemma 2.2 we get We finish the proof by showing that the latter is equivalent to F (X), using the Yoneda lemma in the category M. Indeed, let m ∈ M be any object, then where the first and second step use the exponential adjunction, the third uses the free-forgetful adjunction C → C ⊗ M, the fourth uses the ordinary Yoneda lemma for C 0 and the last step uses that the action of M is level-wise.
We use the same notation よ M K : C 0 → P M K (C 0 ) at to denote the factorization. Recall from Proposition 2.39 that the Yoneda map gives a natural transformation よ M K : ι K ⇒ P M K of functors Cat K → Cat. Since taking the atomics lands in Cat K by Proposition 2.24, together with Corollary 2.41, we obtain a natural transformation よ M K : id ⇒ P M (−) at of functors Cat K → Cat K . Proposition 2.42. For any C 0 ∈ Cat K the category P M K (C 0 ) is molecular, with atomic generators よ M K (X) for X ∈ C 0 . Proof. We first show the result for P M (C 0 ), i.e. for the case K = ∅. Recall that P M (C 0 ) ∼ = P(C 0 )⊗M is generated under colimits from the image of P(C 0 )×M, i.e. from objects of the form F ⊗m. Second, P(C 0 ) is generated under colimits from objects of the form よ(X) for X ∈ C. Therefore, P M (C 0 ) is generated under colimits and the action of M from objects of the form よ M (X) for X ∈ C, which are indeed atomic by Corollary 2.41. For the general case, recall that L K : is an internally left adjoint functor so it sends atomic objects to atomic objects by Proposition 2.13, thus よ M K (X) is atomic for any X ∈ C. Since it preserves colimits and the action of M, and よ M (X) generate P M (C 0 ) under these operations, their images よ M K (X) generate the essential image of L K under these operations. In addition, L K is essentially surjective, so that よ M K (X) are atomic generators of P M K (C 0 ) as needed.
Proposition 2.43. There is an adjunction Proof. To check that the data in the theorem supports an adjunction, it suffices to check that for any C 0 ∈ Cat K and D ∈ Mod iL M , the canonical map is an equivalence (in fact, it suffices to show this for the hom spaces, rather then the functor categories, but we show that the stronger statement holds). Note that Furthermore, both the first and last categories in Eq.
(1) are full subcategories of the first and last categories in Eq.
(2), showing that the composition in Eq.
To finish the argument, we need to show that Eq. (1) is essentially surjective. To that end, let F : C 0 → D at be a functor preserving I op -shaped colimits for I op ∈ K. We can post-compose it with the inclusion D at → D, and using Eq. (2) we get a left adjoint functorF : P M K (C 0 ) → D, and we need to show that it is in fact internally left adjoint. By construction, for any X ∈ C 0 we have thatF (よ M K (X)) ∼ = F (X) ∈ D at is atomic. Proposition 2.42 shows that these are atomic generators for P M K (C 0 ), so Proposition 2.14 shows thatF is indeed internally left adjoint.

Tensor Product of Atomics
Proposition 2.44. The symmetric monoidal structure on Mod M restricts to a symmetric monoidal structure on the subcategory Mod iL M .
Proof. Since Mod iL M is a wide subcategory of Mod M , all we need to show is that if Let R i be the right adjoints of L i , which by assumption are themselves left adjoints. Because they are left adjoints, we can tensor them to obtain another left adjoint functor R 1 ⊗ R 2 : D 1 ⊗ D 2 → C 1 ⊗ C 2 . It is then straightforward to check that tensoring the unit and counit of L i ⊣ R i exhibit an adjunction L 1 ⊗ L 2 ⊣ R 1 ⊗ R 2 , showing that L 1 ⊗ L 2 is an internally left adjoint functor.
We recall that the category Cat K has a symmetric monoidal structure, developed in [ Note that for any operad O we get an induced adjunction whose unit is an enhancement of the Yoneda map (landing in the atomics) to O-algebras. Furthermore, for any C ∈ Alg O (Mod iL M ) we see that C at ⊂ C is in fact an O-monoidal subcategory. We therefore get the following corollary, which generalizes [Gla16, Section 3] and [Lur17, Corollary 4.8.1.12] from the case of M = S, K = ∅ and O = E ∞ and makes them natural.
Corollary 2.47. The Yoneda natural transformation lifts to a natural transformation よ Recall that in Proposition 2.15 we showed that if X ∈ C at and m ∈ M at then m⊗X ∈ C at . Using Theorem 2.46, we strengthen this into a module structure, using the fact that any lax symmetric monoidal functor lands in modules over the image of the unit.
Corollary 2.48. The functor of atomic objects factors as a lax symmetric monoidal functor We also mention the following easy corollary of Proposition 2.44.
Lemma 2.49. Let L : M 1 → M 2 be a smashing localization of modes and let N be another mode

Atomic Modules
In the remainder of the section we show that the atomic objects in LMod R for R ∈ Alg We now recall the following result about left dualizability and adjunctions.
Proof. We explain how this follows from [Lur17, Proposition 4.6.2.18], with C = M, A = R rev and the roles of X and Y reversed (see also [Lur17, Remark 4.6.3.16]). For the first direction, assume that there is an adjunction and let η : id M ⇒ Y ⊗ R X ⊗ 1 M − be the unit. By the adjunction, we know that for each P ∈ M and Q ∈ LMod R the composition is an equivalence. Since both functors in the adjunction preserve colimits, and the categories are in the mode M, the adjunction is M-linear. Therefore the two maps coincide. This shows that c = η 1 M satisfies condition ( * ) of the cited proposition.
Similarly, for the other direction, if Y is left dual to X then the coevaluation map c : which is a unit of an adjunction by condition ( * ) of the cited proposition.
Proof. Consider [Lur17, Corollary 4.2.3.7 (2)] where both C and M in the reference's notation are our M, A = 1 C and B = R. Then, the functor which therefore commutes with all colimits, showing that R is atomic. The second part follows from Proposition 2.15.
Proposition 2.53. Let R ∈ Alg(M). Then LMod R is molecular with R as an atomic generator.
Proof. The previous lemma shows that R is indeed atomic, and we need to show that it generates LMod R under colimits and the action of M. Specifically, we will show that LMod R is generated under colimits from R ⊗ m for m ∈ M. Let X ∈ LMod R , and consider the functor hom M (X, −) : commutes with colimits, we get that X is atomic. Now assume that X is atomic. The two functors hom M (X, −) and X ∨ ⊗ R − are colimit preserving, i.e. morphisms in Mod M , thus also commute with tensor from M. Proposition 2.53 shows that LMod R is generated from R by these operations, and by the construction of X ∨ , they agree on R, so the canonical map between the two is an equivalence. This shows that X ⊗ 1 M − is left adjoint to hom M (X, −) ∼ = X ∨ ⊗ R −, concluding by Proposition 2.51.
Remark 2.55. If R ∈ CAlg(M), then left and right R-modules coincide, and the category of modules Mod R is equipped with a symmetric monoidal structure for which dualizable modules coincide with left dualizable modules, thus also with atomic objects, that is Mod dbl R = Mod at R . Combining Theorem 2.46, Theorem 2.50 and Proposition 2.54 we get the following main result.
Corollary 2.56. There is a lax symmetric monoidal functor LMod at (−) : Alg(M) → Cat K , and LMod at R = LMod ldbl R . As a by product, we also obtain the following result.
Lemma 2.57. Let F : M → N be a map of modes, then it sends M-atomic objects to N-atomic objects.
Proof. By Proposition 2.54, the M-atomic objects in M are the dualizable objects. Since F is symmetric monoidal it sends dualizable objects to dualizable objects. Thus the M-atomic objects are sent to dualizable objects in N. Again by Proposition 2.54, the dualizable objects in N are N-atomic.

Day Convolution
The Day Convolution on functor categories was developed in [Gla16,Lur17]. In this section we prove results about the Day convolution, specifically its functoriality in the source and target. The results of this section are used in Theorem 4.26 to show that the mode symmetric monoidal structure on higher commutative monoids coincides with the localization of the Day convolution. This is subsequently used in Theorem 6.18 to endow higher semiadditive algebraic K-theory with a lax symmetric monoidal structure.
We begin by recalling the universal property of the Day convolution: Theorem 3.1 ([Lur17, Remark 2.2.6.8]). Let I, C be symmetric monoidal categories, and assume that C has all colimits and that its tensor product preserves colimits in each coordinate separately. Then, there is a symmetric monoidal structure on Fun(I, C), called the Day convolution denoted by ⊛, satisfying the following universal property: There is an equivalence of functors CMon(Cat) → Cat which lifts the equivalence of functors Cat → Cat Example 3.2. Let I be a symmetric monoidal category. Then I op is also endowed with a symmetric monoidal structure, and S can be endowed with the cartesian structure, yielding the Day convolution on P(I) = Fun(I op , S Proposition 3.3. Let I, C and D be symmetric monoidal categories, and assume that C and D have all colimits and that their tensor product preserve colimits in each coordinate. Let F : C → D be a functor and letF : Fun(I, C) → Fun(I, D) be the functor induced by post-composition. If F is lax symmetric monoidal, then so isF . If F is colimit preserving, then so isF . If F is both colimit preserving and symmetric monoidal, then so isF .
Proof. We begin with the first part. The identity functor of Fun(I, C) is (lax) symmetric monoidal, therefore by the universal property of the Day convolution, the corresponding functor Fun(I, C) × I → C is also lax symmetric monoidal. Post-composition of this functor with the lax symmetric monoidal functor F gives a lax symmetric monoidal functor Fun(I, C) × I → D. Using the universal property again, we get thatF : Fun(I, C) → Fun(I, D) is also lax symmetric monoidal.
For the second part, if F is colimit preserving, then since colimits in functor categories are computed level-wise,F is colimit preserving.
Lastly, we assume that F is both colimit preserving and symmetric monoidal. We already know from the second part thatF is colimit preserving. We show that the lax symmetric monoidal structure from the first part is in fact symmetric monoidal. Recall that by [Lur17, Example 2.2.6.17], the Day convolution of X, Y ∈ Fun(I, C) is given on objects by The lax symmetric monoidal structure ofF is then given by the canonical map: where map (1) uses the fact F is lax symmetric monoidal, and (2) is the assembly map. Since F is symmetric monoidal (1) is an equivalence, and since F is colimit preserving (2) is an equivalence, showing thatF is in fact symmetric monoidal.
Our next goal is to study the behavior of the Day convolution under the change of the source I, namely given a symmetric monoidal functor p : I → J, what can we say about p ! : Fun(I, C) → Fun(J, C) and p * : Fun(J, C) → Fun(I, C).
We wish to thank Lior Yanovski for suggesting the following argument to prove Proposition 3.6. Applying this to the special case where p is the unit map * → I we get Remark 3.9. One can directly use the universal property of the Day convolution to show that p * is lax symmetric monoidal, even only assuming that p is lax symmetric monoidal. In fact, one can use the main result of [HHLN20a] to construct an oplax symmetric monoidal structure on p ! in this way while only assuming that p is lax symmetric monoidal, and prove that it is symmetric monoidal in case p is. However, we have not shown that the lax symmetric monoidal structure on p * obtained in the above corollary coincides with the one obtained directly from the universal property of the Day convolution.

Higher Commutative Monoids
In this section we recall the notion of (p-typical) m-commutative monoids as developed in [Har20] and [CSY21a] (see Definition 4.2), and their relationship to higher semiadditivity (see Theorem 4.9), which feature prominently in the definition of higher semiadditive algebraic K-theory in Definition 6.5. A key result of this section is Theorem 4.26, which shows that for C ∈ CAlg(Pr L ), the symmetric monoidal structures on CMon (p) m (C) coming from the mode structure on CMon (p) m (S) and from the Day convolution coincide. This result is used in Theorem 6.18 to endow higher semiadditive algebraic K-theory with a lax symmetric monoidal structure.

Definition and Properties
Definition 4.1. Let C ∈ Pr L be a presentable category. We define the category of (p-typical) prem-commutative monoids in C by PCMon   Lemma 4.5. Let C ∈ Pr L be a κ-presentable category. Then, µ-filtered colimits commute with µ-small limits in C, for any µ ≥ κ.
Proof. First, the case C = S is [Lur09, Proposition 5.3.3.3]. Second, the case C = P(C 0 ) follows from the previous case, since limits and colimits are computed level-wise in functor categories. Lastly, for the general case we have that C ∼ = Ind κ (C κ ). By [Lur09, Proposition 5.3.5.3], C ⊆ P(C κ ) is closed under κ-filtered colimits. Additionally, [Lur09, Corollary 5.3.5.4 (3)] shows that it is also closed under limits, since limits commute with limits. To conclude, C is closed under µ-filtered colimits and µ-small limits in P(C κ ) for any µ ≥ κ, and by the second case the result holds for P(C κ ) for any µ.   Proof. Follows immediately from the characterization given in Remark 4.3 and the fact that F commutes with these limits.

Higher Commutative Monoids and Semiadditivity
The   We recall the following:   In particular, this shows that K, the collection of all finite categories and m-finite p-spaces, is a collection of absolute limits of M. Theorem 2.46 then shows that there is a lax symmetric monoidal functor (−) at : Mod iL M → Cat K . Recall that there is a fully faithful functor (−) ⊗ : CMon(Cat) lax → Op from the category of symmetric monoidal categories and lax symmetric monoidal functors to operads. Note that Cat st m-fin ⊂ Cat K is the full subcategory on those categories which are in addition stable, but it is not a sub-symmetric monoidal category, since the unit of Cat K is not stable. However, it is true that the tensor product of a family of categories in either category is the same, in particular Cat st,⊗ m-fin is a sub-operad of Cat ⊗ K . Therefore, we get that the map of operads

Tensor Product of Higher Commutative Monoids
Let C ∈ CAlg(Pr L ) be a presentably symmetric monoidal category. In this subsection, we endow CMon (p) m (C) with two symmetric monoidal structures, and show that they coincide. The first, which we call the mode symmetric monoidal structure (see Definition 4.16), comes from the fact that CMon (p) m (S) is a mode. The second, which we call the localized Day convolution (see Definition 4.25), is obtained by localizing the Day convolution on PCMon (p) m (C). Finally, in Theorem 4.26 we show that the two structures coincide.
Recall that by Theorem 4.9, CMon (p) m (S) is a mode, and in particular it is equipped with a symmetric monoidal structure.
Definition 4.16. Let C ∈ CAlg(Pr L ) be a presentably symmetric monoidal category. The equivalence CMon (p) m (C) ∼ = CMon (p) m (S) ⊗ C of Theorem 4.9 endows CMon (p) m (C) with a presentably symmetric monoidal structure which we call the mode symmetric monoidal structure and denote by ⊗. Furthermore, by construction, F seg : C → CMon (p) m (C) is endowed with a symmetric monoidal structure.
In a different direction, consider the category Span(S (p) m ). Since S (p) m is closed under products, it has a cartesian monoidal structure. By [Hau17, Theorem 1.2 (iv)], its span category Span(S (p) m ) is endowed with a symmetric monoidal structure given on objects by their cartesian product in S (p) m . Therefore, the opposite category Span(S (p) m ) op is also endowed with a symmetric monoidal structure.
Remark 4.17. The symmetric monoidal structure on Span(S (p) m ) that we use is not the cartesian or cocartesian structure. In fact, the cartesian and cocartesian structures coincide (since products and coproducts coincide in Span(S (p) m ), being a semiadditive category), and are given on objects by the disjoint union of spaces, whereas the symmetric monoidal structure we use is given on objects by the product of spaces.    Proof. We let L ′ = L ⊗ id. First note that L ′ is indeed a reflective localization by [CSY21a, Lemma 5.2.1]. Using [Lur17, Proposition 2.2.1.9] we endow D 0 with the localized symmetric monoidal structure, making L into a symmetric monoidal functor. Since ⊗ is the coproduct of CAlg(Pr L ), this makes the categories and the map L ′ : D⊗E → D 0 ⊗E symmetric monoidal. Now let X → Y ∈ D⊗E be an L ′ -equivalence. For any Z ∈ D ⊗ E, we have which is an equivalence.
Note that by the Yoneda lemma, for any A ∈ S (p) m , the object よ(A) ∈ PCMon (p) m (S) corepresents the evaluation at A functor PCMon (p) m (S) → S given by X → X(A). We also note that these functors over all A ∈ S In particular, if X ∈ CMon(S), then so is hom PCMon   Proof. Consider the following commutative diagram in Pr L : The bottom map is an equivalence by Theorem 4.9. The top map is a symmetric monoidal equivalence by Proposition 3.10. By Lemma 4.23, L seg : PCMon (p) m (S) → CMon (p) m (S) is compatible with the Day convolution, so by Lemma 4.21 the left map is also compatible with the symmetric monoidal structure. Therefore, the right map is also compatible with the Day convolution. The main result of this subsection is the following: Theorem 4.26. Let C ∈ CAlg(Pr L ), then the mode symmetric monoidal structure and the localized Day convolution on CMon (p) m (C) coincide, making the following diagram in CAlg(Pr L ) commute: We begin by proving the result for C = S. Therefore, 1⊛ also represents (−), so that 1⊛ ∼ = 1 ⊗ . Since CMon (p) m (S) is a mode, it has a unique presentably symmetric monoidal structure with the given unit as in [CSY21a, Proposition 5.1.6], so that localized Day convolution and the mode symmetric monoidal structure on CMon (p) m (S) coincide. Since there is a unique map of modes S → CMon (p) m (S), the functors L seg F and F seg coincide.
Proof of Theorem 4.26. Consider the following diagram in CAlg(Pr L ) where we endow CMon (p) m (C) with the localized Day convolution structure (and the rest of the categories are endowed with a single symmetric monoidal structure, as we have shown that the two structures on CMon (p) m (S) coincide.) The bottom map is an equivalence by Theorem 4.9, and we wish to upgrade it to a symmetric monoidal equivalence. The top map is a symmetric monoidal equivalence by Proposition 3.10. As in the proof of Proposition 4.24, both the left and the right maps are symmetric monoidal. This shows that the bottom map is the localization of the top map, and thus inherits the structure of a symmetric monoidal equivalence.

Higher Cocartesian Structure
Endowing a category C ∈ Cat with a symmetric monoidal structure is the same as providing a lift C ⊗ ∈ CMon(Cat). If C has finite coproducts, it has a cocartesian structure C ⊔ given by the coproduct. An Eckmann-Hilton style argument characterizes it as the unique symmetric monoidal structure that commutes with coproducts (in all coordinates together), namely satisfying Building on [Har20, Theorem 5.23], in this section we define the (p-typical) m-cocartesian structure as an m-commutative monoid structure, and in Theorem 5.3 we show that it enjoys the expected properties, which in particular gives a construction of the ordinary cocartesian structure. The results of this section feature in the definition of higher semiadditive algebraic K-theory in Definition 6.5, by preserving the m-commutative monoid structure afforded by the m-cocartesian structure.
Definition 5.1. The category of categories with a (p-typical) m-symmetric monoidal structure is CMon (p) m (Cat). That is, an m-symmetric monoidal structure on C ∈ Cat is a lift C ⊗ ∈ CMon (p) m (Cat).
In [Har20,Theorem 5.23] and [CSY21a, Proposition 2.2.7] it is shown that the category Cat m-fin of categories admitting colimits indexed by m-finite p-spaces is itself an (p-typically) m-semiadditive category for any −2 ≤ m ≤ ∞ (the proofs in the cited papers are not in the p-typical case, but the same proofs work in the p-typical case). In other words, the underlying functor is an equivalence. We denote its inverse by (−) ⊔m : Cat m-fin ∼ − → CMon (p) m (Cat m-fin ). We recall from Corollary 4.14 that there is an inclusion CMon (p) m (Cat m-fin ) → CMon (p) m (Cat).
Definition 5.2. For every C ∈ Cat m-fin , we call C ⊔m ∈ CMon (p) m (Cat) the m-cocartesian structure on C. When m is clear from the context, we shall write C ⊔ for C ⊔m .
Our next goal is to justify this name. In particular, we will show that for every m-finite p-space A, the map C A → C induced by evaluating C ⊔ at A → * is given by taking the colimit over A. More precisely, for any C ∈ Cat, let C * ∈ Fun((S (p) m ) op , Cat) be the functor Fun(−, C), given by sending m-finite p-space A to C A and q : A → B to q * : C B → C A . If we assume that C ∈ Cat m-fin , then q * : C B → C A has a left adjoint q ! : C A → C B . By passing to the left adjoints, we obtain a functor C ! ∈ Fun(S To prove this, we first note that not only each q * has a left adjoint q ! , but they also satisfy the Beck-Chevalley condition. This means that C * is in fact in Fun BC ((S (p) m ) op , Cat) (where Fun BC are functors such that each morphism is mapped to a right adjoint, such that the Beck-Chevalley condition is satisfied). We will use Barwick's unfurling construction [Bar17,Definition 11.3]. Barwick works in a more general context, allowing to prescribe only certain right-and wrong-way morphisms, but we shall not use this generality. After straightening, the unfurling construction for (S Using this result, we our now in position to prove Theorem 5.3. Proof of Theorem 5.3. By Barwick's theorem, Υ(C * ) has the properties we ought to prove for C ⊔ , so it suffices to show that C ⊔ ∼ = Υ(C * ). Furthermore, recall that that the underlying functor m ) is C * , it follows that it satisfies the m-Segal condition as well, thus Υ(C * ) ∈ CMon (p) m (Cat). Second, we need to show that Υ(C * ) lands in Cat m-fin . By assumption C ∈ Cat m-fin , thus the same holds for C A for all m-finite p-space A. For morphisms, we need to show they are sent to functors that commute with colimits indexed by any m-finite p-space A. Any morphism in Span(S (p) m ) is the composition of a right-way and a wrong-way map, so we can check these separately. So let q : A → B be a morphism of m-finite p-spaces. Since q ! is a left adjoint, it commutes with colimits indexed by any m-finite p-space A, so it is a morphism in Cat m-fin . Since colimits in functor categories are computed level-wise, the functor q * commutes with them, so it is also a morphism in Cat m-fin .
Remark 5.5. In light of Barwick's construction, one could define the m-cocartesian structure simply by C ⊔ = Υ(C * ). The reason why we define it via the equivalence (−) ⊔ : Cat m-fin ∼ − → CMon (p) m (Cat m-fin ) is two fold. First, this construction characterizes C ⊔ in a universal way. Second, Barwick's unfurling construction, although much more general then our definition, is not shown to be functorial in F , which will be used crucially for C ⊔ in our definition of semiadditive algebraic K-theory. Proof. Let C ∈ Cat st m-fin . We know that C ⊔ ∈ CMon (p) m (Cat m-fin ). By Proposition 4.13, for any mfinite p-space A, C A is computed the same in Cat, Cat m-fin and Cat st m-fin , and in particular it is stable. Furthermore, for any q : A → B, both q ! and q * are exact. Thus C ⊔ ∈ Fun(Span(S

Semiadditive Algebraic K-Theory
In this section we define an m-semiadditive version of algebraic K-theory. We begin by recalling the construction of ordinary algebraic K-theory, and present it in a way which is amenable to generalizations. We then generalize the definition to construct m-semiadditive algebraic K-theory in Definition 6.5, and connect it to ordinary algebraic K-theory in Corollary 6.10. We leverage this connection in Theorem 6.18 to endow the functor of m-semiadditive algebraic K-theory with a lax symmetric monoidal structure. This is later used to prove Theorem 8.10 and Theorem 8.23, two of the main results of this paper.

Ordinary Algebraic K-Theory
We recall the definition of the S • -construction for stable categories and exact functors. One defines the functor S • : Cat st → S ∆ op by letting S n C be the subspace of those functors X : [n] [1] → C that satisfy: (1) X ii = 0, (2) For all i ≤ j ≤ k the following is a bicartesian square The algebraic K-theory space functor K : Cat st → S is then defined as the composition K(C) = Ω|S • C|. One then proceeds to lift to (connective) spectra, e.g. by means of iterated S • -construction. We will give an equivalent construction of the spectrum structure, which will be easier to generalize. To that end, we show the following: Proof. For each n, the functor S n : Cat st → S is equivalent to hom([n − 1], −), and in particular it commutes with limits. Since limits in the functor category S ∆ op are computed level-wise, this implies that S • commutes with limits as well.
This together with Proposition 4.8 implies that we get an induced functor S • : CMon(Cat st ) → CMon(S) ∆ op . Employing Theorem 5.6, we give the following definition.
Definition 6.2. We define algebraic K-theory K : Cat st → Sp by K(C) = Ω|(S • (C ⊔ )) gpc |, that is, as the following composition Lemma 6.3. The composition of K : Cat st → Sp with Ω ∞ : Sp → S is K.
Proof. First note that (−) gpc : CMon(S) → Sp is a left adjoint, and therefore commutes with the colimit | − |, and that Ω((−) gpc ) ∼ = Ω as functors CMon(S) → Sp. This shows that our definition of algebraic K-theory is equivalent to the composition Consider the following diagram: Square (1) commutes because (−) ⊔ and (−) are inverses by Theorem 5.6. Square (2) commutes by the definition of the extension of S • to CMon. Square (3) commutes since the underlying commutes with geometric realizations. Square (4) commutes because Ω is a limit and the underlying is a right adjoint functor. Finally, the top-right composition is Ω ∞ K, whereas the left-bottom composition is K.
We now claim that the above definition of the spectrum structure coincides with the standard one. Note that by construction K in fact lands in connective spectra. is a full subcategory. Therefore, the forgetful is fully faithful, meaning that product preserving functors C → D have unique or no lifts to CMon gl (D). In particular, for D = S, using the equivalence CMon gl (S) ∼ = Sp ≥0 , we get that the forgetful is fully faithful. Applying this to the case C = Cat st , the result follows since K has a lift, which is therefore unique.

Definition of Semiadditive Algebraic K-Theory
We restrict the S • -construction to Cat st m-fin , and use the same notation i.e. S • : Cat st m-fin → S ∆ op . Proposition 4.13 shows that Cat st m-fin → Cat st preserve limits, thus by Lemma 6.1, the restriction S • : Cat st m-fin → S ∆ op preserves limits as well, so using Proposition 4.8 again we get an induced functor S • : CMon (p) m (Cat st m-fin ) → CMon (p) m (S) ∆ op . Employing Theorem 5.6, we give the following definition.
Example 6.6. Proposition 6.4 shows that the case m = 0 recovers the p-localization of the ordinary K-theory of stable categories. All of the functors either preserve all limits (in the case of (−) ⊔ , S • and Ω) or preserve all colimits (in the case of (−) gpc and | − |). In particular they are all msemiadditive functors, thus the composition is an m-semiadditive functor as well.

Relationship to Ordinary Algebraic K-Theory
Proposition 6.7 shows that K [m] is an m-semiadditive functor, and in particular satisfies K [m] (C A ) ∼ = K [m] (C) A for any m-finite p-space A. One may wonder if K [m] can be obtained by forcing ordinary algebraic K-theory to satisfy this condition. In this subsection we show a more general result of this sort. To be more specific, let m 0 ≤ m, then Definition 6.8 introduces a functor K [m] , which associates to C ∈ Cat st m-fin the pre-m-commutative monoid given on objects by A → K [m0] (C A ). The main result of this subsection is Theorem 6.9, which shows that forcing the m-Segal condition on K [m0] [m] is indeed K [m] . In particular, the case m 0 = 0 yields an alternative definition of msemiadditive algebraic K-theory, by forcing A → K(C A ) to satisfy the m-Segal condition.
Consider the inclusion i : . Using this we are lead to the main definition.
Definition 6.8. We define the functor K We recall that for any D we have an equivalence CMon (p) m (D) ∼ = CMon (p) m (CMon (p) m0 (D)) (which is given by sending X ∈ CMon (p) m (D) to the iterated commutative monoid given on objects by A → (B → X(A × B))). In particular, we can consider it as a full subcategory CMon (p) m (D) ⊆ PCMon (p) m (CMon (p) m0 (D)), and this inclusion has a left adjoint L seg .
Applying the above for D = Sp The following square commutes because all maps are left adjoints and the square of right adjoints commutes because they are all forgetfuls.
The following square commutes because L seg is a left adjoint, thus commutes with colimits.
Lastly, L seg is an exact functor between stable categories, thus it commutes with finite limits, so the following square commutes.
In particular, restricting to the case m 0 = 0, we get that the functor K [m] given by A → K(C A ) satisfies the following:  is also smashing localization of modes.
Proposition 6.13. The following square commutes:

‫צ‬ [m]
Sp T(n) Proof. First recall that by definition ‫צ‬ [m] = CMon (p) m (Sp (p) ), and as explained above, Sp T(n) ∼ = CMon (p) m (Sp T(n) ). All of the morphisms in the square in the statement are left adjoints. Using the two identifications and passing to the right adjoints we obtain the square: This square commutes as all morphisms are inclusions, thus the original square of left adjoints commutes as well.
Corollary 6.14. There is an equivalence

Multiplicative Structure
Using Corollary 6.10 we leverage the lax symmetric monoidal structure on algebraic K-theory developed in [BGT14, Corollary 1.6] and [Bar15, Proposition 3.8] to construct a lax symmetric monoidal structure on m-semiadditive algebraic K-theory.
Recall that for any collection of indexing categories K, Cat K has a symmetric monoidal structure constructed in [Lur17, §4.8.1]. If K contains all finite categories, then Cat st K is the full subcategory on those categories that are in addition stable, which is also endowed with a symmetric monoidal structure (but is not a sub-symmetric monoidal category of Cat K , whose unit is not stable).
The first functor is symmetric monoidal by Theorem 4.26, which also shows that the second map is lax symmetric monoidal as the right adjoint of the symmetric monoidal functor L seg . The third and fourth maps are post-composition with the lax symmetric monoidal functors Cat st m-fin → Cat st and K, which are therefore also lax symmetric monoidal by Proposition 3.3.

Redshift
Recall that the redshift philosophy predicts that algebraic K-theory increases height by 1. In this section we prove some results concerning the interplay between semiadditive height and higher semiadditive algebraic K-theory. An immediate application of the redshift result of [CSY21a,Theorem B], gives an upper bound, showing that if R ∈ Alg(‫צ‬ [m] ) has semiadditive height ≤ n for some finite n < m, then K [m] (R) has semiadditive height ≤ n + 1 (see Theorem 7.12). Furthermore, in Theorem 7.25 we show that if R has semiadditive height exactly n, and has (height n) p-th roots of unity (see Definition 7.17), then K [m] (R) has semiadditive height exactly n + 1, i.e. lands in ‫צ‬ n+1 . In particular, the Lubin-Tate spectrum E n has this property, so we conclude that K [m] (E n ) ∈ ‫צ‬ n+1 (see Corollary 7.26).

Semiadditive Height
We begin by recalling the notion of (semiadditive) height from [CSY21a, Definition 3.1.6] and making a few observations which will be used to study the interaction between height and semiadditive algebraic K-theory. We recall from [CSY21a, Definition 3.1.3] that for every m-semiadditive category D, and finite n ≤ m, there is a natural transformation of the identity p (n) : id D ⇒ id D , also denoted by |B n C p |, which is given on an object Y ∈ D by using the fact that the norm map is an equivalence. Alternatively, as D is m-semiadditive, its objects have a canonical m-commutative monoid structure in D, so that the map is given by q ! q * where q : B n C p → * is the unique map.
We denote by D ≤n the full subcategory of objects Y ∈ D with ht(Y ) ≤ n, and similarly D >n for objects of height > n and D n for object of height exactly n.
Proposition 7.2 ([CSY21a, Theorem A]). Let D be an m-semiadditive category which admits all limits and colimits indexed by π-finite p-spaces, and let n ≤ m be a finite number, then D ≤n is ∞-semiadditive. Proof. Since G and F preserves limits and colimits respectively, they are m-semiadditive. By Proposition 7.3, their restrictions to objects of height ≤ n land in objects of height ≤ n. Since by Proposition 7.2 D ≤n and E ≤n are ∞-semiadditive, and the restricted functors preserve limits or colimits, they are in fact ∞-semiadditive.
Proposition 7.5. Let n ≤ m be a finite number, then the mode ‫צ‬ [m] ≤n ∼ = ‫צ‬ ≤n is independent of m, and is the mode classifying the property of being stable p-local ∞-semiadditive and having all objects of height ≤ n. Furthermore, it decomposes as a product where ‫צ‬ k is the mode classifying the property of being stable p-local ∞-semiadditive and having all objects of height exactly n. >n ) ≤n = 0, so the result follows upon taking objects of height ≤ n.
Consider the case D = Cat st m-fin . In this case, the objects are themselves categories C ∈ D on which p (n) acts, and can have heights ht(C) as objects of Cat st m-fin .
Proposition 7.6. Let C ∈ Cat st m-fin . For any m-finite p-space A, the map |A| : C → C is given by |A|(X) ∼ = colim A X. In particular, p (n) (X) ∼ = colim B n Cp X.
Proof. Recall that if we consider the objects of Cat st m-fin as equipped with the canonical CMon (p) m structure, then p (n) ∼ = q ! q * where q : A → * is the unique map. Theorem 5.3 and Theorem 5.6 then show that q * : C → C A is taking the constant diagram and that q ! : C A → C is computing the colimit.

Upper Bound
Proposition 7.7. Let C ∈ Cat st m-fin and assume that ht(C) ≤ n as an object of Cat st m-fin for some finite n ≤ m, then ht(K [m] (C)) ≤ n. [m] is restricted to Cat st m-fin,≤n it already satisfies the m-Segal condition and is thus equivalent to K [m] .
We recall the following redshift result, which we view as the step along the construction at which redshift happens.  Proof. Combine Corollary 7.10 and Proposition 7.7.
Proposition 7.7 shows that p n : K [m] (C) → K [m] (C) is invertible, but in fact we can prove the following stronger result if we assume that C is m-semiadditive. Note that as we know now that K [m] (C) ∈ ‫צ‬ ≤n , it is an object of an ∞-semiadditive category, so that p (k) is defined for all k.
Proposition 7.13. Let C ∈ Cat st m-fin be an m-semiadditive category with ht(C) ≤ n + 1 as an object of Cat st m-fin for some finite n < m. Then p (k) : K [m] (C) → K [m] (C) is the identity for every k ≥ n + 1. In particular, for C ∈ Mod ‫צ‬ ≤n , the map p (k) : K [m] (C at ) → K [m] (C at ) is the identity for every k ≥ n + 1.
Proof. Recall from Theorem 7.9 that for every X ∈ C we have ht(X) ≤ n, i.e. |B n C p | : X → X is invertible. [CSY21a, Proposition 2.4.7 (1)] applied to the case A = B n+1 C p shows that colim B n+1 Cp X ∇ − → X is an equivalence. By Proposition 7.6, p (n+1) : C → C is given by p (n+1) (X) ∼ = colim B n+1 Cp X, which by the above is X itself, i.e. p (n+1) is the identity. By [CSY21a, Proposition 2.4.7], if p (k) is invertible then p (k+1) is also invertible and is its inverse, finishing by induction. For the second part apply Corollary 7.10.
Proof. Follows from Example 7.20 and Theorem 7.25.

Relationship to Chromatically Localized K-Theory
In Section 7 we have shown that higher semiadditive algebraic K-theory interacts well with semiadditive height. For example, ht(K [m] (E n )) = n + 1 when m > n by Corollary 7.26. Note that the assumption m > n is necessary to even define semiadditive height n + 1. In this section we study the connection between higher semiadditive algebraic K-theory and chromatic localizations of ordinary algebraic K-theory by other means, while also dropping the assumption m > n.
(2) Is the comparison map an equivalence?
A positive answer to both questions will imply that K [m] (R) ∼ = L T(n+1) K(R), see Conjecture 1.4. In Proposition 8.4 we show that the first question is closely related to the Quillen-Lichtenbaum conjecture for R, in the guise of having a non-zero finite spectrum X such that K(R) ⊗ X is bounded above. By Corollary 6.14, the second question is equivalent to L T(n+1) K [m] (R) satisfying the m-Segal condition. More informally, having descent properties for T(n + 1)-localized K-theory. Using the Galois descent results for T(n + 1)-localized K-theory of [CMNN20], the second question is answered in the affirmative for m = 1 in Proposition 8.6.
We then study the case where R has height 0. The main result is Theorem 8.10, showing that for any p-invertible algebra R ∈ Alg(Sp[p −1 ]) and m ≥ 1, there is an equivalence This is first proved for R = S[p −1 ] by employing the Quillen-Lichtenbaum property of S[p −1 ] together with Proposition 8.6 mentioned above. The general case then follows via the lax symmetric monoidal structure on K [m] .
Finally, we study the completed Johnson-Wilson spectrum E(n) at height n ≥ 1, endowed with the Hahn-Wilson [HW22] E 3 -algebra structure (see Theorem 8.12) and, more generally, any R ∈ Alg(LMod E(n) ). In Theorem 8.23 we show that for any m ≥ 1, strengthening Theorem 7.25 for E(n)-algebras. In the case m = 1, Proposition 8.6 implies that To prove Theorem 8.23, we first use the Quillen-Lichtenbaum result for BP⟨n⟩ of [HW22] and the lax symmetric monoidal structure on K [m] to show that K [m] ( E(n)) ∈ Sp T(0) × · · · × Sp T(n+1) . We would like to thank the anonymous referee for suggesting this argument. Then, we compute the cardinality of the classifying space of the k-fold wreath product of C p at each chromatic height in two different ways. We observe that they are compatible only in chromatic height n + 1, concluding that K [m] ( E(n)) ∈ Sp T(n+1) . Using the lax symmetric monoidal structure on K [m] , this is generalized to any E(n)-algebra. Throughout this section F (n) denotes a type n finite spectrum (for example, the generalized Moore spectrum S/(p i0 , v i1 1 , . . . , v in−1 n−1 )). Without loss of generality, we may assume that F (n) is an algebra, i.e. F (n) ∈ Alg(Sp), by replacing it by F (n) ⊗ DF (n) ∼ = End(F (n)).

General Results
We begin this subsection by recalling and slightly generalizing some results from [CSY22] and [CSY21a] that will be used in the rest of the section.
Recall from [CSY21a, Proposition 5.3.9] that, similarly to the K(n)-and T(n)-localizations, the map of modes Sp → ‫צ‬ n vanishes on all bounded above spectra when n ≥ 1. Here we prove a slight generalization of this result. >0 vanishes on all bounded above spectra.
Proof. We follow closely the argument of [CSY21a, Proposition 5.3.9], diverging only the case of F p . The class of spectra on which G >0 vanishes is closed under colimits and desuspensions in Sp. Hence, by a standard devissage argument, it suffices to show that G >0 vanishes on Q and F ℓ for all primes ℓ. First, Q and F ℓ for ℓ ̸ = p are p-divisible. Since G >0 is 0-semiadditive, G >0 (Q) and G >0 (F ℓ ) are p-divisible as well, but all objects of ‫צ‬ [m] >0 are p-complete, and so G >0 (Q) = G >0 (F ℓ ) = 0. It remains to show that G >0 (F p ) = 0. Since F p ∈ CAlg(Sp) is an E ∞ -algebra, and G >0 is a map of modes, >0 ) is an E ∞ -algebra as well. Similarly, since p = 0 in F p , the same holds in π 0 G >0 (F p ). Thus, by Lemma 8.1 with C = ‫צ‬ [m] >0 and R = G >0 (F p ), we know that G >0 (F p ) = 0 which concludes the proof.
We now move on to proving the two main results of this subsection.
Recall from Corollary 6.19 that since C is an algebra, we get an algebra map K(C) → K >0 ). Since the functor G >0 is a functor between stable modes, it commutes with the action of Sp. Therefore, tensoring the map with the algebra F (n + 2) yields G >0 (K(C) ⊗ F (n + 2)) → K [m] (C) >0 ⊗ F (n + 2) ∈ Alg(‫צ‬ [m] >0 ). We have shown that the source is 0, and since this is an algebra map, so is the target, which concludes the proof.
In the next proposition we would like to use [CMNN20, Theorem C], which applies to L f n Slinear stable categories. We recall that an L f n S-linear stable category is, by definition, a module over Perf(L f n S) = Mod dbl L f n S in Cat st . Note that since L f n Sp is a smashing localization of Sp we have that Mod L f n S = L f n Sp. In particular, for R ∈ Alg(Sp T(n) ), we have that LMod at R is L f n Slinear, since LMod R ∈ Mod Sp T(n) ⊂ Mod L f n Sp and left dualizable modules coincide with atomics by Proposition 2.54. Thus LMod at R is an example for C in the following proposition.
Proof. By Corollary 6.14 it suffices to show that L T(n+1) K [1] (C) satisfies the 1-Segal condition, that is, for any 1-finite p-space A, the canonical map is an equivalence. As both sides take coproducts in A to direct sums, we may assume that A is connected, i.e. A = BG for a finite p-group G. This is exactly [CMNN20, Theorem C].

Height 0
We ) is an isomorphism on high enough p-local homotopy groups. Tensoring with a finite spectrum preserves the property of a map being an isomorphism on high enough homotopy groups, and p = 0 in F (2), so it suffices to show that the right hand side vanishes after tensoring with F (2). The tensor product of spectra commutes with finite limits, so it suffices to show that each term on the right hand side vanishes after tensoring with F (2).
By definition, any L f 1 -local spectrum vanishes after tensoring with F (2), which shows that both   Proof. By Proposition 7.8, K [m] (C ω ) is independent of m ≥ 1, so we may assume that m = 1. Therefore, the result follows immediately from the combination of Corollary 8.9 and Proposition 8.6.
Proof. The combination of [Sus84, Corollary 4.7] and [Sus83,Main Theorem] shows that there is an equivalence K(Q) p ∼ = K(C) p ∼ = ku p . As KU p is T(1)-local, and T(1)-localization is insensitive to connective covers, L T(1) ku p ∼ = KU p , which shows that L T(1) K(Q) ∼ = KU p , and the result follows by Theorem 8.10.
Henceforth, we shall consider BP⟨n⟩ as an E 3 -algebra with the structure from Theorem 8.12, which also endows the localization E(n) with a compatible E 3 -algebra structure. An immediate corollary of this result is the following: f ′ (a)f (a) for all a ∈ C p . Applying this inductively, we see that f ′ is uniquely determined by f and Proof. By [HKR00, Theorem E], the group ≀ k C p is good in the sense of [HKR00, Definition 7.1], and in particular K(n) 1 (B(≀ k C p )) = 0. Thus, by [HKR00, Theorem B and Lemma 4.13], we know that dim Fp (K(n) 0 (≀ k C p )) = L n (≀ k C p ). By Proposition 8.20, we conclude that E(n)[B(≀ k C p )] is a free E(n)-module of dimension L n (≀ k C p ). Recall from Proposition 7.6 that the action of |B(≀ k C p )| on LMod at E(n) as an object of Cat st m-fin is by E(n)[B(≀ k C p )] ⊗ (−), namely by multiplication by L n (≀ k C p ). Since K [m] is a 1-semiadditive functor by Proposition 6.7, the same holds for the action of |B(≀ k C p )| on K [m] ( E(n)) by [CSY22, Corollary 3.2.7]. By Proposition 8.19, this number is indeed a non-invertible p-adic number.
Proof. By Lemma 8.14, we know that K [m] ( E(n)) ∈ Sp T(0) × · · · Sp T(n+1) . It remains to show that the T(k)-local part, which for brevity we denote by A k ∈ Sp T(k) , vanishes for every 0 ≤ k ≤ n.
We first deal with the case 1 ≤ k ≤ n. We now prove the remaining case k = 0. As above, Proposition 8.21 shows that |BC p | acts on A 0 by L n (C p ) = p. On the other hand, |BC p | 0 = p −1 by [CSY22, Lemma 5.3.3]. Namely p = p −1 on the rational spectrum A 0 , thus A 0 = 0.
Proof. LMod R is a right module over LMod E(n) . Recall from Theorem 2.46 that taking the atomics is a lax symmetric monoidal functor, and from Theorem 6.18 that K [m] is lax symmetric monoidal.
Thus, we get that K [m] (R) is a right module over K [m] ( E(n)). In addition, by Lemma 6.12, ‫צ‬ [m] → Sp T(n+1) is a smashing localization, and since K [m] ( E(n)) lands in the smashing localization by Lemma 8.22, so does K [m] (R).
Proof. This follows immediately from the combination of Theorem 8.23 and Proposition 8.6.
In work in progress with Carmeli and Yanovski [BMCSY] we show that Corollary 8.24 holds for m-semiadditive K-theory for any m ≥ 1.