Multiplicative Structures on Algebraic K-Theory

Clark Barwick

Received: April 27, 2013

Communicated by Lars Hesselholt

Abstract. The algebraic K-theory of Waldhausen ∞-categories is the functor corepresented by the unit object for a natural symmetric monoidal structure. We therefore regard it as the stable homotopy theory of homotopy theories. In particular, it respects all algebraic structures, and as a result, we obtain the Deligne Conjecture for this form of K-theory.

2010 Mathematics Subject Classification: 19D10, 19D55 Keywords and Phrases: Keywords and Phrases: algebraic K-theory, Waldhausen ∞-categories, multiplicative structures, Deligne conjecture

0. Introduction

Dan Kan playfully described the theory of ∞-categories as the homotopy theory of homotopy theories. The aim of this paper, which is a sequel to [2], is to show that algebraic K-theory is the stable homotopy theory of homotopy theories, and it interacts with algebraic structures accordingly. To explain this assertion, let’s recap the contents of [2].

0.1. The kinds of homotopy theories under consideration in this paper are Waldhausen ∞-categories [2, Df. 2.7]. (We employ the quasicategory model of ∞-categories for technical convenience.) These are ∞-categories with a zero object and a distinguished class of morphisms (called cofibrations or ingressive morphisms) that satisfies the following conditions. (0.1.1) Any equivalence is ingressive. (0.1.2) Any morphism from the zero object is ingressive. (0.1.3) Any composite of ingressive morphisms is ingressive. (0.1.4) The (homotopy) pushout of an ingressive morphism along any mor-

phism exists and is ingressive.

A pushout of a cofibration X Y along the map to the zero object is to be viewed as a cofiber sequence

X  Y  Y/X.

Examples of this structure abound: pointed ∞-categories with all finite colimits, exact categories in the sense of Quillen, and many categories with cofibrations and weak equivalences in the sense of Waldhausen all provide examples of Waldhausen ∞-categories. Write Wald_{∞} for the ∞-category whose objects are Waldhausen ∞-categories and whose morphisms are functors that are exact in the sense that they preserve the cofiber sequences. This is a compactly generated ∞-category [2, Pr. 4.7] that admits direct sums [2, Pr. 4.6].

0.2. We will be interested in invariants that split cofiber sequences in Waldhausen ∞-categories. To make this precise, for any Waldhausen ∞-category C, let E(C) denote the ∞-category of cofiber sequences

X  Y  Y/X.

This is a Waldhausen ∞-category [2, Pr. 5.11] in which a morphism

\begin{matrix}U & V & V/U \\ X & Y & Y/X \end{matrix}

is ingressive just in case each of

U  X, V  Y, and V/U  Y/X

is ingressive. We have an exact functor m: E(C) C defined by the assignment

[X Y Y/X] Y.

We also have an exact functor i: C ⊕ C E(C) defined by the assignment

(X, Z)  [X  X ∨ Z  Z]

as well as a retraction r: E(C) C ⊕ C defined by the assignment

[X  Y  Y/X]  (X, Y/X).

0.3. Now let S denote the ∞-category of spaces. The homotopy theory D_{fiss}(Wald_{∞}) constructed in [2, §6] has the property that homology theories (i.e., reduced, 1-excisive functors)

D_{fiss}(Wald_{∞}) S

are essentially the same data [2, Th. 7.4] as functors ϕ: Wald_{∞} S with the following properties. (0.3.1) ϕ is finitary in the sense that it preserves filtered colimits. (0.3.2) ϕ is reduced in the sense that ϕ(0) = ∗.

Multiplicative Structures on Algebraic K-Theory 861

(0.3.3) ϕ splits cofiber sequences in the sense that the exact functor r induces an equivalence

ϕ(E(C)) ∼ ϕ(C) × ϕ(C).

(0.3.4) ϕ is grouplike in the sense that the multiplication

ϕ(m ∘ i): ϕ(C) × ϕ(C) ≃ ϕ(E(C)) ϕ(C)

defines a grouplike H-space structure on ϕ(C). D_{fiss}(Wald_{∞}) is called the fissile derived ∞-category of Waldhausen ∞- categories, and its objects are called fissile virtual Waldhausen ∞-categories. It is possible to be quite explicit about these objects: fissile virtual Waldhausen ∞-categories are functors Wald_{∞}^{op} S that satisfy the dual conditions to (0.3.1–3). The homotopy theory D_{fiss}(Wald_{∞}) is compactly generated and it admits all direct sums; furthermore, suspension in D_{fiss}(Wald_{∞}) is given by the geometric realization of Waldhausen’s S_{•} construction [2, Cor. 6.9.1].

0.4. From any finitary reduced functor F: D_{fiss}(Wald_{∞}) S one may extract the Goodwillie differential P_{1}F, which is the nearest excisive approximation to F, or, in other words, the best approximation to F by a homology theory [7]. This approximation is given explicitly by the colimit of the sequence

F  Ω ∘ F ∘ Σ  ⋯  Ω^{n} ∘ F ∘ Σ^{n}  · · · .

Since P_{1}F is excisive, it factors naturally through the functor Ω^{∞}: Sp S. Consequently, for any fissile virtual Waldhausen ∞-category X, we obtain a homology theory

P_{1}F (X): S_{∗} S.

Unwinding the definitions, we find that this homology theory is itself the Goodwillie differential of the functor

T F (T ⊗ X),

where ⊗ denotes the tensor product

S_{∗} × D_{fiss}(Wald_{∞})  D_{fiss}(Wald_{∞})

guaranteed by the identification of presentable pointed ∞-categories with modules over S_{∗} [10, Pr. 6.3.2.11].

0.5. The main result of [2, §10] can now be stated as follows. We have a Waldhausen ∞-category Fin_{∗} of pointed finite sets, in which the cofibrations are injective (pointed) maps; if I: D_{fiss}(Wald_{∞}) S denotes evaluation at Fin_{∗} (so that I(X) = X(Fin_{∗})), then algebraic K-theory may be identified as

K ≃ P_{1}I.

This gives a “local” universal property for algebraic K-theory: for any fissile virtual Waldhausen ∞-category X, the homology theory K(X) is the Goodwillie differential of the functor

T I(T ⊗ X).

Our version of Waldhausen’s Additivity Theorem states that this differential only requires a single delooping: K(X) is simply the homology theory S_{∗} S given by

T ΩI(ΣT ⊗ X),

or, equivalently, since suspension in D_{fiss}(Wald_{∞}) is given by Waldhausen’s S_{•} construction, the assignment

T ΩI(T ⊗ S_{•}(X)),

0.6. In this paper, we construct (Pr. 1.9) a symmetric monoidal structure on Wald_{∞} in which the tensor product C ⊗ D represents “bi-exact” functors

C × D E,

i.e., functors that preserve cofiber sequences separately in each variable. The unit therein is simply the Waldhausen ∞-category Fin_{∗}. We then descend this symmetric monoidal structure to one on D_{fiss}(Wald_{∞}) with the property that it preserves colimits separately in each variable (Pr. 2.5). Now the functor represented by Fin_{∗} is the unit for the Day convolution symmetric monoidal structure constructed by Saul Glasman [6] on the ∞-category of functors from D_{fiss} to spaces. Its differential — i.e., algebraic K-theory — is therefore the unit among homology theories on D_{fiss}(Wald_{∞}). That is, it plays precisely the same role among homology theories on D_{fiss}(Wald_{∞}) that is played by the sphere spectrum in the ∞-category of spectra. It therefore has earned the mantle the stable homotopy theory of homotopy theories. Just as one may describe the stable homotopy groups of a pointed space X as Ext groups out of the unit:

π_{n}^{s}(X) ≅ Ext^{−n}(Σ^{∞}S^{0}, Σ^{∞}X)

in the stable homotopy category, so too may one describe the algebraic Ktheory groups of a Waldhausen ∞-category C as Ext groups out of the unit:

K_{n}(C) ≅ Ext^{−n}(Σ^{∞}Fin_{∗}, Σ^{∞}C)

in the stable homotopy category of Waldhausen ∞-categories.

0.7. Algebraic K-theory is therefore naturally multiplicative, and so it inherits homotopy-coherent algebraic structures on Waldhausen ∞-categories. That is, we show (Cor. 3.8.2) that if an ∞-operad O acts on a Waldhausen ∞-category C via functors that are exact separately in each variable, then there is an induced action of O on both the space K(C) and the spectrum K(C). As a corollary (Ex. 3.9), we deduce that for any 1 ≤ n ≤ ∞, the algebraic K-theory of an E_{n}-algebra in a suitable symmetric monoidal ∞-category is an E_{n−1} ring spectrum. In particular, we note that the K-theory of an E_{n} ring is an E_{n−1} ring, and the A-theory of any n-fold loopspace is an E_{n−1} ring spectrum. These sorts of results are K-theoretic analogues of the so-called (homological) Deligne Conjecture [11, 8].

0.8. In fact, the main result (Th. 3.6) is much more general: we actually show that any additive theory that can be expressed as the additivization of a multiplicative theory is itself multiplicative in a canonical fashion. This yields a uniform way of reproducing conjectures “of Deligne type” for theories that are both additive and multiplicative. Of course since algebraic K-theory is the stable homotopy theory of Waldhausen ∞-categories, it is initial as a theory that is both additive and multiplicative. The sphere spectrum is the initial E_{∞} object of the stable ∞-category of spectra; similarly, the object of the stable ∞-category of Waldhausen ∞- categories that represents algebraic K-theory is the initial E_{∞} object. Just as the universal property of algebraic K-theory [2] gives a uniform construction of trace maps, this result gives a uniform construction of multiplicative trace maps — in particular, any additive and multiplicative theory accepts a unique (up to a contractible choice) multiplicative trace map (Th. 3.13). The passage to higher categories is a sine qua non of this result. Indeed, note that when n ≥ 3, an E_{n} structure on an ordinary category is tantamount to a symmetric monoidal structure. As a result, it is difficult to identify, e.g., E_{3} structures on the K-theory spectrum of an E_{4}-algebra without employing some higher categorical machinery.

0.9. Remark. Note that the form of algebraic K-theory we study has an exceptionally strong compatibility with the tensor product. For example, the Barratt–Priddy–Quillen–Segal theorem implies the endomorphism spectrum End(Σ^{∞}Fin_{∗}) is the sphere spectrum. That is, the form of algebraic K-theory studied here is strongly unital. This is of course false for any form of algebraic K-theory that applies only to Waldhausen ∞-categories in which every morphism is ingressive. We intend to return to this observation in future work.

0.10. Remark. The heart of the proof is to use the description of K-theory as a Goodwillie differential. The result actually follows from a quite general fact about the interaction between the Goodwillie calculus and symmetric monoidal structures. Namely, the Goodwillie differential of a multiplicative functor between suitable symmetric monoidal ∞-categories inherits a canonical multiplicative structure. This fact, which may be of independent interest, doesn’t seem to be recorded anywhere in the literature. So we do so in this paper (Pr. 3.5).

0.11. Remark. Some variants of some of these results can be found in the literature. Elmendorf and Mandell [5] constructed an algebraic K-theory of multicategories, which the show is lax monoidal as a functor to symmetric spectra. Consequently, they deduce that any operad that acts on a permutative category will also act on its K-theory. In later work of Blumberg and Mandell [4, Th. 2.6], it is shown that the Ktheory of an ordinary Waldhausen category equipped with an action of a categorical operad inherits an action of the nerve of that operad.

In independent work, Blumberg, Gepner, and Tabuada [3] have proved that algebraic K-theory is initial among additive and multiplicative functors from an ∞-category of idempotent complete stable ∞-categories. (They used this to uniquely characterize the cyclotomic trace.) The result here shows that algebraic K-theory is initial among additive and multiplicative functors on all Waldhausen ∞-categories.

0.12. Remark. Finally, let’s emphasize that work of Saul Glasman [6] made it possible for me to sharpen the results of this paper significantly. Previous versions of this paper did not contain the full strength of the universality of algebraic K-theory as an additive and multiplicative theory.

1. Tensor products of Waldhausen ∞-categories

The first thing we need to understand is the symmetric monoidal structure on the ∞-category of Waldhausen ∞-categories. As in [2, §4], we will regard Wald_{∞} as formally analogous to the nerve of the ordinary category V (k) of vector spaces over a field k. The tensor product V ⊗ W of vector spaces is defined as the vector space that represents multilinear maps V × W X, i.e., maps that are linear separately in each variable. In perfect analogy with this, the tensor product C ⊗ D of Waldhausen ∞-categories is defined as the Waldhausen ∞-category that represents functors C × D E that are exact separately in each variable.

1.1. Notation. Let Λ(F) denote the following ordinary category. The objects

will be finite sets, and a morphism J  I will be a map J  I_{+}; one composes
ψ: K  J_{+} with ϕ: J  I_{+} by forming the composite
K ψ J_{+} ϕ_{+} I_{++} μ I_{+},

where μ: I_{++} I_{+} is the map that simply identifies the two added points. (Of course Λ(F) is equivalent to the category Fin_{∗} of pointed finite sets, but we prefer to think of the objects of Λ(F) as unpointed. This is the natural perspective on this category from the theory of operator categories [1].) For any morphism ϕ : J I of Λ(F) and any i ∈ I, write J_{i} for the fiber ϕ^{−1}({i}).

1.2. One way to write down a symmetric monoidal ∞-category [10, Ch. 2] is to give the data of the space of maps out of any tensor product of any finite collection of objects. More precisely, a symmetric monoidal ∞-category is a cocartesian fibration p : C^{⊗} NΛ(F) such that for any finite set I, the various maps χ_{i} : I {i}_{+} such that χ_{i}^{−1}({i}) = {i} together specify an equivalence of ∞-categories

C_{I}^{⊗} ∼ ∏_{i∈I} C_{{i}}^{⊗}.

The objects of C^{⊗} are, in effect, (I, X_{I}) consisting of a finite set I and a collection X_{I} = {X_{i}}_{i∈I} of objects of C. Morphisms (J, Y_{J}) (I, X_{I}) of C^{⊗} are essentially pairs (ω, ϕ_{I}) consisting of morphisms ω : J I of Λ(F) and families of morphisms

{ϕ_{i} : ⨂_{j∈J_{i}} Y_{j} X_{i}}_{i∈I} .

1.3. Example. For any ∞-category that admits all finite products, there is a corresponding symmetric monoidal ∞-category C^{×} called the cartesian symmetric monoidal ∞-category.

We will be particularly interested in identifying a suitable subcategory of the ∞- category Pair_{∞}^{×}, where Pair_{∞} denotes the ∞-category of pairs of ∞-categories [2, Df. 1.11].

1.4. Definition. Suppose I a finite set, and suppose C_{I} := (C_{i})_{i∈I} an I-tuple of Waldhausen ∞-categories. For any Waldhausen ∞-category D, a functor of pairs ∏ C_{I} D is said to be exact separately in each variable if, for any element i ∈ I and any collection of objects (X_{j})_{j∈I\{i}} ∈ ∏ C_{I\{i}}, the functor

C_{i} ≅ C_{i} × ∏_{j∈I\{i}}{X_{j}}  ∏ C_{I}  D

carries cofibrations to cofibrations and is exact as a functor of pairs between Waldhausen ∞-categories.

1.5. Note that we do not assume the cubical cofibrancy criterion that appears in Blumberg–Mandell [4, Df. 2.4]. It seems that the authors of this paper required it to guarantee a compatibility with the “all at once” iterated S_{•} construction. We will not use such a construction here; the only compatibility we will need to find with the S_{•} construction is Pr. 2.3, which deals with one tensor factor at a time.

1.6. Notation. Denote by Wald_{∞}^{⊗} ⊂ Pair_{∞}^{×} the following subcategory. The objects of Wald_{∞}^{⊗} are those objects (I, C_{I}), where for any i ∈ I, the pair C_{i} is a Waldhausen ∞-category. A morphism (J, D_{J}) (I, C_{I}) of Pair_{∞}^{×} is a morphism of Wald_{∞}^{⊗} if and only if, for every element i ∈ I, the functor

∏ D_{J_{i}}  C_{i}

is exact separately in each variable.

We now identify the tensor product of Waldhausen ∞-categories.

1.7. Lemma. Suppose I a finite set, and suppose C_{I} := (C_{i})_{i∈I} an I-tuple of Waldhausen ∞-categories. Then there exist a Waldhausen ∞-category ⨂ C_{I} and a functor of pairs

f: ∏ C_{I} ⨂ C_{I}

such that for every Waldhausen ∞-category D, composition with f induces an equivalence between the ∞-category Fun_{Wald_{∞}}(⨂ C_{I}, D) and the full subcategory of Fun_{Pair_{∞}}(∏ C_{I}, D) spanned by the functors of pairs that are exact separately in each variable.

Proof. We construct ⨂ C_{I} as a colimit in Wald_{∞} in the following manner. First, recall that the forgetful functor Wald_{∞} Cat_{∞} admits a left adjoint W. Consider the pushout K = (I × Δ^{1}) ∪^{I×Δ^{{0}}} (I × Δ^{{0}})^{⊳} and the obvious functor

F: K Wald_{∞}

that carries each object of the form (i, 0) to the coproduct

∐ ; (X_{j})_{j∈I\{i}}∈∏_{j∈I\{i}} C_{j} W (C_{i}),

each object of the form (i, 1) to the coproduct

∐ ; (X_{j})_{j∈I\{i}}∈∏_{j∈I\{i}} C_{j} C_{i},

and the cone point +∞ to W (∏_{i∈I} C_{i}). Now the desired Waldhausen ∞- category ⨂ C_{I} can be constructed as the colimit of F. □

1.8. The construction of this proof is of course the natural analogue of the construction of tensor products of abelian groups. From this description, it is clear that when I = ∅, then the Waldhausen ∞-category ⊗^{0} ≃ W (Δ^{0}) ≃ NFin_{∗}, the nerve of the ordinary category of finite pointed sets, in which the cofibrations are the monomorphisms.

1.9. Proposition. The functor Wald_{∞}^{⊗} NΛ(F) is a symmetric monoidal ∞-category.

Proof. We first claim that the functor p: Wald_{∞}^{⊗} NΛ(F) is a cocartesian fibration; it is an inner fibration because Wald_{∞}^{⊗} is an ∞-category [9, Pr. 2.3.1.5]. Now suppose ϕ: J I an edge of NΛ, and suppose D_{J} a J-tuple of pairs of ∞-categories. We want to find a p-cocartesian edge of Pair_{∞}^{⊗} covering ϕ. For this, for any i ∈ I, consider a pair ⨂ D_{J_{i}} along with a functor ∏ D_{J_{i}} ⨂ D_{J_{i}} satisfying the property described in Lemma 1.7. These fit together to yield a morphism

(J, D_{J})  (I, ( ⨂ D_{J_{i}})_{i∈I})

of Wald_{∞}^{⊗} covering ϕ. The property described in Lemma 1.7 guarantees that this a locally p-cocartesian edge of Wald_{∞}^{⊗}, so p is a locally cocartesian fibration. Now to conclude that p is a cocartesian fibration, it is enough to note that for any 2-simplex

\begin{aligned}(K, E_{K}) & & (I, C_{I}), \\ & (J, D_{J}) \end{aligned}_{\begin{matrix}h \\ g \\ f \end{matrix}}

if f and g are locally p-cocartesian edges, then so is h; this follows directly from our construction of ⨂ C_{I}. □

1.10. Proposition. The tensor product functor

⊗: Wald_{∞} × Wald_{∞} Wald_{∞}

preserves filtered colimits and direct sums separately in each variable.

Proof. Suppose C a Waldhausen ∞-category. We will show that the endofunctor − ⊗ C preserves filtered colimits and direct sums. Suppose Λ a filtered simplicial set, and suppose D: Λ^{⊳} Wald_{∞} a colimit diagram. Since filtered colimits in Wald_{∞} are preserved under the forgetful functor Wald_{∞} Pair_{∞}, it follows that for any Waldhausen ∞-category A, there is an equivalence of ∞-categories

Fun_{Pair_{∞}} (D_{+∞} × C, A) ∼ lim_{α∈Λ} Fun_{Pair_{∞}} (D_{α} × C, A).

To show that the tensor product preserves filtered colimits separately in each variable, it remains to note that, under this equivalence, a functor D_{+∞} × C A that is exact separately in each variable correspond to compatible families of functors D_{α} × C A that are exact separately in each variable. Note that if 0 is the zero Waldhausen ∞-category, then for any Waldhausen ∞- category A, any functor 0 × C A that is exact separately in each variable is essentially constant, whence 0 ⊗ C ≃ 0. Moreover, if D and D′ are Waldhausen ∞-categories, then the two inclusions

D × C  (D ⊕ D′) × C and D′ × C  (D ⊕ D′) × C
given by (y, x)  (y, 0, x) and (y′, x)  (0, y′, x) together induce a functor
Fun_{Pair_{∞}}((D ⊕ D′) × C, A)
Fun_{Pair_{∞}}(D × C, A) × Fun_{Pair_{∞}}(D′ × C, A).

On the other hand, the coproduct induces a functor

Fun_{Pair_{∞}}(D × C, A) × Fun_{Pair_{∞}}(D′ × C, A)
Fun_{Pair_{∞}}((D ⊕ D′) × C, A).

It is not hard to see that these functors carry (pairs of) functors that are exact in each variable separately to (pairs of) functors that are exact in each variable separately. Moreover, if F: (D ⊕ D′) × C A is exact separately in each variable, then

F ≃ F |_{D×C} ∨ F |_{D′×C},
and if G: D × C  A and G′: D′ × C  A are each exact separately in each

variable, then

G ≃ (G ∨ G′)|_{D×C} and G′ ≃ (G ∨ G′)|_{D′×C}.

Hence these two functors exhibit an equivalence

(D ⊕ D′) ⊗ C ≃ (D ⊗ C) ⊕ (D′ ⊗ C),

as desired. □

1.11. For any integer m ≥ 0 and any Waldhausen ∞-category C, write mC for the iterated direct sum C⊕⋯⊕C, and write C^{m} for the iterated tensor product C ⊗⋯⊗C. It follows from the previous proposition that we may form “polynomials” in Waldhausen ∞-categories (with coefficients in the natural numbers), and the usual formulas hold, such as

(C ⊗ D)^{m} ≃ ⨁_{i=0}^{m} \binom{m}{i}C^{i} ⊗ D^{m−i}.

1.12. There is a natural generalization [10, Df. 2.1.1.10] of the notion of a symmetric monoidal ∞-category to that of an ∞-operad. This is an inner fibration p : O^{⊗} NΛ(F) satisfying properties that ensure that the objects of O^{⊗} are, in effect, pairs (I, X_{I}) consisting of a finite set I and a collection X_{I} = {X_{i}}_{i∈I} of objects of O_{{∗}}^{⊗}, and that morphisms (J, Y_{J}) (I, X_{I}) of C^{⊗} are essentially determined by pairs (ω, ϕ_{I}) consisting of morphisms ω : J I of Λ(F) and families of “multi-morphisms”

{ϕ_{i} : Y_{J_{i}} X_{i}}_{i∈I} .

An O^{⊗}-algebra in a symmetric monoidal ∞-category is simply a morphism of ∞-operads [10, Df. 2.1.2.7].

1.13. Definition. For any ∞-operad O^{⊗}, an O^{⊗}-monoidal Waldhausen ∞- category is an O^{⊗}-algebra in Wald_{∞}^{⊗}.

1.14. Example. In particular, a monoidal Waldhausen ∞-category is simply an O^{⊗}-monoidal Waldhausen ∞-category, where O^{⊗} is the associative ∞-operad [10, Df. 4.1.1.3]. Similarly, a symmetric monoidal Waldhausen ∞-category will be a O^{⊗}-monoidal Waldhausen ∞-category, where O^{⊗} is the commutative ∞- operad [10, Ex. 2.1.1.18].

1.15. Example. Suppose that Φ is a perfect operator category [1], and suppose that X NΛ(Φ) is a pair cocartesian fibration such that for any object I of Λ(Φ), the inert morphisms I {i} induce an equivalence of pairs

X_{I} ∼ ∏_{i∈|I|} X_{{i}}.

Then we might call X a Φ-monoidal Waldhausen ∞-category if, for any object I of Λ(Φ), the pair X_{I} is a Waldhausen ∞-category, and the morphism

∏_{i∈|I|} X_{{i}} ≃ X_{I}  X_{{ξ}}

induced by the unique active morphism I {ξ} is exact separately in each variable. One may show that this notion is essentially equivalent to the notion of U_{Φ}^{⊗}-monoidal Waldhausen ∞-category, where U_{Φ}^{⊗} is the symmetrization of the terminal ∞-operad over Φ. In particular, an O^{(n)}-monoidal Waldhausen ∞-category is essentially the same thing as an E_{n}-monoidal Waldhausen ∞- category.

2. Tensor products of virtual Waldhausen ∞-categories

The derived ∞-category D_{≥0}(k) of complexes of vector spaces over a field k with vanishing negative homology inherits a symmetric monoidal structure from the ordinary category of vector spaces. In precisely the same manner, the derived ∞-category of Waldhausen ∞-categories D(Wald_{∞}) — which is the ∞-category of functors Wald_{∞}^{op} S that preserve all filtered limits and all finite products [2, Nt. 4.10] — inherits a symmetric monoidal structure from Wald_{∞}.

2.1. Proposition. There exists a symmetric monoidal ∞-category D(Wald_{∞})^{⊗} and a fully faithful symmetric monoidal functor Wald_{∞}^{⊗} D(Wald_{∞})^{⊗} with the following properties. (2.1.1) The underlying ∞-category of D(Wald_{∞})^{⊗} is the ∞-category

D(Wald_{∞}) of virtual Waldhausen ∞-categories, and the underlying functor is the inclusion Wald_{∞} D(Wald_{∞}).

(2.1.2) For any symmetric monoidal ∞-category E^{⊗} whose underlying ∞- category admits all sifted colimits, the induced functor

Fun_{NFin_{∗}}(D(Wald_{∞}), E)  Fun_{NFin_{∗}}(Wald_{∞}, E)

exhibits an equivalence from the full subcategory of spanned by those morphisms of ∞-operads A whose underlying functor A: D(Wald_{∞}) E preserves sifted colimits to the full subcategory of spanned by those morphisms of ∞-operads B whose underlying functor B: Wald_{∞} E preserves filtered colimits.

(2.1.3) The tensor product functor ⊗: D(Wald_{∞}) × D(Wald_{∞}) D(Wald_{∞}) preserves all colimits separately in each variable.

Proof. The only part that is not a consequence of [10, Pr. 6.3.1.10 and Var. 6.3.1.11] is the assertion that the tensor product functor

⊗: D(Wald_{∞}) × D(Wald_{∞}) D(Wald_{∞})

preserves direct sums separately in each variable. Pr. 1.10 states that this holds among Waldhausen ∞-categories; the general case follows by exhibiting all the virtual Waldhausen ∞-categories as colimits of simplicial diagrams of Waldhausen ∞-categories and using the fact that both the tensor product and the direct sum commute with sifted colimits. □

Since our goal is to study multiplicative structures on additive theories, we should see to it that the symmetric monoidal structure on the derived ∞- category of Waldhausen ∞-categories gives rise to one on the fissile derived ∞-category described in the introduction 0.3. It is not the case that the tensor product of two fissile virtual Waldhausen ∞-categories is still fissile; however, D_{fiss}(Wald_{∞}) is the accessible localization of D(Wald_{∞}) with respect to the set of morphisms

{i : C ⊕ C E(C) | C ∈ Wald_{∞}^{ω}} ,

where i is the functor defined in 0.2. We can therefore ask whether the localization functor is compatible with the resulting localization functor

L^{fiss} : D(Wald_{∞}) D_{fiss}(Wald_{∞}).

That is, we wish to show that the assignment

(X, Y) L^{fiss}(X ⊗ Y);

defines a symmetric monoidal structure on D_{fiss}(Wald_{∞}).

2.2. Construction. The symmetric monoidal ∞-category D(Wald_{∞})^{⊗} can be described in the following manner. An object (I, X_{I}) thereof is a finite set I and an I-tuple of virtual Waldhausen ∞-categories X_{I} := (X_{i})_{i∈I}. A morphism (J, Y_{J}) (I, X_{I}) is a morphism J I of Λ(F) along with the data, for every element i ∈ I, of a functor of pairs

⨂ Y_{J_{i}} X_{i}.

We denote by D_{fiss}(Wald_{∞})^{⊗} the full subcategory of D(Wald_{∞})^{⊗} spanned by those objects (I, X_{I}) such that for every i ∈ I, X_{i} is a distributive virtual Waldhausen ∞-category.

2.3. Lemma. The localization functor L^{fiss} on D(Wald_{∞}) of [2, Pr. 6.7] is compatible with the symmetric monoidal structure on D(Wald_{∞}) is the sense of [10, Df. 2.2.1.6].

Proof. As in [10, Ex. 2.2.1.7], our claim is that for any L^{fiss}-equvalence U V and any virtual Waldhausen ∞-category Z, the morphism U ⊗ Z V ⊗ Z is an L^{fiss}-equivalence. Since the tensor product preserves all colimits separately in each variable, we may assume that Z is a compact Waldhausen ∞-category D. Furthermore, since the set of L^{fiss}-equivalences is generated as a strongly saturated class by the set of maps of the form

i: C ⊕ C E(C)

in which C is compact, we may assume that U V is of this form. Our claim is thus that for any compact Waldhausen ∞-categories C and D, the map

i ⊗ id_{D} : (C ⊕ C) ⊗ D E(C) ⊗ D

is a L^{fiss}-equivalence. We have a retraction

r ⊗ id_{D}: E(C) ⊗ D (C ⊕ C) ⊗ D

of this map, and the composition E(C) ⊗ D E(C) ⊗ D is given by the multi-exact functor that carries a pair (S T T/S, X) to the “simple tensor”

(S  S ∨ (T/S)  T/S) ⊗ X.

That is, the composition E(C) ⊗ D E(C) ⊗ D is given by the direct sum of the functor

(S  T  T/S, X)  (S  S  0) ⊗ X

and the functor

(S  T  T/S, X)  \frac{(S  T  T/S) ⊗ X}{(S  S  0) ⊗ X} .

But in D_{fiss}(Wald_{∞}), this is homotopic to the identity. □

2.4. Note that as a corollary, we find that, for any integer m ≥ 0, we obtain L^{fiss}-equivalences

F_{m}(C) ⊗ D ≃ mC ⊗ D ≃ F_{m}(C ⊗ D),

where F_{m}(C) is the Waldhausen ∞-category of filtered objects

X_{0}  ⋯  X_{m} [2, Nt. 5.5]. Consequently, we obtain an L^{fiss}-equivalence
F_{m}(C) ≃ F_{m}(NFin_{∗}) ⊗ C.

This observation yields another way to think about the suspension functor in D_{fiss}(Wald_{∞}). Just as suspension is smashing with a circle in the homotopy theory of spaces, we have

ΣC ≃ S(C) ≃ S(NFin_{∗} ⊗ C) ≃ S(NFin_{∗}) ⊗ L^{fiss}(C)

in D_{fiss}(Wald_{∞}). Here the fissile virtual Waldhausen ∞-category S(NFin_{∗}) is playing the role of a circle. The algebraic K-theory of a Waldhausen ∞-category C can thus be described as the space

ΩIS(C) ≃ ΩIS(NFin_{∗} ⊗ C) ≃ ΩI(S(NFin_{∗}) ⊗ L^{fiss}(C)),

where I is the left derived functor of ι. More generally, for any pre-additive theory ϕ, the additivization Dϕ can be computed by the formula

Dϕ(C) ≃ ΩΦS(C) ≃ ΩΦS(NFin_{∗} ⊗ C) ≃ ΩΦ(S(NFin_{∗}) ⊗ L^{fiss}(C)),

where Φ is the left derived functor of ϕ.

2.5. Proposition. The functor D_{fiss}(Wald_{∞})^{⊗} NΛ(F) is a symmetric monoidal ∞-category with the property that the tensor product

⊗: D_{fiss}(Wald_{∞}) × D_{fiss}(Wald_{∞}) D_{fiss}(Wald_{∞})

preserves all colimits separately in each variable.

Proof. That D_{fiss}(Wald_{∞})^{⊗} is symmetric monoidal follows from [10, Pr. 2.2.1.9]. Observe that the functor

⊗: D_{fiss}(Wald_{∞}) × D_{fiss}(Wald_{∞}) D_{fiss}(Wald_{∞})

can be identified with the functor given by the assignment

(X, Y) L^{fiss}(X ⊗ Y);

hence it follows from [2, Cor. 6.7.2] it preserves colimits in each variable. □

2.6. Construction. The stabilization

Sp(D_{fiss}(Wald_{∞}))

of the fissile derived ∞-category D_{fiss}(Wald_{∞}) inherits a canonical symmetric monoidal structure Sp(D_{fiss}(Wald_{∞}))^{⊗}, given by applying the symmetric monoidal functor

L^{⊗}: Pr^{L,⊗} Pr_{St}^{L,⊗}

induced by the localization L = Sp⊗− of [10, Pr. 6.3.2.17] to the presentable ∞- category D_{fiss}(Wald_{∞}). By construction, the tensor product functor ⊗ preserves colimits separately in each variable. Furthermore, the functor

Σ^{∞}: D_{fiss}(Wald_{∞}) Sp(D_{fiss}(Wald_{∞}))

is symmetric monoidal. We call the ∞-category Sp(D_{fiss}(Wald_{∞})) the stable ∞-category of Waldhausen ∞-categories, and we call its homotopy category hSp(D_{fiss}(Wald_{∞})) the stable homotopy category of Waldhausen ∞-categories. Note that Sp(D_{fiss}(Wald_{∞})) is equivalent to the full subcategory of the ∞- category Fun(Wald_{∞}^{ω,op}, Sp) spanned by those functors X: Wald_{∞}^{ω,op} Sp such that for any compact Waldhausen ∞-category C, the morphisms induced by the exact functor i: C ⊕ C E(C) exhibits X(E(C)) as the direct sum of X(C) and X(C).

2.7. Observe that the K-groups of a Waldhausen ∞-category C are given by Ext groups in the stable homotopy category of Waldhausen ∞-categories. Indeed, just as one may describe the stable homotopy groups of a pointed space X as Ext groups out of the sphere spectrum:

π_{n}^{s}(X) ≅ ExtSp^{−n}(Σ^{∞}S^{0}, Σ^{∞}X),

so too may one describe the algebraic K-theory groups of a Waldhausen ∞- category C as Ext groups out of the suspension spectrum of Fin_{∗}:

K_{n}(C) ≅ Ext_{Sp(D_{fiss} (Wald_{∞}))}^{−n}(Σ^{∞}Fin_{∗}, Σ^{∞}C).

3. Multiplicative theories

Now we are in a position to study theories that are compatible with the monoidal structure on Waldhausen ∞-categories.

3.1. Recall [2, Df. 7.1] that for any ∞-topos E, an E-valued theory is a reduced functor

ϕ: Wald_{∞} E

that preserves filtered colimits.

3.2. Definition. Suppose E an ∞-topos. A multiplicative theory valued in E is a morphism of ∞-operads

ϕ^{⊗}: Wald_{∞}^{⊗} E^{×}

such that the underlying functor ϕ: Wald_{∞} E preserves all filtered colimits and carries the zero object to the terminal object. We will say that the multiplicative theory ϕ^{⊗} extends the theory ϕ. We denote by

Thy^{⊗}(E) ⊂ Fun_{NΛ(F)}(Wald_{∞}^{⊗}, E^{×})

the full subcategory spanned by the multiplicative theories, and we denote by Add^{⊗}(E) the full subcategory spanned by those multiplicative theories such that the underlying theory Wald_{∞} E is additive.

3.3. It follows from Pr. 2.1 that any multiplicative theory

ϕ^{⊗}: Wald_{∞}^{⊗} E^{×}

can be extended to a reduced functor Φ^{⊗}: D(Wald_{∞})^{⊗} E^{×} of ∞-operads such that the underlying functor D(Wald_{∞}) E_{∗} preserves sifted colimits. This is the multiplicative left derived functor of ϕ^{⊗}.

3.4. Example. The theory ι: Wald_{∞} Kan [2, Nt. 1.7] can be extended to a multiplicative theory ι^{⊗} in the following manner. The right adjoint Cat_{∞} Kan of the inclusion can be given the structure of a symmetric monoidal functor

Cat_{∞}^{×}  Kan^{×}

for the cartesian symmetric monoidal structures in an essentially unique manner. The desired morphism ι^{⊗}: Wald_{∞}^{⊗} Kan^{×} of ∞-operads is now the composite

Wald_{∞}^{⊗}  Pair_{∞}^{×}  Cat_{∞}^{×}  Kan^{×}.

We now wish to show that additivizations of multiplicative theories are naturally multiplicative (Th. 3.6). Since additivizations are constructed via Goodwillie differentials, we will prove a general result about these. What follows is surely not the most general result one can prove, but it’s enough for our purposes.

3.5. Proposition. Suppose C^{⊗} and D^{⊗} symmetric monoidal ∞-categories. Suppose that the underlying ∞-category D is presentable, and suppose that the underlying ∞-category C is small and that it admits a terminal object and all finite colimits. Finally, assume that C^{⊗} is compatible with all finite colimits, and assume that D^{⊗} is compatible with all colimits. Then the inclusion

Exc(C, D) ×_{Fun(C,D)} Alg_{C^{⊗}}(D^{⊗})  Alg_{C^{⊗}}(D^{⊗})

admits a left adjoint.

Proof A. The Adjoint Functor Theorem [9, Cor. 5.5.2.9] and the existence of excisive approximation [10, Th. 7.1.10] implies that Exc(C, D) ⊂ Fun(C, D) is stable under both limits and κ-filtered colimits for some regular cardinal κ. The fiber product

Exc(C, D) ×_{Fun(C,D)} Alg_{C^{⊗}}(D^{⊗})

can be identified with the full subcategory of Alg_{C^{⊗}}(D^{⊗}) spanned by those functors F^{⊗}: C^{⊗} D^{⊗} over NΛ(F) with the property that the underlying functor F: C D is excisive. It now follows from [10, Cor. 3.2.2.5 and Pr. 3.2.3.1(4)] that this subcategory is stable under both limits and κ-filtered colimits. A second coat of the Adjoint Functor Theorem [9, Cor. 5.5.2.9] now yields the result. □

Proof B. Alternately, one may prove this result (in fact a more general version thereof) in an explicit fashion by applying a variant of [9, Cor. 5.2.7.11]. Note that for the argument there to go through, one does not need the full strength of the condition that the forgetful functor

p: Alg_{C^{⊗}}(D^{⊗}) Fun(C, D)

be a cocartesian fibration. (It obviously isn’t in our case.) One need only ensure that for every multiplicative functor F^{⊗}: C^{⊗} D^{⊗} with underlying functor F: C D, there exist a localization θ: F P_{1}F with respect to Exc(C, D) and a p-cocartesian edge F^{⊗} (P_{1}F)^{⊗} that covers θ. Recall [10, Cnstr. 7.1.1.27] that P_{1}F can be obtained as the sequential colimit of the sequence

F T_{1}F T_{1}^{2}F ⋯,

where T_{1}F = Ω ∘ F ∘ Σ as in 0.4. Let N^{+} denote the following ordinary category. An object is a pair (I, k_{I}) consisting of a finite set I and an I-tuple k_{I} = (k_{i})_{i∈I} of natural numbers. A morphism (J, ℓ_{J}) (I, k_{I}) is a map of finite sets J I_{+} such that for any element i ∈ I, one has

∑_{j∈J_{i}} ℓ_{j} ≤ k_{i}.

The forgetful functor N^{+} Λ(F) is the Grothendieck opfibration that corresponds to the functor Λ(F) Cat that exhibits the ordered set of natural numbers as a symmetric monoidal category under addition. Hence the cocartesian fibration NN^{+} NΛ(F) is a symmetric monoidal ∞-category. Now one may define a functor

(P_{1}F)^{⊠}: NN^{+} ×_{NFin_{∗}} C^{⊗} D^{⊗}

such that the formula

(P_{1}F)^{⊠}(I, k_{I}, X_{I}) := (T_{1}^{k_{i}}X_{i})_{i∈I}

holds. Then our desired functor (P_{1}F)^{⊗} will be the left Kan extension of (P_{1}F)^{⊠} along the projection NN^{+} ×_{NΛ(F)} C^{⊗} C^{⊗}. We also have the functor F^{⊠}: NN^{+} ×_{NΛ(F)} C^{⊗} D^{⊗}, which is the projec-

tion NN^{+} ×_{NΛ(F)} C^{⊗}  C^{⊗} composed with F^{⊗}: C^{⊗}  D^{⊗}. The natural
transformations F  T_{1}^{k}F induce a natural transformation F^{⊠}  (P_{1}F)^{⊠}

and thus an induced natural transformation θ^{⊗}: F^{⊗} (P_{1}F)^{⊗}. A quick computation shows that θ^{⊗} is a cocartesian edge covering θ. □

3.5.1. Corollary. Suppose C^{⊗} and D^{⊗} symmetric monoidal ∞-categories. Suppose that the underlying ∞-category D is presentable, and suppose that the underlying ∞-category C is compactly generated. Finally, assume that C^{⊗} and D^{⊗} are compatible with all colimits. Then the inclusion

Exc_{F} (C, D) ×_{Fun(C,D)} Alg_{C^{⊗}}(D^{⊗})  Alg_{C^{⊗}}(D^{⊗})

admits a left adjoint, where Exc_{F} (C, D) denotes the full subcategory of Fun(C, D) spanned by excisive functors C D that preserve all filtered colimits.

With this result in hand, we easily prove our main theorem.

3.6. Theorem. The inclusion Add^{⊗}(E) Thy^{⊗}(E) admits a left adjoint that covers the left adjoint of the inclusion Add(E) Thy(E).

Proof. In light of [2, Th. 7.6], the claim is that the inclusion

Exc_{∗}^{G}(D_{fiss}(Wald_{∞}), E) ×_{Fun(D_{fiss} (Wald_{∞}),E)} Alg_{D_{fiss} (Wald_{∞})^{⊗}}(E^{×})
Fun_{∗}^{G}(D_{fiss}(Wald_{∞}), E) ×_{Fun(D_{fiss} (Wald_{∞}),E)} Alg_{D_{fiss} (Wald_{∞})^{⊗}}(E^{×})

admits a left adjoint. We now appeal to Cor. 3.5.1, and the proof is completed by the observation that if a functor D_{fiss}(Wald_{∞}) E preserves geometric realizations, then so does Goodwillie differential. (This follows from [2, Lm. 7.7]; see the proof of [2, Th. 7.8].) □

3.7. Notation. Write D^{⊗} for the left adjoint of the inclusion

Add^{⊗}(E) Thy^{⊗}(E).

We may now define K^{⊗} := D^{⊗}ι^{⊗}, whence we deduce that algebraic K-theory is naturally a multiplicative theory.

3.8. Proposition. There exists a canonical multiplicative extension K^{⊗} of algebraic K-theory.

In light of [10, Pr. 7.2.4.14 and Pr. 7.2.6.2], we also find the following.

3.8.1. Corollary. There exists a canonical multiplicative extension K^{⊗} of the connective algebraic K-theory functor K: Wald_{∞} Sp.

3.8.2. Corollary. For any ∞-operad O^{⊗}, composition with the multiplicative extensions K^{⊗} and K^{⊗} induce functors

K^{⊗}: Alg_{O^{⊗}}(Wald_{∞}^{⊗}) Alg_{O^{⊗}}(Kan^{×})

and

K^{⊗}: Alg_{O^{⊗}}(Wald_{∞}^{⊗}) Alg_{O^{⊗}}(Sp^{∧}).

As a special case of this, we obtain the following.

3.8.3. Corollary. Suppose A^{⊗} a monoidal Waldhausen ∞-category. Then composition with the multiplicative extensions K^{⊗} and K^{⊗} induce functors

K^{⊗}: LMod_{A^{⊗}}(Wald_{∞}^{⊗}) LMod_{K^{⊗}(A^{⊗})}(Kan^{×}) and

K^{⊗}: LMod_{A^{⊗}}(Wald_{∞}^{⊗}) LMod_{K^{⊗}(A^{⊗})}(Sp^{∧}).

3.9. Example (Deligne Conjecture for algebraic K-theory). Suppose C^{⊗} a pointed, symmetric monoidal ∞-category that admits all finite colimits. Assume that the tensor product ⊗: C × C C preserves finite colimits separately in each variable. Then C^{⊗} may be viewed as an E_{∞} object of Wald_{∞}^{⊗}, and the K-theory spectrum of C is naturally endowed with an E_{∞}-structure given by K^{⊗}(C^{⊗}) [Cor. 3.8.2]. Carrying an E_{n}-algebra in C^{⊗} to its ∞-category of right modules defines a functor Θ_{C} [10, Rk. 6.3.5.15], which factors through a functor

Alg_{E_{n}}(C^{⊗}) Alg_{E_{n−1}}(Wald_{∞}^{⊗}).

Composing this with K^{⊗} and K^{⊗}, we obtain functors

Alg_{E_{n}}(C^{⊗})  Alg_{E_{n−1}}(Kan^{×}) and Alg_{E_{n}}(C^{⊗})  Alg_{E_{n−1}}(Sp^{∧}).

3.10. Example. The previous example makes an assortment of iterated Ktheories possible. If n ≥ 1, then by forming iterated compositions of the various functors Alg_{E_{k}}(Sp^{∧}) Alg_{E_{k−1}}(Sp^{∧}) constructed above, we obtain n-fold algebraic K-theory functor

K^{(n)} : Alg_{E_{n}}(Sp^{∧}) Sp^{∧}

as well as an infinite hierarchy of functors

K^{(n)} : Alg_{E_{∞}}(Sp^{∧}) Alg_{E_{∞}}(Sp^{∧}).

The Chromatic Red Shift Conjectures of Ausoni and Rognes (presaged by Waldhausen and Hopkins) implies that K^{(n)} should, in effect, carry E_{n}-rings of telescopic complexity m to spectra of telescopic complexity m + n. We hope to investigate these phenomena in future work.

3.11. Example. We also find that Waldhausen’s A-theory of an n-fold loopspace X (defined as in [2, Ex. 2.10]) carries a canonical E_{n−1}-monoidal structure.

3.12. Note that, although this result ensures that algebraic K-theory is merely lax symmetric monoidal, as a functor to spectra it’s actually slightly better: by Barratt–Priddy–Quillen, K^{⊗}: Wald_{∞}^{⊗} Sp^{∧} carries the unit NFin_{∗} to the unit S^{0}.

Let us conclude by appealing to the recent work [6] of Saul Glasman. Glasman identifies the ∞-category AlgWald^{⊗}(E^{×}) with the ∞-category of E_{∞} algebras in Fun(Wald_{∞}, E), equipped with the Day convolution structure. When E = Kan, one sees that the functor ι is corepresented by the unit NFin_{∗}; consequently it is the unit for the Day convolution product. Hence it admits a unique E_{∞} structure, which under Glasman’s equivalence must coincide with the multiplicative theory ι^{⊗}. Moreover, ι^{⊗} is initial in AlgWald^{⊗}(E^{×}). Meanwhile, the universal property of K^{⊗} ensures that for any additive multiplicative theory ϕ^{⊗}, we have

Map(K^{⊗}, ϕ^{⊗}) ≃ Map(ι^{⊗}, ϕ^{⊗}) ≃ ∗.

That is, there is an essentially unique multiplicative “trace map” from K-theory to any additive and multiplicative theory. In other words, we have the following.

3.13. Theorem. Algebraic K-theory of Waldhausen ∞-categories is the initial additive and multiplicative theory.

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Clark Barwick Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 USA clarkbar@gmail.com