2 Lawvere theories and their algebraic K-theory

Lawvere theories are a fundamental tool for encoding algebraic structures, first introduced in Lawvere’s thesis [Reference Lawvere30] and now widely used throughout algebra, logic and related disciplines. We review the basic notions and our notation for Lawvere theories, using the same language as in the prequel [Reference Bohmann and Szymik5]. In addition, we also discuss Kronecker products. Some textbook references for Lawvere theories are [Reference Adámek, Rosický and Vitale1, Reference Borceux8, Reference Pareigis37, Reference Schubert42]. Proposition 2.11 singles out a property of Lawvere theories that distinguishes them from the more general symmetric monoidal categories and spares us higher categories throughout.

We also recall the definition of the algebraic K-theory spectrum $\mathrm {K}(T)$ of a Lawvere theory T from [Reference Bohmann and Szymik5]. Our primary approach is to view Lawvere theories as a special case of symmetric monoidal categories and apply classic constructions of K-theory for the latter.

2.1 Lawvere theories

We choose a skeleton $\mathbf {E}$ of the category of finite sets and (all) maps between them. For each integer $r\geqslant 0$ , such a category has a unique object with precisely r elements, and there are no other objects. For the sake of explicitness, let us choose the model $\underline {r}=\{a\in \mathbb {Z}\,|\,1\leqslant a\leqslant r\}$ for such a set. A set with $r+s$ elements is the (categorical) sum (or coproduct) of a set with r elements and a set with s elements.

Definition 2.1. A Lawvere theory $T=(\mathbf {F}_T,\mathrm {F}_T)$ is a pair consisting of a small category $\mathbf {F}_T$ together with a functor

$$\begin{align*}\mathrm{F}_T\colon\mathbf{E}\longrightarrow\mathbf{F}_T \end{align*}$$

that is bijective on sets of objects and that preserves sums. The second condition means that the canonical map $\mathrm {F}_T(\underline {r})+\mathrm {F}_T(\underline {s})\to \mathrm {F}_T(\underline {r}+\underline {s})$ induced by the canonical injections is an isomorphism for all sets $\underline {r}$ and $\underline {s}$ in $\mathbf {E}$ .

The image of the set $\underline {r}$ with r elements under the functor $\mathrm {F}_T\colon \mathbf {E}\to \mathbf {F}_T$ will be written $T_r$ , so that the object $T_r$ is the sum in the category $\mathbf {F}_T$ of r copies of the object $T_1$ .

Remark 2.2. Some authors prefer to work with the opposite category $\mathbf {F}_T^{\mathrm {op}}$ , so that the object $T_r$ is the product (rather than the coproduct) of r copies of the object $T_1$ . For example, this was Lawvere’s convention when he introduced this notion in [Reference Lawvere30]. Our convention reflects the point of view that the object $T_r$ should be thought of as the free T–model (or T–algebra) on r generators, covariantly in r (or rather in $\mathbf {E}$ ). To make this precise, recall the definition of a model (or algebra) for a theory T.

Given a Lawvere theory T, a T–model (or T–algebra) is a presheaf X (of sets) on the category $\mathbf {F}_T$ that sends (categorical) sums in $\mathbf {F}_T$ to (categorical, i.e. Cartesian) products of sets (this means that the canonical map $X(T_r+T_s)\to X(T_r)\times X(T_s)$ induced by the injections is a bijection for all sets $\underline {r}$ and $\underline {s}$ in $\mathbf {E}$ ). We write $\mathbf {M}_T$ for the category of T–models, and we write $\mathbf {M}_T(X,Y)$ to denote the set of morphisms $X\to Y$ between T–models. Such a morphism is defined to be a map of presheaves, that is, a natural transformation, so that $\mathbf {M}_T$ is a full subcategory of the category of presheaves on $\mathbf {F}_T$ .

Remark 2.3. The values of a T–model are determined up to isomorphism by the value at $T_1$ , and we often use the same notation for a model and its value at $T_1$ .

For any Lawvere theory T, the category $\mathbf {M}_T$ of T–models is complete and cocomplete. Limits are constructed levelwise, and the existence of colimits follows from the adjoint functor theorem. The category $\mathbf {M}_T$ becomes symmetric monoidal with respect to the (categorical) sum, and the unit object $T_0$ for this structure is also an initial object in the category $\mathbf {M}_T$ . The covariant Yoneda embedding $\mathbf {F}_T\to \operatorname {\mathrm {Pre}}(\mathbf {F}_T) $ sends the object $T_r$ of the category $\mathbf {F}_T$ to the presheaf $T_s\mapsto \mathbf {F}_T(T_s,T_r)$ represented by it. Such a presheaf is readily checked to be a T–model. We refer to a T–model of this form as free. The definitions unravel to give natural bijections $\mathbf {M}_T(T_r,X)\cong X^r$ for T–models X, so that $T_r$ is indeed a free T–model on r generators.

Definition 2.4. A morphism $S\to T$ between Lawvere theories is a functor $L\colon \mathbf {F}_S\to \mathbf {F}_T$ that preserves sums and free models. This is equivalent to the condition that $\mathrm {F}_T\cong L\circ \mathrm {F}_S$ , that is, that L is a map under $\mathbf {E}$ .

Often, a morphism $S\to T$ between Lawvere theories is described by giving a functor $R\colon \mathbf {M}_T\to \mathbf {M}_S$ that is compatible with the forgetful functors to the category $\mathbf {M}_E$ of sets. Then L is induced by the left adjoint to R, which exists for abstract reasons, namely, by Freyd’s adjoint functor theorem.

Example 2.5. Two of the most important classes of examples of Lawvere theories are given by the theories of A–modules over a fixed ring A, and the theory of G–sets for a fixed group G. In particular, for the trivial group $G=\{e\}$ , we have the Lawvere theory E of sets.

In a slight generalisation, we can define T–models not only in the category of sets but in any category with finite (categorical) products. In particular, we may then consider T–models in other categories of models; this is what we are going to do now.

Given Lawvere theories S and T, their Kronecker product $S\otimes T$ is a Lawvere theory that represents T–models in the category of S–models or, equivalently, S–models in the category of T–models. These theories are described by Freyd [Reference Freyd19], and in Lawvere’s thesis [Reference Lawvere31]. It follows from this description that there are morphisms

$$\begin{align*}S\longrightarrow S\otimes T\longleftarrow T \end{align*}$$

of Lawvere theories.

Example 2.6. If S and T are the theories of modules over rings A and B, respectively, then $S\otimes T$ is the theory of $(A\otimes B)$ –modules [Reference Borceux8, Example 3.11.7b]. If S and T are the theories of G–sets and H–sets for groups G and H, respectively, then $S\otimes T$ is the theory of $(G\times H)$ –sets: sets with commuting actions by G and H.

Example 2.7. We can pair $S=\mathbb {Z}$ , the Lawvere theory of abelian groups, with any Lawvere theory T to obtain a new Lawvere theory $\mathbb {Z}\otimes T$ whose models are the abelian group objects in the category of T–models. This theory $\mathbb {Z}\otimes T$ comes with a morphism

(2.1) $$ \begin{align} T\longrightarrow\mathbb{Z}\otimes T, \end{align} $$

the linearisation. Via the discussion following Definition 2.4, we can view the linearisation morphism as induced by the left adjoint L to the forgetful functor that takes an abelian group object in T–models to its underlying T–model. Thus, this left adjoint is an abelianisation functor. The models of $\mathbb {Z}\otimes T$ are essentially the modules over a ring [Reference Wraith51, Theorem 13.2]. Indeed, it follows from Morita theory that $\mathbb {Z}\otimes T$ –models can be described as modules over the endomorphism ring of of the linearisation $L(T_1)$ of the free T–model $T_1$ on one generator. Continuing to write $\mathbf {M}_T(X,Y)$ for the set of morphisms $X\to Y$ of T-models, we see that for each abelian group object A in T–models there are isomorphisms $A\cong \mathbf {M}_T(T_1,A)\cong \mathbf {M}_{\mathbb {Z}\otimes T}(L(T_1),A)$ . Hence, via precomposition, every abelian group object A is a module over the endomorphism ring $\mathbf {M}_{\mathbb {Z}\otimes T}(L(T_1), L(T_1))\cong L(T_1)$ . This identification yields a functor from $\mathbb {Z}\otimes T$ –models to $L(T_1)$ –modules, and this functor turns out to be an equivalence.

The description of the Kronecker product $S\otimes T$ in terms of its models is not the most convenient for our purpose. We shall give another description of it following Hyland and Power [Reference Hyland and Power25]. Since the category of natural numbers (our skeleton $\mathbf {E}$ of the category of finite sets) has finite products as well as sums, for $\underline {r},\underline {s}\in \mathbf {E}$ , we have the product $\underline {r}\times \underline {s}$ . Since $\mathbf {E}$ is skeletal, this product is the set $\underline {rs}$ ; we will consider $\underline {rs}$ as the r–fold sum of $\underline {s}$ with itself. Under this identification, a morphism of sets $f\colon \underline {s}\to \underline {s}'$ in $\mathbf {E}$ induces a morphism $\underline {r}\times f\colon \underline {r}\times \underline {s}\to \underline {r}\times \underline {s}'$ .

For any Lawvere theory S, we can extend this construction to the category $\mathbf {F}_S$ . Given $r\in \mathbb {Z}$ and $S_s\in \mathbf {F}_S$ , we define

$$\begin{align*}\underline{r}\times S_s=\underbrace{S_s+\dots+ S_s}_r \end{align*}$$

to be the r–fold sum in $\mathbf {F}_S$ of $S_s$ with itself. The definition of the category $\mathbf {F}_S$ means this sum must be the object $S_{r\times s}$ , but this identification provides a corresponding construction on morphisms. If $f\colon S_{s}\to S_{s'}$ is a morphism in $\mathbf {F}_S$ , the functoriality of sums produces a morphism

$$\begin{align*}\underline{r}\times f\colon \underline{r}\times S_{s}\longrightarrow \underline{r}\times S_{s'}. \end{align*}$$

Conjugating by the symmetry $\underline {r}\times \underline {s}\to \underline {s}\times \underline {r}$ , we similarly can construct the object $S_s\times \underline {r}$ and a morphism $f\times \underline {r}\colon S_{s}\times \underline {r}\to S_{s'}\times \underline {r}$ . As an object in $\mathbf {F}_S$ , we have $S_{s}\times \underline {r}=S_{r\times s}=\underline {r}\times S_{s}$ .

Remark 2.8. For fixed r, the construction $S_s\mapsto \underline {r}\times S_s$ yields a functor $\mathbf {F}_S\to \mathbf {F}_S$ that is strong monoidal, as does the construction $S_s\mapsto S_s\times \underline {r}$ .

Definition 2.9. Given two Lawvere theories S and T, their Kronecker (or tensor) product Lawvere theory $S\otimes T$ is defined by the universal property of admitting maps of Lawvere theories $S\to S\otimes T$ and $T\to S\otimes T$ so that the operations of S commute with the operations of T in the sense that for all $f\colon S_r\to S_s$ in $\mathbf {F}_S$ and $f'\colon T_{r'}\to T_{s'}$ in $\mathbf {F}_T$ , the diagram

commutes (in the category $\mathbf {F}_{S\otimes T}$ ). The vertical maps, here, should be interpreted as the image of the maps $f\times r'\colon S_{r\times r'}\to S_{s\times r'}$ in $\mathbf {F}_{S}$ under the map of Lawvere theories $S\to S\otimes T$ , and similarly for the horizontal maps, mutatis mutandis.

Proposition 2.10 ([Reference Hyland and Power25, Proposition 3.3])

The Kronecker product extends to a symmetric monoidal structure on the category of Lawvere theories with the theory E of sets as the unit.

Hyland and Power remark that the construction of $S\otimes T$ can be done by hand, or it can be viewed as a special case of their work on pseudo-commutativity and, in particular, on the pseudo-closed structure of the $2$ –category of symmetric monoidal categories [Reference Hyland and Power24]. However, the following proposition shows that this generality is not necessary for Lawvere theories because equality between any two of them is a logical proposition: it either has a unique proof or none.

Proposition 2.11. Lawvere theories naturally form a $1$ –category rather than a $2$ –category. More precisely, given two maps of Lawvere theories $L_1,L_2\colon S\to T$ , the set—and, hence, the space—of natural transformations between $L_1$ and $L_2$ is empty, unless $L_1=L_2$ , in which case, it contains only the identity transformation.

Proof. A morphism of Lawvere theories is a map under $\textbf {E}$ , that is, a strictly commuting diagram of the following form:

Thus, a natural transformation between such L must restrict to the identity natural transformation on $\mathbf {E}$ . Since all objects in $\mathbf {F}_S$ are in the image of $\mathbf {E}$ , this forces all natural transformations to be the identity.

For Lawvere theories, there is no room for the ‘psubtlety’ of pseudoness.

2.2 Algebraic K-theory

Let $\mathbf {S}$ denote a symmetric monoidal groupoid. To build a K-theory spectrum $\mathrm {K}(\mathbf {S})$ , we can use Segal’s definition [Reference Segal43] of the algebraic K-theory of a symmetric monoidal category in terms of $\Gamma $ –spaces. For the multiplicativity properties we need in the following Section 3, we in fact use a variant of Segal’s construction given by Elmendorf–Mandell [Reference Elmendorf and Mandell16] (see also [Reference Bohmann and Osorno4]) which takes values in the category of symmetric spectra and builds the spaces of the K-theory spectrum ‘all at once’ instead of iteratively.

Definition 2.12. Let T be a Lawvere theory. The algebraic K-theory of T is the spectrum

(2.2) $$ \begin{align} \mathrm{K}(T)=\mathrm{K}(\mathbf{F}_T^{\times}), \end{align} $$

that is, the spectrum corresponding to the symmetric monoidal groupoid $\mathbf {F}_T^{\times }$ of isomorphisms in the symmetric monoidal category $\mathbf {F}_T$ of finitely generated free T–models, where the monoidal structure is given by the categorical sum.

Remark 2.13. Since the category $\mathbf {F}_T$ can be identified with the symmetric monoidal category of finitely generated free T–models, Definition 2.12 concerns the algebraic K-theory of finitely generated free T–models. In particular, the group $\mathrm {K}_0(T)=\pi _0\mathrm {K}(T)$ is the Grothendieck group of isomorphism classes of finitely generated free T–models. This group is always cyclic, generated by the isomorphism class $[\,T_1\,]$ of the free T–model on one generator. However, the group $\mathrm {K}_0(T)$ does not have to be infinite cyclic. This happens, for instance, for the theory of Cantor algebras (see Section 5.1).

Remark 2.14. A morphism $S\to T$ of Lawvere theories as in Definition 2.4 induces, via the left-adjoint functor $\mathbf {F}_S\to \mathbf {F}_T$ , a morphism $\mathrm {K}(S)\to \mathrm {K}(T)$ of algebraic K-theory spectra. The left adjoint $\mathbf {F}_S\to \mathbf {F}_T$ sends the free S–model $S_1$ on one generator to the free T–model $T_1$ on one generator. It follows that the induced homomorphism $\mathrm {K}_0(S)\to \mathrm {K}_0(T)$ between cyclic groups is surjective, being the identity on representatives.

Example 2.15. If A is a ring, denote by $\mathrm {K}(A)$ the K-theory of the Lawvere theory of A–modules. This spectrum is the ‘free’ version of the usual algebraic K-theory of the ring A, that is, Quillen’s algebraic K-theory $\mathrm {K}^{\mathrm {free}}(A)$ of the category of finitely generated free A–modules. It is perhaps more common to consider the algebraic K-theory spectrum $\mathrm {K}^{\mathrm {proj}}(A)$ of the category of finitely generated projective A–modules as the ‘algebraic K-theory of A’. However, the inclusion of free modules into projective modules induces a map

$$\begin{align*}\mathrm{K}^{\mathrm{free}}(A)\longrightarrow\mathrm{K}^{\mathrm{proj}}(A) \end{align*}$$

which is an equivalence whenever projective modules are free, including for fields and for principal ideal domains. In fact, this map is an equivalence on components and so induces an equivalence on higher-homotopy groups $\pi _n$ for $n\geqslant 1$ .

In particular, the usual K-theory spectrum $\mathrm {K}(\mathbb {Z})$ is the K-theory spectrum of the Lawvere theory of abelian groups in the guise of $\mathbb {Z}$ –modules. We refer to [Reference Bohmann and Szymik5] for an extensive supply of examples of algebraic K-theory spectra $\mathrm {K}(T)$ of Lawvere theories T that are not of this form. For the moment, we only mention the initial theory E of sets, where $\mathrm {K}(E)\simeq \mathbb {S}$ is the sphere spectrum. We will discuss other examples, which are arguably even more interesting, in the later sections to illustrate our results.

3 Multiplicative structure

The generalised assembly maps are a consequence of multiplicative structure on algebraic K-theory. In order to define and understand these maps, we first isolate a part of a general multiplicativity statement by Elmendorf–Mandell that we can then use to produce assembly-type maps (see Theorem 3.6). The presentation elides a number of the category-theoretical considerations but tells us precisely what kind of functors we shall need to produce assembly maps. Afterwards, we give a more categorically sophisticated and higher-level discussion of multiplicativity, which, in particular, shows that K-theory is a lax symmetric monoidal on Lawvere theories, as in Theorem 3.9 and Corollary 3.10. Because the proofs for these results are somewhat technical, we have largely postponed them to the end of this section.

In this section, boldface uppercase letters $\mathbf {A},\mathbf {B},\mathbf {C},\dots $ will denote symmetric monoidal categories. Our default notation for the monoidal product is $\oplus $ and $0$ typically denotes the monoidal unit, with indices as in $\oplus =\oplus _{\mathbf {A}}$ and $0=0_{\mathbf {A}}$ if needed. By convention, we use ‘symmetric monoidal category’ in this section for symmetric monoidal categories with strict unit, as our primary references are written for this case. This strictness should be viewed as a basepoint condition. All symmetric monoidal categories can be strictified, so this does not represent a loss of generality.

We use the language of multicategories to describe the constructions in this section. All the multicategories we use are implicitly symmetric. Multicategories may be more familiar to some readers under the term operad, implied to allow several colours. Our choice of terminology reflects that of our primary references [Reference Elmendorf and Mandell16, Reference Elmendorf and Mandell17] for this work. The terminological distinction is partly philosophical. In this work, the multicategories appear as generalisations of categories instead of as parameter spaces of operations. Of course, these roles are intimately linked, and we invite the reader to use their preferred term.

One way to formulate multiplicativity is in terms of ‘bilinear functors’. This formulation is analogous to thinking about bilinear maps between vector spaces, rather than the tensor product of vector spaces.

Definition 3.1. A bilinear functor of symmetric monoidal categories is a functor

$$\begin{align*}P\colon \mathbf {A}\times \mathbf {B}\to \mathbf {C} \end{align*}$$

together with natural distributivity isomorphisms

$$\begin{align*}\delta_l\colon P(a,b)\oplus P(a',b)\to P(a\oplus a',b) \quad \text{and} \quad \delta_r\colon P(a,b)\oplus P(a,b')\to P(a,b\oplus b') \end{align*}$$

satisfying some unitality and compatibility conditions which are spelled out in [Reference Bohmann and Osorno4, Definition 7.1].

Observe that the distributivity transformations mean, in particular, that P is a strong monoidal ‘in each variable separately’ in the sense that if we fix an object $a\in \mathbf {A}$ , the functor $P(a,-)$ is a strong monoidal, and if we fix $b\in \mathbf {B}$ , the functor $P(-,b)$ is a strong monoidal.

Example 3.2. For any symmetric monoidal category $\mathbf {C}$ , there is a ‘left unit’ bilinear functor

$$\begin{align*}u\colon \mathbf{E}^{\times}\times \mathbf{C}\to \mathbf{C} \end{align*}$$

given on objects by $u(\underline {n},c)=c^{\oplus n}$ . The components of the distributivity natural transformation $\delta _l$ are the identity maps

$$\begin{align*}c^{\oplus n}\oplus c^{\oplus n'}=c^{\oplus n+n'} \end{align*}$$

and the components of the distributivity natural transformation $\delta _r$ are the reordering isomorphisms

$$\begin{align*}c^{\oplus n}\oplus {c'}^{\oplus n}\to (c\oplus c')^{\oplus n}. \end{align*}$$

One can similarly define a ‘right unit’ bilinear functor $\mathbf {C}\times \mathbf {E}^{\times }\to \mathbf {C}$ ; here, the left distributivity is given by reordering.

The following result is a consequence of the fact that Elmendorf–Mandell’s K-theory is an enriched multifunctor from permutative categories to spectra [Reference Elmendorf and Mandell16, Theorem 6.1]. The slight extension to symmetric monoidal categories is in [Reference Bohmann and Osorno4, Theorem 7.4]. Since our Lawvere theories $\mathbf {F}_T^{\times }$ form permutative categories, this extension is not strictly necessary for our work.

Theorem 3.5 ([Reference Bohmann and Osorno4, Theorem 7.4], [Reference Elmendorf and Mandell16, Theorem 6.1])

Let $\mathbf {A}$ , $\mathbf {B}$ and $\mathbf {C}$ be symmetric monoidal categories (with strict units). A bilinear functor $\mathbf {A}\times \mathbf {B}\to \mathbf {C}$ of symmetric monoidal categories induces a morphism

$$\begin{align*}\mathrm{K}(\mathbf{A}) \wedge \mathrm{K}(\mathbf{B}) \to \mathrm{K}(\mathbf{C}) \end{align*}$$

of spectra. This structure is associative and unital.

In the case where the bilinear functor $\mathbf {A}\times \mathbf {A}\to \mathbf {A}$ is the multiplication of a ring category as in Example 3.4, the induced map

$$\begin{align*}\mathrm{K}(\mathbf{A})\wedge\mathrm{K}(\mathbf{A})\to \mathrm{K}(\mathbf{A}) \end{align*}$$

is the multiplication of a ring structure on $\mathrm {K}(\mathbf {A})$ .

In the case where the bilinear functor $\mathbf {E}^{\times }\times \mathbf {C}\to \mathbf {C}$ is the left unit bilinear functor of Example 3.2, the induced map

$$\begin{align*}\mathbb{S}\wedge \mathrm{K}(\mathbf{C})\simeq\mathrm{K}(\mathbf{E}^{\times})\wedge \mathrm{K}(\mathbf{C})\to\mathrm{K}(\mathbf{C}) \end{align*}$$

is the left unit map for the smash product $\wedge $ of spectra.

One way to think about Theorem 3.5 is that it tells us that K-theory of symmetric monoidal categories is ‘morally lax symmetric monoidal’, in the following sense. Symmetric monoidal categories do not form a symmetric monoidal category because there is, in general, no representing ‘tensor product’ symmetric monoidal category ‘ $\mathbf {A}\otimes \mathbf {B}$ ’ for bilinear functors [Reference Hyland and Power24]. If such a tensor product exists, then there is a universal bilinear functor $\mathbf {A}\times \mathbf {B}\to \mathbf {A}\otimes \mathbf {B}$ and Theorem 3.5 provides the type of map of spectra $\mathrm {K}(\mathbf {A})\wedge \mathrm {K}(\mathbf {B})\to \mathrm {K}(\mathbf {A}\otimes \mathbf {B})$ needed to make the K-functor lax symmetric monoidal. Since the tensor product doesn’t always exist, Elmendorf and Mandell’s approach is to work with the multicategory of permutative categories, in which n–ary maps are given by n–multilinear functors. They show that K-theory is a multifunctor from this multicategory to the category of spectra. Theorem 3.5 is an explicit statement of the fact that a map of multicategories takes binary maps to binary maps.

Using Theorem 3.5, applied to the case of Lawvere theories, we can prove the following result.

Theorem 3.6. For each pair of Lawvere theories S and T, there is a morphism

(3.1) $$ \begin{align} \mathrm{K}(S)\wedge\mathrm{K}(T)\longrightarrow\mathrm{K}(S\otimes T) \end{align} $$

of spectra that is natural in S and T and that is induced by the multiplication of integers at the level of $\pi _0$ .

We give a detailed proof of Theorem 3.6 at the end of this section. As a consequence of having this result, a Lawvere theory T that has a multiplication $T\otimes T\to T$ produces a multiplication $\mathrm {K}(T)\wedge \mathrm {K}(T)\to \mathrm {K}(T)$ in spectra. We discuss this rather restrictive, but still important, situation further in the following Section 4. Similarly, the left unit map $E\otimes T\to T$ of a Lawvere theory T yields the left unit map $\mathbb {S}\wedge \mathrm {K}(T)\simeq \mathrm {K}(E)\wedge \mathrm {K}(T)\to \mathrm {K}(T)$ in spectra.

The category of Lawvere theories does have a symmetric monoidal structure, with tensor product given by the Kronecker product, and Theorem 3.6 is singling out the natural transformation that makes K-theory into a lax symmetric monoidal functor from the symmetric monoidal category of Lawvere theories to the category of spectra. In fact, the remainder of this section focuses on showing that K-theory of Lawvere theories is a lax symmetric monoidal functor (see Corollary 3.10).

Definition 2.12 constructs the algebraic K-theory of Lawvere theories as a composite functor

$$\begin{align*}\mathbf{Lawvere}\longrightarrow \mathbf{PermCat}\xrightarrow{\ \mathrm{K}\ } \mathbf{Spectra}.\end{align*}$$

Since $\mathbf {PermCat}$ isn’t a symmetric monoidal category, we cannot use this factorisation to prove that the composite is a lax symmetric monoidal functor. Instead, we use a further factorisation of the K-theory construction which relies on a second phrasing of multiplicativity, due to Elmendorf and Mandell [Reference Elmendorf and Mandell17] with revision by Johnson and Yau [Reference Johnson and Yau26]. Their work, which we recall in Theorem 3.8, factors the functor $\mathrm {K}$ above through a symmetric monoidal category of based multicategories that have the structure of modules over a particular multicategory denoted $\mathcal {M}\underline {1}$ (see Definition 3.11). This allows us to factor the algebraic K-theory of Lawvere theories as follows:

(3.2)

All the categories in the lower left path are symmetric monoidal, which allows us to use this factorisation to show that the composite functor across the top is a lax symmetric monoidal.

The factorisation in Diagram (3.2) and the lax symmetric monoidality of the horizontal functor $\mathrm {K}$ are the result of the Elmendorf–Mandell and Johnson–Yau phrasing of the multiplicativity of K-theory, which we now recall. In what follows, we let $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ denote the category of $\mathcal {M}\underline {1}$ –modules in the category of symmetric small based multicategories. This category of modules is in fact a full $2$ –subcategory of the $2$ –category of symmetric small based multicategories [Reference Johnson and Yau26, Proposition III.10.1.28], although this is not necessary for our work.

Theorem 3.8 ([Reference Elmendorf and Mandell17, Theorem 1.3], [Reference Johnson and Yau26, Theorem III.10.3.17])

Based multicategories form a symmetric monoidal category $\mathbf {Mult}_*$ and the multicategory $\mathcal {M}\underline {1}$ of Definition 3.11 is naturally a commutative monoid [Reference Johnson and Yau26, Proposition III.10.1.16]. The K-theory construction of [Reference Elmendorf and Mandell16] factors as the ‘underlying multicategory’ functor U and a lax symmetric monoidal functor from $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ to spectra:

Note that the underlying multicategory $U\mathbf {C}$ of a permutative category $\mathbf {C}$ has a natural basepoint given by the unit object for the monoidal product. This structure extends to a canonical $\mathcal {M}\underline {1}$ –module structure on $U\mathbf {C}$ [Reference Johnson and Yau26, Definition III.10.2.13] (see [Reference Johnson and Yau26, Definition III.10.3.25] for a clear and concise discussion of the factorisation in the above diagram).

With this result in hand, it suffices to prove that the composite of the embedding of Lawvere theories into permutative categories and the underlying multicategory functor is a lax symmetric monoidal functor from Lawvere theories to multicategories.

Theorem 3.9. Let $\iota \colon \mathbf {Lawvere}\to \mathbf {PermCat}$ denote the embedding of Lawvere theories into permutative categories via $T\mapsto \mathbf {F}_T$ . Let $\iota ^{\times }$ denote the embedding $\mathbf {Lawvere}\to \mathbf {PermCat}$ via $T\mapsto \mathbf {F}_T^{\times }$ ; we can view $\iota ^{\times }$ as the composite of $\iota $ and the functor taking a permutative category to its subcategory of isomorphisms. Then the composite functors $U\iota $ and $U\iota ^{\times }$ in the diagrams

are both lax symmetric monoidal.

We give a detailed proof of Theorem 3.9 further down.

Corollary 3.10. The Elmendorf–Mandell construction of K-theory gives a lax symmetric monoidal functor

$$\begin{align*}\mathbf{Lawvere}\longrightarrow\mathbf{Spectra}. \end{align*}$$

We now prove Theorems 3.6 and 3.9 and end this section with some higher-categorical remarks.

Proof of Theorem 3.6

In light of Theorem 3.5, it is sufficient to show that there is a bilinear functor $\mathbf {F}_S^{\times }\times \mathbf {F}_T^{\times }\to \mathbf {F}_{S\otimes T}^{\times }$ of symmetric monoidal categories. Essentially, this is the universal map that comes from the definition of the Kronecker product. However, since general symmetric monoidal categories don’t have such a monoidal product, it is worthwhile to be fairly explicit.

Let S and T be Lawvere theories. We show that there is a bilinear functor

$$\begin{align*}P\colon \mathbf{F}_S\times\mathbf{F}_T\to \mathbf{F}_{S\otimes T} \end{align*}$$

of symmetric monoidal categories with strict unit. Observe that a bilinear functor $\mathbf {A}\times \mathbf {B}\to \mathbf {C}$ restricts to a bilinear functor $\mathbf {A}^{\times } \times \mathbf {B}^{\times } \to \mathbf {C}^{\times }$ on the subcategories of isomorphisms in $\mathbf {A}$ , $\mathbf {B}$ and $\mathbf {C}$ because functors preserve isomorphisms and the natural distributivity maps are isomorphisms by definition. Hence, it suffices to produce the bilinear functor P.

The functor P is defined on objects by

(3.3) $$ \begin{align} P(S_m,T_n)=(S\otimes T)_{m\times n}. \end{align} $$

On morphisms, the arrow $P(f,g)$ is defined as either of the composites in the commuting diagram in $\mathbf {F}_{S\otimes T}$ that we obtain from Definition 2.9:

The fact that these composites agree implies that this assignment is functorial.

The distributivity natural transformations are in fact given by the identity morphisms:

$$\begin{align*}\delta_l\colon P(S_m,T_n)\oplus P(S_{m'},T_{n})=(S\otimes T)_{(m'\times n)+(m\times n)}=(S\otimes T)_{(m+m')\times n} \end{align*}$$

and similarly for $\delta _r$ . It is thus straightforward to check that the required unitality and compatibility conditions hold.

By construction, the monoids of connected components in the categories $\mathbf {F}_S^{\times }$ , $\mathbf {F}_T^{\times }$ and $\mathbf {F}_{S\otimes T}^{\times }$ are all quotients of $\mathbb {N}$ , and by (3.3), the map on connected components is induced by the multiplication $\mathbb {N}\times \mathbb {N}\to \mathbb {N}$ , $(m,n)\mapsto m\times n$ , of natural numbers. Hence, the map

$$\begin{align*}\pi_0\mathrm{K}(S)\otimes \pi_0\mathrm{K}(T)\cong\pi_0(\mathrm{K}(S)\wedge\mathrm{K}(T))\longrightarrow \pi_0\mathrm{K}(S\otimes T) \end{align*}$$

is also induced by multiplication at the level of representatives, using that for any theory U, the abelian group $\pi _0\mathrm {K}(U)$ is canonically a quotient of $\pi _0\mathrm {K}(E)=\mathbb {Z}$ via the unit map $\mathrm {K}(E)\to \mathrm {K}(U)$ .

Before proving Theorem 3.9, we define the multicategory $\mathcal {M}\underline {1}$ and discuss the category $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ in more detail.

Definition 3.11 ([Reference Elmendorf and Mandell17, Definition 5.7], [Reference Johnson and Yau26, Example III.8.4.5])

The multicategory $\mathcal {M}\underline {1}$ has two objects $0$ and $1$ . The n–ary morphism sets are defined by

$$ \begin{align*} \mathcal{M}\underline{1}(0,\dots,0; 0)=\ast \qquad &\text{where the source is the string of } n \ 0'\text{s}\\ \mathcal{M}\underline{1}(0,\dots,1,\dots, 0; 1)=\ast \qquad &\text{where the source is any string containing exactly one } 1. \end{align*} $$

and all other morphism sets are empty. Composition is uniquely determined by these assignments. The multicategory $\mathcal {M}\underline {1}$ may be viewed as a based multicategory with $0$ as the basepoint.

The multicategory $\mathcal {M}\underline {1}$ parametrises a choice of commutative algebra and a module over that algebra in the sense that a multifunctor $F\colon \mathcal {M}\underline {1}\to \textbf {M}$ , where $\textbf {M}$ is an arbitrary multicategory, picks out a commutative algebra object as $F(0)$ and a module over $F(0)$ as $F(1)$ . The unique morphism $(F(0),F(1))\to F(1)$ then encodes the action of $F(0)$ on $F(1)$ , for instance.

Recall from [Reference Elmendorf and Mandell17] the definition of the functor $U\colon \mathbf {PermCat}\to \mathbf {Mult}_*$ , where $\mathbf {Mult}_*$ is the category of based multicategories. For a permutative category $\mathbf {C}$ , the underlying multicategory $U\mathbf {C}$ has the same objects as the category $\mathbf {C}$ , and for any $c_1,\dots ,c_n,d$ , the set of n–ary morphisms $U\mathbf {C}(c_1,\dots ,c_n;d)$ is defined to be the morphism set $\mathbf {C}(c_1\oplus \dots \oplus c_n,d)$ . Composition is defined in the evident way.

In a permutative category $\mathbf {C}$ with strict unit $0$ , the unit map $0\oplus 0\to 0$ makes this unit object a commutative monoid. In fact, for any $c\in \mathbf {C}$ , the unit map $0\oplus c\to c$ makes c a module over $0$ . This means that for each choice of $c\in \mathbf {C}$ , we have a map of based multicategories $\mathcal {M}\underline {1}\to \mathbf {C}$ sending the unit $0\in \mathcal {M}\underline {1}$ to the unit and $1\in \mathcal {M}\underline {1}$ to c. These maps assemble to give a canonical $\mathcal {M}\underline {1}$ –module structure on $U\mathbf {C}$ ; in other words, a map $\mathcal {M}\underline {1}\otimes U\mathbf {C}\to U\mathbf {C}$ (see [Reference Johnson and Yau26, Definition III.10.2.13]). We may thus view U as factoring through the subcategory $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ of $\mathbf {Mult}_*$ . Johnson and Yau [Reference Johnson and Yau26, Definition III.10.1.36] show that $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ is a symmetric monoidal category with unit $\mathcal {M}\underline {1}$ and that the morphisms in $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ are simply underlying based multifunctors. If one views based multicategories as a generalisation of symmetric monoidal categories, then $\mathcal {M}\underline {1}$ –modules are those multicategories where the base object—which is required to have a commutative monoid structure—acts on every object in a uniform way.

Proof of Theorem 3.9

By definition, a functor $F\colon \mathbf {C}\to \mathbf {D}$ is a lax symmetric monoidal if we have a map $0_{\mathbf {D}}\to F(0_{\mathbf {C}})$ and natural maps $F(c_1)\oplus _{\mathbf {D}}F(c_2)\to F(c_1\oplus _{\mathbf {C}}c_2)$ that satisfy some standard axioms. We show that $U\iota $ satisfies this definition.

First, we construct the map of unit objects. The unit object in $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ is the multicategory $\mathcal {M}\underline {1}$ defined in Definition 3.11. The unit object in $\mathbf {Lawvere}$ is the Lawvere theory E of sets. We thus require a functor (of small based multicategories)

$$\begin{align*}\mathcal{M}\underline{1} \to U\iota(E). \end{align*}$$

Since this functor is required to be based, it must send $0\in \mathcal {M}\underline {1}$ to the unit object $\underline {0}\in \mathbf {E}$ , which is the basepoint in $U(\mathbf {E})$ . Thus, the data of this functor is equivalent to picking out a single object $\underline {n}$ in $U(\mathbf {E})$ , together with a map $\underline {0}+\underline {n}\to \underline {n}$ defining an action of $\underline {0}$ on it. The clear choice is $1\mapsto \underline {1}$ , together with the identity map.

Next, we need the maps $U\iota (S)\otimes U\iota (T)\to U\iota (S\otimes T)$ , where we are overloading the symbol $\otimes $ to represent both tensor product of $\mathcal {M}\underline {1}$ –modules and tensor product of Lawvere theories. These maps arise from the universal property of the tensor product of based multicategories. To be more precise, Elmendorf and Mandell define the tensor product of based multicategories so that a morphism of based multicategories $M_1\otimes M_2\to N$ is precisely the data of a based bilinear map $(M_1,M_2)\to N$ . These bilinear maps are a multicategorical generalisation of the bilinear functors in Definition 3.1 (see [Reference Elmendorf and Mandell17, Definition 2.3] for the precise definition).

Since $U\colon \mathbf {PermCat}\to \mathbf {Mod}^{\mathcal {M}\underline {1}}$ is a multifunctor [Reference Johnson and Yau26, Lemma III.10.2.14] and the binary morphisms in the category $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ are the based bilinear maps, it suffices to show that there is a binary morphism $\varphi \colon (\iota S,\iota T)\to \iota (S\otimes T)$ in the multicategory $\mathbf {PermCat}$ . In this case, the composition $U\varphi $ must be a bilinear based map of multicategories $(U\iota S,U\iota T)\to U\iota (S\otimes T)$ , and, thus, $U\varphi $ induces a morphism of based multicategories $(U\iota S)\otimes (U\iota T)\to U\iota (S\otimes T)$ , as required.

For Lawvere theories S and T, we’ve thus reduced our problem to finding a binary morphism of permutative categories $\mathbf {F}_S\times \mathbf {F}_T\to \mathbf {F}_{S\otimes T}$ . By definition, this is just a bilinear functor with strict unit and is precisely what we constructed in the Proof of Theorem 3.6, above.

We’ve thus constructed all the data making $U\iota \colon \mathbf {Lawvere}\to \mathbf {Mod}^{\mathcal {M}\underline {1}}$ a lax monoidal functor. To complete this argument, one must show that the coherence maps are associative and unital. These are fairly straightforward to check using the universal property of the tensor product of multicategories: since the tensor product represents based bilinear maps of multicategories, it suffices to check that the required morphisms agree on that level. For example, to check the commutativity of the left unit diagram

we simply check that the two bilinear maps $\mathcal {M}\underline {1}\times U\iota (S)\to U\iota (S)$ agree. On objects, both are given by sending $(1,S_n)\mapsto S_n$ . On morphisms, a (based) bilinear map out of $\mathcal {M}\underline {1}\otimes U\iota (S)$ is defined by where it sends morphisms of two forms. The first form is $(\operatorname {\mathrm {id}}_1,f\colon S_{n_1}\oplus \dots \oplus S_{n_j}\to S_n)$ , and both bilinear maps send this to f. In the map along the right-hand side, this follows because $\operatorname {\mathrm {id}}_{\underline {1}}\times f\in E\otimes S$ is sent to f under the unit map $E\otimes S\to S$ of Lawvere theories. The second form is $((0,\dots ,1,\dots , 0)\to 1, \operatorname {\mathrm {id}}_{S_n})$ , and both bilinear maps send this to the identity on $S_n$ . On the left side, this is by definition of the $\mathcal {M}\underline {1}$ –modules structure, and in the composite along the right-hand side, this follows from the fact that the map of unit objects $\mathcal {M}\underline {1}\to U\iota (E)$ sends $(0,\dots ,1,\dots ,0)\to 1$ to $\operatorname {\mathrm {id}}_{\underline {1}}$ .

For the associativity diagram, we can use the fact that the underlying multicategory functor U is full and faithful, and so it suffices to check that the two trilinear maps represented by the two composites

are given by the same functor $\mathbf {F}_S\times \mathbf {F}_T\times \mathbf {F}_V\to \mathbf {F}_{S\otimes T\otimes V}$ . Inspecting the definitions shows that both functors send an object $(S_l,T_m,V_n)$ to $(S\otimes (T\otimes V))_{l\times m\times n}$ and both send a morphism $(f,g,h)$ to the composite

$$\begin{align*}(f\times (\operatorname{\mathrm{id}}\times \operatorname{\mathrm{id}}))\circ (\operatorname{\mathrm{id}}\times (g\times \operatorname{\mathrm{id}}))\circ (\operatorname{\mathrm{id}}\times (\operatorname{\mathrm{id}}\times h)), \end{align*}$$

which by the definition of the Kronecker product agrees with the composite of these three maps in any other order.

Finally, observe that $U\iota $ respects the symmetries in $\mathbf {Lawvere}$ and $\mathbf {Mult}_*$ in the sense that for any Lawvere theories S and T, the diagram

commutes. This follows by direct inspection of the definitions: both symmetry maps are determined by the fact that they send morphisms of the form $S_m\otimes g$ to $g\otimes S_m$ and of the form $f\otimes T_n$ to $T_n\otimes f$ .

To see that $U\iota ^{\times }$ is also a lax symmetric monoidal, it suffices to observe that the functor $\mathcal {M}\underline {1}\to U\iota (E)$ factors through $U\iota ^{\times }(E)$ and, as observed in the above proof of Theorem 3.6, the binary morphism of permutative categories $\mathbf {F}_S\times \mathbf {F}_T\to \mathbf {F}_{S\otimes T}$ restricts to a binary morphism $\mathbf {F}_S^{\times } \times \mathbf {F}_T^{\times }\to \mathbf {F}_{S\otimes T}^{\times }$ .

Remark 3.12. We offer, here, an alternative argument to show the lax symmetric monoidality of both of the two functors $U\iota \colon \mathbf {Lawvere}\to \mathbf {Mod}^{\mathcal {M}\underline {1}}$ and $U\iota ^{\times }\colon \mathbf {Lawvere}\to \mathbf {Mod}^{\mathcal {M}\underline {1}}$ . As discussed in [Reference Elmendorf and Mandell17, Section 3], a lax symmetric monoidal functor between symmetric monoidal categories is simply a map between their underlying (symmetric) multicategories. Hence, it suffices to show that $U\iota $ and $U\iota ^{\times }$ are multifunctors between the underlying multicategories of $\mathbf {Lawvere}$ and $\mathbf {Mod}^{\mathcal {M}\underline {1}}$ . In the commutative diagram

the maps U are multifunctors [Reference Johnson and Yau26, Lemma III.10.2.14] and, we have already observed that $(-)^{\times }$ is a multifunctor. Hence, both composites $\mathbf {Lawvere}\to \mathbf {Mod}^{\mathcal {M}\underline {1}}$ are multifunctors if $\iota $ is. Elmendorf–Mandell’s work ([Reference Elmendorf and Mandell16, Theorem 1.1] or the results of [Reference Elmendorf and Mandell17] and [Reference Johnson and Yau26] stated as Theorem 3.8 above) then implies that K-theory of Lawvere theories, which is given by the composite

$$\begin{align*}\mathbf{Lawvere}\overset{\iota^{\times}}{\longrightarrow} \mathbf{PermCat}\overset{U}{\longrightarrow} \mathbf{Mod}^{\mathcal{M}\underline{1}}\overset{\mathrm{K}}{\longrightarrow} \mathbf{Spectra}, \end{align*}$$

is multiplicative.

Multifunctoriality of $\iota $ requires that for any map of Lawvere theories $S_1\otimes \dots \otimes S_k\to T$ , we must have a k–linear functor of permutative categories $\mathbf {F}_{S_1}\times \dots \times \mathbf {F}_{S_k}\to \mathbf {F}_{T}$ . The universality of Kronecker products means we can reduce the construction of any such map to constructing a multilinear functor $\mathbf {F}_{S_1}\times \dots \times \mathbf {F}_{S_k}\to \mathbf {F}_{S_1\otimes \dots \otimes S_k}$ , extending our construction of the bilinear functor $\mathbf {F}_S\times \mathbf {F}_T\to \mathbf {F}_{S\otimes T}$ from Theorem 3.9.

These remarks bring us to a peak of abstraction in our thinking about the multiplicativity of K-theory of Lawvere theories. In the next section, we return to the down-to-earth realm of applications of the concrete maps that multiplicativity produces.

4 Monoids in the category of Lawvere theories

This section contains a brief but complete discussion of monoids in the category of Lawvere theories.

Which Lawvere theories T support such monoidal structures, and how many?

We see that being a monoidal Lawvere theory is a property, not a structure, and theories that have this property are also called commutative. Such structures have been considered by Freyd [Reference Freyd19], Kock [Reference Kock27, Reference Kock28, Reference Kock29] and, much more recently, in Durov’s thesis [Reference Durov15].

5 Assembly for Lawvere theories

In this section, we apply our results from Section 3 to produce assembly maps in the general context of Lawvere theories.

Theorem 5.1. For each Lawvere theory T, the map

(5.1) $$ \begin{align} \mathrm{K}(\mathbb{Z})\wedge\mathrm{K}(T)\longrightarrow\mathrm{K}(\mathbb{Z}\otimes T) \end{align} $$

of K-theory spectra arising from Theorem 3.6 is the unique $\mathrm {K}(\mathbb {Z})$ –linear extension of the morphism $\mathrm {K}(T)\to \mathrm {K}(\mathbb {Z}\otimes T)$ induced by the linearisation $T\to \mathbb {Z}\otimes T$ of the Lawvere theory T as defined in Example 2.7.

Proof. Since $\mathbb {Z}$ is a commutative ring, the theory of $\mathbb {Z}$ –modules is a commutative monoid in the symmetric monoidal category $\mathbf {Lawvere}$ of Lawvere theores, as in Section 4. Furthermore, the theory $\mathbb {Z}\otimes T$ is a module over the theory of $\mathbb {Z}$ –modules. Since the functor $\mathrm {K}$ is lax symmetric monoidal by Corollary 3.10, it follows that $\mathrm {K}(\mathbb {Z})$ is a commutative ring spectrum and the spectrum $\mathrm {K}(\mathbb {Z}\otimes T)$ is a module spectrum over it. Now, the smash product $\mathrm {K}(\mathbb {Z})\wedge \mathrm {K}(T)$ is the free $\mathrm {K}(\mathbb {Z})$ –module spectrum on $\mathrm {K}(T)$ , and so there is a unique dashed extension of the linearisation map $\mathrm {K}(T)\to \mathrm {K}(\mathbb {Z}\otimes T)$ to a horizontal map making the following diagram commute:

In this diagram, the left vertical map comes from the unit map $\mathbb {S}\to \mathrm {K}(\mathbb {Z})$ , which is realised by applying K-theory to the map $E\to \mathbb {Z}$ of Lawvere theories and recalling that the unit $\mathbb {S}\to \mathrm {K}(E)$ is an equivalence. Therefore, to show the claim, we have to verify that the map from Theorem 3.6 in place of the dashed arrow makes the square above commute. This follows from the commutativity of the diagram

(5.2)

Here, the top two horizonal maps are those of Theorem 3.6 and the naturality of these maps implies that the top square commutes. The bottom square is the left unitality condition of the lax symmetric monoidal functor $\mathrm {K}$ and, hence, commutes. The commutativity of the right part of the diagram follows from applying the functor $\mathrm {K}$ to the commutative diagram

of Lawvere theories. To see that this diagram commutes, note that for any Lawvere theory S, the unit $E\otimes T\to S\otimes T$ arises from the forgetful functor taking T–models in S–models to T-models in E–models, that is T–models in sets. So the linearisation $T\to \mathbb {Z}\otimes T$ arises from the same forgetful functor, once we observe that T–models in sets are just T–models. The commutativity of the left part of the diagram follows from the discussion of the unit in the paragraph before the diagram (5.2).

We explain the relationship to the classical assembly maps in the following section. Before doing so, we determine the assembly maps for the theories of Cantor algebras and Boolean algebras, which can be done directly with no more background than the definition (5.1). We see that the assembly map can, in general, be next to anything, from an equivalence to the zero morphism.

5.1 Cantor algebras

Let $a\geqslant 2$ be an integer. A Cantor algebra of arity a is a set X together with a bijection $X^a\to X$ . The Cantor algebras of arity a are the models for a Lawvere theory $\mathrm {Cantor}_a$ , and its algebraic K-theory has been computed in [Reference Szymik and Wahl48]:

(5.3) $$ \begin{align} \mathrm{K}(\mathrm{Cantor}_a)\simeq\mathbb{S}/(a-1), \end{align} $$

the Moore spectrum mod $a-1$ . In particular, the spectrum $\mathrm {K}(\mathrm {Cantor}_2)$ is contractible.

Remark 5.3. We note that the definition makes sense for $a=1$ as well. In that case, we have an isomorphism between $\mathrm {Cantor}_1$ and the Lawvere theory $\mathbb {Z}$ –Sets of permutations, and the equivalence (5.3) is still true, as we shall see in Example 6.2.

Theorem 5.4. The assembly map (5.1)

$$\begin{align*}\mathrm{K}(\mathbb{Z})\wedge\mathrm{K}(\mathrm{Cantor}_a)\longrightarrow\mathrm{K}(\mathbb{Z}\otimes\mathrm{Cantor}_a) \end{align*}$$

for the Lawvere theory $\mathrm {Cantor}_a$ of Cantor algebras of arity a is an equivalence.

Proof. We start from [Reference Szymik and Wahl48], where the algebraic K-theory of $\mathrm {Cantor}_a$ is identified with the Moore spectrum $\mathbb {S}/(a-1)$ . That Moore spectrum is the the cofibre of multiplication with $a-1$ on the sphere spectrum, so that $\mathrm {K}(\mathbb {Z})\wedge \mathrm {K}(\mathrm {Cantor}_a)$ is the cofibre of multiplication with $a-1$ on $\mathrm {K}(\mathbb {Z})$ .

Then we have the observation that $\mathbb {Z}\otimes \mathrm {Cantor}_a$ is the theory of modules over the Leavitt algebra $\mathrm {L}_a$ , the quotient of the free associative ring with unit on $2a$ generators, given as two vectors $R=(R_1,\dots ,R_a)$ and $C=(C_1,\dots ,C_a)$ , modulo the ideal defined by the $a^2+1$ relations that ensure that the two square matrices $R^{\mathrm {t}}C$ and $RC^{\mathrm {t}}$ are the identity matrices. In other words, the modules over the ring $\mathrm {L}_a$ are the abelian groups M together with linear bijections $M^a\to M$ , and these are precisely the models for $\mathbb {Z}\otimes \mathrm {Cantor}_a$ .

Finally, the algebraic K-theory $\mathrm {K}(\mathrm {L}_a)$ has been computed in [Reference Ara, Brustenga and Cortiñas2], and the result shows that it is also the cofibre of multiplication with $a-1$ on $\mathrm {K}(\mathbb {Z})$ . The obvious comparison maps are equivalences.

Remark 5.5. Brin [Reference Brin9] has introduced the higher-dimensional Higman–Thompson groups. These are the automorphism groups of the free models for the Lawvere theories

$$\begin{align*}\mathrm{Cantor}_{a(1)}\otimes\mathrm{Cantor}_{a(2)}\otimes\dots\otimes\mathrm{Cantor}_{a(n)}, \end{align*}$$

for a finite sequence of integers $a(j)\geqslant 2$ (see [Reference Dicks and Martínez-Pérez14, Reference Funar and Neretin20, Reference Martínez-Pérez, Matucci and Nucinkis35, Reference Martínez-Pérez and Nucinkis36]). It follows from our earlier work [Reference Bohmann and Szymik5, Theorem 2.7] that the stable homology of these groups is described by the algebraic K-theory spectrum of that Kronecker product. From Theorem 3.6, we get a map

(5.4) $$ \begin{align} \mathrm{K}(\mathrm{Cantor}_{a(1)})\wedge\mathrm{K}(\mathrm{Cantor}_{a(2)})\wedge\dots\wedge\mathrm{K}(\mathrm{Cantor}_{a(n)})\longrightarrow\mathrm{K}(\mathrm{Cantor}_{a(1)}\otimes\dots\otimes\mathrm{Cantor}_{a(n)}) \end{align} $$

from the smash product of the algebraic K-theory spectra into it. The homotopy type of this smash product can be worked out, because the algebraic K-theory spectra are Moore spectra by [Reference Szymik and Wahl48], but it is not known whether the map (5.4) is an equivalence. In fact, it is not known whether the spectra are equivalent at all.

5.2 Boolean algebras

There are Lawvere theories for which the assembly map is trivially trivial because the target is contractible. For instance, in any abelian T–model M, all constants of T need to be equal, because there is a unique homomorphism $0=M^0\to M$ of abelian groups. This happens for rings with unit ( $1=0$ implies $a=1\cdot a=0\cdot a=0$ for all a) but also for Boolean algebras:

Proposition 5.6. Any abelian group object in the category of Boolean algebras is trivial.

Proof. We first note that $0\wedge x=0$ and $1\wedge x=x$ hold in every Boolean algebra. If, in addition, we have $0=1$ , then this implies $x=1\wedge x=0\wedge x=0$ for all x, and we are done.

Theorem 5.7. The assembly map for the Lawvere theory $\mathrm {Boole}$ is zero and, in particular, not rationally injective.

Proof. It follows from Proposition 5.6 that $\mathbb {Z}\otimes \mathrm {Boole}$ is the theory of modules over the trivial ring, and the algebraic K-theory spectrum $\mathrm {K}(\mathbb {Z}\otimes \mathrm {Boole})\simeq \star $ is contractible.

On the other hand, we can use the computation of the algebraic K-theory of the theory of Boolean algebras in [Reference Bohmann and Szymik5, Corollary 5.2]. The result implies that the source $\mathrm {K}(\mathbb {Z})\wedge \mathrm {K}(\mathrm {Boole})\not \simeq \star $ of the assembly map is not contractible because

$$\begin{align*} \pi _0(\mathrm {K}(\mathbb {Z}) \wedge \mathrm {K}(\mathrm {Boole}))\cong \mathrm {K}_0(\mathbb {Z}) \otimes \mathrm {K}_0(\mathrm {Boole})\cong \mathbb {Z}\otimes \mathbb {Z}\cong \mathbb {Z}. \end{align*}$$

It is easy to generalise the preceding result from Boolean algebras to v–valued Post algebras as in [Reference Bohmann and Szymik5, Theorem 5.1]; we omit the details.

6 Classical assembly maps via the theories of group actions

We can now see how to recover one of the classical assembly maps in algebraic K-theory as a special case of our general assembly map (5.1).

Theorem 6.1. For each group G, we can identify the assembly map (5.1) with the unique $\mathrm {K}(\mathbb {Z})$ –linear morphism

(6.1) $$ \begin{align} \mathrm{K}(\mathbb{Z})\wedge\Sigma^{\infty}_+(\mathrm{B} G)\longrightarrow\mathrm{K}(\mathbb{Z} G) \end{align} $$

between $\mathrm {K}(\mathbb {Z})$ –module spectra that extends the map given by realising the elements of G as units in the group ring $\mathbb {Z} G$ . These maps are natural in the group G.

This is Loday’s version of the assembly map in algebraic K-theory [Reference Loday32]. There are obvious extensions to other coefficient rings than $\mathbb {Z}$ .

Proof. For any discrete group G, we can consider the Lawvere theory T of G–sets and apply Theorem 5.1.

The algebraic K-theory of the theory of G–sets is

$$\begin{align*}\mathrm{K}(G\text{--}\mathrm{Sets})\simeq\Sigma^{\infty}_+(\mathrm{B} G), \end{align*}$$

the suspension spectrum of the classifying space $\mathrm {B} G$ (with a disjoint base point $+$ ). This observation is Segal’s extension of the Barratt–Priddy–Quillen theorem, given by applying Proposition 3.6 of [Reference Segal43] to the example discussed on pages 299–300 of that paper. This identifies the source of the assembly map (5.1) as $\mathrm{K}(\mathbb {Z})\wedge \Sigma ^{\infty }_+(\mathrm {B} G)$ . As for the target, if T is the theory of G–sets for a group G, then $\mathbb {Z}\otimes T$ is the theory of abelian groups with a linear G–action. These are precisely the modules over the group ring $\mathbb {Z} G$ , giving the target of the assembly map (5.1).

Example 6.2. The assembly map (6.1) is obviously an equivalence for the trivial group $G=e$ . Less obviously, it is also an equivalence when $G\cong \mathrm {C}_{\infty }$ is infinite cyclic: The theory $\mathrm {C}_{\infty }$ –sets is the theory of permutations: a model is a set together with a permutation of that set. The Barratt–Priddy–Quillen–Segal theorem gives

$$\begin{align*}\mathrm{K}(\mathrm{C}_{\infty}\text{--}\mathrm{Sets})\simeq\Sigma^{\infty}_+ (\mathrm{B}\mathrm{C}_{\infty})\simeq\Sigma^{\infty}_+(\mathrm{S}^1)\simeq\mathbb{S}\vee\Sigma\mathbb{S}. \end{align*}$$

On the other hand, we have $\mathbb {Z}\mathrm {C}_{\infty }\cong \mathbb {Z}[q^{\pm 1}]$ , and Quillen’s work on the algebraic K-theory of Laurent polynomial rings gives an equivalence

$$\begin{align*}\mathrm{K}(\mathbb{Z}[q^{\pm1}])\simeq\mathrm{K}(\mathbb{Z})\vee\Sigma\mathrm{K}(\mathbb{Z})\simeq\mathrm{K}(\mathbb{Z})\wedge(\mathrm{S}^0\vee\mathrm{S}^1)\simeq\mathrm{K}(\mathbb{Z})\wedge\Sigma^{\infty}_+(\mathrm{B}\mathrm{C}_{\infty}) \end{align*}$$

of spectra (see Grayson’s paper [Reference Grayson22]).

There is one important situation where the map (3.1) from Theorem 3.6 is an equivalence, and this is useful in computations:

Proposition 6.5. For any groups G and H, the map

$$\begin{align*}\mathrm{K}(G\text{--}\mathrm{Sets})\wedge\mathrm{K}(H\text{--}\mathrm{Sets}) \longrightarrow \mathrm{K}(G\text{--}\mathrm{Sets}\otimes H\text{--}\mathrm{Sets}) \end{align*}$$

from Theorem 3.6 is an equivalence.

Proof. Recall from Example 2.6 that the two Lawvere theories $G\text{--Sets}\otimes H\text{--Sets}$ and $(G\times H)\text{--Sets}$ are the same. This gives an equivalence

$$\begin{align*}\mathrm{K}(G\text{--Sets}\otimes H\text{--Sets})\simeq\mathrm{K}((G\times H)\text{--Sets}). \end{align*}$$

Now we can use the Barratt–Priddy–Quillen–Segal equivalence $\mathrm {K}(G\text {--Sets})\simeq \Sigma _+^{\infty }\mathrm {B} G$ of spectra, for $G\times H$ instead of G, and the equivalence $\mathrm {B}(G\times H)\simeq \mathrm {B}(G)\times \mathrm {B}(H)$ of classifying spaces which induces the equivalence $\Sigma _+^{\infty }\mathrm {B}(G\times H)\simeq \Sigma _+^{\infty }\mathrm {B} G\wedge \Sigma _+^{\infty }\mathrm {B} H$ of suspension spectra.

For instance, this result allows us to factor the $\mathbb {Z}$ –linear assembly maps for products $G\times H$ as the $\mathbb {Z}[G]$ –linear assembly map for the factor H and the $\mathbb {Z}$ –linear assembly map for the other factor G smashed with the identity on $\mathrm {K}(H\text {--Sets})$ :

7 Nonabelian groups and a nilpotent interpolation

The algebraic K-theory spectrum $\mathrm {K}(\mathbb {Z})$ of the integers enters our assembly map (5.1) through the Lawvere theory of abelian groups. In this section, we show what happens when we relax the commutativity hypothesis. At the same time, we prove results about the assembly maps for various theories of groups, starting with the following.

Theorem 7.1. The assembly map (5.1) for the theory $T=\mathrm {Groups}$ of all groups,

$$\begin{align*}\mathrm{K}(\mathbb{Z})\wedge\mathrm{K}(\mathrm{Groups})\longrightarrow\mathrm{K}(\mathbb{Z}\otimes\mathrm{Groups}) \end{align*}$$

is an equivalence.

Conversely, we can now also consider assembly-like maps

(7.1) $$ \begin{align} \mathrm{K}(\mathrm{Groups})\wedge\mathrm{K}(T)\longrightarrow\mathrm{K}(\mathrm{Groups}\otimes T). \end{align} $$

By Galatius’s theorem, this is equivalent to the map $\mathrm {K}(T)\to \mathrm {K}(\mathrm {Groups}\otimes T)$ of spectra induced by the canonical morphism $T\to \mathrm {Groups}\otimes T$ in the sense that the obvious triangle commutes.

Moreover, there is an interpolation between the theory of all groups and the theory of all abelian groups by the theories $\mathrm {Nil}_c$ of nilpotent groups of a certain class c, with $1\leqslant c\leqslant \infty $ . There is a corresponding diagram

of algebraic K-theory spectra. This tower has been studied from the point of view of homological stability and stable homology in [Reference Szymik46] and [Reference Szymik47], respectively. We refer to Zeman’s recent work [Reference Zeman52] for related questions in a more geometric direction.

The assembly map (5.1) for the theory of abelian groups is not an equivalence. In fact, more generally, we have the following result.

Theorem 7.2. The assembly map

(7.2) $$ \begin{align} \mathrm{K}(\mathbb{Z})\wedge\mathrm{K}(\mathrm{Nil}_c)\longrightarrow\mathrm{K}(\mathbb{Z}\otimes\mathrm{Nil}_c)=\mathrm{K}(\mathbb{Z}) \end{align} $$

for the theory $\mathrm {Nil}_c$ of nilpotent groups of any given class $c\geqslant 1$ is not rationally injective.

Therefore, a generalisation of the Novikov conjecture to algebraic theories is impossible.

Proof. To see this, assume that it is rationally injective, and base change the $\mathrm {K}(\mathbb {Z})$ –linear assembly map along the composition $\mathrm {K}(\mathbb {Z})\to \mathrm {H}\mathbb {Z}\to \mathrm {H}\mathbb {Q}$ to get a $\mathrm {H}\mathbb {Q}$ –linear map

$$\begin{align*}\mathrm{H}\mathbb{Q}\wedge\mathrm{K}(\mathrm{Nil}_c)\longrightarrow\mathrm{H}\mathbb{Q}, \end{align*}$$

which would then also be rationally injective, contradicting the results in [Reference Szymik47]: the rational homology of $\mathrm {K}(\mathrm {Nil}_c)$ is nontrivial for all positive integers c.

Remark 7.3. There is an entire interpolation of assembly-style maps between Loday’s assembly map (5.1) and the map (7.1): there is a tower

(7.3) $$ \begin{align} \mathrm{K}(\mathrm{Nil}_c)\wedge\mathrm{K}(T)\longrightarrow\mathrm{K}(\mathrm{Nil}_c\otimes T) \end{align} $$

of maps of spectra, indexed by the integer $c\geqslant 1$ . Note how this differs from (7.2) in the way the theory $\mathrm {Nil}_c$ enters. At the time of writing, it is not known to us whether the tower of spectra $\mathrm {K}(\mathrm {Nil}_c)$ converges (as $c\to \infty $ ) to the spectrum $\mathrm {K}(\mathrm {Groups})\simeq \mathbb {S}$ or not. More generally, one may wonder whether or not the tower $\mathrm {K}(\mathrm {Nil}_c)\wedge \mathrm {K}(T)$ converges to $\mathrm {K}(\mathrm {Groups})\wedge \mathrm {K}(T)$ , or whether or not the tower $\mathrm {K}(\mathrm {Nil}_c\otimes T)$ converges to $\mathrm {K}(\mathrm {Groups}\otimes T)$ . It would be interesting to pursue the question for which Lawvere theories T one or both of these is the case.