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Higher Lawvere theories

We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives. Our main result establishes a universal property for the infinity category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its infinity category of models. Many familiar properties of Lawvere theories follow directly. As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.


Introduction
The primary goal of this paper is to lay the foundations for a Lawvere theoretic approach to higher algebra. As such, we organize the paper as a survey, combining others' work with our own. In this introduction, we summarize the new results for those who are familiar with the subject; however, the casual reader may be better served by reading Section 2, followed by this introduction.
A Lawvere theory L encodes a particular type of algebraic theory (for example, commutative monoids). A model of L is an instance of that algebraic structure. As we are working with 8-categories, we take models by default relative to the 8-category Top of spaces.
Specifically, L is an 8-category with finite products, generated by a single object, and models are functors L Ñ Top which preserve finite products. The assignment from L to the 8-category Mdl L of models is functorial where an object of Pr L is a presentable 8-category, and a morphism is a left adjoint functor. In fact, since any Lawvere theory L has a distinguished object 1, Mdl L also has a distinguished object, the model Map L p1,´q. Therefore, Mdl may be promoted to a functor Lwv Ñ Pr L .
Theorem (Theorems 2.10, 3.3). The functor Mdl : Lwv Ñ Pr L is fully faithful and symmetric monoidal, and it has a right adjoint which sends C P Pr L to C op fgf , where C fgf Ď C is the full subcategory of finitely generated free objects.
In this way, Lwv is a symmetric monoidal colocalization of the better understood 8-category Pr L , which allows us to study Lawvere theories using familiar tools like the adjoint functor theorem and Lurie's tensor products of categories ( [17] 4.8).
The math here is not new, but the packaging is. That is, the theorem encapsulates the following facts about Lawvere theories: • (the adjunction) Mdl L is the free cocompletion of L op , regarding the latter as an 8-category with finite coproducts ( [18] 5.3.6.10); • (Mdl is fully faithful) L can be recovered from Mdl L as the full subcategory of finitely generated free objects (Proposition 2.7); The fact that Mdl is symmetric monoidal may be less familiar, but it is a consequence of Lurie's 'tensor products of categories' machinery in [17] 4.8.
In Section 3, we explore two direct corollaries of this monoidality: • (Corollary 3.4) If a Lawvere theory L admits a symmetric monoidal structure compatible with finite products, then Mdl L inherits a closed symmetric monoidal structure called Day convolution; • (Corollary 3.8) If L is as above, then L is a semiring 8-category. Any module over L is naturally enriched in Mdl L with its Day convolution.
The first of these is well-known, but the second we believe is new. It also admits a partial converse: Theorem (Theorem 3.11). If L admits a symmetric monoidal structure compatible with finite products, and another Lawvere theory K is enriched in Mdl L , then K is naturally tensored over L via a map of Lawvere theories This theorem suggests that there may be a strong converse to Corollary 3.8: Conjecture (Conjecture 3.12). If L is as above, the 8-categories of Mdl Lenriched Lawvere theories and L-module Lawvere theories are equivalent.

Sections 4 and 5 are devoted to examples and applications:
For any E 1 -semiring space R, there is a Lawvere theory whose models are R-modules. In Section 4, we show that these are the only semiadditive Lawvere theories; that is, semiadditive Lawvere theories can be identified with semiring spaces. This is an easy result for classical Lawvere theories. We record it here because it is slightly more subtle for 8-categories. It also suggests a philosophy that we are fond of: Lawvere theories may be regarded as generalized (non-additive) rings.
In Section 5.1, we use our ideas relating module Lawvere theories to enriched Lawvere theories, proving a result promised in the author's earlier paper [5]: Theorem (Theorem 5.2). The Burnside 8-category is a commutative semiring 8-category, and there is an equivalence where AddCat 8 denotes the 8-category of additive 8-categories.
Finally, in Section 5.2, we describe an application to equivariant homotopy theory. There are no results; at this point, the application is just motivational.

Acknowledgment
This paper is largely drawn from the author's thesis [4], of which it is the second part. It has been in the works for years, and benefitted from conversations with many others, including Ben Antieau, Clark Barwick [16], and they have been thoroughly studied since then; see [1]. In the setting of 8-categories, Lawvere theories have been studied by just a few authors, notably by Cranch [6] and Gepner-Groth-Nikolaus [9], and in the prequel to this paper [5].
The literature in this area is sparse in part because Lurie's book Higher Algebra [17] founds the subject on operads, instead. That approach is now well-developed, and it has many benefits, but it suffers from the drawback of being pedagogically unwieldy. Even for those who are already invested in operads, like many homotopy theorists, there is a high barrier of entry: Even the definitions of fundamental objects like 8-operads, commutative algebras, and symmetric monoidal 8-categories presuppose a deep understanding of the quasicategory model.
On the other hand, there are surely those who would like to use this machinery without requiring operads at all.
In contrast, the Lawvere theoretic approach can be described in two sentences, and independent of our model of 8-category: Definition 2.1. A Lawvere theory is an 8-category L which is closed under finite products and generated under finite products by a single distinguished object 1. An algebra or model of L in C is a functor L Ñ C which preserves finite products.
Typically, we want to take C to be an 8-category which is presentable and cartesian closed, such as Set, Top, or Cat 8 , but there is no such requirement. By default, we take C to be the 8-category Top of CW complexes, or homotopy types, which is the initial object among 8-categories which are presentable and closed symmetric monoidal. We write Mdl L " FunˆpL, Topq.
Example 2.2. Let Fin denote the category of finite sets. Then Fin op is a Lawvere theory, with product given by disjoint union of sets, and evaluation at the singleton Mdl Fin op Ñ Top is an equivalence of 8-categories. We say Fin op is the trivial Lawvere theory.
Example 2.3. Let Burn eff denote the effective Burnside 2-category, whose objects are finite sets. For finite sets X, Y , the groupoid of morphisms from X to Y is the groupoid of span diagrams X Ð T Ñ Y , and composition is via pullback. Then Burn eff is a Lawvere theory, with product given by disjoint union of sets.
If f : Burn eff Ñ Top is a model, then the spans 0 " Ð Ý 0 Ñ 1, respectively 2 " Ð Ý 2 Ñ 1 endow f p1q with a distinguished point, respectively a binary operation f p1qˆf p1q Ñ f p1q. Composition in Burn eff precisely enforces the structure of a commutative (or E 8 ) monoid on f p1q, and evaluation at 1 is an equivalence of 8-categories. See [5] Remark 3.6.
We say Burn eff is the commutative (or E 8 ) Lawvere theory.
In fact, for any cartesian monoidal Cˆ, commutative monoids in C are equivalent to models of Burn eff in C ( It may appear that Lawvere theories are less general than operads because they apply only to cartesian monoidal 8-categories. This is apparently a significant obstacle: a major application of operads is to understanding multiplicative structure on rings. For example, a ring spectrum is an algebra in spectra under smash product. However, this problem can be easily overcome, as long as we restrict attention to connective ring spectra: Example 2.5. For any connective ring spectrum R, there are Lawvere theories whose models are equivalent to Mod ě0 R , Alg ě0 R , and CAlg ě0 R . This example and others like it follow from a result of Gepner, Groth, and Nikolaus [9] which describes exactly which 8-categories are equivalent to Mdl L for some Lawvere theory L (Theorem 2.6 below).
Before stating their result, we recall the theory of presentable 8-categories. By the adjoint functor theorem ([18] 5.5.2.9), a functor C Ñ D between presentable 8-categories preserves small colimits if and only if it has a right adjoint. We write Pr L for the 8-category of presentable 8-categories along with these left adjoint functors.
If C P Pr L , then since Top is freely generated by one object under colimits, the following data are equivalent: • an object X P C (we say C is pointed); • a left adjoint functor L "´b X : Top Ñ C (we say LpSq is the free object on S); • a right adjoint functor R " MappX,´q : C Ñ Top (we say RpY q is the underlying space of Y ).
We will denote by Pr L the 8-category of these pointed presentable 8categories, along with left adjoint basepoint-preserving functors. (Formally, Pr L -Pr L Top{ .) If L is a Lawvere theory, generated by the distinguished object 1, then we regard Mdl L as canonically pointed by the right adjoint forgetful functor evaluate at 1 : Mdl L Ñ Top.
By the Yoneda lemma, the corresponding basepoint is the model Theorem 2.6 (Gepner-Groth-Nikolaus [9] Theorem B.7). A pointed presentable 8-category C is equivalent to Mdl L for some Lawvere theory L if and only if the forgetful functor C Ñ Top is conservative and preserves geometric realizations.

Reconstructing Lawvere theories from their models
We have just seen that many 8-categories M can be described as models over a Lawvere theory (roughly, those which are presentable and algebraic in nature). We may now ask: is that Lawvere theory unique, and to what extent can it be recovered from M?
If L admits finite products, then Mdl L " FunˆpL, Topq is a full subcategory, by definition, of the 8-category of presheaves, PpL op q " FunpL, Topq. Moreover, every representable presheaf MappX,´q : L Ñ Top preserves any limits that exist in L, and therefore is in Mdl L .
By the Yoneda lemma, then L op is a full subcategory of Mdl L . (We called this a cartesian monoidal Yoneda lemma in [5] 3.7.) If L is a Lawvere theory, we can explicitly identify L op as a subcategory of Mdl L : The embedding L op Ď Mdl L identifies 1 >n P L op with I >n P Mdl L . Here I is the distinguished object of Mdl L , so that I >n can also be regarded as the free model on n generators. In conclusion: Proposition 2.7. Suppose M is a pointed, presentable 8-category with distinguished object I. Let M fgf be the full subcategory of finitely generated free objects; that is, those of the form I >n for integers n ě 0. If M -Mdl L for some Lawvere theory L, then L -M op fgf . Theoretically, this proposition combined with Theorem 2.6 allow us to describe the Lawvere theories associated to any operad, modules over a ring, algebras over a ring, etc. -provided we already understand the 8-category of models.
However, we may instead seek to describe the Lawvere theory first, and use it to construct some new 8-category of models. (For example, we might want to do this for pedagogical purposes as in Example 2.4.) Principle 2.8. Lawvere theories often have combinatorial descriptions, in which their objects are finite sets, morphisms are given by diagrams of finite sets, and products are given by disjoint union.
Example 2.9. The commutative Lawvere theory Burn eff is equivalent to the 2-category of spans of finite sets.
We may revisit this principle in a future paper on combinatorial Lawvere theories; for now, we will not emphasize it.

Main theorem
We have described how to pass back and forth between a Lawvere theory and its 8-category of models. We will show this relation is exceptionally robust.
Let Lwv denote the 8-category whose objects are Lawvere theories and morphisms are functors which preserve finite products and the distinguished object 1 . In other words, Lwv is a colocalization of Pr L . Theorem 2.6 described explicitly which colocalization by providing a testable criterion to determine the essential image of Lwv in Pr L .
Proof. Call L " Mdl and R " p´q op fgf , suggestive of left and right adjoints. The composition RL is equivalent to the identity by Proposition 2.7. Therefore, applying R induces a map of spaces natural in both L P Lwv and C P Pr L , and this R˚is a natural isomorphism by [18] 5.3.6.10. Therefore, L and R are adjoint.
For L, K P Lwv, applying L induces a map of spaces Since RL -Id, R˚L˚is equivalent to the identity, and because R˚is an equivalence (as above), so is L˚. Therefore, L is fully faithful.
Remark 2.11. As seen in the proof, Theorem 2.10 combines two facts in one: the adjunction asserts [18] 5.3.6.10, that Mdl L is the free cocompletion of L op (regarding the latter as an 8-category which already has finite coproducts).
That the left adjoint is fully faithful asserts Proposition 2.7, that L op is a full subcategory of Mdl L . This set-theoretic problem arises because of the definition of presentable 8-categories: they are required to have a small set of generating objects, but these objects are not remembered as part of the data. Theorem 2.10 solves the problem by introducing a framing; that is, by remembering these objects (or in this case, a single object).

Algebra of Lawvere theories
In Section 3.1, we show that the functor Mdl L : Lwv Ñ Pr L is symmetric monoidal. Then we study its behavior on commutative algebras in 3.2, constructing Day convolution products of models. Finally, we study its behavior on modules in Section 3.3, showing that many Lawvere theories have canonical enrichments. In fact, we provide evidence that L-module Lawvere theories can be identified with Mdl L -enriched Lawvere theories (Conjecture 3.12).

Kronecker products of Lawvere theories
The primary technical contribution of this paper is first to cast the relationship between Lawvere theories and their models as a colocalization (Theorem 2.10) and second that this colocalization is compatible with symmetric monoidal structures on Lwv (the Kronecker product) and Pr L (Lurie's tensor product). We review each of these: Remark 3.1 (Lurie's tensor product of presentable 8-categories). There is a closed symmetric monoidal tensor product on Pr L with the following universal property: left adjoint functors C b D Ñ E can be identified with functors CˆD Ñ E which preserve small colimits in each variable separately. This is constructed by Lurie in [17] 4.8.1, and the unit is Top.
If V is presentable, to endow V with the structure of a commutative algebra in Pr L,b is precisely to endow V with its own closed symmetric monoidal structure 2 .
If C, D are two pointed presentable 8-categories, C b D is also canonically pointed (for example, by the free functor Top -Top b Top Ñ C b D), so that Pr L is also symmetric monoidal via Lurie's tensor product.
A commutative algebra object in Pr L is a presentable 8-category with a closed symmetric monoidal structure b, pointed by the unit of b.

Remark 3.2 (Kronecker tensor product of Lawvere theories).
There is a closed symmetric monoidal tensor product of cartesian monoidal 8-categories with the following universal property: functors C b D Ñ E which preserve finite products can be identified with functors CˆD Ñ E which preserve finite products in each variable separately. This can be made precise in two equivalent ways: • by Lurie's general framework of tensor products of categories ( [17] 4.8.1); • as in [5]: cartesian monoidal 8-categories can be identified with modules over the commutative semiring category Fin op , which admit a relative tensor product b Fin op .
If C, D are Lawvere theories (that is, generated by a single object under ), then C b D is also a Lawvere theory, so that Lwv inherits a symmetric monoidal operation b called Kronecker product, and the unit is Fin op . See [5] for details.
For classical Lawvere theories, the Kronecker product goes back to Freyd [8], and it is also compatible with the Boardman-Vogt tensor product (  That is, Lwv is a symmetric monoidal colocalization of Pr L . In the next two sections, we will explore the consequences of this theorem when applied to (first) commutative algebras and (second) modules with respect to the two symmetric monoidal structures. Taking E 0 -algebras (that is, pointed objects) on each side, we see that Mdl : CartMonCat˚Ñ Pr L is symmetric monoidal, and this restricts to the symmetric monoidal full subcategory Lwv Ď CartMonCat˚.

Algebra Lawvere theories and Day convolution
For suitable Lawvere theories L, we can use Theorem 3.3 to construct tensor products of L-models. A commutative algebra structure on L P Lwv b amounts to a symmetric monoidal structure on L which preserves finite products independently in each variable, and such that the unit is the distinguished object 1.
Such Lawvere theories are sometimes called commutative algebraic theories in the classical literature [15]. Conversely, if Mdl L has a closed symmetric monoidal structure with unit I, then L inherits a commutative algebra structure in Lwv.
Proof. Since Mdl is symmetric monoidal, it takes commutative algebras to commutative algebras. Therefore, if L P CAlgpLwv b q, then Mdl L has a commutative algebra structure in Pr L , which is to say a closed symmetric monoidal structure with unit I.
Conversely, the right adjoint to a symmetric monoidal functor is lax symmetric monoidal [10]. If Mdl L has a closed symmetric monoidal structure with unit I, it is a commutative algebra in Pr L , so L P CAlgpLwv b q.
Example 3.5. The effective Burnside 2-category is symmetric monoidal under cartesian product, which makes it a commutative algebraic theory. Therefore, Day convolution provides a closed symmetric monoidal smash product for E 8 -spaces. Remark 3.6 (Models in other 8-categories). More generally, suppose L is a Lawvere theory, and V is a presentable 8-category. By general theory of presentable 8-categories, Lurie proves ([17] 4.8.1.17) for two presentable 8-categories C and D, that C b D -Fun R pC op , Dq. Therefore, models of L in V can be identified with the tensor product This equivalence is due to [9] Proposition B.3.
In particular, if L P CAlgpLwv b q and V has a closed symmetric monoidal structure, then Mdl L pVq also has a closed symmetric monoidal structure 4 (Day convolution).

Module Lawvere theories and enrichment
Let V be presentable and closed symmetric monoidal; i.e., V P CAlgpPr L,b q. Conjecture 3.7. Conversely, we may think of V-modules in Pr L as precisely those V-enriched categories which are presentable in an enriched sense. As far as the author is aware, the notion of 'presentable in an enriched sense ' has not yet been made rigorous for 8-categories, but this is a philosophy already familiar to some.
We have a second corollary of Theorem 3.3: Example 3.9. If L " Burn eff is the Lawvere theory for E 8 -spaces, then Burn eff -modules can be identified with semiadditive 8-categories ([5] Theorem 1.2). By corollary 3.8, any semiadditive 8-category is naturally enriched in E 8 -spaces. This is the homotopical analogue of a classical fact: semiadditive categories are naturally enriched in commutative monoids.
Conversely, if L P CAlgpLwv b q, then any Lawvere theory enriched in Mdl L is naturally tensored over L: Definition 3.10. If V is presentable and closed symmetric monoidal, a Venriched Lawvere theory is a V-enriched category for which the underlying 8-category is a Lawvere theory.
Theorem 3.11. Suppose L P CAlgpLwv b q, and K enr is a Mdl L -enriched Lawvere theory with underlying 8-category K. For any X P L, K P K, there is some object X b K P K with a natural isomorphism Map enr p´, KqpXq -Mapp´, X b Kq, and this b arises from a morphism of Lawvere theories Proof. Because K enr is Mdl L -enriched, there is a functor Map enr K p´,´q : K opˆK Ñ Mdl L , and the composite with the forgetful functor ev 1 : Mdl L Ñ Top is the ordinary mapping space Map K . In particular, that composite preserves finite products in the second variable.
By Theorem 2.6, the forgetful functor ev 1 (evaluation at 1) is conservative. It also preserves finite products because it is a right adjoint. Therefore, the functor Map enr p´,´q preserves finite products in the second variable.
It follows that the adjoint LˆK Ñ FunpK op , Topq preserves finite products independently in each variable, thereby inducing a functor φ : L b K Ñ FunpK op , Topq which preserves finite products. By construction, φpXbKq -Map enr p´, KqpXq. In particular, if n denotes 1ˆn in a Lawvere theory, φpnq -Map enr K p´, nqp1qev 1˝M ap enr K p´, nq -Map K p´, nq, so that φ factors through the full subcategory K Ď FunpK op , Topq. This completes the proof. Together, Corollary 3.8 and Theorem 3.11 are suggestive of:

Additive Lawvere theories
Suppose that L is semiadditive as an 8-category; essentially, finite products are also finite coproducts. By Example 3.9, L is enriched in E 8 -spaces.
Therefore, Endp1q is an E 1 -semiring space, where 1 is the distinguished object of L, and there is a functor Endp1q : SemiaddLwv Ñ E 1 Semiring. The proposition should not be surprising; if L is semiadditive, then we know Mapp1ˆm, 1ˆnq -Mapp1, 1qˆm n , which depends only on the semiring Endp1q. Hence, we expect Endp1q to encode all the data of the Lawvere theory.
Proof. If R is an E 1 -semiring space, write Burn R for the Lawvere theory whose models are Burn R , which exists by Theorem 2.6. We claim that any semiadditive Lawvere theory L is equivalent to Burn Endp1q .
Fix L and write R " Endp1q. Since R acts on Map Mdl L p1,´q, we have Mapp1,´q : Mdl L Ñ Mod R which maps 1 to 1, preserves finite direct sums, and therefore restricts to a map of Lawvere theories α : L Ñ Burn R . Certainly α is essentially surjective, so we show it is full and faithful. Given objects m " 1 >m and n " 1 >n in L, we wish to prove that is an equivalence. When m " n " 1, this is true by construction. Otherwise, since L and Burn R are semiadditive, we know on both sides that Mappm, nq -R mn , so α˚is an equivalence.
Therefore, every semiadditive Lawvere theory is of the form Burn R . By Theorem 2.10, SemiaddLwv is the symmetric monoidal full subcategory of Pr L spanned by the 8-categories Mod R , as R ranges over E 1 -semiring spaces. On the other hand, the functor E 1 Semiring Ñ Pr L which sends R to Mod R is also fully faithful and symmetric monoidal, which completes the proof.
From the proposition, we deduce an important philosophy: Lawvere theories are more like algebraic objects than categorical objects. We might even regard them as generalized (non-additive) rings.
Finally, we apply Theorem 2.6 to deduce: If M is presentable and semiadditive, and M Ñ Top is a right adjoint functor which is conservative and preserves geometric realizations, then M -Mod R for some E 1 -semiring space R, compatibly with the forgetful functor Mod R Ñ Top.

Applications of Lawvere theories
We will end with two applications. The first is to the commutative algebra of semiring 8-categories, and the second to equivariant homotopy theory. That is, an additive 8-category is semiadditive and enriched in Ab 8 -Sp ě0 (grouplike E 8 -spaces, or connective spectra). As promised in [5], we prove: Proof. As in [9], the product map Sp ě0 b Sp ě0 Ñ Sp ě0 is an equivalence. Identifying Sp ě0 with Mdl Burn and noting that Mdl is symmetric monoidal, we find that Burn b Burn Ñ Burn is also an equivalence. This means that the forgetful functor Mod Burn Ñ SymMonCat 8 is fully faithful. We need to show that a symmetric monoidal 8-category admits the structure of a Burn-module if and only if it is additive.

Commutative algebra of categories
First, if C is a Burn-module, it is a Burn eff -module and therefore semiadditive (Example 3.9), but it is also Mdl Burn -enriched by Corollary 3.8. By definition, it is therefore additive.
Conversely, suppose C is additive, so that PpCq " FunpC op , Topq is additive and presentable. By [9], PpCq is a Mdl Burn -module in Pr L , and therefore also in SymMonCat 8 , because the functor Pr L Ñ SymMonCat 8 is lax symmetric monoidal which forgets everything except the cocartesian monoidal structure.
The embedding Burn -Burn op Ď Mdl Burn respects both symmetric monoidal structures (' and b), so PpCq is a Burn-module, as a cocartesian monoidal 8-category. Moreover, the full subcategory C P PpCq is closed under direct sum, so it inherits a Burn-module structure. This completes the proof.

Equivariant homotopy theory
Throughout this section, G is a finite group. We write Fin G for the category of finite G-sets, and Burn G for the associated 8-category of virtual spans, often referred to as the Burnside 8-category without mention of the particular group. See [3] for more on this.
All group actions will be on the right.
There are two classical model categories of equivariant G-spaces: the 'naive' model structure has weak equivalences those maps which are weak equivalences of the underlying space. The corresponding 8-category is FunpBG, Topq, because equivalences in a functor 8-category are likewise checked objectwise, and BG has only one object (up to equivalence).
On the other hand, the 'genuine' model structure has weak equivalences those maps which have inverses up to homotopy. This model category corresponds to an 8-category Top G which is certainly not equivalent to FunpBG, Topq! For example, the map EG Ñ˚is an equivalence in the former but not in the latter model structure.
For spectra as well, there is a distinction between FunpBG, Spq and the 8-category Sp G of genuine equivariant spectra. Consult [12] for a classical survey.
We might ask how to describe Top G and Sp G in higher categorical terms. For this, we have the two theorems: • (Elmendorf's Theorem: [7] Theorem 1) Top G -MdlpFin op G q; • (Guillou-May's Theorem: [13] Theorem 0.1, [3] Example B.6) Sp ě0 G -MdlpBurn G q. Recall that we have used the notation MdlpLq " FunˆpL, Topq whenever L admits finite products, even if it is not a Lawvere theory. However, Fin G and Burn eff G are not far from being Lawvere theories: although they do not have single generating objects, they are generated freely by the set of orbits G{H, as H ranges over subgroups of G.
We call them colored Lawvere theories, with set of colors tG{Hu, or equivariant Lawvere theories, because they admit essentially surjective, productpreserving maps from the groupoid of finite G-sets, Fin iso G .
Remark 5.4. The word 'genuine', used to describe equivariant spaces and spectra, can be misleading. Frequently, group actions on spectra arise via abstract homotopy-theoretic means, such as when the spectra themselves are algebraic in nature (as in chromatic homotopy theory). In these cases, we typically do not expect 'genuine' equivariant structures.
However, when our spaces or spectra arise geometrically out of point-set constructions, group actions will be 'genuine'. This is because we can pass through the model category of genuine equivariant objects, on our way to the abstract 8-categories Top G and Sp G .
It would almost be better to regard the 'genuine' actions as 'geometric', and the 'naive' actions as 'homotopical'.
The theorems of Elmendorf and Guillou-May may be combined with Corollary 3.8 as follows: Corollary 5.5. Regarding Fin G and Burn eff G as commutative semiring 8categories, any Fin G -module is naturally enriched in genuine G-spaces, and any Burn eff G -module is naturally enriched in (connective) genuine G-spectra. More generally, suppose we have some algebraic structure, whose homotopical instances form an 8-category C. For example, C " Sp ě0 corresponds to the structure 'abelian group'. We might ask: what kind of structure does a genuine equivariant G-object of C have? This is a question which is not entirely idle. Following Remark 5.4, if an object of C has an action of G at some sufficiently concrete point-set level, we might expect additional structure to carry over to the 8-category C, beyond a naive G-action.
By analogy with the theorems of Elmendorf and Guillou-May, we propose addressing this question via a 3-step procedure: 1. check whether C is of the form Mdl L for some Lawvere theory L (possibly by means of Theorem 2.6); 2. check whether L can be described combinatorially, by applying some construction M to Fin (as in Principle 2.8); 3. FunˆpMpFin G q, Cq is a candidate for genuine G-objects of C.
Example 5.6. When C " Top, L " Fin op and the combinatorial construction M is the opposite category construction, so that (3) is Elmendorf 's Theorem.
When C " Sp ě0 , L " Burn and the combinatorial construction M is the virtual span construction, so that (3) is Guillou-May's Theorem.
One goal is to use this strategy to understand equivariant E 8 -ring spectra via the Lawvere theory of bispans of finite G-sets, by analogy with the construction of Tambara functors [19].
We hope to address these problems in a sequel, in which we will discuss combinatorial constructions of Lawvere theories (as in Principle 2.8).