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E-infinity structure in hyperoctahedral homology

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.


Introduction
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group [FL91,Section 3]. It was introduced by Fiedorowicz [Fie, Section 2] and developed by the author [Gra22]. The hyperoctahedral crossed simplicial group is the largest of the fundamental crossed simplicial groups, meaning that every other crossed simplicial group is either a subobject or an extension of a subobject [FL91,Theorem 3.6]. It is therefore of interest to study the properties of the associated homology theory. It has applications in equivariant stable homotopy theory. In particular, the author showed [Gra22,Theorem 8.8] that for a group of odd order G and a commutative ground ring k, there is an isomorphism of graded k-modules HO ⋆ (k[G]) ∼ = H ⋆ ΦΩ C 2 Q C 2 BG where BG is the classifying space of the group, Q C 2 is the C 2 -equivariant free infinite loop space functor, Ω C 2 is the C 2equivariant based loops functor and Φ is the C 2 -fixed point functor.
In this paper we show that the simplicial k-module used to compute the hyperoctahedral homology of an involutive algebra has the structure of an E ∞ -algebra in the category of simplicial k-modules.
The notion of E ∞ -algebra first appeared in work of Boardman and Vogt [BV68] and was developed by May [May72] amongst others. An E ∞ -algebra structure on an object means that is has a multiplication operation that is both associative and commutative up to all higher homotopies. One important implication of an E ∞ -algebra structure is that it yields tools for calculation. We will show that the E ∞ -algebra structure on the simplicial k-module which calculates hyperoctahedral homology in this paper gives rise to two such tools: Dyer-Lashof homology operations and a Pontryagin product.
Dyer-Lashof homology operations are an important tool for calculation in algebraic topology and homological algebra. They were introduced in [DL62] and used to calculate the homology of infinite loop spaces. Consider the isomorphism of graded k-modules HO ⋆ (k[G]) ∼ = H ⋆ ΦΩ C 2 Q C 2 BG . When we take k[G] = F p [Z], the right hand side has an action of the Dyer-Lashof algebra. The result of this paper shows that the left hand side also has an action of the Dyer-Lashof algebra in this case. The author hopes that this is the first stage in showing that this isomorphism of graded k-modules is in fact more structured. In particular, it is hoped that it preserves the Dyer-Lashof actions. This is a non-trivial statement and should certainly be investigated in the case of symmetric homology first.
The notion of a Pontryagin product has its roots in [Pon39]. In its simplest formulation it provides a graded product structure on the singular homology of topological spaces although there are many variants and generalizations. For example, there is a Pontryagin product on Hspaces [Hat02,3.C]. The Pontryagin product in this paper gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. This is a remarkable property that hyperoctahedral homology shares with symmetric homology: there is a graded-commutative structure on hyperoctahedral homology, even when the algebra under consideration is not commutative! In light of this it is natural to ask whether the homology theory associated to an arbitrary crossed simplicial group has the structure of a graded ring or a graded-commutative ring. However, in general the homology theory associated to a crossed simplicial group is not known to have any structure beyond that of a graded module.
The method of proof in this paper closely follows [Aul14] in which an E ∞ -algebra structure is found for symmetric homology. Ault's proof has four main stages: • construct an analogue of the Barratt-Eccles operad in the category of small categories; • construct a left operad module, in the sense of [Fre04, 2.1.6], over this operad using under-categories of the symmetric category; • use this left operad module structure to deduce an E ∞ -algebra structure on an object closely related to the concept of a Schur functor [MSS02, Definition 1.24]; • deduce that this induces an E ∞ -algebra structure on the simplicial k-module that computes symmetric homology.
We will see that hyperoctahedral homology shares many of the properties of symmetric homology. The keys to finding an E ∞ -algebra structure for hyperoctahedral homology are as follows.
Firstly we define the tuple category in Section 2. In some sense this can be seen as an "involutive" version of the material in [Aul14, Section 2.2]. In Section 3 we define a left operad module over the categorical version of the Barratt-Eccles operad using under-categories of the hyperoctahedral category. The main work in this section is identifying the correct functors required to verify the left operad module structure. These are found in Lemma 3.7 and Definition 3.9. Once these have been established, the technical details of the proofs of Lemma 3.7, Lemma 3.10 and Theorem 4.4 pass over from the symmetric case mutatis mutandis. The paper is organized as follows. In Section 1 we collate the background material that we will require for the remainder of the paper. In Subsection 1.1 we define the categories that we will use throughout the paper. In Subsection 1.2 we recall constructions from functor homology, in particular the tensor product of functors, Tor functors over a small category and simplicial kmodules used to compute them. Subsection 1.3 recalls the definition of hyperoctahedral homology for an involutive algebra, in terms of functor homology. We recall the hyperoctahedral category and the hyperoctahedral bar construction. Finally, in Subsection 1.4 we recall the necessary material from the theory of operads. We recall the notion of an E ∞ -operad in a symmetric monoidal model category. As examples we recall analogues of the Barratt-Eccles operad in the category of small categories and in the category of simplicial k-modules. We recall the definition of a left operad module in the sense of [Fre04, 2.1.6].
In Section 2 we define the tuple category. For an involutive monoid M , the tuple category is constructed from tuples of elements in M , morphisms in the hyperoctahedral category and the hyperoctahedral bar construction. We show that the tuple category is symmetric strict monoidal, giving it the structure of an E ∞ -algebra in the category of small categories.
In Section 3 we construct a left operad module over the Barratt-Eccles operad in the category of small categories, using under-categories in the hyperoctahedral category. In Section 4 we will use the left operad module structure to deduce that there is an E ∞ -algebra structure on the simplicial k-module which computes hyperoctahedral homology. As a corollary we obtain Dyer-Lashof homology operations and a Pontryagin product for hyperoctahedral homology.
In Section 5, we give an explicit description of hyperoctahedral homology in degree zero. We prove that for an involutive k-algebra A, the degree zero hyperoctahedral homology is isomorphic to the quotient of A by the ideal generated by the relation a 0 a 1 a 2 − a 2 a 1 a 0 . In particular degree zero hyperoctahedral homology has a ring structure induced from the one in A. We show that this description of degree zero hyperoctahedral homology together with the Pontryagin product implies that hyperoctahedral homology does not preserve Morita equivalence.
Acknowledgements. I am very grateful to Sarah Whitehouse for her helpful comments and suggestions. I would like to thank the anonymous referee for their suggestions and for their very interesting comments and questions.
Conventions. Throughout the paper, k will denote a unital commutative ground ring. All the algebras considered are unital.

Background material
We begin this section by collating definitions and notation for the categories that we will use throughout the rest of the paper. In Subsection 1.2 we recall the constructions from functor homology necessary to define hyperoctahedral homology in Subsection 1.3. We recall the hyperoctahedral category and the hyperoctahedral bar construction. We recall the simplicial k-module used to compute hyperoctahedral homology, which we will later show bears the structure of an E ∞ -algebra. Finally in Subsection 1.4 we recall the required material on operad theory, specifically the notions of E ∞ -operad in a symmetric monoidal model category and the notion of a left operad module.
1.1. Categories. We begin by defining the categories that we will require throughout the paper.
Definition 1.1. The following categories will be used throughout the paper.
• Let Cat be the category of small categories and functors.
• Let sSet denote the category of simplicial sets.
• Let Mod k and sMod k denote the categories of k-modules and simplicial k-modules respectively. • Let Top denote the category of compactly-generated Hausdorff topological spaces and continuous maps. • For a finite group G, let EG denote the category whose objects are the elements g ∈ G with a unique morphism g 2 g −1 1 ∈ Hom EG (g 1 , g 2 ). • Let IMon denote the category of involutive monoids and involution-preserving morphisms. • Let S denote the symmetric groupoid. The objects are the sets n = {1, . . . , n} for n 0 and Hom S (n, n) = Σ n . • For a category C and an object C ∈ C, we denote the under-category by C ↓ C, see [ML98, II.6] for instance. • Let {C i } be a set of small categories whose object sets are pairwise disjoint. Let i C i denote the category whose object set is i Ob (C i ) and whose set of morphisms is i Mor (C i ). 1.2. Functor homology. Throughout the paper we will make use of concepts from functor homology. We recall the necessary constructions from [PR02] and [GZ67]. Definition 1.2. Let C be a small category. We define the category of left C-modules, denoted CMod, to be the functor category Fun (C, Mod k ). We define the category of right C-modules, denoted ModC, to be the functor category Fun (C op , Mod k ).
It is well-known that the categories CMod and ModC are abelian with enough projectives and injectives, see for example [PR02, Section 1.6]. Definition 1.3. Let k ⋆ denote the right C-module that is constant at the trivial k-module. We will refer to this functor as the k-constant right C-module. Definition 1.4. Let G be an object of ModC and F be an object of CMod. We define the tensor product G ⊗ C F to be the k-module This quotient module is spanned k-linearly by equivalence classes of elementary tensors in Definition 1.5. One constructs the tensor product of C-modules as a bifunctor It is well-known that the bifunctor − ⊗ C − is right exact with respect to both variables and preserves direct sums, see for example [PR02, Section 1.6].
Recall the nerve of a small category C [Lod98, B.12]. N ⋆ C is the simplicial set such that N n C for n 1 consists of all strings of composable morphisms of length n in C and N 0 C is the set of objects in C. The face maps are defined to either compose adjacent morphisms in the string or truncate the string and the degeneracy maps insert identity morphisms into the string. We will denote an element of N n C by (f n , . . . , We will be particularly interested in the case where G is a functor of the form k N n (− ↓ C) . That is, we take the n th level of the nerve of an under-category and then take the free k-module on this set. We can view this construction as a resolution of the k-constant right C-module k ⋆ in the category ModC. We observe that in this case k N ⋆ (− ↓ C) ⊗ C F has the structure of a simplicial k-module, where the face and degeneracy maps are induced from nerve construction, and the homology of this simplicial k-module is Tor C ⋆ (k ⋆ , F ). There is an isomorphic construction due to Gabriel and Zisman [GZ67, Appendix 2].
where the sum runs through all elements (f n , . . . , The degeneracy maps insert identity maps into the string. We will denote the homology of the associated chain complex by H ⋆ (C, F ).
Remark 1.8. As mentioned above, for a left C-module F there is an isomorphism of simplicial  Remark 1.11. Let ∆S + and ∆H + denote the categories formed from ∆S and ∆H by appending an initial object, which will be denoted by [−1]. Both ∆S + and ∆H + are symmetric strict monoidal categories, as proved in [Aul10, Proposition 9] and [Gra22, Proposition 6.2] respectively.
Remark 1.12. Fiedorowicz and Loday provide a classification theorem for crossed simplicial groups [FL91,Theorem 3.6]. This shows that there are seven fundamental crossed simplicial groups. The largest of these is the hyperoctahedral crossed simplicial group; the remainder being subgroups of hyperoctahedral groups. Any other crossed simplicial group is an extension of a fundamental crossed simplicial group. From this perspective it is important to understand the properties of hyperoctahedral homology. However it also raises interesting questions. Fiedorowicz and Loday prove that any semi-direct extension of the symmetric groups is a crossed simplical group [FL91, Theorem 3.10]. The author plans to investigate the homology theories associated to these crossed simplicial groups.
on elementary tensors and extend k-linearly, where the product is ordered according to the map ϕ and Note that an empty product is defined to be the multiplicative unit 1 A . Definition 1.14. Let A be an involutive, associative algebra. For n 0, we define the n th hyperoctahedral homology of A to be   where M n+1 denotes the (n + 1)-fold Cartesian product. H M is defined on morphisms in ∆H analogously to H A . We define H M (i n ) to be the unique map ∅ → M n+1 . We call H M the hyperoctahedral bar construction for monoids. • for each n 0, O(n) is weakly equivalent to ½ in the model structure; • for each n 0, the action of Σ n on O(n) is free.
The right action of Σ m of EΣ m is given by composition of group elements and is therefore a free action. The category EΣ m is contractible since the nerve of the category EΣ m is a simplicial model of the total space of Σ m . Therefore D Cat is an E ∞ -operad in Cat.  Proof. This is stated as [Aul14, Proposition 3.7]. It can be proved analogously to [MSS02, Proposition 1.25] using the left operad module structure maps in place of the operad structure maps.

The tuple category
In this section we define the tuple category. Given an involutive monoid M we construct a category whose objects are tuples of elements in M and whose morphisms are constructed from the hyperoctahedral category ∆H + . We show that the tuple category is symmetric strict monoidal, making it an E ∞ -algebra in Cat. This structure is key in the proof of Lemma 3.10. Proof. The product of two tuples in T (M ) is given by concatenation. The fact this product is symmetric strict monoidal follows from the fact ∆H + is symmetric strict monoidal under the disjoint union.

A left operad module
In this section we will define a left operad module over D Cat in terms of under-categories of the hyperoctahedral groups. We will use the left operad module defined in this section, in conjunction with Proposition 1.25, to give an E ∞ -algebra structure on C ⋆ (∆H + , H A+ ), the simplicial k-module which computes hyperoctahedral homology. Remark 3.2. Henceforth, for the sake of neatness, we will omit the underline on an object when we apply the functor IF.
We define a family of functors That is, f i acts on a block of size j i and we permute the m blocks according to σ ∈ Σ m . The functor λ is defined on morphisms by That is, we compose g i with f i , set this to act on a block of size j i and then permute the blocks according to the permutation τ ∈ Σ m . The functoriality of λ follows from the functoriality of the composition of morphisms in the under-category.
We will show in Lemma 3.10 that the structure from Definitions 3.1 and 3.3 define the structure of a left D Cat -module on IF(−). Remark 3.5. Observe that the set X has a C 2 -action given by x i → x i . Recall that a category C is called discrete if the only morphisms in C are identity morphisms. Note that for each m 0 we can consider the Cartesian product X m as a discrete category. Furthermore, this category has a left action of the hyperoctahedral group H m given by applying the C 2 -action and permuting the factors.  Proof. The proof is analogous to [Aul14, Lemma 3.4]. One shows that the evaluation maps factor through IF(m) × Hm X m using the functoriality of H M . One then constructs the inverse functor, which is induced from the maps that completely factor the monomials in a tuple in T (F (X)).
• For 1 i m, let I i : IF(j i ) → IF(j i ) × H j i X j i be the inclusion functor f → (f, y i ).
• Let I : IF(j) → IF(j) × H j X j be the inclusion functor f → (f, y).
• For i 0, let be the inclusion of categories.
• Let J = J j 1 × · · · × J jm . Proof. Recall from Lemma 2.3 that the tuple category T (F (X)) has the structure of a D Catalgebra. Therefore, for m 0, we have operad algebra structure maps as defined in [Aul14, Section 2.5].
Recall the isomorphism of categories from Lemma 3.7 and the functors of Definition 3.9. Analogously to [Aul14,Diagram (16)], there is a commutative diagram The associativity condition of the functors λ is induced from the associativity conditions of the operad algebra structure maps θ, since all the vertical functors are injective. The left unit condition is satisfied straightforwardly. The equivariance conditions are satisfied similarly to the diagrams labelled Equivariance A and Equivariance B in [Aul14, Section 3.3].  Proof. The required structure is induced from the D Cat -module structure on IF(−) by the composite of the nerve functor and the free k-module functor.

E-infinity structure
In this section we apply Proposition 1.25 to obtain an E ∞ -algebra in the category of simplicial k-modules, constructed from the left D Mod -module IF of Definition 3.11 and the hyperoctahedral bar construction. We prove that the simplicial k-module C ⋆ (∆H + , H A+ ) is a quotient of this and that the quotient map induces the structure of an E ∞ -algebra structure on C ⋆ (∆H + , H A+ ). As a corollary we observe that hyperoctahedral homology of an involutive algebra admits Dyer-Lashof operations and a Pontryagin product. Theorem 4.4. Let A be an involutive k-algebra. The simplicial k-module C ⋆ (∆H + , H A+ ), which computes the hyperoctahedral homology HO ⋆ (A), bears the structure of an E ∞ -algebra in the category sMod k .
Proof. Analogously to [Aul14,Lemma 3.10], one can show that the E ∞ -algebra structure maps ν of Definition 4.3 and the quotient map q of Definition 4.2 are compatible in the sense that the diagram commutes. That is, the structure map ν remains well defined upon passing to the quotient and we deduce that C ⋆ (∆H + , H A+ ) has the structure of a D Mod -algebra.
Corollary 4.5. Let A be an involutive k-algebra.
(1) When k = F p for a prime p, the hyperoctahedral homology HO ⋆ (A) admits Dyer-Lashof homology operations.
where ν is the D Mod -algebra structure map from Theorem 4.4. The fact that the product is unital follows from the fact that our analogues of the Barratt-Eccles operad are unital operads.

Hyperoctahedral homology in degree zero
In this section we give an explicit description of hyperoctahedral homology in degree zero. We will show that for a unital involutive k algebra A, HO 0 (A) is isomorphic to the quotient of A by the ideal (a 0 a 1 a 2 − a 2 a 1 a 0 ). In particular, we deduce that HO 0 (A) has a ring structure.
The material in this section is best presented using an isomorphic variant of the category ∆H, namely the category of involutive non-commutative sets IF(as). The isomorphism of [Gra22, Theorem 1.7] tells that a morphism in Hom ∆H [n], [m] can be uniquely described as a map of sets f : [n] → [m] such that for each element i ∈ [m] the preimage f −1 (i) has a total ordering and each element comes adorned with a superscript label from the group C 2 . We call such a set a C 2 -labelled ordered set. We also recall that a C 2 -labelled ordered set X has a C 2 -action, X t that reverses the total ordering and multiplies each of the labels by t ∈ C 2 .
We will require two maps in particular.
We begin by constructing a partial resolution of the k-constant right IF(as)-module k ⋆ .
Lemma 5.3. For each n 0 there is an exact sequence of k-modules [2] where ε is defined by ε(f ) = 1 k on generators and extended k-linearly. The map ρ is determined on generators by ρ(g) = µ 2 • g − ν • g and extended k-linearly.
Proof. The augmentation map ε is surjective by definition. The composite ε • ρ is zero since the image of ρ on generators consists of two morphisms with differing signs. Therefore Im(ρ) ⊆ Ker(ε). It remains to show that Ker(ε) ⊆ Im(ρ). We note that Ker(ε) is spanned by Consider a morphism g ∈ Hom IF (as) [n], [2] . This map is determined by g −1 (0) = X, g −1 (1) = Y and g −1 (2) = Z where X, Y and Z are C 2 -labelled, ordered sets. By definition In order to show that the differences f − µ n lie in the image of ρ it suffices to show that there is a finite sequence of morphisms f p , . . . , f 0 in Hom IF (as) [n], [0] such that f p = f , f 0 = µ n and for each pair (f i , f i−1 ) there exists a g ∈ Hom IF (as) [n], [2] such that µ 2 • g = f i and We construct this sequence of morphisms as follows.
(7) If k m = m for 0 m n then we have obtained µ n and we are done.
(8) If not, by repeatedly choosing g such that Y = ∅ and Z is a singleton we cyclically permute the elements of the total ordering without changing any of the labels. We do this until k n = n. (9) If k m = m for 0 m n then we have obtained µ n and we are done. (10) If not, suppose k m = m for m > j. Choose g with X = k 1 0 < · · · k 1 j−1 , Y = k 1 j , Z = k 1 j+1 < · · · < k 1 n .
(11) By choosing g such that Y = ∅ and Z is singleton, we can cyclically permute this total ordering until we obtain k t j < k 1 0 < · · · k 1 j−1 < k 1 j+1 < · · · k n n .
(12) Choose g with X = ∅, Y = j t k and Z = k 1 0 < · · · k 1 j−1 < k 1 j+1 < · · · k n n . (13) By choosing g with Y = ∅ and Z a singleton, we cyclically permute the resulting total ordering to get k 1 j < k 1 0 < · · · < k 1 j−1 < k 1 j+1 < · · · k n n . (14) The result of Steps 10 to 13 is to fix k m for m > j and to cyclically permute k 0 up to k j whilst leaving the labels unchanged. Repeating this process a finite number of times we will obtain µ n as required. This shows that all the differences of the form f −µ n lie in the image of ρ and so Ker(ε) ⊆ Im(ρ). Therefore the sequence is exact.  [2] where ε is defined by ε (ϕ, g) = 1 k on generators and extended k-linearly. The map ρ is determined on generators by ρ (ψ, h) = F (µ 2 ) • (ψ, h) − F (ν) • (ψ, h) and extended k-linearly.
Proof. This follows directly from Lemma 5.3 and [Gra22, Theorem 1.7]. where ε and ρ are the natural transformations formed from the k-module maps of Corollary 5.4.
Proof. The exactness of a sequence of functors is checked object-wise, so this follows directly from Corollary 5.4.
Corollary 5.6. Let A be a unital, involutive k-algebra. The homology of the partial chain complex where d(a 0 ⊗ a 1 ⊗ a 2 ) = a 0 a 1 a 2 − a 2 a 1 a 0 is isomorphic to HO 0 (A).
Proof. We apply the functor − ⊗ ∆H H A to the partial resolution of k ⋆ in Corollary 5.5 and take homology.
Lemma 5.7. The k-submodule k {a 0 a 1 a 2 − a 2 a 1 a 0 } of A is, in fact, an ideal of A.
Proof. We observe that given a product of four elements a 0 a 1 a 2 a 3 we can use the relation to obtain any product of four elements given by applying the involution to any subset of the four factors, then permuting the four factors. This follows from the algorithm of Lemma 5.3 by showing that we can use the relation to obtain µ 3 from any map in Hom IF (as) [3], [0] .
Theorem 5.8. Let A be a unital, involutive k-algebra. There is an isomorphism a 1 a 2 − a 2 a 1 a 0 ) .
In particular, HO 0 (A) has a ring structure induced from the ring structure of A.