2. Some redefinitions

One of the ingredients in the definition of ad theory in [Reference Laures and McClure11] is the target ‘${\mathbb Z}$-graded category’ ${\mathcal A}$ (see [Reference Laures and McClure11, definitions 3.3 and 3.10]). For the purposes of that article, there was no reason to allow morphisms in ${\mathcal A}$ between objects of the same dimension (except for identity maps). For the present article, we do need such morphisms (see definition 3.1 and the proof of theorem 1.1). We therefore begin by giving modified versions of some of the definitions of [Reference Laures and McClure11].

The definition of a strict monoidal structure on a ${\mathbb Z}$-graded category [Reference Laures and McClure11, definition 18.1] needs no change, provided that one uses the new definition of ${\mathbb Z}$-graded category.

Next we explain how to modify the specific examples of target categories in [Reference Laures and McClure11] by adding morphisms which preserve dimension.

For the category ${\mathcal A}_{\mathrm{STop}}$ [Reference Laures and McClure11, example 3.5], the morphisms between objects of the same dimension are the orientation-preserving homeomorphisms.

For the category ${\mathcal A}_{\pi,Z,w}$ [Reference Laures and McClure11, definition 7.3], the morphisms between objects $(X,f,\xi)$ and $(X',f',\xi')$ of the same dimension are the maps $g:X\to X'$ such that $f'\circ g=f$ and $g_*(\xi)=\xi'$.

We do not need the analogous modification for the category ${\mathcal A}^R$ [Reference Laures and McClure11, definition 9.5] because we will be using the version in §12.

The definition of an ad theory [Reference Laures and McClure11, definition 3.10] stays the same. In §14, however, we will restrict our attention to ad theories which are only defined on strict ball complexes. A ball complex is called strict if each component of the intersection of two cells is a single cell. This assumption does not affect the constructions or results of [Reference Laures and McClure11].

3. Commutative ad theories

Definition 3.1. Let ${\mathcal A}$ be a ${\mathbb Z}$-graded category. A permutative structure on ${\mathcal A}$ is a strict monoidal structure $(\boxtimes, \varepsilon)$ [Reference Laures and McClure11, definition 18.1] together with a natural isomorphism

\begin{align*} \gamma_{x,y}:x\boxtimes y\to i^{|x||y|} y\boxtimes x \end{align*}

such that

(a) $i\gamma_{x,y}=\gamma_{ix,y}=\gamma_{x,iy}$,

(b) each of the maps $\gamma_{\emptyset,y}$, $\gamma_{x,\emptyset}$ is the identity map of $\emptyset$,

(c) the composite

is the identity.

(d) $\gamma_{x,\varepsilon}$ is the identity, and

(e) the diagram

commutes. A strict map of ${\mathbb Z}$-graded categories with permutative structures is a map $f:{\mathcal A} \rightarrow \mathcal{B}$ of ${\mathbb Z}$-graded categories for which $f(x\boxtimes y )= f(x)\boxtimes f(y)$ holds for objects and morphisms. Moreover, such a functor takes ϵ to ϵ and the diagram

commutes.

Examples are ${\mathrm{ad}}_C$ when C is a commutative DGA (see [Reference Laures and McClure11, example 3.12]), ${\mathrm{ad}}_{\mathrm{STop}}$ (see [Reference Laures and McClure11, §6]), ${\mathrm{ad}}_{e,*,1}$ (see [Reference Laures and McClure11, §7]), ${\mathrm{ad}}_{\mathrm{STopFun}}$ (see [Reference Laures and McClure11, end of §8]), ${\mathrm{ad}}_{\text{rel}}^R$ when R is commutative (see §12), ${\mathrm{ad}}_{\mathrm{IP}}$ [Reference Banagl, Laures and McClure3, §4], and ${\mathrm{ad}}_{\mathrm{IPFun}}$ [Reference Banagl, Laures and McClure3, §6.1].

For later use, we record some notation for iterated products.

Definition 3.5. (i) For a permutation $\eta\in\Sigma_j$, let $\epsilon(\eta)$ denote 0 if η is even and 1 if η is odd.

(ii) Let ${\mathcal A}$ be a $\mathbb Z$-graded category with a permutative structure. Let $\eta\in \Sigma_j$. Define a functor

\begin{align*} \eta_{\bigstar}:{\mathcal A}^{\times j}\to {\mathcal A} \end{align*}

(where ${\mathcal A}^{\times j}$ is the j-fold Cartesian product) by

\begin{align*} \eta_{\bigstar}(x_1,\ldots,x_j) = i^{\epsilon(\bar{\eta})} (x_{\eta^{-1}(1)}\boxtimes\cdots\boxtimes x_{\eta^{-1}(j)}) \end{align*}

where $\bar{\eta}$ is the block permutation that takes blocks ${\mathbf b}_1$, …, ${\mathbf b}_j$ of size $|x_1|$, …, $|x_j|$ into the order ${\mathbf b}_{\eta^{-1}(1)}$, …, ${\mathbf b}_{\eta^{-1}(j)}$.

4. Multisemisimplicial symmetric spectra

In this section, we define a category ${\mathcal S} p_{{mss }}$ (the ss stands for ‘semisimplicial’) which is a multisemisimplicial version of the category ${\mathcal S} p$ of symmetric spectra. The motivation for the definition is that the sequence R_k in [Reference Laures and McClure11, definition 17.2] should give an object of ${\mathcal S} p_{{mss }}$.

Recall that we write $\Delta_{\mathrm{inj}}$ for the category whose objects are the sets $\{0,\ldots,n\}$ and whose morphisms are the monotonically increasing injections.

A based k-fold multisemisimplicial set is a contravariant functor from the Cartesian product $(\Delta_{\mathrm{inj}})^{\times k}$ to the category ${\mathcal S}_*$ of based sets. In particular, a based 0-fold multisemisimplicial set is just a based set.

Next note that given a category ${\mathcal C}$ with a left action of a group G one can define a category $G \lt imes {\mathcal C}$ whose objects are those of ${\mathcal C}$ and whose morphisms are pairs $(\alpha,f)$ with $\alpha\in G$ and f a morphism of ${\mathcal C}$; the domain of $(\alpha,f)$ is the domain of f and the target is α ⁻¹ applied to the target of f. Composition is defined by

\begin{align*} (\alpha,f)\circ(\beta,g)=(\alpha\beta,\beta^{-1}(f)\circ g). \end{align*}

By remark 4.1, an object of $H{{ss}} {\mathcal S}_k$ can be thought of as a based k-fold multisemisimplicial set with a left ‘action’ of H in which H also acts on the multidegrees.

The category of multisemisimplicial symmetric sequences will be denoted by $\Sigma{{ss}}{\mathcal S} $.

For our next definition, recall [Reference Laures and McClure11, definitions 17.2 and 17.3].

Definition 4.4. (i) For each $k\geq 0$ extend the object $R_k={\mathrm{ad}}^k(\Delta^\bullet)$ of ${{ss}} {\mathcal S}_k$ to an object of $\Sigma_k{{ss}} {\mathcal S}_k$ by letting

\begin{align*} (\alpha,\mathrm{id})_*(F)=i^{\epsilon(\alpha)}\circ F\circ \alpha_\# \end{align*}

(where $\alpha\in \Sigma_k$ and $F\in{\mathrm{ad}}^k(\Delta^{\mathbf n})$).

(ii) Let ${\mathbf R}$ denote the object of $\Sigma{{ss}}{\mathcal S} $ whose k-th term is R_k.

Next we assemble the ingredients needed to define a symmetric monoidal structure on $\Sigma{{ss}}{\mathcal S} $.

Definition 4.5. Given $A\in \Sigma_k{{ss}} {\mathcal S}_k$ and $B\in \Sigma_l{ss}{\mathcal S}_l$, define the object $A\wedge B\in (\Sigma_k\times\Sigma_l){{ss}}{\mathcal S}_{k+l}$ by

\begin{align*} (A\wedge B)_{{\mathbf m},{\mathbf n}}= A_{\mathbf m}\wedge B_{{\mathbf n}} \end{align*}

(where $\mathbf m$ is a k-fold multi-index and ${\mathbf n}$ is an l-fold multi-index).

Definition 4.6. Given $H\subset G\subset \Sigma_k$, define a functor

\begin{align*} I_H^G: H{{ss}} {\mathcal S}_k\to G{{ss}} {\mathcal S}_k \end{align*}

by letting $I_H^G A$ be the left Kan extension of A along $H \lt imes (\Delta_{\mathrm{inj}}^{\text{op}})^{\times k} \to G \lt imes (\Delta_{\mathrm{inj}}^{\text{op}})^{\times k}$.

Remark 4.7. For later use, we give an explicit description of $I_H^G A$. For each multi-index ${\mathbf n}$, we have

\begin{align*} (I_H^G A)_{\mathbf n}= \Bigl( \bigvee_{\alpha\in G} A_{\alpha^{-1}({\mathbf n})} \Bigr) /H \end{align*}

where the action of H is defined as follows: if $\beta\in H$ and x is an element in the α-summand then β takes x to the element $(\beta,\mathrm{id})_*(x)$ in the $\alpha\beta^{-1}$-summand.

Definition 4.9. Given ${\mathbf X},{\mathbf Y}\in \Sigma{{ss}}{\mathcal S} $, define ${\mathbf X}\otimes {\mathbf Y}\in \Sigma{{ss}}{\mathcal S} $ by

\begin{align*} ({\mathbf X}\otimes {\mathbf Y})_k=\bigvee_{j_1+j_2=k}\, I^{\Sigma_k}_{\Sigma_{j_1}\times \Sigma_{j_2}}\, \bigl({\mathbf X}_{j_1}\wedge {\mathbf Y}_{j_2}\bigr). \end{align*}

The proof that ⊗ is a symmetric monoidal product is essentially the same as the corresponding proof in [Reference Hovey, Shipley and Smith9, §2.1]. The symmetry map

\begin{align*} \tau:{\mathbf X}\otimes{\mathbf Y}\to {\mathbf Y}\otimes{\mathbf X} \end{align*}

is given (as in [Reference Hovey, Shipley and Smith9]) by

(4.1)\begin{align} \tau([\alpha,x\wedge y]) =[\alpha\rho_{l,k},y\wedge x] \end{align}

where $x\in (X_k)_{\mathbf{m}}$, $y\in (Y_l)_{\mathbf n}$, and $\rho_{l,k}$ is the permutation of $\{1,\ldots,k+l\}$ which moves the first l elements to the end and the last k elements to the front.

Next we will give the definition of the category ${\mathcal S} p_{{mss }}$ and its symmetric monoidal product. First we need a sphere object.

Definition 4.10. (i) Let S ¹ be the based semisimplicial set that consists of the base point together with a 1-simplex s.

(ii) Let S^k be the object of $\Sigma_k{{ss}} {\mathcal S}_k$ obtained from $(S^1)^{\wedge k}$ by letting

\begin{align*} (\alpha,\mathrm{id})_*(s\wedge\cdots\wedge s)= s\wedge\cdots\wedge s \end{align*}

(iii) Let ${\mathbf S}$ be the object of $\Sigma{{ss}}{\mathcal S} $ whose k-th term is S^k.

It is easy to check that ${\mathbf S}$ is a commutative monoid in $\Sigma{{ss}}{\mathcal S} $.

Remark 4.12. One can give a more explicit version of this definition: an object of ${\mathcal S} p_{{mss }}$ consists of an object ${\mathbf X}$ of $\Sigma{{ss}}{\mathcal S} $ together with suspension maps

\begin{align*} \omega:S^1\wedge X_k\to X_{k+1} \end{align*}

for each k, such that the iterates of the ω’s satisfy appropriate equivariance conditions.

Example 4.13. The object ${\mathbf R}$ of definition 4.4 can be given suspension maps as follows: with the notation of [Reference Laures and McClure11, definition 17.4(i)], define

\begin{align*} \omega:S^1\wedge R_k \to R_{k+1} \end{align*}

by

\begin{align*} \omega(s\wedge F)=\lambda^*(F) \end{align*}

(where s is the 1-simplex of S ¹ and $F\in{\mathrm{ad}}^k(\Delta^{\mathbf n})$). The resulting object of ${\mathcal S} p_{{mss }}$ will also be denoted ${\mathbf R}$.

Definition 4.14. (cf. [Reference Hovey, Shipley and Smith9, definition 2.2.3])

For ${\mathbf X},{\mathbf Y}\in {\mathcal S} p_{{mss }}$, define the smash product ${\mathbf X}\wedge {\mathbf Y}$ to be the coequalizer of the diagram

\begin{align*} {\mathbf X}\otimes {\mathbf S}\otimes {\mathbf Y} \rightrightarrows {\mathbf X}\otimes {\mathbf Y} \end{align*}

where the right action of ${\mathbf S}$ on ${\mathbf X}$ is the composite

\begin{align*} {\mathbf X}\otimes {\mathbf S}\to {\mathbf S}\otimes {\mathbf X}\to {\mathbf X}. \end{align*}

The proof that $\wedge$ is a symmetric monoidal product is essentially the same as the corresponding proof in [Reference Hovey, Shipley and Smith9, §2.2].

5. Geometric realization

Let G be a subgroup of $\Sigma_k$. By remark 4.1(i), an object of $G{{ss}} {\mathcal S}_k$ has an underlying k-fold multisemisimplicial set.

Proposition 5.3. For $A\in \Sigma_k{{ss}} {\mathcal S}_k$, the following formula gives a natural left $\Sigma_k$ action on $|A|$:

\begin{align*} \alpha([u_1,\ldots,u_k,a]) = [u_{\alpha^{-1}(1)}, \ldots, u_{\alpha^{-1}(k)}, (\alpha,\mathrm{id})_*(a)]; \end{align*}

here $(u_1,\ldots,u_k)\in \Delta^{\mathbf n}$, $a\in A_{\mathbf n}$, and $[u_1,\ldots,u_k,a]$ denotes the class of $(u_1,\ldots,u_k,a)$ in $|A|$.

Proposition 5.4. For $H\subset G\subset \Sigma_k$ and $A\in H{{ss}} {\mathcal S}_k$, there is a natural isomorphism of based G-spaces

\begin{align*} |I_H^G A|\cong G_+\wedge_H\, |A|. \end{align*}

6. A family of multiplication maps

In this section, we begin the proof of theorem 1.1. We shall construct a monad ${\mathbb P}$ together with maps

\begin{align*}{\mathbb A}={\mathbb A} ssoc\rightarrow {\mathbb P} \rightarrow {\mathbb C} omm={\mathbb P}'\end{align*}

such that (a) the functor ${\mathbf R}$ factors over ${\mathbb P}$-algebras, (b) the transformation ${\mathbb P} \rightarrow {\mathbb P}' $ is a weak equivalence in a suitable sense so ${\mathbf R}$ is weakly equivalent as a functor to ${\mathbb P}$-algebras to another functor that factors over ${\mathbb P}'$-algebras.

From now until the end of §10, we fix a ${\mathbb Z}$-graded permutative category ${\mathcal A}$ and a commutative ad theory with values in ${\mathcal A}$. Let ${\mathbf R}$ be the object of ${\mathcal S} p_{{mss }}$ constructed from this ad theory as in example 4.13.

Let ${\mathbf M}$ be the symmetric ring spectrum associated with the ad theory [Reference Laures and McClure11, proposition 17.5 and theorem 18.5]. By definition, $M_k=|R_k|$. The multiplication of ${\mathbf M}$ is induced by the collection of maps

\begin{align*} \mu:(R_k)_{\mathbf{m}}\wedge(R_l)_{\mathbf n}\to (R_{k+l})_{\mathbf{m},{\mathbf n}} \end{align*}

defined by

(6.1)\begin{align} \mu(F\wedge G)(\sigma_1\times \sigma_2,o_1\times o_2) = i^{l \dim \sigma_1}F(\sigma_1,o_1)\boxtimes G(\sigma_2,o_2) \end{align}

(this is well-defined because, by [Reference Laures and McClure11, definition 18.1(b)], reversing the orientations o ₁ and o ₂ does not change the right-hand side). These maps give ${\mathbf R}$ the structure of a monoid in ${\mathcal S} p_{{mss }}$ (the proof is essentially the same as for [Reference Laures and McClure11, theorem 18.5]).

In general, even though the ad theory is commutative, ${\mathbf R}$ is not a commutative monoid (this would require the product in the target category ${\mathcal A}$ to be strictly graded commutative). Instead we have the following. Recall definition 3.5 and notation 4.8.

Lemma 6.1. Let $m:{\mathbf R}\wedge{\mathbf R}\to{\mathbf R}$ be the product and let $\eta\in\Sigma_j$. Then the composite

\begin{align*} m_\eta:{\mathbf R}^{\wedge j} \xrightarrow{\eta} {\mathbf R}^{\wedge j} \xrightarrow{m} {\mathbf R} \end{align*}

is determined by the formula

\begin{align*} & m_\eta([e,F_1\wedge\cdots\wedge F_j]) (\sigma_1\times\cdots\times\sigma_j,o_1\times\cdots\times o_j) \\ & \quad = i^{\epsilon(\zeta)} \eta_\bigstar (F_1(\sigma_1,o_1), \ldots, F_j(\sigma_j,o_j)), \end{align*}

where e is the identity element of the relevant symmetric group and ζ is the block permutation that takes blocks ${\mathbf b}_1$, …, ${\mathbf b}_j$, ${\mathbf c}_1$, …, ${\mathbf c}_j$ of size $\deg F_1$, …, $\deg F_j$, $\dim \sigma_1$, …, $\dim \sigma_j$ into the order ${\mathbf b}_1$, ${\mathbf c}_1$, …, ${\mathbf b}_j$, ${\mathbf c}_j$.

The key idea in the proof of theorem 1.1 is that there is a family of operations which can be used to interpolate between the various m_η. To construct this family, we allow a different permutation of the factors for each cell of $\Delta^{{\mathbf n}_1}\times\cdots\times \Delta^{{\mathbf n}_j}$, as explained in our next definition. We begin by defining the operations for pre-ads (see [Reference Laures and McClure11, definitions 3.8(i) and 3.10(ii)]).

Definition 6.2. (i) Given a ball complex K, let U(K) denote the set of all cells of K.

(ii) Let $k_1,\ldots,k_j$ be non-negative integers and let ${\mathbf n}_i$ be a k_i-fold multi-index for $1\leq i\leq j$. For any map

\begin{align*} a: U(\Delta^{{\mathbf n}_1}\times\cdots\times \Delta^{{\mathbf n}_j})\to \Sigma_j \end{align*}

define a map on preads (i.e., functors F from the oriented cells $(\sigma, o)$ to the ambient category)

\begin{align*} a_*: {\mathrm{pre}}^{k_1}(\Delta^{{\mathbf n}_1}) \times \cdots \times {\mathrm{pre}}^{k_j}(\Delta^{{\mathbf n}_j}) \to {\mathrm{pre}}^{k_1+\cdots+k_j}(\Delta^{({\mathbf n}_1,\ldots,{\mathbf n}_j)}) \end{align*}

by

\begin{align*} & a_*(F_1,\ldots, F_j)(\sigma_1\times\cdots\times\sigma_j,o_1\times\cdots\times o_j)) \\ & \quad = i^{\epsilon(\zeta)} (a(\sigma_1\times\cdots\times\sigma_j))_\bigstar (F_1(\sigma_1,o_1), \ldots, F_j(\sigma_j,o_j)), \end{align*}

where ζ is the block permutation that takes blocks ${\mathbf b}_1$, …, ${\mathbf b}_j$, ${\mathbf c}_1$, …, ${\mathbf c}_j$ of size k ₁, …, k_j, $\dim \sigma_1$, …, $\dim \sigma_j$ into the order ${\mathbf b}_1$, ${\mathbf c}_1$, …, ${\mathbf b}_j$, ${\mathbf c}_j$.

Recalling that $(R_k)_{{\mathbf n}}={\mathrm{ad}}^k(\Delta^{\mathbf n})$ with basepoint at the trivial ad (see [Reference Laures and McClure11, definitions 3.8(ii), 3.10(b) and 18.1(c)]), we have now constructed an operation

\begin{align*} a_*: (R_{k_1})_{{\mathbf n}_1} \wedge\cdots\wedge (R_{k_j})_{{\mathbf n}_j} \to (R_{k_1+\cdots+k_j})_{{\mathbf n}_1,\ldots,{\mathbf n}_j} \end{align*}

for each

\begin{align*} a: U(\Delta^{{\mathbf n}_1}\times\cdots\times \Delta^{{\mathbf n}_j})\to \Sigma_j. \end{align*}

For later use, we give the relation between $a_*$ and the suspension map $\omega:S^1\wedge R_k\to R_{k+1}$.

Definition 6.4. For a ball complex K, let

\begin{align*} \Pi:U(\Delta^1\times K)\to U(K) \end{align*}

be the map which takes $\sigma\times\tau$ to τ, where σ is a simplex of Δ¹ and τ is a simplex of K.

Lemma 6.5. Let s be the 1-simplex of S ¹, let $F_i\in (R_{k_i})_{{\mathbf n}_i}$ for $1\leq i\leq j$, and let $a: U(\Delta^{{\mathbf n}_1}\times\cdots\times \Delta^{{\mathbf n}_j})\to \Sigma_j$. Then

\begin{align*} \omega(s\wedge a_*(F_1\wedge\cdots\wedge F_j)) = (a\circ \Pi)_*(\omega(s\wedge F_1)\wedge\cdots\wedge F_j). \end{align*}

In the remainder of this section, we show that the action of the operations $a_*$ can be described in a way that begins to resemble the action of an operad; this resemblance will be developed further in the next two sections.

Definition 6.6. (i) For $j,k\geq 0$ define an object ${\mathcal O}(j)_k$ of $\Sigma_k{{ss}} {\mathcal S}_k$ by

\begin{align*} ({\mathcal O}(j)_k)_{\mathbf n}=\operatorname{Map}(U(\Delta^{\mathbf n}),\Sigma_j)_+ \end{align*}

(where the + denotes a disjoint basepoint); the morphisms in $(\Delta_{\mathrm{inj}}^{\text{op}})^{\times k}$ act in the evident way, and the morphisms of the form $(\alpha,\mathrm{id})$ with $\alpha\in \Sigma_k$ act by permuting the factors in $\Delta^{\mathbf n}$.

(ii) For $j\geq 0$ define ${\mathcal O}(j)$ to be the object of $\Sigma{{ss}}{\mathcal S} $ with k-th term ${\mathcal O}(j)_k$.

Definition 6.7. (i) For $A,B\in \Sigma_k{{ss}} {\mathcal S}_k$, define the degreewise smash product

\begin{align*} A\mathbin{{\overline{\wedge}}} B \in \Sigma_k{{ss}} {\mathcal S}_k \end{align*}

by

\begin{align*} (A\mathbin{{\overline{\wedge}}} B)_{\mathbf n}=A_{\mathbf n}\wedge B_{\mathbf n}, \end{align*}

with the diagonal action of $\Sigma_k$.

(ii) For ${\mathbf X},{\mathbf Y}\in\Sigma{{ss}}{\mathcal S} $, define ${\mathbf X}\mathbin{{\overline{\wedge}}} {\mathbf Y}\in\Sigma{{ss}}{\mathcal S} $ by

\begin{align*} ({\mathbf X}\mathbin{{\overline{\wedge}}} {\mathbf Y})_k=X_k\mathbin{{\overline{\wedge}}} Y_k. \end{align*}

Our next definition assembles the operations $a_*$ for a given j into a single map.

Definition 6.9. Let $j\geq 0$. Define a map

\begin{align*} \phi_j: {\mathcal O}(j) \mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j} \to {\mathbf R} \end{align*}

in $\Sigma{{ss}}{\mathcal S} $ by the formulas

\begin{align*} \phi_j(a\wedge [e, F_1\wedge\cdots\wedge F_j]) = a_*(F_1\wedge\cdots\wedge F_j) \end{align*}

(where e denotes the identity element of the relevant symmetric group) and

\begin{align*} \phi_j(a\wedge [\alpha, F_1\wedge\cdots\wedge F_j]) = (\alpha,\mathrm{id})_*\phi_j((\alpha^{-1},\mathrm{id})_*a\wedge [e, F_1\wedge\cdots\wedge F_j]). \end{align*}

Lemma 6.10. The map ϕ_j induces a map

\begin{align*} \psi_j: {\mathcal O}(j) \mathbin{{\overline{\wedge}}} {\mathbf R}^{\wedge j} \to {\mathbf R} \end{align*}

in $\Sigma{{ss}}{\mathcal S} $.

7. A monad in $\Sigma{{ss}}{\mathcal S} $

In the next section, we will show that there is a monad ${\mathbb P}$ in ${\mathcal S} p_{{mss }}$ with the property that the maps ψ_j constructed in lemma 6.10 give an action of ${\mathbb P}$ on ${\mathbf R}$. As preparation, in this section, we prove the analogous result in $\Sigma{{ss}}{\mathcal S} $; that is, we show that there is a monad ${\mathbb O}$ in $\Sigma{{ss}}{\mathcal S} $ for which the maps ϕ_j of definition 6.9 give an action of ${\mathbb O}$ on ${\mathbf R}$.

First we observe that the collection of objects ${\mathcal O}(j)$ has a composition map analogous to that of an operad. Recall that May defines an operad $\mathcal M$ in the category of sets with ${\mathcal M}(j)=\Sigma_j$ [Reference Peter May17, definition 3.1(i)]. Let $\gamma_{\mathcal M}$ denote the composition operation in $\mathcal M$. Also recall definition 4.9 and notation 4.8.

Definition 7.1. Given $j_1,\ldots,j_i\geq 0$ define a map

\begin{align*} \gamma: {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \bigl({\mathcal O}(j_1)\otimes\cdots\otimes O(j_i)\bigr) \to {\mathcal O}({j_1+\cdots+j_i}) \end{align*}

in $\Sigma{{ss}}{\mathcal S} $ by the formulas

\begin{align*} \gamma(a\wedge [e, b_1\wedge \cdots\wedge b_i]) (\sigma_1\times\cdots\times \sigma_i) = \gamma_{\mathcal M}(a(\sigma_1\times\cdots\times \sigma_i),b_1(\sigma_1),\ldots, b_i(\sigma_i)) \end{align*}

(where e is the identity element of the relevant symmetric group) and

\begin{align*} \gamma(a\wedge [\alpha, b_1\wedge \ldots\wedge b_i]) = (\alpha,\mathrm{id})_*\gamma((\alpha^{-1},\mathrm{id})_*a\wedge [e, b_1\wedge \ldots\wedge b_i]). \end{align*}

In order to formulate the associativity property of γ, we note that for ${\mathbf X}_1,\ldots,{\mathbf X}_i, {\mathbf Y}_1,\ldots,{\mathbf Y}_i \in \Sigma{{ss}}{\mathcal S} $ there is a natural map

\begin{align*} \chi:({\mathbf X}_1\mathbin{{\overline{\wedge}}} {\mathbf Y}_1)\otimes\cdots\otimes({\mathbf X}_i\mathbin{{\overline{\wedge}}} {\mathbf Y}_i) \to ({\mathbf X}_1\otimes\cdots\otimes {\mathbf X}_i)\mathbin{{\overline{\wedge}}} ({\mathbf Y}_1\otimes\cdots\otimes {\mathbf Y}_i) \end{align*}

given by

\begin{align*} \chi([\alpha,x_1\wedge y_1\wedge \cdots\wedge x_i\wedge y_i]) = [\alpha,x_1\wedge \cdots\wedge x_i]\wedge[\alpha,y_1\wedge\cdots\wedge y_i]. \end{align*}

Lemma 7.2. The operation γ has the following associativity property: the composite

\begin{align*} & {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{11})\otimes\cdots\otimes{\mathcal O}(l_{1j_1}) \bigr) \bigr) \otimes\cdots \\ & \quad \otimes \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{i1})\otimes\cdots\otimes{\mathcal O}(l_{ij_i}) \bigr) \bigr) \Bigr) \\ & \xrightarrow{1\mathbin{{\overline{\wedge}}}(\gamma\otimes\cdots\gamma)} {\mathcal O}(i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{11}+\cdots+l_{1j_1}) \otimes \cdots\otimes {\mathcal O}(l_{i1}+\cdots+l_{ij_i}) \bigr) \\ & \xrightarrow{\gamma} {\mathcal O}(l_{11}+\cdots+l_{ij_i}) \end{align*}

is the same as the composite

\begin{align*} & {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{11})\otimes\cdots\otimes{\mathcal O}(l_{1j_1}) \bigr) \bigr) \otimes\cdots\\ & \quad \otimes \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{i1})\otimes\cdots\otimes{\mathcal O}(l_{ij_i}) \bigr) \bigr) \Bigr) \\ & \xrightarrow{1\mathbin{{\overline{\wedge}}} \chi} {\mathcal O}(i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(j_1)\otimes \cdots\otimes {\mathcal O}(j_i) \bigr) \mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{11})\otimes\cdots\otimes{\mathcal O}(l_{i,j_i}) \bigr) \\ & \xrightarrow{\gamma\mathbin{{\overline{\wedge}}} 1} {\mathcal O}(j_1+\cdots+j_i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(l_{11})\otimes\cdots\otimes{\mathcal O}(l_{i,j_i}) \bigr) \\ & \xrightarrow{\gamma} {\mathcal O}(l_{11}+\cdots+l_{ij_i}). \end{align*}

To formulate the unital property of γ, we first need to consider the unit object for the operation $\mathbin{{\overline{\wedge}}}$.

Remark 7.4. (i) ${\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf X}\cong {\mathbf X}$ for any ${\mathbf X}\in\Sigma{{ss}}{\mathcal S} $.

(ii) ${\mathcal O}(0)$ and ${\mathcal O}(1)$ are both equal to ${\overline{\mathbf{S}}}$.

(iii) ${\overline{\mathbf{S}}}$ is a commutative monoid in $\Sigma{{ss}}{\mathcal S} $ with multiplication

\begin{align*} m:{\overline{\mathbf{S}}}\otimes{\overline{\mathbf{S}}}\to {\overline{\mathbf{S}}} \end{align*}

given by

\begin{align*} m([\alpha,s_1\wedge s_2])=t, \end{align*}

where s ₁ and s ₂ are any nontrivial simplices and t is the non-trivial simplex in the relevant multidegree.

Lemma 7.5. The operation γ has the following unital property: the diagrams

and

commute.

To complete the analogy between γ and the composition map of an operad, we need an equivariance property.

Definition 7.6. Define a right action of $\Sigma_j$ on ${\mathcal O}(j)$ by

\begin{align*} (a\alpha)(\sigma)=a(\sigma)\cdot\alpha, \end{align*}

where $a\in \operatorname{Map}(U(\Delta^{\mathbf n}),\Sigma_j)_+$, $\sigma\in U(\Delta^{\mathbf n})$, and · is multiplication in $\Sigma_j$.

Lemma 7.7. (i) The following diagram commutes for all $\alpha\in \Sigma_i$.

Now we use the data defined so far to construct a monad in the category $\Sigma{{ss}}{\mathcal S} $.

Definition 7.8. (i) For ${\mathbf X}\in\Sigma{{ss}}{\mathcal S} $, give ${\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes j}$ the diagonal right $\Sigma_j$ action.

(ii) Define a functor ${\mathbb O}:\Sigma{{ss}}{\mathcal S} \to\Sigma{{ss}}{\mathcal S} $ by

\begin{align*} {\mathbb O}({\mathbf X})= \bigvee_{j\geq 0} \bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes j} \bigr) / \Sigma_j. \end{align*}

(iii) Define a natural transformation

\begin{align*} \iota:{\mathbf X}\to {\mathbb O} {\mathbf X} \end{align*}

to be the composite

\begin{align*} {\mathbf X}\xrightarrow{\cong} {\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf X} = {\mathcal O}(1)\mathbin{{\overline{\wedge}}} {\mathbf X} \hookrightarrow {\mathbb O}({\mathbf X}). \end{align*}

(iv) Define

\begin{align*} \mu:{\mathbb O}{\mathbb O} {\mathbf X}\to {\mathbb O} {\mathbf X} \end{align*}

to be the natural transformation induced by the maps

\begin{align*} & {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes j_1} \bigr) \otimes\cdots\otimes \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes j_i} \bigr) \Bigr) \\ & \qquad\qquad\qquad \xrightarrow{1\mathbin{{\overline{\wedge}}}\chi} \Bigl({\mathcal O}(i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(j_1)\otimes \cdots\otimes {\mathcal O}(j_i) \bigr) \Bigr) \mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes (j_1+\cdots+j_i)} \\ &\qquad\qquad\qquad\qquad\qquad \xrightarrow{\gamma\mathbin{{\overline{\wedge}}} 1} {\mathcal O}(j_1+\cdots+j_i)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes (j_1+\cdots+j_i)}. \end{align*}

We conclude this section by giving the action of ${\mathbb O}$ on ${\mathbf R}$. Observe that the map

\begin{align*} \phi_j: {\mathcal O}(j) \mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j} \to {\mathbf R} \end{align*}

of definition 6.9 induces a map

(7.1)\begin{align} \bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j} \bigr) / \Sigma_j \to {\mathbf R}. \end{align}

Definition 7.10. Define

\begin{align*} \nu: {\mathbb O} {\mathbf R}\to {\mathbf R} \end{align*}

to be the map whose restriction to $\bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\otimes j} \bigr) / \Sigma_j $ is the map (7.1).

Proof. We need to show that the diagrams

and

commute. The first is obvious and for the second it suffices to check that the composite

\begin{align*} & {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j_1} \bigr) \otimes\cdots\otimes \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j_i} \bigr) \Bigr) \\ & \qquad\qquad\qquad \xrightarrow{1\mathbin{{\overline{\wedge}}}\chi} {\mathcal O}(i)\mathbin{{\overline{\wedge}}} \bigl( {\mathcal O}(j_1)\otimes \cdots\otimes {\mathcal O}(j_i) \bigr) \mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes (j_1+\cdots+j_i)} \\ & \qquad\qquad \qquad\qquad\qquad \xrightarrow{\gamma\mathbin{{\overline{\wedge}}} 1} {\mathcal O}(j_1+\cdots+j_i)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes (j_1+\cdots+j_i)} \xrightarrow{\phi} {\mathbf R} \end{align*}

is the same as the composite

\begin{align*} {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j_1} \bigr) \otimes\cdots\otimes \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes j_i} \bigr) \Bigr) \xrightarrow{1\mathbin{{\overline{\wedge}}}(\phi\otimes\cdots\phi)} {\mathcal O}(i)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\otimes i} \xrightarrow{\phi} {\mathbf R}. \end{align*}

8. A monad in ${\mathcal S} p_{{mss }}$

We begin by giving ${\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}$ the structure of a multisemisimplicial symmetric spectrum when ${\mathbf X}\in {\mathcal S} p_{{mss }}$. The definition is motivated by lemma 6.5. Recall definition 6.4.

Definition 8.1. Let $j,k\geq 0$. Let s be the 1-simplex of S ¹. Define

\begin{align*} \omega:S^1\wedge ({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j})_k \to ({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j})_{k+1} \end{align*}

as follows: for $a\in ({\mathcal O}(j)_k)_{\mathbf n}$ and $x\in ((X^{\wedge j})_k)_{\mathbf n}$, let

\begin{align*} \omega(s\wedge (a\wedge x)) = (a\circ \Pi)\wedge \omega(s\wedge x). \end{align*}

The identification given in definition 4.14 was omitted in this notation.

Definition 8.2. (i) For ${\mathbf X}\in{\mathcal S} p_{{mss }}$, give ${\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}$ the diagonal right $\Sigma_j$ action.

(ii) Define a functor ${\mathbb P}:{\mathcal S} p_{{mss }}\to{\mathcal S} p_{{mss }}$ by

\begin{align*} {\mathbb P}({\mathbf X})= \bigvee_{j\geq 0} \bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j} \bigr) / \Sigma_j. \end{align*}

To give ${\mathbb P}$ a monad structure we need

Lemma 8.3. The composite in definition 7.8(iv) induces a map

\begin{align*} {\mathcal O}(i) \mathbin{{\overline{\wedge}}} \Bigl( \bigl( {\mathcal O}(j_1)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j_1} \bigr) \wedge\cdots\wedge \bigl({\mathcal O}(j_i)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j_i} \bigr) \Bigr) \\ \qquad \qquad \to {\mathcal O}(j_1+\cdots+j_i)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge (j_1+\cdots+j_i)} \end{align*}

in ${\mathcal S} p_{{mss }}$.

Definition 8.4. (i) Define a natural transformation

\begin{align*} \iota:{\mathbf X}\to {\mathbb P} {\mathbf X} \end{align*}

to be the composite

\begin{align*} {\mathbf X}\xrightarrow{\cong} {\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf X} = {\mathcal O}(1)\mathbin{{\overline{\wedge}}} {\mathbf X} \hookrightarrow {\mathbb P}({\mathbf X}). \end{align*}

(ii) Define

\begin{align*} \mu:{\mathbb P}{\mathbb P} {\mathbf X}\to {\mathbb P} {\mathbf X} \end{align*}

to be the natural transformation induced by the maps constructed in lemma 8.3.

Next we give the action of ${\mathbb P}$ on ${\mathbf R}$. By definition 8.1 and lemma 6.5, the map

\begin{align*} \psi_j: {\mathcal O}(j) \mathbin{{\overline{\wedge}}} {\mathbf R}^{\wedge j} \to {\mathbf R} \end{align*}

of lemma 6.10 is a map in ${\mathcal S} p_{{mss }}$. It induces a map

(8.1)\begin{align} \bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf R}^{\wedge j} \bigr) / \Sigma_j \to {\mathbf R} \end{align}

in ${\mathcal S} p_{{mss }}$.

Definition 8.6. Define

\begin{align*} \nu: {\mathbb P} {\mathbf R}\to {\mathbf R} \end{align*}

to be the map whose restriction to $\bigl( {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j} \bigr) / \Sigma_j $ is the map (8.1).

For use in §10, we record a lemma.

Proof. Part (i). Let ${\mathbb A}$ be the monad

\begin{align*} {\mathbb A}({\mathbf X})=\bigvee_{j\geq 0} {\mathbf X}^{\wedge j}. \end{align*}

Then a monoid in ${\mathcal S} p_{{mss }}$ is the same thing as an $\mathbb A$-algebra, so it suffices to give a map of monads from $\mathbb A$ to ${\mathbb P}$.

For each $j,k\geq 0$ and each k-fold multi-index ${\mathbf n}$, define an element

\begin{align*} e_{j,k,{\mathbf n}}\in ({\mathcal O}(j)_k)_{\mathbf n} \end{align*}

to be the constant function $U(\Delta^{\mathbf n})\to\Sigma_j$ whose value is the identity element of $\Sigma_j$. Next define a map

\begin{align*} {\overline{\mathbf{S}}}\to {\mathcal O}(j) \end{align*}

by taking the non-trivial simplex of $({\overline{\mathbf{S}}}_k)_{\mathbf n}$ to $e_{j,k,{\mathbf n}}$.

Now the composite

\begin{align*} {\mathbb A}({\mathbf X})=\bigvee_{j\geq 0} {\mathbf X}^{\wedge j} \cong \bigvee_{j\geq 0} {\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j} \to \bigvee_{j\geq 0} \bigl({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}\bigr)/\Sigma_j ={\mathbb P}({\mathbf X}) \end{align*}

is a map of monads.

Part (ii) is an easy consequence of the definitions.

It is worth mentioning that the map from ${\mathbb A}$ to ${\mathbb P}$ does not factor over the commutative monad because $e_{j,k,n}$ is not a fixed point for the $\Sigma_n$-action.

9. Degreewise smash product and geometric realization

For the proof of theorem 1.1, we need to know the relation between $\mathbin{{\overline{\wedge}}}$ and geometric realization.

There is a natural map

\begin{align*} \kappa:|A\mathbin{{\overline{\wedge}}} B|\to |A|\wedge |B| \end{align*}

defined by

\begin{align*} \kappa([u,x\wedge y])=[u,x]\wedge[u,y]. \end{align*}

The analogous map for multisimplicial sets is a homeomorphism, but the situation for multisemisimplicial sets is more delicate.

Proof. Let $\tilde{A}$ and $\tilde{B}$ be multisimplicial sets whose underlying multisemisimplicial sets are A and B. Then the underlying multisemisimplicial set of the degreewise smash product $\tilde{A}\mathbin{{\overline{\wedge}}}\tilde{B}$ is $A\mathbin{{\overline{\wedge}}} B$. Consider the following commutative diagram, where $|\ |$ in the bottom row denotes realization of multisimplicial sets, $\tilde{\kappa}$ is defined analogously to κ, and the vertical maps collapse the degeneracies:

The map $\tilde{\kappa}$ is a homeomorphism, and the vertical arrows are weak equivalences by the multisimplicial analogue of [Reference Segal24, lemma A.5], so κ is a weak equivalence.

Next we give a sufficient condition for a multisemisimplicial set to have compatible degeneracies. Let Dⁿ denote the semisimplicial set consisting of the nondegenerate simplices of the standard simplicial n-simplex. For a multi-index ${\mathbf n}$, let $D^{\mathbf n}$ denote the k-fold multisemisimplicial set

\begin{align*} D^{n_1}\times \cdots \times D^{n_k}. \end{align*}

The following result is proved in [Reference McClure16].

Our next result is proved in the same way as [Reference Laures and McClure11, lemma 15.12] and does not require the ad theory to be commutative.

10. Rectification

In this section, we complete the proof of theorem 1.1.

First we consider a monad in ${\mathcal S} p_{{mss }}$ which is simpler than ${\mathbb P}$.

Definition 10.1. (i) Define ${\mathbb P}'({\mathbf X})$ to be $\bigvee_{j\geq 0} \, {\mathbf X}^{\wedge j}/\Sigma_j$.

(ii) For each $j\geq 0$, let

\begin{align*} \xi_j:{\mathcal O}(j)\to {\overline{\mathbf{S}}} \end{align*}

be the map which takes each non-trivial simplex of ${\mathcal O}(j)_k$ to the non-trivial simplex of ${\overline{S}}_k$ in the same multidegree. Define a natural transformation

\begin{align*} \Xi: {\mathbb P}\to{\mathbb P}' \end{align*}

to be the wedge of the composites

\begin{align*} \bigl({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}\bigr)/\Sigma_j \xrightarrow{\xi_j\mathbin{{\overline{\wedge}}} 1} \bigl({\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}\bigr)/\Sigma_j \xrightarrow{\cong} {\mathbf X}^{\wedge j}/\Sigma_j. \end{align*}

Proposition 10.2. (i) An algebra over ${\mathbb P}'$ is the same thing as a commutative monoid in ${\mathcal S} p_{{mss }}$.

(ii) Ξ is a map of monads.

(iii) Suppose that each X_k has compatible degeneracies (see definition 9.1). Let ${\mathbb P}^q$ denote the q-th iterate of ${\mathbb P}$. Then each map

\begin{align*} \Xi:{\mathbb P}^q({\mathbf X})\to {\mathbb P}'{\mathbb P}^{q-1}({\mathbf X}) \end{align*}

is a weak equivalence.

Parts (i) and (ii) are immediate from the definitions. Part (iii) will be proved at the end of this section.

Proof of theorem 1.1 We apply the monadic bar construction [Reference Peter May17, construction 9.6] to obtain simplicial objects $B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})$ and $B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})$ in ${\mathcal S} p_{{mss }}$. We write ${\mathbf R}_\bullet$ for the constant simplicial object which is ${\mathbf R}$ in each simplicial degree. There are maps of simplicial ${\mathbb P}$-algebras

(10.1)\begin{align} {\mathbf R}_\bullet \xleftarrow{\varepsilon} B_\bullet({\mathbb P},{\mathbb P},{\mathbf R}) \xrightarrow{\Xi_\bullet} B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R}), \end{align}

where ɛ is induced by the action of ${\mathbb P}$ on ${\mathbf R}$ (see [Reference Peter May17, lemma 9.2(ii)]). The map ɛ is a homotopy equivalence of simplicial objects [Reference Peter May17, Proposition 9.8] and the map $\Xi_\bullet$ is a weak equivalence in each simplicial degree by propositions 9.5, 9.6, and 10.2(iii). $B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})$ is a simplicial algebra over ${\mathbb P}'$, which by proposition 10.2(i) is the same thing as a simplicial commutative monoid in ${\mathcal S} p_{{mss }}$. Moreover, by lemma 8.8(i), ${\mathbf R}_\bullet$ and $B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})$ are simplicial monoids, and ɛ and $\Xi_\bullet$ are maps of simplicial monoids.

The objects of the diagram (10.1) are simplicial objects in ${\mathcal S} p_{{mss }}$. We obtain a diagram

(10.2)\begin{align} |{\mathbf R}_\bullet| \xleftarrow{|\varepsilon|} |B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})| \xrightarrow{|\Xi_\bullet|} |B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})| \end{align}

of simplicial objects in ${\mathcal S} p$ (the category of symmetric spectra) by applying the geometric realization functor ${\mathcal S} p_{{mss }}\to {\mathcal S} p$ to the diagram (10.1) in each simplicial degree. The map $|\varepsilon|$ is a homotopy equivalence of simplicial objects and the map $|\Xi_\bullet|$ is a weak equivalence in each simplicial degree. The object $|B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})|$ is a simplicial commutative symmetric ring spectrum, the objects $|{\mathbf R}_\bullet|$ and $|B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})|$ are simplicial symmetric ring spectra, and the maps $|\varepsilon|$ and $|\Xi_\bullet|$ are maps of simplicial symmetric ring spectra.

Finally, we apply geometric realization to the diagram (10.2). We define ${\mathbf M}^{\mathrm{comm}}$ to be $||B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})||$. Now we have a diagram

(10.3)\begin{align} {\mathbf M}=|{\mathbf R}| \xleftarrow{||\varepsilon||} ||B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})|| \xrightarrow{||\Xi_\bullet||} ||B_\bullet({\mathbb P}',{\mathbb P},{\mathbf R})||={\mathbf M}^{\mathrm{comm}} \end{align}

in ${\mathcal S} p$. The map $||\varepsilon||$ is a homotopy equivalence (cf. [Reference Peter May17, corollary 11.9]) and the map $||\Xi_\bullet||$ is a weak equivalence by [Reference Leonard Reedy12, theorem E]. ${\mathbf M}^{\mathrm{comm}}$ is a commutative symmetric ring spectrum, ${\mathbf M}$ is the symmetric ring spectrum of [Reference Laures and McClure11, theorem 18.5] (by lemma 8.8(ii)), $||B_\bullet({\mathbb P},{\mathbb P},{\mathbf R})||$ is a symmetric ring spectrum, and $||\varepsilon||$ and $||\Xi_\bullet||$ are maps of symmetric ring spectra.

We conclude this section with the proof of part (iii) of proposition 10.2. First we need a lemma (which for later use we state in more generality than we immediately need). Recall that a preorder is a set with a reflexive and transitive relation ≤. Examples are $\Sigma_j$, with every element ≤ every other, and U(K) (see definition 6.2(i)), with ≤ induced by inclusions of cells.

Lemma 10.3. Let P be a preorder with an element which is ≥ all other elements, and let $k\geq 0$. Define a k-fold multisemisimplicial set A by

\begin{align*} A_{\mathbf n}=\operatorname{Map}_{\mathrm{preorder}}(U(\Delta^{\mathbf n}),P). \end{align*}

Then

(i) A has compatible degeneracies, and

(ii) A is weakly equivalent to a point.

Note that if P is $\Sigma_j$ with the preorder described above then $A_+$ is $\tilde{{\mathcal O}}(j)_k$.

Proof of 10.2(iii) We begin with the case q = 1, so we want to show that the map $\Xi:{\mathbb P}({\mathbf X})\to{\mathbb P}'({\mathbf X})$ is a weak equivalence. It suffices to show that the map

\begin{align*} ({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j})/\Sigma_j \to {\mathbf X}^{\wedge j}/\Sigma_j \end{align*}

is a weak equivalence for each j. Proposition A.2 and remark A.3 show that the $\Sigma_j$ actions are free away from the basepoint. We will now employ the following fact: if X is a cellular G-spectrum with a free action away from the base point then the canonical map from $EG_+\wedge_{G}X$ to $X/G$ is a weak equivalence. Moreover, $(EG)_+\wedge_{G}X$ sits in a fibration with base BG and fibre X. Hence, it suffices to show that each map

\begin{align*} {\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j} \to {\mathbf X}^{\wedge j} \end{align*}

is a weak equivalence. Now the object $\bigl({\mathcal O}(j)\mathbin{{\overline{\wedge}}} {\mathbf X}^{\wedge j}\bigr)_k$ comes from

\begin{align*} \bigvee_{k_1+\cdots+k_j=k}\, {\mathcal O}(j)_k\mathbin{{\overline{\wedge}}} I_{\Sigma_{k_1}\times\cdots\times\Sigma_{k_j}}^{\Sigma_k} \bigl( {\mathbf X}_{k_1}\wedge\cdots\wedge {\mathbf X}_{k_j} \bigr) \end{align*}

and we have

\begin{align*} & {\mathcal O}(j)_k\mathbin{{\overline{\wedge}}} I_{\Sigma_{k_1}\times\cdots\times\Sigma_{k_j}}^{\Sigma_k} \bigl( {\mathbf X}_{k_1}\wedge\cdots\wedge {\mathbf X}_{k_j} \bigr)\\ & \quad \cong I_{\Sigma_{k_1}\times\cdots\times\Sigma_{k_j}}^{\Sigma_k} \Bigl({\mathcal O}(j)_k\mathbin{{\overline{\wedge}}} \bigl( {\mathbf X}_{k_1}\wedge\cdots\wedge {\mathbf X}_{k_j} \bigr) \Bigr), \end{align*}

so it suffices by proposition 5.4 to show that each map

\begin{align*} {\mathcal O}(j)_k\mathbin{{\overline{\wedge}}} ({\mathbf X}_{k_1}\wedge\cdots\wedge {\mathbf X}_{k_j}) \to {\mathbf X}_{k_1}\wedge\cdots\wedge {\mathbf X}_{k_j} \end{align*}

is a weak equivalence, and this follows from example 9.2, proposition 9.3, and lemma 10.3.

The general case follows from the case q = 1 and example 9.2(ii).

11. Proof of theorem 1.2

It is well-known that Thom spectra are commutative symmetric ring spectra (see for example [Reference Schlichtkrull23]; we recall this below). In this section, we show that the Thom spectrum $M{\mathrm{STop}}$ obtained from the bar construction is weakly equivalent, in the category of commutative symmetric ring spectra, to the commutative symmetric ring spectrum $({\mathbf M}_{\mathrm{STop}})^{\mathrm{comm}}$ given by theorem 1.1.

Our first task is to construct the following chain of weak equivalences in the category of symmetric spectra

(11.1)\begin{align} {\mathbf M}_{{\mathrm{STop}}} \xleftarrow{f_1} {\mathbf Y} \xrightarrow{f_2} {\mathbf X} \xrightarrow{f_3} M{\mathrm{STop}}. \end{align}

First recall that $M{\mathrm{STop}}$ has as kth space the Thom space $T({\mathrm{STop}}(k))$. The $\Sigma_k$ action on $T({\mathrm{STop}}(k))$ is induced by the conjugation action on ${\mathrm{STop}}(k)$.

For the construction of ${\mathbf X}$, we need some facts about multisimplicial sets. Given a space Z and $k\geq 1$, let $S_\bullet^{k\text{-multi}}(Z)$ be the k-fold multisimplicial set whose simplices in multidegree ${\mathbf n}$ are the maps $\Delta^{\mathbf n}\to Z$. There is a natural map

\begin{align*} |S_\bullet^{k-\mathrm{multi}}(Z)|\to Z \end{align*}

(where $|\ |$ denotes realization of the underlying multisimplicial set) which is a weak equivalence by [Reference Andrade1] and the multisimplicial analogue of [Reference Segal24, lemma A.5]. If Z is a based space, there are natural maps

\begin{align*} \lambda: \Sigma|S_\bullet^{k-\mathrm{multi}}(Z)| \to |S^1\wedge S_\bullet^{k-\mathrm{multi}}(Z)| \end{align*}

and

\begin{align*} \kappa: S^1\wedge S_\bullet^{k-\mathrm{multi}}(Z) \to S_\bullet^{(k+1)-\mathrm{multi}}(\Sigma Z) \end{align*}

defined as follows. Given $t\in[0,1]$, $u\in\Delta^{{\mathbf n}}$, and $g:\Delta^{{\mathbf n}}\to Z$, let $\bar{t}$ denote the image of t under the oriented affine homeomorphism $[0,1]\to \Delta^1$, and define

\begin{align*} \lambda(t\wedge [u,g]) = [(\bar{t},u),s\wedge g], \end{align*}

where s is the non-trivial simplex of S ¹. Define

\begin{align*} \kappa(s\wedge g)(\bar{t},u)=t\wedge g(u). \end{align*}

Then the diagram

commutes.

Now let $X_k=|S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))|$. We define the $\Sigma_k$ action on X_k as follows. For $\alpha\in\Sigma_k$ and $g:\Delta^{\mathbf n}\to T({\mathrm{STop}}(k)))$, let $\alpha({\mathbf n})=(n_{\alpha^{-1}(1)},\ldots,n_{\alpha^{-1}(k)})$ and let $\alpha(g)$ be the composite

\begin{align*} \Delta^{\alpha({\mathbf n})} \xrightarrow{\alpha^{-1}} \Delta^{{\mathbf n}} \xrightarrow{g} T({\mathrm{STop}}(k)) \xrightarrow{\alpha} T({\mathrm{STop}}(k)). \end{align*}

This makes $S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))$ an object of $\Sigma_k{{ss}} {\mathcal S}_k$, and now proposition 5.3 gives the $\Sigma_k$ action on X_k. Next define the structure map

\begin{align*} \Sigma X_k\to X_{k+1} \end{align*}

to be the composite

\begin{multline*} \Sigma|S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))| \xrightarrow{\lambda} |S^1\wedge S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))| \\ \xrightarrow{|\kappa|} |S_\bullet^{(k+1)-\mathrm{multi}}(\Sigma T({\mathrm{STop}}(k)))| \to |S_\bullet^{(k+1)-\mathrm{multi}}(T({\mathrm{STop}}(k+1)))|, \end{multline*}

where the last map is induced by the structure map of $M{\mathrm{STop}}$. Let ${\mathbf X}$ be the symmetric spectrum consisting of the spaces X_k with these structure maps. Define $f_3:{\mathbf X}\to M{\mathrm{STop}}$ to be the sequence of weak equivalences

\begin{align*} |S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))| \to T({\mathrm{STop}}(k)). \end{align*}

The commutativity of diagram (11.2) shows that f ₃ is a map of symmetric spectra.

Next let $S_\bullet^{k-\mathrm{multi,}{\mathord{\pitchfork}}}(T({\mathrm{STop}}(k)))$ be the sub-multisemisimplicial set of $S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))$ consisting of maps whose restrictions to each face of $\Delta^{{\mathbf n}}$ are transverse to the zero section (see [Reference Freedman and Quinn5] for topological transversality). Let ${\mathbf Y}$ be the subspectrum of ${\mathbf X}$ with kth space $|S_\bullet^{k-\mathrm{multi,}{\mathord{\pitchfork}}}(T({\mathrm{STop}}(k)))|$, and let $f_2:{\mathbf Y}\to{\mathbf X}$ be the inclusion.

It remains to construct f ₁. Let $S\subset T({\mathrm{STop}}(k))$ be the zero section. First we observe that, if $g:\Delta^{\mathbf n}\to T({\mathrm{STop}}(k))$ is a map whose restriction to each face is transverse to S, we obtain an element $F\in{\mathrm{ad}}_{\mathrm{STop}}(\Delta^{\mathbf n})$ by letting $F(\sigma,o) $ be $g^{-1}(S)\cap \sigma$ with the orientation determined by o. This construction gives a map

\begin{align*} S_\bullet^{k-\mathrm{multi,}{\mathord{\pitchfork}}}(T({\mathrm{STop}}(k))) \to ({\mathbf R}_{\mathrm{STop}})_k, \end{align*}

in $\Sigma_k{{ss}} {\mathcal S}_k$, and applying geometric realization gives a $\Sigma_k$ equivariant map $Y_k\to ({\mathbf M}_{\mathrm{STop}})_k$; we let f ₁ be the sequence of these maps.

Proof. For a k-fold multisemisimplicial set A, let A ^ʹ be the semisimplicial set whose nth set is $A_{0,\ldots,0,n}$. There is an evident map

\begin{align*} \phi:|A'|\to|A|. \end{align*}

If A is $S_\bullet^{k-\mathrm{multi}}(Z)$, then A ^ʹ is $S_\bullet(Z)$, and if A is R_k then A ^ʹ is the semisimplicial set P_k of [Reference Laures and McClure11, definition 15.4(i)], with realization $({\mathbf Q}_{\mathrm{STop}})_k$ [Reference Laures and McClure11, definitions 15.4(ii) and 15.8]. Now we have a commutative diagram

Here g ₃ is the usual weak equivalence, and g ₂ is a weak equivalence by [Reference Freedman and Quinn5, §9.6] and the definition of homotopy groups [Reference Peter May18, definition 3.6], so ϕ ₂ is a weak equivalence. g ₁ was shown to be a weak equivalence in [Reference Laures and McClure11, Appendix B], and ϕ ₁ was shown to be a weak equivalence in [Reference Laures and McClure11, §15], so f ₁ is a weak equivalence as required.

This completes the construction of diagram (11.1).

Next we recall that $M{\mathrm{STop}}$ is a commutative symmetric ring spectrum with product

\begin{align*} T({\mathrm{STop}}(k))\wedge T({\mathrm{STop}}(l)) \to T({\mathrm{STop}}(k+l)). \end{align*}

${\mathbf X}$ is also a commutative symmetric ring spectrum, with the product

\begin{align*} & |S_\bullet^{k-\mathrm{multi}}(T({\mathrm{STop}}(k)))| \wedge |S_\bullet^{l-\mathrm{multi}}(T({\mathrm{STop}}(l)))| \\ & \to |S_\bullet^{(k+l)-\mathrm{multi}}(T({\mathrm{STop}}(k))\wedge T({\mathrm{STop}}(l)))| \to |S_\bullet^{(k+l)-\mathrm{multi}}(T({\mathrm{STop}}(k+l)))|, \end{align*}

and ${\mathbf Y}$ is a commutative symmetric ring spectrum with the product it inherits from ${\mathbf X}$. The maps f ₂ and f ₃ are maps of symmetric ring spectra, so to complete the proof of theorem 1.2, it suffices to show

The proof of lemma 11.3 is outsourced to Appendix B. The results of Appendix B use material from §17 and 18.

12. Relaxed symmetric Poincaré complexes

For a commutative ring R with the trivial involution, we would like to apply theorem 1.1 to obtain a commutative model for the symmetric L-spectrum of R. However, the ad theory ${\mathrm{ad}}^R$ defined in §9 of [Reference Laures and McClure11], with the product defined in [Reference Laures and McClure11, definition 9.12], is not commutative. The difficulty is that this product is defined using a noncommutative coproduct

\begin{align*} \Delta:W\to W\otimes W \end{align*}

for the standard resolution W of ${\mathbb Z}$ by ${\mathbb Z} [{\mathbb Z} /2]$-modules. In this section and the next, we give an equivalent ad theory which is commutative.

Fix a ring R with involution. For a complex C of left R-modules, let C^t be the complex of right R-modules obtained from C by applying the involution of R. As usual, give $C^t\otimes_RC$ the ${\mathbb Z}/2$-action which switches the factors. Write ${\mathcal C}h_{hf}$ for the category of homotopy finite chain complexes as in [Reference Laures and McClure11, definition 9.2(v)].

${\mathcal A}_{\text{rel}}^R$ is a ${\mathbb Z}$-graded category, where d was defined above, i takes $(C,D,\beta, \varphi)$ to $(C,D,\beta, -\varphi)$ and $\emptyset_n$ is the n-dimensional object for which C and D are zero in all degrees. Since the set of morphisms between objects of different dimensions is independent of the chains φ the category ${\mathcal A}_{\text{rel}}^R$ is a balanced in the obvious way [Reference Laures and McClure11, definition 5.1].

Remark 12.5. The construction of example 12.2(i) gives a morphism

\begin{align*} {\mathcal A}^R\to {\mathcal A}_{\text{rel}}^R \end{align*}

of ${\mathbb Z}$-graded categories.

Next we must say what the K-ads with values in ${\mathcal A}_{\text{rel}}^R$ are. We need some preliminary definitions and a lemma. For a balanced pre K-ad F we will use the notation

\begin{align*} F(\sigma,o)=(C_\sigma,D_\sigma,\beta_\sigma,\varphi_{\sigma,o}). \end{align*}

Recall [Reference Laures and McClure11, definition 9.7].

Next recall [Reference Laures and McClure11, definition 12.2].

Lemma 12.7. Let F be a well-behaved pre K-ad. Then

(i) $C^t\otimes_R C$ is well-behaved, and

(ii) the map

\begin{align*} (\beta_\sigma)_*: H_*((C^t\otimes_R C)_\sigma/(C^t\otimes_R C)_{\partial\sigma}) \to H_*(D_\sigma/D_{\partial\sigma}) \end{align*}

is an isomorphism for each σ.

Proof. For part (i), first recall (by [Reference Laures and McClure11, definitions 9.7(b) and 9.6(ii)]) that C takes morphism to cofibrations, that is, split monomorphisms in each degree. Moreover, the canonical map from $C_{\partial \sigma}$ to C_σ is a cofibration for each σ. Let us fix a cell σ of K for the rest of the proof. For each degree n, we will construct a set S_τ and a basis $(b_s)_{s\in S_\tau}$ of C_τ by induction for all $\tau \subset \sigma$ in a functorial way. We will omit the degree from the notation for this part. For points τ, we simply choose a basis. These add up to a basis of $C_{\partial \rho}$ for 1-cells ρ. Given a cell τ of dimension k, we may suppose that we have already constructed a basis of $C_{\partial \tau}$. Since $C_{\partial \tau}\rightarrow C_\tau $ is split injective and all modules, including the quotient module, are free we may extend this basis to a basis of C_τ. This way, we constructed a basis of $C_{\partial \rho}$ for all ρ of dimension k + 1 as well: the set of all b_s with $s\in \bigcup_{\tau \subsetneq \rho} S_\tau$ gives a basis because the free functor commutes with colimits. We have to show that the map

\begin{align*} (C^t\otimes_RC)_{\partial \sigma}= \text{colim}_{\tau \subsetneq \sigma} C^t_\tau \otimes_R C_\tau \longrightarrow C_\sigma^t \otimes C_\sigma \end{align*}

is a cofibration. A basis of the target in degree n is indexed by the union $\bigcup_{p+q=n}S_{\sigma,p }\times S_{\sigma,q}$ where we now have to take care of the different degrees in the notation. A basis for the source is the subset given by the union of all pairs coming from $S_{\tau,p} \times S_{\tau,q}$ for $\tau \subsetneq \sigma$. Since the map is induced by the free functor it is split injective.

For part (ii), first observe that the fact that $C^t\otimes_R C$ and D are well-behaved implies that they are Reedy cofibrant [Reference Hirschhorn8, definition 15.3.3(2)]. The colim that defines $(C^t\otimes_R C)_{\partial\sigma}$ is a hocolim by [Reference Hirschhorn8, theorem 19.9.1(1) and proposition 15.10.2(2)], and similarly for $D_{\partial\sigma}$, so the map

\begin{align*} (\beta_\sigma)_*:H_*((C^t\otimes_R C)_{\partial\sigma})\to H_*(D_{\partial\sigma}) \end{align*}

is an isomorphism by [Reference Hirschhorn8, theorem 19.4.2(1)], and this implies the lemma.

Recall [Reference Laures and McClure11, example 3.12 ].

Note that if F is closed then $\varphi_{\sigma,o}$ represents an element $[\varphi_{\sigma,o}]\in H_*(D_\sigma/D_{\partial\sigma})$.

Notation 12.9.

For a balanced pre K-ad F and a cell σ of K, let

\begin{align*} j_\sigma: (C^t\otimes_R C)_\sigma/(C^t\otimes_R C)_{\partial\sigma} \to (C_\sigma/C_{\partial\sigma})^t\otimes_R C_\sigma \end{align*}

be the map that takes $[c \otimes c']$ to $[c]\otimes c'$.

The next definition is a little more complicated than the corresponding definition in [Reference Laures and McClure11, §9] in order to satisfy the extra condition in definition 3.3.

Definition 12.10. (i) A balanced K-ad is a pre K-ad F with the following properties:

(a) it is balanced, well-behaved, and closed, and

(b) for each σ the slant product with $(j_\sigma)_*(\beta_\sigma)_*^{-1}([\varphi_{\sigma,o}])$ is an isomorphism

\begin{align*} H^*(\operatorname{Hom}_R(C_\sigma,R)) \to H_{\dim \sigma-\deg F-*}(C_\sigma/C_{\partial\sigma}). \end{align*}

(ii)A K-ad is a pre K-ad which is naturally quasi-isomorphic to a balanced K-ad.

We mention that, by the definition of an ad theory, a (K, L)-ad is just a (K, L)-pread which is a K-ad.

Write ${\mathrm{ad}}_{\text{rel}}^R$ for the set of K ads with values in ${\mathcal A}_{\text{rel}}^R$.

For the proof of theorem 12.12, we need a product operation.

Definition 12.13. (i) For $i=1,2$, let R_i be a ring with involution and let $(C^i,D^i,\beta^i,\varphi^i)$ be an object of ${\mathcal A}_{\text{rel}}^R$. Define

\begin{align*} (C^1,D^1,\beta^1,\varphi^1)\otimes (C^2,D^2,\beta^2,\varphi^2) \end{align*}

to be the following object of ${\mathcal A}_{\text{rel}}^{R_1\otimes R_2}$:

\begin{align*} (C^1\otimes C^2,D^1\otimes D^2, \gamma, \varphi^1\otimes\varphi^2), \end{align*}

where γ is the composite

\begin{align*} (C^1\otimes C^2)^t\otimes_{R_1\otimes R_2}\,(C^1\otimes C^2) \cong ((C^1)^t\otimes_{R_1} C^1) \otimes ((C^2)^t\otimes_{R_2} C^2) \\ \xrightarrow{\beta^1\otimes \beta^2} D^1\otimes D^2. \end{align*}

(ii) For $i=1,2$, suppose given a ball complex K_i and a pre K_i-ad F_i of degree k_i with values in ${\mathcal A}_{\text{rel}}^{R_i}$. Define a pre $(K_1\times K_2)$-ad $F_1\otimes F_2$ with values in ${\mathcal A}_{\text{rel}}^{R_1\otimes R_2}$ by

\begin{align*} (F_1\otimes F_2)(\sigma\times\tau,o_1\times o_2) = i^{k_2\dim \sigma} F_1(\sigma,o_1)\otimes F_2(\tau,o_2). \end{align*}

Proof of 12.12 We only need to verify parts (d), (f), and (g) of [Reference Laures and McClure11, definition 3.10].

For part (d), we have to show that a pre K-ad is a K-ad if it restricts to a σ-ad for each closed cell σ of K. It suffices to consider the case of a K-pread F with $K=L\sqcup_{\partial \sigma } \sigma$ whose restriction to L is naturally quasi-isomorphic to a balanced L-ad G and to a balanced σ-ad H on σ. We have to give a quasi-isomorphism of F to a balanced K-ad I. In the following, we will concentrate on the first complex in the datum of a pread and we will use the same letter for this complex. The second complex and all other entries will be clear then. Define the restriction of I to L be G. It remains to define I_σ, a map to F_σ and the maps from its lower dimensional cells. For each $\tau\subset \partial \sigma$, we are given a quasi-isomorphism g_τ from G_τ to F_τ and an h_τ from H_τ to F_τ. Since G_τ and H_τ are cofibrant we find a map $f_\tau:G_\tau\rightarrow H_\tau$ such that $h_\tau f_\tau$ is homotopic to g_τ. This only uses the fact that isomorphisms in the homotopy category between cofibrant objects can be represented by chain maps up to homotopy. Similarly, using the fact that the restriction of G to $\partial \sigma$ is balanced we find a map $f_{\partial \sigma}:G_{\partial \sigma} \rightarrow H_{\partial \sigma}$ whose restriction to $\tau\subset \partial \sigma$ is homotopic to $H_{\tau \subset \partial \sigma}f_\tau$. We also find a system of compatible homotopies, that is, a homotopy between $h_{\partial \sigma} f_{\partial \sigma}$ and $g_{\partial \sigma}$. Let I_σ be the mapping cylinder of

\begin{align*}H_{\partial \sigma \subset \sigma}f_{\partial \sigma} : G_{\partial\sigma } \longrightarrow H_\sigma.\end{align*}

Then the constructed homotopy and the map h_σ complete the required quasi-isomorphism $I_\sigma\rightarrow F_\sigma$. The other maps are obvious.

For part (f), let F be a K ^ʹ-ad. We may assume it is balanced. Let

\begin{align*} F(\sigma,o)=(C_\sigma,D_\sigma,\beta_\sigma,\varphi_{\sigma,o}). \end{align*}

We need to define a K-ad E which agrees with F on each residual subcomplex of K. As in the proof of [Reference Laures and McClure11, theorem 6.5], we may assume by induction that K is a ball complex structure for the n disk with one n cell τ, and that K ^ʹ is a subdivision of K which agrees with K on the boundary. We only need to define E on the top cell τ of K. We define $E(\tau,o)$ to be $(C_\tau, D_\tau, \beta_\tau, \varphi_{\tau,o})$, where

• • $C_\tau= \text{colim}_{\sigma\in K'} C_\sigma$,

• • $D_\tau= \text{colim}_{\sigma\in K'} D_\sigma$,

• • $\beta= \text{colim}_{\sigma\in K'} \beta_\sigma$, and

• • $ \varphi_{\tau,o}=\sum \varphi_{\sigma,o'} $, where $(\sigma,o')$ runs through the n-dimensional cells of K ^ʹ with orientations induced by o.

The fact that E satisfies part (a) of definition 12.10 is a consequence of [Reference Laures and McClure11, proposition A.1(ii)]. We will deduce the isomorphism in part (b) of definition 12.10 from [Reference Laures and McClure11, proposition 12.4], and for this we need some facts from [Reference Laures and McClure11, §12].

First recall that for a well-behaved functor

\begin{align*} B:{\mathcal C}ell^\flat(K')\to{\mathcal C}h_{hf}, \end{align*}

we write

\begin{align*} \operatorname{Nat}(\mathrm{cl},B) \end{align*}

for the chain complex of natural transformations of graded abelian groups; the differential is given by

\begin{align*} \partial(\nu)=\partial\circ \nu-(-1)^{|\nu|}\nu\circ \partial. \end{align*}

Recall [Reference Laures and McClure11, definition 12.3] and also the map Φ defined just before the statement of [Reference Laures and McClure11, lemma 12.6]. Consider the diagram

The horizontal maps are isomorphisms by [Reference Laures and McClure11, lemma 12.6], and the right-hand vertical map is an isomorphism by the proof of lemma 12.7(ii). Hence the left-hand vertical map is an isomorphism.

The collection $\{\varphi_{\sigma,o}\}$ gives a cycle ν in $\operatorname{Nat}(\mathrm{cl},D)$. Let $\mu\in \operatorname{Nat}(\mathrm{cl},C^t\otimes_R C)$ be a representative for $\beta^{-1}([\nu])$. Now fix an orientation o for τ. Let $\psi\in C^t_\tau\otimes_R C_\tau$ be $\sum \mu(\langle\sigma,o' \rangle)$, where $(\sigma,o')$ runs through the n-dimensional cells of K ^ʹ with orientations induced by o. Then ψ is a representative of $(j_\sigma)_*\beta^{-1}([\varphi_{\tau,o}])$, so it suffices to show that the cap product with ψ is an isomorphism $H^*(\operatorname{Hom}_R(C_\tau,R)) \to H_{n-\deg F-*}(C_\tau/C_{\partial\tau})$, and this follows from [Reference Laures and McClure11, proposition 12.4].

It remains to verify part (g) of [Reference Laures and McClure11, definition 3.10]. Let $0,1,\iota$ denote the three cells of the unit interval I, with their standard orientations. As in the proof of [Reference Laures and McClure11, theorem 9.11], it suffices to construct a relaxed symmetric Poincaré I-ad H over ${\mathbb Z}$ which takes both 0 and 1 to the object $({\mathbb Z},{\mathbb Z},\gamma,1)$, where γ is the isomorphism ${\mathbb Z}\otimes {\mathbb Z}\to {\mathbb Z}$. The proof of [Reference Laures and McClure11, theorem 9.11] gives a symmetric Poincaré I-ad G with $G(0)=G(1)=({\mathbb Z},\epsilon)$, where $\epsilon:W\to {\mathbb Z}\otimes {\mathbb Z}$ is the composite of the augmentation with γ ⁻¹. Let us denote the object $G(\iota)$ by $(C,\varphi)$. Applying remark 12.11 to G gives a relaxed symmetric Poincaré I-ad G ^ʹ with $G'(\iota)=(C,(C\otimes C)^W,\beta,\varphi)$, where β is induced by the augmentation. Let e ₀ (resp., e ₁) be the inclusion $0\hookrightarrow\iota$ (resp., $1\hookrightarrow \iota$) and for $i=0,1$ let $g_i=G(e_i):{\mathbb Z}\to C$. Then

\begin{align*} \partial\varphi= (g_1\otimes g_1)\circ \epsilon-(g_0\otimes g_0)\circ\epsilon \end{align*}

because G is closed. We can therefore construct the required I-ad H from G ^ʹ by replacing $G'(0)$ and $G'(1)$ by $({\mathbb Z},{\mathbb Z},\gamma,1)$.

13. Equivalence of the spectra associated with ${\mathrm{ad}}^R$ and ${\mathrm{ad}}_{\text{rel}}^R$

By remark 12.11, the morphism

\begin{align*} {\mathcal A}^R\to {\mathcal A}_{\text{rel}}^R \end{align*}

of remark 12.5 induces a map of spectra

\begin{align*} {\mathbf Q}^R\to {\mathbf Q}_{\text{rel}}^R \end{align*}

(see [Reference Laures and McClure11, §15]) and a map of symmetric spectra

\begin{align*} {\mathbf M}^R\to {\mathbf M}_{\text{rel}}^R \end{align*}

(see [Reference Laures and McClure11, §17]).

Theorem 13.1 The maps

\begin{align*} {\mathbf Q}^R\to {\mathbf Q}_{\mathrm{rel}}^R \end{align*}

and

\begin{align*} {\mathbf M}^R\to {\mathbf M}_{\mathrm{rel}}^R \end{align*}

are weak equivalences.

Recall [Reference Laures and McClure11, definitions 4.1 and 4.2]. By [Reference Laures and McClure11, theorem 16.1, remark 14.2(i), and corollary 17.9(iii)], theorem 13.1 follows from

Proposition 13.3. The map of bordism groups

\begin{align*} \Omega^R_*\to(\Omega_{\mathrm{rel}}^R)_* \end{align*}

is an isomorphism.

The proof of proposition 13.3 will occupy the rest of this section. The following lemma proves surjectivity. As usual, for a chain complex A with a ${\mathbb Z}/2$ action, we write $A^{h{\mathbb Z}/2}$ for $(A^W)^{{\mathbb Z}/2}$. The augmentation induces a map $A^{{\mathbb Z}/2}\to A^{h{\mathbb Z}/2}$.

Lemma 13.4. Let

\begin{align*} (C,D,\beta,\varphi) \end{align*}

be a relaxed symmetric Poincaré $*$-ad and let

\begin{align*} \psi\in (C^t\otimes_R C)^{h{\mathbb Z}/2} \end{align*}

represent the image of φ under the map

\begin{align*} H_*(D^{{\mathbb Z}/2})\to H_*(D^{h{\mathbb Z}/2}) \xleftarrow{\cong} H_*((C^t\otimes_R C)^{h{\mathbb Z}/2}) \end{align*}

(where the isomorphism is induced by β). Then $(C,\psi)$ is a symmetric Poincaré $*$-ad, and $(C,D,\beta,\varphi)$ is bordant to

\begin{align*} (C,(C^t\otimes_R C)^W,\gamma,\psi), \end{align*}

where γ is induced by the augmentation.

For the proof, we need another lemma.

Lemma 13.5. Let $(C,D,\beta,\varphi)$ be a relaxed symmetric Poincaré $*$-ad.

(i) If $\psi\in D^{{\mathbb Z}/2}$ is any representative for the homology class $[\varphi]\in H_*(D^{{\mathbb Z}/2})$ then $(C,D,\beta,\psi)$ is bordant to $(C,D,\beta,\varphi)$.

(ii) If

\begin{align*} (f,g):(C,D,\beta,\varphi)\to (C',D',\beta',\varphi') \end{align*}

is a map of $*$-ads of the same dimension for which f (and hence also g) is a quasi-isomorphism then $(C,D,\beta,\varphi)$ and $(C',D',\beta',\varphi')$ are bordant.

The proof of lemma 13.5 is deferred to the end of the section.

Proof of lemma 13.4 The fact that $(C,\psi)$ satisfies [Reference Laures and McClure11, definition 9.9] (only part (b) is relevant) is immediate from definition 12.10(b).

To see that $(C,D,\beta,\varphi)$ and $(C,(C^t\otimes_R C)^W,\gamma,\psi)$ are bordant, let δ denote the composite

\begin{align*} C^t\otimes_R C\xrightarrow{\beta} D\to D^W, \end{align*}

let $\omega\in D^{h{\mathbb Z}/2}$ be the image of φ, and let $\omega'\in D^{h{\mathbb Z}/2}$ be the image of ψ under the map $(C\otimes C)^{h{\mathbb Z}/2}\to D^{h{\mathbb Z}/2}$ induced by β. Part (ii) of lemma 13.5 shows that $ (C,D,\beta,\varphi)$ and $(C,D^W,\delta,\omega)$ are bordant, and also that $(C,(C^t\otimes_R C)^W,\gamma,\psi)$ and $(C,D^W,\delta,\omega')$ are bordant. But $[\omega]=[\omega']$ in $H_*(D^{h{\mathbb Z}/2})$, so the result follows from part (i) of lemma 13.5.

Next we show that the map in proposition 13.3 is 1-1. So let $(C_0,\varphi_0)$ and $(C_1,\varphi_1)$ be symmetric Poincaré $*$-ads and let F be a relaxed symmetric Poincaré bordism between them. Let $0,1,\iota$ denote the three cells of the unit interval I, with their standard orientations. Denote the object $F(\iota)$ by $(C,D,\beta,\varphi)$. It suffices to show that there is a symmetric Poincaré I-ad G with

(13.1)\begin{align} G(0)=(C_0,\varphi_0),\quad G(1)=(C_1,\varphi_1),\quad G(\iota)=(C,\chi) \end{align}

for an element χ which we will now construct.

φ represents an element

\begin{align*} [\varphi]\in H_*(D^{{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}). \end{align*}

The map $D\to D^W$ induced by the augmentation gives a map

\begin{align*} & H_*(D^{{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ & \qquad\qquad \to H_*(D^{h{\mathbb Z}/2}, ((C_0^t \otimes_R C_0)^W)^{h{\mathbb Z}/2} \oplus ((C_1^t \otimes_R C_1)^W)^{h{\mathbb Z}/2}); \end{align*}

let x be the image of φ under this map. The map $\beta:C^t \otimes_R C\to D$ gives an isomorphism

\begin{align*} & (\beta^{h{\mathbb Z}/2})_*:H_*((C^t \otimes_R C)^{h{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ & \qquad\qquad \to H_*(D^{h{\mathbb Z}/2}, ((C_0^t \otimes_R C_0)^W)^{h{\mathbb Z}/2} \oplus ((C_1^t \otimes_R C_1)^W)^{h{\mathbb Z}/2}); \end{align*}

let $y=(\beta^{h{\mathbb Z}/2})_*^{-1}(x)$.

Lemma 13.6. The image of y under the boundary map

\begin{align*} H_*((C^t \otimes_R C)^{h{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ \qquad\qquad \xrightarrow{\partial} H_{*-1}((C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \end{align*}

is $-[\varphi_0]+[\varphi_1]$.

Before proving this we conclude the proof of proposition 13.3. The lemma implies that there is a representative χ of y with

(13.2)\begin{align} \partial\chi=-\varphi_0+\varphi_1. \end{align}

It suffices to show that, with this choice of χ, the symmetric Poincaré pre I-ad G given by Eq. (13.1) is an ad. Equation (13.2) says that G is closed, and part (b) of [Reference Laures and McClure11, definition 9.9] follows from definition 12.10(b) and the fact that the image of $[\chi]$ under the map

\begin{align*} & H_*((C^t \otimes_R C)^{h{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ & \qquad\qquad \to H_*((C^t \otimes_R C)^W,(C_0^t \otimes_R C_0)^W\oplus (C_1^t \otimes_R C_1)^W) \\ & \qquad\qquad \qquad \xleftarrow{\cong} H_*(C^t \otimes_R C,(C_0^t \otimes_R C_0)\oplus (C_1^t \otimes_R C_1)) \end{align*}

is the same as the image of $[\varphi]$ under the map

\begin{align*} & H_*(D^{{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ & \qquad\qquad \to H_*(D,(C_0^t \otimes_R C_0)^W\oplus (C_1^t \otimes_R C_1)^W) \\ & \qquad\qquad \qquad \xrightarrow{\beta_*^{-1}} H_*(C^t \otimes_R C,(C_0^t \otimes_R C_0)\oplus (C_1^t \otimes_R C_1)). \end{align*}

Proof of lemma 13.6 We know that the image of $[\varphi]$ under the boundary map

\begin{align*} H_*(D^{{\mathbb Z}/2},(C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \\ \qquad\qquad \xrightarrow{\partial} H_{*-1}((C_0^t \otimes_R C_0)^{h{\mathbb Z}/2}\oplus (C_1^t \otimes_R C_1)^{h{\mathbb Z}/2}) \end{align*}

is $-[\varphi_0]+[\varphi_1]$, so it suffices to show that for $i=0,1$ the maps

\begin{align*} (C_i^t \otimes_R C_i)^{h{\mathbb Z}/2} \to ((C_i^t \otimes_R C_i)^W)^{h{\mathbb Z}/2} \end{align*}

induced by $D\to D^W$ and by β give the same map in homology. If we think of these as maps

\begin{align*} a_i,b_i:((C_i^t \otimes_R C_i)^W)^{{\mathbb Z}/2} \to ((C_i^t \otimes_R C_i)^{W\otimes W})^{{\mathbb Z}/2} \end{align*}

(with diagonal ${\mathbb Z}/2$ action on $W\otimes W)$ then a_i and b_i are induced by the maps

\begin{align*} e_1,e_2:W\otimes W\to W \end{align*}

given by the augmentations on the two factors. Now the ${\mathbb Z}/2$ equivariant map

\begin{align*} \Delta:W\to W\otimes W \end{align*}

of [Reference Ranicki20, p. 175] has the property that $e_1\circ \Delta$ and $e_2\circ \Delta$ are both the identity map, so if

\begin{align*} d:((C_i^t \otimes_R C_i)^{W\otimes W})^{{\mathbb Z}/2} \to ((C_i^t \otimes_R C_i)^W)^{{\mathbb Z}/2} \end{align*}

is the map induced by Δ then $d\circ a_i$ and $d \circ b_i$ are both the identity map. But Δ is a ${\mathbb Z}/2$ chain homotopy equivalence, so d is a homology isomorphism and it follows that a_i and b_i induce the same map in homology as required.

It remains to prove lemma 13.5. Let F be the cylinder of $(C,D,\beta,\varphi)$ (which was constructed in the last paragraph of the proof of theorem 12.12). Then F(0) and F(1) are both $(C,D,\beta,\varphi)$. Write

\begin{align*} F(\iota)=(C_\iota,D_\iota,\beta_\iota,\varphi_\iota) \end{align*}

and let

\begin{align*} (h,k):(C,D,\beta,\varphi) \to (C_\iota,D_\iota,\beta_\iota,\varphi_\iota) \end{align*}

be the map $F(1)\to F(\iota)$.

For part (i), the hypothesis gives an element $\rho\in D^{{\mathbb Z}/2}$ with $\partial\rho=\psi-\varphi$. Let $\rho'\in D_\iota^{{\mathbb Z}/2}$ be the image of ρ under $k:D\to D_\iota$. Define an I-ad G by

\begin{align*} G(0)=(C,D,\beta,\varphi),\quad G(1)=(C,D,\beta,\psi),\quad G(\iota)=(C_\iota,D_\iota,\beta_\iota,\varphi_\iota+\rho'). \end{align*}

Then G is the desired bordism.

For part (ii), we first show that $(C,D,\beta,\varphi)$ is bordant to $(C,D',\beta_1,\varphi')$, where β ₁ is the composite

\begin{align*} C^t \otimes_R C\xrightarrow{\beta} D\xrightarrow{g} D' \end{align*}

The idea is to construct a suitable mapping cylinder. Let D ₁ be the pushout of the diagram

Let φ ₁ be the image of φ_ι in D ₁ and let β ₂ be the composite

\begin{align*} C_\iota^t\otimes_R C_\iota\to D_\iota\to D_1. \end{align*}

Define an I-ad H by

\begin{align*} H(0)=(C,D,\beta,\varphi),\quad H(1)=(C,D',\beta_1,\varphi'),\quad H(\iota)=(C_\iota,D_1,\beta_2,\varphi_1). \end{align*}

Then H is the desired bordism.

To conclude the proof we show that $(C,D',\beta_1,\varphi')$ is bordant to $(C',D',\beta',\varphi')$. Let C ₁ be the pushout of the diagram

Let $(C'_\iota,D'_\iota,\beta'_\iota,\varphi'_\iota)$ be the cylinder of $(C',D',\beta',\varphi')$. Let β ₃ be the map

\begin{align*} C_1^t\otimes_R C_1 \to D'_\iota. \end{align*}

Define an I-ad H ₁ by

\begin{align*} H_1(0)=(C,D',\beta_1,\varphi'),\quad H_1(1)=(C',D',\beta',\varphi'),\quad H_1(\iota)=(C_1,D'_\iota,\beta_3,\varphi'_\iota). \end{align*}

Then H ₁ is the desired bordism.

14. The symmetric signature revisited

Fix a group π, a simply connected free π-space Z, and a homomorphism $w:\pi\to \{\pm 1\}$, and recall the symmetric spectrum ${\mathbf M}_{\pi, Z,w}$ [Reference Laures and McClure11, §7 and 17] which represents w-twisted Poincaré bordism over $Z/\pi$.

Let R denote the group ring ${\mathbb Z}[\pi]$ with the w-twisted involution [Reference Ranicki21, p. 196].

We now restrict our attention to strict ball complexes. Recall from §2 that a ball complex is strict if each component of the intersection of two cells is a single cell. In [Reference Laures and McClure11, §10], we gave a functor

\begin{align*} {\mathrm{sig}}:{\mathcal A}_{\pi,Z,w}\to {\mathcal A}^R \end{align*}

which induces a natural transformation

\begin{align*} {\mathrm{sig}}: {\mathrm{ad}}_{\pi,Z,w}(K) \to {\mathrm{ad}}^R(K) \end{align*}

for strict ball complexes K.

Remark 14.1. (i) The restriction to strict ball complexes was not mentioned in [Reference Laures and McClure11] but is necessary, because if K has a cell τ whose boundary is not strict and if F is a K-ad $(X_\sigma, f_\sigma,\xi_{\sigma,o})$ then the map

(14.1)\begin{align} \text{colim}_{\sigma\subset\partial\tau} S_*(X_\sigma) \to S_*(X_{\partial\tau}) \end{align}

is not a monomorphism (because simplices with support in $\sigma\cap\sigma'$ but not in a cell of $\sigma\cap\sigma'$ will have two representatives in the colimit), and hence ${\mathrm{sig}}\circ F$ is not well-behaved.

(ii) If K is strict then the map (14.1) has a left inverse for all τ (because its image is the subcomplex of $S_*(X_\tau)$ generated by the simplices that land in some X_σ with $\sigma\subset \partial\tau$, and the left inverse takes each such simplex to a representative for it in the colimit system; all such representatives are identified because K is strict). Hence the map (14.1) is the inclusion of a direct summand, as required for ${\mathrm{sig}}\circ F$ to be well-behaved.

(iii) The restriction to strict ball complexes does not affect the results about the symmetric signature in [Reference Laures and McClure11] because the only ball complexes that occur in [Reference Laures and McClure11, §15, 17–19] are products of simplices, and these are strict.

In this section, we give a functor

\begin{align*} {\mathrm{sig}_{\mathrm{rel}}}:{\mathcal A}_{\pi,Z,w}\to{\mathcal A}_{\text{rel}}^R \end{align*}

which induces a natural transformation

\begin{align*} {\mathrm{sig}_{\mathrm{rel}}}:{\mathrm{ad}}_{\pi,Z,w}(K) \to {\mathrm{ad}}_{\text{rel}}^R(K) \end{align*}

for strict ball complexes K.

Let $(X,f,\xi)$ be an object of ${\mathcal A}_{\pi,Z,w}$ [Reference Laures and McClure11, definition 7.3].

In the special case where π is the trivial group and Z is a point, the definition is easy:

\begin{align*} {\mathrm{sig}_{\mathrm{rel}}}(X,f,\xi)=(S_*X,S_*(X\times X),\beta,\varphi), \end{align*}

where β is the cross product $S_*X\otimes S_*X\xrightarrow{\times} S_*(X\times X)$ and φ is the image of ξ under the diagonal map.

The definition in the general case is similar. Recall that we write $\widetilde{X}$ for the pullback of Z along f and ${\mathbb Z}^w$ for ${\mathbb Z}$ with the right R action determined by w. Also recall [Reference Laures and McClure11, definition 7.1].

Definition 14.2. (i) Give $S_*(\widetilde{X})$ the left R module structure determined by the action of π on $\widetilde{X}$ and give $S_*(\widetilde{X}\times \widetilde{X})$ and $S_*(\widetilde{X})\otimes S_*(\widetilde{X})$ the left R module structures determined by the diagonal actions of π.

(ii) Define

\begin{align*} {\mathrm{sig}_{\mathrm{rel}}}(X,f,\xi)= (S_*(\widetilde{X}), {\mathbb Z}^w\otimes_R S_*(\widetilde{X}\times \widetilde{X}), \beta, \varphi), \end{align*}

where β is the composite

\begin{align*} S_*(\widetilde{X})^t\otimes_R S_*(\widetilde{X}) \cong {\mathbb Z}^w\otimes_R (S_*(\widetilde{X})\otimes S_*(\widetilde{X})) \xrightarrow{1\otimes \times} {\mathbb Z}^w\otimes_R S_*(\widetilde{X} \times \widetilde{X} ) \end{align*}

and φ is the image of ξ under the map

\begin{align*} S_*(X,{\mathbb Z}^f) = {\mathbb Z}^w \otimes_R S_*(\widetilde{X}) \to {\mathbb Z}^w \otimes_R S_*(\widetilde{X}\times \widetilde{X}) \end{align*}

(where the unmarked arrow is induced by the diagonal map).

Next we compare ${\mathrm{sig}}$ to ${\mathrm{sig}_{\mathrm{rel}}}$.

Let us denote by δ both the map ${\mathcal A}^R\to {\mathcal A}_{\text{rel}}^R$ of remark 12.5 and the map ${\mathbf M}^R\to {\mathbf M}_{\text{rel}}^R$ which it induces.

Proposition 14.4. The diagram

commutes in the homotopy category of symmetric spectra.

We will derive this from a more general result. Recall definition 12.4.

Proof of proposition 14.4 The extended Eilenberg–Zilber map

\begin{align*} W\otimes S_*(Y\times Z) \to S_*(Y)\otimes S_*(Z) \end{align*}

[Reference Friedman and McClure6, proof of proposition 5.8] gives a map

\begin{align*} S_*(Y\times Z) \to ((S_*(Y)\otimes S_*(Z))^W, \end{align*}

and this gives a natural quasi-isomorphism

\begin{align*} \nu: {\mathrm{sig}_{\mathrm{rel}}}\to\delta\circ{\mathrm{sig}}. \end{align*}

The rest of this section is devoted to the proof of proposition 14.5. The basic idea is similar to the proof of theorem 1.1.

Recall definition 6.2(i) and note that U(K) has a poset structure given by inclusions of cells. Our next definition is analogous to definition 6.2(ii).

Definition 14.7. Let $k\geq 0$, let ${\mathbf n}$ be a k-fold multi-index, and let $F\in {\mathrm{pre}}_{\Theta}^k(\Delta^{\mathbf n})$. Let

\begin{align*} b:U(\Delta^{{\mathbf n}})\to P \end{align*}

be a map of posets.

(i) For an object $(\sigma,o)$ of ${\mathcal C}ell(\Delta^{{\mathbf n}})$ define the object $ b_*(F)(\sigma,o) $ of ${\mathcal A}_{\text{rel}}^R$ to be $ b(\sigma)(F(\sigma,o))$.

(ii) For a morphism $f:(\sigma,o)\to (\sigma',o')$ of ${\mathcal C}ell(\Delta^{{\mathbf n}})$ define the morphism

\begin{align*} b_*(F)(f): b_*(F)(\sigma,o)\to b_*(F)(\sigma',o') \end{align*}

to be

\begin{align*} \begin{cases} S_1(f) & \text{if $b(\sigma',o')=S_1$}, \\ S_2(f) & \text{if $b(\sigma,o)=S_2$}, \\ \nu\circ S_1(f) &\text{otherwise}. \end{cases} \end{align*}

Recall that we write ${\mathbf R}$ for the object of ${\mathcal S} p_{{mss }}$ associated with an ad theory (example 4.13). Then $b_*$ gives a map

\begin{align*} (({\mathbf R}_\Theta)_k)_{\mathbf n} \to (({\mathbf R}_{\text{rel}}^R)_k)_{\mathbf n}. \end{align*}

Definition 14.9. (i) For $k\geq 0$ define an object ${\mathcal P}_k$ of $\Sigma_k{{ss}} {\mathcal S}_k$ by

\begin{align*} ({\mathcal P}_k)_{\mathbf n}=\operatorname{Map}_{\text{posets}}(U(\Delta^{\mathbf n}),P)_+ \end{align*}

(where the + denotes a disjoint basepoint); the morphisms in $(\Delta_{\mathrm{inj}}^{\text{op}})^{\times k}$ act in the evident way, and the morphisms of the form $(\alpha,\mathrm{id})$ with $\alpha\in \Sigma_k$ act by permuting the factors in $\Delta^{\mathbf n}$.

(ii) Define ${\mathcal P}$ to be the object of $\Sigma{{ss}}{\mathcal S} $ with k-th term ${\mathcal P}_k$.

Next we give ${\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta}$ the structure of a multisemisimplicial symmetric spectrum (cf. definition 8.1). Recall definition 6.4 and let s be the 1-simplex of S ¹. Define

\begin{align*} \omega:S^1\wedge ({\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta})_k \to ({\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta}) \end{align*}

as follows: for $b\in ({\mathcal P}_k)_{\mathbf n}$ and $x\in (({\mathbf R}_{\Theta})_k)_{\mathbf n}$, let

\begin{align*} \omega(s\wedge (b\wedge x)) = (b\circ \Pi)\wedge \omega(s\wedge x). \end{align*}

It follows from the definitions that we obtain a map

\begin{align*} \beta:{\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \to {\mathbf R}_{\text{rel}}^R \end{align*}

in ${\mathcal S} p_{{mss }}$ by

\begin{align*} \beta(b\wedge F)=b_*(F). \end{align*}

For each k and ${\mathbf n}$, define elements $c_{k,{\mathbf n}},d_{k,{\mathbf n}}\in ({\mathcal P}_k)_{\mathbf n}$ to be the constant functions $U(\Delta^{\mathbf n})\to P$ whose values are respectively S ₁ and S ₂. Then define maps

\begin{align*} c,d:{\overline{\mathbf{S}}}\to {\mathcal P} \end{align*}

in $\Sigma{{ss}}{\mathcal S} $ by taking the non-trivial simplex of $({\overline{\mathbf{S}}}_k)_{\mathbf n}$ to $c_{k,{\mathbf n}}$, resp., $d_{k,{\mathbf n}}$. Finally, define maps

\begin{align*} {\mathbf c}, {\mathbf d}: {\mathbf R}_{\Theta}\to {\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \end{align*}

in ${\mathcal S} p_{{mss }}$ by letting $\mathbf c$ be the composite

\begin{align*} {\mathbf R}_{\Theta}\cong {\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \xrightarrow{c\wedge 1} {\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \end{align*}

and similarly for $\mathbf d$.

Now $\beta\circ\mathbf c$ is the map S ₁ and $\beta\circ\mathbf d$ is the map S ₂, so to complete the proof of proposition 14.5 it suffices to show:

Proof of lemma 14.10 For each $k\geq 0$ and each ${\mathbf n}$ let $e_{k,{\mathbf n}}:({\mathcal P}_k)_{\mathbf n}\to S^0$ be the map which takes every simplex except the basepoint to the non-trivial element of S ⁰, and let

\begin{align*} e:{\mathcal P}\to{\overline{\mathbf{S}}} \end{align*}

be the map given by the $e_{k,{\mathbf n}}$. Let

\begin{align*} {\mathbf e}: {\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \to {\mathbf R}_{\Theta} \end{align*}

be the composite

\begin{align*} {\mathcal P}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \xrightarrow{e\wedge 1} {\overline{\mathbf{S}}}\mathbin{{\overline{\wedge}}} {\mathbf R}_{\Theta} \cong {\mathbf R}_{\Theta}. \end{align*}

Then ${\mathbf e}\circ{\mathbf c}$ and ${\mathbf e}\circ{\mathbf d}$ are both equal to the identity. But $\mathbf e$ is a weak equivalence by proposition 9.3 and lemma 10.3, and the result follows.

15. Background for the proof of theorem 1.3

We now turn to the proof of theorem 1.3, which will follow the general outline of the proof of theorem 1.1. The key ingredient in that proof was the action of the monad ${\mathbb P}$ on ${\mathbf R}$. That action was constructed from the family of operations given in definition 6.2(ii), and this family in turn was constructed from the family of functors $\eta_\bigstar$ given in definition 3.5(ii). For our present purpose, we need the functors $\eta_\bigstar$ and also a family of functors

\begin{align*} {\mathbf d}_\blacksquare:{\mathcal A}_1\times\cdots\times{\mathcal A}_j \to {{\mathcal A}_{\text{rel}}^{\mathbb Z}}, \end{align*}

where each ${\mathcal A}_i$ is equal to ${{\mathcal A}_{e,*,1}}$ or ${{\mathcal A}_{\text{rel}}^{\mathbb Z}}$; these will be built from the symmetric monoidal structures of ${{\mathcal A}_{e,*,1}}$ and ${{\mathcal A}_{\text{rel}}^{\mathbb Z}}$ and the functor

\begin{align*} {\mathrm{sig}_{\mathrm{rel}}}: {\mathcal A}_{e,*,1}\to {{\mathcal A}_{\text{rel}}^{\mathbb Z}}. \end{align*}

It is convenient to represent this situation by a function r from $\{1,\ldots,j\}$ to a two element set $\{u,v\}$, with ${\mathcal A}_i={{\mathcal A}_{e,*,1}}$ if $r(i)=u$ and ${\mathcal A}_i={{\mathcal A}_{\text{rel}}^{\mathbb Z}}$ if $r(i)=v$.

Example 15.2. A typical example is the functor

\begin{align*} {{\mathcal A}_{\text{rel}}^{\mathbb Z}}\times({{\mathcal A}_{e,*,1}})^{\times 5}\times {{\mathcal A}_{\text{rel}}^{\mathbb Z}} \to {{\mathcal A}_{\text{rel}}^{\mathbb Z}} \end{align*}

which takes $(x_1,\ldots,x_7)$ to

\begin{align*} i^\epsilon {\mathrm{sig}_{\mathrm{rel}}}(x_4\boxtimes x_3)\otimes x_7\otimes {\mathrm{sig}_{\mathrm{rel}}}(x_6\boxtimes x_2\boxtimes x_5)\otimes x_1, \end{align*}

where i ^ϵ is the sign that arises from permuting $(x_1,\ldots,x_7)$ into the order $(x_4,x_3,x_7,x_6,x_2,x_5,x_1)$. In definition 15.4(iv), we will represent such a functor by a surjection h which keeps track of which inputs go to which output factors and a permutation η which keeps track of the order in which the inputs to each ${\mathrm{sig}_{\mathrm{rel}}}$ factor are multiplied. In the present example, h is the surjection $\{1,\ldots,7\}\to \{1,2,3,4\}$ with

\begin{align*} h^{-1}(1)=\{3,4\}, h^{-1}(2)=7, h^{-1}(3)=\{2,5,6\}, h^{-1}(4)=1 \end{align*}

and η is the permutation $(256)(34)$.

In order to get the signs right we need a preliminary definition.

Definition 15.3. (i) For totally ordered sets $S_1,\ldots,S_m$, define

\begin{align*} \coprod_{i=1}^n S_i \end{align*}

to be the disjoint union with the order relation given as follows: s < t if either $s\in S_i$ and $t\in S_j$ with i < j, or $s,t\in S_i$ with s < t in the order of S_i.

(ii) For a surjection

\begin{align*} h:\{1,\ldots,j\}\to\{1,\ldots,m\} \end{align*}

define $\theta(h)$ to be the permutation

\begin{align*} \{1,\ldots,j\} \cong h^{-1}(1)\coprod\cdots\coprod h^{-1}(m) \cong \{1,\ldots,j\}; \end{align*}

here the first map restricts to the identity on each $h^{-1}(i)$ and the second is the unique ordered bijection.

In example 15.2, $\theta(h)$ takes $1,\ldots,7$ respectively to $7,4,1,2,5,6,3$.

Definition 15.4. Let $j\geq 0$ and let $r:\{1,\ldots,j\}\to \{u,v\}$. Let ${\mathcal A}_i$ denote ${{\mathcal A}_{e,*,1}}$ if $r(i)=u$ and ${{\mathcal A}_{\text{rel}}^{\mathbb Z}}$ if $r(i)=v$.

(i) Let $1\leq m\leq j$. A surjection

\begin{align*} h:\{1,\ldots,j\}\to\{1,\ldots,m\} \end{align*}

is adapted to r if r is constant on each set $h^{-1}(i)$ and h is monic on $r^{-1}(v)$.

(ii) Given a surjection

\begin{align*} h:\{1,\ldots,j\}\to\{1,\ldots,m\} \end{align*}

which is adapted to r, define

\begin{align*} h_\blacklozenge: {\mathcal A}_1\times\cdots\times{\mathcal A}_j \to ({{\mathcal A}_{\text{rel}}^{\mathbb Z}})^{\times m} \end{align*}

by

\begin{align*} h_\blacklozenge(x_1,\ldots,x_j) =(i^{\epsilon}y_1,\ldots,y_m), \end{align*}

where i ^ϵ is the sign that arises from putting the objects $x_1,\ldots,x_j$ into the order $x_{\theta(h)^{-1}(1)},\ldots, x_{\theta(h)^{-1}(j)}$ and

\begin{align*} y_i= \begin{cases} {\mathrm{sig}_{\mathrm{rel}}}(\boxtimes_{l\in h^{-1}(i)} \,x_l) & \text{if $h^{-1}(i)\subset r^{-1}(u)$}, \\ x_{h^{-1}(i)} & \text{if $h^{-1}(i)\in r^{-1}(v)$}. \end{cases} \end{align*}

(iii) A datum of type r is a pair

\begin{align*} (h,\eta), \end{align*}

where h is a surjection which is adapted to r and η is an element of $\Sigma_j$ with the property that $h\circ\eta=h$.

(iv) Given a datum

\begin{align*} {\mathbf d}=(h,\eta), \end{align*}

of type r, define

\begin{align*} {\mathbf d}_\blacksquare : {\mathcal A}_1\times\cdots\times{\mathcal A}_j \to {{\mathcal A}_{\text{rel}}^{\mathbb Z}} \end{align*}

to be the composite

\begin{align*} {\mathcal A}_1\times\cdots\times{\mathcal A}_j \xrightarrow{\eta} {\mathcal A}_{\eta^{-1}(1)}\times\cdots\times{\mathcal A}_{\eta^{-1}(j)} = {\mathcal A}_1\times\cdots\times{\mathcal A}_j \xrightarrow{h_\blacklozenge} ({{\mathcal A}_{\text{rel}}^{\mathbb Z}})^{\times m} \xrightarrow{\otimes} {{\mathcal A}_{\text{rel}}^{\mathbb Z}}, \end{align*}

where η permutes the factors with the usual sign.

We also need natural transformations between the functors ${\mathbf d}_\blacksquare$. First observe that $\mbox{sig}_{rel}$ is lax monoidal: there is a natural transformation from the functor

\begin{align*} ({{\mathcal A}_{e,*,1}})^{\times l} \xrightarrow{{\mathrm{sig}_{\mathrm{rel}}}^{\times l}} ({{\mathcal A}_{\text{rel}}^{\mathbb Z}})^{\times l} \xrightarrow{\otimes} {{\mathcal A}_{\text{rel}}^{\mathbb Z}} \end{align*}

to the functor

\begin{align*} ({{\mathcal A}_{e,*,1}})^{\times l} \xrightarrow{\boxtimes} {{\mathcal A}_{e,*,1}} \xrightarrow{{\mathrm{sig}_{\mathrm{rel}}}} {{\mathcal A}_{\text{rel}}^{\mathbb Z}} \end{align*}

given by the maps

\begin{align*} S_*({X_1}) \otimes\cdots\otimes S_*({X_l}) \xrightarrow{\times} S_*({X_1}\times\cdots\times {X_l}) \end{align*}

and

\begin{align*} S_*(X_1\times X_1) \otimes\cdots\otimes S_*(X_l\times X_l) & \xrightarrow{\times} S_*(X_1\times X_1\times\cdots\times X_l\times X_l) \\ & \cong S_*((X_1\times\cdots\times X_l)\times (X_1\times\cdots\times X_l)). \end{align*}

Combining this with the symmetric monoidal structures of ${{\mathcal A}_{e,*,1}}$ and ${{\mathcal A}_{\text{rel}}^{\mathbb Z}}$, we obtain a natural transformation ${\mathbf d}_\blacksquare\to{\mathbf d}'_\blacksquare$ whenever ${\mathbf d}\leq{\mathbf d}'$, as defined in:

Definition 15.5. For data of type r, define

\begin{align*} (h,\eta) \leq (h',\eta') \end{align*}

if each set $h^{-1}(i)$ is contained in some set ${h'}^{-1}(l)$.

Our next definition is analogous to definition 6.6 (the presence of the letter v in the symbols $P_{r;v}$ and ${\mathcal O}(r;v)$ will be explained in a moment).

Definition 15.6. Let $r:\{1\ldots,j\}\to \{u,v\}$.

(i) Let $P_{r;v}$ be the preorder whose elements are the data of type r, with the order relation given by definition 15.5.

(ii) Define an object ${\mathcal O}(r;v)_k$ of $\Sigma_k{{ss}} {\mathcal S}_k$ by

\begin{align*} ({\mathcal O}(r;v)_k)_{\mathbf n}=\operatorname{Map}_{\mathrm{preorder}}(U(\Delta^{\mathbf n}),P_{r;v})_+ \end{align*}

(where the + denotes a disjoint basepoint); the morphisms in $(\Delta_{\mathrm{inj}}^{\text{op}})^{\times k}$ act in the evident way, and the morphisms of the form $(\alpha,\mathrm{id})$ with $\alpha\in \Sigma_k$ act by permuting the factors in $\Delta^{\mathbf n}$.

(iii) Define ${\mathcal O}(r;v)$ to be the object of $\Sigma{{ss}}{\mathcal S} $ with k-th term ${\mathcal O}(r;v)_k$.

Remark 15.7. (i) Given $r:\{1\ldots,j\}\to \{u,v\}$, let $m=|r^{-1}(v)|+1$, let

\begin{align*} h:\{1,\ldots,j\}\to \{1,\ldots,m\} \end{align*}

be any surjection which is adapted to r, and let e be the identity element of $\Sigma_j$. Then the datum (h, e) is ≥ every element in $P_{r;v}$.

(ii) Lemma 10.3 shows that each of the objects ${\mathcal O}(r;v)_k$ has compatible degeneracies and is weakly equivalent to a point.

We also need a preorder corresponding to the family of functors

\begin{align*} \eta_\bigstar:({{\mathcal A}_{e,*,1}})^{\times j}\to {{\mathcal A}_{e,*,1}}: \end{align*}

In the next section, we will use the objects ${\mathcal O}(r;v)$ and ${\mathcal O}(r;u)$ to construct a monad. In preparation for that, we show that the collection of preorders $P_{r;v}$ and $P_{r;u}$ has suitable composition maps. Specifically, we show that it is a coloured operad (also called a multicategory) in the category of preorders.

We refer the reader to [Reference Elmendorf and Mandell4, §2] for the definition of multicategory; we will mostly follow the notation and terminology given there. In our case, there are two objects u and v, and we think of a function $r:\{1,\ldots,j\}\to \{u,v\}$ as a sequence of u’s and v’s.

Remark 15.10. Let $r:\{1,\ldots,j\}\to \{u,v\}$ and let ${\mathcal A}_i$ denote ${{\mathcal A}_{e,*,1}}$ if $r(i)=u$ and ${{\mathcal A}_{\text{rel}}^{\mathbb Z}}$ if $r(i)=v$. Let us write ${\mathcal A}_r$ for the category ${\mathcal A}_1\times\cdots\times{\mathcal A}_j$ and ${\mathcal A}_{r;u}$ (resp., ${\mathcal A}_{r;v}$) for the category of functors ${\mathcal A}_r\to {{\mathcal A}_{e,*,1}}$ (resp., ${\mathcal A}_r\to {{\mathcal A}_{\text{rel}}^{\mathbb Z}}$). Define a multicategory with objects u and v as follows. The multimorphisms from r to u are given by ${\mathcal A}_{r;u}$ and from r to v are given by ${\mathcal A}_{r;v}$. The actions of the symmetric groups are the obvious ones and the composition is provided by the composition of functors. Moreover, definitions 3.5(ii) and 15.4(iv) give inclusion functors

\begin{align*} \Phi_{r;u}:P_{r;u}\to {\mathcal A}_{r;u} \end{align*}

and

\begin{align*} \Phi_{r;v}:P_{r;v}\to {\mathcal A}_{r;v}. \end{align*}

We like to promote these to a functor between multicategories. The source also has two objects, u, v say and the functor is the identity on objects. The set of multimorphisms with target u are empty unless the source only involves u’s in which case it is $P_{r;u}$ with $r=r_u(j)$. The set of multimorphisms with target v and source r is the $P_{r;v}$. The following definitions are chosen so that the inclusion functors preserve the $\Sigma_j$ actions and the composition operations.

We define the right $\Sigma_j$ action on a multimorphism with j sources as follows. Let $\alpha\in\Sigma_j$ and $r:\{1,\ldots,j\}\to \{u,v\}$. Define r ^α to be the composite