# Models of $G$-spectra as presheaves of spectra

Bertrand Guillou<EMAIL_ADDRESS>Department of Mathematics, University of
Kentucky, Lexington, KY 40506 USA  and  J.P. May<EMAIL_ADDRESS>Department of Mathematics, The University of Chicago, Chicago, IL 60637 USA

(Date: August 21, 2011)

###### Abstract.

Let $G$ be a finite group. We give Quillen equivalent models for the category
of $G$-spectra as categories of spectrally enriched functors from explicitly
described domain categories to nonequivariant spectra. Our preferred model is
based on equivariant infinite loop space theory applied to elementary
categorical data. It recasts equivariant stable homotopy theory in terms of
point-set level categories of $G$-spans and nonequivariant spectra. We also
give a more topologically grounded model based on equivariant Atiyah duality.

###### Contents

  1. 1 The $\scr{S}$-category $G\scr{B}$ and the $\scr{S}_{G}$-category $\scr{B}_{G}$
    1. 1.1 The bicategory $G\scr{E}$ of $G$-spans
    2. 1.2 The precise statement of the main theorem
    3. 1.3 The $G$-bicategory $\scr{E}_{G}$ of spans: intuitive definition
    4. 1.4 The $G$-bicategory $\scr{E}_{G}$ of spans: working definition
    5. 1.5 The categorical duality maps
  2. 2 The proof of the main theorem
    1. 2.1 The equivariant approach to 1.9
    2. 2.2 Results from equivariant infinite loop space theory
    3. 2.3 The self-duality of $\Sigma^{\infty}_{G}(A_{+})$
    4. 2.4 The proof that $\scr{B}_{G}$ is equivalent to $\scr{D}_{G}$
    5. 2.5 Identifications of suspension $G$-spectra and of tensors with spectra
  3. 3 Atiyah duality for finite $G$-sets
    1. 3.1 The categories $G\scr{Z}$, $G\scr{D}$, and $\scr{D}_{G}$
    2. 3.2 Space level Atiyah duality for finite $G$-sets
    3. 3.3 The weakly unital categories $G\scr{A}$ and $\scr{A}_{G}$
    4. 3.4 The category of presheaves with domain $G\scr{A}$

## Introduction

The equivariant stable homotopy category is of fundamental importance in
algebraic topology. It is the natural home in which to study equivariant
stable homotopy theory, a subject that has powerful and unexpected
nonequivariant applications. For recent examples, it plays a central role in
the solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel,
it is central to calculations of topological cyclic homology and therefore to
calculations in algebraic K-theory made by Angeltveit, Gerhardt, Hesselholt,
Lindenstrauss, Madsen, and others, and it plays an interesting role by analogy
and comparision in the work of Voevodsky and others in motivic stable homotopy
theory. It is also of great intrinsic interest.

Setting up the equivariant stable homotopy category with its attendant model
structures takes a fair amount of work. The original version was due to Lewis
and May [11] and more modern versions that we shall start from are given in
[12]. A result of Schwede and Shipley [20], reproven in [5], asserts that any
stable model category $\scr{M}$ is equivalent to a category
$\mathbf{Pre}(\scr{D},\scr{S})$ of spectrally enriched presheaves with values
in a chosen category $\scr{S}$ of spectra. However, the domain
$\scr{S}$-category $\scr{D}$ is a full $\scr{S}$-subcategory of $\scr{M}$ and
typically is as inexplicit and mysterious as $\scr{M}$ itself. From the point
of view of applications and calculations, this is therefore only a starting
point. One wants a more concrete understanding of the category $\scr{D}$. We
shall give explicit equivalents to the domain category $\scr{D}$ in the case
when $\scr{M}=G\scr{S}$ is the category of $G$-spectra for a finite group $G$,
and we fix a finite group $G$ throughout.

We shall define an $\scr{S}$-category (or spectral category) $G\scr{B}$ by
applying a suitable infinite loop space machine to simply defined categories
of finite $G$-sets. The letter $\scr{B}$ stands for “Burnside”, and $G\scr{B}$
is a spectrally enriched version of the Burnside category of $G$. We shall
prove the following result.

######  Theorem 0.1 (Main theorem).

There is a zig-zag of Quillen equivalences

$G\scr{S}\simeq\mathbf{Pre}(G\scr{B},\scr{S})$

relating the category of $G$-spectra to the category of spectrally enriched
contravariant functors $G\scr{B}\longrightarrow\scr{S}$.

As usual, we call such functors presheaves. We reemphasize the simplicity of
our spectral category $G\scr{B}$: no prior knowledge of $G$-spectra is
required to define it.

We give a precise description of the relevant categorical input and restate
the main theorem more precisely in §1. The central point of the proof is to
use equivariant infinite loop space theory to construct the spectral category
$G\scr{B}$ from elementary categories of finite $G$-sets. We prove our main
theorem in §2, using the equivariant Barratt-Priddy-Quillen (BPQ) theorem to
compare $G\scr{B}$ to the spectral category $G\scr{D}$ given by the suspension
$G$-spectra $\Sigma^{\infty}_{G}(A_{+})$ of based finite $G$-sets $A_{+}$. It
is crucial to our work that these $G$-spectra are self-dual. Our original
proof (§3.2) took this as a special case of equivariant Atiyah duality,
thinking of $A$ as a trivial example of a smooth closed $G$-manifold. We later
found a direct categorical proof (§2.3) of this duality based on equivariant
infinite loop space theory and the equivariant BPQ theorem. This allows us to
give an illuminating new proof of the required self-duality as we go along. We
give an alternative model for the category of $G$-spectra in terms of
classical Atiyah duality in §3.

We take what we need from equivariant infinite loop space theory as a black
box in this paper, deferring the proofs of all but one detail to a sequel [7],
with that detail deferred to another sequel [18].

We thank a diligent referee for demanding a reorganization of our original
paper. We also thank Angelica Osorno and Inna Zakharevich for very helpful
comments.

##  1\. The $\scr{S}$-category $G\scr{B}$ and the $\scr{S}_{G}$-category
$\scr{B}_{G}$

We first define the $\scr{S}$-category $G\scr{B}$ and restate our main
theorem. We shall avoid categorical apparatus, but conceptually $G\scr{B}$ is
obtained by applying a nonequivariant infinite loop space machine $\mathbb{K}$
to a category $G\scr{E}$ “enriched in permutative categories”. The term in
quotes can be made categorically precise [4, 9, 19], but we shall use it just
as an informal slogan since no real categorical background is necessary to our
work: we shall give direct elementary definitions of the examples we use, and
they do satisfy the axioms specified in the cited sources. We then define a
$G$-category $\scr{E}_{G}$ “enriched in permutative $G$-categories”, from
which $G\scr{E}$ is obtain by passage to $G$-fixed subcategories. Finally, we
outline the proof of the main theorem, which is obtained by applying an
equivariant infinite loop space machine $\mathbb{K}_{G}$ to $\scr{E}_{G}$.

###  1.1. The bicategory $G\scr{E}$ of $G$-spans

In any category $\scr{C}$ with pullbacks, the bicategory of spans in $\scr{C}$
has $0$-cells the objects of $\scr{C}$. The $1$-cells and $2$-cells
$A\longrightarrow B$ are the diagrams

$\textstyle{B}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$

and

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{B}$$\textstyle{A}$$\textstyle{E.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Composites of $1$-cells are given by (chosen) pullbacks

(1.1)
$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C}$$\textstyle{B}$$\textstyle{A.}$

The identity $1$-cells are the diagrams
$\textstyle{A}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{=}$$\textstyle{A}$.
The associativity and unit constraints are determined by the universal
property of pullbacks. Observe that the $1$-cells $A\longrightarrow B$ can
just as well be viewed as objects over $B\times A$. Viewed this way, the
identity $1$-cells are given by the diagonal maps $A\longrightarrow A\times
A$.

Our starting point is the bicategory of spans of finite $G$-sets. Here the
disjoint union of $G$-sets over $B\times A$ gives us a symmetric monoidal
structure on the category of $1$-cells and $2$-cells $A\longrightarrow B$ for
each pair $(A,B)$. We can think of the bicategory of spans as a category
“enriched in the category of symmetric monoidal categories”. Again, the notion
in quotes does not make obvious mathematical sense since there is no obvious
monoidal structure on the category of symmetric monoidal categories, but
category theory due to the first author [4] (see also [9, 19]) explains what
these objects are and how to rigidify them to categories enriched in
permutative categories. We repeat that we have no need to go into such
categorical detail. Rather than apply such category theory, we give a direct
elementary construction of a strict structure that is equivalent to the
intuitive notion of the category “enriched in symmetric monoidal categories”
of spans of finite $G$-sets.

######  Definition 1.1.

We first define a bipermutative category $G\scr{E}(1)$ equivalent to the
symmmetric bimonoidal category of finite $G$-sets. Any finite $G$-set is
isomorphic to a finite $G$-set of the form $A=(\mathbf{n},\alpha)$, where
$\mathbf{n}=\\{1,\cdots,n\\}$, $\alpha$ is a homomorphism
$G\longrightarrow\Sigma_{n}$, and $G$ acts on $\mathbf{n}$ by $g\cdot
i=\alpha(g)(i)$ for $1\leq i\leq n$. We understand finite $G$-sets to be of
this specific restricted form from now on. A $G$-map
$f\colon(\mathbf{m},\alpha)\longrightarrow(\mathbf{n},\beta)$ is a function
$f\colon\mathbf{m}\longrightarrow\mathbf{n}$ such that
$f\circ\alpha(g)=\beta(g)\circ f$ for $g\in G$. The morphisms of $G\scr{E}(1)$
are the isomorphisms $(\mathbf{n},\alpha)\longrightarrow(\mathbf{n},\beta)$ of
$G$-sets. The disjoint union $D\amalg E$ of finite $G$-sets
$D=(\mathbf{s},\sigma)$ and $E=(\mathbf{t},\tau)$ is
$(\mathbf{s+t},\sigma+\tau)$, with $\sigma+\tau$ being the evident block sum
$G\longrightarrow\Sigma_{s+t}$. With the evident commutativity isomorphism,
this gives the permutative category $G\scr{E}(1)$ of finite $G$-sets; the
empty finite $G$-set is the unit for $\amalg$. Similarly, the cartesian
product $D\times E$ of $D$ and $E$ is $(\mathbf{st},\sigma\times\tau)$ where
the set $\mathbf{st}$ is identified with $\mathbf{{s}\times{t}}$, ordered
lexicographically, and $\sigma\times\tau$ is the evident block product. There
is again an evident commutativity isomorphism, and $\amalg$ and $\times$ give
$G\scr{E}(\ast)$ a structure of bipermutative category in the sense of [17];
the multiplicative unit is the trivial $G$-set $1=(\mathbf{1},\varepsilon)$,
where $\varepsilon(g)=1$ for $g\in G$.

We may view $G\scr{E}(1)$ as the category of finite $G$-sets over the one
point $G$-set $1$, and we generalize the definition as follows.

######  Definition 1.2.

For a finite $G$-set $A$, we define a permutative category $G\scr{E}(A)$ of
finite $G$-sets over $A$. The objects of $G\scr{E}(A)$ are the $G$-maps
$p\colon D\longrightarrow A$. The morphisms $p\longrightarrow q$, $q\colon
E\longrightarrow A$, are the $G$-isomorphisms $f\colon D\longrightarrow E$
such that $q\circ f=p$. Disjoint union of $G$-sets over $A$ gives
$G\scr{E}(A)$ a structure of permutative category; its unit is the empty set
over $A$. When $A=1$, $G\scr{E}(A)$ is the (“additive”) permutative category
of the previous definition.

######  Remark 1.3.

There is also a product $\times\colon G\scr{E}(A)\times
G\scr{E}(B)\longrightarrow G\scr{E}(A\times B)$. It takes $(D,E)$ to $D\times
E$, where $D$ and $E$ are finite $G$-sets over $A$ and $B$, respectively. This
product is also strictly associative and unital, with unit the unit of
$G\scr{E}(1)$, and it has an evident commutativity isomorphism. Restriction to
the object $1$ gives the “multiplicative” permutative category of 1.1. This
product distributes over $\amalg$ and makes the enriched category $G\scr{E}$
of the next definition into a “strict symmetric monoidal category enriched in
permutative categories” in a sense defined in [4].

######  Definition 1.4.

We define a category $G\scr{E}$ “enriched in permutative categories” as
follows. The $0$-cells of $G\scr{E}$ are the finite $G$-sets, which may be
thought of as the categories $G\scr{E}(A)$. The permutative category
$G\scr{E}(A,B)$ of $1$-cells and $2$-cells $A\longrightarrow B$ is
$G\scr{E}(B\times A)$, as defined in 1.2. The composition

$\circ\colon G\scr{E}(B,C)\times G\scr{E}(A,B)\longrightarrow G\scr{E}(A,C)$

is defined via pullbacks, as in the diagram (1.1). Precisely, the pullback $F$
is the sub $G$-set of $E\times D$ consisting of the elements $(e,d)$ such that
$d$ and $e$ map to the same element $b\in B$. This composition is strictly
associative and unital.

######  Remark 1.5.

We are suppressing some categorical details. The composition distributes over
coproducts, and it should be defined on a “tensor product” rather than a
cartesian product of permutative categories. Such a tensor product does in
fact exist [9], but we shall not use the relevant category theory. Rather we
will change notation to $\wedge$ since the composition is a pairing that gives
rise to a pairing defined on the smash product of the spectra constructed from
$G\scr{E}(B,C)$ and $G\scr{E}(A,B)$.

######  Remark 1.6.

It is helpful to observe that the composition just defined can be viewed as a
composite of maps of finite $G$-sets induced contravariantly and covariantly
by the maps of finite $G$-sets

$\textstyle{C\times B\times B\times
A}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces C\times
B\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times\Delta\times\operatorname{id}}$$\scriptstyle{\pi}$$\textstyle{C\times
A,}$

where $\pi:C\times B\times A\longrightarrow C\times A$ is the projection.

Before beginning work, we recall an old result that motivated this paper. The
category $[G\scr{E}]$ of $G$-spans is obtained from the bicategory $G\scr{E}$
of $G$-spans by identifying spans from $A$ to $B$ if there is an isomorphism
between them. Composition is again by pullbacks. We add spans from $A$ to $B$
by taking disjoint unions, and that gives the morphism set $[G\scr{E}](A,B)$ a
structure of abelian monoid. We apply the Grothendieck construction to obtain
an abelian group of morphisms $A\longrightarrow B$. This gives an additive
category $\scr{A}\\!b[G\scr{E}]$. The following result is [11, V.9.6]. Let
$\text{Ho}G\scr{D}$ denote the full subcategory of the homotopy category
$\text{Ho}G\scr{S}$ of $G$-spectra whose objects are the $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs over the finite $G$-sets.

######  Theorem 1.7.

The categories $\text{Ho}G\scr{D}$ and $\scr{A}\\!b[G\scr{E}]$ are isomorphic.

###  1.2. The precise statement of the main theorem

Infinite loop space theory associates a spectrum $\mathbb{K}\scr{A}$ to a
permutative category $\scr{A}$. There are several equivalent machines
available. For definiteness, and because we have used it in working out the
details, we use a modernized version of [14, 16] that lands in the category
$\scr{S}$ of orthogonal spectra [13]. Precise details are given in [7]. With
this choice, the zeroth space of $\mathbb{K}\scr{A}$ is the classifying space
$B\scr{A}$. The objects $a\in\scr{A}$ are the vertices of the nerve of
$\scr{A}$ and thus are points of $B\scr{A}$. Therefore each $a$ determines a
map $S\longrightarrow\mathbb{K}\scr{A}$, where $S$ is the sphere spectrum. For
any $\scr{A}$, $\mathbb{K}\scr{A}$ is a positive $\Omega$-spectrum ([13, §14])
such that its structure map
$B\scr{A}\longrightarrow\Omega(\mathbb{K}\scr{A})_{1}$ is a group completion.

Since $\scr{S}$ is closed symmetric monoidal under the smash product, it makes
sense to enrich categories in $\scr{S}$. Our preferred version of spectral
categories is categories enriched in $\scr{S}$, abbreviated
$\scr{S}$-categories. Model theoretically, $\scr{S}$ is a particularly nice
enriching category since its unit $S$ is cofibrant in the stable model
structure and $\scr{S}$ satisfies the monoid axiom [13, 12.5].

When a spectral category $\scr{D}$ is used as the domain category of a
presheaf category, the objects and maps of the underlying category are
unimportant. The important data are the morphism spectra $\scr{D}(A,B)$, the
unit maps $S\longrightarrow\scr{D}(A,A)$, and the composition maps

$\scr{D}(B,C)\wedge\scr{D}(A,B)\longrightarrow\scr{D}(A,C).$

The presheaves $\scr{D}^{op}\longrightarrow\scr{S}$ can be thought of as
(right) $\scr{D}$-modules.

######  Definition 1.8.

We define a spectral category $G\scr{B}$. Its objects are the finite $G$-sets
$A$, which may be viewed as the spectra $\mathbb{K}G\scr{E}(A)$. Its morphism
spectra $G\scr{B}(A,B)$ are the spectra $\mathbb{K}G\scr{E}(B\times A)$. Its
unit maps $S\longrightarrow G\scr{B}(A,A)$ are induced by the points
$\operatorname{id}_{A}\in G\scr{E}(A,A)$ and its composition

$G\scr{B}(B,C)\wedge G\scr{B}(A,B)\longrightarrow G\scr{B}(A,C)$

is induced by composition in $G\scr{E}$.

As written, the definition makes little sense: to make the word “induced”
meaningful requires properties of the infinite loop space machine $\mathbb{K}$
that we will spell out in §2.2. Once this is done, we will have the presheaf
category $\mathbf{Pre}(G\scr{B},\scr{S})$ of $\scr{S}$-functors
$(G\scr{B})^{op}\longrightarrow\scr{S}$ and and $\scr{S}$-natural
transformations. As shown for example in [5], it is a cofibrantly generated
model category enriched in $\scr{S}$, or $\scr{S}$-model category for short.
As shown in [12], the category $G\scr{S}$ of (genuine) orthogonal $G$-spectra
is also an $\scr{S}$-model category. Our main theorem can be restated as
follows.

######  Theorem 1.9 (Main theorem).

There is a zigzag of enriched Quillen equivalences connecting the
$\scr{S}$-model categories $G\scr{S}$ and $\mathbf{Pre}(G\scr{B},\scr{S})$.

Therefore $G$-spectra can be thought of as constructed from the very
elementary category $G\scr{E}$ enriched in permutative categories, ordinary
nonequivariant spectra, and the black box of infinite loop space theory. The
following reassuring result falls out of the proof. Let $\scr{O}\\!rb$ denote
the orbit category of $G$. For a $G$-spectrum $X$, passage to $H$-fixed point
spectra for $H\subset G$ defines a functor
$X^{\bullet}\colon\scr{O}\\!rb^{op}\longrightarrow\scr{S}$. Analogously, a
presheaf $Y\in\mathbf{Pre}(G\scr{B},\scr{S})$ restricts to a functor
$\scr{O}\\!rb^{op}\longrightarrow\scr{S}$.

######  Corollary 1.10.

The zigzag of equivalences induces a natural zigzag of equivalences between
the fixed point orbit functor on $G$-spectra and the restriction to orbits of
presheaves; thus, if $X$ corresponds to $Y$, then $X^{H}$ is equivalent to
$Y(G/H)$.

######  Remark 1.11.

There is an important missing ingredient needed for a fully satisfactory
theory: we have not described the behavior of smash products under the
equivalences of 1.9. This problem deserves study both in our work and in
related work of others. The obvious guess is that
$\mathbf{Pre}(G\scr{B},\scr{S})$ is symmetric monoidal and the zigzag
connecting it to $G\scr{S}$ is a zigzag of symmetric monoidal Quillen
equivalences. We see how the problem can be attacked, but we also have reason
to believe that the obvious guess may be wrong. We intend to return to this
question elsewhere.

######  Remark 1.12.

Much of what we do applies to $G$-spectra indexed on an incomplete universe,
provided that we restrict attention to those finite $G$-sets $A$ that embed in
that universe, so that Atiyah duality applies to the orbit $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$. By [10], duality fails for orbits that do not
embed in the universe. Unfortunately, however, the cited restriction leads to
the wrong weak equivalences, since we are then only entitled to see the
homotopy groups of $H$-fixed point spectra for those $H$ that embed in the
given universe.

###  1.3. The $G$-bicategory $\scr{E}_{G}$ of spans: intuitive definition

Everything we do depends on first working equivariantly and then passing to
fixed points. Following [6, §1.2], we fix some generic notations. For a
category $\scr{C}$, let $G\scr{C}$ be the category of $G$-objects in $\scr{C}$
and $G$-maps between them. Let $\scr{C}_{G}$ be the $G$-category of
$G$-objects and nonequivariant maps, with $G$ acting by conjugation. The two
categories are related conceptually by $G\scr{C}=(\scr{C}_{G})^{G}$. The
objects, being $G$-objects, are already $G$-fixed; we apply the $G$-fixed
point functor to hom sets. More generally, we can start with a category
$\scr{C}$ with actions by $G$ on its objects and again define a category
$G\scr{C}$ of $G$-maps and a $G$-category $\scr{C}_{G}$ with $G$-fixed
category $G\scr{C}$.

We apply this framework to the category of finite $G$-sets. We have already
defined the $G$-fixed bicategory $G\scr{E}$, and we shall give two definitions
of $G$-bicategories $\scr{E}_{G}$ with fixed point bicategories equivalent to
$G\scr{E}$. The first, given in this section, is more intuitive, but the
second is more convenient for the proof of our main theorem.

Let $U$ be a countable $G$-set that contains all orbit types $G/H$ infinitely
many times. Again let $A$, $B$, and $C$ denote finite $G$-sets, but now let
the $D$, $E$ and $F$ of §1.1 be finite subsets of the $G$-set $U$; these
subsets need not be $G$-subsets. The action of $G$ on $U$ gives rise to an
action of $G$ on the finite subsets of $U$: for a finite subset $D$ of $U$ and
$g\in G$, $gD$ is another finite subset of $U$.

######  Definition 1.13.

We define a $G$-category $\scr{E}_{G}(A)$. The objects of $\scr{E}_{G}(A)$ are
the nonequivariant maps $p\colon D\longrightarrow A$, where $A$ is a finite
$G$-set and $D$ is a finite subset of $U$. The morphisms $f\colon
p\longrightarrow q$, $q\colon E\longrightarrow A$, are the bijections $f\colon
D\longrightarrow E$ such that $q\circ f=p$. The group $G$ acts on morphisms
via the maps $g\colon D\longrightarrow gD$ and the formula $(gf)(gd)=gf(d)$.

######  Definition 1.14.

We define a bicategory $\scr{E}_{G}$ with objects the finite $G$-sets and with
$G$-categories of morphisms between objects specified by
$\scr{E}_{G}(A,B)=\scr{E}_{G}(B\times A)$. Thinking of the objects of
$\scr{E}_{G}(A,B)$ as nonequivariant spans $B\longleftarrow D\longrightarrow
A$, composition and units are defined as in 1.4.

Observe that taking disjoint unions of finite sets over $A$ will not keep us
in $U$ and is thus not well-defined. Therefore the $\scr{E}_{G}(A)$ are not
symmetric monoidal (let alone permutative) $G$-categories in the naive sense
of symmetric monoidal categories with $G$ acting compatibly on all data. In
fact, the notion of a genuine permutative $G$-category, one that provides
input for an equivariant infinite loop space machine, is subtle. We shall give
two solutions to that categorical problem in [7]. In both, genuine permutative
$G$-categories are described in terms of actions by an $E_{\infty}$ operad of
$G$-categories, to which equivariant infinite loop space theory applies. One
solution gives each of the $\scr{E}_{G}(A)$ such a structure, but that is not
the solution we shall use.

###  1.4. The $G$-bicategory $\scr{E}_{G}$ of spans: working definition

The other solution starts from a less intuitive definition of $\scr{E}_{G}$
and gives an equivalent way of solving that categorical problem. It uses a
more convenient $E_{\infty}$ operad of $G$-categories, denoted $\scr{O}_{G}$.
We give details of this operad in [7], where we define a genuine permutative
$G$-category to be an algebra over $\scr{O}_{G}$. To give the idea, we apply
our general point of view on equivariant categories to the category
$\scr{C}\\!at$ of small categories. Thus, for $G$-categories $\scr{A}$ and
$\scr{B}$, let $\scr{C}\\!at_{G}(\scr{A},\scr{B})$ be the $G$-category of
functors $\scr{A}\longrightarrow\scr{B}$ and natural transformations, with $G$
acting by conjugation, and let $G\scr{C}\\!at(\scr{A},\scr{B})$ be the
category of $G$-functors and $G$-natural transformations.

######  Definition 1.15.

Let $\tilde{G}$ (sometimes denoted $EG$ in the literature111While $\tilde{G}$
is isomorphic as a $G$-category to the translation category of $G$, the action
of $G$ on that category is defined differently, as is explained in [8, Lemma
1.7].) be the groupoid with object set $G$ and a unique morphism, denoted
$(h,k)$, from $k$ to $h$ for each pair of objects. Let $G$ act from the right
on $\tilde{G}$ by $h\cdot g=hg$ on objects and $(h,k)\cdot g=(hg,kg)$ on
morphisms. The objects of $\scr{E}_{G}$ are the finite $G$-sets
$A=(\mathbf{n},\alpha)$, regarded as discrete (identity morphisms only)
$G$-categories. Define $\scr{O}(j)=\tilde{\Sigma}_{j}$; this is the $j$th
category of an $E_{\infty}$ operad of categories whose algebras are the
permutative categories [16]. Define $\scr{O}_{G}(j)$ to be the $G$-category

$\scr{C}\\!at_{G}(\tilde{G},\tilde{\Sigma}_{j})=\scr{C}\\!at_{G}(\tilde{G},\scr{O}(j)).$

Here $G$ acts trivially on $\tilde{\Sigma}_{j}$. The left action of $G$ on
$\scr{O}_{G}(j)$ is induced by the right action of $G$ on $\tilde{G}$, and the
right action of $\Sigma_{j}$ is induced by the right action of $\Sigma_{j}$ on
$\tilde{\Sigma}_{j}$. The functor $\scr{C}\\!at_{G}(\tilde{G},-)$ is product
preserving and the operad structure maps are induced from those of $\scr{O}$.
We interpret $\scr{O}(0)$ and $\scr{O}_{G}(0)$ to be trivial categories;
$\scr{O}_{G}(1)$ is also trivial, with unique object denoted
$\operatorname{id}$.

######  Definition 1.16.

Define the $G$-category $\scr{E}_{G}(A)$ by

(1.2) $\scr{E}_{G}(A)=\coprod_{n\geq
0}\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n}=(\coprod_{n\geq
1}\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n})_{+}.$

We interpret the term with $n=0$ to be a trivial base category $\ast$, which
explains the second equality, and we identify the term with $n=1$ with $A$. An
alternative formulation is $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$, where
$\mathbb{O}_{G}$ denotes the monad in the category of based $G$-categories
whose algebras are the same as the $\scr{O}_{G}$-algebras. Thus
$\mathbb{O}_{G}(A_{+})$ is the free $\scr{O}_{G}$-algebra (= genuine
permutative $G$-category) generated by the based $G$-category $A_{+}$, with
unit given by a disjoint trivial base category added to $A$.

The following result is neither obvious nor difficult. It is proven in [7].

######  Theorem 1.17.

The $G$-fixed permutative category $\scr{E}_{G}(A)^{G}$ is naturally
isomorphic to the permutative category $G\scr{E}(A)$.

The starting point of the proof is the observation that a functor
$\tilde{G}\longrightarrow\tilde{\Sigma}_{n}$ is uniquely determined by its
object function $G\longrightarrow\Sigma_{n}$. In particular, for a finite
$G$-set $B=(\mathbf{n},\beta)$ we may view the $G$-map $\beta\colon
G\longrightarrow\Sigma_{n}$ as a $G$-fixed object of the category
$\scr{O}_{G}(n)$, and all $G$-fixed objects of $\scr{O}_{G}(n)$ are of this
form. With a little care, we see that a $G$-fixed object
$(\beta;a_{1},\cdots,a_{n})$ of $\scr{O}_{G}(n)\times_{\Sigma_{n}}A^{n}$ can
be interpreted as a $G$-map $B\longrightarrow A$ and that all finite $G$-sets
over $A$ are of this form.

The following is a sketch definition whose details will be fleshed out below.

######  Definition 1.18.

The $G$-category $\scr{E}_{G}$ “enriched in permutative $G$-categories” has
$0$-cells the finite $G$-sets $A$, which may be thought of as the
$G$-categories $\scr{E}_{G}(A)$. The permutative $G$-category
$\scr{E}_{G}(A,B)$ of $1$-cells and $2$-cells $A\longrightarrow B$ is
$\scr{E}_{G}(B\times A)$. The unit $\operatorname{id}_{A}$ of
$A=(\mathbf{n},\alpha)$ is the object $(\alpha;(1,1),\cdots,(n,n))$ of
$\scr{O}_{G}(n)\times_{\Sigma_{n}}(A\times A)^{n}$; it can be thought of as a
$G$-map $1\longrightarrow\scr{E}_{G}(A,A)$ of $G$-categories, where $1$ is the
trivial $G$-category. Composition is given by the following composite; its
first map is a specialization of a pairing of free $\scr{O}_{G}$-algebras, and
its second and third maps are specializations of contravariant functoriality
of the free $\scr{O}_{G}$-algebra functor on inclusions and covariant
functoriality on surjections that we shall shortly make precise.

$\textstyle{\scr{E}_{G}(C\times B)\wedge\scr{E}_{G}(B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\scr{E}_{G}(C\times
B\times B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id}\times\Delta\times\operatorname{id})^{*}}$$\textstyle{\scr{E}_{G}(C\times
B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{!}}$$\textstyle{\scr{E}_{G}(C\times
A).}$

We shall place the following ad hoc definition of the required pairing
$\omega$ in a suitable general context in [7], modernizing part of [15]. We
first comment on its domain; compare 1.5.

######  Remark 1.19.

We can define the smash product of based $G$-categories in the same way as the
smash product of based $G$-spaces. We are most interested in examples of the
form $\scr{A}_{+}$ and $\scr{B}_{+}$ for unbased $G$-categories $\scr{A}$ and
$\scr{B}$, and then $\scr{A}_{+}\wedge\scr{B}_{+}$ can be identified with
$(\scr{A}\times\scr{B})_{+}$. In particular,

$(\coprod_{m\geq
1}\scr{O}_{G}(m)\times_{\Sigma_{m}}A^{m})_{+}\wedge(\coprod_{n\geq
1}\scr{O}_{G}(n)\times_{\Sigma_{n}}B^{n})_{+}$

is isomorphic to

$(\coprod_{m\geq 1,n\geq
1}\scr{O}_{G}(m)\times\scr{O}_{G}(n)\times_{\Sigma_{m}\times\Sigma_{n}}A^{m}\times
B^{n})_{+}.$

We do not claim that this is an $\scr{O}_{G}$-category, but an equivariant
infinite loop space machine nevertheless constructs from it the smash product
of the spectra constructed from $\scr{E}_{G}(A)$ and $\scr{E}_{G}(B)$.

######  Definition 1.20.

Identify the ordered set $\mathbf{mn}$ with the set of pairs $(i,j)$, $1\leq
i\leq m$ and $1\leq j\leq n$, ordered lexicographically. This fixes a
homomorphism $\Sigma_{m}\times\Sigma_{n}\longrightarrow\Sigma_{mn}$ and
therefore a functor
$\tilde{\Sigma}_{m}\times\tilde{\Sigma}_{n}\longrightarrow\tilde{\Sigma}_{mn}$.
Applying the functor $\scr{C}\\!at_{G}(\tilde{G},-)$, we obtain pairings
$\omega_{m,n}\colon\scr{O}_{G}(m)\times\scr{O}_{G}(n)\longrightarrow\scr{O}_{G}(mn)$.
For finite $G$-sets $A$ and $B$, we have the injection $A^{m}\times
B^{n}\longrightarrow(A\times B)^{mn}$ that sends
$(a_{1},\cdots,a_{m})\times(b_{1},\cdots,b_{n})$ to the set of pairs
$(a_{i},b_{j})$, ordered lexicographically. Combining, there result functors

$\omega_{m,n}\colon(\scr{O}_{G}(m)\times_{\Sigma_{m}}A^{m})\times(\scr{O}_{G}(n)\times_{\Sigma_{n}}B^{n})\longrightarrow\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(A\times
B)^{mn}.$

Distributing products over disjoint unions, these specify pairings of
$G$-categories

$\omega\colon\scr{E}_{G}(A)\wedge\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A\times
B).$

The naturality maps in 1.18 are both applications of the free
$\scr{O}_{G}$-category functor to maps $f$ of based finite $G$-sets.
Conceptually, the definition (1.2) hides an extension of functors from
$\scr{E}_{G}(A)$, which a priori appears to be a functor on unbased finite
$G$-sets, to $\mathbb{O}_{G}(A_{+})$, which is a functor on based finite
$G$-sets.

######  Definition 1.21.

For a map $f\colon A_{+}\longrightarrow B_{+}$ of based finite $G$-sets, we
obtain a functor $f_{!}\colon\scr{E}_{G}(A)\longrightarrow\scr{E}_{G}(B)$ by
taking the disjoint union over $n$ of the functors
$\operatorname{id}\times_{\Sigma_{n}}f^{n}$. This is unproblematical if $f$ is
obtained from a map $A\longrightarrow B$ of unbased finite $G$-sets, so that
$f^{-1}(\ast)=\ast$.222With the intuitive version of $\scr{E}_{G}$,
$f_{!}\colon\scr{E}_{G}(A)\longrightarrow\scr{E}_{G}(B)$ is just the
pushforward functor obtained by composing $f$ with maps over $A$. In general,
however, the specification of $f_{!}$ depends on implicit basepoint
identifications that are invisible to (1.2) but become visible when evaluating
$\scr{E}_{G}f$. Because $\scr{O}_{G}(0)$ is the trivial category $\ast$, there
is a degeneracy $G$-functor
$\sigma_{i}\colon\scr{O}_{G}(n)\longrightarrow\scr{O}_{G}(n-1)$ associated to
the ordered inclusion $\mathbf{n-1}\colon\longrightarrow\mathbf{n}$ that
misses $i$. As in [14, 2.3], if $\gamma$ is the structural map of the operad
and $\nu\in\scr{O}_{G}(n)$,

$\sigma_{i}(\nu)=\gamma(\nu;\operatorname{id}^{i-1},\ast,\operatorname{id}^{n-i}).$

If $a_{i}=\ast$, then $(\nu,a_{1},\cdots,a_{n})$ must be identified with
$(\sigma_{i}(\nu),a_{1},\cdots,\hat{a}_{i},\cdots,a_{n})$, where $\hat{a}_{i}$
means delete $a_{i}$. In particular, if $i\colon A\longrightarrow B$ is an
inclusion of unbased finite $G$-sets, define an associated retraction $r\colon
B_{+}\longrightarrow A_{+}$ of based finite $G$-sets by setting $ri(a)=a$ and
$r(b)=\ast$ if $b\notin\operatorname{im}(A)$. Then define
$i^{*}=r_{!}\colon\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A)$.333With the
intuitive version of $\scr{E}_{G}$,
$i^{*}\colon\scr{E}_{G}(B)\longrightarrow\scr{E}_{G}(A)$ is just the functor
obtained by pulling back maps over $B$ to maps over $A$. By 2.14 below, we may
think of $i^{*}$ as the dual of $i$.

The associativity of the composition defined in 1.18 is an easy diagram chase,
starting from the associativity of the pairing on $\scr{O}_{G}$. The
verification that composition with the prescribed unit objects
${\operatorname{id}_{A}}$ gives identity functors illustrates how 1.21 works.
Set $B=A$ and consider the composite

$(\mu;(c_{1},a_{1}),\cdots,(c_{m},a_{m}))\circ\operatorname{id}_{A}.$

We are focusing on objects, and $\mu\in\scr{O}_{G}(m)$, $c_{i}\in C$,
$a_{i}\in A$, and $A=(\mathbf{n},\alpha)$. Applying the pairing we get the
object

$(\omega_{m,n}(\mu,\alpha);(c_{i},a_{i},j,j))\in\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(C\times
A\times A\times A)^{mn}.$

The four-tuple $(c_{i},a_{i},j,j)$ is in the image of
$\operatorname{id}\times\Delta\times\operatorname{id}$ if and only if
$a_{i}=j$. The $r$ corresponding to this inclusion maps all other
$(c_{i},a_{i},j,j)$ to the basepoint, and we have an accompanying iterated
degeneracy $\sigma\colon\scr{O}_{G}(mn)\longrightarrow\scr{O}_{G}(m)$ such
that $\sigma(\omega_{m,n}(\mu,\alpha))=\mu$. Therefore our composite is
$(\mu;(c_{1},a_{1}),\cdots,(c_{m},a_{m}))$, as required. The proof that
composition on the left with $\operatorname{id}_{A}$ is the identity functor
is similar.

1.17 has the following corollary by direct comparison of definitions.

######  Corollary 1.22.

The $G$-fixed category $(\scr{E}_{G})^{G}$ enriched in permutative categories
is isomorphic to the category $G\scr{E}$ enriched in permutative categories.

###  1.5. The categorical duality maps

Since various specializations are central to our work, we briefly recall how
duality works categorically, following [11, III§1] for example. We then define
maps of $\scr{O}_{G}$-algebras that will lead in §2.3 to the proof that the
objects of $G\scr{B}$ are self-dual.

Let $\scr{V}$ be a closed symmetric monoidal category with product $\wedge$,
unit $S$, and hom objects $F(X,Y)$; write $DX=F(X,S)$. A pair of objects
$(X,Y)$ in $\scr{V}$ is a dual pair if there are maps $\eta\colon
S\longrightarrow X\wedge Y$ and $\varepsilon\colon Y\wedge X\longrightarrow S$
such that the composites

$\textstyle{X\cong S\wedge
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{X\wedge
Y\wedge
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{X\wedge
S\cong X}$ $\textstyle{Y\cong Y\wedge
S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{Y\wedge
X\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\textstyle{S\wedge
Y\cong Y}$

are identity maps. For any such pair, the adjoint $\tilde{\varepsilon}\colon
Y\longrightarrow DX$ of $\varepsilon$ is an isomorphism. We have a natural map

(1.3) $\zeta\colon Y\wedge DX=Y\wedge F(X,S)\longrightarrow F(X,Y)$

in $\scr{V}$, namely the adjoint of

$\operatorname{id}\wedge\varepsilon\colon Y\wedge DX\wedge X\longrightarrow
Y\wedge S\cong Y,$

where $\varepsilon$ is the evident evaluation map. The map $\zeta$ is an
isomorphism when either $X$ or $Y$ is dualizable [11, III.1.3]. When $X$ is
dualizable and $Y$ is arbitrary, we have the composite isomorphism

(1.4) $\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon
Y\wedge X\longrightarrow Y\wedge DX\longrightarrow F(X,Y).$

This map in various categories will play a central role in our work. When
$(X,Y)$ and $(X^{\prime},Y^{\prime})$ are dual pairs, the dual of a map
$f\colon X\longrightarrow X^{\prime}$ is the composite

(1.5) $\textstyle{Y^{\prime}\cong Y^{\prime}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{Y^{\prime}\wedge
X\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge
f\wedge\operatorname{id}}$$\textstyle{Y^{\prime}\wedge X^{\prime}\wedge
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{\prime}\wedge\operatorname{id}}$$\textstyle{S_{G}\wedge
Y\cong Y.}$

There are two maps of $\scr{O}_{G}$-algebras that are central to duality and
therefore to everything we do. Let $S^{0}=\\{\ast,1\\}$, where $\ast$ is the
basepoint and ${1}$ is not. We think of $S^{0}$ as $1_{+}$, where $1$ is the
one-point $G$-set. Remember that $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$ is the
free $\scr{O}_{G}$-algebra generated by $A_{+}$, where we view finite $G$-sets
as categories with only identity morphisms. We have already seen the first map
implicitly.

######  Definition 1.23.

For a finite $G$-set $A$, define based $G$-maps

$\varepsilon\colon(A\times A)_{+}\longrightarrow S^{0},\ \ r\colon(A\times
A)_{+}\longrightarrow A_{+}\ \ \text{and}\ \ \pi\colon A_{+}\longrightarrow
S^{0}$

by $r(a,b)=\ast$ if $a\neq b$ and $r(a,a)=a$, $\pi(a)=1$, and
$\varepsilon=\pi\circ r$, so that $\varepsilon(a,b)=\ast$ if $a\neq b$ and
$\varepsilon(a,a)=1$. Note that $r\circ\Delta=\operatorname{id}$ and that
$\varepsilon$ is just an example of a Kronecker $\delta$-function. We agree to
again write $\varepsilon$ for the induced map of $\scr{O}_{G}$-algebras

$\varepsilon=\scr{E}_{G}\varepsilon\colon\scr{E}_{G}(A\times
A)\longrightarrow\scr{E}_{G}(1).$

######  Definition 1.24.

For a finite $G$-set $A$, regard the object
$\operatorname{id}_{A}\in\scr{E}_{G}(A)$ as the map of $G$-categories
$i_{A}\colon 1\longrightarrow\scr{E}_{G}(A)$ that sends the object $1$ to the
object $\operatorname{id}_{A}$. By freeness, there results a map of
$\scr{O}_{G}$-algebras

$\eta\colon\scr{E}_{G}(1)\longrightarrow\scr{E}_{G}(A\times A).$

If $A=(\mathbf{n},\alpha)$, then $\eta$ is the disjoint union of maps

$\scr{O}_{G}(m)/\Sigma_{m}\cong\scr{O}_{G}(m)\times_{\Sigma_{m}}1^{m}\longrightarrow\scr{O}_{G}(mn)\times_{\Sigma_{mn}}(A\times
A)^{mn}.$

These are obtained by composing $\scr{O}_{G}(m)\times i_{A}^{m}$ with the map
induced on passage to orbits from the maps

$\textstyle{\scr{O}_{G}(m)\times(\scr{O}_{G}(n)\times(A\times
A)^{n})^{m}\cong(\scr{O}_{G}(m)\times\scr{O}_{G}(n)^{m})\times((A\times
A)^{n})^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\scr{O}_{G}(mn)\times(A\times
A)^{mn}}$

given by shuffling and applying the structure map
$\gamma\colon\scr{O}_{G}(m)\times\scr{O}_{G}(n)^{m}\longrightarrow\scr{O}_{G}(mn)$.

The following categorical observation will lead to our proof in §2.3 that the
$G$-spectra $\Sigma^{\infty}_{G}(A_{+})$ are self-dual. Since care of
basepoints is crucial, we use the alternative notation
$\mathbb{O}_{G}(A_{+})$. Remember that $(A\times A)_{+}$ can be identified
with $A_{+}\wedge A_{+}$. We identify $1_{+}\wedge A_{+}$ and $A_{+}\wedge
1_{+}$ with $A_{+}$ at the bottom center of our diagrams.

######  Proposition 1.25.

The left and right squares commute in the following diagrams, and

(1.6)
$\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)\circ\zeta_{\ell}=\operatorname{id}=\mathbb{O}_{G}(\varepsilon\wedge\operatorname{id})\circ\zeta_{r}.$

Therefore the diagrams obtained by removing the maps $\zeta_{\ell}$ and
$\zeta_{r}$ commute.

$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(1_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\ell}}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(1_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$

$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{O}_{G}(\varepsilon\wedge\operatorname{id})}$$\textstyle{\mathbb{O}_{G}(A_{+}\wedge
A_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\scriptstyle{\omega}$$\textstyle{\mathbb{O}_{G}(A_{+})\wedge\mathbb{O}_{G}(1_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{r}}$$\textstyle{\mathbb{O}_{G}(1_{+})\wedge\mathbb{O}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$

###### Proof.

In the right vertical arrows, $\varepsilon$ means
$\mathbb{O}_{G}(\varepsilon)$. Since the right squares are just naturality
diagrams, they clearly commute. For the rest, we must first define the maps
$\zeta_{\ell}$ and $\zeta_{r}$. Remember that the elements of $A$ are the
elements of $\mathbf{n}=\\{1,\cdots,n\\}$, permuted according to $\alpha\colon
G\longrightarrow\Sigma_{n}$. Define $j_{\ell}:A\longrightarrow(A\times A\times
A)^{n}$ by $j_{\ell}(a)=\big{(}(1,1,a),\cdots,(n,n,a)\big{)}$. Then define
$J_{\ell}\colon A_{+}\longrightarrow\scr{O}_{G}(A_{+}\wedge A_{+}\wedge
A_{+})$ by

$J_{\ell}(a)=(\alpha,j_{\ell}(a))\in\scr{O}_{G}(n)\times_{\Sigma_{n}}(A\times
A\times A)^{n}.$

Define

$\zeta_{\ell}\colon\mathbb{O}_{G}(A_{+})\longrightarrow\scr{O}_{G}(A_{+}\wedge
A_{+}\wedge A_{+})$

to be the map of $\scr{O}_{G}$-algebras induced by freeness. For
$\mu\in\scr{O}_{G}(m)$ and $\nu\in\scr{O}_{G}(q)$,

(1.7)
$\zeta_{\ell}(\omega(\mu,\nu);(a_{1},\cdots,a_{q})^{m})=(\gamma(\omega(\mu,\nu);\alpha^{mq});(j_{\ell}(a_{1}),\cdots,j_{\ell}(a_{q}))^{m})$

where $\gamma$ is the structural map of the operad $\scr{O}_{G}$. Define
$j_{r}$, $J_{r}$, and $\zeta_{r}$ by symmetry.

Clearly $\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)$ sends
$J_{\ell}(a)$ to $a$. Indeed, $a$ is one of the elements $j\in\mathbf{n}$ and
$\operatorname{id}\wedge\varepsilon$ sends the coordinates $(i,i,a)$ with
$i\neq j$ to the basepoint and the coordinate $(j,j,a)$ to $a$. Since
$\mathbb{O}_{G}(\operatorname{id}\wedge\varepsilon)\circ\zeta_{\ell}$ is a map
of $\scr{O}_{G}$-algebras with domain the free $\scr{O}_{G}$-algebra
$\mathbb{O}_{G}(A_{+})$, this implies the first equality in (1.6); the
symmetric argument proves the second equality. It remains to prove that the
left squares of our diagrams commute; by symmetry it suffices to consider the
first diagram. Consider an element

$x=((\mu;1^{m}),(\nu;a_{1},\cdots,a_{q}))\in(\scr{O}_{G}(m)\times_{\Sigma_{m}}1^{m})\times(\scr{O}_{G}(q)\times_{\Sigma_{q}}A^{q}),$

where $m\geq 1$, $q\geq 1$, $\mu\in\scr{O}_{G}(m)$, $\nu\in\scr{O}_{G}(q)$,
and $a_{k}\in A$ for $1\leq k\leq q$. Write $[j,j,a_{k}]$ for the element of
$(A^{3})^{mnq}$ with $(i,j,k)th$ coordinate $(j,j,a_{k})$, $1\leq i\leq m$,
$1\leq j\leq n$, and $1\leq k\leq q$. Then

(1.8)
$\omega\circ(\eta\wedge\operatorname{id})(x)=(\omega(\gamma(\mu;\alpha^{m}),\nu);[j,j,a_{k}])\in\scr{O}_{G}(mnq)\times_{\Sigma_{mnq}}(A^{3})^{mnq}.$

On the other hand,

$\omega(x)=(\omega(\mu,\nu);(a_{1},\cdots,a_{q})^{m})\in\scr{O}_{G}(mq)\times_{\Sigma_{mq}}A^{mq}$

and therefore

(1.9)
$\zeta_{\ell}\omega(x)=(\gamma(\omega(\mu,\nu);\alpha^{mq});(j_{\ell}(a_{1}),\cdots,j_{\ell}(a_{q}))^{m})\in\scr{O}_{G}(mnq)\times_{\Sigma_{mnq}}(A^{3})^{mnq}.$

The coordinates in $A^{3}$ of the element on the right side of (1.9) differ
from those of the right side of (1.8) by a permutation
$\sigma\in\Sigma_{mnq}$, and it is a special case of the formula relating the
pairing $\omega$ to the structure map $\gamma$ of the operad $\scr{O}_{G}$
that

(1.10)
$\gamma(\omega(\mu,\nu);\alpha^{mq})\sigma=\omega(\gamma(\mu;\alpha^{m}),\nu).$

Therefore the right sides of (1.8) and (1.9) are equal and
$\omega\circ(\eta\wedge\operatorname{id})=\zeta_{\ell}\circ\omega$. ∎

######  Remark 1.26.

A more general form of (1.10) is the key defining property [15, 1.4(ii)] of a
pairing of operads, such as $\omega$. We have proven that the left and right
squares of our diagrams are examples of maps of pairings of algebras over a
permutative operad, as defined in [17, IX.1.3] and [15, 1.1]. Those sources
are nonequivariant and outdated, but a modern treatment of equivariant
pairings will be included in [18].

##  2\. The proof of the main theorem

###  2.1. The equivariant approach to 1.9

As we will explain in [7], equivariant infinite loop space theory associates
an orthogonal $G$-spectrum $\mathbb{K}_{G}\scr{A}_{G}$ to a (genuine)
permutative $G$-category $\scr{A}_{G}$. The $0$th space of
$\mathbb{K}_{G}\scr{A}_{G}$ is the classifying $G$-space $B\scr{A}_{G}$. The
$0$th structure map
$B\scr{A}_{G}\longrightarrow\Omega(\scr{B}_{G}\scr{A}_{G})_{1}$ is an
equivariant group completion.444The papers from around 1990, such as [2, 21]
are not adequate for our purposes, in part because the target category of
$G$-spectra was not yet well understood then. A full dress modern treatment of
equivariant infinite loop space theory, complementing [7], is in progress
[18]. The category $G\scr{S}$ of orthogonal $G$-spectra is the $G$-fixed
category of a $G$-category $\scr{S}_{G}$ of $G$-spectra and non-equivariant
maps with the same objects as $\scr{S}_{G}$ and with $G$ acting by
conjugation. Applying the functor $\mathbb{K}_{G}$ to $\scr{E}_{G}$, we obtain
the following equivariant analogue of 1.8.

######  Definition 2.1.

We define a $G$-spectral category, or $\scr{S}_{G}$-category555There is a
slight abuse of language here since the notion of a category enriched in
$\scr{S}_{G}$ (alias a $G$-spectral category) does not quite make sense in
classical enriched category theory because the smash product of $G$-spectra is
only functorial on $G$-maps, not on the more general maps in $\scr{S}_{G}$.
The terminology is explained and justified in [6, 1.9]. $\scr{B}_{G}$. Its
objects are the finite $G$-sets $A$, which may be viewed as the $G$-spectra
$\mathbb{K}_{G}\scr{E}_{G}(A)$. Its morphism $G$-spectra $\scr{B}_{G}(A,B)$
are the $G$-spectra $\mathbb{K}_{G}\scr{E}_{G}(B\times A)$. Its unit $G$-maps
$S_{G}\longrightarrow\scr{B}_{G}(A,A)$ are induced by the points
$\operatorname{id}_{A}\in G\scr{E}(A,A)$ and its composition $G$-maps

$\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,C)$

are induced by composition in $\scr{E}_{G}$.

Again, as written, the definition makes little sense: to make the word
“induced” meaningful requires properties of the equivariant infinite loop
space machine $\mathbb{K}_{G}$ that we will spell out in §2.2. This depends on
having a functor that takes pairings of free $\scr{O}_{G}$-algebras to
pairings of $G$-spectra.

The equivariant and non-equivariant infinite loop space functors are related
by the following result.

######  Theorem 2.2 ([7]).

There is a natural equivalence of spectra

$\iota\colon\mathbb{K}(G\scr{A})\longrightarrow(\mathbb{K}_{G}\scr{A}_{G})^{G}$

for permutative $G$-categories $\scr{A}_{G}$ with $G$-fixed permutative
categories $G\scr{A}$.

In view of 1.22, there results an equivalence of $\scr{S}$-categories

$\textstyle{G\scr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(\scr{B}_{G})^{G}.}$

The proof of 1.9 goes as follows. We start with the following specialization
of a general result about stable model categories; it is discussed in [6,
§3.1].

######  Theorem 2.3.

Let $G\scr{D}$ be the full $\scr{S}$-subcategory of $G\scr{S}$ whose objects
are fibrant approximations of the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs through the finite $G$-sets. Then
there is an enriched Quillen adjunction

$\textstyle{\mathbf{Pre}(G\scr{D},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{T}}$$\textstyle{G\scr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathbb{U}}$

and it is a Quillen equivalence.

Here $G\scr{D}$ is isomorphic to $(\scr{D}_{G})^{G}$, where $\scr{D}_{G}$ is a
full $\scr{S}_{G}$-subcategory $\scr{D}_{G}$ of $\scr{S}_{G}$.

######  Theorem 2.4 (Equivariant version of the main theorem).

There is a zigzag of weak equivalences connecting the $\scr{S}_{G}$-categories
$\scr{B}_{G}$ and $\scr{D}_{G}$.

A weak equivalence between $\scr{S}_{G}$-categories with the same object sets
is just an $\scr{S}_{G}$-functor that induces weak equivalences on morphism
$G$-spectra.666A more general definition is given in [5, 2.3]. On passage to
$G$-fixed categories, this equivariant zigzag induces a zigzag of weak
$\scr{S}$-equivalences connecting the $\scr{S}$-categories $G\scr{B}$ and
$G\scr{D}$. In turn, by [5, 2.4], this zigzag induces a zigzag of Quillen
equivalences between $\mathbf{Pre}(G\scr{B},\scr{S})$ and
$\mathbf{Pre}(G\scr{D},\scr{S})$. Since $\mathbf{Pre}(G\scr{D},\scr{S})$ is
Quillen equivalent to $G\scr{S}$, it follows that 2.4 implies 1.9.

######  Remark 2.5.

The functor $\mathbb{U}$ sends $G/H$ to
$F_{G}(\Sigma^{\infty}_{G}G/H_{+},X)^{G}\cong X^{H}$. Keeping that fact in
mind shows why 1.10 follows from the proof of 1.9.

To understand $G\scr{S}$ as an $\scr{S}$-category, we must first understand
$\scr{S}_{G}$ as an $\scr{S}_{G}$-category. That is, to understand the
$G$-fixed spectra $F_{G}(X,Y)^{G}$, we must first understand the function
$G$-spectra $F_{G}(X,Y)$. Using infinite loop space theory to model function
spectra implicitly raises a conceptual issue: there is no known infinite loop
space machine that knows about function spectra. That is, given input data $X$
and $Y$ (permutative $G$-categories, $E_{\infty}$-$G$-spaces,
$\Gamma$-$G$-spaces, etc) for an infinite loop space machine $\mathbb{K}_{G}$,
we do not know what input data will have as output the function $G$-spectra
$F_{G}(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$. The problem does not even make sense
as just stated because the output $G$-spectra $\mathbb{K}_{G}X$ are always
connective, whereas $F_{G}(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$ is generally not.
The most that one could hope for in general is to detect the connective cover
of $F(\mathbb{K}_{G}X,\mathbb{K}_{G}Y)$. In our case, the relevant function
$G$-spectra are connective since the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$ are self-dual, as we shall reprove in §2.3.

###  2.2. Results from equivariant infinite loop space theory

The proof of 2.4 is the heart of this paper, and of course it depends on
equivariant infinite loop space theory and in particular on the relationship
between the $G$-spectra $\scr{B}_{G}(A)=\mathbb{K}_{G}\scr{E}_{G}(A)$ and the
suspension $G$-spectra $\Sigma^{\infty}_{G}(A_{+})$. We collect the results
that we need from [7] in this section, making Definitions 1.8 and 2.1 precise
and expanding on Theorems 1.17 and 2.2. We warn the skeptical reader that the
results of this paper depend on the two results just cited and on Theorems 2.6
and 2.7 below. The knowledgable expert will immediately accept the
plausibility of these results, especially since those of the results which
make sense when $G=e$ have been known for decades. However, their proofs
require work that is far afield from the applications in this paper.

In fact, 2.4 is an application of a categorical version of the equivariant
Barratt-Priddy-Quillen (BPQ) theorem for the identification of suspension
$G$-spectra.777For $A=\ast$, Carlsson [1, p.6] mentions a space level version
of the BPQ theorem. Shimakawa [21, p. 242] states and gives an incomplete
sketch proof of a $G$-spectrum level version. We state the theorem in full
generality before restricting attention to finite $G$-sets. We shall find use
for the full generality in §2.5.

Recall from 1.16 that $\scr{E}_{G}(A)$ is the category
$\mathbb{O}_{G}(A_{+})$, where $\mathbb{O}_{G}$ is the free
$\scr{O}_{G}$-category functor. We may view any based $G$-space $X$ as a
topological category888We understand a topological category to mean an
internal category in the category of spaces, not just a category enriched in
spaces. that is discrete in the categorical sense: its morphism and object
spaces are both $X$, and its source, target, identity, and composition maps
are all just the identity map of $X$. The functor $\mathbb{O}_{G}$ applies
equally well to based topological $G$-categories, hence we have the
topological $\scr{O}_{G}$-category $\mathbb{O}_{G}(X)$. The geometric
realization of its nerve is the free $E_{\infty}$ $G$-space generated by $X$.

Henceforward, we use the term stable equivalence, rather than weak
equivalence, for the weak equivalences in our model categories of spectra and
$G$-spectra.

######  Theorem 2.6 (Equivariant Barratt-Quillen Theorem, [7]).

For based $G$-spaces $X$, there is a natural stable equivalence

$\alpha\colon\Sigma^{\infty}_{G}X\longrightarrow\mathbb{K}_{G}\mathbb{O}_{G}(X).$

Of course, the naturality statement says that the following diagram commutes
for a map $f\colon X\longrightarrow Y$ of based $G$-spaces.

(2.1)
$\textstyle{\Sigma^{\infty}_{G}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}f}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}(f)}$$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(Y)}$

There is a companion theorem that relates $\alpha$ to smash products. The
pairing $\omega$ of 1.20 generalizes to give a natural pairing

$\omega\colon\mathbb{O}_{G}(X)\wedge\mathbb{O}_{G}(Y)\longrightarrow\mathbb{O}_{G}(X\wedge
Y)$

for based $G$-spaces $X$ and $Y$.

######  Theorem 2.7.

[7] The pairing $\omega$ induces a natural stable equivalence

$\wedge\colon\mathbb{K}_{G}\mathbb{O}_{G}(X)\wedge\mathbb{K}_{G}\mathbb{O}_{G}(Y)\longrightarrow\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
Y)$

such that the following diagram commutes.

(2.2)
$\textstyle{\Sigma^{\infty}_{G}X\wedge\Sigma^{\infty}_{G}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X)\wedge\mathbb{K}_{G}\mathbb{O}_{G}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
Y)}$

The left map $\wedge$ in (2.2) is a canonical natural isomorphism, and this
diagram says that the natural map $\alpha$ is lax monoidal. The result that we
need to prove 2.4 is an immediate specialization.

######  Theorem 2.8.

For finite $G$-sets $A$, there is a lax monoidal natural stable equivalence

$\alpha\colon\Sigma^{\infty}_{G}(A_{+})\longrightarrow\mathbb{K}_{G}\scr{E}_{G}(A).$

Identifying $A_{+}\wedge B_{+}$ with $(A\times B)_{+}$, (2.2) specializes to
the commutative diagram

(2.3)
$\textstyle{\Sigma^{\infty}_{G}(A_{+})\wedge\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)\wedge\mathbb{K}_{G}\scr{E}_{G}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}_{G}(A\times
B)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
B).}$

We restate the naturality of $\alpha$ with respect to $G$-maps $f\colon
A\longrightarrow B$ in the diagram

(2.4)
$\textstyle{\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}f_{+}}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}f_{!}}$$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(B).}$

If $i\colon A\longrightarrow B$ is an inclusion with retraction $r\colon
B_{+}\longrightarrow A_{+}$, we have the induced map of $G$-spectra

$\mathbb{K}_{G}i^{*}=\mathbb{K}_{G}r_{!}\colon\mathbb{K}_{G}\scr{E}_{G}(B)\longrightarrow\mathbb{K}_{G}\scr{E}_{G}(A),$

and (2.4) specializes to

(2.5)
$\textstyle{\Sigma^{\infty}_{G}(B_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}i^{*}}$$\textstyle{\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A)}$

By 2.14 below, we may identify $\mathbb{K}_{G}i^{*}$ as the dual of
$\mathbb{K}_{G}i$ and thus $\Sigma^{\infty}_{G}r$ as the dual of
$\Sigma^{\infty}_{G}i_{+}$.

We combine these diagrams to construct those that we need to prove 2.4. Let
$A$, $B$, and $C$ be finite $G$-sets and recall 1.18.

######  Proposition 2.9.

The following diagram of $G$-spectra commutes.

(2.6) $\textstyle{\Sigma^{\infty}_{G}(C\times
B)_{+}\wedge\Sigma^{\infty}_{G}(B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B)\wedge\mathbb{K}_{G}\scr{E}_{G}(B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}(C\times
B\times B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B\times B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}(\operatorname{id}\times\Delta\times\operatorname{id})^{*}}$$\textstyle{\Sigma^{\infty}(C\times
B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}\pi}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
B\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\pi_{!}}$$\textstyle{\Sigma^{\infty}_{G}(C\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(C\times
A)}$

Here $r$ is the retraction which sends the complement of the image of
$\operatorname{id}\times\Delta\times\operatorname{id}$ to the basepoint.

######  Definition 2.10.

To make 2.1 and therefore 1.8 precise, define the composition

$\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,C)$

to be the right vertical composite in the diagram (2.6).

The diagram (2.6) relates the composition pairing of the
$\scr{S}_{G}$-category $\scr{B}_{G}$ to remarkably simple and explicit maps
between suspension $G$-spectra. In fact, recalling 1.23 and again writing
$\varepsilon=\Sigma^{\infty}_{G}\varepsilon$, we see that the left vertical
composite in (2.6) can be identified with
$\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}$. We have proven
the following result.

######  Theorem 2.11.

The following diagram of $G$-spectra commutes.

$\textstyle{\Sigma^{\infty}_{G}(C\times
B)_{+}\wedge\Sigma^{\infty}_{G}(B\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\scr{B}_{G}(B,C)\wedge\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\Sigma^{\infty}_{G}(C_{+})\wedge\Sigma^{\infty}_{G}(B\times
B)_{+}\wedge\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{\Sigma^{\infty}_{G}(C_{+})\wedge
S_{G}\wedge\Sigma^{\infty}_{G}(A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\Sigma^{\infty}_{G}(C\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\scr{B}_{G}(A,C)}$

###  2.3. The self-duality of $\Sigma^{\infty}_{G}(A_{+})$

Let $A$ be a finite $G$-set and write $\mathbb{A}=\Sigma^{\infty}_{G}(A_{+})$
for brevity of notation. As recalled in §1.5, we must define maps $\eta\colon
S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}$ and
$\varepsilon\colon\mathbb{A}\wedge\mathbb{A}\longrightarrow S_{G}$ in the
stable homotopy category $HoG\scr{S}$ such that the composites

(2.7)
$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\mathbb{A}}$
and
$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\wedge\operatorname{id}}$$\textstyle{\mathbb{A}}$

are the identity map in $HoG\scr{S}$. Using the stable equivalence $\alpha$
and the definitions of $\eta$ and $\varepsilon$ from Definitions 1.23 and
1.24, we let $\eta$ and $\varepsilon$ be the composites

$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\eta}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{\Sigma^{\infty}_{G}(A\times
A)_{+}\cong\mathbb{A}\wedge\mathbb{A}}$

and

$\textstyle{\mathbb{A}\wedge\mathbb{A}\cong\Sigma^{\infty}_{G}(A\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(A\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\varepsilon}$$\textstyle{\mathbb{K}_{G}\scr{E}_{G}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{-1}}$$\textstyle{S_{G}.}$

The following commutative diagram proves that the first composite in (2.7) is
the identity map in $HoG\scr{S}$; the second is dealt with similarly. We
abbreviate notation by setting $\scr{B}_{G}A=\mathbb{K}_{G}\scr{E}_{G}(A)$.
Remember that $\scr{E}_{G}(A)=\mathbb{O}_{G}(A_{+})$. The center two squares
are derived by use of the diagrams from 1.25.

$\textstyle{\scr{B}_{G}(A^{2})\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\textstyle{(\mathbf{A^{2}})\wedge\mathbb{A}\cong\mathbf{A^{3}}\cong\mathbb{A}\wedge(\mathbf{A^{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{A}\wedge\scr{B}_{G}(A^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\scr{B}_{G}(A^{2})\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\scr{B}_{G}(A^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times\varepsilon}$$\textstyle{\scr{B}_{G}A\wedge\scr{B}_{G}(A^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\scr{B}_{G}1\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\alpha}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\scr{B}_{G}1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\scr{B}_{G}1\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\scriptstyle{\wedge}$$\textstyle{\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\ell}}$$\textstyle{\scr{B}_{G}A\wedge\scr{B}_{G}1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{S_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{A}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{id}\wedge\alpha}$

Given 2.8, it is trivial that the outer parts of the diagram commute. We
comment on the passage from the diagrams of 1.25 to the central squares of the
diagram; compare 1.26.

######  Remark 2.12.

Nonequivariantly, the passage from pairings on the category level to pairings
on the spectrum level is worked out in [15], implicitly using orthogonal
spectra. The sequel [7] to this paper constructs the pairing $\wedge$ from the
pairing $\omega$ of free $\scr{O}_{G}$-categories used here, but it does not
treat its naturality with respect to maps of pairings that are not induced by
maps of finite $G$-sets. Modernized generalizations and details will be
supplied in [18].

Specializing general observations about duality recalled in §1.5, we have the
following corollary. This homotopical input is the crux of the proof of 2.4.

######  Corollary 2.13.

For finite $G$-sets $A$ and $B$, the canonical map

$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{B}\wedge
D\mathbb{A}\longrightarrow F_{G}(\mathbb{A},\mathbb{B})$

of (1.4) is a stable equivalence.

We insert a mild digression concerning the identification of some of our maps.

######  Remark 2.14.

For an inclusion $i\colon A\longrightarrow B$ of finite $G$-sets, (1.5) and
the precise constructions of $\eta$ and $\varepsilon$ starting from
Definitions 1.23 and 1.24 imply that the dual of $i$ is the map
$\mathbb{B}\longrightarrow\mathbb{A}$ induced by the evident retraction
$r\colon B_{+}\longrightarrow A_{+}$. A $G$-map $\pi\colon G/H\longrightarrow
G/K$ is a bundle, and the dual of $\Sigma^{\infty}\pi_{+}$ is the associated
transfer map (see e.g. [11, IV.pp 182 and 192]). It can be identified
explicitly by a similar (but not especially illuminating) inspection of
definitions.

###  2.4. The proof that $\scr{B}_{G}$ is equivalent to $\scr{D}_{G}$

We will have to chase large diagrams, and we again abbreviate notations by
writing

$\mathbb{A}=\Sigma^{\infty}_{G}(A_{+}),\ \ \
\mathbb{B}=\Sigma^{\infty}_{G}(B_{+}),\ \ \ \text{and}\ \ \
\mathbb{C}=\Sigma^{\infty}_{G}(C_{+})$

for finite $G$-sets $A$, $B$, and $C$. We also abbreviate notation by writing

$\scr{B}_{G}(A)=\scr{B}_{G}(\ast,A).$

It is the $G$-spectrum $\scr{B}_{G}(A)=\mathbb{K}_{G}\scr{E}_{G}(A)$, which is
equivalent to $\mathbb{A}$ by 2.8. Remember that we are free to choose any
bifibrant equivalents of the $G$-spectra $\mathbb{A}$ as the objects of
$\scr{D}_{G}$.

###### Proof of 2.4.

We use model categorical arguments, and we work with the stable model
structure on $G\scr{S}$. We use [5, §2.4] to obtain a model structure on the
category $G\scr{S}\mathbb{O}$-$\scr{C}\\!at$ of $G\scr{S}$-categories with the
same object set $\mathbb{O}$ as $G\scr{E}$. Maps are weak equivalences or
fibrations if they induce weak equivalences or fibrations on hom objects in
$G\scr{S}$. Here the nature of the objects is irrelevant; we are concerned
with $G\scr{S}$-categories with one object for each finite $G$-set $A$.

Let $\lambda\colon Q{\scr{B}_{G}}\longrightarrow\scr{B}_{G}$ be a cofibrant
approximation of $\scr{B}_{G}$. By [5, 2.16], since $S_{G}$ is cofibrant in
the stable model structure each morphism $G$-spectrum $Q\scr{B}_{G}(A,B)$ is
cofibrant in $G\scr{S}$. The maps $\lambda\colon
Q\scr{B}_{G}(A,B)\longrightarrow\scr{B}_{G}(A,B)$ are stable acyclic
fibrations. Digressively, since the $\scr{B}_{G}(A,B)$ are fibrant in the
positive stable model structure, that is also true of the $Q\scr{B}_{G}(A,B)$;
we will use this fact later, in §2.5.

Let $\rho\colon Q\scr{B}_{G}\longrightarrow RQ\scr{B}_{G}$ be a fibrant
approximation of $Q\scr{B}_{G}$. The morphism $G$-spectra $RQ\scr{B}_{G}(A,B)$
are then bifibrant in the stable model structure. Therefore $RQ\scr{B}_{G}(A)$
is bifibrant for each $A$, and it is stably equivalent to $\mathbb{A}$. We
take the $RQ\scr{B}_{G}(A)$ as the bifibrant approximations of the
$\mathbb{A}$ that we use to define the full $G\scr{S}$-subcategory
$\scr{D}_{G}$ of $G\scr{S}$.

We define $\scr{C}_{G}$ to be the full $G\scr{S}$-subcategory of $G\scr{S}$
with objects the $Q\scr{B}_{G}(A)$. To abbreviate notation, we agree to write

$Q\scr{B}_{G}(\ast,A)=Q\scr{B}_{G}A\ \ \ \text{and}\ \ \
RQ\scr{B}_{G}(\ast,A)=RQ\scr{B}_{G}A.$

With our notational conventions, it is consistent to write
$Q\scr{B}_{G}(B\times A)=Q\scr{B}_{G}(A,B)$.

For finite $G$-sets $A$ and $B$, let

$\beta\colon
Q\scr{B}_{G}(A,B)\longrightarrow\scr{C}_{G}(A,B)=F_{G}(Q\scr{B}_{G}A,Q\scr{B}_{G}B)$

be the adjoint of the composition map

$\circ\colon Q\scr{B}_{G}(A,B)\wedge Q\scr{B}_{G}A\longrightarrow
Q\scr{B}_{G}B$

and let

$\gamma\colon
RQ\scr{B}_{G}(A,B)\longrightarrow\scr{D}_{G}(A,B)=F_{G}(RQ\scr{B}_{G}A,RQ\scr{B}_{G}B)$

be the adjoint of the composition map

$\circ\colon RQ\scr{B}_{G}(A,B)\wedge RQ\scr{B}_{G}A\longrightarrow
RQ\scr{B}_{G}B.$

By [5, 5.6], these define $G\scr{S}$-functors

$\beta\colon Q\scr{B}_{G}\longrightarrow\scr{C}_{G}\ \ \ \text{\ \ and\ \ }\ \
\ \gamma\colon RQ\scr{B}_{G}\longrightarrow\scr{D}_{G}.$

It suffices to prove that each of the maps $\gamma$ is a stable equivalence.
For each finite $G$-set $A$, $\mathbb{A}$ is cofibrant and $\lambda\colon
Q\scr{B}_{G}A\longrightarrow\scr{B}_{G}A$ is an acyclic fibration in the
stable model structure. Therefore there is a map
$\mu\colon\mathbb{A}\longrightarrow Q\scr{B}_{G}A$ such that the diagram

$\textstyle{Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\alpha}$$\textstyle{\scr{B}_{G}A}$

commutes. Since $\alpha$ and $\lambda$ are stable equivalences, so is $\mu$.

We claim that the following diagram of $G$-spectra commutes in $HoG\scr{S}$.
Indeed, modulo inversion of maps which are stable equivalences, it commutes on
the nose. As before, we identify
$\mathbb{B}\wedge\mathbb{A}=\Sigma^{\infty}_{G}B_{+}\wedge\Sigma^{\infty}_{G}A_{+}$
with $\Sigma^{\infty}_{G}(B\times A)_{+}$.

$\textstyle{RQ\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{F_{G}(RQ\scr{B}_{G}A,RQ\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\rho,\operatorname{id})}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(Q\scr{B}_{G}A,RQ\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\scriptstyle{\simeq}$$\textstyle{Q\scr{B}_{G}(A,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\simeq}$$\scriptstyle{\beta}$$\textstyle{F_{G}(Q\scr{B}_{G}A,Q\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\rho)}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\textstyle{F_{G}(\mathbb{A},RQ\scr{B}_{G}B)}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\simeq}$$\scriptstyle{\delta}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(\mathbb{A},\mathbb{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\mu)}$$\scriptstyle{\simeq}$$\textstyle{F_{G}(\mathbb{A},Q\scr{B}_{G}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\rho)}$$\scriptstyle{\simeq}$

The map $\delta$ is the stable equivalence of 2.13. The maps $\mu$ and $\rho$
are also stable equivalences. The maps $F_{G}(\rho,\operatorname{id})$ and
$F_{G}(\mu,\operatorname{id})$ that are labeled $\simeq$ are stable
equivalences by [5, 1.22] since $\rho$ and $\mu$ are maps between cofibrant
objects and $RQ\scr{B}_{G}B$ is fibrant. The maps
$F_{G}(\operatorname{id},\mu)$ and $F_{G}(\operatorname{id},\rho)$ that are
labeled $\simeq$ are stable equivalences by [12, III.3.9], which shows that
the functor $F_{G}(\mathbb{A},-)$ preserves stable equivalences. Granting that
the diagram commutes, it follows that $\gamma$ is a stable equivalence since
all of the other outer arrows of the diagram are stable equivalences.

To prove that the diagram commutes in $HoG\scr{S}$, we consider its adjoint.
Remembering that $\lambda\circ\mu=\alpha$, we see that the adjoint can be
written in the following expanded form. Here we have inserted the map
$\circ\colon\scr{B}_{G}(A,B)\wedge\scr{B}_{G}A\longrightarrow\scr{B}_{G}B$ and
wrong way arrows into its source and target for purposes of proof.

---
$\textstyle{RQ\scr{B}_{G}(A,B)\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\textstyle{RQ\scr{B}_{G}(A,B)\wedge
Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\rho}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
RQ\scr{B}_{G}(A,B)\wedge
RQ\scr{B}_{G}A}$$\scriptstyle{\circ}$$\textstyle{Q\scr{B}_{G}(A,B)\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\scriptstyle{\rho\wedge\operatorname{id}}$$\scriptstyle{\rho\wedge\mu}$$\textstyle{Q\scr{B}_{G}(A,B)\wedge
Q\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\scriptstyle{\rho\wedge\operatorname{id}}$$\scriptstyle{\rho\wedge\rho}$$\scriptstyle{\lambda\wedge\lambda}$$\textstyle{Q\scr{B}_{G}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\lambda}$$\textstyle{RQ\scr{B}_{G}B}$$\textstyle{\scr{B}_{G}(A,B)\wedge\scr{B}_{G}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\scr{B}_{G}B}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\Sigma^{\infty}_{G}\varepsilon}$$\scriptstyle{\alpha\wedge\alpha}$$\scriptstyle{\mu\wedge\operatorname{id}}$$\scriptstyle{\mu\wedge\mu}$$\textstyle{\mathbb{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\mu}$$\scriptstyle{\rho\mu}$

Since $\lambda$ and $\rho$ are maps of $G\scr{S}$-categories, it is apparent
that all parts of the diagram commute except for the bottom trapezoid. Taking
$(A,B,C)=(\ast,A,B)$ in 2.11, we see that the trapezoid commutes. Since the
wrong way maps $\alpha$ and $\lambda$ are stable equivalences and can be
inverted upon passage to the homotopy category, this diagram and its adjoint
commute there. ∎

###  2.5. Identifications of suspension $G$-spectra and of tensors with
spectra

We expand the adjoint $\scr{S}$-equivalences in 1.9 more explicitly as
follows.

(2.8)
$\textstyle{G\scr{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{U}}$$\textstyle{\mathbf{Pre}(G\scr{D},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\mathbb{T}}$$\textstyle{\mathbf{Pre}((RQ\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{*}}$$\scriptstyle{\gamma_{!}}$$\textstyle{\mathbf{Pre}(G\scr{B},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{!}}$$\textstyle{\mathbf{Pre}((\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{*}}$$\scriptstyle{\iota^{*}}$$\textstyle{\mathbf{Pre}((Q\scr{B}_{G})^{G},\scr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{!}}$$\scriptstyle{\rho_{!}}$

The map $\iota:G\scr{B}\longrightarrow(\scr{B}_{G})^{G}$ is the equivalence of
2.2. Before passage to $G$-fixed points, the proof in §2.4 gives stable
equivalences of $\scr{S}_{G}$-categories

$\rho\colon Q\scr{B}_{G}\longrightarrow RQ\scr{B}_{G},\ \
\gamma:RQ\scr{B}_{G}\longrightarrow\scr{D}_{G},\ \text{and}\ \lambda\colon
Q\scr{B}_{G}\longrightarrow\scr{B}_{G},$

and these maps give stable equivalences of $\scr{S}$-categories after passage
to fixed points.

For a finite $G$-set $B$, $\Sigma^{\infty}_{G}B_{+}$ corresponds under this
zigzag to the presheaf $\mathbf{B}$ that sends $A$ to $G\scr{B}(A,B)$. This is
almost a tautology since, for $E\in G\scr{S}$, $\mathbb{U}(E)$ is the presheaf
represented by $E$, while $G\scr{E}(-,B)$ is the functor represented by $B$.
In the proof of 2.4, we chose the bifibrant approximation of
$\Sigma^{\infty}_{G}B_{+}$ in $G\scr{D}_{G}$ to be $RQ\scr{B}_{G}(B)$. With
$B$ fixed, that proof shows that $\gamma$ gives an equivalence of presheaves

$RQ\scr{B}_{G}(-,B)\longrightarrow\gamma^{*}\mathbb{U}RQ\scr{B}_{G}(B)$

(before passage to $G$-fixed points). The functors $\rho^{*}$ and
$\lambda_{!}$ and the isomorphism $\iota^{*}$ preserve representable functors,
and therefore $\iota^{*}\lambda_{!}\rho^{*}RQ\scr{B}_{G}(-,B)\simeq
K_{G}\scr{E}_{G}(-,B)$.

This observation can be generalized from finite based $G$-sets $B_{+}$ to
arbitrary based $G$-spaces $X$. To see this, we mix general based $G$-spaces
$X$ with finite based $G$-sets $A$ to obtain a functorial construction of a
presheaf $\scr{P}_{G}(X)$.

######  Definition 2.15.

Define a presheaf
$\scr{P}_{G}(X)\colon(\scr{B}_{G})^{op}\longrightarrow\scr{S}_{G}$ by letting

$\scr{P}_{G}(X)(A)=\scr{K}_{G}\mathbb{O}_{G}(X\wedge A_{+}).$

The contravariant functoriality map

$\scr{P}_{G}(X)\colon\scr{B}_{G}(A,B)\longrightarrow
F_{G}(\scr{B}_{G}(X)(B),\scr{B}_{G}(X)(A))$

is the adjoint of the right vertical composite in the commutative diagram

(2.9) $\textstyle{\Sigma^{\infty}_{G}(X\wedge
B_{+})\wedge\Sigma^{\infty}_{G}(B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\scriptstyle{\cong}$$\scriptstyle{\alpha\wedge\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+})\wedge\mathbb{K}_{G}\mathbb{O}_{G}(B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wedge}$$\textstyle{\Sigma^{\infty}(X\wedge
B_{+}\wedge B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}_{G}r}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+}\wedge B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}(r)}$$\textstyle{\Sigma^{\infty}(X\wedge
B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\Sigma^{\infty}\pi}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
B_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{K}_{G}\mathbb{O}_{G}\pi}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge
A_{+}).}$

Here $r$ is the evident left inverse of
$\operatorname{id}\wedge\Delta\wedge\operatorname{id}$ and $\pi$ is the
projection. The diagram commutes by the same concatenation of commutative
diagrams as in 2.9.

######  Theorem 2.16.

Let $X$ be a based $G$-space. Under our zigzag of equivalences,
$\Sigma^{\infty}_{G}X$ corresponds naturally to the presheaf
$(\scr{P}_{G}(X))^{G}$ that sends $A$ to
$\mathbb{K}\big{(}\mathbb{O}_{G}(X\wedge A_{+})^{G}\big{)}$.

###### Proof.

Note that $\mathbb{K}_{G}\mathbb{O}_{G}(X\wedge-_{+})$ is no longer a
representable presheaf. We again work with $G$-spectra and obtain the
conclusion after passage to $G$-fixed spectra. According to 2.6, we may
replace $\Sigma_{G}^{\infty}X$ by the positive fibrant $G$-spectrum
$\mathbb{K}_{G}\mathbb{O}_{G}(X)$, which we abbreviate to $\scr{B}_{G}(X)$ by
a slight abuse of notation. After this replacement, the presheaf
$\mathbb{U}(\Sigma_{G}^{\infty}X)$ may be computed as

$\mathbb{U}(\Sigma_{G}^{\infty}X)(A)=F_{G}(RQ\scr{B}_{G}(A),\scr{B}_{G}(X)).$

Therefore, following the chain of (2.8), we may compute
$\rho^{*}\gamma^{*}\mathbb{U}(\Sigma_{G}^{\infty}X)$ as

$\rho^{*}\gamma^{*}\mathbb{U}(\Sigma_{G}^{\infty}X)\simeq
F_{G}(Q\scr{B}_{G}(-),\scr{B}_{G}(X)).$

Thinking of $(B,A)$ above replaced by $(A,\ast)$, the adjoint to the composite

(2.10) $\scr{P}_{G}(X)(A)\wedge
Q\scr{B}_{G}(A)\xrightarrow{\operatorname{id}\wedge\lambda}\scr{P}_{G}(X)(A)\wedge\scr{B}_{G}(A)\xrightarrow{\circ}\scr{P}_{G}(X)(\ast)=\scr{B}_{G}(X)$

defines a map of presheaves

(2.11) $\lambda^{*}\scr{P}_{G}(X)\longrightarrow
F_{G}(Q\scr{B}_{G}(-),\scr{B}_{G}(X))$

with domain $Q\scr{B}_{G}$. It remains to show that this map is an
equivalence. To compute the adjoint (2.11), observe that the composite (2.10)
is the top horizontal composite in the commutative diagram

---
$\textstyle{\scr{P}_{G}(X)(A)\wedge
Q\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\lambda}$$\textstyle{\scr{P}_{G}(X)(A)\wedge\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\scr{B}_{G}(X)}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\wedge
Q\scr{B}_{G}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\textstyle{\scr{B}_{G}(A,X)\wedge\Sigma^{\infty}_{G}A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\alpha}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\wedge\Sigma^{\infty}_{G}A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mu}$$\scriptstyle{\alpha\wedge\operatorname{id}}$$\scriptstyle{\cong}$$\textstyle{\Sigma^{\infty}_{G}X\wedge\Sigma^{\infty}_{G}(A_{+}\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{\Sigma^{\infty}_{G}X.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$

We have used that $\lambda\circ\mu=\alpha$. The pentagon on the right is a
special case of (2.9).

Therefore the map (2.11) is the top horizontal composite in the diagram

$\textstyle{\scr{P}_{G}(X)(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{G}(\scr{B}_{G}(A),\scr{B}_{G}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\lambda,\operatorname{id})}$$\textstyle{F_{G}(Q\scr{B}_{G}(A),\scr{B}_{G}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\mu,\operatorname{id})}$$\textstyle{\Sigma^{\infty}_{G}(X\wedge
A_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\delta}$$\textstyle{F_{G}(\Sigma^{\infty}_{G}A_{+},\Sigma^{\infty}_{G}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\operatorname{id},\alpha)}$$\textstyle{F_{G}(\Sigma^{\infty}_{G}A_{+},\scr{B}_{G}(X)).}$

The map $\alpha$ is a stable equivalence by 2.6. The map $\delta$ is the
stable equivalence of (1.4). The map $F_{G}(\operatorname{id},\alpha)$ is a
stable equivalence by [12, III.3.9]. Finally, the map
$F_{G}(\mu,\operatorname{id})$ is a stable equivalence by [5, 1.22]. ∎

There is another visible identification. The category $G\scr{S}$ and our
presheaf categories are $\scr{S}$-complete, so that they have tensors and
cotensors over $\scr{S}$ (see [5, §5.1]). It is formal that the left adjoint
of an $\scr{S}$-adjunction preserves tensors and the right adjoint preserves
cotensors. A quick chase of our zigzag of Quillen $\scr{S}$-equivalences gives
the following conclusion.

######  Theorem 2.17.

For $G$-spectra $Y$ and spectra $X$, if $Y$ corresponds to a presheaf
$\scr{P}Y$ under our zigzag of weak equivalences, then the tensor $Y\odot X$
corresponds to the tensor $\scr{P}Y\odot X$.

##  3\. Atiyah duality for finite $G$-sets

It is illuminating to see that we can come very close to constructing an
alternative model for the spectrally enriched category $G\scr{D}$ just by
applying the suspension $G$-spectrum functor $\Sigma^{\infty}_{G}$ to the
category of based $G$-spaces and $G$-maps and then passing to $G$-fixed
points. This is based on a close inspection of classical Atiyah duality
specialized to finite $G$-sets. However, it depends on working in the
alternative category $G\scr{Z}$ of $S_{G}$-modules [3, 12] rather than in the
category $G\scr{S}$ of orthogonal $G$-spectra. Because every object of
$G\scr{Z}$ is fibrant and its suspension $G$-spectra are easily understood, it
is more convenient than $G\scr{S}$ for comparison with space level
constructions. This leads us to a variant, 3.6, of 0.1 that does not invoke
infinite loop space theory. It is more topological and less categorical. It is
also more elementary.

###  3.1. The categories $G\scr{Z}$, $G\scr{D}$, and $\scr{D}_{G}$

Relevant background about $G\scr{Z}$ appears in [6, §3.4] and we just give a
minimum of notation here. In analogy with 2.3, we have the following
specialization of the same general result about stable model categories. It is
discussed in [6, §3.1].

######  Theorem 3.1.

Let $G\scr{D}$ be the full $\scr{Z}$-subcategory of $G\scr{Z}$ whose objects
are cofibrant approximations of the suspension $G$-spectra
$\Sigma^{\infty}_{G}(A_{+})$, where $A$ runs through the finite $G$-sets. Then
there is an enriched Quillen adjunction

$\textstyle{\mathbf{Pre}(G\scr{D},\scr{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{T}}$$\textstyle{G\scr{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathbb{U}}$

and it is a Quillen equivalence.

Here $G\scr{D}$ is isomorphic to $(\scr{D}_{G})^{G}$, where $\scr{D}_{G}$ is a
full $\scr{Z}_{G}$-subcategory $\scr{D}_{G}$ of $\scr{S}_{G}$. All objects of
$G\scr{Z}$ are fibrant, and we need to choose cofibrant approximations of the
$\Sigma^{\infty}_{G}(A_{+})$. The construction of $G\scr{Z}$ starts from the
Lewis-May category $G\scr{S}\\!p$ of $G$-spectra, and $S_{G}$-modules are
$G$-spectra with additional structure. We have an elementary suspension
$G$-spectrum functor $\Sigma^{\infty}_{G}\colon G\scr{T}\longrightarrow
G\scr{S}\\!p.$ There is a left adjoint $\mathbb{F}\colon
G\scr{S}\\!p\longrightarrow G\scr{Z}$, which is a Quillen equivalence [3, 12].
Define $\mathbf{\Sigma}^{\infty}_{G}\colon G\scr{T}\longrightarrow G\scr{Z}$
to be the composite $\mathbb{F}\circ\Sigma^{\infty}_{G}$. Suspension
$G$-spectra have natural structures as $S_{G}$-modules, and there is a natural
stable equivalence of $S_{G}$-modules
$\gamma\colon\mathbf{\Sigma^{\infty}_{G}}X\longrightarrow\Sigma^{\infty}_{G}X.$
Viewing $\Sigma^{\infty}_{G}$ as a functor $G\scr{T}\longrightarrow G\scr{Z}$,
it is strong symmetric monoidal. However, the $\Sigma^{\infty}_{G}X$ are not
cofibrant. The functor $\mathbf{\Sigma^{\infty}_{G}}$ takes based $G$-CW
complexes $X$, such as $A_{+}$ for a finite $G$-set $A$, to cofibrant
$S_{G}$-modules. Therefore $\mathbf{\Sigma^{\infty}_{G}}$ may be viewed as a
cofibrant replacement functor for $\Sigma^{\infty}_{G}$. In particular, we
write $\mathbf{S_{G}}=\mathbf{\Sigma^{\infty}_{G}}S^{0}$ and have a cofibrant
approximation $\gamma\colon\mathbf{S_{G}}\longrightarrow S_{G}$ of the unit
object $S_{G}$. Moreover, the cofibrant approximation
$\mathbf{\Sigma^{\infty}_{G}}(A_{+})$ is isomorphic to
$\mathbf{S_{G}}\wedge\Sigma^{\infty}_{G}(A_{+})$ over
$\Sigma^{\infty}_{G}(A_{+})$.

As before, we consider finite $G$-sets $A$, $B$, and $C$, but we now agree to
write

$\mathbb{A}=\mathbf{\Sigma^{\infty}_{G}}A_{+},\ \ \
\mathbb{B}=\mathbf{\Sigma^{\infty}_{G}}B_{+},\ \ \ \text{and}\ \ \
\mathbb{C}=\mathbf{\Sigma^{\infty}_{G}}C_{+}.$

The $\mathbb{A}$ are bifibrant objects of $G\scr{Z}$ and we let $G\scr{D}$ and
$\scr{D}_{G}$ be the full subcategories of $G\scr{Z}$ and $\scr{Z}_{G}$ whose
objects are the $S_{G}$-modules $\mathbb{A}$, where $A$ runs over the finite
$G$-sets. Then $\scr{D}_{G}$ is enriched in $G\scr{Z}$ and
$G\scr{D}=(\scr{D}_{G})^{G}$ is enriched in the category $\scr{Z}$ of
$S$-modules. The functor $\mathbf{\Sigma^{\infty}_{G}}$ is almost strong
symmetric monoidal. Precisely, by [6, 3.9] there is a natural isomorphism

(3.1)
$\mathbb{A}\wedge\mathbb{B}\cong\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
B)_{+}$

with appropriate coherence properties with respect to associativity and
commutativity. Since $S_{G}$ is the unit for the smash product, we can compose
with

$\gamma\wedge\operatorname{id}\colon\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
B)_{+}\longrightarrow\mathbf{\Sigma^{\infty}_{G}}(A\wedge B)_{+}$

to give a pairing as if $\mathbf{\Sigma^{\infty}_{G}}$ were a lax symmetric
monoidal functor. However, the map $\gamma\colon\mathbf{S_{G}}\longrightarrow
S_{G}$ points the wrong way for the unit map of such a functor.

###  3.2. Space level Atiyah duality for finite $G$-sets

To lift the self-duality of $Ho\scr{D}_{G}$ to obtain a new model for
$\scr{D}_{G}$, we need representatives in $G\scr{Z}$ for the maps

$\eta\colon S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}\ \ \ \text{and}\ \
\ \varepsilon\colon\mathbb{A}\wedge\mathbb{A}\longrightarrow S_{G}$

in $\text{Ho}G\scr{Z}$ that express the duality there. The map $\varepsilon$
is induced from the elementary map $\varepsilon$ of 1.23. The observation that
it plays a key role in Atiyah duality seems to be new. The definition of
$\eta$ requires desuspension by representation spheres.

Let $A$ be a finite $G$-set and let $V=\mathbb{R}[A]$ be the real
representation generated by $A$, with its standard inner product, so that
$|a|=1$ for $a\in A$. Since we are working on the space level, we may view
$A_{+}\wedge S^{V}$ as the wedge over $a\in A$ of the spaces (not $G$-spaces)
$\\{a\\}_{+}\wedge S^{V}$, with $G$ acting by $g(a,v)=(ga,gv)$. There is no
such wedge decomposition after passage to $G$-spectra.

######  Definition 3.2.

Recall that $\varepsilon\colon(A\times A)_{+}\longrightarrow S^{0}$ is the
$G$-map defined by $\varepsilon(a,b)=\ast$ if $a\neq b$ and
$\varepsilon(a,a)=1$. Recall too that $(A\times B)_{+}$ can be identified with
$A_{+}\wedge B_{+}$ and that the functor $\mathbf{\Sigma^{\infty}_{G}}$ is
almost strong symmetric monoidal. We shall also write $\varepsilon$ for the
composite map of $S_{G}$-modules

(3.2)
$\textstyle{\mathbb{A}\wedge\mathbb{A}\cong\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}(A\times
A)_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\mathbf{\Sigma^{\infty}_{G}}\varepsilon}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{S_{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\gamma}$$\textstyle{S_{G}\wedge
S_{G}\cong S_{G},}$

where the unlabeled isomorphisms are two instances of (3.1).

######  Definition 3.3.

Embed $A$ as the basis of the real representation $V=\mathbb{R}[A]$. The
normal bundle of the embedding is just $A\times V$, and its Thom complex is
$A_{+}\wedge S^{V}$. We obtain an explicit tubular embedding $\nu\colon
A\times V\longrightarrow V$ by setting

$\nu(a,v)=a+(\rho(|v|)/|v|)v,$

where $\rho\colon[0,\infty)\longrightarrow[0,d)$ is a homeomorphism for some
$d<1/2$; $\nu$ is a $G$-map since $|gv|=|v|$ for all $g$ and $v$. Applying the
Pontryagin-Thom construction, we obtain a $G$-map $t\colon
S^{V}\longrightarrow A_{+}\wedge S^{V}$, which is an equivariant pinch map

$S^{V}\longrightarrow\vee_{a\in A}S^{V}\cong A_{+}\wedge S^{V}.$

To be more precise, after collapsing the complement of the tubular embedding
to a point, we use $\nu^{-1}$ to expand each small homeomorphic copy of
$S^{V}$ to the canonical full-sized one; explicitly, if $|w|<d$, then

$\nu^{-1}(a+w)=(a,(\rho^{-1}(|w|)/|w|)w).$

The diagonal map on $A$ induces the Thom diagonal $\Delta\colon A_{+}\wedge
S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}$, and we let

(3.3) $\eta\colon S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}$

be the composite $\Delta\circ t$. Explicitly,

(3.4) $\eta(v)=\left\\{\begin{array}[]{ll}(a,a,(\rho^{-1}(|w|)/|w|)w)&\mbox{if
$v=a+w$ where $a\in A$ and $|w|<d$}\\\
\ast&\mbox{otherwise.}\end{array}\right.$

The negative sphere $G$-spectrum $S^{-V}$ in $G\scr{S}\\!p$ is obtained by
applying the left adjoint of the $V^{th}$-space functor to $S^{0}$, and
$S_{G}$ is isomorphic to $S^{V}\odot S^{-V}$ (see [11, I.4.2] and [12,
IV.2.2]). Taking the tensor of $\eta$ with $S^{-V}$ we obtain a map of
$G$-spectra

$S_{G}\cong S^{V}\odot S^{-V}\longrightarrow(A_{+}\wedge A_{+}\wedge
S^{V})\odot S^{-V}\cong(A_{+}\wedge A_{+})\odot
S_{G}\cong\Sigma^{\infty}_{G}(A_{+}\wedge A_{+}).$

Applying the functor $\mathbb{F}$ to this map and smashing with
$\mathbf{S_{G}}$ we obtain the second map in the diagram

(3.5) $\textstyle{S_{G}\cong S_{G}\wedge
S_{G}}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{S_{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\gamma}$$\scriptstyle{\eta}$$\textstyle{\mathbf{S_{G}}\wedge\mathbf{\Sigma^{\infty}_{G}}{(A\times
A)_{+}}\cong\mathbb{A}\wedge\mathbb{A}.}$

The following result is a reminder about space level Atiyah duality. The
notion of a $V$-duality was defined and explained for smooth $G$-manifolds in
[11, §III.5].

######  Proposition 3.4.

The maps

$\eta\colon S^{V}\longrightarrow A_{+}\wedge A_{+}\wedge S^{V}\ \ \text{and}\
\ \varepsilon\wedge\operatorname{id}\colon A_{+}\wedge A_{+}\wedge
S^{V}\longrightarrow S^{V}$

specify a $V$-duality between $A_{+}$ and itself.

###### Proof.

This could be proven from scratch by proving the required triangle identities,
but in fact it is a special case of equivariant Atiyah duality for smooth
$G$-manifolds, $A$ being a $0$-dimensional example. Our specification of
$\eta$ is a specialization of the description of $\eta$ for a general smooth
$G$-manifold $M$ given in [11, p. 152]. We claim that our
$\varepsilon\wedge\operatorname{id}$ is a specialization of the definition of
$\varepsilon$ for a general smooth $G$-manifold given there. Indeed, letting
$s$ be the zero section of the normal bundle $\nu$ of the embedding
$A\subset\mathbb{R}[A]=V$, we have the composite embedding

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{A\times
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s\times\mathrm{id}}$$\textstyle{(A\times
V)\times A\cong A\times A\times V.}$

The normal bundle of this embedding is $A\times V$, and we may view

$\Delta\times{\mathrm{id}}\colon A\times V\longrightarrow A\times A\times V$

as giving a big tubular neighborhood. The Pontryagin-Thom map here is obtained
by smashing the map $r\colon(A\times A)_{+}\longrightarrow A_{+}$ that sends
$(a,b)$ to $a$ if $a=b$ and to $\ast$ if $a\neq b$ with the identity map of
$S^{V}$. Composing with the map induced by the projection $\pi\colon
A_{+}\longrightarrow S^{0}$ that sends $a$ to $1$, this gives
$\varepsilon\wedge\operatorname{id}$. We observed this factorization of
$\varepsilon$ in 1.23 and we have used it before, in the proof of 2.11. ∎

Tensoring with $S^{-V}$, applying the functor
$\mathbf{S_{G}}\wedge\mathbb{F}$, and composing with $\gamma$, we obtain the
explicit duality maps in $G\scr{Z}$ displayed in (3.2) and (3.5).

###  3.3. The weakly unital categories $G\scr{A}$ and $\scr{A}_{G}$

Since the $G$-spectra $\mathbb{A}$ are self-dual,
$F_{G}(\mathbb{A},\mathbb{B})$ is naturally isomorphic to
$\mathbb{B}\wedge\mathbb{A}$ in $\text{Ho}G\scr{Z}$, and the composition and
unit

(3.6) $F_{G}(\mathbb{B},\mathbb{C})\wedge
F_{G}(\mathbb{A},\mathbb{B})\longrightarrow F_{G}(\mathbb{A},\mathbb{C})\ \ \
\text{and}\ \ \ S_{G}\longrightarrow F_{G}(\mathbb{B},\mathbb{B})$

can be expressed as maps

(3.7)
$\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{C}\wedge\mathbb{A}\
\ \ \text{and}\ \ \ S_{G}\longrightarrow\mathbb{A}\wedge\mathbb{A}$

in $\text{Ho}G\scr{Z}$. We want to understand these maps in terms of duality
in $G\scr{Z}$, without use of infinite loop space theory. However, since we
are working in $G\scr{Z}$, we must take the isomorphisms (3.1) and the
cofibrant approximation $\gamma\colon\mathbf{S}_{G}\longrightarrow S_{G}$ into
account, and we cannot expect to have strict units. The notion of a weakly
unital enriched category was introduced in [5, §3.5] to formalize what we see
here.

Thus we shall construct a weakly unital $G\scr{Z}$-category $\scr{A}_{G}$ and
compare it with $\scr{D}_{G}$. The $G$-fixed category $G\scr{A}$ will be a
weakly unital $\scr{Z}$-category.999Mnemonically, the $\scr{A}$ stands for
Atiyah.

The objects of $\scr{A}_{G}$ and $G\scr{A}$ are the $S_{G}$-modules
$\mathbb{A}$ for finite $G$-sets $A$. The morphism $S_{G}$-modules of
$\scr{A}_{G}$ are
$\scr{A}_{G}(\mathbb{A},\mathbb{B})=\mathbb{B}\wedge\mathbb{A}$. Composition
is given by the maps

(3.8)
$\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}\colon\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{C}\wedge\mathbb{A},$

where $\varepsilon$ is the map of (3.2); compare 2.11.

As recalled in §1.5, the adjoint
$\tilde{\varepsilon}\colon\mathbb{A}\longrightarrow
D\mathbb{A}=F_{G}(\mathbb{A},S_{G})$ of $\varepsilon$ is a stable equivalence,
and we have the composite stable equivalence

(3.9)
$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})\colon\mathbb{B}\wedge\mathbb{A}\longrightarrow\mathbb{B}\wedge
D\mathbb{A}\longrightarrow F_{G}(\mathbb{A},\mathbb{B}).$

Formal properties of the adjunction ($\wedge$,$F_{G}$) give the following
commutative diagram in $G\scr{Z}$, which uses $\delta$ to compare composition
in $\scr{A}_{G}$ with composition in $\scr{D}_{G}$.

(3.10)
$\textstyle{\mathbb{C}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\scriptstyle{\mathrm{id}\wedge\tilde{\varepsilon}\wedge\mathrm{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{C}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{C}\wedge
D\mathbb{B}\wedge\mathbb{B}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\wedge\varepsilon\wedge\mathrm{id}}$$\scriptstyle{\zeta\wedge\zeta}$$\textstyle{\mathbb{C}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{F_{G}(\mathbb{B},\mathbb{C})\wedge
F_{G}(\mathbb{A},\mathbb{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{F_{G}(\mathbb{A},\mathbb{C})}$

At the bottom, we do not know that the function $S_{G}$-modules or their smash
product are cofibrant, but all objects at the top are cofibrant and thus
bifibrant. In general, to compute the smash product of $G$-spectra $X$ and $Y$
in the homotopy category, we should take the smash product of cofibrant
approximations $QX$ and $QY$ of $X$ and $Y$. Since all objects of $G\scr{Z}$
are fibrant, to compute a map $X\wedge Y\longrightarrow Z$ in the homotopy
category, we should represent it by a map $QX\wedge QY\longrightarrow QZ$ and
take its homotopy class. The diagram displays such a cofibrant approximation
of the composition in $\scr{D}_{G}$.

The unit $S_{G}\longrightarrow F_{G}(\mathbb{A},\mathbb{A})$ of $\scr{A}_{G}$
is represented by the (formal) composite

(3.11)
$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\tilde{\varepsilon}}$$\textstyle{\mathbb{A}\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})}$

that is obtained by inverting the map $\gamma\wedge\gamma$ in (3.5) to obtain
the map denoted $\eta$. The weak unital property is a way of expressing the
unital property by maps in $\scr{Z}_{G}$, without use of inverses in
$Ho\scr{Z}_{G}$. This is a bit tedious. Here are the details.

######  Definition 3.5.

Let $V=\mathbb{R}[A]$. For $a\in A$, define $\xi_{a}\colon\\{a\\}_{+}\wedge
S^{V}\longrightarrow\\{a\\}_{+}\wedge S^{V}$ by

(3.12)
$\xi_{a}(a,v)=\left\\{\begin{array}[]{ll}(a,(\rho^{-1}(|w|)/|w|)w)&\mbox{if
$v=a+w$ and $|w|<d$}\\\ \ast&\mbox{otherwise,}\end{array}\right.$

where $\rho$ is as in 3.3. Then the wedge of the $\xi_{a}$ is a $G$-map

(3.13) $\xi\colon A_{+}\wedge S^{V}\longrightarrow A_{+}\wedge S^{V};$

$\xi$ is $G$-homotopic to the identity map of $A_{+}\wedge S^{V}$ via the
explicit $G$-homotopy

$h(a,v,t)=\left\\{\begin{array}[]{ll}(a,v)&\mbox{if $t=0$ or $v=a$}\\\
(a,(1-t)v+t(\rho^{-1}(t|w|)/|w|)w)&\mbox{if $v=a+w$ and $t|w|<d$}\\\
\ast&\mbox{otherwise.}\end{array}\right.$

With $\eta$ as specified in (3.3), easy and perhaps illuminating inspections
show that the following unit diagrams already commute in $G\scr{T}$, before
passage to homotopy. In both, $A$ and $B$ are finite $G$-sets. In the first,
$V=\mathbb{R}[A]$. In the second, $V=\mathbb{R}[B]$ and we move $S^{V}$ from
the right to the left for clarity.

$\textstyle{B_{+}\wedge A_{+}\wedge
S^{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\eta_{A}}$$\scriptstyle{\operatorname{id}\wedge\xi_{A}}$$\textstyle{B_{+}\wedge
A^{3}_{+}\wedge
S^{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{B_{+}\wedge
A_{+}\wedge S^{V}}$ and
---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces S^{V}\wedge
B_{+}\wedge
A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{B}\wedge\operatorname{id}_{A}}$$\scriptstyle{\eta_{B}\wedge\operatorname{id}}$$\textstyle{S^{V}\wedge
B^{3}_{+}\wedge
A_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\operatorname{id}\wedge\varepsilon\wedge\operatorname{id}}$$\textstyle{S^{V}\wedge
B_{+}\wedge A_{+}}$

Tensoring with $S^{-V}$ and using the natural isomorphisms

$(X\wedge S^{V})\odot S^{-V}\cong X\odot S_{G}\cong\Sigma^{\infty}_{G}X$

for based $G$-spaces $X$, we see that the space level $G$-equivalence $\xi$
induces a spectrum level $G$-equivalence
$\xi\colon\mathbb{A}\longrightarrow\mathbb{A}$.

Tensoring with $S^{-V}$ and using (3.1) to pass to smash products of
$S_{G}$-modules, a little diagram chase shows that the previous pair of
diagrams in $G\scr{T}$ gives rise to the following pair of commutative
diagrams in $G\scr{Z}$. These express the unit laws for a weakly unital
$G\scr{Z}$-category $\scr{A}_{G}$ [5, §3.5] with objects the $\mathbb{A}$ and
composition as specified in (3.8). The cited unit laws allow us to start with
any chosen cofibrant approximation $\gamma\colon QS_{G}\longrightarrow S_{G}$
of the unit $S_{G}$, and we are led by (3.5) to choose our cofibrant
approximation to be
$\gamma\wedge\gamma\colon\mathbf{S_{G}}\wedge\mathbf{S_{G}}\longrightarrow
S_{G}\wedge S_{G}\cong S_{G}.$ Using the notation $\gamma\colon
QS_{G}\longrightarrow S_{G}$ for this map, we obtain the required diagrams

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{B}\wedge\mathbb{A}\wedge
QS_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\xi\wedge\gamma}$$\scriptstyle{\operatorname{id}\wedge\eta}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{\mathbb{B}\wedge\mathbb{A}\wedge
S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{B}\wedge\mathbb{A}}$
and $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
QS_{G}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\xi\wedge\operatorname{id}}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ}$$\textstyle{S_{G}\wedge\mathbb{B}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{B}\wedge\mathbb{A}.}$

Taking $A=S^{0}$ in our second space level diagram and changing $B$ to $A$, we
also obtain the following commutative diagrams in $G\scr{Z}$, where the second
diagram is adjoint to the first.

(3.14) $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
QS_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\wedge\xi}$$\scriptstyle{\eta\wedge\operatorname{id}}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\varepsilon}$$\textstyle{S_{G}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{A}}$
and
$\textstyle{QS_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\eta}$$\textstyle{\mathbb{A}\wedge\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\wedge\tilde{\varepsilon}}$$\textstyle{bA\wedge
D\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{S_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{G}(\xi,\operatorname{id})}$$\textstyle{F_{G}(\mathbb{A},\mathbb{A})}$

Here $\eta$ at the bottom right is adjoint to the identity map of
$\mathbb{A}$. In effect, this uses
$\delta=\zeta\circ(\operatorname{id}\wedge\tilde{\varepsilon})$ to compare the
actual unit $\eta$ in $\scr{D}_{G}$ at the top with the weak unit in
$\scr{A}_{G}$, which is given by the interrelated maps $\eta$, $\gamma$, and
$\xi$.

###  3.4. The category of presheaves with domain $G\scr{A}$

The diagrams (3.10) and (3.14) show that the maps
$\delta\colon\mathbb{A}\wedge\mathbb{B}\longrightarrow
F_{G}(\mathbb{A},\mathbb{B})$ specify a map of weakly unital
$\scr{Z}_{G}$-categories from the weakly unital $\scr{Z}_{G}$-category
$\scr{A}_{G}$ to the (unital) $\scr{Z}_{G}$-category $\scr{D}_{G}$. Passing to
$G$-fixed points, we obtain a weakly unital $\scr{Z}$-category $G\scr{A}$ and
a map $\delta\colon G\scr{A}\longrightarrow G\scr{D}$ of weakly unital
$\scr{Z}$-categories. Weakly unital presheaves and presheaf categories are
defined in [5, 3.25]. By [5, 3.26], we obtain the same category of presheaves
$\scr{Z}^{G\scr{D}}$ using unital or weakly unital presheaves. Since $\delta$
is an equivalence, we can adapt the methodology of [5, §2] to prove the
following result. However, since we find the use of weakly unital categories
unpleasant and our main result 1.9 more satisfactory, we shall leave the
details to the interested reader. Nevertheless, it is this equivalence that
best captures the geometric intuition behind our results.

######  Theorem 3.6.

The categories $\mathbf{Pre}(G\scr{A},\scr{Z})$ and
$\mathbf{Pre}(G\scr{D},\scr{Z})$ are Quillen equivalent.

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