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\title{Notes from the dark ages}

\author{J.P. May}

\begin{document}

\tableofcontents

Since I don't have time to think right now, this will be a direct transcription
of my notes from the early 1980's on permutative $G$-categories. No changes from
what I wrote then, minor updating comments in brackets. 

\section{The categorical $E_{\infty}$ $G$-operad $\sD_G$}

We begin by constructing a monstrous big $G$-monoid of canonical $G$-sets.

[Draft one(?)] Thus let $M$ be the free monoid generated by 
$\coprod_{H\subset G}^{\infty}(G/H)$,
the disjoint union of a countably infinite number of copies of each $G/H$, 
with unit element the first copy of $G/G$. Let $^{j}M$ denote the disjoint
union of $j$ copies of $M$. 

[Draft two(?)] Thus let $M$ be the disjoint union of countably many copies of each
finite product $G/H_1\times \cdots \times G/H_r$, where the $H_i$ run over all 
finite sets of subgroups of $G$. The product on $M$ is given by the identity maps
\[\xymatrix{ (G/H_1\times \cdots \times G/H_r)_m\times G/J_1\times \cdots \times G/J_s)_n
\ar[d]\\
 (G/H_1\times \cdots \times G/H_r \times G/J_1\times \cdots \times G/J_s)_{m+n}\\} \]
If $r$ and $m$ (or $s$ and $n$) are $1$ and $H_1$ (or $J_1$) is $G/G$, delete the 
corresponding point from the product on the right. Here copies are indexed on the 
positive integers and the subscripts $m$ and $n$ specify which copies we are 
considering. The first copy  of the point $G/G$ is a two-sided identity element,
and this multiplication is evidently associative. Let $^{j}M$ denote the disjoint
union of $j$ copies of $M$. 

[There is a Draft 3 below that is perhaps the definitive version of $M$.]

Define a category $\sS_G(j)$ as follows. The objects are ordered $j$-tuples
$\langle m_1,\cdots,m_j \rangle$ of distinct points of the ambient $G$-space
$^{j}M$. [$i^{\text{th}}$ point in $i^{\text{th}}$ copy ?] We note that $M$ is a $G$-monoid with action inherited from that on
the generating set via the requirement that the multiplication $M\times M\rtarr M$
be a $G$-map, where $G$ acts diagonally on $M\times M$. The morphisms of $\sS_G(j)$
are bijections of (ordered) tuples, so that there is precisely one morphism from any
one object to any other. In particular, every object is initial and terminal, so
the category is contractible. Define a left $G$-action and a right $\SI_j$-action
on $\sS_G(j)$ by the specifications
\[  g\langle m_1,\cdots,m_j\rangle = \langle gm_1,\cdots,gm_j\rangle \ \ \ \text{and}
\ \ \ \langle m_1,\cdots,m_j\rangle \si = \langle m_\si(1),\cdots,m_\si(j) \rangle \]
on objects. Visibly, these actions commute and so (by use of inverses in $\SI_j$) 
specify a left action of $G\times \SI_j$. Let $G\times \SI_j$ act on morphisms the
only way possible (since there is a unique morphism between each pair of objects).

For a representation $\rh\colon H\rtarr \SI_j$, $H\subset G$, the fixed subcategory
$\sS_G(j)^{(H,\rh)}$ is the full subcategory with objects those 
$\langle m_1,\cdots,m_j\rangle$ with
\[ h\langle m_1,\cdots,hm_j\rangle = \langle m_{\rh(h)(1)},\cdots,m_{\rh(h)(j)}\rangle \]
for all $h\in H$. This category is nonempty and, again, every object is initial. Thus
each $\sS_G(j)^{(H,\rh)}$ is contractible. The action by $\SI_j$ is free.

Define functors 
\[ \tilde{\ga}\colon \sS_G(k) \times \sS_G(j_1)\times \sS_G(j_k) \rtarr
\sS_G(j_1 + \cdots + j_k) \]
by 
\[ \tilde{\ga}( \langle m_1,\cdots,m_k\rangle ; \langle n_{1,1}, \cdots, n_{1,j_1}\rangle ,
\cdots \langle n_{k,1}, \cdots, n_{k,j_k}\rangle) \]
\[= \langle m_1n_{1,1}, \cdots, m_1n_{1,j_1}, \cdots, m_kn_{k,1}, \cdots, m_kn_{k,j_k}\rangle. \]
This requires interpretation. Each $m_r$ is a point of $^{k}M$ and so lies in a copy of $M$.
{\em We pay no attention to which copy.} Each $n_{r,s}$ is a point of $^{j_r}M$. We use the
evident ``sum'' identification
\[ {^{j_1}M} \amalg \cdots \amalg  {^{j_k}M} = ^{j_1 + \cdots + j_k}M \]
to regard $n_{r,s}$ as a point in some particular copy of $M$ in $^{j_1 + \cdots + j_k}$.
Then $m_rn_{r,s}$ is interpreted as lying in that same copy of $M$, and is of course 
obtained by the monoid structure. The use of blocks ensures that we obtain a
$(j_1+\cdots + j_k)$-tuple of distinct points.

The unity and associativity formulae for an operad are direct consequences of the fact
that $M$ is a monoid, and the equivariance formulae are formal trivialities (?!).

We define $\sD_G(j) = B\sS_G(j)$ and have an $E_{\infty}$ $G$-operad. 

Note that we have an orbit category $\sS_G(j)/\SI_j$ specified simply by use of 
unordered $j$-tuples (or $j$-pointed sets).  Let $\sE_G$ be the coproduct over $j$
of these categories. We call $\sE_G$ the Burnside category of $G$. It is $G$-permutative,
or will be so as soon as I define what that means, and $B\sE_G$ will equal $D_GS^0$ as a
$\sD_G$-space. The associated $G$-spectrum will be $S_G = \SI^{\infty}_G S^0$. 

The category $\tilde{\SI}_j$ may be thought of as having as objects all ordered $j$-tuples
of $^{j}\{e\}$, where $\{e\}$ is the unit monoid. $M\{e\}$ above is bigger, 
$\{e\}\subset M\{e\}$ embedding the old $\sD$ in $\sD_{\{e\}}$ as an equivalent sub
$E_{\infty}$ operad.

\section{Equivariant algebraic $K$-theory}

Let $R$ be a commutative ring, $G$ a group (possibly both topologized?) and let $R[G]$ 
denote the group ring. Let $\otimes$ mean $\otimes_R$ and let $R[G/H]$ be the evident
left $R[G]$-module. Let $\sG\sL_G(R)$ be the disjoint union over $j\geq 0$ of the
following categories. Let $R[M]$ be the monoid ring of the monoid $M$. Thus $R[M]$ is the
direct sum of countably many copies of each left $R[G]$-module
\[ R[G/H_1]\otimes \cdots \otimes R[G/H_r].\]
Let $^{j}R[M]$ denote the direct sum of
$j$ copies of $R[M]$. The objects of the $j^{\text{th}}$ category $\sG\sL_G(R,j)$ are the 
$j$-dimensional free sub $R$-modules of  $^{j}R[M]$ with specified {\em unordered} basis
in $^{j}M$. The morphisms are the isomorphisms of $R$-modules. The category $\sG\sL_G(R,j)$
is given the evident left $G$-action, induced from that on $^{j}M$ by conjugation.

It follows that $\sG\sL_G(R,0)$ is the trivial $G$-category with one object $0$. There is 
an evident strictly associative and unital sum operation
\[ \oplus: \sG\sL_G(R,j)\times \sG\sL_G(R,k) \rtarr \sG\sL_G(j+k). \]
It admits the natural commutativity isomorphism $c$ specified on $(V,W)$ in the obvious
way, $c(v+w) = w+v$. All of this structure is $G$-equivariant.

Define functors 
\[ \tilde{\tha}_k\colon \sS_G(k)\times \sG\sL_G(R)^k\rtarr \sG\sL_G(R) \]
as follows. Write objects of $\sG\sL_G(R)$ as free $R$-modules on the objects 
of $\amalg \sS_G(j)/\SI_j$. Indeed we may think of these categories as having
the same objects. Due to the diagonals implicit in $\tilde{\ga}$, the $\tilde{\ga}$
induce $\tilde{\tha}_k$ in an evident way on objects, and then $\tilde{\tha}_k$ is
$G\times \SI_k$-equivariant. On morphisms
\[ (f;g_1,\cdots,g_k)\colon (m;n_1,\cdots,n_k) \rtarr (m';n'_1,\cdots,n'_k), \]
define $\tilde{\tha}_k$ as the evident composite
\[ \xymatrix{
R[\tilde{\ga}(m;n_1\cdots, n_k)] \ar[d]^{\iso}\\
R[n_1]\oplus\cdots \oplus R[n_k]\ar[d]^{g_1\oplus \cdots \oplus g_k}\\
R[n'_1]\oplus\cdots \oplus R[n_k] \ar[d]^{\iso}\\
R[\tilde{\ga}(m';n'_1\cdots, n'_k)].\\}\]
The unlabelled isomorphisms are given on bases in the in the obvious way [Huh??]
$(m_1n_1\amalg\cdots \amalg m_kn_k \leftrightarrow n_1\amalg \cdots n_k$, etc).
This is $G\times \SI_k$-equivariant, and easy verifications show that we obtain
an action of $\sD_G$ on $B\sG\sS_G(R)$ via $B\tilde{\tha}$. 

\section{The Burnside cate�gory $\sE_G = \amalg _{j\geq 0}\sE_G(j)$: bipermutative version}

[Draft 3: not integrated with drafts above, a bit different notation,
maybe better.]

Let $S$ be the disjoint union over $H\subset G$ of the disjoint union of countably
many copies of $G/H$. Let $M$ be the free monoid generated by the set $S$. The 
evident $G$ action on $S$ and the trivial $G$ action on the unit $1$ induce a $G$
action on $M$ such that the product is a $G$-map. Let $^{j}M$ be the disjoint union
of $j$ copies of $M$. 

The objects of $\sE_G(j)$ are the (unordered) $j$-point subsets of $^{j}M$ with
one point in each copy of $M$. The morphisms are the bijections. $G$ acts on $\sE_G(j)$
by translation of objects and conjugation of morphisms. The sum
\[ \oplus \colon \sE_G(j) \times \sE_G(k)\rtarr \sE_G(j+k) \]
is obtained by identifying $^{j}M\amalg {^{k}\!M}$ with $^{j+k}M$ in the obvious way and
taking disjoint unions of sets and maps. The category $\sE_G(0)$ is 
the trivial category with one object $0$ (thought of as the empty set) and is a two-sided
unit for $\oplus$. Clearly $\oplus$ is associative, and the obvious transposition 
reordering of copies of $M$ as $^{j+k}M$ gives a natural commutativity isomorphism.

Define 
\[ \otimes\colon \sE_G(j) \times \sE_G(k)\rtarr \sE_G(jk) \] 
as follows.
Identify $\{1,2,\cdots, jk\}$ with $\{(a,b)| 1\leq a\leq j, 1\leq b\leq k\}$,
ordered lexicographically. For objects $P$ of $\sE_G(j)$ and $Q$ of $\sE_G(k)$, 
the element of $P\otimes Q$ in the $(a,b)^{\text{th}}$ copy of $M$ is the product
in $M$ of the element of $P$ in the $(a,b)^{\text{th}}$ copy of $M$ is the product
in $M$ of the element of $P$ in the $a^{\text{th}}$ copy of $M$ and the element 
of $Q$ in the $b^{\text{th}}$ copy of $M$. The definition of $\otimes$ on maps
is then evident, $(f\otimes g)(pq) = f(p)g(q)$. The set $\{1\}$ regarded as an
object of $\sE_G(1)$ is a two-sided unit for $\otimes$. Again, $\otimes$ is associative,
and the commutativity isomorphism $P\otimes Q\rtarr Q\otimes P$ sends $pq$ in the 
$(a,b)^{\text{th}}$ copy to $qp$ in the $(b,a)^{\text{th}}$ copy of $M$; its
naturality is clear. 

The right distributive law $(P\oplus Q)\otimes R = (P\otimes R)\oplus (Q\otimes R)$ 
is satisfied and the coherence diagrams required of a bipermutative category are 
easily seen to commute. The key to understanding this definition is to keep firmly
in mind that pigeonholing of points in copies of $M$ is not to be regarded as 
prescribing an ordering of the points of objects. Thus $\sE_G(j)$ is the
$\SI_j$-orbit category of an analogous category of ordered sets. The size of $S$
ensures that $B\sE_G(j)$ is a model for $B(G,\SI_j)$. 

\section{Permutative $G$-categories}

\subsection{Segalic definition}
Let $(\sA,\Box,\ast,c)$ be a weakly permutative $G$-category [undefined, 
naive permutative category with $G$ action] and let $\tilde{\sA}_n$ be its
weakly permutative $G$-category of $n$-fold sums as specified in Step 1
of Construction 10. [Reference to: J.P. May. The spectra associated to 
permutative categories.  Topology 17(1978), 225--228.] Step 2 of Construction 10
gives $\bar{\sA} = \{\bar{\sA}_n\}$ a structure of weak $\sF G$-category [naive
version of Segal $\GA$-$G$-space]. To ensure that this is actually an 
$\sF G$-category, so that $B\bar{\sA}$ is an $\sF G$-space, we need functors
$\nu_{\rh}\colon (\sA^n)^{\ph}\rtarr (\bar{\sA}_n)^{\ph}$ which provide 
inverse equivalences to the restrictions 
$\de^{\rh} \colon (\bar{\sA}_n)^{\ph}\rtarr (\sA^n)^{\ph}$ for each 
$\rh\colon H\rtarr \SI_n$. If $\rh$ is trivial, we can take 
$\nu_{\rh} = \nu^{\rh}$. [Notation $\nu$ from op cit]. Otherwise $\nu$ will not be 
$\rh$-equivariant and this fails. 

The existence of the $\nu_{\rh}$ is of importance. They should not be regarded
as part of the structure: we are really concerned only with the properties of $\de$.
This condition, the existence of the $\nu_{\rh}$, provides our first way of 
specifying the notion of a permutative $G$-category.

\subsection{Operadic definition}  
Our second way is simply to require a categorical action of the categorical
$E_{\infty}$ $G$-operad on the $G$-category $\sA$. We do not have to presuppose
a weakly permutative structure on $\sA$, since this results via the embedding
of the translation operad $\tilde{\SI}$ in $\sS_G$. Here we are given the 
commutative diagrams
\[ \xymatrix{
\sS_G(k) \times \sS_G(j_1) \times \cdots \times \sS_G(j_k) \times\sA^j \ar[dd]_{\iso}
\ar[rr]^-{\tilde{\ga}\times \id} & & \sS_G(j)\times \sA^j \ar[d]^{\tilde{\tha}_j} \\
& &  \sA \\
\sS_G(k) \times \sS_G(j_1) \times \sA^{j_1} \cdots \times \cdots\times  
\sS_G(j_k) \times \sA^{j_k} 
\ar[rr]_-{\id\times \tilde{\tha}_{j_1}\times\cdots\times \tilde{\tha}_{j_k}  } 
& & \sS_G(k)\times \sA^k \ar[u]_{\tilde{\tha}_k},  \\} \]
where $j= j_1+\cdots + j_k$. 

Let $e_k$ be the canonical unit object in $\sS_G(k)$, the ordered $k$-tuple in
$^{k}M$ given by the units of the $k$ copies of $M$. Then $e_0$ gives the unit
$\ast\in \sA$, $e_1$ induces the identity of $\sA$, and $e_2$ induces a product
$\Box\colon \sA\times \sA\rtarr \sA$.  The unique isomorphism $e_2\rtarr e_2\ta$
in $\sS_G(2)$, where $\ta\in \SI_2$ is the transposition, induces 
$c\colon\Box\rtarr \Box\com\ta$. The diagram and easy calculations with $\tilde{\ga}$
demonstrate that $(\sA,\Box,\ast,c)$ is a weakly permutative $G$-category.

\subsection{Operadic vs Segalic}
I claim that the operadic structure implies the Segalic properties. That is,
I can construct the $\nu_{\rh}$ if I am given the $\tilde{\tha}_k$.  [Take 
$k=n$ to reconcile notations.] Given $\rh\colon H\rtarr \SI_n$, we use $\rh$ to regard
$\ul{n}$ as an ordered ``$n$-pointed'' $H$-set . Write $\ul{n} = \amalg_{r} n_r H/K_R$
as an $H$-set. Form 
\[ G\times_{H_r} \ul{n} \iso \amalg_{r} n_r G\times_H (H/K_R) \iso \amalg{_r} n_r G/K_r. \]
We have $\ul{n} = H\times_{H} \ul{n}\subset G\times_H \ul{n}$ as a sub $H$-set. Fix an
embedding of $G\times _H \ul{n}$ as a sub $G$-set of $M$ and regard $M$ as the first copy
of $M$ in $^{n}M$.  Define 
\[\nu_{\rh}\colon (\sA^n)^{\rh}\rtarr (\bar{\sA}_n)^{\rh}\]
as follows. [Notations op cit. and/or suspect.] For $s\subset \ul{n}$ with $|s-\{0\}| = q$ and for $A_i\in (\sA^n)^{\rh}$, so that
$hA_i = A_{\rh(h)(i)}$ for all $h\in H$, set
\[ A_s = \tilde{\tha}_q(s) (A_{s_1},\cdots, A_{s_q})\in \sA, \]
where $0 < s_1 < \cdots < s_q$, $s_a\in s$. This makes sense both on objects and, with
$s$ interpreted as $\id_s$, on morphisms. We then have 
\begin{eqnarray*} 
hA_s & = & h\tilde{\tha}_q(s)(A_{s_1},\cdots, A_{s_q}) \\
     & = & \tilde{\tha}_q(hs)(hA_{s_1},\cdots, hA_{s_q})\\
     & = & \tilde{\tha}_q(\rh(h)(s))(A_{\rh(h)(s_1)},\cdots, A_{\rh(h)(s_q)}) = A_{\rh(h)(s)},
\end{eqnarray*}
where the last equality is obtained by using a permutation to rearrange $\rh(h)(s)$ in
increasing order and exploiting the $\SI_q$-equivariance of the the functor $\tilde{\tha}_q$.
To obtain the coherent equivalences $i_{(s,t)}\colon A_{s\cup t} \rtarr A_s\Box A_t$ for 
$s\cup t\subset \ul{n}$, consider the following instance of the basic diagram, 
$|t-\{0\}| = r.$ [|t| = r as written, sic?]
\[ \xymatrix{
\sS_G(2) \times \sS_G(q) \times \sS_G(r) \times\sA^{q+r} \ar[dd]_{\iso}
\ar[rr]^-{\tilde{\ga}\times \id} & & \sS_G(q+r)\times \sA^{q+r} \ar[d]^{\tilde{\tha}_{q+r}} \\
& &  \sA, \\
\sS_G(2) \times \sS_G(q) \times \sA^{q} \times  
\sS_G(r) \times \sA^{r} 
\ar[rr]_-{\id\times \tilde{\tha}_{q}\times\tilde{\tha}_{r}  } 
& & \sS_G(2)\times \sA^2 \ar[u]_{\tilde{\tha}_2} \\} \]
Start with $e_2\in \sS_G(2)$, $s\in \sS_G(q)$, $t\in \sS_G(r)$. Bottom yields
$A_s\Box A_t$. $\tilde{\ga}(e_2; s, t)$ is $s\amalg t$, a $(q+r)$-pointed subset
of $^{q+r}M$, with the points of $s$ in the first copy of $M$ and the points of $t$
in the $(r+1)^{\text{st}}$ copy. If $(u_1,\cdots,u_{q+r})$ is $s\cup t$ arranged
in increasing order, then
\[ A_{s\cup t} = \tilde{\tha}_{q+r}(s\cup t)(A_{u_1},\cdots, A_{u_{q_r}}). \]
Let $\si\in \SI_{q+r}$ be the permutation which when applied to
$( s_1, \cdots, s_q,t_1,\cdots, t_r )$ yields $(u_1,\cdots,u_{q+r})$. There is a unique
morphism in $\sS_G(q+r)$ from $s\cup t$ to $(s\cup t)\si$. Here
$s\cup t\subset M\subset ^{q+r}M$, $M$ being the first copy. This isomorphism, the diagram,
and the equivariance of $\tilde{\tha}_{q+r}$ yield $i_{(s,t)}$ as 
$\tilde{\tha}_{q_r}(j_{(s,t)})$, $j_{(s,t)}\colon s\cup t \iso (s\cup t)\si$.

The coherence of the $i_{(s,t)}$ and their $\rh$-invariance are easy verifications.
Since $\tilde{\tha}_1(s)$ is equivalent to the identity for any $1$-point set $s$,
$\nu_{\rh}\com \de^{\rh}$  is certainly equivlaent to the identity. The requisite
equivalence $\xi\colon \de^{\rh}\com \nu_{\rh} \rtarr \id$ is ``tautological''. 
Given $\langle A_s, i_{(s,t)}\rangle\in (\sA_n)^{\rh}$, the sum structures specify
equivalences
\[ \xymatrix@1{
A_s \ar[r]^-{i} & A_{s_1}\Box \cdots \Box A_{s_q} \ar[r]^-{j^{-1}} 
& \tilde{\tha}_q(s)(A_{s_1},\cdots, A_{s_q}),\\} \]
and these yield the required 
natural equivalence $\xi$ by easy verifications.

\subsection{Note the logical situation}
Nonequivariantly, $\tilde{\SI}$-categories and permutative categories were one and the same
concept.  We can take that as an equivariant definition, via $\sS_G$, but here the notion
becomes vastly more structured than can be captured by the derived $\sF G$-space, for which
the $\nu_{\rh}$'s are not part of the structure. That is, an operad action makes explicit
what is implicit in an $\sF G$-category. Concomitantly, $\sF G$-categories are
presumably much more common. [Not so!]. The ``correct'' notion of permutative category should
be $(\sA,\Box,\ast, c)$ plus ``?'', and it still remains to give the simplest possible ``?'',
with a concomitant symmetric monoidal notion.

\subsection{Reminder on the uniqueness theorem}

However defined, it is vital that $\bar{\sA}_n$ should be permutative when $\sA$ is.
Otherwise uniqueness [of op cit above] fails. This actually seems a bit delicate, and is the 
heart of the theory. No problem with stronger second definition. We define 
$\tilde{\tha}_k\colon \sS_G(k)\times \bar{\sA}_n^k\rtarr \bar{\sA}_n$ termwise. That is,
we let [notations op cit]
\[ \tilde{\tha}_k(m,\langle A^1_s\rangle, \cdots, \langle A^k_s\rangle)_s = 
\tilde{\tha}_k(m, A^1_s,\cdots, A_s^k). \]
$\SI_k$-equivariance, unit, and Adem diagram are clearly inherited. Similarly on morphisms.
The $i^r_{(s,t)}\colon A_{s\cup t}^r \rtarr A^r_s\Box A^r_t$ yield
\[ 
\tilde{\tha}_k(m, i^1_{(s,t)},\cdots, i^k_{(s,t)})\colon 
\tilde{\tha}_k(m,A^1_{(s,t)}, \cdots, A^k_{(s,t)}) \rtarr
\tilde{\tha}_k(m,A^1_s\Box A^1_t, \cdots, A^k_s\Box A^k_t). \]
The target is equal to 
\[ \tilde{\tha}_{2k} (\tilde{\ga}(m; e_2,\cdots, e_2), A^1_s,A^1_t, \cdots, A^k_s, A^k_t).\]
We must compare this with 
\[ \tilde{\tha}_k(m,A^1_s, \cdots, A^k_s) \Box \tilde{\tha}_k(m,A^1_t, \cdots, A^k_t), \]
which is equal to 
\[ \tilde{\tha}_{2k} (\tilde{\ga}(e_2;m,m), A^1_s,\cdots A^k_s, A^1_t,\cdots, A^k_t).\]
We de this via the morhphism
\[ \tilde{\ga}(m;e_2,\cdots,e_2) \rtarr \tilde{\ga}(e_2;m,m)\mu \]
where $\mu$ is the evident shuffle isomorphism. This gives the new $i_{(s,t)}$, and
the coherence follows by standard diagram chases. With this definition, the permutative
$G$-category uniqueness theorem is now fully established. With the first definition,
$\bar{\sA}_n$ seems {\em not} to inherit permutativity. For that, one would need a 
commutativity relation between $\nu_{\rh}\colon (\sA^n)^{\rh}\rtarr (\bar{\sA}_n)^{\rh}$
and $\Box$ sums, and this appears not to drop out automatically.

\subsection{Permutative $G$-category definition and discussion}
[This may have motivated and preceded the introduction of $M$ above.]

Let $S$ be a finite $G$-set and let $\sP(j,S)$ be the category of $j$-pointed ordered
subsets of $S$ and bijective functions. Recall that a weakly permutative $G$-category 
$(\sA,\Box,\ast, c)$ is equivalent to a $G$-action of the category level translation
operad $\tilde{\SI}$. Let $\epz_j$ be the trivial $j$-pointed $G$-set and note that
$\tilde{\SI}_j$ may be identified with $\sP(j,\epz_j)$. $\sA$ is a permutative 
$G$-category if it comes with $G\times \SI_j$-equivariant functors
\[ \tha_{j,s}\colon \sP(j,S)\times \sA^j\rtarr \sA \]
[sometimes abbreviated $\tha_j$] such that 
\begin{enumerate}[(i)]
\item $\sP(j,e_j)\times \sA^j\rtarr \sA $ specifies $(\sA,\Box,\ast, c)$
\item If $S\subset T$, $\tha_{j,S}$ is the composite
\[ \xymatrix@1{ \sP(j,S)\times \sA^j \ar[r] 
& \sP(j,T)\times \sA^j \ar[r]^-{\tha_{j,T}} & \sA.\\}\]
\item The following diagrams commute.
\[ \xymatrix{
\sP(k,T) \times \sP(j_1,S_1) \times \cdots \times \sP(j_k,S_k) \times \sA^j \ar[dd]_{\iso}
\ar[rr]^-{\tilde{\ga}\times \id} & & \sP(j,\coprod_{1\leq r\leq k} T\times S_r)\times \sA^j \ar[d]^{\tilde{\tha}_j} \\
& &  \sA \\
\sP(k, T)\times\sP(j_1,S_1)\times \sA^{j_1} \cdots \times \cdots\times  
\sP(j_k,S_k) \times \sA^{j_k} 
\ar[rr]_-{\id\times \tilde{\tha}_{j_1}\times\cdots\times \tilde{\tha}_{j_k}  } 
& & \sP(k,T)\times \sA^k  \ar[u]_{\tilde{\tha}_k},  \\} \]
where $j= j_1+\cdots + j_k$. Here
\[ \tilde{\ga}(\langle t_1,\cdots, t_k\rangle; \langle s_{1,1}, \cdots, s_{1,j_1}\rangle, \cdots,
 \langle s_{k,1}, \cdots, s_{k,j_k}\rangle) = \langle t_1s_{1,j_1},\cdots, t_1s_{1,j_1},
 \cdots t_ks_{k,1},\cdots, t_ks_{k,j_k} \rangle. \]
\end{enumerate} 
(i) and (ii) applied to $S\subset S\coprod \epz_j$ make clear that we are looking at 
``underlying $j$-fold sums'' with suitably twisted group actions.

Intuition: think of $\tilde{\tha}_j(\langle s_1,\cdots, s_j\rangle, A_1,\cdots, A_j)$ as
a copy of $A_1\Box \cdots \Box A_j$ indexed on $(s_1,\cdots, s_j)$. $G$ acts by sending
$A_1\Box \cdots \Box A_j$ indexed on $(s_1,\cdots, s_j)$ to $gA_1\Box \cdots \Box gA_j$ 
indexed on $(gs_1,\cdots, gs_j)$. In particular, if $|S| = j$, then
\[ (gs_1,\cdots,gs_j) = (s_{\rh(g)(1)},\cdots, s_{\rh(g)(j)})\]
for some representation $\rh\colon G\rtarr \SI_j$. Here
\[\tilde{\tha}_j(\langle gs_1,\cdots, gs_j\rangle, gA_1,\cdots,gA_j)
= g\tilde{\tha}_j(\langle s_1,\cdots, s_j\rangle, A_1,\cdots,A_j).  \]
$gA_1\Box\cdots \Box gA_j$ is indexed on $(s_{\rh(g)(1)},\cdots, s_{\rh(g)(j)})$,
but $\SI_j$-equivariance says this is $gA_{\rh(g)^{-1}(1)}\Box\cdots \Box gA_{\rh(g)^{-1}(j)}$
indexed on $(s_1,\cdots, s_j)$. In other words, taking $S$ given as ordered and thinking of
\[ \tilde{\tha}_j(\langle s_1,\cdots, s_j\rangle, A_1,\cdots,A_j)   \]
as ``$A_1 \Box \cdots^{\rh} \Box A_j$", we see that we are giving these objects of $\sA$ a
$\rh$-twisted $G$-action
\[ g(``{A_1 \Box \cdots^{\rh} \Box A_j}") = ``{gA_{\rh(g)^{-1}(1)}\Box\cdots^{\rh} \Box gA_{\rh(g)^{-1}(j)}}"  \]
To explain [more clearly], perhaps introduce some such notation as 
$A_1(s_1)\Box\cdots \Box A_j(s_j)$.

Actually, this is perhaps an ersatz definition on categorical [size] grounds. Use of all
finite sets makes (ii) and (iii) suspect: they should hold only up to coherent natural
isomorphism perhaps. That is, it is too much to expect that they will hold on the nose.
In particular, verification of (iii) almost demands 
\[ \coprod_{1\leq r\leq k} T\times S_r = T\times (\coprod_{1\leq r\leq k} S_r), \]
whereas they are really canonically isomorphic. Picky, but still it is perhaps better 
to fix one big infinite $S$ and use only it. This can't quite work since $\ga$ cannot
be defined for (iii), but it does work if we use $^j\!S = S\amalg \cdots \amalg S$ and
$\sP(j,{^j\!}S)$ for the $j^{\text{th}}$ ambient category. This leads to equivalent 
(essentially) operad action definiton of $G$-permutative category and thus leads
into such a definition nicely. No problem with exposition that way. [Or so I wrote then.]










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