! , we de*
*fine the
composite (fi, t) O (ff, q) to be (fi O t*ff, t O q).
We define the category of G*-categories as a certain category built out of t*
*he category of
functors from G into the category of small categories. In order to avoid possib*
*le confusion
as to the meaning of "functor" and "natural transformation" where they occur be*
*low, we
define G*-objects in an arbitrary bicomplete category C. We write * for a chos*
*en final
object in this category, and C* for the category of objects of C equipped with *
*a structure
map from *. We think of C* as the category of based objects in C. In our applic*
*ations
C is always either Cat , the category of small categories, or SS , the category*
* of simplicial
sets.
Definition 5.2. A G*-object in C consists of a functor F : G ! C together with *
*a map
* ! F () such that F (m1, . .,.mr) = * whenever any mi = 0 and such that the fo*
*llowing
diagram commutes,
* __=___//_F (0)
| |
| |
fflffl| fflffl|
F ()_____//F (1),
where the left hand map is the given map, the top map is the unique map, the ri*
*ght hand
map is induced by the unique map (0) ! (1) in G, and the bottom map is induced *
*by the
map () ! (1) in G from the unique map 0_! 1_in Inj and the identity map on 1 in*
* F.
24 A. D. ELMENDORF AND M. A. MANDELL
A map of G*-objects F ! G is a natural transformation f: F ! G making the follo*
*wing
diagram commute:
* B
"" BBB
""" BBB
""""" B!!
F ()_____f()____//_G().
We denote the category of G*-objects as G*-C.
We remark that for a G*-object F , the objects F of C are based: the map*
* from *
is the explicitly given one for = (), and the map * = F (0, . .,.0) ! F (m1*
*, . .,.mr)
is induced from the unique map (0, . .,.0) ! (m1, . .,.mr) in Fr for r > 0. It *
*is easy to
see from the universal property of the terminal object and the diagram in the d*
*efinition,
that any map ! in G induces a based map F ! F . Likewise, for a *
*map
f: F ! G in G*-C, the maps F ! G are based for all in G. The follow*
*ing
proposition is now clear.
Proposition 5.3. The category G*-C is the full subcategory of the category of f*
*unctors
G ! C* consisting of those functors F with F (m1, . .,.mr) = * whenever any mi=*
* 0.
In order to define the multicategory structure on G*-objects, we take advant*
*age of
additional structure on the category G: it is actually a permutative category. *
*The product
operation is given on objects by concatenation of tuples, with the obvious exte*
*nsion to
morphisms. We denote this operation as . This allows us to regard a G*-object *
*G as a
functor from Gk = Gx. .x.G to C* by the formula G(, . .,.) = G( . .*
* .).
We will also exploit the smash product (written ^) in C*. For X and Y object*
*s of
C*, the smash product X ^ Y is the pushout
(X x *) q (* x Y )____//X x Y
| |
| |
fflffl| fflffl|
*_____________//X ^ Y.
It is well-known that ^ is a closed symmetric monoidal product on SS *and Cat*,*
* and more
generally, when C is bicomplete and Cartesian closed, ^ is a closed symmetric m*
*onoidal
product on C*.
Given G*-objects F1, . .,.Fk, and G, the set of k-morphisms in G*-C from (F1*
*, . .,.Fk)
to G is the set of natural transformations f: F1 x . .x.Fk ! G of functors Gk !*
* C which
take the map F1()x. .x.Fi-1()x*xFi+1()x. .x.Fk() ! F1()x. .x.Fk() induced by the
given map * ! Fi() to the given map * ! G(). Equivalently and more concisely in*
* the case
when C is Cartesian closed, this is the set of natural transformations f: F1 ^ *
*. .^.Fk ! G
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 25
of functors Gk ! C*:
F1x...xFk k
Gk ________//_C*
|| f |^|
fflffl| fflffl|
G ___G____//_C*.
We obtain an action of k from the symmetry isomorphism of , and we obtain a m*
*ulti-
product from composition in C*.
Proposition 5.4. G* -C forms a multicategory with the definitions above.
When C is enriched over small categories or simplicial sets, the conditions *
*defining the
objects and k-morphisms of G*-C translate into limits on the categories or simp*
*licial sets
of maps, and the multicategory G*-C therefore inherits an enrichment. In the ca*
*se when C
is the category of simplicial sets, the description of simplicial sets of k-mor*
*phisms is clear.
In the case when C is the category of small categories, we can describe the enr*
*ichment of
G*-Cat over Cat explicitly as follows.
First, since * is the trivial category with one object and one morphism, the*
* map * ! G()
in the definition of G*-category is equivalent to specifying a distinguished "b*
*asepoint"
object of G(). For G*-categories F1, . .,.Fk and G, the category of k-morphism*
*s from
(F1, . .,.Fk) to G has as its objects the natural transformations f: F1 ^ . .^.*
*Fk ! G of
functors from Gk to Cat *, i.e. collections of based functors f : F1 ^ .*
* .^.Fk !
G( . . .) natural in Gk. A map of such k-morphisms OE: f ! g assigns *
*to each
object = (, . .,.) of Gk a natural transformation OE: f ! g such that
the value of OE() at the basepoint object of F1() ^ . .^.Fk() is the identity m*
*ap on the
basepoint object of G() and such that for any morphism (h1, . .,.hk): (, . *
*.,.) !
(, . .,.) in Gk, the transformations given by the following two pasting*
* diagrams
26 A. D. ELMENDORF AND M. A. MANDELL
coincide:
___________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*___________________________________________________f
________________________________________________*
*_____________________________________________________________________________*
*____**OO
F1 ^ . .^.Fk____________OOff'OEG( . . .)
________________________________________________*
*_____________________________________________________________________________*
*_________________________________44
| _________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____________|
| g |
F1(h1)^...^Fk(hk)|| G(h1|...|hk)
| |
fflffl| fflffl|
F1 ^ . .^.Fk_____g______//_G( . . .)
F1 ^ . .^.Fk _____f_____//_G( . . .)
| |
| |
= F1(h1)^...^Fk(hk)|| G(h1|...|hk)
| __________________________|___________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*___________f
fflffl|__________________________fflffl|_____________*
*___________________________________________________**OO
F1 ^ . .^.Fk____________OOff'OEG( . . .).
________________________________________________*
*_____________________________________________________________________________*
*_________________________________44
_________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*____________
g
Note, however, that since the left vertical arrows in both diagrams are induced*
* by maps
using Cartesian products rather than smash products, the coincidence of the tra*
*nsforma-
tions given in the diagrams could equally well be specified by replacing the sm*
*ash products
with Cartesian products. This will be of use to us in the next section. A colle*
*ction of nat-
ural transformations satisfying the coherence condition in the display above is*
* called a
modification.
It turns out that in the case when C is the category of simplicial sets or t*
*he category of
small categories, or more generally, a bicomplete Cartesian closed category, th*
*en G*-C is
a bicomplete closed symmetric monoidal category, and the multicategory structur*
*e asso-
ciated to the symmetric monoidal structure is the one considered above. Since t*
*his is not
needed in the remainder of the paper, we give only a brief sketch of the argume*
*nt.
Let F(0)be the category with objects * and () where * is a null object (both*
* initial and
final) and the set of maps from () to itself consists of just the zero map and *
*the identity.
For r > 0, let F(r)be the r-th smash power of the based category F. (Note that*
* F(0)
is not the usual zeroth smash power of based categories, although it is the zer*
*oth smash
power in the full subcategory of Cat *of categories whose base object is null.)*
* As above,
we write = (m1, . .,.mr) but now (for r > 0), = *, the basepoint object*
*, if any
mi= 0. The categories F(r)have a based action of Injinduced from the action des*
*cribed
above for the Cartesian powers Fr; in particular, the object () of F(0)gets sen*
*t to the
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 27
R
constant string (1, . .,.1). We define G* to be the wreath product Inj F(-) f*
*ormed in
based categories. Specifically, the set of objects of G* is
` i j
Ob F(r)
r_2Ob(Inj)
and the set of maps from to in G* form the based set
` `^r ' ` ^ '
F(mi, nq(i)) ^ nj
q:r_!s_i=1 q-1(j)=;
where the wedge is over the maps q in Inj. The empty wedge is of course the one*
* point
set, and the empty smash is 1. Note that the basepoint object * of G* is a null*
* object, and
the basepoint in each mapping set is the unique map that factors through *.
We have a canonical functor G ! G*. In fact, we can identify the category G**
*as the cate-
gory obtained from G*by attaching a new null object * and identifying with *
** whenever
any mi= 0. In particular, every map in G*(, ) is either the trivial morph*
*ism (factor-
ing through *) or in the image of G(, ). The function G(, ) ! G*(, ) is
in fact one-to-one on the subset of G(, ) that does not map to the trivia*
*l morphism.
A based functor G* ! C* is a functor that takes the null object * of G* to the *
*null object
* of C*. The following proposition is now clear from the discussion and Proposi*
*tion 5.3.
Proposition 5.5. The category G*-C is isomorphic to the category of based funct*
*ors G* !
C*.
Concatenation again makes G* into a permutative category (where concatenatio*
*n with
* on either side yields *). It follows from theorems of Day ([3], Theorems 3.3 *
*and 3.6) that
the category of based functors from G* to C* has a closed symmetric monoidal st*
*ructure,
enriched over C*, in which the product of functors F1 and F2 is given by the le*
*ft Kan
extension F1 ^ F2 in the diagram on the left below. The universal property of *
*the Kan
extension is that maps from F1 ^ F2 to G are in one-to-one correspondence with *
*natural
transformations f as in the diagram on the right below.
G*x G* F1xF2//_C* x C*_^__//C*44ii G*x G* __F1xF2_//_C* x C*
i i
|| i i i || iiiiffi ^||
fflffl|iiiiF1^F2 fflffl| fflffl|
G* G* ______G______//_C*
Because G is the final object * whenever any mi = 0, a natural transformatio*
*n f in
the diagram above is precisely the same as a 2-morphism in G*-C under the ident*
*ification
of the previous proposition. The analogous observation for iterated smash produ*
*cts and
consistency with the multiproduct and symmetric group actions then imply the fo*
*llowing
theorem.
28 A. D. ELMENDORF AND M. A. MANDELL
Theorem 5.6. Let C be a bicomplete Cartesian closed category. Then G*-C is a cl*
*osed
symmetric monoidal category enriched over C. The multicategory structure of Pr*
*oposi-
tion 5.4 coincides with the multicategory structure inherited from the symmetri*
*c monoidal
structure.
Although we have no need for it in this paper, the discussion of this sectio*
*n may be
generalized to the context of a bicomplete symmetric monoidal closed category C*
*, where
x in C is replaced with the symmetric monoidal product in C; however, * remains*
* the
final object in C and not the unit object.
6. From Permutative Categories to G*-categories
We turn next to the description of our enriched multifunctor from permutativ*
*e categories
to G*-categories. This section is devoted to the proof of the following theorem.
Theorem 6.1. Construction 4.4 extends to an enriched multifunctor J from permut*
*ative
categories to G*-categories.
To each permutative category C we need to associate a G*-category JC : G ! C*
*at *.
Since this construction is_to_extend Construction 4.4, we must define it on obj*
*ects =
(n1, . .,.ns) by JC := C. We then have JC = * if any ni= 0. We take JC*
*() = C_
and we use the 0-object of C as the basepoint object. The canonical isomorphism*
* C ~=C(1)
takes the unit of C to the image of the single object of C(0).
Next we specify JC on morphisms of G. Given (ff, q) : ! , where =
(m1 , . .,.mr) and = (n1, . .,.ns), the functor
__ __
JC(ff, q) : C ! C
is obtained by composing the isomorphism
__ __
C ~=Cq*
induced by the permutation and extension functors described in Section 4 with t*
*he functor
__ __
ff* : Cq* ! C
described in the proof of Theorem 4.2. This constructs a G*-category JC.
Next we describe the functor J on k-morphisms. For this we need to describe *
*functors
J: Pk(C1, . .,.Ck; D) ! G*-Cat(JC1, . .,.JCk; JD)
between categories of k-morphisms that preserve the symmetric group actions and*
* the
multiproduct.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 29
We begin by giving J on objects of Pk(C1, . .,.Ck; D); for this fix a k-line*
*ar map f: C1x
. .x.Ck ! D. The k-morphism Jf: (JC1, . .,.JCk) ! JD in G*-Cat then consists *
*of a
natural transformation of functors Jf as in the following diagram:
JC1x...xJCk k
Gk _________//_Cat*
|| fflflfllJf^||
fflffl| fflffl|
G ____JD____//Cat*.
This means that for each k-tuple of objects of G, (, . .,.), where we h*
*ave =
(ni1, . .,.nisi), we need to specify a functor
Jf(, . .,.) : JC1 x . .x.JCk ! JD( . . .)
which returns 0 or id0 whenever any of the input objects or morphisms are 0 or *
*id0,
respectively.
An object of JCi is a system of objects Cof Ci, where = (Si1, .*
* .,.Sisi)
and Sijruns over all subsets of {1, . .,.nij}, together with structure maps as *
*specified in
Construction 4.4. Given a k-tuple of such systems (C, . .,.C), we need *
*to construct
an object of JD( . . .), which is a system of objects Dof D, wher*
*e runs
over all s1 + . .+.sk-tuples of subsets of the sets {1, . .,.nij} for 1 i k*
* and 1 j si.
But such a is simply the concatenation of a collection of lists for 1*
* i k, each
of which determines an object Cof Ci. We therefore define the object D a*
*s simply
f(C, . .,.C) for the component sublists of ; note that this ob*
*ject is 0 if
any of the inputs are 0, by the k-linearity of f. It is now a lengthy but strai*
*ghtforward
exercise to check that the evident structure maps satisfy the requirements for *
*an object of
JD( . . .). The definition easily extends to morphisms of JC1 x . .x*
*.JCk.
We need to check that this construction is natural in morphisms of Gk. But t*
*he mor-
phisms of Gk are generated by the morphisms in each factor of G, which in turn *
*are
generated by maps in the component Fs's and induced maps ! q* for inject*
*ions
q : r_! s_. For maps ffi: ! in Fsi, we need the following diagram to*
* commute:
JC1 x . .x.JCk _Jf__//_JD( . . .)
JC|| |JD|
fflffl| fflffl|
JC1 x . .x.JCkJf__//JD( . . .).
However, going around the square either way sends a k-tuple of systems (C, *
*. .,.C)
to the system D, where runs over s1+. .+.sk-tuples of subsets of the se*
*ts {1, . .,.n0ij}
for 1 i k and 1 j si, and D is defined by breaking up into comp*
*onent
30 A. D. ELMENDORF AND M. A. MANDELL
lists = where has length si. The subsets in the list <*
*Ti> are then
pulled back along the ffi's to give a list of subsets ff-1i of {1, . .,.nij*
*}, and D is
then f(Cff-11, . .,.Cff-1k). A similar formula gives the composite in e*
*ither direction
on morphisms.
The induced morphisms from the maps ! q* for an injection q : r_!_s_a*
*re the
isomorphisms induced by the permutation and extension isomorphisms in the C's. *
*Again,
given injections qi: ri_! si_, we need the following diagram to commute:
JC1 x . .x.JCk ______Jf______//JD( . . .)
JC|| |JD|
fflffl| fflffl|
JC1(q1)* x . .x.JCk(qk)* _Jf__//JD((q1)* . . .(qk)*).
Again, an object in the upper left category is a k-tuple of systems (C, . .*
*,.C), and we
produce a Dfrom going around the square in either direction. But is a c*
*oncatenation
of lists of subsets of {1, . .,.miq-1i(j)}, where if q-1i(j) is empty, we *
*set miq-1i(j)= 1
for consistency with our construction of JCi as a G*-category. If any of the c*
*omponent
subsets Sijare empty, then Dmust be 0, while if all of the Sij's for q-1i(j)*
* = ; are {1},
then the 1's can be dropped and we get a permutation of lists indexing the give*
*n object
(C, . .,.C), say (C, . .,.C). The corresponding obj*
*ect under either
composition is then f(C, . .,.C). A similar (but easier) ch*
*eck shows that
the diagram commutes on morphisms as well. We have therefore specified a k-morp*
*hism
Jf in G*-Cat from (JC1, . .,.JCk) to JD.
We also need to specify a modification JOE from Jf to Jg whenever we have a *
*morphism
OE : f ! g in Pk(C1, . .,.Ck; D). This means that for each object (, . .,.<*
*nk>) of Gk, we
need to specify a natural transformation
____________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________________Jf
__________________________________________________*
*_____________________________________________________________________________*
*__**OO
JC1 x . .x.JCk____________OOff'JOEJD( . . .).
_________________________________________________*
*_____________________________________________________________________________*
*________________________________44
___________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________
Jg
This means in turn that for each object > = (C, . .,.C) of JC1 x . .x.
JCk, we need a morphism in JD( . . .) from Jf> to Jg*
*>. But
Jf> is a system of objects of the form f>, and Jg> is a syste*
*m of objects
of the form g>, and OE provides a natural transformation from one system *
*of objects
to the other. We leave to the reader the tedious but straightforward verificati*
*ons necessary
to show that we have, in fact, specified a map J of multicategories enriched ov*
*er Cat .
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 31
7. From G*-categories to Symmetric Spectra
We turn next to the description of our multifunctor from G*-categories to sy*
*mmetric
spectra. As before, to avoid the confusion of the different levels of functors*
* and natural
transformations, it is convenient to work as long as possible with G*-objects i*
*n a general
Cartesian closed bicomplete category C; we are only really interested in the ca*
*se when C
is the category of small categories Cat and the case when C is the category of *
*simplicial
sets SS . As before, let C* be the category of based objects in C. The constr*
*uctionoinp
this generality is a multifunctor into the multicategory of symmetric spectra i*
*n C* . We
begin with a review of this multicategory.
The standard definition of the category of symmetric spectra in C*op in the *
*case when
C is the category of sets is usually phrased in terms of the smash product of b*
*ased sim-
plicial sets, which is a special case of the smash product in C* introduced in *
*Section 5.
The formulation of the category of symmetric spectra that follows is therefore *
*a simple
generalization of the category of symmetric spectra of [9].
Definition 7.1. Let C be bicomplete and Cartesian closed. Let C*op denote the c*
*ategory
of simplicial objects in C*, i.e., contravariant functors from the simplicial c*
*ategory to
theocategorypC*. We use * to denote both the null objectoofpC* and also the nul*
*l object in
C* , the constant simplicial object on *. For X in C* and K a finite based s*
*implicial
set, write X ^ K for the tensor of X with K; concretely, X ^ K has n-simplices
`
(X ^ K)n = Xn,
Kn\{*}
W op
where denotes theocoproductpin C*. A symmetric spectrum in C* consists of
objects X(p) in C* for all non-negative integers p, an action of the symmetri*
*c group p
on X(p), and "suspension" maps
oep : X(p) ^ S1o! X(p + 1),
such that for each q 1 the composite X(p) ^ (S1o)q ! X(p + q) preserves the (*
* p x q)-
action.
A k-morphism in symmetric spectra in C*op from (X1, . .,.Xk) to Y consists o*
*f maps
X1(p1) ^ . .^.Xk(pk) ! Y (p1 + . .+.pk)
for all p1, . .,.pk that preserve the p1 x . .x. pk action and that make the f*
*ollowing
diagram commute for 1 i k:
(X1(p1) ^ . .^.Xk(pk)) ^ S1o__________//Y (p1 + . .+.pk) ^ S1o
|~=| ||
fflffl| fflffl|
(7.1) X1(p1) x . .^.(Xi(pi) ^ S1o) ^ . .^.Xk(pk) Y (p1 + . .+.pk + 1)
| |
| |ci
fflffl| fflffl|
X1(p1) ^ . .^.Xi(pi+ 1) ^ . .^.Xk(pk)______//Y (p1 + . .+.pk + 1),
32 A. D. ELMENDORF AND M. A. MANDELL
where ci denotes the permutation in p1+...+pk+1that moves the last element to *
*the (p1+
. .+.pi+ 1)-st position but otherwise preserves the order, i.e., the cycle (q +*
* 1, . .,.p, p + 1)
where q = p1 + . .+.pi and p = p1 + . .+.pk. The k action on the k-morphisms *
*is
induced by permuting the product factors and the symmetric group action on the *
*target,
permuting blocks. The multiproduct is induced by smash products and compositio*
*ns in
C*.
By the simplicial nature of the construction, the multicategory is enriched *
*over simplicial
sets. When C* is enriched over small categories or simplicial sets, the condit*
*ions in the
previous definition translate into limitsoonpthe categories or simplicial sets *
*of maps, and the
multicategory of symmetric spectra in C* becomes enriched over simplicial cat*
*egories
or bisimplicial sets.
Proposition 7.2. The multicategory of symmetric spectra in based simplicial set*
*s as de-
fined above is isomorphic to the multicategory associated to the symmetric mono*
*idal cate-
gory of symmetric spectra of [9].
Proof. This is an easy consequence of the external formulation of the smash pro*
*duct of
symmetric spectra. Technically, the paper [9] considers the category of "left S*
*-modules"
whereas the (external) formulation above specifies the category of right S-modu*
*les, but the
identity isomorphism S ~=Sop induces a strong symmetric monoidal isomorphism be*
*tween
these categories.
Now we describe our multifunctor I from G*-objects in C to symmetric spectra*
* in C*op.
Recall from Section 4 that we have defined our model of the circle S1oso that i*
*ts based set
of n-simplices is n, giving S1oas a functor from op to F.
Construction 7.3. For F a G*-object in C and for p 0 let IF (p) be given by t*
*he
composite in the following diagram:
(S1o)p F
op __D__//( op)p____//Fp_____//G____//C*,
where D is the diagonal, and the unlabelled arrow is the canonical inclusion of*
* Fp into G.
In particular, IF (0) is the constant simplicial object F (). We give IF (p) t*
*he p action
arising from the action of p on Fp. We have maps
IF (p) ^ S1o! IF (p + 1)
induced by the maps in G
(n1, . .,.np) ! (n1, . .,.np, np+1)
indexed by the nonzero elements of np+1, with the map indexed by x being the ma*
*p given
by the injection including {1, . .,.p} in {1, . .,.p + 1} and the map in the p *
*+ 1'st copy of
F sending 1 to np+1 by the unique based map sending 1 to x. The composite map
IF (p) ^ (S1o)q ! IF (p + q)
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 33
has a similar description and so is easily seen to be p x q equivariant. It f*
*ollows that
these objects and maps assemble to a symmetric spectrum which we write as IF .
Theorem 7.4. I extends to a multifunctor fromothepmulticategory of G*-objects i*
*n C to
the multicategory of symmetric spectra in C* .
Proof. Let F1, . .,.Fk and G be G*-objects in C, and consider a k-morphism f f*
*rom
(F1, . .,.Fk) to G, which consists of a natural transformation as indicated in *
*the following
diagram:
F1x...xFk k
Gk ________//_C*
|| f |^|
fflffl| fflffl|
G ___G____//_C*.
It is straightforward to verify that the following diagram commutes:
Fp1 x . .x.Fpk_____//Gk
~=|| ||
fflffl| fflffl|
Fp1+...+pk________//G,
so by pasting we obtain the following composite diagram, which gives the induce*
*d map of
symmetric spectra. (For reasons of space, we have written F(p1,...,pk)for Fp1 x*
* . .x.Fpk,
and similarly for other superscripted k-tuples.)
k (S1o)(p1,...,pk) F1x...xFk
( op)kO____D_____//_(Oop)(p1,...,pk)_____//_F(p1,...,pk)_____//_Gk_________//_*
*Ck*
D | ~=|| ~=|| || ffififiif*
*^||
|| D fflffl|(S1o)p1+...+pk fflffl| fflffl|G f*
*flffl|
op ___________//_( op)p1+...+pk_______//_Fp1+...+pk________//G__________//C*
The suspension diagram 7.1 commutes by naturality of f and the definition of th*
*e sus-
pension maps and symmetric group action because the following diagram of maps i*
*n G
commutes for all i, all objects , . .,., and all based maps 1 ! n. For *
*reasons of
space, let = . . ., and let = . . . Then we *
*have
(1)___// (n)
id o|| |id|o
fflffl| fflffl|
(1) __// (n) ,
where the horizontal maps are induced by the given map 1 ! n. We leave to the r*
*eader
the exercise of correlating definitions to check that this association preserve*
*s the symmetric
group action on the k-morphisms, the units, and the multiproduct.
34 A. D. ELMENDORF AND M. A. MANDELL
When we regard the k-morphisms of G*-objects as discrete simplicial sets, th*
*e multicat-
egory G*-C is enriched over simplicial sets and the multifunctor described abov*
*e is enriched
(for trivial reasons). When C is enriched over small categories or simplicial s*
*ets, we can
regard the multicategory of G*-objects as enriched over simplicial categories o*
*r bisimpli-
cial sets by taking the (other) simplicial direction to be discrete. A straight*
*forward check
then shows that the multifunctor described above is enriched over simplicial ca*
*tegories or
bisimplicial sets.
Composing the multifunctor J from the previous section, the multifunctor I, *
*the nerve
functor, and the diagonal functor (from bisimplicial sets to simplicial sets), *
*we obtain a
multifunctor K from the multicategory of small permutative categories to the mu*
*lticate-
gory of symmetric spectra. By inspection, the underlying functor is naturally i*
*somorphic
to the functor Knew described in Definition 4.5. This completes the proof of Th*
*eorem 1.1.
8. Ring Categories, Bipermutative
Categories, and the Operads * and E *
This section is devoted to the proofs of Theorems 3.4 and 3.8.
Proof of Theorem 3.4. First, suppose we are given a small ring category A; we m*
*ust
produce a multifunctor * ! P sending the single object of * to A. In this ca*
*se, a
multifunctor as specified in the theorem is precisely a map of operads (in Cat *
*) from *
to the endomorphism operad of A in P, whose component categories are the k-line*
*ar
maps Pk(A, . .,.A; A). In other words, we must define a sequence of functors Tk*
*: k !
Pk(A, . .,.A; A), and show that they specify a map of operads. Since k is a d*
*iscrete
category, specifying the functor Tk is equivalent to specifying a k-morphism Tk*
*oe for every
element oe in the group k. As per Definition 3.2, the k-morphism T oe consists*
* of a functor
foe: Ak ! A and natural distributivity maps ffioeifor 1 i k.
We define foeby
foe(a1, . .,.ak) = aoe-1(1) . . .aoe-1(k).
For notational convenience in defining ffioei, let P = aoe-1(1) . . .aoe-1(oe(*
*i)-1), and Q =
aoe-1(oe(i)+1) . . .aoe-1(k). We then define ffioeias the common diagonal of t*
*he following
square, which commutes by Definition 3.3, condition (e):
(P ai Q) (P a0i Q)__dl_//((P ai) (P a0i)) Q
dr || |dr|1
fflffl| fflffl|
P ((ai Q) (a0i Q))___1_dl__//_P (ai a0i) Q.
The reader may now verify that the requirements for distributivity maps are sat*
*isfied.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 35
We must verify that the Tk's give a map of operads. Equivariance is element*
*ary; we
check preservation of the multiproduct. This follows as a consequence of the f*
*ollowing
commutative diagram, where oe 2 k and OEi2 jifor 1 i k:
Aj1x . .x.AjkQQ
OE x...xOE| QQQQfOE1x...xfOEkQQ
1 | k QQQ
fflffl| QQQ((
Aj1x . .x.Ajk _____k___//AkQQQ
QQQQ oe
oe|| |oe| QfQQQQ
fflffl| fflffl| QQQQ((
Ajoe-1(1)x . .x.Ajoe-1(k)_k//_Ak______________//A.
We must also check that the distributivity maps of (T oe; T OE1, . .,.T OEk) c*
*oincide with
those of T (oe; OE1, . .,.OEk). However, both distribute to the same ending po*
*int, which may
be written
P1 P2 (ai a0i) Q2 Q1,
where P1 is the tensor product of blocks preceding the one in which ai a0iappe*
*ars, and
P2 is the tensor product of the terms in the same block which precede ai a0i. *
* Q1 and
Q2 are described analogously. Now (T oe; T OE1, . .,.T OEk) distributes first *
*P1 and Q1, and
then P2 and Q2, while T (oe; OE1, . .,.OEk) does it all at once. The resulting*
* maps coincide
by property (d) of the distributivity maps in Definition 3.3. Therefore T pres*
*erves the
multiproduct, and we get a map of operads, i.e., a multifunctor T : * ! P.
Now suppose given a map of operads T : * ! {Pk(Ak; A)}; we must produce a r*
*ing
structure on A. First, the tensor product functor : A2 ! A is the functor par*
*t of the
image of 1 2 2, and the unit object is the image of the unique element of 0. *
*Write 1n
for the identity element of n. Then the strict associativity of follows from*
* the fact that
(12; 12, 11) = 13 = (12; 11, 12), and the unit condition follows from (12; 1*
*1, 10) = 11 =
(12; 10, 11).
The distributivity maps dland dr arise as part of the structure of the targe*
*t of 12 2 2.
Properties (a), (b), (c), and (f) follow immediately from requirements for k-mo*
*rphisms in
P. Properties (d) and (e) follow from the facts that T is a map of operads, and*
* also that
(12; 11, 12) = (12; 12, 11). The distributivity maps for the images of these*
* composites
must therefore coincide, and both (d) and (e) follow. We therefore have a ring *
*structure
whenever we have a map of operads * ! {Pk(Ak; A)}.
Finally, we must verify that these correspondences are inverse to each other*
*. First
suppose given a ring structure on A, and let T : * ! {Pk(Ak; A)} be the induce*
*d map of
operads. By definition, T (12) is the tensor product on A, together with both d*
*istributivity
maps, and the multiplicative unit is given by T (10). We therefore recover the*
* original
structure from its induced map of operads.
36 A. D. ELMENDORF AND M. A. MANDELL
Now suppose we start with a map of operads T : * ! {Pk(Ak; A)}, and give A *
*the
induced ring structure. By induction using the fact that (12; 1k-1, 11) = 1k, *
*we find that
f1k(a1, . .,.ak) = a1 . . .ak,
and from equivariance it follows that, for oe 2 k,
foe(a1, . .,.ak) = aoe-1(1) . . .aoe-1(k).
We therefore recover the map of operads T on underlying functors f, and we are *
*left
with the recovery of the distributivity maps. By equivariance, it suffices to *
*recover the
distributivity maps ffi1ki, which we do by induction on k. This is trivial if k*
* 2. Since T
is a map of operads, we have
(T (12); T (1i), T (1k-i)) = T (1k).
If i < k, assume by induction that ffi1iiis given by
(P ai) (P a0i)dr_//P (ai a0i).
Then by the definition of distributivity maps in the multiproduct (T (12); T (*
*1i), T (1k-i)),
we have ffi1kigiven by the composite
(P ai Q) (P a0i Q)__dl_//((P ai) (P a0i))_drQ_1//P (ai a0i) *
* Q,
as required. In the remaining case, where i = k, we use the fact that the (sing*
*le) distribu-
tivity map of T (11) is the identity, together with
(T (12); T (1k-1), T (11)) = T (1k),
to exhibit ffi1kkas simply
(P ak) (P a0k)dr_//P (ak a0k),
as required. This completes the proof.
Proof of Theorem 3.8. First suppose given a map of operads E * ! {Pk(Rk; R)}. T*
*hen
we have the composite multifunctor
* _____//E *__R__//P,
so by Theorem 3.4, R is associative. We therefore get all of the bipermutative *
*structure
except for:
(1) fl ,
(2) The coherence diagram for fl from the requirement that (R, , 1) form a*
* permu-
tative category, and
(3) Diagram (e0).
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 37
The symmetry isomorphism fl is the image of the isomorphism between the two ob*
*jects
of E 2. The coherence diagram
fl
a b Mc___________________//c88 a b
MMM qqqq
MMM qqq
1 fl MMM&& qqq fl 1
a c b
now follows as a consequence of there being exactly one isomorphism in E 3 betw*
*een
13 2 3 and the permutation sending (abc) to (cab). Diagram (e0) is simply the *
*requirement
that fl , being the image of a morphism in E 2, must be a morphism in P2(R2; R*
*). A
map of operads E * ! {Pk(Rk; R)} therefore determines a bipermutative structure*
* on R.
Suppose now that we are given that R is a small bipermutative category; we n*
*eed to
construct the multifunctor T : E * ! P. From Theorem 3.4, we get the map of ope*
*rads on
the objects * once we know that R is a ring category, and the only issue here *
*is diagram
(e) in Definition 3.3, which we have replaced with (e0). However, diagram (e) f*
*ollows as a
consequence of the commutativity of the diagram in Figure 1 (see page 38), all *
*of whose
subdiagrams are instances of the coherence requirements for a bipermutative cat*
*egory.
We therefore get a map of operads T : * ! {Pk(Rk; R)}, and it remains to ex*
*tend this
to the morphisms in the E k's. These consist of one isomorphism between each pa*
*ir of
objects. Given any pair of elements oe and OE in k, the permutative structure *
*on (R, , 1)
gives a canonical isomorphism
aoe-1(1) . . .aoe-1(k)~=//_aOE-1(1) . . .aOE-1(k),
as a composite of the maps fl ; we take this as the image of the unique morphi*
*sm from
oe to OE. The coherence condition for fl implies that any ways of composing *
*various
instances of fl that lead to the same permutation of the tensor factors give *
*the same
isomorphism; we use this fact multiple times below, and refer to it as "uniquen*
*ess of the
permutation isomorphisms". Compatibility of these permutation isomorphisms wit*
*h the
given distributivity maps follows from coherence of the bipermutative structure*
*, specifically
property (e0) using the fact that k is generated by transpositions. The unique*
*ness of the
permutation isomorphisms implies that Tk is a functor E k ! Pk(Rk; R). In orde*
*r to
see that T defines a map of operads on the morphisms, we apply a little more co*
*herence
theory. Given objects (oe; OE1, . .,.OEk) and (oe0; OE01, . .,.OE0k) of E k x *
*E j1x . .x.E jk,
there is a unique isomorphism from one to the other in E k x E j1x . .x.E jk. T*
*he
target of this morphism under T first permutes within blocks, and then permute*
*s the
blocks, while the target under T does this all at once; these are the same is*
*omorphism
by the uniqueness of the permutation isomorphisms. This concludes the proof tha*
*t T is
a map of operads, and consequently the given data determine a multifunctor E * *
*! P.
The proof that the passages back and forth are inverse to each other is exactly*
* as in the
proof of Theorem 3.4.
38 A. D. ELMENDORF AND M. A. MANDELL
| |
(a b c)|_E(a__ b0 _c)______________dl_________________//((a b)| (a b0*
*)) c
|____________ mm66 |
|__EEE______________________ flmmmm |
||___EEE_________________________ mmmmm ||
| ____EEEflEfl_______________________ m 0 |
| _______EE____________________c 6((a6 b) (a b )) |
| ________EEE____________________drmm|mmm |
| _________EEE____________________mm||mmm |
| __________EE""__________________________|mmm |
|| __________________(c a b) |1(drc a b0) ||
| (1 fl)_2________________________________________|QQQQ| |
| _______________________drQQQ|| |
| _______________________QQQQ|| |
| _______________________((QQfflffl|| |
| ________________________ 2 | 0
| ________________________(fl|1)c1 a (b |b )
________________________||1
dr|| _______________________||111| |dr|1
|| __AEAE_____________________fflffl|111|| ||
0)
| (a c b) (aQQ c b fl 1|| 11 |
| | QQQdrQ | 11 |
| | QQQQ | 11 |
| | QQ((Q fflffl| 11 |
| | 01fl1 |
| dr| a c (b b )1 |
| | mm66m BBB 11 |
| | mmmmm BB 11 |
| | mmmm1 dr BBB 11 |
| fflffl|m BB 11 |
| a ((c b) (c b0)) BB 1 |
| ll 1 fl BBB 11 |
| 1 (fllfl)lll BB 11|
| llll BB 1|1
fflffl|vvlll B!fflffl|,,!
a ((b c) (b0 c))________________1_dl_________________//a (b b0) c
Figure 1
9. Modules and Algebras in Permutative Categories
In this section, we describe some of the module and algebra structures in P,*
* the multi-
category of permutative categories. We first define each structure in terms of *
*functors and
natural transformations; we then reinterpret the structure in terms of paramete*
*r multi-
categories. All of the parameter multicategories we describe below have contrac*
*tible com-
ponents in their k-morphism categories, so collapsing each component to a singl*
*e point
gives a map of multicategories that is the identity on objects and a weak equiv*
*alence on
k-morphisms. From Theorems 1.3 and 1.4, it follows that the structures we descr*
*ibe pass
to structures on K-theory spectra equivalent to the associated strict structure*
*s.
9.1. Modules.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 39
Definition 9.1.1. Let A be a ring category and D a permutative category. A lef*
*t A-
module structure on D consists of a functor : A x D ! D that is strictly assoc*
*iative in
the sense that the diagram
A x A x D _1x__//A x D
x1 || ||
fflffl| fflffl|
A x D _________//_D
commutes, strictly unital in the sense that the composite
D ~={1} x D _____//A x D_____//D
coincides with the identity, together with natural distributivity maps
dl: (a d) (a0 d) ! (a a0) d
and
dr: (a d) (a d0) ! a (d d0)
subject to the commutativity of all the diagrams in Definition 3.3.
Left module structure over a ring category can be described in terms of a pa*
*rameter
multicategory; recall that a ring category structure is given by a map out of t*
*he operad
* (Theorem 3.4).
Definition 9.1.2. The multicategory `M * is the following parameter multicatego*
*ry for
modules: It has two objects, A (the "ring") and M (the "module"). In the case i*
*n which
all inputs and the output are A, we have `Mk*(Ak; A) = k, and if exactly one i*
*nput is
M and the output is also M, we set `Mk*(Aj-1, M, Ak-j; M) = {oe 2 k : oe(j) = *
*k}.
All other k-morphism sets are required to be empty. The multiproduct and *-ac*
*tion
are defined in exactly the same way as in the operad *; see the discussion fol*
*lowing
Theorem 3.4.
Note that restricting our attention to the single object A gives a multifunc*
*tor
* ! `M *,
so if we have a multifunctor `M * ! P, the image of A is a ring category. The f*
*undamental
theorem about left module structures on permutative categories is the following:
Theorem 9.1.3. Left A-module structures on D determine and are determined by mu*
*lti-
functors `M * ! P sending A to A and M to D such that the restriction
* ! `M * ! P
40 A. D. ELMENDORF AND M. A. MANDELL
gives the structure map for A as a ring category.
Proof. First suppose given a left A-module structure on D; we must produce a mu*
*ltifunctor
T : `M * ! P. The ring structure on A gives us the multifunctor on the k-morph*
*isms
of `M * involving only A, so consider oe 2 `Mk*(Aj-1, M, Ak-j; M), i.e., oe 2 *
*k and
oe(j) = k. We define
T oe: Aj-1 x D x Ak-j ! D
by the formula
T oe(a1, . .,.aj-1, d, aj+1, . .,.ak) = aoe-1(1) . . .aoe-1(k-1) *
*d.
Since oe(j) = k, all of the objects aoe-1(1), . .,.aoe-1(k-1)are indeed objects*
* of A, and this
formula is simply a special instance of the usual formula
T oe(b1, . .,.bk) = boe-1(1) . . .boe-1(k).
The proof that this formula determines a multifunctor now proceeds exactly as i*
*n the proof
of Theorem 3.4.
On the other hand, given a multifunctor `M * ! P sending A to A and M to D,
and which restricts on A to the ring category structure map for A, we must prod*
*uce a
left A-module structure on D. The tensor pairing : A x D ! D is the image of *
*the
single element of `M *(A, M; M), and the distributivity maps are part of the st*
*ructure
of the target of this element. The rest of the proof now follows exactly as in *
*the proof of
Theorem 3.4.
We have the following immediate consequence.
Corollary 9.1.4. If D is a left A-module, then KD is a left KA module.
When A is not just ring but actually bipermutative, we can describe a parame*
*ter multi-
category that captures this further structure using the translation category co*
*nstruction E
applied to `M *: for a multicategory of sets M, let EM denote the multicategory*
* enriched
over small categories for which EMk(B1, . .,.Bk; C) is the category obtained by*
* applying
E to Mk(B1, . .,.Bk; C). There is an obvious inclusion of multicategories M ! *
*EM,
where we consider M enriched over small categories with all the categories disc*
*rete.
Lemma 9.1.5. Let * ! `M * be the inclusion of the k-morphisms of `M * involving
only A. Then the diagram of multicategories
* ______//_`M *
| |
| |
fflffl| fflffl|
E * _____//E`M *
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 41
is a pushout. In other words, making the k-morphisms in * all canonically iso*
*morphic
forces all the other k-morphisms in `M * to be canonically isomorphic as well.
Proof. Let Q be another multicategory, and suppose we have a commutative diagram
* ______//`M *
| |
| |
fflffl| fflffl|
E * _______//Q
of multicategories. We must show that there is a unique dashed arrow making the*
* diagram
of multicategories
* ______//_`M4*
| | 444
| | 4
fflffl| fflffl|444
E * _____//TTTTE`MG44*
TTTTT GG 44
TTTTT G44
TTTT4aeae))T##G
Q
commute. Certainly there is no choice about the values on the objects of the k-*
*morphism
category E`Mk*(B1, . .,.Bk; C), since the objects are the same as the objects o*
*f `M *.
The values on morphisms of E * are also determined. We show that whenever oe1 *
*and
oe2 are objects in E`Mk*(Aj-1, M, Ak-j; M), the image of the map from oe1 to oe*
*2 is also
determined. Since oe2 O oe-11fixes k, we can think of it as an element of k-1,*
* and let OE be
the unique map in E k-1 from the identity permutation to oe2O oe-11. Then we ca*
*n express
the unique map from oe1 to oe2 in E`Mk*(Aj-1, M, Ak-j; M) by the formula
(id,; OE, 1M ) . oe1
where , is the single object of `M2*(A, M; M). This establishes uniqueness of *
*such a
multifunctor, and it remains to show existence. Using the formula above to def*
*ine the
functors, it is straightforward to show that they preserve the symmetric group *
*action and
the multiproduct and therefore define a multifunctor E`M * ! Q.
Corollary 9.1.6. Let R be a small bipermutative category, D a small permutative*
* cate-
gory. Then left R-module structures on D determine and are determined by multif*
*unctors
E`M * ! P sending M to D and restricting on A to the bipermutative structure map
E * ! P for R.
Proof. This follows immediately from Lemma 9.1.5 with Q replaced by P.
Applying Theorem 1.4 we obtain the following corollary.
42 A. D. ELMENDORF AND M. A. MANDELL
Corollary 9.1.7. If D is a left module over a bipermutative category R, then KD*
* is weakly
equivalent to a strict module over a strictly commutative ring spectrum weakly *
*equivalent
to KR.
For right modules, the relevant definitions are as follows.
Definition 9.1.8. Let A be a ring category, D a permutative category. Then the *
*structure
of a right A-module on D consists of a functor : D x A ! D that is strictly as*
*sociative
and unital in the analogous sense as in Definition 9.1.1, together with distrib*
*utivity maps
again defined analogously and satisfying the corresponding diagrams.
Definition 9.1.9. The multicategory rM * is the following parameter multicatego*
*ry for
modules: It has two objects, A and M, with k-morphism sets being empty unless a*
*ll inputs
are A and the output is A or exactly one input is M and the output is M. In th*
*e first
case, the k-morphisms are k, so the endomorphism operad of A is * (as in `M **
*), but
we set
rMk*(Aj-1, M, Ak-j; M) = {oe 2 k : oe(j) = 1}.
The *-action and multiproduct are defined exactly as in *.
Theorem 9.1.10. Let A be a small ring category and D a small permutative catego*
*ry.
Then right A-module structures on D determine and are determined by multifuncto*
*rs
rM * ! P sending M to D and restricting on A to the structure map for A as a ri*
*ng
category.
The proof is safely left to the reader, given the proof of Theorem 9.1.3. T*
*he obvious
analogs to Corollaries 9.1.4, 9.1.6, and 9.1.7 also hold.
Just as in ordinary algebra, a right module over A is the same thing as a le*
*ft module
over the opposite structure "Aop", which we now define.
Definition 9.1.11. The opposite map is the particular map of operads op: * ! *
defined as follows. For k 0, define rk 2 k by rk(j) = k + 1 - j, so rk rever*
*ses order.
We then define
op: k ! k
by op(oe) = rk O oe.
We leave to the reader the check that op defines a map of operads.
Definition 9.1.12. Let A be a ring category. The opposite of A, written Aop, i*
*s the
ring category given by the composite
op A
* _____// *____//P.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 43
Corollary 9.1.13. Right A-module structures on a small permutative category D d*
*eter-
mine and are determined by left Aop-module structures on D.
Proof. The automorphism * -op! * extends to an isomorphism `M * -op! rM * for
which the diagram
op
* ________// *
| |
| |
|fflffl fflffl|
`M * _op__//rM *
commutes. The extension is given by exactly the same formula: using the element*
*s rk 2 k
defined by rk(j) = k + 1 - j, we define op(oe) = rk O oe, and clearly if oe(j) *
*= k, then
op(oe)(j) = 1. The result now follows immediately.
Corollary 9.1.14. If R is bipermutative, so is Rop.
Proof. The map "op" of operads extends to the map of operads
E(op): E * ! E *.
9.2. Bimodules.
The following is the explicit definition of a bimodule in the context of per*
*mutative
categories.
Definition 9.2.1. Let A and B be ring categories, and D a permutative category.*
* We say
that D is an A-B bimodule if D is a left A-module and a right B-module, the ass*
*ociativity
diagram
A x D x B __x1_//D x B
1x || ||
fflffl| fflffl|
A x D _________//D
commutes, and diagrams (e) and (f) from Definition 3.3 commute in all situation*
*s in which
the maps are defined.
For bimodule structures, the fundamental parameter multicategory is as follo*
*ws.
Definition 9.2.2. The bimodule parameter multicategory B * has objects A, B (the
"rings", with A acting on the left and B on the right) and M (the "module"). A*
*ll sets
of k-maps are empty with the exception of those in which M appears exactly once*
* in the
input and is the output, those where all inputs and the output are A, and those*
* where
all inputs and the output are B. In the latter two cases the set of k-maps is *
*k. In the
44 A. D. ELMENDORF AND M. A. MANDELL
case of Bk*(C1, . .,.Ck; D) with Cj = D = M and all other entries either A or B*
*, we set
Bk* = {oe 2 k : oe(i) < oe(j) , Ci = A}. These are precisely the oe's for whic*
*h the list
Coe-1(1), . .,.Coe-1(k)is the list Aoe(j)-1, M, Bk-oe(j). In particular, oe(j) *
*must always be one
plus the number of A's occurring in the input. The k action and the multiprodu*
*ct are
defined exactly as for the operad *.
Note in particular that restriction to either of the single objects A or B d*
*etermines a
multifunctor * ! B *.
Theorem 9.2.3. Let A and B be small ring categories. Then an A-B bimodule struc*
*ture
on a small permutative category D determines and is determined by a multifuncto*
*r B * !
P sending M to D, restricting on the single object A to the structure multifunc*
*tor * ! P
for A and on the single object B to the structure multifunctor for B.
Proof. Given a bimodule structure on D and an element oe 2 Bk*(C1, . .,.Ck; D),*
* we need
to define a functor T oe, and we use the usual formula
T oe(c1, . .,.ck) = coe-1(1) . . .coe-1(k).
The proof that this gives a multifunctor B * ! P now proceeds in exactly the sa*
*me
way as in the proof of Theorem 3.4. Conversely, suppose we are given a multifu*
*nctor
T : B * ! P satisfying the conditions in the theorem. Restricting to pairs of *
*objects
(A, M) or (B, M) gives us restriction multifunctors `M * ! B * and rM * ! B *, *
*and
we immediately obtain a left A-module structure on D and a right B-module struc*
*ture on
D. The associativity diagram commutes because B3*(A, M, B; M) has only one elem*
*ent,
and diagrams (e) and (f) commute exactly as in the proof of Theorem 3.4. This c*
*oncludes
the proof.
Corollary 9.2.4. If D is an A-B bimodule for ring categories A and B, then KD i*
*s a
KA-KB bimodule in symmetric spectra.
In the case where A = B, we can collapse the parameter multicategory further*
* using a
special case of the parameter multicategory in the second example after Definit*
*ion 2.4:
Definition 9.2.5. The parameter multicategory bM * has two objects, A and M, and
is a parameter multicategory for modules, so there are no k-morphisms unless M *
*is the
output and appears exactly once in the input, or else A is the output and only *
*A appears
in the input. In these cases the k-morphisms are k, with the multiproduct defi*
*ned as in
*.
To compare this multicategory with the previous one, we use the following le*
*mma:
Lemma 9.2.6. Consider the diagram of multicategories
* _____////_B_*__//bM *
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 45
where the two arrows on the left are the inclusions of the endomorphism operads*
* of the ob-
jects A and B, and the arrow on the right sends both A and B to A, and sends pe*
*rmutations
in B * to corresponding ones in bM *. This is a coequalizer diagram of multicat*
*egories.
Proof. The key point here is that each permutation in bMk*(Aj-1, M, Ak-j; M) ha*
*s ex-
actly one preimage in B *. Once we realize this, extending an equalizing multif*
*unctor to
bM * is simply a matter of sending all permutations to their images under the m*
*ultifunc-
tor.
The characterization of A-A bimodules in terms of a parameter multicategory *
*now
follows immediately.
Corollary 9.2.7. If A is a small ring category and D is a small permutative cat*
*egory,
then an A-A bimodule structure on D determines and is determined by a multifunc*
*tor
bM * ! P sending M to D and restricting on A to the ring category structure mul*
*tifunctor
* ! P for A.
The analog of Corollary 9.2.4 now follows as well.
If one or both of A and B are bipermutative, one can also describe A-B bimod*
*ules with
this extra structure in terms of parameter multicategories. We leave this to th*
*e interested
reader.
We can also ask for an analogous characterization of A-A bimodules as in Cor*
*ollary 9.2.7
in the case where A is bipermutative. The answer is NOT to apply E to all the m*
*ulticat-
egories in the diagram in Lemma 9.2.6. (This illustrates the fact that E does n*
*ot preserve
coequalizers). Instead, we get a multicategory described as follows.
Definition 9.2.8. The multicategory bE M * is a parameter multicategory for mod*
*ules,
so has objects A and M, with the k-morphisms empty except in the cases where M *
*appears
exactly once in the input and is the output, or else all inputs and the output *
*are A. We
set bE Mk*(Ak; A) = E k. The objects of bE Mk*(Aj-1, M, Ak-j; M) are the elemen*
*ts of
k, but the objects are not all isomorphic. Instead, we look at the equivalence*
* relation on
k in which oe ~ oe0 if and only if oe(j) = oe0(j) and oe and oe0 are in the sa*
*me coset of the
left action of oe(j)-1x k-oe(j)on k. Equivalently, we could say that oe ~ oe*
*0 means that
oe(i) < oe(j) , oe0(i) < oe0(j) whenever 1 i k. There is exactly one morphi*
*sm from oe
to oe0 when oe and oe0 are equivalent and no morphisms when they are not equiva*
*lent. We
leave it to the reader to check that the same formula for the multiproduct in *
** extends
to give multicategory structure on bE M *.
Lemma 9.2.9. Consider the diagram of multicategories
E * _____////_EB_*__//bE M *
where the two arrows on the left are the inclusions of the endomorphism operads*
* of the ob-
jects A and B, and the arrow on the right sends both A and B to A, and sends pe*
*rmutations
to themselves. This is a coequalizer diagram of multicategories.
46 A. D. ELMENDORF AND M. A. MANDELL
Proof. Given Lemma 9.2.6, the only issue is the morphisms. However, the definit*
*ion of the
morphisms in bE M * is precisely the requirement that two k-morphisms are isomo*
*rphic
in bE M * if and only if they come from isomorphic k-morphisms in EB *. The re*
*sult
follows.
Corollary 9.2.10. Let R be a small bipermutative category. Then R-R bimodule st*
*ruc-
tures on a small permutative category D determine and are determined by multifu*
*nctors
bE M * ! P sending A to R and M to D, and which restrict on A to the bipermutat*
*ive
structure map E * ! P for R. Consequently, the K-theory spectrum KD is equivale*
*nt to
a bimodule over a strictly commutative ring spectrum equivalent to KR.
This still leaves the question of what sort of bimodule structure is paramet*
*erized by
EbM *. The relevant definition is as follows.
Definition 9.2.11. Let R be a bipermutative category. The structure of a symmet*
*ric
bimodule over R on a permutative category D consists of an R-R bimodule structu*
*re
together with a natural isomorphism
fl: r d ~=d r
for r an object of R and d an object of D. The isomorphism fl must be compatibl*
*e with
the multiplicative symmetry isomorphism fl for R, in the sense that all possib*
*le diagrams
of the form given in part 3 of Definition 3.1 must commute (with the 's replac*
*ed with
's). We also require diagram (e0) given in Definition 3.6 to commute.
Theorem 9.2.12. Let R be a small bipermutative category and D a small permutati*
*ve
category. Then symmetric bimodule structures for D over R determine and are det*
*ermined
by multifunctors EbM * ! P sending M to D and restricting on A to the structure*
* map
E * ! P for R as a bipermutative category. Consequently, the K-theory spectrum *
*KD
is equivalent to a module over a strictly commutative ring spectrum equivalent *
*to KR.
The proof is the same as the proof of Theorem 3.8 with bM * in place of *.
9.3. Algebras.
We turn our attention next to algebras. The parameter multicategories we wi*
*ll be
interested in here are of the following form.
Definition 9.3.1. A parameter multicategory for algebras is a multicategory A w*
*ith two
objects, R (the "ring") and A (the "algebra"), subject to the following conditi*
*on. Suppose
given inputs B1, . .,.Bk with at least one of the Bj's being equal to A. Then w*
*e require
that Ak(B1, . .,.Bk; R) = ;. If all the other k-morphism spaces are contractibl*
*e, then we
say that A is a parameter multicategory for E1 algebras.
Again, we can look at the example in which all the nonempty k-morphism space*
*s are a
single point, and we map to a symmetric monoidal category. Then the images of b*
*oth R
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 47
and A are commutative monoids, and the rest of the structure is induced by a st*
*rict map
of monoids from R to A given by the single element of A1(R; A).
A more interesting example is given by letting S = {j : Bj = A} in the expre*
*ssion
Ak(B1, . .,.Bk; C) and, if not required to be empty, setting this k-morphism sp*
*ace equal
to k= ~, where ~ is the equivalence relation on k given by requiring oe ~ oe0*
* if and only
if, for all elements i and j of S, oe(i) < oe(j) , oe0(i) < oe0(j). Then a mul*
*tifunctor to
a symmetric monoidal category makes the image of R again a commutative monoid, *
*the
image of A is now a noncommutative monoid, and the map induced by the single el*
*ement
of A1(R; A) is central in the obvious sense.
For a third example, let O be an operad. Then we can let Ak(B1, . .,.Bk; C)*
* = Ok
whenever it is not required to be empty. Then the images of both R and A are O-*
*rings,
and there is a map of O-rings given by the identity element of O1 = A1(R; A) wh*
*ich
determines the entire algebra structure.
The explicit characterization of a central algebra over a bipermutative cate*
*gory depends
on the following notion of a central map from a bipermutative category to a rin*
*g category.
Definition 9.3.2. Let R be a bipermutative category and A a ring category. A ce*
*ntral
map from R to A is a lax map OE: R ! A (i.e., (OE, ~) 2 Ob (P1(R; A))) and a na*
*tural
isomorphism fl: OE(r) a ~=a OE(r) for r an object of R and a an object of A*
*, satisfying
the following conditions:
(1) OE preserves the tensor product in the sense that the diagram
OExOE
R x R _____//A x A
|| ||
fflffl| fflffl|
R ____OE___//_A
commutes strictly and OE(1) = 1.
(2) The lax structure map ~ preserves the distributivity maps in the sense t*
*hat the
diagram
(OEr1 OEr2) (OEr1 _OEr3)dr//_OEr1 (OEr2 OEr3)
= || 1|~|
fflffl| fflffl|
OE(r1 r2) OE(r1 r3) OEr1 OE(r2 r3)
~|| =||
fflffl| fflffl|
OE[(r1 r2) (r1 r3)]OE(dr)//_OE(r1 (r2 r3))
and a similar diagram involving dl commute.
48 A. D. ELMENDORF AND M. A. MANDELL
(3) fl must be consistent with the symmetry isomorphism fl in R in the sens*
*e for all
objects r1, r2 of R, the diagram
OE(r1) OE(r2)fl_//OE(r2) OE(r1)
= || |=|
fflffl| fflffl|
OE(r1 r2)OE(fl_/)/OE(r2 r1)
commutes.
(4) fl satisfies all instances of the diagrams in part (3) of Definition 3.1*
*, and diagram
(e0) of Definition 3.6.
An R-algebra structure on A consists of a central map from R to A.
Definition 9.3.3. Let A * be the multicategory with two objects, R (the ground *
*ring)
and A (the algebra). The category Ak*(B1, . .,.Bk; C) is empty if C = R and one*
* or more
of the Bj's are A. Otherwise, Ak*(B1, . .,.Bk; C) has k as its set of objects*
*, and has
morphisms as follows. Let S = {j : Bj = A} and consider the equivalence relatio*
*n on the
elements of k where oe ~ oe0 means that for all i and j in S, oe(i) < oe(j) , *
*oe0(i) < oe0(j).
We have precisely one morphism from oe to oe0 when oe ~ oe0, and no morphisms b*
*etween
inequivalent elements.
In the previous definition, if we restrict our attention to the object R, we*
* get E *,
while if we restrict our attention to the object A, we get *. We wish to show*
* that R-
algebra structures on a small ring category A correspond to multifunctors from *
*A * to P
extending the structure multifunctors for both R and A. To do this, we need the*
* following
combinatorial lemma about permutations.
Lemma 9.3.4. Suppose T k_= {1, . .,.k} and that ae 2 k is order-preserving o*
*n T
in the sense that if i and j are elements of T with i < j, then ae(i) < ae(j). *
*Then ae can
be written as a product of transpositions of consecutive integers in k_, say ae*
* = t1 . .t.m, in
such a way that for 1 n m, tn does not transpose two elements of tn+1 . .t.*
*mT .
Proof. Let the elements of T be written in order as {a1, . .,.aq}. First, we us*
*e transposi-
tions of the required form to map T to {1, . .,.q}; we do this by first transpo*
*sing a1 with
its predecessors, in order, and then repeating the process with a2 through aq. *
*Then use
transpositions of adjacent elements of {q + 1, . .,.k} to rearrange this set in*
* the same order
that ae rearranges k_\ T . Finally, start with q and transpose it with its succ*
*essors, in order,
until it reaches ae(aq), and repeat the process with q - 1 back through 1. The *
*result is ae,
with the transpositions involved having the required property.
Theorem 9.3.5. Let R be a small bipermutative category and A a small ring categ*
*ory.
Then R-algebra structures on A determine and are determined by multifunctors fr*
*om A *
to P restricting on the object R to the structure multifunctor for R as a biper*
*mutative
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 49
category and on the object A to the structure multifunctor for A as a ring cate*
*gory. Con-
sequently, KA is equivalent to a central algebra over a strictly commutative ri*
*ng spectrum
equivalent to KR.
Proof. Suppose we are given a multifunctor from A * restricting as required. T*
*hen we
obtain a functor OE: R ! A as the image of the unique element 11 of A1*(R; A); *
*we claim
that this functor is a central map. First, we have the formula (11; 12) = (12*
*; 11, 11) = 12
in A *, which we can express by saying that the diagram in A *
(11,11)
(R, R)_____//(A, A)
12|| |12|
fflffl| fflffl|
R ___11____//A
commutes, and consequently its image in P
OExOE
R x R _____//A x A
|| ||
fflffl| fflffl|
R ____OE___//_A
commutes as well. A similar argument shows that OE(1) = 1. Since the commutativ*
*ity of
this diagram in P also requires that the distributivity maps coincide, we get t*
*he diagrams
showing that ~ preserves the distributivity maps. The natural isomorphism fl: O*
*E(r) a ~=
a OE(r) is the image of the isomorphism between the two elements of A2*(R, A;*
* A) = 2.
Because the diagram
(11,11)
(R, R)_____//(R, A)
| |
| |
fflffl| fflffl|
R ___11____//A
in A * commutes when the downward arrows are both one of the two elements of 2,
the isomorphism between the two possible elements on the left gets taken by OE *
*to the
isomorphism between the two possible elements on the right, i.e., fl = OE(fl )*
*, as required.
Further, diagram (e0) of Definition 3.6 is satisfied because fl is a morphism i*
*n P2(R, A; A).
We therefore get a central map OE: R ! A given a multifunctor A * ! P restricti*
*ng to
the structure multifunctors of R and A on the objects R and A, respectively.
Now suppose we are given a central map OE: R ! A; we must show that this ext*
*ends
uniquely to a multifunctor A * ! P by requiring the multifunctor to restrict to*
* the
structure multifunctors for R and A and also by requiring the single element of*
* A1*(R; A)
50 A. D. ELMENDORF AND M. A. MANDELL
to map to OE. The functor on Ak*(B1, . .,.Bk; C) is already determined when C =*
* R or
when C = A and all the Bj's are A. In the other cases, set S = {i : Bi = A} as*
* in
the definition. It remains to determine the images of the categories Ak*(B1, .*
* .,.Bk; A)
with S 6= ; and S 6= {1, . .,.k}. By equivariance, it suffices to consider the*
* special case
S = {1, . .,.q} for q < k. The objects are the elements of k, and it is clear *
*that the image
of 1k is the composite
1xOEk-q
Aq x Rk-q ______//_Ak_____//_A,
and the images of the rest of the objects are determined by equivariance. We mu*
*st also
determine the images of the isomorphisms in Ak*(B1, . .,.Bk; A). For this, not*
*e that
when oe ~ oe0 as in the definition, oe0oe-1 is order-preserving on oeS, so by L*
*emma 9.3.4,
can be written as a product of transpositions of adjacent integers which are no*
*t both
elements of oeS. Now the image of a typical k-tuple (b1, . .,.bk) under the el*
*ement oe is
boe-1(1) . . .boe-1(k), and we need to produce an isomorphism between this and *
*the image
under oe0. Write oe0oe-1 as t1 . .t.m, where tj is a transposition of adjacent*
* integers not
both in tj+1 . .t.moeS, and say tm transposes i and i + 1. Then the term boe-1(*
*i) boe-1(i+1)
appears as part of the image under oe, and since oe-1 (i) and oe-1 (i+1) are no*
*t both elements
of S, the two b's are not both objects of A, so they can be transposed using fl*
*. We get an
isomorphism between a tensor product of elements of the form
boe-1(i)= boe0-1oe0oe-1(i)= boe0-1t1...tm (i)
and elements of the form
boe0-1t1...tm-1(i).
By iterating the process m times, we get an isomorphism between the image under*
* oe
and the image under oe0. The isomorphism is uniquely determined by oe0oe-1 an*
*d not
its presentation, because the fl's satisfy the relations among transpositions i*
*n k. This
completes the proof.
In the special case where A is also a bipermutative category and the symmetr*
*y isomor-
phism is given by the isomorphism already present in A, we can give a somewhat *
*simpler
description.
Definition 9.3.6. Let R and A be bipermutative categories. A map of bipermutat*
*ive
categories OE: R ! A is a lax map that preserves the tensor product, distributi*
*vity maps,
and multiplicative unit in the same sense that a central map does, and for whic*
*h also
OE(flR ) = flA .
The corresponding definition in terms of a parameter multicategory is as fol*
*lows.
Definition 9.3.7. The multicategory AE * is a parameter multicategory for algeb*
*ras, so
by Definition 9.3.1 has two objects, A and R, and with AEk* (B1, . .,.Bk; C) = *
*; if S 6= ;
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 51
and C = R, where S = {i : Bi = A}. Otherwise, we set AEk* (B1, . .,.Bk; C) = E *
*k, so
this is an example of the sort discussed as the third example following Definit*
*ion 9.3.1.
The proof of the following theorem can now be safely left to the reader.
Theorem 9.3.8. Let R and A be small bipermutative categories. Then a map of bip*
*er-
mutative categories OE: R ! A determines and is determined by a multifunctor AE*
* * ! P
which restricts on the object R to the structure multifunctor for R and on the *
*object A
to the structure multifunctor for A. Consequently, KOE is equivalent to a map o*
*f strictly
commutative ring spectra.
10. Free Permutative Categories
This section is devoted to the construction of additional examples of both r*
*ing and
bipermutative categories via the "free permutative category" construction. This*
* associates
to any small category C a small permutative category PC as follows. Let E k be*
* the
translation category of k. Then we define
a
PC = E k x k Ck.
k 0
The objects of PC are the elements of the free monoid on the objects of C, with*
* 0 given
by the empty string and the direct sum given by concatenation, which is the mon*
*oid
operation. The symmetry isomorphism arises from the isomorphism in E 2 between *
*the
two elements of 2. Although implicit in [16], Dunn [5] apparently first obser*
*ved that
P defines a monad in Cat whose algebras are precisely the small permutative cat*
*egories.
The resulting morphisms are called the strict morphisms and are even more restr*
*ictive
than the strong morphisms. In fact, they are too restrictive to form a multicat*
*egory.
The following theorem shows how additional structure on C gives rise to addi*
*tional
structure on PC.
Theorem 10.1. Let C be a small strict monoidal category (i.e., one equipped wit*
*h a strictly
associative and unital "tensor product" operation). Then PC supports the struc*
*ture of a
ring category. If C is permutative, then PC becomes a bipermutative category.
Proof. There are actually uncountably many different ways of constructing such *
*struc-
ture, depending on one's choice of what we call a priority order. Let m_ denot*
*e the
set {1, . .,.m} for positive integers m. Then a priority order is a choice of *
*bijection
!m,n: mn__! m_ x n_for each m and n that is coherent in the sense that all diag*
*rams
of the form
mnp___!mn,p_//mn_x p_
!m,np || !m,nx1||
fflffl| fflffl|
m_ x np_1x!_//_m_x n_x p_
n,p
52 A. D. ELMENDORF AND M. A. MANDELL
commute. By ordering m_ x n_using lexicographic order and taking the inverse o*
*f the
resulting bijection, we get a priority order, as we do using reverse lexicograp*
*hic order, but
there are uncountably many other choices as well. For example, we can use lexic*
*ographic
order to define a bijection m_ ! 2_(m)_x ^m_, where ^m is odd, and then for any*
* m and n,
use the inverse of the bijection
m_ x n_____//2_(m)_x ^m_x 2_(n)_x ^n_
_1xox1_//_
2_(m)_x 2_(n)_x ^m_x_^n__//2_(m)2_(n)^m^n_= mn__,
where the unlabelled arrows are given by lexicographic order or its inverse. W*
*e can use
the same sort of trick for any set of primes, not just 2, to get uncountably ma*
*ny additional
priority orders. In any case, pick one, and call it !. Let !1 and !2 denote ! f*
*ollowed by
projection onto the first or second factor, respectively. Then we define a rin*
*g structure
on PC as follows. Write a typical object (a1, . .,.am ) of PC as mi=1(ai), an*
*d write the
monoidal operation in C as . Then we define the tensor product on PC by the fo*
*rmula
Mm Mn mnM
(ai) (bj) := (a!1(k) b!2(k)).
i=1 j=1 k=1
In the case where C is permutative, we can then use the symmetry isomorphism in*
* C to
map this to
Mmn
(b!2(k) a!1(k)),
k=1
and then shuffle inside of PC to map this to
Mmn
(b!1(k) a!2(k)),
k=1
defining the multiplicative symmetry isomorphism necessary for a bipermutative *
*category.
The reader can check that one needs only the associativity condition on a prior*
*ity order to
show that these definitions satisfy the requirements for a ring or a bipermutat*
*ive category,
respectively.
An example of particular importance of this form is the free permutative cat*
*egory P(*)
on a one point category, which becomes a bipermutative category via this constr*
*uction.
The reader should be aware, however, that modules over P(*) depend strongly on *
*the
priority order chosen. We leave as an exercise to the reader that if we use lex*
*icographic
order, then any permutative category is a left module over P(*), while if we us*
*e reverse
lexicographic order, every permutative category is a right module over P(*). Of*
* course, the
two orders give opposite bipermutative structures on P(*), so the duality is to*
* be expected.
Other choices of priority order seem to give far fewer modules over P(*).
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 53
11. Model Categories of Rings, Modules,
and Algebras in Symmetric Spectra
In this section we prove Theorem 1.3. Fix a small multicategory M enriched *
*over
simplicial sets, and let O denote its set of objects. Let SO denote the catego*
*ry obtained
as the product of copies of the category S of symmetric spectra indexed on the *
*set O.
As a product category, SO inherits a simplicial closed model structure for each*
* simplicial
closed model structure on S, precisely, one with its fibrations, cofibrations, *
*and weak
equivalences formed objectwise (i.e., coordinatewise). Our goal is to prove tha*
*t the category
SM of simplicial multifunctors from M to S has a simplicial closed model struc*
*ture with
the fibrations and weak equivalences the maps that are fibrations and weak equi*
*valences
respectively in SO for the positive stable model structure on S. Throughout thi*
*s section,
we use the terminology stable equivalence, positive stable fibration, and acycl*
*ic
positive stable fibration in SM to indicate those maps in SM whose underlying*
* maps
in SO are weak equivalences, fibrations, and acyclic fibrations in the positive*
* stable model
structure.
The first step is to show that the category SM has limits and colimits. For*
* this, it is
convenient to observe that SM is the category of algebras over a monad M on SO*
* .
Definition 11.1. For b 2 O, and T in SO , let
0 1
` `
(MT )b = @ M(a1, . .,.an; b)+ ^ (Ta1^ . .^.Tan)A= n,
n 0 a1,...,an2O
let j: T ! MT be the map
Tb ~={idb}+ ^ Tb ! M(b; b)+ ^ Tb ! (MT )b,
and ~: MMT ! MT the map induced by the multiproduct of M.
The proof of the following theorem in the special case of operads [15] easil*
*y generalizes
to multicategories.
Theorem 11.2. M is a simplicial monad on the category SO . An M-algebra structu*
*re on
an object of SO is equivalent to an M-multifunctor structure, and the simplici*
*al category
of M-algebras is isomorphic to SM .
Corollary 11.3. M, viewed as a functor SO ! SM , is left adjoint to the forgetf*
*ul functor
SM ! SO .
Corollary 11.4. The category SM is complete and cocomplete (has all small limi*
*ts and
colimits), and is tensored and cotensored over simplicial sets.
54 A. D. ELMENDORF AND M. A. MANDELL
Proof. As a category of algebras over a monad on a complete category, SM is co*
*mplete,
with limits and cotensors formed in SO . Since M preserves reflexive coequalize*
*rs (by the
argument of [7] Proposition II.7.2), SM is cocomplete with reflexive coequaliz*
*ers created
in SO by [7] Proposition II.7.4. General colimits are formed by rewriting the c*
*olimit as a
reflexive coequalizer, and the tensor of an object A of SM and a simplicial se*
*t X is formed
as a (reflexive) coequalizer of the form
M((MA) ^ X+ ) _____////_M(A ^ X+_)__//A X.
In order to prove the required factorization and lifting properties, we need*
* to review
briefly the positive stable model structure on S. Recall that in any category C*
* with small
colimits, for any set I of maps, a relative I-complex ([14] Definition 5.4) is *
*a map X ! Y
in C where Y = Colim Xk, with X0 = X, and Xk+1 is formed from Xk as a pushout o*
*f a
coproduct of maps in I. In this terminology, a map of symmetric spectra is a co*
*fibration
in the positive stable model structure if and only if it is a retract of a rela*
*tive I+ -complex,
where
I+ = {Fm @ [n]+ ! Fm [n]+ | m > 0, n 0},
and Fm is the functor from simplicial sets to symmetric spectra left adjoint t*
*o the m-th
space functor. A map is an acyclic cofibration if and only if it is a retract *
*of a relative
J+ -complex for a certain set of maps J+ (q.v. [9] Definition 3.4.9 and [14] Se*
*ction 14). A
complete description of the maps in J+ is not difficult but would require an u*
*nnecessary
digression; all we need to know about the maps is that the domain and codomain *
*are small,
meaning that the sets of maps out of them commute with sequential colimits.
For a 2 O, let 'a denote the functor S ! SO that is left adjoint to the proj*
*ection functor
ssa: SO ! S. For a symmetric spectrum T , the object 'aT of SO satisfies
aeT b = a
('aT )b =
* b 6= a.
The positive stable model structure on SO then has a similar description of its*
* cofibrations
and acyclic cofibrations: Let
'*I+ = {'af | f 2 I+ , a 2 O}
'*J+ = {'af | f 2 J+ , a 2 O}.
A map in SO is cofibration if and only if it is the retract of a relative '*I+ *
*-complex and is
an acyclic cofibration if and only if it is a retract of a relative '*J+ -compl*
*ex. Let
I+= M'*I+ = {M'af | f 2 I+ , a 2 O} = {Mf | f 2 '*I+ }
J+= M'*J+ = {M'af | f 2 J+ , a 2 O} = {Mf | f 2 '*J+ }.
The adjunction of Corollary 11.3 and the lifting properties in SO then imply th*
*e following.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 55
Proposition 11.5. A map in SM is an acyclic positive stable fibration if and o*
*nly if it
has the right lifting property with respect to I+, if and only if it has the ri*
*ght lifting property
with respect to retracts of relative I+-complexes. It is a positive stable fibr*
*ation if and only
if it has the right lifting property with respect to J+, if and only if it has *
*the right lifting
property with respect to retracts of relative J+-complexes.
Because the domains and codomains of the maps in I+ and J+ are small in symm*
*etric
spectra, the domains and codomains of the maps in I+ and J+ are small in SM . *
*The
Quillen small object argument then gives the following.
Proposition 11.6. A map in SM can be factored as a relative I+-complex followe*
*d by
an acyclic positive stable fibration or as a relative J+-complex followed by a *
*positive stable
fibration.
The proof of the following lemma is complicated but similar to the analogous*
* lemma
in the case of commutative ring symmetric spectra. Since we need some specific*
*s of the
argument in the next section, we provide the proof at the end of that section.
Lemma 11.7. A relative J+-complex is a stable equivalence.
The usual lifting and retract argument then gives the following.
Proposition 11.8. A map in SM has the left lifting property with respect to th*
*e acyclic
positive stable fibrations if and only if it is a retract of a relative I+-comp*
*lex. A map in
SM has the left lifting property with respect to the positive stable fibration*
*s if and only if
it is a retract of a relative J+-complex.
We have now collected all the facts we need to prove Theorem 1.3.
Proof of Theorem 1.3. We have shown (in Corollary 11.4) that SM has all finite*
* limits and
colimits. It is clear by their definition that weak equivalences (the stable e*
*quivalences)
are closed under retracts and have the two-out-of-three property. Also clear f*
*rom the
definition is that the fibrations (the positive stable fibrations) are closed u*
*nder retracts,
and if we define the cofibrations in terms of the left lifting property, then i*
*t is clear that
these are closed under retracts. The lifting properties follow from Propositio*
*n 11.5 and
Proposition 11.8, and the factorization properties follow from Proposition 11.6*
*. Thus, all
that remain is SM7.
We need to show that when i: T ! U is a cofibration and p: X ! Y is a fibrat*
*ion, the
map of simplicial sets
SM (U, X) -! SM (U, Y ) xSM (T,Y )SM (T, X)
is a fibration, and a weak equivalence if either i or p is. Using the characte*
*rization in
Proposition 11.8 of cofibrations and acyclic cofibrations as the maps that are *
*retracts of
56 A. D. ELMENDORF AND M. A. MANDELL
relative I+- and J+-complexes respectively, this easily reduces to the case whe*
*n i is a map
in I+ or a map in J+. Using the adjunction of Corollary 11.3, this reduces to S*
*M7 in SO ,
which reduces to SM7 in S, proved in [9].
12. Multifunctors and Quillen Adjunctions
In this section we prove Theorem 1.4. Before we can begin the proof, we need*
* to complete
the statement, by giving the full definition of weak equivalence of multicatego*
*ries.
The definition of weak equivalence of multicategories is a generalization of*
* the definition
of a weak equivalence of categories enriched over simplicial sets from [6], and*
* for this, we
need to recall the category of components. When C is a category enriched over s*
*implicial
sets, the sets of components ss0C(x, y) for objects x, y have the composition
ss0C(y, z) x ss0C(x, y) ! ss0C(x, z)
induced by the composition in C. This composition and the identity components *
*make
ss0C into a category, called the category of components. A simplicial functor f*
*: C ! C0
is a weak equivalence when the induced functor ss0f is an equivalence of catego*
*ries of
components and for all objects x, y in C, the map of simplicial sets C(x, y) ! *
*C0(fx, fy) is
a weak equivalence. In the following definition, we understand the category of *
*components
of a enriched multicategory to be the category of components of its underlying *
*enriched
category.
Definition 12.1. A simplicial multifunctor f: M ! M0 is a weak equivalence whe*
*n the
induced functor ss0f is an equivalence of categories of components and for all *
*a1, . .,.an, b
in O, the map of simplicial sets M(a1, . .,.an; b) ! M0 (fa1, . .,.fan; fb) is *
*a weak equiv-
alence.
We now begin the proof of Theorem 1.4 by constructing the Quillen adjunction*
* asso-
ciated to a simplicial multifunctor. Let f: M ! M0 be a simplicial multifunctor*
* between
small multicategories enriched over simplicial sets. Let O denote the set of ob*
*jects of M
and O0 the set of0objects of M0 . The multifunctor0f in particular induces a p*
*rojection
functor ssf: SO ! SO . Let 'f: SO ! SO be the left adjoint: For T an object i*
*n SO and b
in O0, `
('fT )b = Ta.
a2f-1(b)
The multifunctor f induces a natural transformation
'fM ! M0'f,
where M0 is the monad on SO0 from Definition 11.1. For an object A of SM ,0we u*
*se this
natural transformation and the structure map MA ! A to construct f*A in SM by*
* the
(reflexive) coequalizer diagram
M0'fMA _____////_M0'fA__//f*A.
Unwinding the universal property and the adjunctions, we obtain the following r*
*esult.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 57
Proposition 12.2. f*: SM ! SM0 is left adjoint to the pullback functor f*: SM*
*0 ! SM .
Since the functor f* clearly preserves weak equivalences and fibrations, the*
* first state-
ment of Theorem 1.4 is an immediate consequence of the previous proposition.
Corollary 12.3. Given small multicategories M and M0 , enriched over0simplicial*
* sets
and f: M ! M0 a simplicial multifunctor, the induced functor f* : SM ! SM is *
*the right
adjoint in a Quillen adjunction.
For the rest of the section, we assume that f is a weak equivalence. We need*
* to show
that (f*, f*) is a Quillen equivalence. The following lemma is the first step.
Lemma 12.4. A map OE: T ! U is a stable equivalence in SM0 if and only if f*OE*
* is a
stable equivalence in SM .
Proof. By definition, f*OE is a stable equivalence in SM if and only if it is *
*a stable equiva-
lence0in SO , i.e., if and only if ssfOE is a stable equivalence.0Since OE is a*
* stable equivalence
in SM if and0only if it is a stable equivalence in SO , it follows that f* tak*
*es stable equiv-
alences in SM to stable equivalences in SM . Thus, it remains to show that OE*
* is a stable
equivalence when f*OE is.
Assume that f*OE is a stable equivalence. Then for any a in O0 in the image*
* of f,
OEa: Ta ! Ua is a stable equivalence. If b is an arbitrary element of O0, then *
*the hypothesis
that f is a weak equivalence implies that we can find an a in the image of f an*
*d an
isomorphism from a to b in the category of components of M0 . Choosing maps in *
*M0 (a, b)
and M0 (b, a) in the components giving such an isomorphism and its inverse, the*
*re are
generalized simplicial intervals connecting the composites with the appropriate*
* identity
map (on a and on b). Using the naturality of OE, it follows that OEb is (level*
*wise) weakly
equivalent to OEa, and is therefore a positive stable equivalence.
We spend much of the rest of the section proving the following theorem.
Theorem 12.5. If A is a cofibrant object of SM , then the unit A ! f*f*A of the*
* (f*, f*)
adjunction is a stable equivalence.
Assuming the previous theorem for the moment, we have all we need to prove T*
*heo-
rem 1.4.
Proof of Theorem 1.4. It remains to show that when f is a weak equivalence, the*
* Quillen
adjunction (f*, f*) is0a Quillen equivalence. Let A be a cofibrant object of SM*
* and B a
fibrant object of SM ; we need to show that a map OE: f*A ! B is a stable equi*
*valence
if and only if the adjoint map _: A ! f*B is a stable equivalence. By Lemma 12.*
*4, we
know that OE is a stable equivalence if and only if f*OE is a stable equivalenc*
*e. Since _ is
the composite *
A -! f*f*A f-OE!f*B,
58 A. D. ELMENDORF AND M. A. MANDELL
Theorem 12.5 implies that _ is a stable equivalence if and only if f*OE is. Thi*
*s concludes
the proof.
We now move on to the proof of Theorem 12.5. The proof requires an analysis *
*of the
pushouts in SM of the form B qM'xX M'xY for a map of symmetric spectra X ! Y a*
*nd a
map 'xX ! B in SO . For this we need to set up two constructions. For the first*
*, for each
x1, . .,.xk in O, construct Ux1,...,xkB as the coequalizer in SO
_ !
` `
M(a1, . .,.an, x1, . .,.xk; -)+ ^ (MB)a1,...,an= n
n 0 _a1,...,an !
____//_//_` ` M(a1, . .,.an, x1, . .,.xk; -)+ ^ Ba
1,...,an= n
n 0 a1,...,an
____//_Ux1,...,xkB.
where Ba1,...,anis shorthand for Ba1 ^ . .^.Ban and similarly for MB. (One map*
* is
induced by the action map MB ! B and the other by the multiproduct.) The purpos*
*e of
introducing U*B is that for any T in SO , the underlying object in SO of the c*
*oproduct
B q MT in SM is
_ !
` `
Ux1,...,xkB ^ Tx1 ^ . .^.Txk= k.
k x1,...,xk
When x1 = . .=.xk = x and x is understood, we write UkB for Ux1,...,xkB.
The second construction is defined for maps of symmetric spectra g: X ! Y .*
* We
construct symmetric spectra Qki(g) (or Qkiwhen g is understood) for k 0, 0 *
*i k
inductively as follows: Qk0= X(k), Qkk= Y (k)(the k-th smash power of X and Y )*
*, and
for 0 < i < k, we define Qkiby the pushout square:
k+ ^ k-ix i X(k-i)^ Qii-1_____// k+ ^ k-ix i X(k-i)^ Y (i)
| |
| |
fflffl| fflffl|
Qki-1__________________________//_Qki
Essentially, Qkiis the k-sub-spectrum of Y (k)of with i factors of Y and k - i*
* factors of
X: The quotient Y (k)=Qkk-1is naturally isomorphic to (Y=X)(k). When g is Fm *
* of an
injection of simplicial sets X ! Y , Qkiis precisely Fmk of the subspace of Y *
*k where at
most i factors are in Y \ X.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 59
Combining these constructions, we get a filtration on B qM'xX M'xY as follo*
*ws. Let
B0 = B, and let Bk be the pushout in SO
UkB ^ k Qkk-1 _____//UkB ^ k 'xY (k)
| |
| |
fflffl| fflffl|
Bk-1 _______________//Bk,
where the map UkB ^ k Qkk-1! Bk-1 is induced by the map 'xX ! B. Let B1 =
Colim Bk.
Proposition 12.6. With notation above, B1 is isomorphic to the underlying obje*
*ct of
B qM'xX M'xY in SO .
In order to use this below, we need to know that the map Bk-1 ! Bk is object*
*wise a
level cofibration of symmetric spectra.
Lemma 12.7. Let T be any right k object in symmetric spectra. If g: X ! Y i*
*s a
cofibration, then T ^ k Qkk-1(g) ! T ^ k Y (k)is a level cofibration, i.e., lev*
*el injection.
Proof. It suffices to consider the case when X ! Y is a relative I+ -complex, a*
*nd a filtered
colimit argument reduces to the case when X ! Y is formed by attaching a singl*
*e cell,
i.e., is the pushout over a map
Fm i: Fm @ [n]+ ! Fm [n]+
in I+ . Then the map in the statement is the pushout over the map
T ^ k Qkk-1(Fm i) ! T ^ k (Fm [n]+ )(k).
We can identify this as T ^ k (-) applied to the map
Fmk @( [n]k)+ ! Fmk [n]k+.
It is easy to check explicitly that this is a level cofibration.
Proof of Theorem 12.5. It suffices to consider the case when A is an I+-complex*
*, i.e., the
map from the initial object M(; -)+ ^S to A is a relative I+-complex. Then A = *
*ColimAn
where A0 = M(; -)+ ^ S, and An+1 is formed from An as a pushout over a coproduct
of maps in I+ . Since f*f*A = Colim f*f*An, it suffices to show that An ! f*f*A*
*n is a
weak equivalence for all n.
We prove this by induction on n for all An. Specifically, we say that an I+-*
*complex B
can be built in n stages if, starting with B0 = M(; -)+ ^ S, we can construct B*
* as a
60 A. D. ELMENDORF AND M. A. MANDELL
sequence of n pushouts over coproducts of maps in I+, B0 ! B1 ! . .!.Bn = B. Our
inductive hypothesis is that for any I+-complex B that can be built in n stages*
*, B ! f*f*B
is a stable equivalence. Since f is a weak equivalence, M(; -)+ ^ S ! M0 (; -)+*
* ^ S is a
stable equivalence, and this gives the base case n = 0. Our argument also needs*
* the base
case n = 1, where we are looking at a map of the form MT ! f*M0'fT for some T i*
*n SO
that is objectwise cofibrant. Using the explicit formula for M and M0 in Defini*
*tion 11.1,
we see that this is a stable equivalence.
For the inductive step from n to n + 1, a filtered colimit argument reduces *
*to the case
of C = B qM'xX M'xY for X ! Y in I+ , where B can be built in n stages. We have*
* the
filtration preceding Proposition 12.6,
B = B0 ! B1 ! . .,. C = B1 = ColimBk,
whose associated graded is `
UkB ^ k (Y=X)(k),
k
which is isomorphic in SO to B q M'x(Y=X), with the coproduct in SM . Let B0 =*
* f*B
and C0 = f*C. Since C0 = B0qM0'fxXM0'fxY , we have the analogous filtration
B0= B00! B01! . .,. C0 = B01 = ColimB0k,
whose associated graded is isomorphic in SO0 to B0q M0'fx(Y=X). The map C ! f*C*
*0 =
ssfC0 preserves the filtrations, and the map of associated gradeds
B q M'x(Y=X) ! ssf(B0q M0'fx(Y=X) ~=f*f*(B q M'x(Y=X))
is a stable equivalence, because B q M'x(Y=X) can be built in n stages (since n*
* 1). By
Lemma 12.7, the maps in the filtration are objectwise level cofibrations, and i*
*t follows that
each map Bk ! ssfBk is a stable equivalence. The map C ! ssfC0 = f*f*C is there*
*fore a
stable equivalence.
The constructions in this section also provide what is needed for the proof *
*of Lem-
ma 11.7.
Proof of Lemma 11.7. A filtered colimit argument reduces to showing that the ma*
*p B !
B qM'xX M'xY is a stable equivalence for X ! Y in J+ . Let B = B0 ! B1 ! . .b.e*
* as
above Proposition 12.6; it suffices to show that each Bk-1 ! Bk is a stable equ*
*ivalence.
The quotient Bk=Bk-1 is naturally isomorphic to UkB ^ k (Y=X)(k). Moreover, Y=X
is positive cofibrant and stably equivalent to the trivial symmetric spectrum **
*, and so
Bk=Bk-1 is stably equivalent to the trivial object * in SO . Since the map Bk-1*
* ! Bk is
objectwise a level cofibration, it follows that it is a stable equivalence.
RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 61
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Department of Mathematics, Purdue University Calumet, Hammond, IN 46323
E-mail address: aelmendo@calumet.purdue.edu
Department of Mathematics, Indiana University, Bloomington, IN 47405
E-mail address: mmandell@indiana.edu