DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S
M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
Contents
1. Symmetric monoidal categories 2
2. Categories of D-spaces 4
3. Symmetric monoidal categories of D-spaces 5
4. Forgetful and prolongation functors on diagram spaces 6
5. Diagram spectra and functors with smash product 8
6. An interpretation of diagram spectra as diagram spaces 11
7. Forgetful and prolongation functors on diagram spectra 12
8. Examples of diagram spectra 13
References 17
This is one of several papers in which we explore the interrelationships among
certain categories that can be taken as models for spectra and (highly structur*
*ed)
ring spectra. In this paper, we study certain categories of "diagram ring spect*
*ra"
that are isomorphic to corresponding categories of "functors with smash product
(FSP's)". The notion of an FSP was introduced by B"okstedt [2], and his use of
FSP's to define topological Hochschild homology established their convenience a*
*nd
importance in stable homotopy theory. Versions of FSP's had been defined earlie*
*r:
in different language, what we call FSP's in the category of symmetric spectra *
*were
defined by Gunnarson [7], and what we call FSP's in the category of orthogonal
spectra were defined by the second author and others [18, 17] in the early 1970*
*'s.
As we shall see, FSP's are defined in terms of "external smash products". It *
*is
a crucial insight of Jeff Smith that external smash products can be internalize*
*d.
We show that each category of generalized FSP's is isomorphic to the category of
monoids in an associated symmetric monoidal category of diagram spectra. Such
monoids are what we mean by diagram ring spectra. Smith introduced the category
of symmetric spectra, showed that it is symmetric monoidal, and observed that i*
*ts
externally defined FSP's and internally defined monoids give isomorphic categor*
*ies.
We study a general form of the construction. In fact, the relevant categorical
framework was already in place by 1970, in work of Day [5].
Hovey, Shipley, and Smith [8] have studied the category of symmetric spectra *
*and
its homotopy theory, and the papers [21] and [23] go further with the homotopic*
*al
study of its ring spectra. There is a coordinate-free analogue of the category *
*of sym-
metric spectra, which we call the category of orthogonal spectra; it was introd*
*uced
by May [17, x5] (who called its objects I*-prespectra). The names come from the
fact that actions by symmetric groups and by orthogonal groups are built into t*
*he
____________
Date: November 14, 1998.
1
2 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
spaces that comprise symmetric spectra and orthogonal spectra. Still another ca*
*t-
egory of the type that we consider is the category of -spaces. That category was
introduced by Segal [22], and its homotopy theory was studied by Anderson [1] a*
*nd
Bousfield and Friedlander [3]. Its symmetric monoidal structure and concomitant
theory of ring spectra have been studied by Lydakis [13] and Schwede [20].
In this paper, we explore the formal structure that is common to such cate-
gories. We recall some standard facts and terminology about symmetric monoidal
categories in Section 1. We introduce functor categories DT of D-spaces in Sect*
*ion
2. Here D is a suitable domain category for the space-valued functors we shall *
*study,
and T is the category of based spaces. We discuss symmetric monoidal functor
categories DT of D-spaces in Section 3. Our focus is on comparisons between such
categories as D varies. In Section 4, we define forgetful and prolongation func*
*tors
U : DT - ! C T and P : C T - ! DT associated to suitable maps C -! D of
domain categories; U and P are right and left adjoint.
We show how functors with smash product, or D-FSP's, fit into this framework
in Section 5. Briefly, there is an external smash product that takes a pair of *
*D-
spaces to a (D x D)-space, and there is a related internal smash product that t*
*akes
a pair of D-spaces to a D-space. A D-FSP is defined in terms of the external sm*
*ash
product, and it determines and is determined by a D-monoid (or D-ring) defined
with respect to the internal smash product.
A D-monoid R has an associated category of R-modules. We refer to the corre-
sponding structure defined in terms of the external smash product as a D-spectr*
*um
over R. In Section 6, we interpret the category of R-modules, alias the category
of D-spectra over R, as the category of DR -spaces for a domain category DR con-
structed from D and R. We discuss adjoint forgetful and prolongation functors U
and P between categories of C -spectra and D-spectra in Section 7.
Finally, in Section 8, we specialize to the examples that we are most interes*
*ted in.
For particular domain categories D, we fix a canonical D-monoid S that is relat*
*ed
to spheres and obtain the category DS of D-spectra over S. It is symmetric
monoidal when S is commutative. In a continuation of this paper [16], we shall
study model structures on such categories and compare them homotopically.
We have chosen to work with functors that take values in based spaces because
some of our motivating examples make little sense simplicially. However, every-
thing in this paper can be adapted without difficulty to functors that take val*
*ues
in the category of based simplicial sets. The simplicially minded reader will *
*un-
derstand "spaces" to mean "simplicial sets" and "continuous" to mean "simplicia*
*l"
throughout. In fact, all of our constructions apply verbatim to functors that t*
*ake
values in any symmetric monoidal category that is tensored and cotensored over
either topological spaces or simplicial sets. Examples of such symmetric monoid*
*al
functor categories arise in other fields, such as algebraic geometry.
1.Symmetric monoidal categories
We first fix some language to avoid later confusion. A monoidal category is a
category D together with a product = D : D x D -! D and a unit object
u = uD such that is associative and unital up to coherent natural isomorphism;*
* D
is symmetric monoidal if is also commutative up to coherent natural isomorphis*
*m.
See [9, 10, 15] for discussions of the precise meaning of coherence. A symmetr*
*ic
monoidal category D is closed if it has internal hom objects F (d; e) with adju*
*nction
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 3
isomorphisms
D(de; f) ~=D(d; F (e; f)):
There are evident notions of monoids in monoidal categories and commutative
monoids in symmetric monoidal categories. The highly structured ring spectra in
any of the modern approaches to stable homotopy theory are exactly the monoids
and commutative monoids in the relevant symmetric monoidal ground category.
To compare such objects in different ground categories, we need language to de-
scribe when functors and natural transformations preserve monoids and commuta-
tive monoids.
Definition 1.1.A functor F : A -! B between monoidal categories is lax
monoidal if there is a map : uB -! F (uA ) and there are maps
OE : F (A)B F (A0) -! F (AA A0)
that specify a natural transformation OE : B O (F x F ) -! F O A ; it is requir*
*ed
that all coherence diagrams relating the associativity and unit isomorphisms of*
* A
and B to the maps and OE commute. If A and B are symmetric monoidal, then
F is lax symmetric monoidal if all coherence diagrams relating the associativit*
*y,
unit, and commutativity isomorphisms of A and B commute. The functor F is
strong monoidal or strong symmetric monoidal if and OE are isomorphisms.
The definition is incomplete in that we have not specified the relevant "cohe*
*rence
diagrams", but the intuition should be clear enough. See [9, 10] for details. T*
*he
direction of the arrows and OE in the definition leads to the following conclu*
*sion.
Lemma 1.2. If F : A -! B is lax monoidal and M is a monoid in A with unit
j : uA -! M and product : MA M -! M, then F (M) is a monoid in B with
unit F (j) O : uB -! F (uA ) -! F (M) and product
F () O OE : F (M)B F (M) -! F (MA M) -! F (M):
If F : A -! B is lax symmetric monoidal and M is a commutative monoid in A ,
then F (M) is a commutative monoid in B.
We also need the concomitant notion of a monoidal natural transformation. Here
we needn't use an adjective "lax" or "strong" since the definition is the same *
*for
either lax or strong monoidal functors.
Definition 1.3.Let F and G be lax monoidal or lax symmetric monoidal functors
A -! B. A natural transformation ff : F - ! G is monoidal if the following
diagrams commute:
uB G F (A)B F (A0)_ffff//_G(A)B G(A0)
F wwww GGGGG
ww GGG and OEF| |OEG
--www G## fflffl|| fflffl||
F (uA )_____ff_____//G(uA ) F (AA A0) ________// 0
ff G(AA A ):
The following assertion is obvious from the definition and the previous lemma.
Lemma 1.4. If ff is monoidal and A is a monoid in A , then ff : F (A) -! G(A)
is a map of monoids in B. If ff is symmetric monoidal and A is a commutative
monoid in A , then ff : F (A) -! G(A) is a map of commutative monoids in B.
4 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
2.Categories of D-spaces
Spaces will mean compactly generated weak Hausdorff spaces throughout. One
reference is [19]; a thorough treatment is given in [11, App]. We let T denote
the category of based spaces. The category T is a closed symmetric monoidal
topological category under the smash product and function space functors, writt*
*en
X ^ Y and F (X; Y ); its unit object is S0. Conceptually, it is a distinctive f*
*eature
of the topological, as opposed to simplicial, setting that the internal hom spa*
*ces
F (X; Y ) and the categorical hom spaces T (X; Y ) coincide.
Let D be a topological category. We assume that D is based, in the sense that
it has a given initial and terminal object *. Thus the space D(d; e) of maps d *
*! e
is based with basepoint d ! * ! e. When D is given as an unbased category, we
implicitly adjoin a base object *; in other words, we then understand D(d; e) to
mean the union of the unbased space of maps d ! e in D and a disjoint basepoint.
The base object of T is a one-point space. By a functor between based categorie*
*s,
we always understand a functor that carries base objects to base objects; that *
*is,
we take this as part of our definition of "functor". A functor F : D -! D0 betw*
*een
topological categories is continuous if F : D(d; e) -! D0(F d; F e) is a contin*
*uous
map for all d and e; a natural transformation is continuous if its evaluation o*
*n each
object is a continuous map.
Definition 2.1.A D-space is a continuous functor T : D -! T . Let DT denote
the category of D-spaces and continuous natural maps between them.
We think of a D-space as a diagram of spaces whose shape is specified by D.
The category of D-spaces is complete and cocomplete, with limits and colimits
constructed levelwise. It is also tensored and cotensored. For a D-space T and
based space X, the tensor T ^ X is given by the levelwise smash product and the
cotensor F (X; T ) is given by the levelwise function space. Thus
(2.2) DT (T ^ X; T 0) ~=T (X; DT (T; T 0)) ~=DT (T; F (X; T 0)):
We define homotopies between maps of D-spaces by use of the cylinders T ^ I+ .
Spaces and D-spaces are related by a system of adjoint pairs of functors.
Definition 2.3.For an object d of D, define the evaluation functor Evd : DT - !
T by EvdT = T (d) and define the shift suspension functor Fd : T -! DT by
(FdX)(e) = D(d; e) ^ X. The functors Fd and Evd are left and right adjoint,
(2.4) DT (FdX; T ) ~=T (X; EvdT ):
Moreover, Evd is covariantly functorial in d and Fd is contravariantly functori*
*al in
d. We write EvDdand FdDwhen necessary to avoid confusion.
Notation 2.5. We use the alternative notation d* = FdS0. Thus d*(e) = D(d; e)
and FdX = d* ^ X; d* is the D-space represented by the object d.
Recall that a skeleton skD of a category D is a full subcategory with one obj*
*ect
in each isomorphism class. The inclusion skD -! D is an equivalence of categori*
*es.
When D is topological and has a small skeleton skD, DT is a topological categor*
*y.
The set DT (T; T 0) of maps T -! T 0is topologized as the equalizer displayed in
the diagram
Q 0__"_//_Q 0
D(T; T 0)____// dF (T (d); T (d))_//_ff:d!eF (T (d); T (e));
"
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 5
where the products run over the objects and morphisms of skD. For f = (fd), the
ffth component of "(f) is T 0(ff) O fd and the ffth component of "(f) is fe O T*
* (ff).
By an immediate comparison of represented functors, this implies that any D-spa*
*ce
T can be written as the coend of the contravariant functor d* of d and the given
covariant functor T .
Lemma 2.6. Let D have a small skeleton skD and let T be a D-space. Then the
evaluation maps " : d* ^ T (d) -! T induce a natural isomorphism
Z d2skD
d* ^ T (d) -! T:
Explicitly, T is isomorphic to the coequalizer of the parallel arrows in the di*
*agram
W * _"^id//_W* "
d;ee ^ D(d; e) ^ T_(d)_//dd ^ T (d)___//_T;
id^"
where the wedges run over pairs of objects and objects of skD and the parallel *
*arrows
are wedges of smash products of identity and evaluation maps.
3. Symmetric monoidal categories of D-spaces
Let D be a symmetric monoidal (based) topological category with unit object
u and continuous product . The reader may want to glance ahead to Section 8,
where the examples we have in mind are displayed.
Definition 3.1.For D-spaces T and T 0, define the "external" smash product T ZT*
* 0
by
T Z T 0= ^ O (T x T 0) : D x D -! T ;
thus, for objects d and e of D, (T Z T 0)(d; e) = T (d) ^ T 0(e). For a D-space*
* T 0and
a (D x D)-space T 00, define the external function D-space F(T 0; T 00) by
F (T 0; T 00)(d) = D(T 0; T 00);
where T 00(e) = T 00(d; e). Then, for D-spaces T and T 0and a (D x D)-space *
*T 00,
(3.2) (D x D)T (T Z T 0; T 00) ~=DT (T; F(T 0; T 00)):
Lemma 3.3. There is a natural isomorphism
FdX Z FeY -! F(d;e)(X ^ Y ):
Proof.Using (3.2), (2.4), and the definitions, we see that
(D x D)T (FdX Z FeY; T ) ~=T (X ^ Y; T (d; e)) ~=(D x D)T (F(d;e)(X ^ Y ); T )
for a (D x D)-space T . |___|
Now assume further that our given symmetric monoidal category D has a small
skeleton skD; skD inherits a symmetric monoidal structure such that the inclusi*
*on
skD D is strong symmetric monoidal.
We internalize the external smash product T Z T 0by taking its topological le*
*ft
Kan extension along [15, Ch.X]. This gives the category of D-spaces a smash
product ^ under which it is a closed symmetric monoidal topological category. F*
*or
an object d of D, let =d denote the category of objects -over d; its objects ar*
*e the
maps ff : ef ! d and its morphisms are the pairs of maps (OE; ) : (e; f) -! (e*
*0; f0)
such that ff0(OE ) = ff. This category inherits a topology from D, and a map
d ! d0induces a continuous functor =d -! =d0.
6 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
Definition 3.4.Let T and T 0be D-spaces. Define the internal smash product
T ^T 0to be the topological left Kan extension of T ZT 0along . It is character*
*ized
by the universal property
(3.5) DT (T ^ T 0; T 00) ~=(D x D)T (T Z T 0; T 00O ):
On an object d, it is specified explicitly as the colimit
(T ^ T 0)(d) = colimef!d T (e) ^ T 0(f)
indexed on =d; this makes sense since =d has a small cofinal subcategory. When
D itself is small, (T ^ T 0)(d) can also be described as the coend
Z (e;f)2DxD
(T ^ T 0)(d) = D(ef; d) ^ (T (e) ^ T 0(f))
with its topology as a quotient of _(e;f)D(ef; d) ^ (T (e) ^ T 0(f)). By the fu*
*ncto-
riality of colimits, maps d ! d0in D induce maps (T ^ T 0)(d) -! (T ^ T 0)(d0) *
*that
make T ^ T 0into a D-space.
Definition 3.6.Let T , T 0, and T 00be D-spaces. Define the internal function D-
space F (T 0; T 00) by
F (T 0; T 00) = F(T 0; T 00O ):
Then (3.1) and (3.5) immediately imply the adjunction
(3.7) D(T ^ T 0; T 00) ~=D(T; F (T 0; T 00)):
Lemma 3.8. There is a natural isomorphism
FdX ^ FeY -! Fde (X ^ Y ):
Proof.Using (3.5), (3.7), (2.4), and the definitions, we see that
DT (FdX ^ FeY; T ) ~=T (X ^ Y; T (de)) ~=DT (Fde (X ^ Y ); T )
for a D-space T . |___|
Theorem 3.9. The category DT is closed symmetric monoidal under ^ and the
internal function D-space functor F ; its unit object is u*.
The proof is formal and will be omitted; see Day [5]. In Section 8, the smash
product of D-spaces is displayed explicitly in some of the most interesting exa*
*mples.
4.Forgetful and prolongation functors on diagram spaces
We wish to compare the categories DT as D varies.
Definition 4.1.Let C and D be (based) topological categories. We say that D is
a category under C if we are given a continuous functor : C -! D. In practice,
is faithful; we often regard it as an embedding of categories and omit it from*
* the
notations. We say that D is a symmetric monoidal category under C if is a stro*
*ng
symmetric monoidal functor.
We assume that D is a category under C and that C is skeletally small in the
rest of this section.
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 7
Definition 4.2.Define the forgetful functor U : DT - ! C T on D-spaces Y by
letting (UY )(c) = Y (c). Define the prolongation functor P : C T -! DT on
C -spaces T by letting PT be the topological left Kan extension of T along . It*
* is
characterized by the universal property
(4.3) DT (PT; Y ) ~=C T (T; UY ):
Let =d be the topological category of objects -over d; its objects are the maps
ff : c -! d in D and its morphisms are the maps : c -! c0 in C such that
ff0( ) = ff. On an object d, PT is specified explicitly as the colimit
PT (d) = colimc!dT (c)
indexed on =d. If C is small, PT (d) can also be described as the coend
Z c2C
(4.4) PT (d) = D(c; d) ^ T (c):
The name "prolongation" is motivated by the following special case.
Lemma 4.5. If : C - ! D is fully faithful, then the unit j : Id -! UP of the
adjunction (4.3) is a natural isomorphism. Thus P prolongs T to a functor defin*
*ed
on D that restricts to T on C .
Proof.For c 2 C , the identity map of c is a terminal object in =c. |*
*___|
The evident relation EvcUY = Y (c) = Evc Y implies the following observation.
Lemma 4.6. For an object c of C , PFcX is naturally isomorphic to Fc X.
By an h-cofibration i : A -! X of D-spaces, we understand a map that satisfies
the Homotopy Extension Property (HEP). That is, for every map f : X -! Y and
homotopy h : A^I+ -! Y such that h0 = fOi, there is a homotopy "h: X^I+ -! Y
such that "h0= f and "hO(i^id) = h. The universal test case is the mapping cyli*
*nder
Y = Mi = X [i(A ^ I+ ), with the evident f and h, in which case "his a retracti*
*on
X ^ I+ -! MI. The following easy lemma is crucial to our work.
Lemma 4.7. The functors U and P preserve colimits, smash products with spaces,
and h-cofibrations.
Proof.Since colimits of C -spaces and D-spaces are defined levelwise, as are sm*
*ash
products with spaces, the first two preservation properties are clear, and the *
*third_
follows by the retract of mapping cylinders criterion. |*
*__|
We are especially interested in the multiplicative properties of U and P. We *
*now
assume that D is a symmetric monoidal category under C and that both C and
D are skeletally small. Note that we require to be strong rather than just lax
symmetric monoidal.
Proposition 4.8.The functor U : DT -! C T is lax symmetric monoidal and
the functor P : C T - ! DT is strong symmetric monoidal. Moreover, the follow-
ing diagrams commute, where T and T 0are C -spaces and Y and Y 0are D-spaces:
T ^ T 0P PUY ^ PUY 0 ____//_PU(Y ^ Y 0):
PPP nnn
j^j|| PPjPPPP "^" || nnnnn
fflffl| PP'' fflffl|vv"nnnn
UPT ^ UPT 0_____//UP(T ^ T 0) T ^ T 0
8 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
Proof.Observe that left Kan extension also gives a functor
P : (C x C )T - ! (D x D)T :
A direct comparison of colimits shows that
(4.9) P(T Z T 0) ~=PT Z PT 0;
and it is trivial to check the analogous isomorphism
(4.10) U(Y Z Y 0) ~=UY Z UY 0:
We have unit and product isomorphisms : uD -! uC and OE : D O ( x ) -!
O C. For (D x D)-spaces Z, OE induces a natural isomorphism
(4.11) U(Z O D ) ~=(UZ) O C:
The unit isomorphism Pu*C~=u*Dis given by Lemma 4.6, and the unit map u*C-!
Uu*Dis its adjoint. The defining universal properties of ^ and P, together with*
* (4.9)
and (4.11), give a natural isomorphism
~= 0
DT (PT ^ PT 0; Y )-! DT (P(T ^ T ); Y );
and this implies the product isomorphism PT ^PT 0~=P(T ^T 0). Note the direction
of the displayed arrow: P would not even be lax monoidal if were only lax, rat*
*her
than strict, monoidal. Similarly, the defining universal properties of ^ and P,
together with (4.10) and (4.11), give a composite natural map
DT (Y ^ Y 0; Y ^ Y 0)~= (D x D)T (Y Z Y 0; (Y ^ Y 0) O D )
-"*! (D x D)T (PU(Y Z Y 0); (Y ^ Y 0) O
D )
~= (C x C )T (U(Y Z Y 0); U((Y ^ Y 0) O D ))
~= (C x C )T (UY Z UY 0; U(Y ^ Y 0) O C)
~= C T (UY ^ UY 0; U(Y ^ Y 0)):
The product map UY ^UY 0-! U(Y ^Y 0) is the image of the identity map of Y ^Y 0
along this composite. Note that one cannot expect this map to be an isomorphism*
*._
The commutativity of the diagrams displayed in the statement is formal. |*
*__|
5.Diagram spectra and functors with smash product
Fix a skeletally small symmetric monoidal category D. We have the symmetric
monoidal category DT of D-spaces, and we consider its monoids and commutative
monoids and their modules and algebras. These are defined in terms of the inter*
*nal
smash product in DT , and we shall explain their reinterpretations in terms of *
*the
more elementary external smash product Z. The proofs of the comparisons are easy
direct applications of the defining universal properties of ^ (3.5) and Fd (2.3*
*).
Recall the definitions in Section 1. We have the category of lax monoidal fun*
*ctors
D -! T and monoidal transformations and its full subcategory of lax symmetric
monoidal functors. These are the structures defined in terms of the external sm*
*ash
product that correspond to monoids and commutative monoids in DT .
Proposition 5.1.The category of monoids in DT is isomorphic to the category
of lax monoidal functors D -! T . The category of commutative monoids in DT
is isomorphic to the category of lax symmetric monoidal functors D -! T .
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 9
Proof.Let R : D -! T be lax monoidal. We have a unit map : S0 -! R(u)
and product maps OE : R(d) ^ R(e) -! R(de) that make all coherence diagrams
commute. We may view OE as a natural transformation R Z R -! R O . By the
defining properties of Fu and ^, and OE determine and are determined by maps
": u* -! R and "OE: R ^ R -! R that give R a structure of monoid in DT . |__*
*_|
Now assume given a lax monoidal functor R : D -! T . Regarding R as a
monoid in DT , we have an evident notion of a (right) R-module M defined in
terms of a map M ^ R -! M. The external version of an R-module is called a
D-spectrum over R.
Definition 5.2.A D-spectrum over R is a D-space T : D -! T together with a
continuous natural transformation oe : T Z R -! T O such that the composite
T (d) ~=T (d) ^_S0id//^_T (d) ^_R(u)//oe_T (du) ~=T (d)
is the identity and the following diagram commutes:
T (d) ^ R(e) ^ R(f)_oe^id___//T (de) ^ R(f)
id^OE|| |oe|
fflffl| fflffl|
T (d) ^ R(ef)______oe______//T (def):
Proposition 5.3.The category of (right) R-modules is isomorphic to the category
of D-spectra over R.
We let DSR denote the category of D-spectra over R, regarding it interchangab*
*ly
as the category of (right) R-modules.
We can mimic the definitions of tensor product and Hom functors in algebra.
Definition 5.4.For a right R-module T and a left R-module T 0, define T ^R T 0
to be the coequalizer in the category of D-spaces (constructed levelwise) displ*
*ayed
in the diagram
___^id_____
T ^ R ^ T 0_________////_T ^_T_0//_T ^R T 0;
id^0
where and 0are the given actions of R on T and T 0.
Definition 5.5.For right R-modules T 0and T 00, define FR (T 0; T 00) to be the*
* equal-
izer displayed in the following diagram of D-spaces:
*
FR (T 0; T_00)//_F (T 0;_T_00)//_//!_F (T 0^ R; T 00):
Here * = F (; id) and ! is the adjoint of the composite
F (T 0; T 00) ^ T"0^^R//id_T 00^_R//_T 00;
where and are the actions of R on T 0and T 00.
In the rest of this section, we assume that our given R is a commutative mono*
*id
in DT ; that is, R is a lax symmetric monoidal functor D -! T . In this case,
the categories of left and right R-modules are isomorphic. Moreover, T ^R T 0and
10 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
FR (T; T 0) inherit R-module structures from T or, equivalently, T 0. For R-mod*
*ules
T , T 0, and T 00,
(5.6) DSR (T ^R T 0; T 00) ~=DSR (T; FR (T 0; T 00)):
Theorem 5.7. When R is commutative, the category DSR of R-modules is closed
symmetric monoidal with unit R under the smash product ^R and the function
R-module functor FR .
Definition 5.8.A (commutative) R-algebra is a (commutative) monoid in DSR .
The external version of an R-algebra is called a D-FSP (functor with smash
product) over R. We write o consistently for symmetry isomorphisms.
Definition 5.9.A D-FSP over R is a D-space T together with a unit map j :
R -! T of D-spaces and a continuous natural product map : T Z T -! T O of
functors D x D -! T such that the composite
id^j
T (d) ~=T (d) ^_S0id//^_T (d) ^_R(u)//_T (d) ^ T_(u)//_T (du) ~=T (d)
is the identity and the following unity, associativity, and centrality of unit *
*diagrams
commute:
R(d) ^ R(e)j^j_//T (d) ^ T (e)
OE|| ||
fflffl| fflffl|
R(de) ___j___//_T (de);
T (d) ^ T (e) ^ T_(f)^id___//T (de) ^ T (f)
id^ || ||
|fflffl fflffl|
T (d) ^ T (ef)_____________//T (def);
and
R(d) ^ T (e)j^id//_T (d) ^ T_(e)//_T (de)
o|| T(o)||
fflffl| fflffl|
T (e) ^ R(d)id^j//_T (e) ^ T_(d)//_T (ed)
A D-FSP is commutative if the following diagram commutes, in which case the
centrality of unit diagram just given commutes automatically:
T (d) ^ T (e)__//_T (de)
o|| |T(o)|
fflffl| fflffl|
T (e) ^ T (d)__//_T (ed):
A D-FSP over R is a D-spectrum over R with additional structure.
Lemma 5.10. A D-FSP over R has an underlying D-spectrum over R with struc-
ture map
oe = O (idZj) : T Z R -! T O :
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 11
Proposition 5.11.The category of R-algebras is isomorphic to the category of
D-FSP's over R. The category of commutative R-algebras is isomorphic to the
category of commutative D-FSP's over R.
6. An interpretation of diagram spectra as diagram spaces
Let D be symmetric monoidal and fix a lax monoidal functor R : D -! T . We
do not require R to be lax symmetric monoidal in general, although that is the *
*case
of greatest interest. We reinterpret the category DSR of D-spectra over R, alias
the category of right R-modules, as the category DR T of DR -spaces, where DR i*
*s a
lax monoidal category constructed from D and R. If R is a lax symmetric monoidal
functor, then DR is a symmetric monoidal category. In this case, we can reinter*
*pret
the smash product ^R of R-modules as the smash product in the category of DR -
spaces. Conceptually, this simplifies the theory by reducing the study of D-spe*
*ctra
over R to a special case of the general study of functor categories.
Just as in algebra, for a D-space T , T ^ R is the free R-module generated by
T . Recall the represented functors d* from Notations 2.5 and remember that they
behave contravariantly with respect to d.
Construction 6.1. We construct a monoidal category DR and a strong monoidal
functor ffi : D -! DR . The objects of DR are the objects of D. For objects d a*
*nd
e of D, the space of morphisms d -! e in DR is
DR (d; e) = DSR (e* ^ R; d* ^ R);
and composition is inherited from composition in DSR . Thus DR may be identified
with the full subcategory of DSRopwhose objects are the free R-modules d*^R. We
specify ffi on morphisms by smashing maps given by the contravariant functorial*
*ity
of d* with the identity map of R. Observe that we have adjunction isomorphisms
DR (d; e) ~=(d* ^ R)(e):
On objects, the product R in DR is the product of D. The unit object is the
unit object u of D. The product fR f0 of morphisms f : e* ^ R -! d* ^ R and
f0 : e0*^ R -! d0*^ R is
f ^R f0 : (ee0)*^ R ~=(e ^ R) ^R (e0*^ R) -! (d ^ R) ^R (d0*^ R) ~=(dd0)*^ R:
Observe that
(6.2) (d* ^ R)(e) ~=colimff:fg-!eD(d; f) ^ R(g):
Taking ff to be the canonical isomorphism eu ~=e and using the unit : S0 -!
R(u), we can identify ffi on morphisms as the evident map
D(d; e) = D(d; e) ^ S0 -! D(d; e) ^ R(u) -! (d* ^ R)(e):
By inspection of definitions, ffi is a strong monoidal functor.
Proposition 6.3.The categories DSR of D-spectra over R and DR T of DR -
spaces are isomorphic.
Proof.Taking ff in (6.2) to be the identity map of de and using the identity map
d -! d, we obtain an inclusion : R(e) -! (d* ^ R)(de). Let T be a DR -space.
Pullback along ffi gives T a structure of D-space. Pullback along of the evalu*
*ation
map DR (d; de) ^ T (d) -! T (de) gives the components T (d) ^ R(e) -! T (de)
of a map T Z R -! T O . Via (3.5), this gives an action of R on T . These
12 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
two actions determine the original action of DR . Indeed, working conversely, *
*if
T is an R-module and ff : fg -! e is a morphism of D, then the composites
displayed in the following diagram pass to colimits to define the evaluation ma*
*ps
(d* ^ R)(e) ^ T (d) -! T (e) of a functor T : DR -! T .
D(d; f) ^ R(g) ^ T (d)
id^o||
fflffl| "^id
D(d; f) ^ T (d) ^ R(g)___//T (f) ^ R(g)
((-) id)^ || ||
fflffl| fflffl|
D(dg; fg) ^ T (dg) ___"__//_T (fg)
|T(ff)|
fflffl|
T (e):
Here " is the evaluation map of T and is the action of R on T . This gives_the
desired isomorphism of categories. |__|
In the commutative case, we have the following important addenda.
Proposition 6.4.If R : D - ! T is a lax symmetric monoidal functor, then
DR is a symmetric monoidal category under R , ffi : D -! DR is a strong sym-
metric monoidal functor, and the isomorphism of categories DSR ~=DR T is an
isomorphism of symmetric monoidal categories.
Proof.Inspections of definitions give the statements about DR and ffi. To show *
*that
the smash products agree under the isomorphism between DSR and DR T , we can
either compare the definitions of the respective smash products directly, or we*
* can
compare the defining universal properties. Note that the unit (uDR)* of the sma*
*sh
product of DR -spaces is isomorphic to R since
(uDR)*(d) = DR (uDR; d) = ((uD )* ^ R)(e) ~=R(e): |___|
Remark 6.5.If R = (uD )*, then ffi : D -! DR is an identification. That is, as *
*in
any symmetric monoidal category, D-spaces admit a unique structure of module
over the unit for the smash product.
7. Forgetful and prolongation functors on diagram spectra
We use the categories DR to reduce comparisons of categories of diagram spect*
*ra
to comparisons of categories of diagram spaces. Return to the context of Section
4 and assume given a symmetric monoidal category D under C , with strong sym-
metric monoidal functor : C -! D, where both C and D are skeletally small.
Proposition 7.1.Let R : D -! T be a lax monoidal functor and let Q = R O :
C -! T . Then : C -! D extends to a strong monoidal functor : CQ -! DR .
If R is lax symmetric monoidal, then is strong symmetric monoidal.
Proof.Using (6.2) to write the morphism sets of CQ and DR as colimits, we see
immediately that smash products of maps : C (a; b) -! D(a; b) and identity *
* __
maps of the spaces Q(c) = R(c) pass to colimits to give the required extension.*
* |__|
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 13
Applied to : CQ -! DR , the results of Section 4 give a prolongation functor
P from CQ -spaces (= C -spectra over Q) to DR -spaces (= D-spectra over R) that
is left adjoint to the evident forgetful functor U. Just as Proposition 6.4 giv*
*es two
equivalent descriptions of the smash product of R-modules (when R is symmetric),
so we have two equivalent descriptions of the prolongation of Q-modules, the se*
*cond
of which does not make use of the categories CQ and DR .
Lemma 7.2. Consider P : C T -! DT . If Q : C -! T is a lax (symmetric)
monoidal functor, then PQ : D -! T is a lax (symmetric) monoidal functor and
P restricts to a functor from the category of C -spectra over Q to the category*
* of
D-spectra over PQ. Moreover, the adjunction (4.3) restricts to an adjunction
(7.3) DSPQ(PT; Y ) ~=C SQ (T; UY ):
Proof.The first statement is immediate from Proposition 4.8. For the second sta*
*te-
ment, we must show that if T is a Q-module and Y is a PQ-module, then a map
f : PT -! Y of D-spaces is a map of PQ-modules if and only if its adjoint
"f: T -! UY is a map of Q-modules. The proof is a pair of diagram chases that
ultimately boil down to use of the pair of triangles displayed in Proposition 4*
*.2. |___|
Lemma 7.4. Let Q = R O = U(R), where R is a lax (symmetric) monoidal
functor D -! T . The adjoint " : PQ = PUR -! R of the identity map of UR is
a map of (commutative) monoids in DT , hence an R-module Y may be regarded
as a PQ-module by pullback of the action along ". Passage to coequalizers speci*
*fies
an extension of scalars functor
(-) ^PQ R : DSPQ -! DSR
such that, for R-modules T and PQ-modules T 0,
DSR (T ^PQ R; T 0) ~=DSPQ(T; T 0):
When R is symmetric, the pullback of action functor is lax symmetric monoidal
and the extension of scalars functor is strong symmetric monoidal.
Proof.The proof is formally the same as for extension of scalars in algebra. *
* |___|
Now the uniqueness of adjoints gives the following result.
Proposition 7.5.Let Q = R O : Then P : CQ T -! DR T agrees under the
isomorphisms of its source and target with the composite of P : C SQ -! DSPQ
and extension of scalars DSPQ -! DSR .
8.Examples of diagram spectra
It is high time that we specialized the general abstract theory to the exampl*
*es
of interest in stable homotopy theory. Here we change our point of view. So far*
*, we
have considered general lax monoidal functors R : D -! T , usually symmetric.
Now we focus on a particular, canonical, choice, which we denote by S, or SD
when necessary for clarity, to suggest spheres. It is a faithful functor in all*
* of our
examples.
We take Sn to be the one-point compactification of Rn; the one-point compact-
ification of {0} is S0, and it is convenient to let Sn = * if n < 0. Similarly*
*, for
a finite dimensional real inner product space V , we take SV to be the one-point
compactification of V . Our first example is elementary, but crucial to the the*
*ory.
14 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
Example 8.1 (Prespectra).Let N be the (unbased) category of non-negative in-
tegers, with only identity morphisms between them. The symmetric monoidal
structure is given by addition, with 0 as unit. An N -space is a sequence of ba*
*sed
spaces. The canonical functor S = SN sends n to Sn. It is strong monoidal, but*
* it
is not symmetric since permutations of spheres are not identity maps. This is t*
*he
source of difficulty in defining the smash product in the stable homotopy categ*
*ory.
A prespectrum is an N -spectrum over S. Let P denote the category of prespectra.
Since Sn is canonically isomorphic to the n-fold smash power of S1, the category
of prespectra defined in this way is isomorphic to the usual category of prespe*
*ctra,
whose objects are sequences of based spaces Tn and based maps Tn -! Tn+1.
The shift suspension functors to N -spectra are given by (Fm X)n = X ^ Sn-m .
The smash product of N -spaces is given by
_n
(T ^ T 0)n = Tp ^ Tn0-p:
p=0
The category NS such that an N -spectrum is an NS-space has morphism spaces
NS(m; n) = Sn-m :
Because SN is not symmetric, the category of N -spectra does not have a smash
product that makes it a symmetric monoidal category. For all other D that we
consider, the functor SD is a strong symmetric monoidal embedding D - ! T .
Therefore the category of D-spectra over S is symmetric monoidal.
Example 8.2 (Symmetric spectra).Let be the (unbased) category of finite sets
n = {1; : :;:n}, n 0, and their permutations; thus there are no maps m -! n
for m 6= n, and the set of maps n ! n is the symmetric group n. The symmetric
monoidal structure is given by concatenation of sets and block sum of permutati*
*ons,
with 0 as unit. The canonical functor S = S sends n to Sn. A symmetric spectrum
is a -spectrum over S. Let S denote the category of symmetric spectra. Define
a strong symmetric monoidal faithful functor : N -! by sending n to n
and observe that SN = S O . In effect, we have made S symmetric by adding
permutations to the morphisms of N . The idea of doing this is due to Jeff Smit*
*h.
The shift suspension functors to symmetric spectra are given by
(Fm X)(n) = n+ ^n-m (X ^ Sn-m ):
The smash product of -spaces is given by
_n
(T ^ T 0)(n) ~= n + ^pxn-p T (p) ^ T (n - p)
p=0
as a n-space. Implicitly, we are considering the set of partitions of the set n*
*. If
we were considering the category of all finite sets k, we could rewrite this as
_
(T ^ T 0)(k) = T (j) ^ T 0(k - j);
jk
and this reinterpretation explains the associativity and commutativity of ^. The
category S such that a -spectrum is a S-space has morphism spaces
S(m; n) = n+ ^n-m Sn-m :
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 15
Example 8.3. The functor S is the case X = S1 of the strong symmetric monoidal
functor SX : -! T that sends n to the n-fold smash power X(n)for a based space
X. Moreover, the SX give all strong symmetric monoidal functors -! T . Ap-
plied to SX , our theory constructs a symmetric monoidal category of "SX -modul*
*es".
The homotopy theory of these categories is relevant to localization theory.
Example 8.4 (Orthogonal spectra).Let I be the (unbased) category of finite
dimensional real inner product spaces and linear isometric isomorphisms; there
are no maps V -! W unless dim V = dim W = n for some n 0, when the
space of morphisms V -! W is homeomorphic to the orthogonal group O(n).
The symmetric monoidal structure is given by direct sums, with {0}as unit. The
canonical functor S = SI sends V to SV . An orthogonal spectrum is an I -
spectrum over S. Let I S denote the category of orthogonal spectra. Define a
strong symmetric monoidal faithful functor : -! I by sending n to Rn and
using the standard inclusions n -! O(n). Observe that S = SI O .
The shift suspension functors to orthogonal spectra are given on W V by
(FV X)(W ) = O(W )+ ^O(W-V )(X ^ SW-V );
where W - V is the orthogonal complement of V in W ; an analogous description
applies whenever dim W dim V , and (FV X)(W ) = * if dim W < dim V . Note
that we can restrict attention to the skeleton {Rn} of I . For an inner product
space V of dimension n, choose a subspace Vp of dimension p for each p n. The
smash product of I -spaces is given by
_n
(T ^ T 0)(V ) ~= O(V )+ ^O(Vp)xO(V -Vp)T (Vp) ^ T 0(V - Vp)
p=0
as an O(V )-space. This describes the topology correctly, but to see the associ*
*ativity
and commutativity of ^, we can rewrite this set-theoretically as
_
(T ^ T 0)(V ) = T (W ) ^ T 0(V - W ):
WV
The category IS such that an I -spectrum is an IS-space has morphism spaces
IS(V; W ) = O(W )+ ^O(W-V )SW-V
for V W .
This example admits several variants. For instance, we can use real vector sp*
*aces
and their isomorphisms, without insisting on inner product structures and isome-
tries, or we can use complex vector spaces.
Example 8.5. Let V have dimension n and let T O(V ) be the Thom space of the
tautological n-plane bundle over the Grassmannian of n-planes in V V . Then
T O(-) is the object function of a lax symmetric monoidal functor T O : I -! T .
The equivariant generalization of this example was exploited in [6]. Our form*
*al
theory applies to functors like T O, but we focus on the canonical functors SD .
Example 8.6 (W -spaces).It is tempting to take D = T , but that does not have
a small skeleton. We can take D to be any skeletally small based subcategory
of T that contains S0 and is closed under smash products, taking SD to be the
inclusion D -! T . In particular, we can take D to be the category W of based
spaces homeomorphic to finite CW complexes. The theory works equally well if
16 M. A. MANDELL, J. P. MAY, S. SCHWEDE, AND B. SHIPLEY
we redefine W in terms of countable rather than finite CW complexes. We have
evident strong symmetric monoidal faithful functors -! W and I -! W under
which SW restricts to S and SI .
The shift suspension functors to W -spaces are given by
(FY X)(Z) = F (Y; Z) ^ X:
The simplicial counterpart of this example is treated in detail by Lydakis [1*
*4].
This example suggests an alternative way of viewing and I .
Remark 8.7.It is sometimes convenient, and sometimes inconvenient, to change
point of view and think of the objects of and I as the spheres Sn and SV ,
thus thinking of and I as subcategories of W . With this point of view, is a
subfunctor of ^ and S is the inclusion of a symmetric monoidal subcategory.
All of our examples so far are categories under N . However, our last example
is not of this type.
Example 8.8 (F -spaces = -spaces).Let F be the category of finite based sets
n+ = {0; 1; : :;:n}and all based maps, where 0 is the basepoint. This is the
opposite of Segal's category [22]. This category is based with base object the*
* one
point set 0+ . Take to be the smash product of finite based sets; to be precis*
*e,
we order the non-zero elements of m+ ^ n+ lexicographically. The unit object is
1+ . The canonical functor SF sends n+ to n+ regarded as a discrete based space;
it is the restriction to F of the functor SW .
In contrast to the cases of symmetric spectra and orthogonal spectra, the act*
*ion
of SD required of D-spectra gives no additional data when D = F or D = W .
Lemma 8.9. Let SD : D -! T be an embedding of D as a full symmetric
monoidal subcategory of T . Then a D-space T admits a unique structure of D-
spectrum, and the categories of D-spaces and D-spectra are isomorphic. In parti*
*c-
ular, this applies to D = F and D = W .
Proof.This is an instance of Remark 6.5, but it is worthwhile to explain it exp*
*licitly.
Let X and Y be spaces in D, omit the embedding SD from the notation, and write
^ for . The action oe : T (X) ^ Y -! T (X ^ Y ) is the adjoint of the composite
Y -ff!T (X; X ^ Y ) = D(X; X ^ Y )-T!T (T (X); T (X ^ Y ));
where ff(y)(x) = x ^ y. The equality holds because D is a full subcategory of
T , and the map T is continuous by our requirement that D-spaces be continuous
functors. For the uniqueness, let S0 = {*; 1}and observe that an element y 2 Y
determines the based map "y: S0 -! Y with "y(1) = y. The naturality and unit
conditions in the definition of an action oe force the relation
oe(t ^ y) = oe O (id^"y)(t ^ 1) = T (id^"y)(t ^ 1)
for t 2 T (X). This agrees with our definition of oe and proves its uniqueness.*
* |___|
Special cases of D-FSP's (over SD ) were first defined by different authors.
Remark 8.10.Up to nomenclature, D-FSP's were first introduced as follows.
1. A T -FSP is an FSP as introduced by B"okstedt in [2], although his definit*
*ion
was simplicial and he imposed convergence and connectivity conditions.
DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP'S 17
2. A commutative I -FSP is an I*-prefunctor as defined by May, Quinn, and
Ray [18]; this was the earliest definition of this general type.
3. A -FSP is a symmetric ring spectrum as defined by Smith [8]. This no-
tion first appeared under the name "strictly associative ring spectrum" in
Gunnarson [7]. The name "FSP defined on spheres" has also been used.
4. An F -FSP is a Gamma-ring as defined by Lydakis and Schwede [13, 20].
Different kinds of FSP's arise naturally in different applications. In B"okst*
*edt's
construction of topological Hochschild homology, it is natural to use -FSP's. In
the early applications of [4, 18, 17] and in recent equivariant applications [6*
*], it is
essential to work with I -FSP's. In other applications, F -FSP's are essential *
*[20].
We shall compare these categories homotopically in the sequel [16].
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