Diagram spaces and symmetric spectra
Christian Schlichtkrull
Department of Mathematics
University of Bergen, Norway
p-Adic Geometry and Homotopy Theory
Loen, August 2–8, 2009
Outline
Symmetric spectra and S-algebras
E∞ spaces and infinite loop spaces
The I-space approach to E∞ spaces
Units of S-algebras
Spectra and cohomology theories
In algebraic topology, homology and cohomology theories are
represented by spectra.
Definition
A spectrum E is a sequence of based spaces E (n) for n ≥ 0,
together with structure maps S 1 ∧ E (n) → E (n + 1).
Here ∧ is the smash product of based spaces:
X ∧ Y = X × Y {∗} × Y ∪ X × {∗}.
A basic example is the sphere spectrum S with S(n) = S n .
Some constructions:
The shift suspension spectrum Σk E has Σk E (n) = E (n + k).
For a based space X , the spectrum E ∧ X has nth space E (n) ∧ X .
The associated homology and cohomology theories on based
spaces are defined by
En (X ) = [Σn S, E ∧ X ]
and E n (X ) = [S ∧ X , Σn E ]
where [−, −] denotes the abelian group of morphism in the
homotopy category of spectra (aka the stable homotopy category).
In particular, the homotopy groups are defined by
πk E = [Σk S, E ].
A multiplicative structure on a cohomology theory corresponds to a
multiplicative structure on the representing spectrum.
In order to make this precise it is convenient to work in a category
of spectra with a strictly associative “smash product”. There are
several good choices for such a category, for instance
I
the category of symmetric spectra (Hovey-Shipley-Smith),
I
the category of orthogonal spectra
(Mandell-May-Schwede-Shipley),
I
the category of simplicial functors (Lydakis),
I
the category of S-modules (Elmendorf-Kriz-Mandell-May).
In this talk we shall concentrate on the category of symmetric
spectra.
Symmetric spectra
Definition (Hovey-Shipley-Smith)
A symmetric spectrum E is a spectrum in which each space E (n)
is equipped with a based Σn -action such that the iterated structure
maps
S m ∧ E (n) → E (m + n)
are Σm × Σn -equivariant.
The sphere spectrum S is an example of a symmetric spectrum.
Let SpΣ be the category of symmetric spectra.
Theorem (Hovey-Shipley-Smith)
The category SpΣ has an associative smash product ∧ such that
(SpΣ , ∧, S) is a symmetric monoidal category.
S-algebras
The universal property of the smash product is such that a map of
symmetric spectra E ∧ E 0 → E 00 amounts to a “compatible” family
of Σm × Σn -equivariant maps
E (n) ∧ E 0 (m) → E 00 (m + n).
A symmetric ring spectrum R is a monoid in (SpΣ , ∧, S). Thus,
there are maps of symmetric spectra S → R and R ∧ R → R,
which amount to maps
S n → R(n)
and R(m) ∧ R(n) → R(m + n).
Since S is the unit for the smash product, we have that
S-modules = SpΣ ,
S-algebras = Symmetric ring spectra,
Comm S-algebras = Comm symmetric ring spectra.
Using the smash product ∧ in analogy with the tensor product ⊗
of abelian groups, one may define topological versions of
constructions from algebra.
The basic idea is to replace the ground ring Z by S.
Forgetting the symmetric monoidal structure, the homotopy theory
associated to SpΣ is equivalent to the homotopy theory for
ordinary (non-symmetric) spectra:
Theorem (Hovey-Shipley-Smith)
There is a model structure on SpΣ that makes it Quillen equivalent
to the category of ordinary spectra. In particular, the homotopy
category of symmetric spectra is equivalent to the stable homotopy
category.
Examples of S-algebras
Example (Eilenberg-Mac Lane spectra)
An ordinary ring R gives rise to the Eilenberg-Mac Lane spectrum
HR defined by
HR(n) = |R(S•n )|,
where
I
S•n is the n-fold smash product of the simplicial circle S•1 ,
I
R(S•n ) is the simplicial set [k] 7→ R[Skn ]/R{∗}.
This is an S-algebra which is commutative if R is. The
multiplication is induced by the simplicial maps
R(S•m ) ∧ R(S•n ) → R[S•m ∧ S•n ] = R[S•m+n ].
The S-algebra HR represents ordinary cohomology with coefficients
in R.
Example (Thom spectra)
The cobordism spectrum MO can naturally be defined as a
commutative S-algebra:
MO(n) = B(∗, O(n), S n )/BO(n),
where S n is the one-point compactification of Rn with the usual
action of the orthogonal group O(n). The multiplication is induced
by
B(∗, O(m), S m ) × B(∗, O(n), S n ) → B(∗, O(m) × O(n), S m ∧ S n )
→ B(∗, O(m + n), S m+n ).
A similar construction applies to MSO. These S-algebras represent
the unoriented and the oriented cobordism theories.
We shall later see more examples of Thom spectra that are S-algebras
(for instance MU).
Example (Topological K -theory)
Commutative S-algebra models of the topological K -theory spectra
KO and KU have been constructed by M. Joachim and M.
Mandell. These S-algebras represent real and complex topological
K -theory.
Example (Spherical monoid rings)
Given a (topological) monoid M, the spherical monoid ring S[M] is
defined by
S[M](n) = S n ∧ M+ ,
(M+ = M ∪ {+}).
This is an S-algebra which is commutative if M is. It is related to
the ordinary monoid ring via π0 S[M] = Z[π0 M].
E∞ spaces and infinite loop spaces
A spectrum E has an associated infinite loop space Ω∞ (E ) defined
by
Ω∞ (E ) = (ho)colim Ωn E (n),
n
Ωn E (n) = Map∗ (S n , E (n)).
The recognition theorem of Boardman-Vogt and May gives a
criterion for when a space has the form Ω∞ (E ).
Recall that an operad E is a sequence of spaces E(n), for n ≥ 0, such
that Σn acts on E(n) and there are structure maps
E(k) × E(j1 ) × · · · × E(jk ) → E(j1 + · · · + jk )
which satisfy certain associativity and equivariance conditions.
An action of an operad E on a space X is given by a compatible sequence
of maps
E(k) × X k → X .
In this way E(k) parametrizes k-fold multiplications in X .
An operad E is an E∞ operad if the space E(n) is Σn -free and
contactible for all n.
Definition
An E∞ space is a space with an action of an E∞ operad.
The operad action on an E∞ space X induces a homotopy
associative and commutative multiplication. In fact, the term
“E∞ ” indicates that X is “homotopy everything” in the sense that
it has all possible coherence homotopies for associativity and
commutativity.
We say that X is grouplike if π0 X is a group.
Theorem (Boardman-Vogt, May)
If X is a grouplike E∞ space, then there exists a spectrum E such
that X ' Ω∞ E .
Example
The Barratt-Eccles operad E is the E∞ operad with
E(n) = E Σn (= B(∗, Σn , Σn )).
If A is a symmetric (strict) monoidal category, then E acts on the
classifying space BA which is therefore an E∞ space.
Example
The linear isometries operad L is the E∞ operad with
L(n) = Linear isometries |R∞ ⊕ ·{z
· · ⊕ R∞} → R∞
n
Let O(n) be the orthogonal group and let BO(∞) be the
stabilization of the classifying spaces BO(n). Then L acts on
BO(∞) which is therefore an E∞ space.
Remark
We may think of an E∞ space as the topological analogy of a
commutative monoid in algebra.
It is important to realize that not every E∞ space is homotopy
equivalent to a strictly commutative topological monoid. In fact,
for this to be the case the space must up to homotopy be a
product of Eilenberg Mac Lane spaces.
The E∞ space BO(∞) is an example of a space which is not
homotopy equivalent to a product of Eilenberg Mac Lane spaces.
In the following we shall describe an approach to E∞ spaces which
overcomes this difficulty and which is convenient when working
with symmetric spectra.
Remark
A. Blumberg has introduced an approach to “structured spaces” which is
convenient for working with the Elmendorf-Kriz-Mandell-May S-modules.
The category of I-spaces
Let I be the category with objects the finite sets n = {1, . . . , n}
(including the empty set 0) and morphisms the injective maps.
The ordered concatenation m t n of ordered sets makes this a
symmetric monoidal category (I, t, 0).
Let S be the category of spaces (or simplicial sets).
Definition
We define an I-space to be a functor X : I → S and we write S I
for the category of such functors.
The symmetric monoidal structure of I induces a symmetric
monoidal structure on S I :
X Y (n) = colim X (n1 ) × Y (n2 ),
n1 tn2 →n
where the colimit is over the comma category (t ↓ n). The unit is
the constant I-diagram ∗.
The universal property of the -product is such that a map of
I-spaces X Y → Z amounts to a natural map of I × I-spaces
X (m) × Y (n) → Z (m + n).
An I-space monoid A is a monoid in (S I , , ∗). Thus, there are
maps ∗ → A and A A → A which amounts to maps
∗ → A(n)
and A(m) × A(n) → A(m + n).
The following notation is often convenient in order to keep track of
the combinatorics involved with I-spaces:
Given a morphism α : m → n in I, we write n − α for the
complement of α(m) in n.
Example
Given an ordinary ring R, the correspondence n 7→ GLn (R) defines
a functor I → Groups: a morphism α : m → n in I gives rise to
the group homomorphism α∗ : GLm (R) → GLn (R) defined by
id⊕f
α∗ (f ) : R n ' R (n−α) ⊕ R m −−−→ R (n−α) ⊕ R m ' R n .
Applying the classifying space functor we get the commutative
I-space monoid BGL(R).
The multiplication is induced by the homomorphisms
A 0
GLm (R) × GLn (R) → GLm+n (R), (A, B) 7→
0 B
For R = R we may remember the topology on GL(R) and get the
commutative I-space monoids BGL(R) with I-space submonids
BO and BSO.
The model category structure on I-spaces
I
A map of I-spaces X → Y is an I-equivalence if the induced
map of homotopy colimits XhI → YhI is a weak homotopy
equivalence.
Here we use the Bousfield-Kan simplicial replacement of the
diagram in order to define the homotopy colimit:
a
XhI = hocolim X = X (nk )
I
n ←...←n
0
I
k
A map of I-spaces X → Y is an I-fibration if it is a level
fibration and if for each morphism α : m → n in I
X (m) −−−−→ Y (m)


α
α
y ∗
y ∗
X (n) −−−−→ Y (n)
is a homotopy cartesian square.
I
A map of I-spaces X → Y is an I-cofibration if for each n
the map
X (n) ∪Ln (X ) Ln (Y ) → Y (n)
is a projective Σn -cofibration. Here Ln (X ) denotes the nth
latching space,
Ln (X ) = colim
X (m).
m→n
m<n
That a map of spaces is a projective Σn -cofibration means that it is
a retract of a relative free Σn -cell complex.
Theorem (Sagave-S)
The I-equivalences, I-fibrations, and I-cofibrations define a
cofibrantly generated proper model category structure on S I .
The fibrant objects are the “homotopy constant” I-spaces.
This is the I-space analogue of the stable model structure on SpΣ .
We shall sometimes refer to it as the absolute model structure.
There are adjoint functors
colim : S I o
/
S : const
These are strong symmetric monoidal functors, hence preserve
multiplicative structures.
Theorem (Sagave-S)
The functors colim and const define a Quillen equivalence between
S I and S. In particular, the homotopy category of I-spaces is
equivalent to the usual homotopy category of spaces.
It follows that an I-space X is equivalent to the constant I-space
XhI . We think of XhI as the underlying space associated with X .
Remark
If X is a commutative I-space monoid, then XhI has a canonical
action of the Barratt-Eccles operad.
Example
Consider the commutative I-space monoid BO : n 7→ BO(n).
In this case the connectivity of the maps BO(m) → BO(n) tends
to infinity with m and n which implies that
BOhI ' colim{BO(0) → BO(1) → BO(2) → . . . } = BO(∞)
Thus, we have “represented” the E∞ space BO(∞) as a strictly
commutative monoid in S I .
Example
Let R be an ordinary ring and consider the commutative I-space
monoid n 7→ BGLn (R). In this case the underlying space can be
identified in terms of Quillen’s plus construction:
BGL(R)hI ' BGL∞ (R)+ .
Thus, we have “represented” the higher algebraic K -theory of R by
a commutative monoid in S I .
Example
Let X be a based space and consider the commutative I-space monoid
X • : n 7→ X n , where α : m → n induces the map
id×{∗}
∼
α∗ : X m −−−−→ X m × X (n−α) −
→ X n.
In this case the underlying space can be identified in terms of the
Barratt-Eccles construction
•
XhI
' Γ+ (X ) =
∞
a
E Σn ×Σn X n
.
∼.
n=0
It follows from the Barratt-Priddy-Quillen Theorem that if X is
•
connected then XhI
' Ω∞ Σ∞ (X ), and by the Dold-Thom Theorem that
the canonical map
XhI = hocolim X • → colim X • = SP ∞ (X )
I
I
induces the Hurewicz homomorphism on homotopy groups.
Example (Symmetric Thom spectra from I-spaces)
Let S I /BO be the category of I-spaces over the commutative I-space
monoid BO : n 7→ BO(n). The Thom space functor has a refinement to
a strong symmetric monoidal Thom spectrum functor
T : S I /BO → SpΣ .
In particular, T takes commutative I-space monoids over BO to
commutative S-algebras.
For instance, the Thom spectra MO and MSO are obtained from the
identity BO → BO and the inclusion BSO → BO.
In order to realize the complex cobordism spectrum MU, notice that the
inclusions
U(n) ⊆ O(2
· · t 2})
| t ·{z
n
make n 7→ BU(n) a diagram on a certain subcategory of I. The left Kan
extension defines a commutative I-space monoid over BO and the
associated Thom spectrum represents MU as a commutative S-algebra.
The model category of commutative I-space monoids
We wish to define a model category of commutative I-space monoids
such that the weak equivalences are the I-equivalences of the underlying
I-spaces.
In such a model structure the fibrant objects will in general not be
homotopy constant: if this was the case, then AhI would be equivalent to
a strictly commutative monoid for any commutative I-space monoid A.
The commutative I-space monoid BO is a counter example to this.
I
A map of I-spaces X → Y is a positive I-fibration if it is a level
fibration in positive degrees and if for any morphism α : m → n
with m > 0 the square
X (m) −−−−→ Y (m)


α
α
y ∗
y ∗
X (n) −−−−→ Y (n)
is homotopy cartesian.
Theorem (Sagave-S)
There is a cofibrantly generated model structure on the category of
commutative I-space monoids in which the weak equivalences are the
I-equivalences and the fibrations are the positive I-fibrations.
We shall refer to this as the positive model structure.
In this model structure the cofibrations are retracts of relative cell
complexes obtained by attaching free commutative cells generated by
I-spaces of the form I(d, −) × D n for d > 0.
Remark
The above theorem is the I-space version of the analogous result for
symmetric spectra due to Mandell-May-Schwede-Shipley.
E∞ spaces and commutative I-space monoids
In the following we fix an E∞ operad (e.g., the Barratt-Eccles
operad) and consider the associated category of E∞ spaces.
Theorem (Sagave-S)
The category of E∞ spaces is Quillen equivalent to the category of
commutative I-space monoids. In particular, these categories have
equivalent homotopy categories.
In order to prove this one first observes that there is a natural notion of
an E∞ I-space, defined using the -product instead of the cartesian
product of spaces.
Furthermore, the category of E∞ I-spaces both has an absolute and a
positive model structure with weak equivalences the I-equivalences of
the underlying I-spaces.
We then have the following chain of Quillen equivalences:
(E∞ spaces, “projective” model structure)
O
const
colim
(E∞ I-spaces, absolute model structure)
O
id
id
(E∞ I-spaces, positive model structure)
O
“induction”
“restriction”
(Commutative I-space monoids, positive model structure)
Remark
The (“induction”,“restriction”) adjunction is the I-space version of the
analogues Quillen equivalence for symmetric spectra established by
Elmendorf-Mandell.
Example
Consider for each d ≥ 0 the I-space Fd (∗) = I(d, −). The “induction”
functor takes the E∞ I-space
a
E(Fd (∗)) =
(E Σk × Fd (∗)k )/Σk
k≥0
to the commutative I-space monoid
a
C(Fd (∗)) =
Fd (∗)k /Σk .
n≥0
k
z }| {
There is a Σk -equivariant isomorphism Fd (∗) = I(d t · · · t d, −)
which shows that the Σk -action is level free except for d = 0. It follows
that the canonical map of E∞ spaces
k
E(Fd (∗)) → C(Fd (∗))
is an I-equivalence for d > 0 but not for d = 0. This corresponds to the
fact that E(Fd (∗)) is positively cofibrant if and only if d > 0.
Units of S-algebras
An ordinary commutative ring R has an underlying commutative
monoid (R, ·).
An ordinary commutative monoid M has an associated
commutative monoid ring Z[M].
These are adjoint functors
Comm monoids o
Z[−]
/
Comm rings
(−,·)
There are adjoint functors
forgetful
Comm groups o
/
(−)∗
Comm monoids
where M ∗ is the submonoid of invertible elements. The units of
the commutative ring R is the commutative group
GL1 (R) = (R, 1)∗ .
A commutative S-algebra R has an “underlying” commutative
I-space monoid ΩI (R):
ΩI (R)(n) = Ωn (R(n)) = Map∗ (S n , R(n))
where α : m → n in I induces α∗ : ΩI (R)(m) → ΩI (R)(n),
id∧f
α∗ (f ) : S n ' S n−α ∧ S m −−−→ S n−α ∧ R(m) → R(n).
A commutative I-space monoid M gives rise to a commutative
S-algebra SI [M]:
SI [M](n) = S n ∧ M(n)+
These are adjoint functors (in fact a Quillen adjunction)
Comm I-space monoids o
SI [−]
ΩI
/
Comm S-algebras
Definition
A commutative I-space monoid M is grouplike if the topological
monoid MhI is grouplike (that is, π0 MhI is a group).
There are adjoint functors
forgetful
Grouplike comm I-space monoids o
/
(−)∗
Comm I-space monoids
where M ∗ (n) ⊆ M(n) is the subspace of components that map
into the homotopy invertible elements in MhI .
Definition
The units of a commutative S-algebra R is the commutative
I-space monoid GL1 (R) = ΩI (R)∗ .
Remark
If R is fibrant (or semistable), then GL1 (R)hI ' (ΩI (R)hI )∗ and
π0 GL1 (R) = π0 GL1 (R)hI = GL1 (π0 (R)).
Remark
There are adjoint functors
I-spaces o
SI [−]
ΩI
/
S-modules
which restrict to the above adjunctions. The main point of the
above is that using the factorization
ΩI
Ω∞ : S-modules −−→ I-spaces → Spaces
we can “represent” Ω∞ (R) as the strictly commutative object
ΩI (R) if R is a commutative S-algebra.