W ALDHAUSEN K - THEORYVIACOMODULES

W ALDHAUSEN K - THEORY VIA COMODULES

Kathryn Hess

MATHGEOM

Ecole Polytechnique Fédérale de Lausanne

Manifolds, K -theory, and related topics Dubrovnik

23 June 2014

Joint work with Brooke Shipley.

O UTLINE

1

O VERVIEW

2

C OMODULES AND RETRACTIVE SPACES

3

M ODEL CATEGORY STRUCTURES

4

C ONSEQUENCES FOR K - THEORY

O VERVIEW

B ACKGROUND

Let K denote the functor from Waldhausen categories to spectra.

Let R

^hf_X

denote the category of homotopically finite retractive spaces over a simplicial set X .

A(X ) is the K -theory spectrum of the Waldhausen category R

^hf_X

, with cofibrations and weak equivalences created in the underlying category of simplicial sets.

It is well known that

A(X ) ' K Mod

^hf_Σ∞(ΩX )+

.

[Blumberg-Mandell, 2010] If X is simply connected, then

A(X ) ' K Mod

^th_DX

(S)
.

T HE MAIN THEOREM

A sort of Koszul dual of the module description of K (X ) on the previous slide...

T HEOREM (H.-S HIPLEY , 2014)

For any connected simplicial set X , the categories Comod

^hf_X+

and Comod

^hf_Σ^∞_X+

of homotopically finite X

₊

-comodules and Σ

^∞

X

₊

-comodules admit Waldhausen category structures such that there are natural weak equivalences of K -theory spectra

A(X ) ' K Comod

^hf_X+

' K Comod

^hf_Σ∞X+

.

P ROOF STRATEGY

1

Establish existence of Quillen equivalence (R

_X

)

_E

_oo ⊥ ^// (Comod

_X+

)

_E

, giving rise, via a result of Dugger and Shipley, to

K (R

_X

)

^hf_E

' K (Comod

_X+

)

^hf_E

.

2

Apply Hovey’s stabilization machine to (Comod

_X+

)

_E

, obtaining a model category structure (Comod

_Σ^∞_X+

)

_E

such that

K (Comod

_X+

)

^hf_E

' K (Comod

_Σ^∞_X+

)

^hf_E

.

P OTENTIAL APPLICATIONS

Work in progres...

A new description of the splitting of

A(X × S

¹

) ' A(X ) × BA(X ) × (nil terms).

A new description of the assembly map

A(∗) ∧ X

₊

→ A(X ).

C ONVENTIONS

X is an unpointed simplicial set.

E

∗

is a generalized, reduced homology theory.

(sSet

∗

)

_E

is the category of pointed simplicial sets endowed with the model category structure such that the weak equivalences are the E

∗

-homology isomorphisms, while the cofibrations are the levelwise injections. The classes of weak equivalences, fibrations and cofibrations in (sSet

∗

)

_E

are denoted

W

_E

, F

_E

, and C

_E

,

respectively.

C OMODULES AND RETRACTIVE SPACES

X + - COMODULES

Comod

_X+

= category of right X

+

-comodules in (sSet

∗

, ∧, S

⁰

) Objects: pairs (Y , ρ), where ρ : Y → Y ∧ X

₊

is

coassociative and counital.

Comod

_X+

is complete and cocomplete, as well as tensored, cotensored, and enriched over sSet

∗

. There is an sSet

∗

-adjunction

Comod

_X+

U

//

⊥ sSet

∗

F_X+

oo ,

where F

_X+

(Y ) = Y ∧ X

₊

, Y ∧ (∆

_X

)

₊

for all Y , and U is

the forgetful functor.

X + - COMODULES

For any simplicial map a : X

⁰

→ X , there is a pushforward functor

a

_∗

: Comod

_X⁰

+

→ Comod

X+

, specified on objects by

a

_∗

(Y , ρ) = Y , (Y ∧ a

₊

)ρ
, and which admits a right adjoint

a

^∗

: Comod

_X+

→ Comod

_X⁰

+

that commutes with colimits.

If (X , µ, x

₀

) is a simplicial monoid, then the monoidal structure (sSet

∗

, ∧, S

⁰

) lifts to a monoidal structure

Comod

_X+

, ⊗, (S

⁰

, ρ

u

)
,

which is symmetric if µ is commutative.

R ETRACTIVE SPACES OVER X

R

_X

= category of retractive objects over X Objects: X − → Z

ⁱ

− → X such that ri = Id

^r _X

. There is an adjunction

R

_X

V

//

⊥ sSet

∗

RetX

oo ,

where

Ret

_X

(Y , y

₀

) = X −→ Y × X

^iy0

x 7→(y0,x )

proj₂

−−−→ X and

V (X − → Z

ⁱ

− → X ) = Z /i(X ), i(X )
.

^r

R ETRACTIVE SPACES OVER X

For any generalized reduced homology theory E

∗

and any X − → Z

ⁱ

− → X , and any choice of basepoint in X ,

^r

E

∗

(Z ) ∼ = E

∗

Z /i(X )
⊕ E

∗

(X ).

For any simplicial map a : X

⁰

→ X , there is an adjunction

R

_X⁰

a∗

//

⊥ R

_X

,

a^∗

oo ,

given by pushout and pullback along a.

T HE KEY ADJUNCTION

T HEOREM (H.-S HIPLEY , 2014)

There is an adjoint pair of functors, natural in X ,

R

_X

−/X

//

⊥ Comod

_X+

−?X

oo ,

preserving E

∗

-equivalences and such that the counit map is a natural isomorphism and the unit map a natural E

∗

-equivalence, for every generalized reduced homology theory E

∗

.

R EMARK

When X = ∗, this is an equivalence of categories: R

∗

and

Comod

_S0

are equivalent to sSet

∗

, and −/∗ and − ? ∗ induce the

identity functors. It is not an equivalence if X 6= ∗.

T HE − ? X FUNCTOR

− ? X : Comod

X+

→ R

X

: (Y , ρ) 7→ (Y ? X , i

_ρ

, r

_ρ

)

where

Y ? X

 //

y

Y × X

πY



Y

^ρ

^// Y ∧ X

₊

is a pullback in sSet, and

i

_ρ

: X → Y ? X : x 7→ (y

₀

, x ) and r

_ρ

: Y ? X → X : (y , x ) 7→ x .

E XAMPLE

F

_X+

(Y ) ? X = Ret

_X

(Y ).

T HE −/X FUNCTOR

−/X : R

_X

→ Comod

_X+

: (Z , i, r ) 7→ Z /i(X ), ρ

_{(i,r )}

,

where

ρ

_{(i,r )}

: Z /i(X ) → Z /i(X )
∧ X

₊

is the unique pointed simplicial map such that

Z

^{(pi×r )∆Z}

^//

pi

Q Q Q Q Q Q ((Q Q Q

Q Q Q Q Q Q

Q Z /i(X )
× X ^// Z /i(X )
∧ X

₊

Z /i(X )

ρ_{(i,r )}

44

commutes, where p

_i

: Z → Z /i(X ) is the quotient map.

M ODEL CATEGORY STRUCTURES

T HE UNSTABLE CASE

T HEOREM (H.-S HIPLEY , 2014)

There are cofibrantly generated, left proper, simplicial model category structures (R

_X

)

_E

and (Comod

_X+

)

_E

such that

(R

_X

)

_E

−/X

//

⊥ (Comod

_X+

)

_E

oo

−?X

is a Quillen equivalence and WE

_Comod

X+

= U

⁻¹

(WE

_E

), Cof

_Comod

X+

= U

⁻¹

(Cof

_E

), and WE

_RX

= V

⁻¹

(WE

_E

), Cof

_RX

= V

⁻¹

(Cof

_E

).

The existence of (R

_X

)

_E

is standard (see below); we prove the

existence of (Comod

_X+

)

_E

in two complementary ways, by right-

and left-induction.

R EMARKS

By a standard “slice” argument, R

_X

inherits a left proper, cofibrantly generated, simplicial model category structure from (sSet)

_E

such that f : (Z , i, r ) → (Z

⁰

, i

⁰

, r

⁰

) is a fibration (respectively, cofibration or weak equivalence) if and only if the underlying morphism of simplicial sets f : Z → Z

⁰

is of the same type.

If a : X

⁰

→ X is a simplicial map, then

(R

_X⁰

)

_E

a∗

//

⊥ (R

_X

)

_E

,

a^∗

oo

and

(Comod

_X⁰

)

_E

a∗

//

⊥ (Comod

_X+

)

_E

,

a^∗

oo

are Quillen pairs that are Quillen equivalences if a is an

E

∗

-equivalence.

P ROOF BY RIGHT - INDUCTION

Apply standard transfer of cofibrantly generated model category structure to

R

_X

−/X

//

⊥ Comod

_X+

−?X

oo ,

where R

_X

is equipped with the model category structure inherited from (sSet)

_E

.

Advantage: know sets of generating (acyclic) cofibrations for

the model category structure on Comod

_X+

U SEFUL NOTATION

N OTATION

Let f and g be morphisms in a category C. If for every commutative diagram in C

·

f



a

// ·

g

 ·

b^c

//

@@

·

the dotted lift c exists, then we write f  g . If X is a class of morphisms C, then

X

^

= {f ∈ Mor C | x  f ∀x ∈ X}.

D EFINITION OF LEFT - INDUCED STRUCTURES

D EFINITION Let C

U

//

⊥ M

F

oo be an adjoint pair of functors, where (M, F, C, W) is a model category, and C is a bicomplete category. If the triple of classes of morphisms in C



U

⁻¹

( C ∩ W)

^

, U

⁻¹

( C), U

⁻¹

( W) 

satisfies the axioms of a model category, then it is a

left-induced model structure on C.

E XISTENCE OF LEFT - INDUCED STRUCTURES

T HEOREM

(B AYEH -H.-K ARPOVA -K E ¸ DZIOREK -R IEHL -S HIPLEY , 2014) Let C

U

//

⊥ M

F

oo be an adjoint pair of functors, where C is locally presentable, and (M, F, C, W) is a combinatorial model category.

If

U

⁻¹

C

^

⊂ U

⁻¹

W,

then the left-induced model structure on C exists and is

cofibrantly generated.

A PPLICATION TO Comod _X

₊

Applying the existence theorem from the previous slide, we get...

T HEOREM (H.-S HIPLEY , 2014)

There is a model category structure (Comod

_X+

)

_E

left-induced from (sSet

∗

)

_E

by the adjunction

Comod

_X+

U

//

⊥ sSet

∗

.

F_X+

oo

Moreover if (X , µ, x

₀

) is a simplicial monoid, then (Comod

_X+

)

_E

, ⊗, (S

⁰

, ρ

u

)

is a monoidal model category satisfying the monoid axiom.

Advantages to this approach become clear when we stabilize.

T HE NON SIMPLY CONNECTED CASE T HEOREM

Let q : e X → X be a universal cover. The adjunctions R

_Xe

q∗

//

⊥ R

_X

q^∗

oo , Comod

_eX+

q∗

//

⊥ Comod

_X+

q^∗

oo

right-induce left proper, cofibrantly generated model category structures (R

_X

)

_Hq^∗

and (Comod

_X+

)

_Hq^∗

from (R

Xe

)

_HZ

and (Comod

Xe

)

_HZ

. In particular, the adjunction (R

_X

)

_Hq^∗

−/X

//

⊥ (Comod

_X+

)

_Hq^∗

oo

−?X

is a Quillen equivalence.

K OSZUL DUALITY

T HEOREM

If X is a reduced simplicial set, and E is any generalized homology theory, then there is a Quillen equivalence

(Mod

_GX

)

_E

−∧_{(GX )+}(PX )+

// ⊥ (Comod

_X+

)

_E

.

oo

T HE STABLE CASE

T HEOREM (H.-S HIPLEY , 2014)

There are combinatorial, left proper, spectral model category structures

Sp

_E

, (Comod

_Σ^∞_X+

)

^st_E

, and (Comod

_Σ^∞_X+

)

^left_E

, where the first two are stabilized from (sSet

∗

)

_E

and (Comod

_X+

)

_E

, and the third is left-induced from the first.

In particular, the functors

(Comod

Σ^∞X+

)

^st_E ^Id

// (Comod

Σ^∞X+

)

^left_E ^U

// Sp

_E

are left Quillen, and weak equivalences and fibrations in

(Comod

_Σ^∞_X+

)

^left_E

are created by U.

R EMARKS

The description of (Comod

_X+

)

_E

as a left-induced structure is crucial for this proof.

(Comod

_Σ^∞_X+

)

^?_HZ

= (Comod

_Σ^∞_X+

)

^?_πs

∗

for ? = st or left If X is a simplicial monoid, then (Comod

_Σ^∞_X+

)

^left_E

admits a monoidal structure satisfying the monoid axiom.

Koszul duality stabilizes: if X is a reduced simplicial set and E any generalized homology theory, then there is a Quillen equivalence

(Mod

_Σ^∞_{(GX )+}

)

^st_E

_oo ⊥ ^// (Comod

_Σ^∞_X+

)

^st_E

,

A LGEBRAIC HOMOTOPY OF COMODULES If H is a simplicial monoid, then

Alg

_Σ^∞_H+

= the category of monoids in Comod

_Σ^∞_H+

. Objects: symmetric ring spectra R endowed with a

coassociative, counital morphism

ρ : R → R ∧ Σ

^∞

H

₊

of symmetric ring spectra.

C OROLLARY

There is a cofibrantly generated model category structure (Alg

_Σ^∞_H+

)

_E

with respect to which the forgetful/cofree adjunction

(Alg

_Σ∞H+

)

_E

U

//

⊥ (Alg)

_E

−∧Σ^∞H+

oo

is a Quillen pair.

C ONSEQUENCE FOR H OPF -G ALOIS THEORY Can now formulate rigorously the notion of the homotopy coinvariants of the Σ

^∞

H

₊

-coaction on an object (R, ρ) in Alg

_Σ∞H+

, which is essential in the definition of a homotopic Hopf-Galois extension [Rognes].

D EFINITION

A model for the homotopy coinvariants of (R, ρ) is the equalizer in Alg

_Σ∞H+

(R, ρ)

^{hco Σ∞H+}

= equal R

^f

ρ^f

⇒

R^f∧η

R

^f

∧ Σ

^∞

H

₊

,

where (R

^f

, ρ

^f

) is a fibrant replacement for (R, ρ) in Alg

_Σ∞H+

,

and η : S → Σ

^∞

H

₊

is the unit of the ring spectrum Σ

^∞

H

₊

.

C ONSEQUENCES FOR K - THEORY

F ROM COMODULES TO K - THEORY

N OTATION

A(X ; E

∗

) denotes the K -theory of R

^hf_X

with the usual

cofibrations, but with E

∗

-equivalences as weak equivalences.

T HEOREM (H.-S HIPLEY , 2014)

For any simplicial set X and any generalized reduced homology theory E

∗

, (Comod

_X+

)

^hf_E

is a Waldhausen category, and there are natural weak equivalences of K -theory spectra

A(X ; E

∗

) − → K (Comod

^' _X+

)

^hf_E

'

← − K (Mod

_ΩX

)

^hf_E

.

T HE NON SIMPLY CONNECTED CASE

Let q : e X → X be a universal cover. Recall (R

_X

)

_Hq^∗

and

(Comod

_X+

)

_Hq^∗

. Note that Hq

^∗

= HZ

∗

if X is simply connected.

L EMMA

A(X ) − → A(X ;

^'

Hq

^∗

) is a weak equivalence for every connected simplicial set X .

C OROLLARY

There is a natural weak equivalence of K -theory spectra A(X ) −

^'

→ K (Comod

X+

)

^hf_Hq^∗

.

In particular, if X is simply connected, then

A(X ) −

^'

→ K (Comod

_X+

)

^hf_HZ

.

T HE STABLE VERSION

C OROLLARY

There is a natural weak equivalence of K -theory spectra A(X ) −

^'

→ K (Comod

_Σ^∞X+

)

^hf

,

where (Comod

_Σ^∞_X+

)

^hf

denotes the category of homotopically

finite comodules over Σ

^∞

X

+

, with cofibrations and weak

equivalences inherited from the stabilization of the model

structure on (Comod

_X+

)

_Hq^∗

.

Happy Birthday, Tom!