W ALDHAUSEN K - THEORYVIACOMODULES

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W ALDHAUSEN K - THEORY VIA COMODULES

Kathryn Hess

MATHGEOM

Ecole Polytechnique Fédérale de Lausanne

Manifolds, K -theory, and related topics Dubrovnik

23 June 2014

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Joint work with Brooke Shipley.

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O UTLINE

1

O VERVIEW

2

C OMODULES AND RETRACTIVE SPACES

3

M ODEL CATEGORY STRUCTURES

4

C ONSEQUENCES FOR K - THEORY

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O VERVIEW

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B ACKGROUND

Let K denote the functor from Waldhausen categories to spectra.

Let R

hfX

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

hfX

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

It is well known that

A(X ) ' K Mod

hfΣ(ΩX )+

.

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

A(X ) ' K Mod

thDX

(S).

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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

hfX+

and Comod

hfΣX+

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

hfX+

 ' K Comod

hfΣX+

.

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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

)

hfE

 ' K (Comod

X+

)

hfE

.

2

Apply Hovey’s stabilization machine to (Comod

X+

)

E

, obtaining a model category structure (Comod

ΣX+

)

E

such that

K (Comod

X+

)

hfE

 ' K (Comod

ΣX+

)

hfE

.

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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 ).

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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.

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C OMODULES AND RETRACTIVE SPACES

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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

FX+

oo ,

where F

X+

(Y ) = Y ∧ X

+

, Y ∧ (∆

X

)

+

 for all Y , and U is

the forgetful functor.

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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.

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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

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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.

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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= ∗.

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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 ).

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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.

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M ODEL CATEGORY STRUCTURES

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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.

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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.

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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+

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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

 ·

bc

//

 @@

  ·

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}.

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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.

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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.

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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

.

FX+

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.

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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.

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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

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T HE STABLE CASE

T HEOREM (H.-S HIPLEY , 2014)

There are combinatorial, left proper, spectral model category structures

Sp

E

, (Comod

ΣX+

)

stE

, and (Comod

ΣX+

)

leftE

, 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

ΣX+

)

stE Id

// (Comod

ΣX+

)

leftE U

// Sp

E

are left Quillen, and weak equivalences and fibrations in

(Comod

ΣX+

)

leftE

are created by U.

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R EMARKS

The description of (Comod

X+

)

E

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

(Comod

ΣX+

)

?HZ

= (Comod

ΣX+

)

?πs

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

ΣX+

)

leftE

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 )+

)

stE

oo// (Comod

ΣX+

)

stE

,

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

Alg

ΣH+

= the category of monoids in Comod

ΣH+

. 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

ΣH+

)

E

with respect to which the forgetful/cofree adjunction

(Alg

ΣH+

)

E

U

//

⊥ (Alg)

E

−∧ΣH+

oo

is a Quillen pair.

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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

ΣH+

, 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

ΣH+

(R, ρ)

hco ΣH+

= equal R

f

ρf

Rf∧η

R

f

∧ Σ

H

+

,

where (R

f

, ρ

f

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

ΣH+

,

and η : S → Σ

H

+

is the unit of the ring spectrum Σ

H

+

.

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C ONSEQUENCES FOR K - THEORY

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F ROM COMODULES TO K - THEORY

N OTATION

A(X ; E

) denotes the K -theory of R

hfX

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+

)

hfE

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

A(X ; E

) − → K (Comod

' X+

)

hfE



'

← − K (Mod

ΩX

)

hfE

.

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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+

)

hfHq

.

In particular, if X is simply connected, then

A(X ) −

'

→ K (Comod

X+

)

hfHZ

.

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T HE STABLE VERSION

C OROLLARY

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

'

→ K (Comod

ΣX+

)

hf

,

where (Comod

ΣX+

)

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

.

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Happy Birthday, Tom!

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