## 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

_{X}

^{0}

+

### → Comod

X+### , specified on objects by

### a

_{∗}

### (Y , ρ) = Y , (Y ∧ a

_{+}

### )ρ, and which admits a right adjoint

### a

^{∗}

### : Comod

_{X}

_{+}

### → Comod

_{X}

^{0}

+

### 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

^{i}

^{y0}

x 7→(y0,x )

proj_{2}

### −−−→ 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

_{X}

^{0}

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

_{S}0

### 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

^{(p}

^{i}

^{×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

_{R}

_{X}

### = V

^{−1}

### (WE

_{E}

### ), Cof

_{R}

_{X}

### = 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

_{X}

^{0}

### )

_{E}

a∗

### //

### ⊥ (R

_{X}

### )

_{E}

### ,

a^{∗}

### oo

### and

### (Comod

_{X}

^{0}

### )

_{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

_{X}

_{e}

q∗

### //

### ⊥ R

_{X}

q^{∗}

### oo , Comod

_{e}

_{X}

_{+}

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}

_{+}

### )

^{?}

_{π}s

∗

### 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}

^{∗}