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Home , Coalgebra, Cofree coalgebra

A Mo del Category Structure for Dierential Graded

Ezra Getzler and Paul Go erss

June

The structure of coalgebras

In this section we lay the groundwork for the calculations of the next The rst main result is this

every is the ltered colimit of its nite dimensional subcoalgebras The pro of we give

is essentially that of Sweedler mo died for the case of dierential graded coalgebras We then

use this to give a formula for the on a dierential graded this will b e

used in the pro of of Theorem The calculation of that result is the only part of the pro of of the

existence of the mo del category structure which is not essentially formal

Let C M denote the category of nonnegatively graded chain complexes over a eld k The

k

b oundary map has degree This category has a symmetric monoidal structure with

C D C D

n p q

p+q =n

and dierential

p

x y x y x y

with x y C D The switch map T C D D C must also have a sign

p q

pq

T x y y x

It is this tensor pro duct we use for coalgebras como dules and so on

Let CA denote the category of coasso ciative and counital coalgebras in C M We do not

k

assume co commutavity although in section we will discuss a restricted category of coalgebras

where we make some assumptions ab out degree zero We also do not assume that the unit map

C k is an in degree zero at this p oint C is simply a coasso ciative coalgebra

0

Given a xed coalgebra C there is a notion of right como dule over C we will write

M M C

M

for the como dule structure map

There are also the auxiliary categories of graded vector spaces coalgebras in graded vector

spaces and como dules in graded vector spaces There are forgetful from the ab ove cate

gories which neglect the b oundary map

The following observation is standard



The authors were partially supp orted by the National Science Foundation

Lemma Let C be a coalgebra in graded vector spaces and let M be a right in graded

vector spaces over C If x M is a homogeneous element then there is a nite dimensional

subcomodule N M so that x N

Pro of Compare x Let fc g b e a homogeneous for C Then we may write

i

X

x x c

i i

i

Note that x except for nitely many i Let N b e the span of the x in M Since x x

i i

one has

X

x x c N

i i

i

To show N is a subcomo dule one computes

X

x c x

i i

i

x

C

X

x c

j j

j

X

x c c

j ij i

ji

for some c C Then

ij

X

x c N C x

j ij i

j

This completes the pro of

If C is any coalgebra in graded vector spaces then C fHom C k g is a nonp ositively

n

k

graded algebra and if M is a right como dule over C in graded vector spaces then M is also a left

C mo dule indeed if C and x M then

X

x x h c i

i i

P

where x x c Not every left C mo dule arises this way b ecause of Lemma

i i

Lemma Let C be a coalgebra in graded vector spaces and x C a homogeneous element Then

there is a nite dimensional subcoalgebra D C so that x D Furthermore it can be assumed

that D for n larger than the degree of x

n

Pro of Again compare x The coalgebra C has an obvious structure as a right como dule

over itself Thus Lemma supplies a nitedimensional subcomo dule N C with x N If C

is co commutative then N is in fact already a subcoalgebra see however in the more general

case one must pro ceed as follows

Let J C b e the annihilator of N note that J is the of the map of rings

C End N

k

from C to the endomorphism of N determined by the left mo dule structure on N Since N is

nite dimensional C J is a nite dimensional vector space Let

D J fy C h y i for all J g

Note that x N D We now show that D is a nite dimensional subcoalgebra of C

Since C J is nite dimensional C J is a nite dimensional coalgebra Since D C

C J C we have that D is nite dimensional therefore this result follows from Lemma

b elow

If we want D for n greater than the degree of x simply take the subvector space of the

n

D constructed ab ove generated by the homogeneous elements of degree less than or equal to that

of x This is also a subcoalgebra

Lemma Let C be a coalgebra in graded vector spaces and I C a twosided ideal Then

I C is a subcoalgebra

Pro of The argument is identical to that of Prop osition of

Lemma has the following immediate consequences which will b e useful later If V C is a

subvector space then we also have an orthogonal complement V C Note that V V

If D C is a subcoalgebra then D C is a twosided ideal

Lemma Let C be a coalgebra either in graded vector spaces or in chain complexes and let

fC g be a of subcoalgebras of C in the appropriate category Then

i

C is a subcoalgebra of C in the appropriate category and

i i

P

C is a subcoalgebra of C in the appropriate category

i

i

P

Pro of If C is a coalgebra in chain complexes then C and C are subchain complexes thus

i i

it is sucient to argue the graded case However

X

C C C

i

i i

and

X

C C

i

i

This brings us to the following result crucial to all that follows

Prop osition Let C CA be a coalgebra in chain complexes and let x C be a homogeneous

element Then there is a nite dimensional dierential graded coalgebra D C so that x C

Pro of Supp ose the degree of x is n We dene an ascending sequence of subcoalgebras

D n D n D C

with the prop erties that

x D n

each D k is nitedimensional

D n for m n and D k D k for m k and

m m m

D k D k

k k 1

Then D D is the desired coalgebra Note that D n is supplied by Lemma If D k has

b een constructed cho ose a basis for fy g for D k and use Lemma to pro duce nite dimensional

i

k

subcoalgebras D y C so that y D y and D y for m k Then set

i i i i m

X

D k D k D y

i

i

This is a subcoalgebra by the previous result and the prop osition follows

This last result immediately implies the next Note that the forgetful from CA to

dierential graded vector spaces makes all colimits

Corollary Every dierential graded coalgebra over a eld k is the right ltered colimit of its

nite dimensional subcoalgebras

Now let Algf b e the category of pronite nonp ositively graded dierential algebras over k

with unit The ob jects of Algf can either b e regarded as ltered diagrams of nonp ositively graded

nite dimensional dierential k algebras or as complete top ological dierential k algebras with a

neighb orho o d base of consisting of twosided ideals of nite co and closed under the

dierential Then morphisms are either promorphisms of diagrams of nite dimensional dgas or

continuous dga morphisms The following is now a formal consequence of Corollary and the

fact that the dual of a nite dimensional algebra is a coalgebra This implies that the continuous

dual of a pronite algebra is a coalgebra Compare

Prop osition Linear duality denes an antiequivalence between the category CA of dierential

graded coalgebras and the category Algf of pronite dierential graded algebras

The functor back takes the continuous dual of a pronite algebra

An implication of the previous result is the following

Prop osition The category CA has al l smal l limits and colimits

Pro of Indeed as mentioned ab ove the forgetful functor from CA to dierential graded vector

spaces makes all colimits To prove the existence of limits it is sucient by the previous result

to show that the category Algf has all colimits

Let A I Algf b e diagram of pronite dgalgebras and let B b e the colimit of this diagram

in the category of dgalgebras This is not yet a pronite algebra Dene a neighb orho o d base of

in B to b e the set of twosided ideals J which can b e realized as the kernel of map of algebras

B C such that C is a nite dimensional dgalgebra and so that for each i I the comp osite

A B C is continuous Then the colimit is the completion of B with resp ect to this top ology

i

In light of Corollary the following technical result will have many applications

Lemma Let fC g be a right ltered diagram of coalgebras in CA and let D CA be nite

dimensional Then the natural map

colim Hom D C Hom D colim C

CA CA

is an isomorphism

Pro of The forgetful functor to dierential graded vector spaces makes colimits so this map is an

injection To prove it is a surjection note that any coalgebra map f D colim C factors as

a morphism of D C of dierential graded vector spaces since D is nite dimensional Since f

is a coalgebra map there must b e a morphism C C so that comp osite D C C is a

coalgebra map again since D is nite dimensional This completes the pro of

For the next result we need the following observation The forgetful functor from Algf to

the category of dierential graded algebras has a left adjoint given by pronite completion to a

dierential graded algebra A one assigns the diagram A fAI g where I runs over all twosided

ideals of A of nite co dimension and closed under the dierential

We can now prove

Prop osition The forgetful functor from CA to dierential graded vector spaces has a right

adjoint S

Pro of Let V b e a dierential graded vector space If V is nite dimensional let S V b e the

continuous dual of the pronite completion of the on V Then

\

Hom C S V Hom T ensV C

CA

Algf

Hom T ensV C

al g



C V Hom

M

k

Hom C V

M

k

is the category of pronite k vector spaces Thus S V has the desired prop erty For where M

k

general V dene

S V colim S V

where V runs over the nite dimensional subvector spaces of V Then if C colim C is a

coalgebra in CA written as the colimit of its nite dimensional subdgcoalgebras one has by

Lemma

Hom C S V lim Hom C S V

CA CA

lim colim Hom C S V

CA

lim colim Hom C V

M

k

Hom C V

M

k

This completes the pro of

The preceding argument gives only mo derate insight into the structure of S V we did obtain

the formula Equation and the fact that if V is nite dimensional then the dual of S V is the

pronite completion of the tensor algebra on V We sp end the rest of this section delving more

deeply into the structure of S V For mo del category theoretic reasons were actually interested

in C S V for an arbitrary coalgebra C Here is a preliminary reduction

Lemma Suppose A and B are dierential graded coalgebras and we have presentations

colim A A and colim B B

of A and B as ltered colimits of subcoalgebras Then the natural map

colim A B A B

is an isomorphism

Pro of If D colim D is written as the colimit of its nite dimensional subdierential graded

coalgebras then by Lemma

Hom D colim A B lim colim Hom D A B

CA CA

lim colim Hom D A Hom D B

CA CA

Now b ecause we are working with subcoalgebras the maps

0

Hom D A Hom D A

CA CA

and

0

Hom D B Hom D B

CA CA

are inclusions so this last colimit is a union of sets and hence one can interchange the pro duct

and colimit and again use Lemma

Hom D colim A B limcolim Hom D A colim Hom D B

CA CA CA

limHom D colim A Hom D colim B

CA CA

lim Hom D A B

CA

Hom D A B

CA

In particular this last result implies that if C is any coalgebra in CA and V is a dierential

graded vector space then there is an isomorphism

colim C S V C S V

where C runs over the nite dimensional subdgcoalgebras of C and V runs over the nite

dimensional subdgvector spaces of V Here we use Equation We reduce further

Let S n denote the dierential graded vector space which is of dimension over k concentrated

in degree n Let D n b e the dierential graded vector space with D n unless p n or n

p

D n D k and the b oundary map is the identity Every nite dimensional dierential

n n+1

vector space can b e written noncanonically as a nite pro duct of dierential vector spaces of the

form S n or D n Since the functor S is a right adjoint it preserves pro ducts thus to understand

C S V we are reduced in some sense to understanding C S S n and C S D n This is

the purp ose of the next result

If V is a dierential graded vector space concentrated in nonnegative degrees then T ensV

has a natural structure of a dierential graded algebra in nonp ositive degrees However more is

true If A is a dierential graded algebra concentrated in nonp ositive degrees and M is a dierential

graded A bimo dule also in nonp ositive degrees then we can form the tensor algebra

T ens M M M M

A A A A

n0

where in the nth summand M app ears n times If n we have only A This has an obvious

structure as a dierential graded Aalgebra In particular if W is a dierential graded vector space

in nonp ositive degrees we can form

T ens A W A A W A A M A

A

n0

where in the nth summand W app ears n times The functor

W T ens A W A

A

is left adjoint to the forgetful functor from A algebras of dgvector spaces hence there is a natural

isomorphism

T ens A W A A t T ensW

A

all in the category of dgalgebras

If A is an algebra let X A denote the twosided ideal of A generated by a subset X of A

Lemma Let C be a nite dimensional coalgebra and V a dierential graded vector space

which is in nonnegative degrees and nite dimensional in each degree



C V C T ens If V then C S V

C 0

If V k with generator x then

0



C S V lim T ens C V C q x q x

C

q (x)

where q x k x runs over al l monic polynomials and there is a projection

 

T ens C V C q x q x T ens C V C px px

C C

if and only if px divides q x

Pro of In either case C S V is the pronite completion of the dgalgebra



C t T ensV T ens C V C

C

In the rst case let



I T ens C V C

n C

b e the ideal of elements of degrees less than n Then the system



T ens C V C I

C n



C V C of nite dimension therefore this algebra is is conal among quotients of T ens

C

its own pronite completion

In the second case the system of algebras



C V C I q x q x T ens

n C

is again conal among quotients of of nite dimension hence

 

S V lim T ens C V C I q x q x lim T ens C V C q x q x

C n C

q (x)

Example One can use Lemma to calculate S S S k where k is in degree zero at

least in the case where k is p erfect Let k b e the algebraic closure of k and G the Galois group of

k over k Then write a for the orbit of a k and k a for k adjoin a Then as pronite algebras

Y

k az S k

[a]

where a runs over the distinct orbits of the action of G on k

The mo del category structure

The purp ose of this section is to prove that the category CA of nonnegatively graded dierential

coalgebras over a eld k has the structure of a closed mo del category where the weak equivalences

are quasi and the cobrations are simply inclusions This is in fact a cobrantly

generated mo del category and we end the section by sp ecifying the generating cobrations

We b egin with the key algebraic result Let S b e the cofree coalgebra functor of section

and let D n b e the nonnegatively graded chain complex with D n unless p n or n

p

D n D k and the b oundary map is the identity This is the same ob ject that was lab elled

n n+1

D n in section

Theorem For al l coalgebras C and al l nonnegative integers n the natural projection

C S D n C

is a quasiisomorphism

Pro of Let fC g b e the ltered system of nite dimensional subcoalgebras of C Then Lemma

supplies an isomorphism

colim C S D n C S D n

Since the colimit is ltered and made in dierential graded vector spaces it is sucient to discuss

the case where C is nite dimensional If n Lemma supplies the answer Thus we are

reduced to the case n Then Lemma writes C S D as a ltered colimit



C S D colim T ens C D C q x q x

C

q (x)

The algebras in this diagram for which q x is divisible by x form a conal subdiagram so can b e

used to compute this colimit We will show that each of these algebras is quasiisomorphic to C

b e a generator and To simplify notation let A C and M A D A Let x D

0

y x b e the generator in degree To b egin D has a chain contraction h given by hy x

and hx This gives a chain contraction for M by the formula

jaj

ha z b a hz b

This extends to a chain homotopy H T ens M T ens M from the identity to the comp osite

A A

T ens M T ens A T ens M

A A A

where is the unit map and is the augmentation induced by the zero map M To dene H

write

T ens M M M

A A A

n0

Then H restricted to A the summand n is the zero map and on the nth summand with

n

H x x hx x x

1 n 1 2 n

Let I b e the augmentation ideal in T ens M that is I is the kernel of Note that we have the

A

formula for a b T ens M

A

H ab a I

H ab

aH b a A

Since every element a T ens M can b e written uniquely as

A

a a a a

with a a I and a A we can can combine this observation into a single formula

H ab H a ab aH b

Now supp ose J I T ens M is twosided ideal contained in the augmentation ideal so

A

that H restricts to a chain homotopy on J Then the induced map on T ens M J is a chain

A

homotopy b etween the identity and

T ens M J T ens A T ens M J

A A A

So let q x b e a p olynomial in x divisible by x Then q x is in the augmentation ideal so the

twosided ideal J q x q x is in augmentation ideal Thus we need only show that H restricts

to J

Every element of J can b e written as a sum

aq xb c q xd

Applying H to this sum and using Equation we see that H restricts to J if and only if H q x

and H q x are in J However q x xq x since q x is divisible by x Because x is in the

0

augmentation ideal Equation implies

H q x H xq x hxq x

0 0

Similarly y is in the augmentation ideal so

H q x H y q x x q x hy q x hx q x xq x q x

0 0 0 0 0

This completes the argument

Theorem has the following corollary

Prop osition Let V be a nonnegatively graded dierential vector space with H V Then

for al l C CA the projection

C S V C

is a quasiisomorphism

Pro of The dgvector space V can b e written as a sum of ob jects isomorphic to some D n Thus

V can b e written as a ltered colimit of V colim V so that H V Then Equation and

Lemma imply that

colim C S V C S V

Since the colimit is ltered and made in vector spaces we are reduced to the case where V is nite

dimensional In that case write down an isomorphism

V D n D n D n

1 2

k

Since S is a right adjoint we have

C S V C S D n S D n

1

k

and the result follows by induction from Theorem

We now formally sp ecify the classes of maps of our mo del category structure

Denition A morphism f C D of dierential graded coalgebras is

a weak equivalence if it is a quasiisomorphism that is if H f H C H D

a cobration if it is a degreewise injection of graded vector spaces

a bration if it has the right lifting prop erty with resp ect to acyclic cobrations

As is customary we use the shorthand acyclic cobration for a morphism which is at once a

weak equivalence and a cobration There is also a notion of acyclic bration A morphism X Y

has the right lifting prop erty with resp ect to a morphism A B if every lifting problem

/ A / X }> }

i }  } 

 /  Y

B /

has a solution so that b oth triangles commute

The following result is why went to such diculty with Theorem

Lemma Let V be a nonnegatively graded dierential vector space so that H V Then

for al l coalgebras C CA the projection

C S V C

is an acyclic bration with the right lifting property with respect to al l cobrations

Any morphism f D C is CA can be factored

j q

D X C

where j is a cobration and q is an acyclic bration with the right lifting property with respect to

al l brations

Pro of We b egin with part By Prop osition the pro jection is a weak equivalence therefore

we need only show it is a bration This will follow if we show that the map has the right lifting

prop erty with resp ect to all cobrations Consider a lifting problem in CA

/

C S V

A / u: u :

i u u  u 

 /  C

B /

where the morphism i is a cobration By an adjointness argument this is equivalent to a lifting

problem in dierential graded vector spaces

/ V A / ~> ~

i ~  ~ 

 / 

B /

Since i is a cobration and V is an acyclic bration in the standard mo del category structure

on dierential graded vector spaces see a solution to the lifting problem exists

For part regard D as a dierential graded vector space and cho ose an inclusion of dgvector

spaces i D V with H V Dene X C S V q X C to b e the pro jection and

j D X to b e

j f i D C S V

where i is adjoint to i D V By part q is an acyclic bration of the required sort thus we

need only show that j is an inclusion Consider the comp osition

j

2

D C S V S V V

where is pro jection onto the second factor and is the counit of the adjunction This comp osition

2

is i hence an injection therefore j is an injection as required

There is another factorization required by the mo del category structure the acylic cobration

bration factorization This is more formal using the nowstandard technique pioneered by Bous

eld The crucial input is supplied by Lemmas and to follow the factorization is done

by the small ob ject argument in Lemma

Lemma Let j C D be an acylic cobration and x D a homogeneous element Then the

is a subcoalgebra B D so that

x B

B has a countable homogeneous basis and

C B B is an acyclic cobration in CA

Pro of We recursively dene subdgcoalgebras

B B D

so that x B each B n is nite dimensional and the induced map of dgvector spaces

B n C B n B nC B n

is zero in homology Then we may set B to b e the union of the B n

The subdgcoalgebra B is supplied by Prop osition Supp ose that B n has b een

constructed Since B n is nite dimensional we may cho ose a nite set

z C B n B n C B n

i

of homogeneous cycles so that the resulting homology classes span H B n C B n

For each index i use Prop osition to select a nite dimensional subcoalgebra of Az D so

i

that z Az Then set

i i

X

B n B n Az

i

i

This is again nite dimensional and has the requisite prop erties

Lemma A morphism q X Y in CA is a bration if and only if q has the right lifting

property with respect to al l acyclic cobrations A B so that B has a countable homogeneous

basis

Pro of The necessity of this lifting prop erty is a consequence of Denition For suciency

supp ose we have a lifting problem

f / X C / }>

} q i }  } 

 /  Y

D /

where j is an arbitrary acyclic cobration We solve this problem by a Zorns Lemma argument

Dene to b e the set of pairs D g where D ts into a sequence of acyclic cobrations in CA

C D D

and g D X is a solution to the restricted lifting argument Order by setting D g

1 1

D g if D D and g restricts to g Since C f the set is nonempty and any chain has

2 2 1 2 2 1

an upp er b ound given by the union Let E g b e a maximal element We show E D Note

that E D is an acyclic cobration

Let x D By the previous lemma there is a subdgcoalgebra B D so that x B B has a

countable basis and E B B is an acyclic cobration Then the induced lifting problem g

/ /

E X B E / / l l6 l l l l q  l l 

 / / 

D Y

B / /

has a solution by hyp othesis Hence g can b e extended over E B Furthermore the Meyer

Vietoris sequence

H E B H E H B H E B H E B

q q q q q 1

shows that E E B is an acylic cobration By the maximality of E g we must then have

E E B and x E as required

For the following result the class of morphisms generated by a stipulated list of morphisms A

is the smallest class containing A and closed under copro ducts cobase change directed colimits

retracts and isomorphisms

Lemma Any morphism C D in CA can be factored

p

i

C X D

where i is an acyclic cobration and p is a bration Furthermore i is in the class or morphisms

generated by the acyclic cobrations A B so that B has a countable homogeneous basis

Pro of Since colimits in CA are made in dierential graded vector spaces the class of acyclic

cobrations is closed under cobase change and directed colimits In light of Lemma the small

ob ject argument over an ordinal whose cardinality larger than that of the rst innite cardinal

now applies See where such arguments rst app eared and among many references

We can now prove the main result

Theorem With the denitions of weak equivalence bration and cobration given in Deni

tion the category CA becomes a model category

Pro of The category CA has all small limits and colimits by Prop osition Weak equivalences

satisfy the twooutofthree by insp ection similarly the classes of weak equivalences cobra

tions and bration are closed under retracts The factorization axiom follows from Lemmas

and The acyclic cobration bration half of the lifting axiom is the denition of bration

Thus we need only show that any acyclic bration has the right lifting prop erty with resp ect to all

cobrations Let p C D b e an acyclic bration By Lemma we may factor p as

j q

C X D

where j is a cobration and q is an acyclic bration with the right lifting prop erty with resp ect

to all cobrations Note that j is also a weak equivalence Thus there is a solution to the lifting problem

= / C C / }>

} p j }

}  q 

 /  D

X /

as p has the right lifting prop erty with resp ect to all acyclic cobrations This solution shows that

p is a retract of q and since q has the right lifting prop erty with resp ect to all cobrations so do es

p

We are also going to prove that CA is a cobrantly generated mo del category See x of

for the denitions and implications We have

Lemma A morphism p C D in CA is an acylic bration if and only if it has the right

lifting property with respect to al l cobrations A B with B nite dimensional

The cobrations A B with B nite dimensional generate the class of cobrations in CA

Pro of Part is proved by the evident variation of the Zorns Lemma argument given in Lemma

Prop osition substitutes from Lemma in the argument

To prove part let i C D b e any cobration CA Use part and the small ob ject argument

to factor i as

j q

C X D

where j is generated by the cobrations A B with B nite dimensional and q is an acyclic

bration Then the solution to the lifting problem

j / C / X }>

} q i }  } 

 = /  D

D /

shows i is a retract of j and hence in the class of morphisms generated by inclusions of nite

dimensional coalgebras

We can now state

Prop osition The class of cobrations in CA is generated by al l cobrations A B with

B nite dimensional and the class of acyclic cobrations is generated by al l acyclic cobrations

C D so that D has a countable homogeneous basis The category CA is cobrantly generated

Pro of The statements ab out generating cobrations and acyclic cobrations follow from Lemma

and Lemma resp ectively Since b oth typ es of morphisms have targets which are small with

resp ect to long enough directed colimits the result follows

We also have the following result which is technically convenient

Prop osition The cofree coalgebra functor S preserves brations and weak equivalences

Pro of The forgetful functor from CA to dgvector spaces preserves weak equivalences and co

brations hence S preserves brations and weak equivalences b etween brant ob jects However

every dgvector space is brant

Spacelike coalgebras

The normalized chains on a top ological space form a coasso ciative dierential graded coalgebra

using the AlexanderWhitney diagonal however the degree zero part of such a coalgebra supp orts

a very sp ecial structure We isolate and study this structure

In the category CA of dierential graded coalgebras the one dimensional coalgebra k concen

trated in degree zero is the terminal ob ject If C CA we dene the set X C of p oints in C by

the equation

X C Hom k C

CA

This is the set of elements x C so that x x x Note that if X is a set then the vector

0 C

space k X generated by X is a coalgebra with diagonal

k k X k X X k X k X

X

and there is a natural isomorphism X X k X

Denition A dierential graded coalgebra C CA is spacelike if the natural map of coalge

bras

k X C C

0

is an isomorphism Let CA denote the full subcategory of CA of spacelike coalgebras

+

For example the chains on a space are spacelike hence the name

Note that any subcoalgebra of a spacelike coalgebra is spacelike Hence the fundamental

structure result of coalgebras that any coalgebra is the ltered colimit of its nite dimensional

subcoalgebras applies equally well to CA

+

Lemma The inclusion functor CA CA has a right adjoint Furthermore commutes

+

with ltered colimits

Pro of In fact one can give a formula for

X

C C

where C runs over the spacelike subdgcoalgebras of C Then since the image of a spacelike

coalgebra is a spacelike coalgebra C has the requisite adjointness prop erties

To see that commutes with ltered colimits let fC g b e a ltered diagram of coalgebras and

D a nite dimensional spacelike coalgebra Then we have using Lemma

Hom D colim C colim Hom D C

CA CA

+ +

colim Hom D C

CA

Hom D colim C

CA

Hom D colim C

CA

+

This immediately implies the following result

Lemma The category CA has al l smal l limits and colimits

+

The forgetful functor from CA to dierential graded vector spaces has a right adjoint S

+ +

Also if V colim V is written as a ltered colimit of nite dimensional subdgvector spaces

then

colim S V S V

+ +

Pro of The forgetful functor to dierential graded vector spaces makes all colimits For limits if

fC g is a diagram in CA then the limit in that category is applied to the limit in CA The right

i +

adjoint is given by the formula S S where S is the right adjoint to the forgetful functor

+

from CA to dgvector spaces

We would now like to give the analog of Lemma in the category of spacelike coalgebras

Part of that Lemma remains unchanged Thus we need only worry ab out part Supp ose that

C is a nite dimensional spacelike coalgebra and V is a dierential graded vector space nite

dimensional in each degree and V k with generator x We dene certain ideals

0



I T ens C V C

C

q (x)

by

I q x q x a x a x

q (x)

where q x k x is a monic p olynomial a C and b z bz z b is the commutator There

0

is an inclusion

I I

q (x) p(x)

if and only if px divides q x

Lemma If C is a nite dimensional spacelike coalgebra and V is a dierential graded vector

space nite dimensional in each degree and V k with generator x then in CA

0 +



C S V lim T ens C V C I

+ C

q (x)

q (x)

where q x runs over the monic polynomials in k x that split completely into distinct factors over

k

Pro of Dene a pronite algebra A lim A where A runs over the nite dimensional algebra

quotients of



T ens C V C C t T ensV

C

so that A is a spacelike coalgebra Then the claim is that the continuous dual of A is C S V

+

This follows from the following calculation where D is a nite dimensional spacelike coalgebra

and A is the continuous dual of A We use Lemma for the rst isomorphism

Hom D A colim Hom D A

CA CA

+ +

colim Hom A D

k al g

Hom C t T ensV D

k al g

D Hom V D Hom C

C M

k al g



k

Hom D C Hom D V

CA C M



+

k

Hom D C S V

CA +

+

So let D b e a nitedimensional spacelike coalgebra and B D Then the degreezero sub

X (D )

algebra of B B is isomorphic to the semisimple commutative algebra k in particular B

0 0

is commutative and and algebra map k x B must factor

0

k x k xq x B

0

Y

where q x is monic and generates the kernel Since every subk algebra of B is of the form k

0

for some quotient set of X D we must have that q x splits completely into distinct factors over

Y

k Conversely if q x so splits k xq x k for some set Y



Let I T ens C V C I denote the elements of degree less than n Then the

n C

q (x)

algebras



T ens C V C I I

C n

q (x)

are conal among all nite dimensional dgalgebra quotients which map to duals of spacelike

coalgebras The result follows

Example Let k b e the onedimensional dierential vector space concentrated in degree zero

Then

k

S k k

+

Compare Example Note this implies that S k k k

+

To pro duce a mo del category structure on the category CA we follow the rubric of the pre

+

vious section The only part argument of that section which was not a formal consequence of the

prop erties of coalgebras that are inherited by spacelike coalgebras is the analog of Theorem

This we now prove

Theorem For al l spacelike coalgebras C and al l nonnegative integers n the natural projection

C S D n C

+

is a quasiisomorphism

Pro of The strategy of the pro of is the same Only the case n requires thought and we may

reduce to the case where C is nite dimensional Then we again set A C M A D A

and x D the generator in degree Then we must show that the chain contraction H

0

on T ens M to the inclusion of the unit A T ens M descends to a chain contraction on

A A

T ens M I where q x is as in Lemma Again we may assume that x divides q x In the

A

q (x)

end we must show that H sends the generators of I back into I However using Equation

q (x) q (x)

we have

H q x and H q x q x

as b efore and

H a x a H x

and

H a x H a x a y a H y a x

We would now like to prop ose the following result

Theorem The category CA of spacelike coalgebras has a closed model category structure

+

where a morphism f C D is

a weak equivalence if it is a quasiisomorphism

a cobration if it is a levelwise inclusion and

a bration if it has the right lifting property with respect to al l acyclic cobrations

Pro of In outline the pro of is the same as that of Theorem We touch on the highlights

The analog of Prop osition is a formal consequence of Theorem and Lemma which

applies equally well to CA The analog of Lemma is now formal Lemma for CA is a

+ +

consequence of Prop osition and the fact that every subcoalgebra of a spacelike coalgebra is

spacelike from this one deduces the analogs of Lemmas and This completes the existence

of the factorizations and the pro of of Theorem go es through verbatim

This mo del category is also cobrantly generated Indeed we have exactly as in Prop osition

Prop osition The class of cobrations in CA is generated by al l cobrations A B with

+

B nite dimensional and the class of acyclic cobrations is generated by al l acyclic cobrations

C D so that D has a countable homogeneous basis The category CA is cobrantly generated

+

One also has exactly as in Prop osition

Prop osition The cofree coalgebra functor S from dierential graded vector spaces to CA

+ +

preserves weak equivalences

Remark There are other varieties of coalgebras one might consider The standard example

see and also although the later is graded over the integers is that of connected coalgebras

A coalgebra C is connected if it has a unique simple subcoalgebra D See for denitions The

dual of graded coalgebra is a complete lo cal k algebra with residue eld a nite extension of k

A dierential graded coalgebra is connected if C is connected It is relatively straightforward to

0

pro duce a mo del category structure on connected dierential coalgebras the metho ds of apply

If one wants to use the ideas prop osed here we note that the forgetful functor from connected dg

coalgebras to dgvector spaces has a right adjoint S and if D has generators x and y x

con

then

n n

S D lim T ens x y x x

con

k

n

From this one easily deduces the analogs of Lemma and Theorem

Another variety of coalgebras that arises naturally are the etale ones Let k b e a p erfect eld

and k its algebraic closure a coalgebra C is etale if k C is spacelike For example if E is a

k

nite eld extension of k then E is etale but spacelike if and only of k E Again there is a

category of dierential graded etale coalgebras we require C b e etale The right adjoint S now

0 et

has

S D lim T ens x y q x q x

et

k

q (x)

where q x is a monic p olynomial which splits into distinct factors over k The analog of Theorem

in this case can b e deduced from Theorem by tensoring up over k or by direct calculation

References

AK Bouseld The lo calization of spaces with resp ect to homology Topology

M Demazure Lectures on pdivisible groups Lecture Notes in Springer

Verlag Berlin

V Hinich DG coalgebras as formal stacks preprint

M Hovey Model Categories Mathematical Surveys and Monographs AMS Providence

RI

D Quillen Homotopical Algebra Lecture Notes in Math SpringerVerlag BerlinHeidel

b ergNew York

D Quillen Ann of Math

M Sweedler Hopf Algebras W A Benjamin New York