Table Of ContentHOMOTOPY THEORY OF COALGEBRAS OVER OPERADS
6
0
JUSTINR.SMITH
0
2
Abstra
t. Thispaper
onstru
tsmodelstru
tures onthe
ategoriesof
oal-
t
c gebras and pointed irredu
ible
oalgebras over an operad whose
omponents
O are proje
tive, (cid:28)nitely generated in ea
h dimension, and satisfy a
ondition
that allows one to take tensor produ
ts with a unit interval. The underly-
3 ing
hain-
omplex is assumed to be unbounded and the results for bounded
oalgebras over anoperadare derivedfromtheunbounded
ase.
]
T
C
.
h
t
a Contents
m
1. Introdu
tion 1
[
2. Notation and
onventions 3
4 3. The general
ase 9
v 4. The bounded
ase 20
7
5. Examples 21
1
Appendix A. Nearly free modules 21
3
5 Appendix B. Category-theoreti
onstru
tions 24
0 B.1. Cofree-
oalgebras 24
3 B.2. Core of a module 27
0
B.3. Categori
al produ
ts 29
/
h B.4. Limits and
olimits 31
t Referen
es 39
a
m
:
v
i
X
1. Introdu
tion
r
a
Althoughtheliterature
ontainsseveralpapersonhomotopytheoriesforalgebras
over operads (cid:22) see [14℄, [17℄, and [18℄ (cid:22) it is more sparse when one pursues
similar results for
oalgebras. In [20℄, Quillen developed a model stru
ture on
the
ategory of 2-
onne
ted
o
ommutative
oalgebras over the rational numbers.
V. Hini
h extended this in [13℄ to
oalgebras whose underlying
hain-
omplexes
were unbounded (i.e., extended into negative dimensions). Expanding on Hini
h's
methods, K. Lefèvre derived a model stru
ture on the
ategory of
oasso
iative
oalgebras (cid:22) see [15℄. In general, these authors use indire
t methods, relating of
oalgebra
ategories to other
ategories with known model stru
tures.
Date:8thFebruary 2008.
1991 Mathemati
s Subje
tClassi(cid:28)
ation. Primary18G55;Se
ondary 55U40.
Key words and phrases. operads,
ofree
oalgebras.
1
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
Ourpaper(cid:28)ndsmodelstru
turesfor
oalgebrasoveranyoperadful(cid:28)llingabasi
requirement (
ondition 3.5). Sin
e operads uniformly en
ode many diverse
oal-
gebra stru
tures (
oasso
iative-, Lie-, Gerstenhaber-
oalgebras, et
.), our results
have wide appli
ability.
Several unique problems arise that require spe
ial te
hniques. For instan
e,
onstru
ting inje
tive resolutions of
oalgebrasnaturally leads into in(cid:28)nitely many
negative dimensions. The resulting model stru
ture (cid:22) and even that on the un-
derlying
hain-
omplexes (cid:22) fails to be
o(cid:28)brantly generated (see [5℄).
Wedevelopthegeneraltheoryforunbounded
oalgebras,andderivethebounded
results by applying a trun
ation fun
tor.
InŸ2,wede(cid:28)neoperadsand
oalgebrasoveroperads. Wealsogiveabasi
ondi-
tion(see3.5)ontheoperadunder
onsiderationthatweassumetoholdthroughout
thepaper. This
onditionissimilarto thatofadmissibility ofBergerandMoerdijk
in [2℄. Co(cid:28)brant operads always satisfy this
ondition and every operad is weakly
equivalent to one that satis(cid:28)es this
ondition.
In Ÿ 3, we brie(cid:29)y re
all the notion of model stru
ture on a
ategory and de(cid:28)ne
model stru
tures on two
ategories of
oalgebras over operads. When the operad
is proje
tive and (cid:28)nitely-generated in all dimensions, we verify that nearly free
oalgebrassatisfy Quillen's axioms of a model stru
ture (see [19℄ or [11℄).
A key step involves proving the existen
e of
o(cid:28)brant and (cid:28)brant repla
ements
for obje
ts. In our model stru
ture, all
oalgebras are
o(cid:28)brant (solving this half
of the problem) and the hard part of is to (cid:28)nd (cid:28)brant repla
ements.
Wedevelopresolutionsof
oalgebrasby
ofree
oalgebrasthatsolvestheproblem
(cid:22) see lemma 3.16 and
orollary 3.17. This
onstru
tion naturally leads into in-
(cid:28)nitelymany negativedimensions and wasthe motivationforassumingunderlying
hain-
omplexes are unbounded.
Fibrant
oalgebrasare
hara
terizedas retra
ts of layered
oalgebras (see de(cid:28)n-
ition 3.18 and
orollary 3.19) (cid:22) an analogue to total spa
es of Postnikov towers.
In the
o
ommutative
ase over the rational numbers, the model stru
ture that
we get is not equivalent to that of Hini
h in [13℄. He gives an example (9.1.2) of a
oalgebrathatisa
y
li
butnot
ontra
tible. Inourtheoryitwould be
ontra
tible,
sin
e it is over the rational numbers and bounded.
InŸ4,wedis
ussthe(minor)
hangestothemethodsinŸ3tohandle
oalgebras
thatareboundedfrombelow. Thisinvolvesrepla
ingthe
ofree
oalgebrasbytheir
trun
ated versions.
In Ÿ 5, we
onsider two examples over the rational numbers. In the rational,
2-
onne
ted,
o
ommutative,
oasso
iative
ase, we re
over the model stru
ture
Quillen de(cid:28)ned in [20℄ (cid:22) see example 5.2.
Z
InappendixA,westudynearly free -modules. Thesearemoduleswhose
ount-
Z
able submodules are all -free. They take the pla
e of free modules in our work,
sin
e the
ofree
oalgebra on a free modules is not free (but is nearly free).
In appendix B, we develop essential
ategory-theoreti
onstru
tions, in
luding
equalizers (Ÿ B.2), produ
ts and (cid:28)bered produ
ts (Ÿ B.3), and
olimits and limits
(Ÿ B.4). The
onstru
tion of limits in Ÿ B.4 was this proje
t's most
hallenging
aspe
t and
onsumed the bulk of the time spent on it. This se
tion's key results
are
orollary B.21, whi
h allows
omputation of inverse limits of
oalgebras and
theorem B.23, whi
h shows that these inverse limits share a basi
property with
those of
hain-
omplexes.
2
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
I am indebted to Professor Bernard Keller for severaluseful dis
ussions.
2. Notation and
onventions
R Z
Throughout this paper, will denote a (cid:28)eld or .
R M
De(cid:28)nition 2.1. An -module will be
alled nearly free if every
ountable
R
submodule is -free.
R=Z
Remark. This
ondition is automati
ally satis(cid:28)ed unless .
Z Zℵ0
Clearly,any -freemoduleisalsonearlyfree. The Baer-Spe
kergroup, , isa
Z
well-known example of a nearly free -module that is not free (cid:22) see [10℄, [1℄, and
ℵ
1
[23℄. Compare this with the notion of -free groups (cid:22) see [4℄.
By abuse of notation, we will often
all
hain-
omplexes nearly free if their
underlying modules are (ignoring grading).
Z
Nearly free -modules enjoy useful properties that free modules do not. For
instan
e, in many interesting
ases, the
ofree
oalgebra of a nearly free
hain-
omplex is nearly free.
De(cid:28)nition 2.2. WeRwill denote the
losed sRymmetri
monoidal
aCtehg(oRr)y of un-
bounded, nearly free -
hain-
omplexes with -tensor produ
ts by .
We make extensive use of the Koszul Convention (see [12℄) regarding signs in
homologi
al
al
ulations:
f:C →D g:C →D a⊗b∈C ⊗C a
1 1 2 2 1 2
De(cid:28)nition2.3. If , aremaps,and (where
(f⊗g)(a⊗b) (−1)deg(g)·deg(a)f(a)⊗
isahomogeneouselement),then isde(cid:28)nedtobe
g(b)
.
f g
i i
Remark 2.4.(fIf ⊗,g )◦ar(efm⊗apgs,)i=t i(s−n'1t)hdeagr(df2)t·odevg(egr1i)f(yfth◦aft t⊗hegK◦osgzu)l
onvention
1 1 2 2 1 2 1 2
implies that .
I
De(cid:28)nition 2.5. The symbol willdenotetheunitinterval, a
hain-
omplexgiven
by
I = R·p ⊕R·p
0 0 1
I = R·q
1
I = 0 k 6=0,1
k
if
A∈Ch(R)
Given , we
an de(cid:28)ne
A⊗I
and
Cone(A)=A⊗I/A⊗p
1
The set of morphisms of
hain-
omplexes is itself a
hain
omplex:
A,B ∈Ch(R)
De(cid:28)nition 2.6. Given
hain-
omplexes de(cid:28)ne
Hom (A,B)
R
R
to be the
hain-
omplex of graded -morphisms where the degree of an element
x∈Hom (A,B)
R
is its degree as a map and with di(cid:27)erential
∂f =f ◦∂ −(−1)degf∂ ◦f
A B
R Hom (A,B) = Hom (A ,B )
As a -module R k j R j j+k .
Q
3
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
Remark. Given A,B ∈Ch(R)Sn, we
an de(cid:28)ne HomRSn(A,B) in a
orresponding
way.
De(cid:28)nition 2.7. De(cid:28)ne:
Set Set2
(1) f tobethe
ategoryof(cid:28)nitesetsandbije
tions. Let f bethe
ategory
of (cid:28)nite sets whose elements are also (cid:28)nite sets. Morphisms are bije
tions
ofsetsthatrespe
tthe(cid:16)(cid:28)nestru
ture(cid:17) ofelementsthatarealsosets. There
is a forgetful fun
tor
f:Set2 →Set
f f
Set2
that simply forgets that the elements of an obje
t of f are, themselves,
(cid:28)nite sets. There is also a (cid:16)(cid:29)attening(cid:17) fun
tor
g:Set2 →Set
f f
that sends a set (of sets) to the union of the elements (regarded as sets).
X Σ =End (X)
(2) FSeotr a−(cid:28)mnoitde set , X Setf . Func(Setop,Ch(R))
(3) f tobethe
ategoryof
ontravariantfun
tors f ,
with morphisms that are natural transformations.
C, D ∈Set −mod Hom(C,D)
f
(4) Given , de(cid:28)ne to be the setof natural trans-
Hom (C,D) X ∈ Set
formations of fun
tors. Also de(cid:28)ne X , where f, to be
C D X
the natural transformations of and restri
ted to sets isomorphi
to
(i.e., of the same
ardinality). Both of these fun
tors are
hain-
omplexes.
Σ−mod {M(n)} m ≥ 1 M(n) ∈
(5) Ch(R) to bMe(tnh)e
ategory of sequen
es S , where
n
and is equipped with a right -a
tion.
[n] n Σ = S
[n] n
Remark. If is the set of the (cid:28)rst positive integers, then , the sym-
M Set X
f
metri
group. If is a -module then, for ea
h (cid:28)nite set, , there is a right
Σ M(X)
X
-a
tion on .
S =S ={1}
0 1
We follow the
onvention that , the trivial group.
Σ−mod
Noate=th{a{tx},{y,z,ti}s,w{hh}a}t ∈is oSfette2n
alledf(tah)e∼=
a[t3e]gory of
olle
tions.
g(aI)f={x,y,z,t,h} f then , a set of three elements, and
.
Set −mod Σ−mod
f
It is well-known that the
ategories and are isomorphi
(cid:22)
see se
tion 1.7 in part I of [18℄. The restri
tion isomorphism
r:Set −mod→Σ−mod
f
[n] n ≥ 1 F ∈
simply involves evaluating fun
tors on the (cid:28)nite sets for all . If
Set −mod r(F) = {F([n])} F F([n])
f
, then . The fun
torial nature of implies that
S Hom (C,D)
is equipped with a natural n-a
tion. The fun
tors n
orrespond to
Hom (C([n]),D([n])) Set
RSn and the fa
t that morphisms in f preserve
ardinality
imply that
Hom(C,D)= Hom (C,D)
n
n≥0
Y
Set
f
Although -modules are equivalent to modules with a symmetri
group a
-
Set −mod
f
tion, it is often easier to formulate operadi
onstru
tions in terms of .
Equivarian
e relations are automati
ally satis(cid:28)ed.
X n X
De(cid:28)nition 2.8. If is a (cid:28)nite set of
ardinality the set of orderings of is
Ord(X)={f|f:X −∼=→[n]}
4
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
Set X
f
Nowwedne(cid:28)nea analogueto themultiCple∈teCnsho(rRp)rodu
t. Given asext∈Xof
x
ardinality , andanassignmentofanobje
t forea
helement ,
g ∈Ord(X)
we
an de(cid:28)ne, for ea
h a produ
t
Cx =Cg−1(1)⊗···⊗Cg−1(n)
g
O
The symmetry of tensor produ
ts determines a morphism
σ¯: C → C
x x
g σ◦g
O O
σ ∈S ±1
n
for ea
h whi
h essentially permutes fa
tors and multiplies by , following
the Koszul Convention in de(cid:28)nition 2.3.
De(cid:28)nition 2.9. The unordered tensor produ
t is de(cid:28)ned by
C =coequalizer σ¯: C → C
x x x
OX σ∈Sn g∈OMrd(X)Og g∈OMrd(X)Og
C ∈Ch(R) X ∈Set CX
If and fthen will denote the unordered tensor produ
t
C
X
O
C X C⊗ Set
f
of
opiesof indexedbyelementsof ,and willdenotethe -modulewhose
X ∈Set CX
f
value on is .
X·C n C n
We use to denote a dire
t sum of
opies of , where isthe
ardinality
X
of a (cid:28)niteXse∈t Se.t2
f
When ,
C
X
O
f(X) X
is regardedas being taken over (cid:22) i.e., we (cid:16)forget(cid:17) that the elements of are
sets themselves.
Ch(R)
Remark. The unordered tensor produ
t is isomorphi
(as an obje
t of to
C x X
x
the tensor produ
t of the , as runs over the elements of . The
oequalizer
onstXru
=tio[nn]determCin[ne]s=hoCwnthe it behavCesXw⊗ithCrYes=peC
tXt⊔oYset-mXor,pYhis∈mSs.et
f
If , then . Note that , for . We
C∅ = =R 0
also follow the
onvention that 1 ,
on
entrated in dimension .
X ∈Set x∈X {f :V →U } Ch(R)
f y y y
De(cid:28)nition 2.10. If , and are morphisms of
y ∈X
indexed by elements then de(cid:28)ne
(U,V)= Z −1−⊗−·−··⊗−−fx−⊗−·−··⊗−→1 U
y y
X,x y∈X y∈X
O O O
to be the unordered tensor produ
t, where
U y 6=x
y
Zy = if
(Vy u=x
if
X
Remark. Given any ordering of the elements of the set , there exists a
anoni
al
isomorphism
(U,V)=U ⊗···⊗V ⊗···⊗U
OX,x x
position
| {z }
5
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
X,Y ∈Set x∈X
f
De(cid:28)nition 2.11. Let and let . De(cid:28)ne
X ⊔ Y =(X \{x})∪Y
x
X ⊔ ∅=X\{x}
x
Remark. Note that .
X, Y, Z ∈Set xx , X ∈X y ∈Y
f 1 2
Proposition. If , and and , then
X ⊔ (Y ⊔ Z) = (X ⊔ Y)⊔ Z
x y x y
(X ⊔ Y)⊔ Z = (X ⊔ Z)⊔ Y
x1 x2 x2 x1
Ch(R) Set C
f
De(cid:28)nition 2.12. An operad in is a -module, equipped with opera-
tions
◦ :C(X)⊗C(Y)→C(X ⊔ Y)
x x
x∈X X, Y ∈Set
f
for all and all and satisfying the two axioms
(1) Asso
iativity:
◦ (1⊗◦ )=◦ (◦ ⊗1):
x y y x
C(X)⊗C(Y)⊗C(Z)→C(X ⊔ (Y ⊔ Z))
x y
◦ (◦ ⊗1)=◦ (◦ ⊗1)(1⊗τ):
x2 x1 x1 x2
C(X)⊗C(Y)⊗C(Z)→C((X ⊔ Y)⊔ Z)
x1 x2
X, Y, Z ∈ Set x, x , x ∈ X y ∈ Y τ:C(Y)⊗
f 1 2
for all and all and , where
C(Z)→C(Z)⊗C(Y)
is the transposition isomorphism.
η : → C({x}) {x} ∈
x
(2) Unit: There exist morphisms 1 for all singleton sets
Set
f
that make the diagrams
C(X)⊗ ∼= //C(X) ⊗C(X) ∼= //C(X)
1⊗ηx (cid:15)(cid:15) q1qqqq◦qxqqqq88 η1x⊗1(cid:15)(cid:15) ssssss◦sxsss99
C(X)⊗C(x) C(X)
X ∈ Set
f
ommute, for all . The operad will be
alled nonunital if the
axioms above only hold for nonempty sets.
Remark. See theorem1.60and 1.61and se
tion1.7.1of [18℄ forthe proofthat this
de(cid:28)nes operads
orre
tly. For more traditional de(cid:28)nitions, see [22℄, [14℄. This is
basi
ally the de(cid:28)nition of a pseudo-operad in [18℄ where we have added the unit
nth
axiom. To translate this de(cid:28)nition into the more traditional ones, set the
C([n])
omponent of the operad to .
Set −mod
f
The use of
auses the equivarian
e
onditions in [14℄ to be automat-
i
ally satis(cid:28)ed.
The operads we
onsider here
orrespond to symmetri
operads in [22℄.
The term (cid:16)unital operad(cid:17) is used in di(cid:27)erent ways by di(cid:27)erent authors. We use
0
itinthesenseofKrizandMayin[14℄,meaningtheoperadhasa -
omponentthat
C(∅)=
a
ts like an arity-loweringaugmentation under
ompositions. This is 1.
A simple example of an operad is:
X C(X)=ZΣ
X
Example 2.13. Forea
h(cid:28)niteset, , , with
ompositionde(cid:28)nedby
S n
0 th
in
lusion of sets. This operad is denoted . In other notation, its
omponent
ZS
n
is the symmetri
group-ring .
6
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
For the purposes of this paper, the
anoni
al example of an operad is
C ∈ Ch(R)
De(cid:28)nition 2.14. Given any , the asso
iated
oendomorphism operad,
CoEnd(C)
is de(cid:28)ned by
CoEnd(C)(X)=Hom (C,CX)
R
X ∈Set CX = C
for f, and X is the unordered tensor produ
t de(cid:28)ned in de(cid:28)ni-
{◦ }
x
tion 2.9. The
ompositionNs are de(cid:28)ned by
◦ :Hom (C,CX)⊗Hom (C,CY)→
x R R
Hom (C,CX\{x}⊗C ⊗Hom (C,CY))−H−o−m−R−(−1−,1−⊗−e→)
R x R
Hom (C,CX\{x}⊗CY))=Hom (C,CX⊔xY)
R R
C C x∈X e:C ⊗Hom (C,CY)→CY
x x R
where isthe
opyof
orrespondingto and C ∈Ch(R)
is the evaluation morphism. This is a non-unital operad, but if has an
ε:C →
augmentation map 1 then we
an set
CoEnd(C)(∅)=
1
and
◦ :Hom (C,CX)⊗Hom (C,C∅)=Hom (C,CX)⊗
x R R R
1
−H−o−m−R−(−1,−1X−−\{−x−}⊗−ε−x→) Hom (C,CX\{x})
R
1 :CX\{x} → CX\{x} ε :C →
X\{x} x x
where is the identity map and 1 is the aug-
C x∈X
mentationC, a∈pCplhie(dRt)o the
opy of ind{eDxed,.b.y.,D } .
1 k
Given withsub
omplexes ,therelative
oendomorphism
CoEnd(C;{D }) CoEnd(C)
i
operadf ∈Hom (C,CX)is de(cid:28)ned tfo(bDe)th⊆eDsuXb-⊆opCerXad of j
onsisting of
maps R su
h that j j for all .
We use the
oendomorphism operad to de(cid:28)ne the main obje
t of this paper:
V C ∈ Ch(R)
De(cid:28)nition 2.15. A
oalgebra over an operad is a
hain-
omplex
α:V → CoEnd(C)
with an operad morphism ,
alled its stru
ture map. We will
sometimes want to de(cid:28)ne
oalgebrasusing the adjoint stru
ture map
α¯:C →Hom(V,C⊗)
Ch(R))
(in or even the set of
hain-maps
α¯ :C →Hom (V(X),CX)
X X
X ∈Set
f
for all .
We
an also de(cid:28)ne the analogueof an ideal:
C U
De(cid:28)nition 2.16. Let be a
oalgebra over the operad with adjoint stru
ture
map
α:C →Hom(U,C⊗)
D ⊆ ⌈C⌉ D
and let be a sub-
hain
omplex that is a dire
t summand. Then will
C
be
alled a
oideal of if the
omposite
α|D:D →Hom(U,C⊗)−H−o−m−(−1U−,−p−⊗→) Hom(U,(C/D)⊗)
p:C →C/D Ch(R)
vanishes, where is the proje
tion to the quotient (in ).
7
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
Remark. Notethatitiseasierforasub-
hain-
omplextobea
oidealofa
oalgebra
than to be an ideal of an algebra. For instan
e, all sub-
oalgebras of a
oalgebra
are also
oideals. Consequently it is easy to form quotients of
oalgebrasand hard
to form sub-
oalgebras. This is dual to what o
urs for algebras.
V
We will sometimeswantto fo
uson aparti
ular
lassof -
oalgebras: the poin-
ted, irredu
ible
oalgebras. We de(cid:28)ne this
on
ept in a way that extends the
on-
ventional de(cid:28)nition in [24℄:
V
De(cid:28)nition 2.17. Givena
oalgebraoveraunitaloperad withadjointstru
ture-
map a :C →Hom (V(X),CX)
X X
c∈C a (c)=f (cX) n>0 cX ∈CX
X X
anelement is
alledgroup-like if forall . Here
n R n X
is the -fold -tensor produ
t, where is the
ardinality of ,
f =Hom (ǫ ,1):Hom ( ,CX)=CX →Hom (V(X),CX)
X R X R 1 X
ǫ :V(X) → V(∅) = = R n
X
and 1 is the augmentation (whi
h is -fold
omposition
V(∅)
with ).
C V
A
oalgebra over an operad is
alled pointed if it has a unique group-like
1
element (denoted ), and pointed irredu
ible if the interse
tion of any two sub-
oalgebras
ontains this unique group-likeelement.
V C
Remark. Notethatagroup-likeelementgeneratesasub -
oalgebraof andmust
0
lie in dimension .
Althoughthisde(cid:28)nitionseems
ontrived,itarisesin(cid:16)nature(cid:17): The
hain-
omplex
of a pointed, simply-
onne
tedredu
ed simpli
ial set is naturallya pointed irredu-
S = {C(K(S ,1))}
n
ible
oalgebra over the Barratt-E
les operad, (see [21℄). In
this
ase, the operad a
tion en
odes the
hain-level e(cid:27)e
t of Steenrod operations.
D V
Proposition 2.18. Let be a pointed, irredu
ible
oalgebra over an operad .
Then the augmentation map
ε:D →R
is naturally split and any morphism of pointed, irredu
ible
oalgebras
f:D →D
1 2
is of the form 1⊕f¯:D =R⊕kerε →D =R⊕kerε
1 D1 2 D2
ε :D →R i=1,2
i i
where , are the augmentations.
R·1⊆D
i
Proof. Thede(cid:28)nition(2.17)ofthesub-
oalgebra isstatedinaninvariant
way, so that any
oalgebra morphism must preserve it. Any morphism must also
0th
preserve augmefntations be
kaeursεe the akuegrmεentation is the -order stru
ture-map(cid:3).
Consequently, must map D1to D2. The
on
lusion follows.
V S
0
De(cid:28)nition 2.19. We denote the
ategory of nearly free
oalgebrasover by .
V V C
If isunital,every -
oalgebra, ,
omesequippedwitha
anoni
alaugmentation
ε:C →R
R V
so the terminal obje
t is . If is not unital, the terminal obje
t in this
ategory
0
is , the null
oalgebra. V I
0
The
ategory of nearly free pointed irredu
ible
oalgebras over is denoted
V
(cid:22) this is only de(cid:28)ned if is unital. Its terminal obje
t is the
oalgebra whose
R 0
underlying
hain
omplex is
on
entrated in dimension .
8
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
We also need:
A ∈ C = I S ⌈A⌉
0 0
De(cid:28)nitionC2h.2(0R.)If or , then denotes the underlying
hain-
omplex in of
kerA→t
t C ⌈∗⌉
where denotes the terminCal obCjeh
(tRin) (cid:22) see de(cid:28)nition 2.19. We will
all
the forgetful fun
tor from to .
We will use the
on
ept of
ofree
oalgebra
ogenerated by a
hain
omplex:
C ∈Ch(R) V V G
De(cid:28)nition 2.21. Let and let be an operad. Then a -
oalgebra
C
will be
alled the
ofree
oalgebra
ogenerated by if
ε:G→C
(1) there exists a morphism of DG-modules
V D f:D → C
(2) given any -
oalgebra and anyVmorphism offˆ:DDG→-mGodules ,
there existsaunique morphismof -
oalgebras, , that makesthe
diagram
D fˆ // G
@
@
@
@
@@ ε
f @ (cid:15)(cid:15)
C
ommute.
This universal property of
ofree
oalgebras implies that they are unique up to
isomorphism if they exist. The paper [22℄ gives a
onstru
tive proof of their exist-
en
e in great generality (under the unne
essary assumption that
hain-
omplexes
R LVC
are -free). In parti
ular, this paper de(cid:28)nes
ofree
oalgebras and pointed
PVC C
irredu
ible
ofree
oalgebras
ogeneratedby a
hain-
omplex .
R
WeRwill denote the
loseCdhsy(Rm)metri
monoidal
ategory of -
hain-
omplexes
with -tensor produ
ts by . These
hain-
omplexes are allowed to extend
R
into arbitrarily many negative dimensions and have underlying graded -modules
that are
• R
arbitrary if is a (cid:28)eld (but they will be free)
• R=Z
nearly free, in the sense of de(cid:28)nition 2.1, if .
3. The general
ase
G
We re
allthe
on
eptof a model stru
ture on a
ategory . Thisinvolvesde(cid:28)n-
ing spe
ialized
lasses of morphisms
alled
o(cid:28)brations, (cid:28)brations, and weak equi-
valen
es (see [19℄ and [11℄). The
ategory and these
lasses of morphisms must
satisfy the
onditions:
G
CM 1: is
losed under all (cid:28)nite limits and
olimits
G
CM 2: Suppose the following diagram
ommutes in :
X@ g //??Y
@ ~
@ ~
@ ~
@ ~
@ ~
h @ ~~ f
Z
f,g,h
If any two of are weak equivalen
es, so is the third.
f g g
CM 3: If isaretra
tof and isaweakequivalen
e,(cid:28)bration,or
o(cid:28)bra-
f
tion, then so is .
9
HOMOTOPY THEORY OF COALGEBRAS JUSTIN R. SMITH
CM 4: Suppose that we are given a
ommutative solid arrowdiagram
U >>//A
i f
(cid:15)(cid:15) (cid:15)(cid:15)
W //B
i p
where is a
o(cid:28)bration and is a (cid:28)bration. Then the dotted arrow exists
i p
making the diagram
ommute if either or is a weak equivalen
e.
f:X →Y G
CM 5: Any morphism in may be fa
tored:
f =p◦i p i
(1) , where is a (cid:28)bration and is a trivial
o(cid:28)bration
f =q◦j q j
(2) , where is a trivial (cid:28)bration and is a
o(cid:28)bration
We also assume that these fa
torizations are fun
torial (cid:22) see [9℄.
X •→X
De(cid:28)nition 3.1. An obje
t, , forwhi
hthe map isa
o(cid:28)bration,is
alled
Y Y →•
o(cid:28)brant. An obje
t, , for whi
h the map is a (cid:28)bration, is
alled (cid:28)brant.
Ch(R)
Example 3.2. The
ategory, , of unbounded
hain
omplexes overthe ring
R
has a model stru
ture in whi
h:
(1) Weak equivalen
es are
hain-homotopy equivalen
es
(2) Fibrations are surje
tions of
hain-
omplexes that are split (as maps of
R
graded -modules).
(3) Co(cid:28)brations are inje
tions of
hain-
omplexes that are split (as maps of
R
graded -modules).
Remark. All
hain
omplexes are (cid:28)brant and
o(cid:28)brant in this model.
Thisistheabsolutemodelstru
turede(cid:28)nedbyChristensenandHoveyin[8℄,and
Cole in [6℄. In this model stru
ture, all unbounded
hain-
omplexes are
o(cid:28)brant
and a quasi-isomorphismmay fail to be a weak equivalen
e.
R R = Z
Remark 3.3. We must allow non- -free
hain
omplexes (when ) be
ause
PV(∗) LV(∗)
the underlying
hain
omplexes of the
ofree
oalgebras and are not
R R R=Z
knowntobe -free. They
ertainlyareif isa(cid:28)eld,butif theirunderlying
Zℵ0 Z
abelian groups are subgroups of the Baer-Spe
ker group, , whi
h is -torsion
free but well-known not to be a free abelian group (see [23℄, [3℄ or the survey [7℄).
Proposition3.4. The forgetfulfun
tor(de(cid:28)nedinde(cid:28)nition2.20)and
ofree
oal-
gebra fun
tors de(cid:28)ne adjoint pairs
PV(∗):Ch(R) ⇆ I0:⌈∗⌉
LV(∗):Ch(R) ⇆ S0:⌈∗⌉
Remark. The adjointness of the fun
tors follows from the universal property of
ofree
oalgebras(cid:22) see [22℄.
V
Condition 3.5. Throughout the rest of this paper, we assume that is an operad
equipped with a morphism of operads
δ:V→V⊗CoEnd(I;{R·p ,R·p })
0 1
(see de(cid:28)nition 2.14) that makes the diagram
VRRRRRRδRRRRRR//RVRRRR⊗RRRRRCRRoRRERRRRnRRdRRR(RIRR;{(cid:15)(cid:15)R·p0,R·p1})
V
10
