BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY OF
MODULES OVER OPERADS
JOHN E. HARPER
1.Introduction
There are many interesting situations in which algebraic structure can be nat*
*u
rally described by operads [28, 31, 32, 33]. In many of these, there is a notio*
*n of
abelianization or stabilization [1, 2, 13, 14, 15, 16, 17, 36, 37, 40, 42] whic*
*h provides
a notion of (derived) homology. In this context, homology is not just a graded *
*col
lection of abelian groups, but a geometric object like a chain complex or spect*
*rum,
and distinct algebraic structures tend to have distinct notions of homology. F*
*or
commutative algebras this is the cotangent complex appearing in AndreQuillen
homology, and for the empty algebraic structure on spaces this is a chain compl*
*ex
calculating the singular homology of spaces.
In this paper we consider algebraic structures parametrized by operads acting
on symmetric sequences of unbounded chain complexes and symmetric spectra [26];
such algebraic structures are called modules over an operad [27, 38], which inc*
*ludes
the more familiar algebras over an operad [12, 28, 29, 33, 38] as a special cas*
*e.
Underlying every operad is a symmetric sequence [3, 9, 10, 12, 27, 38, 45, 46],*
* and
it turns out that symmetric sequences provide a useful setting for studying the
derived homology of algebraic structures. In addition to the classical associa*
*tive
algebra, commutative algebra, and Lie algebra structures, operads can describe *
*in
a useful manner various highly structured homotopy versions of these, as natura*
*lly
appear for example in the algebraic analogs of nfold loop spaces and infinite *
*loop
spaces [2, 13, 17, 21, 23, 28, 29, 31, 32, 33]. In other words, operads arise b*
*ecause
they act on many objects.
Even in the case of a simple algebraic structure such as commutative algebras*
*, ho
mology provides interesting invariants; in [34] Miller proves the Sullivan conj*
*ecture
on maps from classifying spaces, and in his proof derived homology of commutati*
*ve
algebras [15, 16, 36, 37, 39] is a critical ingredient. This suggests that homo*
*logy,
for the larger class of algebraic structures parametrized by an action of an op*
*erad,
will provide interesting and useful invariants.
Consider any catgory C with all small limits, and with terminal object denoted
by *. Let Cabbe the category of abelian group objects in (C, x, *) and define
__Ab_//
C ooU__Cab
abelianization Ab to be the left adjoint of the forgetful functor U, if it exis*
*ts. Then
if C and Cabare equipped with an appropriate homotopy theoretic structure, ho
mology is the total left derived functor of abelianization; i.e., if X 2 C then*
* Quillen
homology of X is defined by QH (X) := L Ab(X). This notion of homology is in
teresting in several contexts, including left modules and algebras over augment*
*ed
1
2 JOHN E. HARPER
operads O in unbounded chain complexes over a commutative ring k. In this con
text, the abelianizationforgetful adjunction takes the form of a "change of op*
*erads"
adjunction
_IOO//_ IOO()//_
LtOoo___LtI= SymSeq = (LtO)ab AlgOoo___AlgI= Chk= (AlgO)ab
with left adjoints on top, provided that O[0] = * and O[1] = k; hence in this
setting, abelianization is the "indecomposables" functor. Using the framework a*
*nd
corresponding homotopy theory established in [19], we show that the desired Qui*
*llen
homology functors are welldefined and can be calculated as realization of simp*
*licial
bar constructions. The theorem is this.
Theorem 1.1. Let k be a field of characteristic zero and let Chk be the category
of unbounded chain complexes over k. If f : O! Iis a morphism of operads in
Chkand X is a left Omodule (resp. Oalgebra), then there is a zigzag of weak
equivalences
I OLOX' R(Bar(I, O, X))
i j
resp. I OLO(X)' R(Bar(I, O, X))
natural in X. In particular, Quillen homology QH (X) ' R(Bar(I, O, X)) provided
that O[0] = * and O[1] = k.
The condition in Theorem 1.1 that k is a field of characteristic zero, ensure*
*s the
appropriate homotopy theoretic structures exist on the category of left Omodul*
*es
and the category of Oalgebras, when O is an arbitrary operad in chain complexes
[19].
When passing from the context of chain complexes to the context of symmetric
spectra, abelian group objects appear less meaningful, and the interesting corr*
*e
sponding notion of homology is derived "indecomposables". If X is a left module
or algebra over an augmented operad O in symmetric spectra, there is a "change
of operads" adjunction
IOO_// IOO()//_
LtO oo___LtI= SymSeq AlgO oo___AlgI= Sp
with left adjoints on top. If O[0] = * and O[1] = S, then Quillen homology of X
is defined by QH (X) := I OLOX for left Omodules and by QH (X) := I OLO(X) for
Oalgebras; hence in this setting, Quillen homology is the total left derived f*
*unctor
of "indecomposables". Using the framework and corresponding homotopy theory
established in [19, 20], we show that the desired Quillen homology functors are
welldefined and can be calculated as realization of simplicial bar constructio*
*ns,
modulo cofibrancy conditions. The theorem is this.
Theorem 1.2. Let Sp be the category of symmetric spectra. If f : O! Iis a
morphism of operads in Sp and X is a left Omodule (resp. Oalgebra) such that
one of the following is true:
(a)the simplicial bar construction Bar(O, O, X) is objectwise cofibrant in L*
*tO
(resp. AlgO), or
(b)the simplicial bar construction Bar(O, O, Xc) is objectwise cofibrant in *
*LtO
(resp. AlgO) for some functorial factorization ;! Xc! X in LtO giving
a cofibration followed by a weak equivalence,
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 3
then there is a zigzag of weak equivalences
I OLOX' R(Bar(I, O, X))
i j
resp. I OLO(X)' R(Bar(I, O, X))
natural in such X. In particular, Quillen homology QH (X) ' R(Bar(I, O, X)) for
such X provided that O[0] = * and O[1] = S.
Remark 1.3. The conditions in (a) are satisfied if O[0] = *, O is cofibrant in *
*SymSeq,
and X is cofibrant in SymSeq (resp. Sp ). The conditions in (b) are satisfied i*
*f the
forgetful functor LtO! SymSeq (resp. AlgO! Sp ) preserves cofibrant objects.
These cofibrancy conditions in Sp and SymSeq are with respect to the stable fl*
*at
positive model structures (Section 3.8).
Working with several model category structures, we give a homotopical proof of
Theorems 1.1 and 1.2, once we have proved that certain homotopy colimits in left
Omodules and Oalgebras can be easily understood. The key result here, which
is at the heart of this paper, is showing that the forgetful functor commutes w*
*ith
certain homotopy colimits. The theorem is this.
Theorem 1.4. Let k be a field of characteristic zero. If O is an operad in Sp *
*or
Chkand X is a simplicial left Omodule (resp. simplicial Oalgebra), then there*
* is
a zigzag of weak equivalences
LtO
hocolimXop'hocolim opX
i AlgO j
resp. hocolimXop'hocolim opX
natural in X, with the forgetful functor.
In this paper we develop results for both chain complexes and symmetric spect*
*ra,
in parallel. It turns out, we can use the techniques developed in [20] in the c*
*ontext
of symmetric spectra to compare homotopy categories of modules (resp. algebras)
over operads in the context of chain complexes. The theorem is this.
Theorem 1.5. Let k be a field of characteristic zero. Suppose O is an operad in
Chkand let LtO (resp. AlgO) be the category of left Omodules (resp. Oalgebras*
*).
If f : O! O0is a map of operads, then the adjunction
_f*_//_ i __f*_// j
LtO oo___LtO0, resp. AlgOoo___AlgO0,
f* f*
is a Quillen adjunction with left adjoint on top and f* the forgetful functor. *
* If
furthermore, f is an objectwise weak equivalence, then the adjunction is a Quil*
*len
equivalence, and hence induces an equivalence on the homotopy categories.
In the last few sections of this paper, we present analogous results for non
operads, operads in chain complexes over a commutative ring, and right modules
over operads.
4 JOHN E. HARPER
1.1. Relationship to previous work. One of the main theorems of Fresse [9]
is that for positive chain complexes over a field of characteristic zero, and f*
*or left
modules and operads which are trivial at zero (e.g., such modules do not specia*
*lize
to algebras over operads), then under additional conditions, the total left der*
*ived
"indecomposables" functor is welldefined, and can be calculated as realization*
* of a
simplicial bar construction in the underlying category. Theorem 1.1 improves th*
*is
result to unbounded chain complexes over a field of characteristic zero, to alg*
*ebras
and arbitrary left modules over operads, and also provides a simplified homotop*
*ical
proof of Fresse's original result.
One of the main theorems of Hinich [22] is that for unbounded chain complexes
over a field of characteristic zero, a morphism of operads which is an objectwi*
*se
weak equivalence induces a Quillen equivalence between categories of algebras o*
*ver
operads. Theorem 1.5 improves this result to the category of left modules over
operads.
Acknowledgments. The author would like to thank Bill Dwyer for his constant
encouragement and invaluable help and advice. The research for part of this pap*
*er
was carried out, while the author was a visiting researcher at the Thematic Pro
gram on Geometric Applications of Homotopy Theory at the Fields Institute for
Mathematics, Toronto.
2.Symmetric sequences
2.1. Two contexts.
Definition 2.1. Let k be a commutative ring.
o (Sp , ^, S) is the category of symmetric spectra.
o (Chk, , k) is the category of unbounded chain complexes over k.
Both are symmetric monoidal closed categories with all small limits and colimit*
*s;
the null object is denoted by *.
Remark 2.2. By closed we mean there exists a functor
op
Sp x Sp ! Sp , (Y, Z) 7! Map(Y, Z),
i j
resp. Chopkx Chk! Chk, (Y, Z) 7! Map(Y, Z)
which we call mapping object, which fits into isomorphisms
hom(X ^Y, Z)~= hom(X, Map(Y, Z))
i j
resp. hom(X Y, Z)~= hom(X, Map(Y, Z))
natural in X, Y, Z.
2.2. Symmetric sequences. Define the sets n := {1, . .,.n} for each n 0, where
0 := ; denotes the empty set. Define the totally ordered sets [n] := {0, 1, . .*
*,.n}
for each n 0, and given their natural ordering. If T is a finite set, define *
*T  to
be the number of elements in T .
Definition 2.3. Let k be a commutative ring. Let n 0.
o is the category of finite sets and their bijections.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 5
o A symmetric sequence in Sp (resp. Chk) is a functor A : op! Sp(resp.
A : op! Chk). SymSeq is the category of symmetric sequences in Sp
(resp. Chk) and their natural transformations.
o A symmetric sequence A is concentrated at n if A[r] = * for all r 6= n.
2.3. Symmetric sequences build functors.
Definition 2.4. Consider symmetric sequences in Sp (resp. in Chk). Each A 2
SymSeqdetermines a corresponding functor defined objectwise by
a
Sp ! Sp Z 7! A(Z):= A[t] ^ tZ ^t
i t 0
a j
resp. Chk! Chk Z 7! A(Z):= A[t] tZ t
t 0
If a symmetric sequence A has the extra structure of an operad (Section 3.6),
then the corresponding functor A() has the extra structure of a monad (or trip*
*le),
and the assignment Z 7! A(Z) fits into a freeforgetful adjunction. Consider
A, B 2 SymSeq. In the next section, the tensor product A ~B 2 SymSeqis presented
and used to define the circle product A O B 2 SymSeq, which has the property th*
*at
(A O B)(Z) ~=A B(Z) .
3.Monoidal structures on SymSeq
To remain consistent with [20], and to avoid confusion with other tensor prod*
*ucts
appearing in this paper, we use the following ~ notation.
3.1. Tensor product.
Definition 3.1. Consider symmetric sequences in Sp (resp. in Chk). Let A1, . .,*
*.At2
SymSeq. The tensor products A1~ . .~.At2 SymSeq are the left Kan extensions of
objectwise smash (resp. objectwise tensor) along coproduct of sets,
xt ^ A1x...xAt xt
( op)xt_A1x...xAt//_Sp ____//_Sp ( op)xt__________//_Chk ____//_Chk
` `
 
fflffl A1~...~At fflfflop A1~...~At
op______left_Kan_extension//_Sp ______left_Kan_extension//_Chk
The following calculations will be useful when working with tensor products.
Proposition 3.2. Consider symmetric sequences in Sp (resp. in Chk). Let
A1, . .,.At2 SymSeq and R 2 , with r := R. There are natural isomorphisms,
a
(A1~ . .~.At)[R]~= A1[ss1(1)] ^. .^.At[ss1(t)],
ss:R!itn Set
~= a A1[r1] ^. .^.At[rt] . r,
r1+...+rt=r r1x...x rt
a
resp. (A1~ . .~.At)[R]~= A1[ss1(1)] . . .At[ss1(t)],
ss:R!itn Set
~= a A1[r1] . . .At[rt] . r,
r1+...+rt=r r1x...x rt
6 JOHN E. HARPER
3.2. Tensor powers. It will be useful to extend the definition of tensor powers
A~t to situations in which the integers t are replaced by a finite set T .
Definition 3.3. Consider symmetric sequences in Sp (resp. in Chk). Let A 2
SymSeqand R, T 2 . The tensor powers A~T 2 SymSeq are defined objectwise
by
a ~ a
(A~;)[R] := S, (A T )[R] := ^ t2TA[ss1(t)] (T 6= ;),
ss:R!i;n Set ss:R!iTn Set
i a a j
resp. (A~;)[R] := k, (A~T )[R] := t2TA[ss1(t)] (T 6= ;).
ss:R!i;n Set ss:R!iTn Set
We will use the abbreviation A~0 := A~;. The smash products (resp. tensor
products) indexed by T are regarded as unordered [19].
3.3. Circle product (composition product).
Definition 3.4. Consider symmetric sequences in Sp (resp. in Chk). Let A, B 2
SymSeq, R 2 , and define r := R. The circle product (or composition product)
A O B 2 SymSeq is defined objectwise by the coend
a ~
(A O B)[R] := A ^ (B ~)[R]~= A[t] ^ t(B t)[r]
i t 0
a ~ j
resp. (A O B)[R] := A (B ~)[R]~=A[t] t(B t)[r] .
t 0
3.4. Monoidal structures.
Proposition 3.5. Consider symmetric sequences in Sp (resp. in Chk).
(a)(SymSeq, ~, 1) has the structure of a symmetric monoidal closed category
with all small limits and colimits. The unit for ~ denoted "1" is the sym
metric sequence concentrated at 0 with value S (resp. k).
(b)(SymSeq, O, I) has the structure of a monoidal closed category with all s*
*mall
limits and colimits. The unit for O denoted "I" is the symmetric sequence
concentrated at 1 with value S (resp. k). Circle product is not symmetric.
3.5. Symmetric sequences build functors (revisited).
Definition 3.6. Let Z 2 Sp (resp. Z 2 Chk). Define ^Z2 SymSeq to be the
symmetric sequence concentrated at 0 with value Z.
The category Sp (resp. Chk) embeds in SymSeq as the full subcategory of
symmetric sequences concentrated at 0, via the functor
Sp ! SymSeq Z 7! ^Z
i j
resp. Chk! SymSeq Z 7! ^Z
Definition 3.7. Consider symmetric sequences in Sp (resp. in Chk). Each A 2
SymSeqdetermines a corresponding functor defined objectwise by
a
Sp ! Sp Z 7! A O (Z):= A[t] ^ tZ ^t~=(A O ^Z)[0]
i t 0
a j
resp. Chk! Chk Z 7! A O (Z):= A[t] tZ t~=(A O ^Z)[0]
t 0
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 7
3.6. Modules and algebras over operads.
Definition 3.8. An operad is a monoid object in (SymSeq, O, I) and a morphism
of operads is a morphism of monoid objects in (SymSeq, O, I).
Definition 3.9. Let O be an operad in Sp (resp. Chk).
o A left Omodule is an object in (SymSeq, O, I) with a left action of O and
a morphism of left Omodules is a map which respects the left Omodule
structure.
o A right Omodule is an object in (SymSeq, O, I) with a right action of O *
*and
a morphism of right Omodules is a map which respects the right Omodule
structure.
o An Oalgebra is an object X 2 Sp (resp. X 2 Chk) with a left Omodule
structure on ^X. Let X and X0be Oalgebras. A morphism of Oalgebras is
a map f : X !X0in Sp (resp. Chk) such that ^f: ^X!X^0is a morphism
of left Omodules.
o LtO is the category of left Omodules and their morphisms.
o RtO is the category of right Omodules and their morphisms.
o AlgO is the category of Oalgebras and their morphisms.
Proposition 3.10. Let O be an operad. The category AlgO embeds in LtO as the
full subcategory of left Omodules concentrated at 0, via the functor
AlgO! LtO Z 7! ^Z
Hence, an Oalgebra is the same as a left Omodule concentrated at 0.
3.7. Freeforgetful adjunctions. It will be useful to summarize the following
basic properties of AlgOand LtO. For an operad O, the assignment Z 7! O O (Z)
given in Definition 3.7, is the free algebra on the underlying object.
Proposition 3.11. Let O be an operad in Sp (resp. Chk).
(a)There are adjunctions
_OO//_ OO()//_ i OO()//_ j
SymSeq oo___LtO Sp oo___AlgO resp. Chkoo___AlgO
U U U
with left adjoints on top and U the forgetful functor.
(b)All small colimits exist in LtOand AlgO, and both reflexive coequalizers *
*and
filtered colimits are preserved (and created) by the forgetful functors.
(c)All small limits exist in LtO and AlgO, and are preserved (and created) by
the forgetful functors.
3.8. Model category structures. We assume the reader is familiar with model
categories. A useful introduction is given in [6]. See also the original arti*
*cles
by Quillen [35, 37], and the more recent [4, 14, 18, 24, 25]. The adjunctions *
*in
Proposition 3.11(a) can be used to create model category structures on LtO and
AlgO[19, 20]. We recall the statements here.
Theorem 3.12. Let k be a field of characteristic zero. Let O be an operad in
Chk. Then LtO and AlgO both have natural model category structures. The weak
equivalences and fibrations in these model structures are inherited in an appro*
*priate
sense from the homology isomorphisms and the dimensionwise surjections in Chk.
8 JOHN E. HARPER
Theorem 3.13. Let O be an operad in Sp . Then LtOand AlgOboth have natural
model category structures. The weak equivalences and fibrations in these model
structures are inherited in an appropriate sense from the stable weak equivalen*
*ces
and the stable flat positive fibrations in Sp .
We have followed Schwede [43] in using the term flat (e.g., stable flat model
structure) for what is called S (e.g., stable Smodel structure) in [26, 41, 44*
*].
Theorem 3.14. Let O be an operad in Sp . Then LtOand AlgOboth have natural
model category structures. The weak equivalences and fibrations in these model
structures are inherited in an appropriate sense from the stable weak equivalen*
*ces
and the stable positive fibrations in Sp .
4. Simplicial objects
We assume the reader is familiar with simplicial objects [7, 11, 14, 18, 47].
Definition 4.1. Let D be a category with all small limits and colimits.
o is the category with objects the totally ordered sets [n] for n 0 and
morphisms the maps of sets , : [n]![n0]which respect the ordering; i.e.,
such that k l implies ,(k) ,(l).
o + is the subcategory of with all the objects and morphisms the surjec
tive maps. op
o A simplicial object in D is a functor A : op! D. sD := D is the
category of simplicial objects in D and their natural transformations.
o A cosimplicial object in D is a functor A : ! D. cD := D is the catego*
*ry
of cosimplicial objects in D and their natural transformations.
o If X 2 sD, we will sometimes use the notation ss0X := colimX : op! D .
o ; denotes an initial object in D and * denotes a terminal object in D.
If X 2 sDand n 0, we usually use the notation Xn := X([n]).
4.1. Model structures on simplicial objects.
Definition 4.2. Let D be a category with all small colimits. If X 2 sD (resp.
X 2 D) and K 2 sSet, then X . K 2 sDis defined objectwise by
a i a j
(X . K)n := Xn resp. (X . K)n := X
Kn Kn
the coproduct in D, indexed over the set Kn, of copies of Xn (resp. X). Let z *
* 0
and define the evaluation functor Evz : sD! Dobjectwise by Evz(X) := Xz.
Theorem 4.3. Let k be a field of characteristic zero. Let O be an operad in Sp
or Chk. Consider LtO(resp. AlgO) with any of the model structures in Section 3.*
*8.
Then sLtO (resp. sAlgO) has a corresponding natural model category structure.
The weak equivalences and fibrations in this model structure are inherited in an
appropriate sense from the weak equivalences and fibrations in LtO (resp. AlgO).
Proof.The model category structure on sLtOis created by the set of adjunctions
. [z]//
LtO _____sLtOoo_, z 0,
Evz
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 9
with left adjoints on top. Define a map f in sLtOto be a weak equivalence (resp.
fibration) if Evz(f) is a weak equivalence (resp. fibration) in LtO for every z*
* 0.
Define a map f in sLtOto be a cofibration if it has the left lifting property w*
*ith
respect to all acyclic fibrations in sLtO. To verify the model category axioms,*
* argue
as in the proof of [19, Theorem 12.2]. Since the right adjoints Evz commute with
filtered colimits, the smallness conditions needed for the (possibly transfinit*
*e) small
object arguments are satisfied. By construction, the model category is cofibran*
*tly
generated.
The model category structure on sAlgOis created by the set of adjunctions
. [z]
AlgO ____//_sAlgOoo_, z 0,
Evz
with left adjoints on top. Argue as in the sLtOcase.
4.2. Homotopy colimit functors. Here we define certain homotopy colimit func
tors.
Proposition 4.4. Let k be a field of characteristic zero. Let O be an operad in
Sp or Chk. The left derived functors
LtO AlgO
colim op colim op
sLtO______//_LtO___//Ho(LtO) sAlgO_______//AlgO___//_Ho(AlgO)
 
 LtO  AlgO
fflffl hocolim op fflffl hocolim op
Ho(sLtO)left_derived_functor//_Ho(LtO)Ho(sAlgO)left_derived/functor/_Ho(AlgO)
exist.
Proof.It is enough to verify that the adjunction
LtO AlgO
colim op i colim op j
sLtO ____//_LtOoo_resp. sAlgO____//_AlgOoo_
with left adjoint on top, is a Quillen pair. Noting that the right adjoint pres*
*erves
fibrations and acyclic fibrations finishes the proof.
4.3. Bar constructions of modules over operads.
Definition 4.5. Let O be an operad, X 2 RtO, and Y 2 LtO. The simplicial bar
construction Bar(X, O, Y ) 2 sSymSeqlooks like (showing the face maps only)
mOidoo_ oo___ oo___oo_
X O Y idOmoXoO_O O Yoo___XoOoO_O O O Yoo__.o.o._
and is defined objectwise by Bar(X, O, Y )k := X O OOkO Y with the obvious face
and degeneracy maps. Similarly, let O be an operad in Sp (resp. Chk), X 2 RtO,
and Y 2 AlgO. The simplicial bar construction Bar(X, O, Y ) in sSp (resp. sChk)
is defined objectwise by Bar(X, O, Y )k := X O OOkO (Y ).
Sometimes the simplicial bar construction has a naturally occurring left (or *
*right)
simplicial Omodule structure.
Remark 4.6. Let O be an operad, X 2 RtO, and Y 2 LtO. Then Bar(O, O, Y ) 2
sLtOand Bar(X, O, O) 2 sRtO.
10 JOHN E. HARPER
Theorem 4.7. Let k be a field of characteristic zero. Let O be an operad in Sp
or Chk and X 2 LtO(resp. X 2 AlgO). There is a zigzag of weak equivalences
LtO
hocolimBopar(O, O,'X)X
i AlgO j
resp. hocolimBopar(O, O,'X)X
in LtO (resp. AlgO), natural in X.
Proof.Consider any X 2 LtOand define BX := Bar(O, O, X) 2 sLtOand X :=
X . [0] 2 sLtO. By Theorem 1.4 there is a commutative diagram
LtO '
hocolimBopX_____R(BX)

(*) (**)
fflffl fflffl
LtO '
hocolim opX_____R( X)
with each row a zigzag of weak equivalences. We know that (**) is a weak equiv
alence (Section 8), hence (*) is a weak equivalence. The map ;! X factors func
torially ;! Xc! X in LtOas a cofibration followed by an acyclic fibration. Th*
*is
gives a natural zigzag of weak equivalences
LtO LtO LtO LtO
hocolimBopX ' hocolim opX ' hocolim op(Xc) ' colim op(Xc) ~=Xc ' X
in LtO, which finishes the proof.
4.4. Total left derived "change of operads" functors. In this section we cal
culate certain left derived functors as realizations of simplicial bar construc*
*tions.
Proposition 4.8. Let k be a field of characteristic zero. Let f : O! O0be a
morphism of operads in Sp or Chk. The left derived functors
0OO O0OO()
LtO _O____//_LtO0__//_Ho(LtO0) AlgO ______//_AlgO0__//_Ho(AlgO0)
 
 
fflffl O0OLO fflffl O0OLO()
Ho(LtO)left_derived_functor//_Ho(LtO0)Ho(AlgO)left_derived/functor/_Ho(AlgO0)
exist.
Proof.It is enough to verify that the adjunction
O0OO//_ i O0OO()//_ j
LtOoo___LtO0 resp. AlgOoo___AlgO0
f* f*
with left adjoint on top, is a Quillen pair. Noting that the forgetful functor*
* f*
preserves fibrations and acyclic fibrations finishes the proof.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 11
Theorem 4.9. Let k be a field of characteristic zero. If f : O! O0is a morphism
of operads in Chk and X 2 LtO (resp. X 2 AlgO), then there is a zigzag of weak
equivalences
O0OLOX ' R(Bar(O0, O, X))
i j
resp.O0OLO(X) ' R(Bar(O0, O, X))
natural in X.
Proof.Consider the case of LtO. For each X 2 LtO, consider the zigzags of weak
equivalences
i LtO j
O0OLOX ' O0OLO hocolimBopar(O, O, X) ' hocolim opO0OLOBar(O, O, X)
0 0
' hocolimOopOO Bar(O, O, X) ' hocolimBopar(O , O, X)
0
' R Bar(O , O, X) .
To verify these weak equivalences, use Theorem 4.7, Proposition 4.11, and Theor*
*em
6.3. We have used the fact that every object in SymSeqis cofibrant. Argue simil*
*arly
for the case of AlgO.
Theorem 4.10. If f : O! O0is a morphism of operads in Sp and X 2 LtO
(resp. X 2 AlgO) such that one of the following is true:
(a)the simplicial bar construction Bar(O, O, X) is objectwise cofibrant in L*
*tO
(resp. AlgO), or
(b)the simplicial bar construction Bar(O, O, Xc) is objectwise cofibrant in *
*LtO
(resp. AlgO) for some functorial factorization ;! Xc! X in LtO giving
a cofibration followed by a weak equivalence,
then there is a zigzag of weak equivalences
O0OLOX ' R(Bar(O0, O, X))
i j
resp.O0OLO(X) ' R(Bar(O0, O, X))
natural in such X.
Proof.Argue as in the proof of Theorem 4.9.
Proposition 4.11. Let k be a field of characteristic zero. Let f : O! O0be a
morphism of operads in Sp or Chk and X 2 sLtO(resp. X 2 sAlgO). There is a
zigzag of weak equivalences
LtO 0 L
O0OLO hocolimXop' hocolimOopOO X
i AlgO j
resp. O0OLO hocolimXop' hocolimOop0OLO(X)
natural in X.
5. Simplicial objects in Sp and Chk
This section is a first step in comparing realization with certain homotopy c*
*ol
imits. Similar homotopy invariance arguments appear in a variety of contexts [5,
Appendix A], [8, Section X.2], [18, Section IV.1], [24, Chapter 18].
12 JOHN E. HARPER
5.1. Model category structures.
Theorem 5.1. Let k be a commutative ring. Consider symmetric sequences in
Chk. Then sChk and sSymSeq have natural model category structures. The weak
equivalences and fibrations in these model structures are inherited in an appro*
*priate
sense from the homology isomorphisms and the dimensionwise surjections in Chk.
Theorem 5.2. Consider symmetric sequences in Sp . Then sSp and sSymSeq
both have natural model category structures. The weak equivalences and fibratio*
*ns
in these model structures are inherited in an appropriate sense from the stable*
* weak
equivalences and the stable flat fibrations in Sp .
Theorem 5.3. Consider symmetric sequences in Sp . Then sSp and sSymSeq
both have natural model category structures. The weak equivalences and fibratio*
*ns
in these model structures are inherited in an appropriate sense from the stable*
* weak
equivalences and the stable fibrations in Sp .
Proof.The model category structures are created by the set of adjunctions
. [z]// . [z]//
Chk_____sChkoo_, z 0, Sp _____sSpoo_, z 0,
Evz Evz
. [z]
SymSeq_____//sSymSeqoo_, z 0.
Evz
with left adjoints on top. Argue as in the case of Theorem 4.3.
5.2. Realization calculates hocolim.
Theorem 5.4. Let k be a commutative ring. Let X 2 sSp (resp. X 2 sChk).
There is a zigzag of weak equivalences
hocolimXop'R(X)
natural in X.
An intermediate step in the proof of Theorem 5.4 is the following homotopy
invariance property.
Theorem 5.5. Let k be a commutative ring. If f : X !Y in sSp (resp. sChk)
is an objectwise weak equivalence, then Rf : RX ! RY is a weak equivalence.
Before proving Theorems 5.5 and 5.4, we establish some notation.
5.3. Realization.
Definition 5.6.
o sSetis the category of simplicial sets.
o sSet*is the category of pointed simplicial sets.
There are adjunctions
()+//_ S_G0//_ __k_//_ __N__// _____//
sSetooU__sSet*oo___Sp sSetoUo__sModkoo___Ch +koo__Ch k
with left adjoints on top, U the forgetful functor, N the normalization functor*
* ap
pearing in the DoldKan correspondence [18, Section III.2], and the righthand *
*func
tor on top the natural inclusion of categories. We will denote by Nk : sSet!Chk
the composition of the left adjoints on the righthand side.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 13
Remark 5.7. The functor S G0 is left adjoint to "evaluation at 0"; the notation
agrees with [20, Section 4.1] and [26, after Definition 2.2.5]. Let X 2 Sp and
K 2 sSet*. There are natural isomorphisms X ^K ~=X ^(S G0K) in Sp .
Remark 5.8. If X 2 sSet*, there are natural isomorphisms Xx [] ~=X ^ []+.
Definition 5.9. Let n 0.
(a)The realization functors R and Rn for simplicial symmetric spectra are
defined objectwise by the coends
R : sSp ! Sp X 7! X ^ []+
Rn : sSp ! Sp X 7! X ^ skn []+
(b)The realization functors R and Rn for simplicial chain complexes are defi*
*ned
objectwise by the coends
R : sChk!Ch k X 7! X Nk []
Rn : sChk!Ch k X 7! X Nkskn []
Proposition 5.10. Let k be a commutative ring. Let n 0. The realization
functors fit into adjuctions
_R__//_ _Rn__//
sSp oo___Sp sSp oo___Sp
_R__//_ _Rn_//_
sChkoo___Chk sChkoo___Chk
with left adjoints on top. Each adjunction is a Quillen pair.
Proof.Consider the case of sSp (resp. sChk). Use the universal property of a
coend to verify that the functor given objectwise by
i j
Map (S G0 []+, Y ) resp. Map (Nk [], Y )
is a right adjoint of R. To check the adjunctions form Quillen pairs, it is eno*
*ugh to
verify the right adjoints preserve fibrations and acyclic fibrations; since the*
* model
structures on Sp and Chkare monoidal model category structures, this follows by
noting that S G0 [m]+ and Nk [m] are cofibrant for each m 0. Argue similarly
for Rn.
Proposition 5.11. Let n 0 and X 2 Sp (resp. X 2 Chk). There are isomor
phisms R(X . [0]) ~=X and Rn(X . [0]) ~=X, natural in X.
Proof.This follows from uniqueness of left adjoints (up to isomorphism).
5.4. Normalization.
Definition 5.12. Let X 2 sChk (resp. X 2 sMod k) and n 0. Define the
subobject NXn Xn by
"
NX0 := X0 NXn := ker(di) Xn (n 1)
0 i n1
Proposition 5.13. Let X 2 sChk(resp. X 2 sModk). There is a natural isomor
phism between X and a simplicial object of the form
oo___ oo___oo_
NX0 oo___NX0oqoNX1_ oo___NX0oqoNX1_q NX1 q NX2 oo___.o.o._
14 JOHN E. HARPER
(showing the face maps only) which is given objectwise by isomorphisms
a
(5.14) Xn ~= NXk.
[n]! [k]
in +
Proof.This follows from the DoldKan correspondence [18, Section III.2] that no*
*r
malization fits into the following
__N_//_+ i __N__//+ j
(5.15) sChkoo___Ch (Chk) resp. sModkoo___Chk
equivalence of categories.
5.5. Skeletal filtration of realization.
Proposition 5.16. Let X 2 sSp (resp. X 2 sChk). The realization R(X) is
naturally isomorphic to a filtered colimit of the form
i j
R(X) ~=colim R0(X) _____//R1(X)____//R2(X)____//. . .
Proof.Consider the case of sChk. We know that [] ~=colimnskn [] in sSet.
Since the functors Nk : sSet !Ch kand X  : Chk! Chkpreserve colimiting
cones, it follows that there are natural isomorphisms
Nk [] ~= colimnNkskn []
X Nk [] ~= colimnX Nkskn [].
Consider the case of sSp . We know that []+ ~= colimnskn []+ in sSet*.
Since the functors
S G0 : sSet*! Sp
X ^  : Sp ! Sp
preserve colimiting cones, a similar argument finishes the proof.
Definition 5.17. Let X 2 sSp (resp. X 2 sChk) and n 0. Define the subobject
DXn Xn by
[
DX0 := * DXn := siXn1 Xn (n 1)
i 0 i n1
X j
resp. DX0 := * DXn := siXn1 Xn (n 1)
0 i n1
We refer to DXn as the degenerate subobject of Xn.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 15
Proposition 5.18. Let X 2 sSp (resp. X 2 sChk) and n 1. There are pushout
diagrams
(5.19) (DXn ^ [n]+) [ (Xn ^@ [n]+)____//_Rn1(X)
 
 
fflffl fflffl
Xn ^ [n]+ ______________//_Rn(X)
(5.20) resp. (DXn Nk [n]) + (Xn Nk@ [n])____//Rn1(X)
 
 
fflffl fflffl
Xn Nk [n]_______________//Rn(X)
in Sp (resp. Chk). The vertical maps in (5.19)and (5.20)are monomorphisms.
The following will be useful.
Proposition 5.21. Let X 2 sSp (resp. X 2 sChk) and n 1. There are pushout
diagrams
(5.22) DXn ^@ [n]+ ____________//_Xn ^@ [n]+
 
 
fflffl fflffl
DXn ^ [n]+ _____//_(DXn ^ [n]+) [ (Xn ^@ [n]+)
(5.23) resp. DXn Nk@ [n] _____________//Xn Nk@ [n]
 
 
fflffl fflffl
DXn Nk [n] _____//_(DXn Nk [n]) + (Xn Nk@ [n])
in Sp (resp. in Chk). The maps in (5.22)and (5.23)are monomorphisms.
Proof.In the case of sSp , the pushout diagrams (5.22)follow from the correspon*
*d
ing pushout diagrams for a bisimplicial set [18, Section IV.1]. In the case of *
*sChk,
use Proposition 5.13 to reduce to verifying that the diagram
DXn Nk@ [n] ______DXn Nk@ [n]
 
 
fflffl fflffl
DXn Nk [n] _______DXn Nk [n]
is a pushout diagram.
Proof of Proposition 5.18.In the case of sSp , the pushout diagrams (5.19)follow
from the corresponding pushout diagrams for a bisimplicial set [18, Section IV.*
*1].
In the case of sChk, use Proposition 5.13 to reduce to verifying that the diagr*
*am
(5.24) NXn Nk@ [n] ____//_Rn1(X)
 
 
fflffl fflffl
NXn Nk [n] ______//Rn(X)
16 JOHN E. HARPER
is a pushout diagram in Chk, which follows from the simplicial identities and t*
*he
property that Nk : sSet!Chkpreserves colimiting cones.
Proposition 5.25. Let k be a commutative ring. If f : X !Y in sSp (resp.
sChk) is a monomorphism, then Rf : RX ! RY is a monomorphism.
Proof.In the case of sSp , this follows from the corresponding property for rea*
*liza
tion of a bisimplicial set [18, Section IV.1]. Consider the case of sChk. Use P*
*ropo
sition 5.13 to argue that N : sChk!Ch kpreserves monomorphisms; either use the
DoldKan correspondence (5.15)and note that right adjoints preserve monomor
phisms, or use (5.14)and note that monomorphisms are preserved under retracts.
To finish the argument, forget differentials and use the pushout diagrams (5.24*
*)to
give a particularly simple filtration of Rf : RX ! RY in the underlying catego*
*ry
of graded kmodules. Since NXn! NYn is a monomorphism for each n 0, it
follows from this filtration that Rf is a monomorphism.
5.6. Homotopy invariance of realization.
Proof of Theorem 5.5.Consider the case of sSp (resp. sChk). Skeletal filtration
gives a commutative diagram
R0(X) ____//_R1(X)___//_R2(X)___//_. . .
R0(f) R1(f) R2(f)
fflffl fflffl fflffl
R0(Y )____//R1(Y_)___//R2(Y_)___//. . .
We know that R0(f) ~=f0 is a weak equivalence. Since the horizontal maps are
monomorphisms and we know that
Rn(X)=Rn1(X) ~=(Xn=DnX) ^( [n]=@ [n])
i j
resp. Rn(X)=Rn1(X) ~=(Xn=DnX) (Nk [n]=Nk@ [n])
it is enough to verify that Dfn : DXn ! DYnis a weak equivalence for each n *
*1,
and Proposition 5.26 finishes the proof.
Proposition 5.26. Let k be a commutative ring. If f : X !Y in sChk is an
objectwise weak equivalence, then Dfn : DXn ! DYnis a weak equivalence for each
n 1.
Before proving this, it will be useful to filter the degenerate subobjects.
Definition 5.27. Let X 2 sSp (resp. X 2 sChk) and n 1. For each 0 r
n  1, define the subobjects s[r]Xn1 Xn by
[
s[r]Xn1 := siXn1 Xn,
i 0 i r
X j
resp. s[r]Xn1 := siXn1 Xn
0 i r
The following proposition is motivated by the corresponding statement for bis*
*im
plicial sets [18, Section IV.1].
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 17
Proposition 5.28. Let k be a commutative ring. Let X 2 sSp (resp. X 2 sChk)
and n 1. For each 0 r n  1, the diagram
(5.29) s[r]Xn1_sr+1//_s[r]Xn
 
fflfflsr+1 fflffl
Xn ______//_s[r+1]Xn
is a pushout diagram. The maps in (5.29)are monomorphisms.
Proof.Consider the case of sSp . This follows from the corresponding pushout
diagrams for a bisimplicial set [18, Section IV.1]. Consider the case of sChk. *
*This
follows from the Proposition 5.13 and the simplicial identities.
Proof of Proposition 5.26.Consider the case of sSp (resp. sChk). Let n = 1. By
Proposition 5.28, Df1 fits into the commutative diagram
s0X0 ___s1___//_WWWWWs0X1VVVVV
 WWWWWWWW VVVVVVVV
 WWWW(a)WW VVVV(d)VV
fflffls1 fflffWWWWWW++l VVVVV++
X1 _________//_VVVVVVDX1VVVVVs0Y0_s1___//_s0Y1
 VVVVVVVV VVVVVVVV 
 VVVVV(b)V VVVDf1VV 
fflffl~= fflffVVVVVVV++lffVVVVVV++lfflfflffl
X1=s0X0 _____//VVVDX1=s0X1V Y1 ____s1___//_DY1
VVVVV VVVVVVVV 
VVVVVVVV(c)VV VVVVV(c)VV 
VVV++V fflfflVVV** fflffl
Y1=s0Y0__~=_//DY1=s0Y1
Since we know the maps (a) and (b) are weak equivalences, it follows that each
map (c) is a weak equivalence. Since we know the map (d) is a weak equivalence,
it follows that Df1 is a weak equivalence. Similarly, use Proposition 5.28 in *
*an
induction argument to verify that Dfn : DXn ! DYn is a weak equivalence for
each n 2.
5.7. Comparing realization and hocolim.
Proof of Theorem 5.4.Consider any map X !Y in sSp (resp. sChk). Use func
torial factorization to obtain a commutative diagram
* ____//_Xc___//_X
  
  
fflffl fflfflfflffl
* ____//_Y_c__//_Y
18 JOHN E. HARPER
in sSp (resp. sChk) such that each row is a cofibration followed by an acyclic
fibration. Hence we get a corresponding commutative diagram
hocolimXopoo__hocolimXopc___//colimXopc(*R(Xc))oo_(**)//_R(X)
    
    
fflffl fflffl fflffl fflffl fflffl
hocolimYopoo__hocolimYopc___//colimYopc(*R(Y)c)oo_(**)//_R(Y )
such that the rows are maps of weak equivalences; the maps (*) and (**) are weak
equivalences by Proposition 5.30 and Theorem 5.5, respectively.
Proposition 5.30. Let k be a commutative ring. If Z 2 sSp (resp. Z 2 sChk) is
cofibrant, then the natural map
R(Z) ____//_R (ss0Z) . [0]
is a weak equivalence.
Proof.Let X !Y be a generating cofibration in Sp (resp. Ch k). Consider the
pushout diagram
(5.31) X . [z]____//Z0
 
 
fflffl fflffl
Y . [z]____//Z1
in sSp (resp. sChk) and the natural maps
(5.32) R(Z0) ____//_R (ss0Z0) . [0]
(5.33) R(Z1) ____//_R (ss0Z1) . [0]
Assume (5.32)is a weak equivalence; let's verify (5.33)is a weak equivalence. C*
*on
sider the commutative diagram
R(Z0)_____________//_R(Z1)_______//R (Y=X) . [z]
  
  
fflffl fflffl fflffl
R (ss0Z0) . [0]___//R (ss0Z1) . [0]_//_R (Y=X) . [0]
The lefthand horizontal maps are monomorphisms, the lefthand vertical map is a
weak equivalence by assumption, and the righthand vertical map is a weak equiv
alence by Section 8, hence the middle vertical map is a weak equivalence. Consi*
*der
a sequence
Z0_____//Z1___//_Z2___//_. . .
of pushouts of maps as in (5.31). Assume Z0 makes (5.32)a weak equivalence; we
want to show that for Z1 := colimkZk the natural map
(5.34) R(Z1 ) ____//_R (ss0Z1 ) . [0]
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 19
is a weak equivalence. Consider the commutative diagram
R(Z0)_____________//_R(Z1)___________//_R(Z2)_______//. . .
  
  
fflffl fflffl fflffl
R (ss0Z0) . [0]__//_R (ss0Z1) . _[0]//_R (ss0Z2) . _[0]//_. . .
We know that the horizontal maps are monomorphisms and the vertical maps are
weak equivalences, hence the induced map (5.34)is a weak equivalence. Noting th*
*at
every cofibration *! Z in sSp (resp. sChk) is a retract of a (possibly transfi*
*nite)
composition of pushouts of maps as in (5.31), starting with Z0 = *, finishes the
proof.
6.Simplicial objects in SymSeq
In this section we verify that realization for simplicial symmetric sequences*
* enjoys
similar properties.
6.1. Realization.
Definition 6.1. Let k be a commutative ring. Consider symmetric sequences in
Sp (resp. in Chk). The realization functor R is defined objectwise by
i j
R : sSymSeq! SymSeq X 7! RX T 7! R X[T ]
Proposition 6.2. Let k be a commutative ring. Consider symmetric sequences in
Sp (resp. in Chk). There is an adjunction
__R__//
sSymSeqoo___SymSeq
with left adjoint on top. The adjunction is a Quillen pair.
6.2. Realization calculates hocolim.
Theorem 6.3. Let k be a commutative ring. Consider symmetric sequences in Sp
(resp. in Chk). Let X 2 SymSeq.
(a)There is a zigzag of weak equivalences hocolimXop'R(X), natural in X.
(b)If f : X !Y in sSymSeq is a weak equivalence, then Rf : RX ! RY is a
weak equivalence.
Proof.Consider part (b). This follows from Definition 6.1 and Theorem 5.5. Con
sider part (a). Use the argument in the proof of Theorem 5.4, except replace the
categories sSp and sChkwith the category sSymSeq; the maps (*) and (**) are
weak equivalences by Proposition 6.4 and part (b), respectively.
6.3. Comparing realization and hocolim.
Proposition 6.4. Let k be a commutative ring. Consider symmetric sequences in
Sp (resp. in Chk). If Z 2 sSymSeqis cofibrant, then the natural map
R(Z) ____//_R (ss0Z) . [0]
is a weak equivalence.
Proof.Use the argument in the proof of Proposition 5.30, except replace the cat*
*e
gories sSp and sChkwith the category sSymSeq.
20 JOHN E. HARPER
7.Simplicial objects in LtO and AlgO
7.1. Realization calculates hocolim.
Theorem 7.1. Let k be a field of characteristic zero. Let O be an operad in Sp
or Chk and X 2 sLtO(resp. X 2 sAlgO). There is a zigzag of weak equivalences
LtO
hocolimXop'R( X)
i AlgO j
resp. hocolimXop'R( X)
natural in X, with the forgetful functor.
Proof.Consider any X 2 sLtO. The map ;! X factors functorially ;! Xc! X
in sLtOas a cofibration followed by an acyclic fibration. This gives a diagram
LtO LtO c (*) c (**)
hocolimXopoo__hocolimXopc___//colimXopoo_R(X ) _____//R(X)
of weak equivalences; the maps (*) and (**) are weak equivalences by Proposition
7.15 and Theorem 6.3(b), respectively.
7.2. Forgetful functor commutes with hocolim.
Proof of Theorem 1.4.Let X 2 sLtO (resp. X 2 sAlgO). By Theorems 7.1 and
6.3(a), there is a zigzag of weak equivalences
LtO
hocolimXop'R( X) ' hocolim opX
i AlgO j
resp. hocolimXop'R( X) ' hocolim opX
natural in X, with the forgetful functor.
7.3. Analysis of pushouts in sLtOand sAlgO.
Definition 7.2. Let (C, , k) be a monoidal category. If X, Y 2 sCthen X Y 2 sC
is defined objectwise by
(X Y )n := Xn Yn.
Definition 7.3. Consider symmetric sequences in Sp (resp. Chk).
o A symmetric array in Sp (resp. Chk) is a symmetric sequence in SymSeq;
i.e. a functor A :opop! SymSeq.
o SymArray:= SymSeq is the category of symmetric arrays in Sp (resp.
Chk) and their natural transformations.
Proposition 7.4. Let O be an operad in Sp or Chk, A 2 sLtO(resp. A 2 LtO),
and Y 2 sSymSeq (resp. Y 2 SymSeq). Consider any coproduct in sLtO (resp.
LtO) of the form
(7.5) A q (O O Y ).
There exists OA 2 sSymArray(resp. OA 2 SymArray) and natural isomorphisms
a ~
A q (O O Y ) ~= OA[q] ~ qY q
q 0
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 21
in the underlying category sSymSeq (resp. SymSeq).
Proof.The case of A 2 LtO and Y 2 SymSeq is given in [19, Proposition 13.8]. If
A 2 sLtOand Y 2 sSymSeq, use the colimit construction in [19, Proposition 13.8]
to define OA : op! SymArrayobjectwise.
Proposition 7.6. Let O be an operad in Sp or Chk, A 2 sLtO, Y 2 sSymSeq,
and t 0. There are natural isomorphisms
LtO i ja ~
(7.7) colim opA q (O O Y~)= Oss0A[q] ~ q(ss0Y ) q,
i j q 0
(7.8) colimOopA[t]~=Oss0A[t],
in the underlying category SymSeq.
Proof.To argue (7.7), use the natural isomorphisms
LtO i j
colim opA q (O O Y ) ~=(ss0A) q ss0(O O Y ) ~=(ss0A) q O O (ss0Y )
in LtO together with Proposition 7.4. To argue (7.8), use the properties of ref*
*lex
ive coequalizers in [19, Section 8.1] together with the colimit construction in*
* [19,
Proposition 13.8].
Definition 7.9. Let i : X !Ybe a morphism in sSymSeq (resp. SymSeq) and
t 1. Define Qt0:= X ~tand Qtt:= Y ~t. For 0 < q < t define Qtqinductively by
the pushout diagrams
pr* t
t. tqx qX ~(tq)~Qqq1___//_Qq1
i* i*
fflffl fflffl
t. tqx qX ~(tq)~Y ~q____//_Qtq
in sSymSeq t(resp. SymSeq t).
Proposition 7.10. Let O be an operad in Sp or Chk, A 2 sLtO(resp. A 2 LtO),
and i : X !Yin sSymSeq (resp. in SymSeq). Consider any pushout diagram in
sLtO(resp. LtO) of the form,
f
(7.11) O O X ___________//_A
idOi j
fflffl fflffl
O O Y ____//_A q(OOX)(O O Y ).
The pushout in (7.11)is naturally isomorphic to a filtered colimit of the form
i j1 j2 j3 j
(7.12) A q(OOX)(O O Y ) ~=colim A0 _____//A1____//A2___//_. . .
22 JOHN E. HARPER
in the underlying category sSymSeq(resp. SymSeq), with A0 := OA[0] ~=A and At
defined inductively by pushout diagrams in sSymSeq (resp. SymSeq) of the form
(7.13) OA[t] ~ tQtt1f*_//_At1
id~ti* jt
fflffl ,t fflffl
OA[t] ~ tY ~t____//_At
Proof.The case of A 2 LtO and i : X !Yin SymSeq is given in [19, Proposition
13.13]. If A 2 sLtOand i : X !Yin sSymSeq, use Proposition 7.4 together with t*
*he
argument in the proof of [19, Proposition 13.13] to construct the pushout diagr*
*ams
(7.13)objectwise.
Proposition 7.14. Let O be an operad in Sp or Chk, A 2 sLtO, and i : X !Yin
sSymSeq. Consider any pushout diagram in sLtOof the form (7.11). Then ss0()
commutes with the filtered diagrams in (7.12); i.e., there are natural isomorph*
*isms
which make the diagram
ss0(j1) ss0(j2) ss0(j3)
ss0(A0)____//ss0(A1)__//_ss0(A2)_//_. . .
~= ~= ~=
fflfflj1 fflfflj2 fflfflj3
(ss0A)0____//(ss0A)1__//_(ss0A)2_//_. . .
commute.
Proof.Use the properties of reflexive coequalizers in [19, Section 8.1] togethe*
*r with
Proposition 7.6 and the arguments in the proof of [19, Proposition 13.13].
7.4. Comparing realization with hocolim.
Proposition 7.15. Let k be a field of characteristic zero. Let O be an operad in
Sp or Chk. If Z 2 sLtO(resp. Z 2 sAlgO) is cofibrant, then the map
R( Z) ____//_R((ss0 Z) . [0])
is a weak equivalence, with the forgetful functor.
Proof.Let X !Y be a generating cofibration in SymSeq and consider the pushout
diagram
(7.16) O O X . [z]___//_Z0
 
 
fflffl fflffl
O O Y . [z]___//_Z1
in sLtO. For each cofibrant Wff2 SymSeq, lff 0, and set A, consider the natural
maps
i a j i a j
(7.17) R Z0 q (O O Wff. [lff])  !R (ss0Z0) q O O Wff . [0] ,
i ff2A ff2A
a j i a j
(7.18) R Z1 q (O O Wff. [lff])  !R (ss0Z1) q O O Wff . [0] ,
ff2A ff2A
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 23
and note that the diagram
`
O O X . [z]____//_Z0 q ff2A(O O Wff. [lff]) =: A
 
 
 fflffl
fflffl ` ~
O O Y . [z]___//_Z1 q ff2A(O O Wff. [lff]) = A1
is a pushout diagram in sLtO. Assume (7.17)is a weak equivalence for each cofib*
*rant
Wff2 SymSeq, lff 0, and set A; let's verify (7.18)is a weak equivalence for ea*
*ch
cofibrant Wff2 SymSeq, lff 0, and set A. Suppose A is a set, Wff2 SymSeq
cofibrant, and lff 0, for each ff 2 A. By Proposition 7.10 there are correspon*
*ding
filtrations together with induced maps ,t (t 1) which make the diagram
R(A0)_____________//_R(A1)___________//_R(A2)_______//_. . .
____ ______
R(,0) _R(,1)______ _R(,2)____
fflffl fflffl___ _fflffl__
R (ss0A0) . [0]___//R (ss0A1) . [0]_//_R (ss0A2) . _[0]//_. . .
in SymSeq commute. Since R() commutes with colimits we get
~=
colimtR(At)_____________//_R(A1 )
 
 
fflffl ~ fflffl
colimtR (ss0At) . [0]=//_R (ss0A1 ) . [0] .
By assumption we know R(,0) is a weak equivalence, and to verify (7.18)is a weak
equivalence, it is enough to check that R(,t) is a weak equivalence for each t *
* 1.
Since the horizontal maps are monomorphisms and we know that there are natural
isomorphisms
~t
R(At)=R(At1)~=R OA[t] ~ t(Y=X . [z]) ,
~t
R (ss0At) . [0] =R (ss0At1) .~=[0]R (Oss0A[t] ~ t(Y=X) ) . [0] ,
it is enough to verify that
i j i j
R A q O O (Y=X) . [z]_____//R (ss0A) q O O (Y=X) . [0]
is a weak equivalence. Noting that Y=X is cofibrant finishes the argument that
(7.18)is a weak equivalence. Consider a sequence
Z0_____//Z1___//_Z2___//_. . .
of pushouts of maps as in (7.16). Assume Z0 makes (7.17)a weak equivalence
for each cofibrant Wff2 SymSeq, lff 0, and set A; we want to show that for
Z1 := colimkZk the natural map
i a j i a j
(7.19) R Z1 q (O O Wff. [lff])  !R (ss0Z1 ) q O O Wff . [0]
ff2A ff2A
24 JOHN E. HARPER
is a weak equivalence for each cofibrant Wff2 SymSeq, lff 0, and set A. Consid*
*er
the diagram
i ` j i ` j
R Z0 q (O O Wff. [lff])_____//_R Z1 q (O O Wff. [lff])__//_. . .
ff2A ff2A
 
 
i fflffl j i fflffl j
` `
R (ss0Z0) q O O Wff . [0]___//R (ss0Z1) q O O Wff . _[0]_//. . .
ff2A ff2A
in SymSeq. The horizontal maps are monomorphisms and the vertical maps are
weak equivalences, hence the induced map (7.19)is a weak equivalence. Noting
that every cofibration O O*. [0]! Z in sLtOis a retract of a (possibly transfi*
*nite)
composition of pushouts of maps as in (7.16), starting with Z0 = O O * . [0],
together with Proposition 8.4, finishes the proof.
8.Simplicial homotopies
Definition 8.1. Let D be a category with all small colimits and consider the le*
*ft
hand diagram
__f__// _id.d1//_ H
(8.2) X __g__//Y X ~=X . [0]____//X . [1]__//_Y
id.d0
in sD. A simplicial homotopy from f to g is any map H : X . [1]!Yin sD such
that the two diagrams in (8.2)are identical. The map f is simplicially homotopic
to g if there exists a simplicial homotopy from f to g.
Remark 8.3. This definition of simplicial homotopy agrees with [18, Section I.6]
and [30, between Propositions 6.2 and 6.3].
Proposition 8.4. Let k be a commutative ring. Let O be an operad in Sp or Chk.
Let A be a set, Wff2 SymSeq, and lff 0, for each ff 2 A. Consider the maps
` ` `
(O O Wff. [lff])r//_ (O O Wff. [0])s_//_ (O O Wff. [lff])
ff2A ff2A ff2A
in sLtO induced by the maps [lff]rff//_ [0]sff//_ [lff]in simplicial sets, such
that each map sffrepresents the vertex 0. Then the map
i ` R(jr) i ` j
R (O O Wff. [lff])__//R (O O Wff. [0])
ff2A ff2A
in SymSeq is a weak equivalence.
Proof.For each ff 2 A, we know that rffsff= idand sffrffis simplicially homotop*
*ic
to the identity map. Hence rs = idand sr is simplicially homotopic to the ident*
*ity
map. In the case of symmetric spectra, since objectwise weak equivalences are
weak equivalences, it follows that R(r) is a weak equivalence. In the case of c*
*hain
complexes, since Tot N takes simplicially homotopic maps to chain homotopic
maps, it follows from Proposition 8.5 that R(r) is a weak equivalence.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 25
Proposition 8.5. Let k be a commutative ring. If X 2 sChk, then there are
isomorphisms
R(X) = X Nk [] ~=Tot N(X)
natural in X.
Proof.If A, B 2 Chk, define the objectwise tensor A `B 2 Ch(Chk) such that
A B = Tot (A `B). It follows that there are natural isomorphisms,
X Nk [] ~=Tot X ` Nk [] ~=Tot Nk [] ` X .
Arguing as in the proof of Proposition 5.16, verify that
Nk [] ` X ~=colimnNkskn [] ` X
and use the pushout diagrams
Nk@ [n] `NXn ____//_Nkskn1 [] ` X
 
 
fflffl fflffl
Nk [n] `NXn ______//Nkskn [] ` X
in Ch(Chk) to verify that Nk [] ` X ~=N(X), which finishes the proof.
9.Modules over Non Operads
In this section we indicate some of the analogous results for non operads.
These follow from essentially the same arguments given in the previous sections,
but using the non filtrations in [19, Proposition 13.35] instead of the filtr*
*ations
in [19, Proposition 13.13].
9.1. Model category structures. Here we recall the model category structures
established in [19, Theorems 1.2 and 1.6] for modules and algebras over non
operads.
Theorem 9.1. Let k be a commutative ring. Let O be a non operad in Chk. Then
LtOand AlgO both have natural model category structures. The weak equivalences
and fibrations in these model structures are inherited in an appropriate sense *
*from
the homology isomorphisms and the dimensionwise surjections in Chk.
Theorem 9.2. Let O be a non operad in Sp . Then LtOand AlgOboth have nat
ural model category structures. The weak equivalences and fibrations in these m*
*odel
structures are inherited in an appropriate sense from the stable weak equivalen*
*ces
and the stable flat fibrations in Sp .
We have followed Schwede [43] in using the term flat (e.g., stable flat model
structure) for what is called S (e.g., stable Smodel structure) in [26, 41, 44*
*].
Theorem 9.3. Let O be a non operad in Sp . Then LtOand AlgOboth have nat
ural model category structures. The weak equivalences and fibrations in these m*
*odel
structures are inherited in an appropriate sense from the stable weak equivalen*
*ces
and the stable fibrations in Sp .
The underlying model category structures for Sp were established in [26, 44].
26 JOHN E. HARPER
9.2. Analogous results for non operads.
Theorem 9.4. Let k be a commutative ring. If f : O! O0is a morphism of non
operads in Sp or Chk and X 2 LtO (resp. X 2 AlgO) such that one of the
following is true:
(a)the simplicial bar construction Bar(O, O, X) is objectwise cofibrant in L*
*tO
(resp. AlgO), or
(b)the simplicial bar construction Bar(O, O, Xc) is objectwise cofibrant in *
*LtO
(resp. AlgO) for some functorial factorization ;! Xc! X in LtO giving
a cofibration followed by a weak equivalence,
then there is a zigzag of weak equivalences
O0OLOX ' R(Bar(O0, O, X))
i j
resp.O0OLO(X) ' R(Bar(O0, O, X))
natural in such X.
Proof.Argue as in the proof of Theorem 1.2.
Theorem 9.5. Let k be a field. If f : O! O0is a morphism of non operads in
Chkand X is an Oalgebra, then there is a zigzag of weak equivalences
O0OLO(X) ' R(Bar(O0, O, X))
natural in X. In particular, Quillen homology QH (X) ' R(Bar(I, O, X)) provided
that O0= I, O[0] = *, and O[1] = k.
Proof.Since k is a field, it follows that every object in Chkis cofibrant. Henc*
*e the
conditions of Theorem 9.4(a) are satisfied.
Theorem 9.6. Let k be a commutative ring. If O is a non operad in Sp or
Chkand X is a simplicial left Omodule (resp. simplicial Oalgebra), then there*
* is
a zigzag of weak equivalences
LtO
hocolimXop'hocolim opX
i AlgO j
resp. hocolimXop'hocolim opX
natural in X, with the forgetful functor.
Proof.Argue as in Section 7 and the proof of Theorem 1.4, together with Proposi
tion 9.10.
Theorem 9.7. Let k be a commutative ring. If O is a non operad in Sp or
Chkand X 2 LtO(resp. X 2 AlgO), then there is a zigzag of weak equivalences
LtO
hocolimBopar(O, O,'X)X
i AlgO j
resp. hocolimBopar(O, O,'X)X
in LtO (resp. AlgO), natural in X.
Proof.Argue as in the proof of Theorem 4.7.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 27
Theorem 9.8. Let k be a commutative ring. If f : O! O0is a map of non
operads in Sp or Chk, then the adjunction
_f*_//_ i __f*_// j
LtO oo___LtO0, resp. AlgOoo___AlgO0,
f* f*
is a Quillen adjunction with left adjoints on top and f* the forgetful functor.*
* If
furthermore, f is an objectwise weak equivalence, then the adjunction is a Quil*
*len
equivalence, and hence induces an equivalence on the homotopy categories.
Proof.Argue as in the proof of [20, Theorem 1.4 and Proposition 4.41], together
with Proposition 9.9.
In the next proposition, we use notation from [19] for non operads and their
underlying sequences.
Proposition 9.9. Let k be a commutative ring. Consider sequences in Sp or Chk.
If Z 2 Seqis cofibrant, then the functor
 ^OZ: Seq! Seq
preserves weak equivalences.
Proof.Consider the case of Sp . We know that smashing with a cofibrant sym
metric spectrum preserves weak equivalences. Consider the case of Chk. We know
that tensoring with a cofibrant chain complex preserves weak equivalences. Use
[19, Proposition 5.4] together with [19, Theorem 12.4] and the assumption that Z
is cofibrant in Seqto finish off the proof.
Proposition 9.10. Let k be a commutative ring. Consider sequences in Sp or
Chk. If the map i : X !Yin [19, Proposition 13.35] is a generating cofibration*
* in
Seq, then jt is a monomorphism for each t 1.
Proof.Consider the case of Sp . This follows from [19, Proposition 13.41]. Cons*
*ider
the case of Chk. After forgetting differentials, the map i : X !Yhas the form
i : S !S q,S0and hence (forgetting differentials) Qtt1!Y ^thas a left inverse
in Seqfor each t 1, which finishes the proof.
10.Operads in chain complexes over a commutative ring
In this section, we indicate some of the analogous results for operads in cha*
*in
complexes over a commutative ring. The main difficulty here is that an appropri*
*ate
model category structure on modules and algebras over an arbitrary operad may
not exist. On the other hand, sometimes it is possible to establish an appropri*
*ate
homotopy theory for particularly nice operads. One approach to this is studied
in [3]. As indicated in the statements below, it is assumed that a suitable mod*
*el
category structure is available.
10.1. Model category structures. Here we recall the model category structure
on symmetric sequences [19, Theorem 2.2] that comes up when using the free
forgetful adjunction in Proposition 3.11 to create a model category structure on
modules over an operad, if such a model structure exists.
28 JOHN E. HARPER
Theorem 10.1. Let k be a commutative ring. Then SymSeq has a natural model
category structure. The weak equivalences and fibrations in this model structu*
*re
are inherited in an appropriate sense from the homology isomorphisms and the
dimensionwise surjections in Chk.
10.2. Analogous results.
Theorem 10.2. Let k be a commutative ring. Let f : O! O0be a morphism of
operads in Chk and suppose the freeforgetful adjunction creates a model catego*
*ry
structure on LtO and LtO0resp. (AlgO and AlgO0). If X 2 LtO (resp. X 2 AlgO)
such that one of the following is true:
(a)the simplicial bar construction Bar(O, O, X) is objectwise cofibrant in L*
*tO
(resp. AlgO), or
(b)the simplicial bar construction Bar(O, O, Xc) is objectwise cofibrant in *
*LtO
(resp. AlgO) for some functorial factorization ;! Xc! X in LtO giving
a cofibration followed by a weak equivalence,
then there is a zigzag of weak equivalences
O0OLOX ' R(Bar(O0, O, X))
i j
resp.O0OLO(X) ' R(Bar(O0, O, X))
natural in such X.
Proof.Argue as in the proof of Theorem 1.2.
Theorem 10.3. Let k be a field. Let f : O! O0be a morphism of operads in
Chkand suppose the freeforgetful adjunction creates a model category structure*
* on
AlgOand AlgO0. Then there is a zigzag of weak equivalences
O0OLO(X) ' R(Bar(O0, O, X))
natural in X. In particular, Quillen homology QH (X) ' R(Bar(I, O, X)) provided
that O0= I, O[0] = *, and O[1] = k.
Proof.Since k is a field, it follows that every object in Chk is cofibrant. The*
*orem
10.2(a) finishes the proof.
Theorem 10.4. Let k be a commutative ring. Let O be an operad in Chk and
suppose the freeforgetful adjunction creates a model category structure on LtO*
*(resp.
AlgO). If X is a simplicial left Omodule (resp. simplicial Oalgebra), then th*
*ere
is a zigzag of weak equivalences
LtO
hocolimXop'hocolim opX
i AlgO j
resp. hocolimXop'hocolim opX
natural in X, with the forgetful functor.
Proof.Argue as in Section 7 and the proof of Theorem 1.4, together with Proposi
tion 10.8.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 29
Theorem 10.5. Let k be a commutative ring. Let O be an operad in Chk and
suppose the freeforgetful adjunction creates a model category structure on LtO*
*(resp.
AlgO). If X 2 LtO(resp. X 2 AlgO), then there is a zigzag of weak equivalences
LtO
hocolimBopar(O, O,'X)X
i AlgO j
resp. hocolimBopar(O, O,'X)X
in LtO (resp. AlgO), natural in X.
Proof.Argue as in the proof of Theorem 4.7.
Theorem 10.6. Let k be a commutative ring. Let f : O! O0be a morphism of
operads in Chk and suppose the freeforgetful adjunction creates a model catego*
*ry
structure on LtO and LtO0(resp. AlgO and AlgO0). Then the adjunction
_f*_//_ i __f*_// j
LtO oo___LtO0, resp. AlgOoo___AlgO0,
f* f*
is a Quillen adjunction with left adjoint on top and f* the forgetful functor. *
* If
furthermore, f is an objectwise weak equivalence and both O and O0 are cofibrant
in the underlying category SymSeq, then the adjunction is a Quillen equivalence,
and hence induces an equivalence on the homotopy categories.
Proof.Argue as in the proof of [20, Theorem 1.4 and Proposition 4.41], together
with Proposition 10.7.
Proposition 10.7. Let k be a commutative ring. Consider symmetric sequences
in Chk. If Z 2 SymSeq is cofibrant, then the functor
 O Z : SymSeq! SymSeq
preserves weak equivalences between cofibrant objects.
Proof.This follows from [19, Proposition 12.16].
Proposition 10.8. Let k be a commutative ring. Consider symmetric sequences
in Chk. If the map i : X !Yin Proposition 7.10 is a generating cofibration in
SymSeq, then jt is a monomorphism for each t 1.
Proof.After forgetting differentials, the map i : X !Yhas the form i : S !S q*
*,S0
and hence (forgetting differentials) Qtt1!Y ~thas a left inverse in SymSeq t *
*for
each t 1, which finishes the proof.
11.Right modules over an operad
In this section we briefly indicate some of the analogous results for RtO. Co*
*m
pared to LtOand AlgO, the corresponding arguments for RtO are substantially less
complicated, since colimits in RtO are calculated in the underlying category [1*
*9,
Section 8.4].
30 JOHN E. HARPER
11.1. Model category structures. First we establish appropriate model category
structures. Similar model structures are considered in [10].
Theorem 11.1. Let k be a commutative ring. Let O be an operad in Chk. Then
RtOhas a natural model category structure. The weak equivalences and fibrations
in this model structure are inherited in an appropriate sense from the homology
isomorphisms and the dimensionwise surjections in Chk.
Proof.Argue as in the proof of [19, Theorem 1.4]. To verify the appropriate RtO*
*ver
sion of [19, Proposition 13.4] (but without the cofibrancy assumption), use Pro*
*posi
tion 11.4 and note that pushouts in RtO are calculated in the underlying catego*
*ry.
Use the property that tensoring with a cofibrant chain complex preserves weak
equivalences, and note that the generating (acyclic) cofibrations have cofibran*
*t do
mains.
Theorem 11.2. Let O be an operad in Sp . Then RtO has a natural model cat
egory structure. The weak equivalences and fibrations in this model structure a*
*re
inherited in an appropriate sense from the stable weak equivalences and the sta*
*ble
flat fibrations in Sp .
Theorem 11.3. Let O be an operad in Sp . Then RtOhas a natural model category
structure. The weak equivalences and fibrations in this model structure are inh*
*erited
in an appropriate sense from the stable weak equivalences and the stable fibrat*
*ions
in Sp .
Proof.Argue as in the proof of [19, Theorem 1.4]. To verify the appropriate RtO*
*ver
sion of [19, Proposition 13.4] (but without the cofibrancy assumption), use Pro*
*po
sition 11.4 and note that pushouts in RtOare calculated in the underlying categ*
*ory.
Use the property that smashing with a cofibrant symmetric spectrum in the sta
ble flat model structure preserves weak equivalences, and note that the generat*
*ing
(acyclic) cofibrations have cofibrant domains.
Proposition 11.4. Let k be a commutative ring. Consider symmetric sequences
in Sp or Chk. Let B 2 SymSeq. If X !Y in SymSeq is a generating (acyclic)
cofibration, then X O B! Y O B is a monomorphism.
Proof.In both cases, this follows from Definition 3.4. In the case of chain co*
*m
plexes, first forget differentials. See the proof of [19, Theorem 12.2] for the*
* particular
form of the generating (acyclic) cofibrations.
11.2. Analogous results for right modules over operads.
Theorem 11.5. Let k be a commutative ring. If O is an operad in Sp or Chk
and X is a simplicial right Omodule, then there is a zigzag of weak equivalen*
*ces
RtO
hocolimXop'hocolim opX
natural in X, with the forgetful functor.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY 31
Proof.Argue as in the proof of Theorems 1.4 and 7.1, and Proposition 5.30, exce*
*pt
replace (5.31)with pushout diagrams of the form
[z] . X O O___//_Z0
 
 
fflffl fflffl
[z] . Y O O___//_Z1
in sRtO, with X !Y a generating cofibration in SymSeq, and note that pushouts
in sRtOare calculated in the underlying category sSymSeq.
Theorem 11.6. Let k be a commutative ring. Let O be an operad in Sp or Chk
and X 2 RtO. There is a zigzag of weak equivalences
RtO
hocolimBopar(X, O,'O)X
in RtO, natural in X.
Proof.Argue as in the proof of Theorem 4.7.
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
Email address: jharper1@nd.edu