Homotopy of Operads & Grothendieck-Teichmüller Groups
Book project in progress, by Benoit Fresse (Université Lille 1)
Chapter II.6
# Date of writing: 6/1/2014
# Latest significant revision: 9/6/2014
# Date of current version: 1/5/2015 (citations and cross-references
updated on 24/11/2014)
# Revisions on 9/6/2014:
* §6.2 removed and transferred to §II.5.3 (together with
some basic definitions of §6.1).
* Minor amendments in §§6.3-6.4 (§§6.4-6.5 in the former version) and typo corrections.
# Further revisions: notation changes on 28/7/2014,
minor style corrections on 1/5/2015
CHAPTER 6
Differential Graded Commutative Algebras and
Cosimplicial Algebras
This chapter is devoted to the study of commutative algebras in dg-modules, in
simplicial modules, and in cosimplicial modules. In the differential graded context,
we mainly consider the case of commutative algebras in the subcategories of chain
and cochain graded dg-modules studied in the previous chapter. For short, we also
use the name of a unitary commutative chain (respectively, cochain) dg-algebra
to refer to a unitary commutative algebra in chain (respectively, cochain) modules.
We similarly use the name of a unitary commutative simplicial (respectively, cosimplicial) algebra to refer to a unitary commutative algebra in simplicial (respectively,
cosimplicial) modules.
We make the definition of these categories of unitary commutative algebras
explicit in the first section of this chapter (§6.1). Then our main goal is to explain the definition of a model structure on the category of unitary commutative
cochain dg-algebras and on the category of cosimplicial unitary commutative algebras. We address this topic in the second section of the chapter (§6.2). The
category of unitary commutative chain dg-algebras and the category of simplicial
unitary commutative algebras also inherit a natural model structure, but we do not
use these model categories in this monograph, and we only give a few remarks on
this construction.
To complete our account, we prove that the Dold-Kan correspondence, studied in the previous chapter, can be used to define a Quillen equivalence between
the model category of cosimplicial unitary commutative algebras and the model
category of unitary commutative cochain dg-algebras.
In a first step, we explain the definition of a bar construction with coefficients
which represents a simplicial resolution of the objects obtained by a cell attachment
in the model category of commutative algebras (in both the cochain dg-module and
the cosimplicial module settings). We use this device to compare of the homotopy
type of cofibrant objects in the category of cosimplicial unitary commutative algebras and in the category of unitary commutative cochain dg-algebras. We devote
the third section of the chapter (§6.3) to this technical issue.
In a second step, we explain the definition of our Quillen equivalence itself.
We mostly rely on our study of the Eilenberg-Zilber equivalence of the previous
chapter. We will observe that the Dold-Kan functor, from cochain graded dgmodules to cosimplicial modules, preserves commutative algebra structures, but
not the conormalization functor. We therefore need to enhance our construction
in order get a functor from cosimplicial unitary commutative algebras to unitary
commutative cochain dg-algebras. We explain this process and the definition of our
Quillen equivalence in the fourth section of the chapter (§6.4).
157
158
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
6.1. Symmetric monoidal structures and commutative algebras
We checked in §5.3 that the category of chain (respectively, cochain) graded
dg-modules inherits a symmetrically monoidal structure. We therefore rely on the
approach of §I.2.0.1, where we explain the general definition of a unitary commutative algebra in a symmetric monoidal category, to define our category of unitary
commutative algebras in chain (respectively, cochain) graded dg-modules. The
main purpose of this section is to revisit this general approach in order to make
this definition more explicit.
We proceed similarly when we deal with unitary commutative algebras in simplicial (respectively, cosimplicial) modules. We just explained in §5.3 that we have
a natural symmetric monoidal structure on any category simplicial (respectively,
cosimplicial) objects in a base symmetric monoidal category M. We therefore also
examine the application of our definition of a unitary commutative simplicial (respectively, cosimplicial) algebra in this general setting. We will moreover observe
that a commutative algebra in the category of simplicial (respectively, cosimplicial)
objects in M is equivalent to a simplicial (respectively, cosimplicial) object in the
category of commutative algebras in M.
6.1.1. Unitary commutative dg-algebras. We first examine the definition of a
unitary commutative algebra in the category of chain (respectively, cochain) graded
dg-modules. We may also consider unitary commutative algebras in general dgmodules, but we will not use such structures.
From our general definition (see §I.2.0.1), we get that a unitary commutative
algebra in the category of cochain graded dg-modules consists of a cochain graded
dg-module A equipped with a unit morphism η : k → A, a product µ : A ⊗ A → A,
both formed in the category of dg-modules, and so that the unit, associativity
and commutativity relations in §I.2.0.1 hold in this category. If we go back to
the definition of the symmetric monoidal structure of dg-modules, then we readily
obtain that:
(1) the unit morphism of a unitary commutative cochain dg-algebra A is equivalent
to a degree 0 element 1 ∈ A0 such that δ(1) = 0;
(2) the product morphism µ : A ⊗ A → A maps any tensor product of homogeneous elements a, b ∈ A, to a homogeneous element µ(a, b) = ab with
deg(ab) = deg(a) + deg(b), and so that the differential δ : A → A satisfies the
derivation formula δ(ab) = δ(a)b + ±aδ(b), where the sign ± is determined by
the commutation of the differential δ with the factor a.
We moreover obtain that the unit, associativity and commutativity relations are
equivalent to the point-wise identities 1 · a = a = a · 1, (a · b) · c = a · (b · c) and
a · b = ±b · a, where a, b, c are (homogeneous) elements in A, and the sign ± in the
commutativity relation is yielded by the permutation of the elements a, b ∈ A.
We have a similar description for the structure of a unitary commutative chain
dg-algebra. We just assume that A is a chain graded dg-module in this case and we
consider lower gradings (rather than upper gradings) in the above definition. We
have in particular 1 ∈ A0 for the unit element.
6.1.2. Forgetful functors and unitary commutative dg-algebras. The underlying
graded module of a unitary commutative cochain dg-algebra A clearly forms a
unitary commutative cochain graded algebra (a unitary commutative algebra in the
symmetric monoidal category of cochain graded modules). Furthermore, defining a
unitary commutative dg-algebra A amounts to giving a unitary commutative graded
6.1. SYMMETRIC MONOIDAL STRUCTURES AND COMMUTATIVE ALGEBRAS
159
algebra A[ , together with a homomorphism δ : A → A, defining a differential on
A, and so that the derivation relation of 6.1.1(2) holds. The relation δ(1) = 0
in 6.1.1(1) is implied by this derivation relation. We have similar observations in
the chain graded context.
For short, we adopt the notation dg ∗ Com + (respectively, gr ∗ Com + ) for the
category of unitary commutative algebras in dg ∗ Mod (respectively, in gr ∗ Mod ),
and similarly in the chain graded context. We have a functor (−)[ : dg ∗ Com + →
gr ∗ Com + mapping a unitary commutative cochain dg-algebra A to the underlying
graded algebra A[ , and a functor in the converse direction gr ∗ Com + ,→ dg ∗ Com +
defined by identifying any graded module with a dg-module equipped with a trivial
differential. In the chain graded context, we have a functor (−)[ : dg ∗ Com + →
gr ∗ Com + and an embedding gr ∗ Com + ,→ dg ∗ Com + .
We checked in §5.3.6 that the cohomology functor H∗ (−) : dg ∗ Mod → gr ∗ Mod
is unit pointed and is equipped with a symmetric monoidal transformation in the
sense of §I.2.3.1. We deduce from this verification (see §2.0.4) that the cohomology functor on cochain graded dg-modules induces a functor H∗ (−) : dg ∗ Com + →
gr ∗ Com + from the category of unitary commutative cochain dg-algebras dg ∗ Com +
towards the category of unitary commutative cochain graded algebras gr ∗ Com + .
We similarly have a functor H∗ (−) : dg ∗ Com + → gr ∗ Com + induced by the homology functor on chain graded dg-modules in the chain graded context.
6.1.3. Unitary commutative simplicial and cosimplicial algebras. We again rely
on the general definition of a unitary commutative algebra in a symmetric monoidal
category §I.2.0.1 to define the notion of a unitary commutative algebra in the category of simplicial (respectively, cosimplicial) objects s M (respectively, c M) in a
base symmetric monoidal category M. For short, we also use the expression of a
unitary commutative simplicial (respectively, cosimplicial) algebra to refer to this
structure when the base symmetric monoidal category M is fixed by the context
(most usually, when M = Mod ).
From our general definition (see §I.2.0.1), we get that a unitary commutative algebra in cosimplicial modules consists of a cosimplicial module A ∈ c Mod ,
equipped with a unit morphism η : k → A, and a product µ : A ⊗ A → A, both
formed in c Mod , and so that the unit, associativity and commutativity relations
of §I.2.0.1 hold in this category. If we go back to our definition of the symmetric
monoidal structure on cosimplicial modules, then we readily obtain that:
(1) the unit morphism of a unitary commutative algebra in cosimplicial modules
is equivalent to a zero dimensional element 1 ∈ A0 such that d0 (1) = d1 (1);
we then have a unit element 1 ∈ An in each dimension of the cosimplicial
algebra A, which is uniquely determined by considering the image of this zero
dimensional unit element 1 ∈ A0 under a cosimplicial operator u∗ : A0 → An ,
for any choice of the map u ∈ Mor∆ (0, n) in the simplicial category ∆;
(2) the product morphism µ : A ⊗ A → A is defined dimension-wise by a collection
of morphisms µ : An ⊗An → An mapping a tensor product of elements a, b ∈ An
to an element µ(a, b) = ab of the same dimension n, and so that we have the
relation di (ab) = di (a)di (b) for any action of a coface di on A, as well as the
relation sj (ab) = sj (a)sj (b) for any codegeneracy action.
We also obtain that the unit, associativity and commutativity relations are equivalent to point-wise identities 1 · a = a = a · 1, (a · b) · c = a · (b · c) and a · b = b · a within
160
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
the module An . We deduce from this description that a cosimplicial unitary commutative algebra is equivalent to a collection of plain unitary commutative algebras
An , n ∈ N, forming a cosimplicial object in the category of unitary commutative
algebras in k-modules.
We have a similar description in the simplicial module case, and we also get that
a unitary commutative algebra in simplicial modules is equivalent to a simplicial
object in the category of unitary commutative algebras in k-modules.
This identity between the structure of a unitary commutative simplicial (respectively, cosimplicial) algebra and the structure of a simplicial (respectively, cosimplicial) object in the category of unitary commutative algebras actually holds for any
base symmetric monoidal category M. Under our notation conventions, we have
category identities (s M) Com + = s(M Com + ) and (c M) Com + = c(M Com + ). In
the module case M = Mod , we also use the short notation s Com + (respectively,
c Com + ) for the category of unitary commutative simplicial (respectively, cosimplicial) algebras.
6.1.4. Unitary commutative algebras and the Eilenberg-Zilber equivalence. We
proved in Theorem 5.3.3 that the normalized complex functor on the category of
simplicial modules N∗ : s Mod → dg ∗ Mod is unit pointed and comes equipped
with a symmetric monoidal transformation. We accordingly get that the normalized complex N∗ (A) ∈ dg ∗ Mod of a simplicial unitary commutative algebra
A ∈ s Com + inherits a natural unitary commutative structure, so that the mapping N∗ : A 7→ N∗ (A) induces a functor N∗ : s Com + → dg ∗ Com + from the category
of simplicial unitary commutative algebras s Com + towards the category of unitary commutative chain dg-algebras dg ∗ Com + . The inverse of the normalization
functor in the Dold-Kan equivalence Γ• : dg ∗ Mod → s Mod is not equipped with
a symmetric monoidal transformation however, and we can not provide the image of a unitary commutative chain dg-algebra under this functor with a natural
unitary commutative algebra structure in general. Nonetheless, we will see that
we can still define a commutative algebra enhancement of the Dold-Kan functor
Γ]• : dg ∗ Com + → s Com + by adjunction from the functor N∗ : s Com + → dg ∗ Com + ,
and we can use this process to give a commutative algebra upgrade of the DoldKan equivalence between simplicial modules and chain graded dg-modules. We will
explain this construction with full details in §6.4.
In the context of cochain graded dg-modules and cosimplicial modules, we symmetrically obtain that the image of a unitary commutative cochain dg-algebra A ∈
dg ∗ Com + under the dual Dold-Kan functor Γ• : dg ∗ Mod → c Mod inherits natural
a unitary commutative structure (we then use the result of Proposition 5.3.5), so
that the mapping Γ• : A 7→ Γ• (A) induces a functor Γ• : dg ∗ Com + → c Com + from
the category of unitary commutative cochain dg-algebras dg ∗ Com + towards the
category of cosimplicial unitary commutative algebras c Com + , while the conormalized complex functor N∗ : c Mod → dg ∗ Mod does not preserve commutative
algebra structures in general. We will again see, nonetheless, that this functor admits a commutative algebra enhancement N∗] : c Com + → dg ∗ Com + . We use this
construction to get a commutative algebra upgrade of the dual Dold-Kan equivalence between cosimplicial modules and cochain graded dg-modules. We will also
explain this construction with full details in §6.4.
6.1.5. Homology (and cohomology) of unitary commutative algebras. We already observed that the homology H∗ (A) of a unitary commutative chain dg-algebra
6.1. SYMMETRIC MONOIDAL STRUCTURES AND COMMUTATIVE ALGEBRAS
161
A ∈ dg ∗ Com + inherits a unitary commutative algebra structure because the homology functor H∗ (−) : dg ∗ Mod → gr ∗ Mod is unit-pointed and is equipped with
a symmetric monoidal transformation, and similarly in the case of the cohomology
H∗ (A) of a unitary commutative cochain dg-algebra A ∈ dg ∗ Com + .
We checked in §5.3.6 that the prolongation of the homology functor to simplicial modules H∗ (−) : s Mod → gr ∗ Mod is also unit-pointed and equipped with
a symmetric monoidal transformation, just because we define this functor by the
composite of the homology functor on chain graded dg-modules H∗ (−) : dg ∗ Mod →
gr ∗ Mod with the normalized complex functor N∗ : s Mod → dg ∗ Mod which is unitpointed and equipped with a symmetric monoidal transformation too. We accordingly get that the homology H∗ (A) = H∗ N∗ (A) of a simplicial unitary commutative
algebra A ∈ s Com + inherits a unitary commutative algebra structure too, so that
the mapping H∗ : A 7→ H∗ N∗ (A) induces a functor H∗ (−) : s Com + → gr ∗ Com ∗
from the category of simplicial unitary commutative algebras s Com + towards the
category of unitary commutative chain graded algebras gr ∗ Com + .
We have a similar result for the prolongation of the cohomology functor of
cochain graded gd-modules to cosimplicial modules H∗ (−) = H∗ N∗ (−) : c Mod →
gr ∗ Mod , though the conormalized functor occurring in this construction is not
equipped with a symmetric monoidal transformation, because the dual EilenbergMacLane map ∆ : N∗ (K ⊗ L) → N∗ (K) ⊗ N∗ (L), which is a symmetric comonoidal
transformation, still gives a natural isomorphism at the cohomology level ∆ :
'
H∗ (N∗ (K ⊗L)) −
→ H∗ (N∗ (K)⊗N∗ (L)), for all K, L ∈ c Mod , and we can compose the
inverse of this isomorphism with the Künneth map µ : H∗ (N∗ (K)) ⊗ H∗ (N∗ (L)) →
H∗ (N∗ (K)⊗N∗ (L)) to get the required symmetric monoidal transformation on the cohomology of cosimplicial modules (see §5.3.6). We accordingly get that the cohomology H∗ (A) = H∗ N∗ (A) of a cosimplicial unitary commutative algebra A ∈ c Com +
also inherits a unitary commutative algebra structure, so that the cohomology functor H∗ : A 7→ H∗ N∗ (A) still gives a functor H∗ (−) : c Com + → gr ∗ Com ∗ from the
category of cosimplicial unitary commutative algebras c Com + towards the category
of unitary commutative cochain graded algebras gr ∗ Com + .
6.1.6. The free commutative algebra adjunction. In our constructions, we use
the existence of a left adjoint of the forgetful functor ω : M Com + → M. This
left adjoint is given by the symmetric algebra functor S : M 7→ S(M ), already
considered in §I.1.3 (in the non-unitary context), as an instance of a free algebra
functor associated to an operad (see §I.1.3.5), and in §I.7.2 (for the study of Hopf
algebras).
For an object M in a symmetric monoidal category M, we define the symmetric
algebra S(M ) as the object of M such that:
S(M ) =
∞
M
(M ⊗r )Σr ,
r=0
where we form the tensor products in M, we define the action of Σr on the tensor
power M ⊗r by using the symmetric structure of this tensor product in M, and we
define the coinvariants (−)Σr (as well as the coproduct) as a particular instance
of a colimit construction. The product of S(M ) is induced by the concatenation
operation M ⊗p ⊗ M ⊗q → M ⊗p+q at the level of tensor products (see §I.7.2.4).
The adjunction augmentation λ : S(A) → A, associated to any unitary commutative algebra A ∈ M Com + , is defined on (A⊗r )Σr ⊂ S(A) by the morphism
162
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
induced by the r-fold product of the algebra A. The adjunction unit ι : M → S(M ),
associated to any M ∈ M, is given by the identity of the object M with a summand
of order r = 1 of the symmetric algebra S(M ). The proof of the adjunction relation
MorM Com + (S(M ), A) = MorM (M, A)
follows from straightforward verifications, by using these explicit definitions of the
adjunction unit and adjunction augmentation.
6.1.7. The commutative algebra adjunction in the differential graded and simplicial contexts. In the module context, and more generally, as soon as we deal with
a category made of point-sets so that the tensor product is identified with an object
spanned by some point-wise tensors, we use the notation x1 · · · xr for the element of
the symmetric algebra represented by the tensor x1 ⊗ · · · ⊗ xr ∈ M ⊗r . This tensor
actually represents the product of the elements x1 , . . . , xr ∈ M in the symmetric
algebra S(M ). We then use the canonical embedding ι : M → S(M ) to identify
any element of the object M with an element in S(M ).
In the dg-module context, the coinvariant relation of the symmetric algebra
reads xs(1) · · · xs(r) ≡ ±x1 · · · xr , where ± is the sign produced by the permutation
of the homogeneous elements x1 , . . . , xr ∈ M involved in this relation. In the
simplicial context, we can identify the symmetric algebra S(M ), formed in the
category of simplicial objects s M, with the result of a dimension-wise application
of the symmetric algebra functor to the components of M ∈ s M, and we have a
similar statement in the cosimplicial context.
6.1.8. The Eilenberg-Zilber equivalence and the Künneth formula for symmetric algebras. We observed in §6.1.4 that the normalized complex functor preserves
unitary commutative algebra structures. In the case of a symmetric algebra, we deduce from this statement that we have a morphism of unitary commutative chain
dg-algebras ∇ : S(N∗ (K)) → N∗ (S(K)), for any simplicial module K ∈ s Mod ,
which is given, at the level of the object N∗ (K) ⊂ S(N∗ (K)), by the image of the
embedding ι : K → S(K) under the normalization functor N∗ (−). We easily see
that this morphism ∇ : S(N∗ (K)) → N∗ (S(K)) is given by a term-wise application of
the Eilenberg-MacLane
map of Theorem 5.3.3 on the summands of the symmetric
L
algebra S(M ) = r≥0 (M ⊗r )Σr . We have a similarly defined morphism of cosimplicial unitary commutative algebras ∇ : S(Γ• (C)) → Γ• (S(C)), for any cochain
graded dg-module C ∈ dg ∗ Mod , which is given by a term-wise application of the
transformation of Proposition 5.3.5(b) on the summands of the symmetric algebra.
We use these observations in our definitions of the commutative algebra upgrade of
the Dold-Kan and dual Dold-Kan correspondences in §6.4.
In the characteristic zero setting, we obtain that the Eilenberg-MacLane equivalence defines a weak-equivalence at the symmetric algebra level:
∼
∇ : S(N∗ (K)) −
→ N∗ (S(K)),
for any simplicial module K ∈ s Mod . Indeed, the quotient map M ⊗r → (M ⊗r )Σr
admitsP
a natural section, defined by the symmetrization map e(x1 · . . . · xr ) =
(1/r!)· σ∈Σr xσ(1) ·. . .·xσ(r) , for any r ∈ N, so that the symmetric algebra S(M ) =
L
L
⊗r
)Σr forms a retract of the tensor algebra functor T(M ) = r≥0 M ⊗r ,
r≥0 (M
and this retraction is preserved by the Eilenberg-MacLane map. We have a similar
result for the natural transformation on the symmetric algebra of a cosimplicial
module Γ• (C) ∈ c Mod , for any C ∈ dg ∗ Mod .
6.1. SYMMETRIC MONOIDAL STRUCTURES AND COMMUTATIVE ALGEBRAS
163
If the ground ring is a field (still of characteristic zero), then we moreover
have a Künneth formula H∗ (S(C)) = S(H∗ (C)) for the symmetric algebra of any
chain graded dg-module C ∈ dg ∗ Mod , and similarly H∗ (S(C)) = S(H∗ (K)) in the
case of a cochain graded dg-module C ∈ dg ∗ Mod . We also have the Künneth
formula H∗ (S(K)) = S(H∗ (K)) for the symmetric algebra of a simplicial module
K ∈ s Mod , and H∗ (S(K)) = S(H∗ (K)) for the symmetric algebra of a cosimplicial
module K ∈ c Mod . We just adapt the above argument to deduce these relations
from a term-wise L
application of the standard Künneth formula on the tensor algebra
functor T(M ) = r≥0 M ⊗r .
6.1.9. Limits and colimits. We observed in Proposition I.1.3.6 that the category
of algebras over any operad inherits limits as soon as limits exist in the base category
M, and colimits as soon as colimits exist in the base category and are preserved
by the tensor product operation (see §I.0.9). We can apply this statement to the
unitary commutative operad case (see Proposition I.2.1.1-2.1.2) to get the definition
of limits and colimits in the category of unitary commutative algebras M Com + .
We more precisely obtain, according to the result of Proposition I.1.3.6, that the
forgetful functor ω : M Com + → M creates limits, the filtered colimits, and the
coequalizers which are reflexive in the base category.
In the case M = dg ∗ Mod , we readily see that the functor (−)[ : dg ∗ Com + →
∗
gr Com + creates (limits and) colimits. In the particular case M = c Mod , we
obtain that the (limits and) colimits of the category c Com + = c(Mod Com + ) are
created dimension-wise in the category of unitary commutative algebras in plain
k-modules Com + = Mod Com + . We will use these observations in our definition of
a model structure on unitary commutative algebras.
6.1.10. Coproducts. We also give a general construction of coproducts for algebras over operads in the proof of Proposition I.1.3.6, but we observed in §I.2.0.2 that,
in the unitary commutative algebra case, the coproduct of objects A, B ∈ M Com +
is identified with the tensor product A ⊗ B ∈ M, which inherits a natural unitary
commutative algebra structure factor-wise from A and B. Thus, under our conventions for the notation of coproducts, we have an identity ∨ = ⊗. In what follows,
we use the tensor product notation ⊗, rather than our general notation ∨ for this
particular instance of a coproduct construction.
j
i
Recall that the morphisms A →
− A⊗B ←
− B, which define the universal morphisms associated with the coproduct construction, are given by the tensor products
i = id ⊗ηB (respectively, j = ηA ⊗ id ), where ηA (respectively, ηB ) refers to unit
morphism of the algebra A (respectively, B).
6.1.11. Pushouts. We also easily see that the outcome of a pushout construction
(1)
R
h
/A
g
/B
f
φ
S
is identified with the relative tensor product B = S ⊗R A, defined by the reflexive
coequalizer
(2)
y
S⊗R⊗A
s0
d0
d1
// S ⊗ A
/ S ⊗R A
164
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
such that:
– the morphism d0 is given by the right action of the algebra R on S through
the morphism φ : R → S (we explicitly have d0 = (µR ⊗ id ) · (id ⊗φ ⊗ id ),
where µS refers to the product on S);
– the morphism d1 is symmetrically given by the left action of the algebra R
on A through the morphism h : R → S (we explicitly have d1 = (id ⊗µA ) ·
(id ⊗h ⊗ id ), where µA refers to the product on A);
– the morphism s0 is given by the insertion of the unit morphism ηR : 1 → R
in the tensor product S ⊗ A (we explicitly take s0 = id ⊗ηR ⊗ id ).
g
f
The universal morphisms S −
→ S ⊗R A ←
− A, in this construction of the pushout,
are given by the composite of the universal morphisms of the coproduct (such as
defined in §6.1.10) with the coequalizer map S ⊗ A → S ⊗R A.
6.2. The model category of commutative algebras
We now check that the category of commutative algebras in cochain graded dgmodules (respectively, cosimplicial modules) inherits a model category structure.
We follow the process of §4.3, where we explain the definition of model categories by
adjunction. We first give the definition of the model structure in the general setting
of commutative algebras in a symmetric monoidal category M. We just assume that
this base category M is equipped with a cofibrantly generated model structure in
order to give a sense to our definition. We check the validity of this definition in
the cochain graded dg-module case M = dg ∗ Mod and in the cosimplicial module
case M = c Mod afterwards.
6.2.1. The definition of the model structure. In the approach of §4.3.2, we define
our model structure on the category of unitary commutative algebras M Com + , by
assuming that the forgetful functor ω : M Com + → M creates weak-equivalences
and fibrations. Thus, we take:
(1) the morphisms of unitary commutative algebras f : A → B which are weakequivalences (respectively, fibrations) in the base category M as class of weakequivalences (respectively, fibrations) in M Com + ;
(2) and the morphisms which have the left lifting property with respect to the class
of acyclic fibrations given by the above definition (1) as class of cofibrations.
Besides:
(3) we take the morphisms of symmetric algebras S(i) : S(K) → S(L), where
i : K → L runs over the generating cofibrations (respectively, the generating
acyclic cofibrations) of the base category M, as set of generating cofibrations
(respectively, generating acyclic cofibrations) in the category of unitary commutative algebras M Com + .
We know from the general observations of §4.3 that the right lifting property
with respect to these morphisms (3) detects the acyclic fibrations (respectively, fibrations) in the category of unitary commutative algebras. We need more, however,
in order to ensure that our definition returns a valid model structure. We formulate
general conditions, ensuring that such a claim holds, in Theorem 4.3.3. We check
that the assumptions of this statement are fulfilled in the case of unitary commutative algebras in cochain graded dg-modules (at least, when the ground ring is
a field of characteristic zero) and in the case of unitary commutative algebras in
cosimplicial modules (at least, when the ground ring is a field).
6.2. THE MODEL CATEGORY OF COMMUTATIVE ALGEBRAS
165
We already reviewed the definition of colimits and limits in the category of
unitary commutative algebras in §6.1.9. We notably checked that, according to
the result of Proposition I.1.3.6, the forgetful functor ω : M Com + → M creates
limits, the filtered colimits, and the coequalizers of unitary commutative algebras
which are reflexive in the base category M (as soon as the tensor product in M
distributes over colimits). We get, according to this general statement, that the
sequential colimit requirement of Theorem 4.3.3(a) is always fulfilled in the category
of unitary commutative algebras, for any base model category M.
The crux of the definition of our model structure therefore lies in the verification
of the pushout condition of Theorem 4.3.3(b). We precisely check that this second
requirement holds when we work in a category of cochain graded dg-modules over a
field of characteristic zero, and when we work in a category of cosimplicial modules
over a field. For this aim, we use that the pushouts are identified with relative
tensor products in the category of unitary commutative algebras (see §6.1.11). We
also use that the functor (−)[ : dg ∗ Com → gr ∗ Com, which forgets about differentials, creates (limits and) colimits in the cochain dg-algebra case, while the (limits
and) colimits of cosimplicial algebras are created dimension-wise in the category of
unitary commutative algebras in plain k-modules (see §6.1.9).
We examine the case of unitary commutative algebras in cochain graded dgmodules first. We use a commutative algebra version of the twisted dg-modules
of §5.1.8 in order to give a description of pushouts along a generating (acyclic)
cofibration of unitary commutative algebras from the relative tensor product construction of §6.1.11. We explain this notion first.
6.2.2. Twisting derivations of commutative algebras. In §5.1.8, we explain the
definition of a dg-module (K, ∂) by the addition of a twisting homomorphism ∂ :
K → K to the internal differential of a given dg-module K. Recall that a twisting
homomorphism ∂ : K → K is a homomorphism of degree 1 (in the cochain graded
case) such that δ∂ + ∂δ + ∂∂ = 0.
In the case of a unitary commutative cochain graded dg-algebra K = A, this
dg-module construction produces a unitary commutative dg-algebra (A, ∂) provided
that the twisted differential δ + ∂ : A → A fulfils the requirements of §6.1.1 for the
differential of dg-algebras. To be explicit, we have to ensure that δ + ∂ defines
a derivation with respect to the product of A. This is the case of the internal
differential δ since we assume that A forms a unitary commutative dg-algebra.
The verification of the derivation equation for δ + ∂ then clearly reduces to the
verification of this relation for the twisting homomorphism ∂. Thus, if we start
with a unitary commutative cochain graded dg-algebra A, then the twisting process
returns a unitary commutative cochain graded dg-algebra (A, ∂) as soon as the
twisting homomorphism ∂ : A → A satisfies the derivation relation of §6.1.1(2)
with respect to the product:
∂(a · b) = ∂(a) · b + ±a · ∂(b),
In this situation, we also say that ∂ defines a twisting derivation on A.
From the form of the generating cofibrations in cochain graded dg-modules
im : Bm → Em , m > 0, we obtain that:
166
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
Proposition 6.2.3. The cell attachments of generating cofibrations in the category of unitary commutative cochain dg-algebras
N
h /
A
α S(Bmα )
S(imα )∗
N α S(Emα )
f
g
/B
are equivalent to twisted objects
B = (S(Ξ) ⊗ A, ∂)
such that:
L
– the generating module Ξ is a free graded module Ξ = α k ξ mα −1 (equipped
with a trivial internal differential),
– the twisting derivation ∂ vanishes on A, and satisfies ∂(ξ mα −1 ) ∈ A, for
any generating element ξ mα −1 ∈ Ξ, where we regard A as a sub-object of
the tensor product S(Ξ) ⊗ A by using the identity between tensor products
and coproducts in the category of unitary commutative algebras.
The morphism f : A → B, in this representation, is given by the obvious inclusion
of the factor A in the twisted tensor product B = (S(Ξ) ⊗ A, ∂).
Explanations. Recall that each Emα is freely generated by homogeneous elements emα −1 , bmα ∈ Emα , and is equipped with the differential such that δ(emα −1 ) =
bmα . The dg-module Bmα is the submodule of Emα spanned by the element bmα
and is equipped with a trivial differential.
We precisely define the graded module Ξ of the proposition as the free graded
module spanned by the elements ξ mα −1 associated with the generating cofibrations of our attachment. We also take ∂(ξ mα −1 ) = h(bmα ) ∈ A, where we considerNthe image of a generating element bmα in Bmα under the attaching map
h : α S(Bmα ) → A, to define the image of each ξ mα −1 ∈ Ξ under our twisting
derivation ∂.
The expression of the derivation relation implies that the twisting derivation
on S(Ξ) ⊗ A is determined by giving this image ∂(ξ mα −1 ) ∈ A, for each generating
element ξ mα −1 , and the requirement that ∂ vanishes on A. To be explicit, by
identifying any tensor ξ mα1 −1 · · · ξ mαr −1 ⊗ a ∈ S(Ξ) ⊗ A with the product of the
elements ξ mα1 −1 , . . . , ξ mαr −1 ∈ Ξ and a ∈ A, we obtain the formula:
P
∂(ξ mα1 −1 · · · ξ mαr −1 ⊗ a) = i ±ξ mα1 −1 · · · ∂(ξ mαi −1 ) · · · ξ mαr −1 ⊗ a,
P
mαi −1
= i ±ξ mα1 −1 · · · ξ\
· · · ξ mαr −1 ⊗ (h(bmαi ) · a).
This mapping is a derivation by construction, and we readily check that the equation of twisting homomorphisms δ∂ + ∂δ + ∂ 2 = 0 holds too, by using that the
commutation of h with differentials is equivalent to the relation δ(h(bmα )) = 0, for
each bmα .
We determine the morphism g by g(emα −1 ) = ξ mα −1 ∈ S(Ξ) and g(bmα ) =
mα
h(b ) ∈ A, for each pair of generating elements emα −1 , bmα ∈ Emα . We immediately check that this morphism commutes with differential and fits a commutative square of the form of the proposition. We use that the functor (−)[ :
dg ∗ Com + → gr ∗ Com + which forgets about differentials creates colimits and the
6.2. THE MODEL CATEGORY OF COMMUTATIVE ALGEBRAS
167
identity S(Emα )[ = S(Ξ) ⊗ S(Bmα )[ , to conclude that this commutative square
defines a pushout, and this verification completes the proof of our proposition. Recall that the generating acyclic cofibrations of cochain graded dg-modules
are the zero maps jm : 0 → Em , m > 0, where we consider the same acyclic cochain
graded dg-modules Em as in the definition of generating cofibrations. The associated generating acyclic cofibrations of unitary commutative cochain dg-algebras
are identified with the initial morphisms η : k → S(Em ) of the symmetric algebras
S(Em ). From the form of these morphisms, we obtain:
Proposition 6.2.4. The cell attachments of generating acyclic cofibrations in
the category of unitary commutative cochain dg-algebras
k
N
S(E
mα )
α
/A
f
/B
are equivalent to tensor products of the form B = S(⊕α Emα ) ⊗ A.
Proof. This claim is an immediate consequence of the definition of coproducts
in the category of unitary commutative algebras.
We then check the following statement:
Lemma 6.2.5.
(a) If we assume that the ground ring is a field, then any morphism f : A → B
obtained by a cell attachment of generating (acyclic) cofibrations of unitary commutative cochain dg-algebras in Proposition 6.2.3-6.2.4) is an injection, and hence,
defines a cofibration in the category of cochain graded dg-modules.
(b) If we assume further that the ground ring is a field of characteristic zero,
then we have H∗ (S(⊕α Emα ) ⊗ A) = H∗ (A) so that the morphisms f : A → B obtained by a cell attachment of generating acyclic cofibrations of unitary commutative
cochain dg-algebras are also weak-equivalences.
Proof. If we forget about differentials, then we can identify the morphism
f : A → B with a factor inclusion A[ → S(Ξ)⊗A[ both in the generating cofibration,
and in the generating acyclic cofibration case. We immediately obtain that such a
morphism is injective as soon as the ground ring is a field.
We prove that H∗ (S(Em )) = k for any dg-module Em . We have an obvious
cochain homotopy h on Em defined by h(bm ) = em−1 for the generating element
bm . We consider a derivation ∂h : S(EmP
) → S(Em ) extending this cochain homotopy, and defined by ∂h (x1 · · · xr ) =
i ±x1 · · · h(xi ) · · · xr for any monomial
x1 · · · xr ∈ S(Em ). We readily check that this derivation satisfies the relation
(∂h δ + δ∂h )(x1 · · · xr ) = r · id (x1 · · · xr ), from which we conclude that the cohomology module H∗ (S(Em )) reduces to the ground ring k when each positive integer
rN> 0 is invertible in k. We then use the Künneth formula H∗ (S(⊕α Emα ) ⊗ A) =
∗
∗
α H (S(Emα )) ⊗ H (A) to obtain the relation asserted in the lemma, and our
conclusion follows.
From the result of Lemma 6.2.5, we conclude that all the assumptions of Theorem 4.3.3 (the definition of adjoint model structures) are fulfilled for the category
168
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
of unitary commutative algebras in cochain graded dg-modules M = dg ∗ Mod , at
least when the ground ring is a field of characteristic zero. Thus:
Theorem 6.2.6. If we take a field of characteristic zero as ground ring, then
the definition of §6.2.1 provides a valid model structure on the category of unitary commutative cochain dg-algebras M Com + = dg ∗ Com + such that the forgetful functor ω : dg ∗ Com + → dg ∗ Mod creates weak-equivalences, creates fibrations, and preserves cofibrations. This model structure is also cofibrantly generated by construction (see §6.2.1), with the morphisms of symmetric algebras
S(i) : S(K) → S(L), where i runs over the generating (acyclic) cofibrations of
the model category of cochain graded dg-modules (see §5.1.2), as set of generating
(acyclic) cofibrations.
In fact, we have a valid model structure on the category of unitary commutative
cochain dg-algebras as soon as we work over a ring k such that Q ⊂ k, but the
forgetful functor towards the base category of cochain graded dg-modules does not
preserve cofibrations in general.
From our theorem, we obtain that the cofibrant objects of the category of unitary commutative cochain dg-algebras are identified with retracts of cell complexes
of generating cofibrations. We give a description of these cell complexes to complete our results. For this aim, we elaborate on the observation of Proposition 6.2.5,
where we give a description of the outcome of a cell attachment in the category
of unitary commutative cochain dg-algebras. For simplicity, we focus on cell complexes formed by a countable sequence of cell attachments in what follows. This
case is sufficient for our purpose, because the adjunction process implies that the
domains of our generating cofibrations of unitary commutative cochain dg-algebras
are countably small as soon as this is so in the category of cochain graded dgmodules (check the arguments of Theorem 4.3.3), and the small object argument
returns relative cell complexes formed by a countable sequence of cell attachments
in this case (see our remarks on Proposition 4.1.2 and Proposition 4.2.1).
6.2.7. Quasi-free objects and cell complexes of unitary commutative cochain
dg-algebras. First, we readily see that a cell complex of generating cofibrations of
unitary commutative cochain dg-algebras, let R, forms a symmetric algebra (a free
object) R[ = S(C) when we forget about the differential. In this situation, we also
say that R is quasi-free as a unitary commutative cochain dg-algebra.
In the case of a cell complex of generating cofibrations, the generating graded
module C of our cochain dg-algebra R is equipped with a filtration
(1)
0 = F0 C ⊂ · · · ⊂ Fs−1 C ⊂ Fs C ⊂ · · · ⊂ colim Fs C = C
s
s
such that S(F C) inherits a twisting differential ∂ : S(Fs C) → S(Fs C), which
extends the twisting differential defined on the previous filtration layer S(Fs−1 C) ⊂
S(Fs C), and so that we have the relation
(2)
∂(Fs C) ⊂ S(Fs−1 C),
for any s > 0. We then have an identity R = colims (S(Fs C), ∂). The cochain
dg-algebras (S(Fs C), ∂) ⊂ R define the layers of our cell complex. The filtration
condition (2) automatically implies that each (S(Fs C), ∂) forms a quasi-free extension (of the form considered in Proposition 6.2.3) of the unitary commutative
cochain dg-algebra (S(Fs−1 C), ∂) defining the previous layer of our filtration. We
6.2. THE MODEL CATEGORY OF COMMUTATIVE ALGEBRAS
169
therefore have an equivalence between such quasi-free object structures and the
cell complexes of generating cofibrations in the category of unitary commutative
cochain dg-algebras.
In our construction, the twisting differential ∂ associated to each symmetric algebra S(Fs C) represents the restriction of the differential of the cochain dg-algebra
R to this subobject S(Fs C) ⊂ R[ . If we forget about the filtration, then we have
an identity R = (S(C), ∂).
6.2.8. Outlook: The example of Chevalley-Eilenberg complexes. In general, when
we have a quasi-free unitary commutative cochain dg-algebra such that R[ = S(C),
we can not ensure that the generating graded module C inherits a filtration such
that the differential of R maps Fs C ⊂ R[ into S(Fs−1 C) ⊂ R[ . For instance, we
will observe in §?? that the Chevalley-Eilenberg complex of a (finite dimensional)
Lie algebra g is equivalent to a quasi-free dg-algebra R = (S(Σ−1 g∨ ), ∂) where the
generating graded module C = Σ−1 g∨ is the dual of the Lie algebra g put in degree
1. In this case, requiring the existence of a filtration Fs C = Σ−1 Fs (g∨ ) such that
condition (2) holds for each s > 0 amounts to assuming that g is nilpotent as a Lie
algebra (see §??).
The Chevalley-Eilenberg complex has a generalization for L∞ -algebras, which
are homotopy versions of Lie algebra structures equivalent to algebras over a cofibrant resolution of the Lie operad Lie in the category of operads in chain graded
dg-modules. The quasi-free objects R[ = S(C) satisfying R0 = k ⇔ C 0 = 0
and where the generating graded module C is degree-wise finite dimensional are
equivalent to the Chevalley-Eilenberg complex of L∞ -algebras g such that C n+1 =
n+1 ∨
∨
) , for each n ≥ 0. In this setting, we can interpret
Σ−1 (g∨
n ) ⇔ gn = (Σ C)n = (C
our filtration condition (2) as the definition of a generalized nilpotence requirement
for our L∞ -algebra g.
We go back to the study of the Chevalley-Eilenberg complex in §??, where we
explain the definition of models for the rational homotopy of classifying spaces in
the category of unitary cochain dg-algebras.
We refer to [190] for the interpretation of quasi-free structures in terms of L∞ algebras (homotopy Lie algebras), and to [133, 134] for the introduction and the
first studies of L∞ -algebra structures in the literature. The operadic interpretation of L∞ -algebras arises from Ginzburg-Kapranov work [93] (see also [150] for a
comprehensive survey of this subject).
6.2.9. Outlook: Minimal models. The structure of quasi-free unitary commutative cochain dg-algebras is studied in Sullivan’s article [209].
The minimal models, introduced in this reference (see also [36, 69]), are quasifree unitary commutative cochain dg-algebras R = (S(C), ∂), equipped with a
filtration as in §6.2.7, and of which twisting differential satisfies
(∗)
∂(C) ⊂ S(C) · S(C),
where S(C) denotes the L
augmentation ideal of the symmetric algebra S(C). We
⊗r
explicitly have S(C) =
)Σr , and the
r≥1 Sr (C), where we set Sr (C) = (C
module S(C) · S(C) ⊂ S(C) in our relation ∂(C)L
⊂ S(C) · S(C) is identified with
the submodule of the symmetric algebra S≥2 = r≥2 Sr (C) ⊂ S(C) spanned by
monomials x1 · . . . · xr such that r ≥ 2. We also assume that the twisting differential
of a minimal model satisfies the filtration condition of §6.2.7, and we moreover follow
the convention of §6.2.7 to assume that C is a graded module equipped a trivial
170
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
internal differential, so that our minimal models are equivalent to cell complexes of
a particular form.
The minimal model of a unitary commutative cochain dg-algebra A precisely
∼
consists of a cofibrant resolution of this form R −
→ A, where we have a quasifree commutative algebra R = (S(C), ∂) equivalent to a cofibrant cell complex in
the category of unitary commutative cochain dg-algebras and of which twisting
differential satisfies the decomposition condition (∗) in addition to the filtration
condition §6.2.7(2). Every unitary commutative cochain dg-algebra A such that
H0 (A) = Q has a minimal model which is also unique up to isomorphism (see [36,
§7] and [209] for a detailed proof of this statement).
We now prove the validity of our model structure §6.2.1 for the unitary commutative algebras in cosimplicial modules. We again check that the pushout condition of Theorem 4.3.3 holds in this setting. We first examine the definition of
the pushouts along generating cofibrations and generating acyclic cofibrations of
unitary commutative algebras. We use that the colimits of cosimplicial unitary
commutative algebras are created dimension-wise in the category of of unitary
commutative algebras in plain k-modules. We get the following statements:
Proposition 6.2.10. The cell attachments of generating cofibrations in the
category of cosimplicial unitary commutative algebras
N
α
S(Γ• (Bmα ))
S(imα )∗
N
α
h
/A
g
/B
f
S(Γ• (Emα ))
are given, in each dimension n ∈ N, by a tensor product of unitary commutative
algebras
B n = S(Ξn ) ⊗ An
L
L
such that Ξn is a supplementary module of K n = α Γn (Bmα ) in Ln = α Γn (Emα ).
The morphism f : A → B is given dimension-wise by the obvious inclusion of
the factor An in this tensor product. The modules Ξn ⊂ B n , n ∈ N, are preserved
by the action of codegeneracies sj and by the action of the cofaces di such that
i > 0.
L
L
Explanations. Let K = α Γ• (Bmα ) and L = α Γ• (Emα ). From the construction of the functor Γ• in §5.2.6, we immediately get an identity Ln = Ξn ⊕ K n
at theL
level of our cosimplicial modules K, L ∈ c Mod , where Ξn is the k-module
n
Ξ = α Ξnα such that:
M
Ξnα =
di1 · · · dik (k ξ mα −1 ),
n≥i1 >···>ik ≥1
n−k=mα −1
for each dimension
n ∈ N.
N
We have α S(Γ• (Bmα )) = S(K), S(Γ• (Emα )) = S(L), B = S(L) ⊗S(K) A, and
from the decomposition Ln = Ξn ⊕ K n ⇒ S(Ln ) = S(Ξn ) ⊗ S(K n ), we readily
obtain the identity:
B n = S(Ξn ⊕ K n ) ⊗S(K n ) An = S(Ξn ) ⊗ An ,
6.2. THE MODEL CATEGORY OF COMMUTATIVE ALGEBRAS
171
for each n ∈ N, which is the claim of the proposition. The identity between the
morphism f : An → B n and the inclusion of the factor An in this tensor product
is immediate.
From the construction of the functor Γ• in §5.2.6, we also obtain that the
codegeneracies sj and the cofaces di such that i > 0 are induced
L termwise by a
mapping between the summands of the objects Ξnα , n ∈ N, in L = α Γ• (Emα ). Proposition 6.2.11. The cell attachements of generating acyclic cofibrations
in the category of cosimplicial unitary commutative algebras
k
N
α
S(Γ• (Emα ))
/A
f
/B
are equivalent to tensor products of the form B = S(⊕α Γ• (Emα )) ⊗ A.
Proof. This assertion is, like the claim of Proposition 6.2.4, an immediate
consequence of the definition of coproducts in the category of unitary commutative
algebras.
We use the result of these propositions to establish the following lemma:
Lemma 6.2.12.
(a) If we assume that the ground ring is a field, then any morphism f : A → B
obtained by a cell attachment of generating (acyclic) cofibrations of cosimplicial
unitary commutative algebras in Proposition 6.2.10-6.2.11) is an injection, and
hence, defines a cofibration in the category of cosimplicial modules.
(b) In the acyclic cofibration case, we moreover have H∗ (S(⊕α Γ• (Emα ))⊗A) =
∗
H (A), so that the morphisms f : A → B obtained by a cell attachment of generating acyclic cofibrations of cosimplicial unitary commutative algebras are also
weak-equivalences.
Proof. If we assume that the ground ring is a field, then the identity between
the morphism f : A → B in Proposition 6.2.10 (respectively, Proposition 6.2.11)
and a dimension-wise inclusion of a tensor product factor immediately implies that
such a morphism is injective and hence, defines a cofibration in the category of
cosimplicial modules.
The dg-module Em , regarded as a coaugmented object over the trivial module 0, admits obvious contracting chain-homotopies which, by Proposition 5.5.5,
gives rise to contracting extra codegeneracies
L at •the level of the cosimplicial module Γ• (EN
m ) associated to Em . Let L =
α Γ (Emα ). The symmetric algebra
S(L) = α S(Γ• (Emα )), regarded as a coaugmented object over k = S(0), inherits contracting extra codegeneracies (by functoriality). The normalized complex of
this symmetric algebra N∗ (S(L)) then inherits a coaugmentation over k and a contracting chain-homotopy (as a cochain graded dg-module). By the Eilenberg-Zilber
equivalence (see Theorem 5.3.4), this is enough to conclude that our morphism
172
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
f : A → B, which fits in a diagram
N∗ (A)
'
k ⊗ N∗ (A)
N∗ (f )
/ N∗ (S(L) ⊗ A)
O
,
∼
/ N∗ (S(L)) ⊗ N∗ (A)
induces an iso in cohomology, and hence, defines a weak-equivalence.
Note that we do not need any characteristic assumption in this statement. If
the ground ring is not a field, then we still obtain that the morphism f : A → B
obtained by a cell attachment of generating acyclic cofibrations in Proposition 6.2.11
is a weak-equivalence. However, this morphism is not a cofibration in general.
From the result of this lemma, we conclude (as in the cochain dg-algebra case)
that all the assumptions of Theorem 4.3.3 (the definition of adjoint model structures) are fulfilled for the category of unitary commutative algebras in cosimplicial
modules M = c Mod , at least when the ground ring is a field (of any characteristic).
Thus:
Theorem 6.2.13. If we take a field as ground ring, then the definition of §6.2.1
provides a valid model structure on the category of cosimplicial unitary commutative
algebras M Com + = c Com + such that the forgetful functor ω : c Com + → c Mod
creates weak-equivalences, creates fibrations, and preserves cofibrations. This model
structure is also cofibrantly generated by construction, with the morphisms of symmetric algebras S(i) : S(K) → S(L), where i runs over the generating (acyclic)
cofibrations of the model category of cosimplicial modules (see §5.2.10), as set of
generating (acyclic) cofibrations.
If the ground ring is not a field, then we still have a model structure on the
category of cosimplicial unitary commutative algebras, but the forgetful functor
towards the base category of cosimplicial modules does not preserve cofibrations in
general.
6.2.14. Outlook: Quasi-free cosimplicial unitary commutative algebras and minimal models. We may also elaborate on the analysis of pushouts in Proposition 6.2.10
to give a description of cell complexes of generating cofibrations in the category of
cosimplicial unitary commutative algebras which parallels our description of the cell
complexes of generating cofibrations of unitary commutative cochain dg-algebras
in §6.2.7. We give a brief outline of this correspondence. We do not use this representation further in this monograph. We still focus on the case of the cell complexes
formed by a countable sequence of cell attachments.
We consider the functor (−)[ : R 7→ R[ which forgets about the action of
the 0th coface d0 on a cosimplicial algebra R. We readily see that a cell complex
of generating cofibrations of unitary commutative cochain dg-algebras is identified
with a symmetric algebra R[ = S(Γ• (C)) when we forget about the action of this
coface, where C is a cochain graded module, identified with a cochain graded dgmodule equipped with a trivial differential, of which we take the image under the
Dold-Kan functor of §5.2.
We generally say that R is quasi-free as a cosimplicial unitary commutative
algebra when R[ has such a symmetric algebra structure. We then get that the 0th
6.3. THE BAR CONSTRUCTION IN THE CATEGORY OF COMMUTATIVE ALGEBRAS 173
coface of our cosimplicial algebra R is determined by a collection of maps
∂
(0)
Cn −
→ Nn+1 S(Γ• (C)) ⊂ S(Γ• (C))n+1 , n ∈ N,
Pn
so that d0 (x) = ∂(x) − i=1 (−1)n di (x) for every generating element x ∈ C n ,
n ∈ N. In this correspondence, we use that the module C n is identified with a direct
summand of the object Γ• (C), and we apply the cosimplicial relations to extend
our map d0 to the whole Γ• (C). We use the preservation of commutative algebra
structures to get the expression of the coface d0 on the symmetric algebra. We
need some extra assumptions on our maps ∂ in order to ensure that the cosimplicial
identity d0 d0 = d1 d0 holds in S(Γ• (C)). We may
easily make explicit an equation
satisfied by our maps ∂ : C n → Nn+1 S(Γ• (C)) and equivalent to this identity. We
leave this exercise to interested readers.
In the case of a cell complex of generating cofibrations, the generating graded
module C is moreover equipped with a filtration
(1)
0 = F0 C ⊂ · · · ⊂ Fs−1 C ⊂ Fs C ⊂ · · · ⊂ colim Fs C = C
s
and our maps (0), which determine the 0th coface on our cosimplicial algebra satisfy
the relation
(2)
∂(Fs C) ⊂ S(Γ• (Fs−1 C)) ∩ N∗ S(Γ• (C)) ,
for every s > 0. The equivalence between this filtration structure and the decomposition of the algebra R such that R[ = S(Γ• (C)) into a sequence of cell attachments
is straightforward.
We may also define minimal cell complexes of cosimplicial unitary commutative
algebras. We then assume that the maps (0) satisfies the decomposition condition
(3)
∂(C) ⊂ S(Γ• (C)) · S(Γ• (C)),
in addition to the filtration relation (2). We may see that every cosimplicial unitary
∼
commutative algebra A satisfying H0 (A) = k has a cofibrant resolution R −
→ A
such that R forms a minimal cell complex. In contrast with the minimal models
of unitary commutative cochain dg-algebras (see §6.2.9), which are only defined in
the characteristic zero setting, we now get a construction of minimal models which
remain valid for every ground field, even when the characteristic is positive.
6.3. The bar construction in the category of commutative algebras
We use that the cell attachments of generating cofibrations of unitary commutative algebras have an effective simplicial resolution which is given by a variant
of the bar complex of classical homological algebra (such as defined in [152, §X.10]
and in [220, §8.6]). We explain the definition of this bar construction in the general
setting of a symmetric monoidal category first.
6.3.1. The bar construction. We fix a symmetric monoidal category M. We
consider a cocartesian square of unitary commutative algebras in M:
R
φ
S
h
/A
/ S ⊗R A
.
174
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
The bar construction of the unitary commutative algebra R with coefficients
in (S, A) is a simplicial object B• (S, R, A) such that
Bn (S, R, A) = S ⊗ R ⊗ · · · ⊗ R ⊗A,
|
{z
}
n
for any dimension n ∈ N. The face di , i = 0, . . . , n, is induced by the appropriate
multiplication operation on the (i, i + 1)th factors of this tensor product, and the
degeneracy sj is induced by the insertion of a unit morphism η : 1 → R between
the (j, j + 1)th factors.
To be more precise, we adopt the convention to number factors from left to
right in the tensor product defining Bn (S, R, A), and we start with the object S in
position 0. In the case i = 0, we use the action of R on S through the morphism
φ : R → S to get the multiplication operation S ⊗ R → S required in the definition
of the face d0 . In the case i = n, we similarly use the action of R on S through the
morphism h : R → A to get the multiplication operation R ⊗ A → A required in
the definition of the face dn .
In the context where point-wise tensors make sense, we have the following
expression for the faces of the bar construction:


vφ(u1 ) ⊗ u2 ⊗ · · · ⊗ un ⊗ a, for i = 0,
di (v ⊗ u1 ⊗ · · · ⊗ un ⊗ a) = v ⊗ u1 ⊗ · · · ⊗ ui ui+1 ⊗ · · · ⊗ un ⊗ a, for 0 < i < n,


v ⊗ u1 ⊗ · · · ⊗ un−1 ⊗ h(un )a, for i = n,
and the following formula for the degeneracies:
sj (v ⊗ u1 ⊗ · · · ⊗ un ⊗ a) = v ⊗ u1 ⊗ · · · ⊗ uj ⊗ 1 ⊗ uj+1 ⊗ · · · ⊗ un ⊗ a,
for any tensor v ⊗ u1 ⊗ · · · ⊗ un ⊗ a ∈ Bn (S, R, A).
The canonical morphism
for all 0 ≤ j ≤ n,
B0 (S, R, A) = S ⊗ A →
− S ⊗R A
satisfies the relation d0 = d1 , and provides the simplicial object B• (S, R, A) with
an augmentation over the unitary commutative algebra B = S ⊗R A.
For the moment, we mainly use the definition of the bar construction B• (S, R, A)
as an augmented simplicial object in the base category M. Nonetheless, we may
observe that the tensor products defining the components of our object Bn (S, R, A),
n ∈ N, inherit a unitary commutative algebra structure, which is preserved by our
face (respectively, degeneracy) morphisms on B• (S, R, A), and the augmentation
: B0 (S, R, A) → S ⊗R A is a morphism of unitary commutative algebras as well.
Hence, the bar construction B• (S, R, A) actually forms an augmented simplicial
object in the category of unitary commutative algebras in M.
We have the following observation:
Lemma 6.3.2. The bar construction B• (S, R, A), regarded as an augmented simplicial object over S ⊗R A, is equipped with contracting extra degeneracies when we
have R = S(M ), S = S(Ξ ⊕ M ), and φ : R → S is the morphism of symmetric
algebras induced by the canonical inclusion i : M → Ξ ⊕ M .
Proof. We have S = S(Ξ) ⊗ R and S ⊗R A = S(Ξ) ⊗ A when the assumptions
of the lemma hold. We use the operation
(S(Ξ) ⊗ R) ⊗ R⊗n ⊗ A = (S(Ξ) ⊗ 1) ⊗ R⊗n+1 ⊗ A → (S(Ξ) ⊗ R) ⊗ R⊗n+1 ⊗ A,
6.3. THE BAR CONSTRUCTION IN THE CATEGORY OF COMMUTATIVE ALGEBRAS 175
given by the insertion of a unit in the tensor product S = S(Ξ) ⊗ R, to define the
extra degeneracy s−1 : Bn (S, R, A) → Bn+1 (S, R, A) in any dimension n ≥ 0, and
we use a similar construction for the section of the augmentation η : S(Ξ) ⊗ A →
B0 (S, R, A). The proof of the lemma reduces to the straightforward verification of
the formulas of §5.5.1.
6.3.3. The bar construction for cochain dg-algebras. In the case M = dg ∗ Mod ,
the construction of §6.3.1 returns an augmented simplicial object in the category
of cochain graded dg-modules:
B• (S, R, A) ∈ s dg ∗ Mod .
To this simplicial object, we associate a chain complex of dg-modules
∂
∂
∂
→ Bn (S, R, A) −
→ ... −
→ B0 (S, R, A),
... −
where the boundaries ∂ are dg-module morphisms, satisfying ∂ 2 = 0, and defined
by the alternate sums of the face operators of §6.3.1:
∂=
n
X
i=0
(−1)i di : Bn (S, R, A) → Bn−1 (S, R, A).
This chain complex of dg-modules is equivalent to a second quadrant bicomplex:
– the module Bn (S, R, A)q , which defines the component of degree q of the
nth dg-module Bn (S, R, A) ∈ dg ∗ Mod , represents the component of degree
(n, q) of this bicomplex, for each n, q ≥ 0;
– the maps
δ : Bn (S, R, A)q → Bn (S, R, A)q+1
yielded by the internal differential of each dg-module Bn (S, R, A), form the
vertical differentials;
– and the maps
∂ : Bn (S, R, A)q → Bn−1 (S, R, A)q
defining the components of our boundary morphisms ∂, form the horizontal
differentials.
To the simplicial object B• (S, R, A), we also associate a (lower graded) dgmodule T∗ (B• (S, R, A)) ∈ dg Mod , the total complex of the bicomplex, such that:
M
T∗ (B• (S, R, A))m =
Bn (S, R, A)q
m=n−q
for any degree m ∈ Z. The differential of this dg-module is defined on any summand
Bn (S, R, A)q ⊂ T∗ (B• (S, R, A))n−q by the (signed) sum ∂ + (−1)n δ of the vertical
differential δ and of the horizontal differential ∂ of the bicomplex B• (S, R, A).
The augmentation of the bar construction : B0 (S, R, A) → S ⊗R A induces
a morphism of dg-modules : T∗ (B• (S, R, A)) → S ⊗R A. In this definition, we
use the equivalence between cochain graded dg-modules and non-positively lower
graded dg-modules to identify the cochain graded dg-module S ⊗R A with a dgmodule in the same category as the object T∗ (B• (S, R, A)).
We have the following result:
176
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
Proposition 6.3.4. For a cell attachment of the form of Proposition 6.2.3
S(C)
/A
S(D)
/ S(D) ⊗S(C) A
,
L
N
L
where
C =
α Bmα ⇒ S(C) =
α S(Bmα ) and D =
α Emα ⇒ S(D) =
N
S(E
),
the
augmentation
mα
α
: T∗ (B• (S(D), S(C), A)) → S(D) ⊗S(C) A
induces an isomorphism in homology.
Proof. Let R = S(C), S = S(D), B = S(D) ⊗S(C) A = S ⊗R A. We consider
the spectral sequence Er∗ ⇒ H∗ (T∗ (B• (S, R, A))) which has the horizontal homology
of our bicomplex as E 1 -page:
E1mq = Hm (B• (S, R, A)q , ∂).
We have a formal identity E1mq = Hm (B• (S[ , R[ , A[ )q , ∂), where (−)[ refers to
the forgetting of the internal differential of algebras. We deduce from Lemma 6.3.2
that the augmented simplicial object B• (S[ , R[ , A[ ) inherits contracting extra degeneracies since we have R[ = S(C[ ), S[ = S(D[ ), and C is a direct summand of D
when we forget about differentials. We then get
(
B q , if m = 0,
E1mq =
0,
otherwise,
for each q ∈ N (see Proposition 5.5.2), and, by standard spectral sequence arguments, we conclude that the morphism
T∗ (B• (S, R, A)) →
− S ⊗R A = B
induces an isomorphism in homology, as claimed in our proposition.
6.3.5. The bar construction for cosimplicial algebras. In the case M = c Mod ,
the construction of §6.3.1 returns an augmented simplicial object in the category
of cosimplicial modules:
B• (S, R, A) ∈ s c Mod .
The operations involved in the construction of §6.3.1 are performed dimensionwise (with respect to the cosimplicial dimension) when we form this simplicial
cosimplicial object B• (S, R, A) ∈ s c Mod . If we consider the cosimplicial dimension first (rather that the simplicial dimension), then we may identify our bar
construction B• (S, R, A) with the cosimplicial module defined by the bar complex B• (S q , Rq , Aq ) of the unitary commutative algebras in k-modules S q , Rq , Aq ∈
Com + in each cosimplicial dimension q ∈ N, together with the coface operators di : B• (S q−1 , Rq−1 , Aq−1 ) → B• (S q , Rq , Aq ) and the codegeneracy operators
sj : B• (S q+1 , Rq+1 , Aq+1 ) → B• (S q , Rq , Aq ) induced by the coface and codegeneracy operators of our algebras S, R, A ∈ c Com + (by using the functoriality of the
bar construction).
We can also apply the conormalized complex construction of cosimplicial modules to the bar construction dimension-wise. This operation returns a simplicial
object in the category of cochain graded dg-modules N∗ (B• (S, R, A)) ∈ s dg ∗ Mod .
6.3. THE BAR CONSTRUCTION IN THE CATEGORY OF COMMUTATIVE ALGEBRAS 177
To this simplicial object, we again associate a chain complex of cochain graded
dg-modules
∂
∂
∂
... −
→ N∗ (Bn (S, R, A)) −
→ ... −
→ N∗ (B0 (S, R, A)),
where in the definition of the boundary operators
∂=
n
X
i=0
(−1)i di : N∗ (Bn (S, R, A)) → N∗ (Bn−1 (S, R, A))
we now consider the morphisms induced by the face operators of §6.3.1 on the
conormalized complex N∗ (B• (S, R, A)).
We again consider the second quadrant bicomplex equivalent to this chain complex of cochain graded dg-modules, with the module Nq (Bn (S, R, A)) as component
of bidegree (n, q), the maps δ : Nq (Bn (S, R, A)) → Nq+1 (Bn (S, R, A)) induced by
the internal differential of the dg-module N∗ (Bn (S, R, A)) ∈ dg ∗ Mod as vertical
differentials, and the maps ∂ : Nq (Bn (S, R, A)) → Nq (Bn−1 (S, R, A)) defined by our
boundary operators as horizontal differentials.
We also write T∗ (B• (S, R, A)) for the total complex of our bicomplex, the dgmodule such that:
M
T∗ (B• (S, R, A))m =
Nq (Bn (S, R, A)),
m=n−q
for any degree m ∈ Z, and of which differential is defined by the (signed) sum
∂ + (−1)n δ on any summand Bn (S, R, A)q ⊂ T∗ (B• (S, R, A))n−q . We still have a
morphism of dg-modules : T∗ (B• (S, R, A)) → N∗ (S ⊗R A) induced by the augmentation of the bar construction : B0 (S, R, A) → S ⊗R A, where we again use
the equivalence between cochain graded dg-modules and non-negative lower graded
dg-modules to identify the conormalized complex N∗ (S ⊗R A) with a dg-module in
the same category as T∗ (B• (S, R, A)).
We have the following cosimplicial analogue of the result of Proposition 6.3.6:
Proposition 6.3.6. For a cell attachment of the form of Proposition 6.2.10
S(K)
/A
,
/ S(L) ⊗S(K) A
L •
N
L •
•
where KN=
α Γ (Bmα ) ⇒ S(K) =
α S(Γ (Bmα )) and L =
α Γ (Emα ) ⇒
S(L) = α S(Γ• (Emα )), the morphism of dg-modules
S(L)
: T∗ (N∗ (B• (S(L), S(K), A))) → N∗ (S(L) ⊗S(K) A)
induces an iso in homology.
Proof. Let R = S(K), S = S(L) and B = S(L) ⊗S(K) A = S ⊗R A. We
consider the spectral sequence Er∗ ⇒ H∗ (T∗ (B• (S, R, A))) which has the horizontal
homology of our bi-complex as E 1 -page:
E1mq = Hm (Nq (B• (S, R, A)), ∂)
(as in the proof of Proposition 6.3.4).
178
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
We have an identity E1mq = Nq (Hm (B• (S, R, A), ∂)) (since, according to Proposition 5.2.3, any conormalized complex is defined by a natural retract of the associated cosimplicial module). We fix a cosimplicial dimension q in the bar construction B• (S, R, A), and, as we explain in §6.3.5, we consider the bar construction
B• (S q , Rq , Aq ) associated to the unitary commutative algebras S q , Rq ,L
Aq ∈ Com + .
•
We already observed (in the proof
α Γ (Bmα )
L of• Proposition 6.2.10) that K =
forms a direct summand of L = α Γ (Emα ) when we fix the cosimplicial dimension
q. We therefore have
(
Nq (B), if m = 0,
E1mq =
0,
otherwise,
by Lemma 6.3.2, and we conclude from our usual spectral sequence argument that
the morphism
T∗ (B• (S, R, A)) →
− N∗ (S ⊗R A) = N∗ (B)
induces an isomorphism in homology as claimed in the proposition.
6.4. The commutative algebra upgrade of the Dold-Kan equivalence
We prove in this section that the cosimplicial Dold-Kan equivalence of §5.2
yields, at the unitary commutative algebra level, a Quillen equivalence between the
category of unitary commutative cochain dg-algebras and the category of cosimplicial unitary commutative algebras. We have a similar result in the simplicial
setting, but we essentially explain the definition of the adjunction in this case and
we just give a few hints on the proof of the Quillen equivalence assertion.
In a preliminary step, we explain the definition of adjoint Dold-Kan functors on
unitary commutative algebra categories. We already observed that the normalized
complex functor N∗ : s Mod → dg ∗ Mod is unit-pointed and equipped with a symmetric monoidal transformation (see Theorem 5.3.3), so that this functor induces
a functor from the category of unitary commutative algebras in simplicial modules
(the category of simplicial unitary commutative algebras) towards the category of
unitary commutative algebras in chain graded dg-modules (the category of unitary
commutative chain dg-algebras), but we do not have such a result for the DoldKan functor Γ• : dg ∗ Mod → s Mod , which defines the inverse of the normalization
functor in the Dold-Kan equivalence. We similarly observed that the cosimplicial
Dold-Kan functor Γ• : dg ∗ Mod → c Mod preserves unitary commutative algebra
structures, but we do not have such a result yet for the conormalized complex functor N∗ : c Mod → dg ∗ Mod which we use in the cosimplicial version of the Dold-Kan
equivalence.
We can fix these problems however. We record our results in the following
proposition:
Proposition 6.4.1.
(a) The normalized complex functor N∗ (−) : s Mod → dg ∗ Mod induces a functor N∗ (−) : s Com + → dg ∗ Com + from the category of simplicial unitary commutative algebras s Com + to the category of unitary commutative chain dg-algebras
dg ∗ Com + . This functor admits a left adjoint
Γ]• (−) : dg ∗ Com + → s Com +
6.4. THE COMMUTATIVE ALGEBRA DOLD-KAN EQUIVALENCE
179
that defines a commutative algebra upgrade of the Dold-Kan functor from chain
graded dg-modules to simplicial modules.
(b) The cosimplicial Dold-Kan functor Γ• (−) : dg ∗ Mod → c Mod , inverse
to the conormalization functor on cosimplicial modules, induces a functor Γ• (−) :
dg ∗ Com + → c Com + from the category of unitary commutative cochain dg-algebras
dg ∗ Com + to the category of cosimplicial unitary commutative algebras c Com + .
This functor admits a left adjoint
N∗] (−) : c Com + → dg ∗ Com +
that defines a commutative algebra upgrade of the conormalization functor from
cosimplicial modules to cochain graded dg-modules.
Proof. We already checked that the normalized complex functor N∗ (−) :
s Mod → dg ∗ Mod induces a functor on unitary commutative algebras (see §6.1.4).
We mainly aim to establish that this functor N∗ (−) : s Com + → dg ∗ Com + admits a left adjoint Γ]• : dg ∗ Com + → s Com + so that we get an adjunction N∗ (−) :
s Com + dg ∗ Com + : Γ• (−) between the category of unitary commutative chain
dg-algebras dg ∗ Com + and the category of simplicial unitary commutative algebras
s Com + as asserted in the proposition.
We use the same argument line to define our dual adjunction N∗] (−) : c Com + ∗
dg Com + : Γ• (−) between the category of unitary commutative cochain dg-algebras
dg ∗ Com + and the category of cosimplicial unitary commutative algebras c Com + .
We then start with the functor Γ• : dg ∗ Com + → c Com + , already defined in §6.1.4
and induced by the cosimplicial Dold-Kan functor on cochain graded dg-modules
Γ• : dg ∗ Mod → c Mod . We mainly aim to establish that this functor Γ• :
dg ∗ Com + → c Com + admits a left adjoint which give our unitary commutative
algebra upgrade of the conormalized complex functor N∗] : c Com + → dg ∗ Com + .
We rather use this adjunction relation in the follow up. We therefore give full
details on the second assertion of the proposition rather that on the first one.
We adapt a standard definition of adjoint functors in the context of algebras
over a monad to complete our construction. We already used this general adjoint
functor construction in the case of operads in the proof of Proposition I.2.1.7.
We define the image of a symmetric algebra under our functor in a first step.
We explicitly set N∗] (S(K)) = S(N∗ (K)), for any K ∈ c Mod , since we have the
relations:
Mordg ∗ Com + (S(N∗ (K)), B) = Mordg ∗ Mod (N∗ (K), B)
= Mordg ∗ Mod (N∗ (K), N∗ Γ• (B))
= Morc Mod (K, Γ• (B))
= Morc Com + (S(K), Γ• (B)).
We then see that any morphism φ : S(K) → S(L) admits, according to the Yoneda
lemma, a cochain dg-algebra counterpart φ] : S(N∗ (K)) → S(N∗ (L)) which corresponds to the natural transformation induced by φ on morphism sets:
Mordg ∗ Com + (S(N∗ (K)), B)
O
'
φ∗
]
Mordg ∗ Com + (S(N∗ (L)), B)
/ Morc Com + (S(K), Γ• (B)) .
O
φ∗
'
/ Morc Com + (S(L), Γ• (B))
180
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
If we assume that φ = φf is associated to a morphism of cosimplicial modules
f : K → S(L), then we can determine φ] as the morphism of unitary commutative
cochain dg-algebras associated to the composite morphism of cochain dg-modules
such that
N∗ (f )
∆
N∗ (K) −−−→ N∗ (S(L)) −→ S(N∗ (L)),
where we consider the comonoidal transformation of Proposition 5.3.5(b).
We finally use that any unitary commutative algebra A, which we identify with
an algebra over the symmetric algebra monad S(−) = S(Com, −) (see §I.1.3.11),
fits a natural reflexive coequalizer of the form
z
S(S(A))
s0
d0
d1
// S(A)
/A
(see [153, §VI.7]). We take the image of this coequalizer under our functor N∗] (−) to
determine the object N∗] (A) ∈ dg ∗ Com + which we associate to any A ∈ c Com + . In what follows, we mainly use that the functor N∗] : c Com + → dg ∗ Com +
maps symmetric algebras in cosimplicial modules to symmetric algebras in cochain
graded dg-modules and preserves colimits.
We also use the following proposition to effectively define morphisms on the
unitary commutative cochain dg-algebra N∗] (A) ∈ dg ∗ Com + , which we associate to
any object A ∈ c Com + :
Proposition 6.4.2. Let A ∈ c Com + . Let B ∈ dg ∗ Com + . Giving a morphism
of unitary commutative cochain dg-algebras φ] : N∗] (A) → B amounts to giving a
morphism of cochain dg-modules φ : N∗ (A) → B such that the diagrams
N∗ (k)
=
N∗ (ηA )
N∗ (A)
/k
and
ηB
φ
/B
N∗ (A ⊗ A)
∆
/ N∗ (A) ⊗ N∗ (A)
N∗ (µA )
N∗ (A)
φ⊗φ
/ B⊗B
µB
φ
/B
commute.
We only deal with the case of this proposition in the follow up, but we may
readily see that a similar result holds for the morphisms on the simplicial unitary
commutative algebra Γ]• (A) ∈ s Com + which we associate to a unitary commutative
algebra in chain graded dg-modules A ∈ dg ∗ Com + .
Proof. The adjunction relation Mordg ∗ Com + (N∗] (A), B) = Morc Com + (A, Γ• (B))
asserts that we have a one-to-one correspondence between the morphisms of unitary
commutative cochain dg-algebras φ] : N∗] (A) → B and the morphisms of cosimplicial unitary commutative algebras ψ : A → Γ• (B), where we take the image of
our unitary commutative cochain dg-algebra B ∈ dg ∗ Com + under the cosimplicial
Dold-Kan functor Γ• : B 7→ Γ• (B). For this morphism ψ : A → Γ• (B), the preservation of commutative algebra structures is expressed by the following commutative
6.4. THE COMMUTATIVE ALGEBRA DOLD-KAN EQUIVALENCE
181
diagrams:
k
=
/ Γ• (k)
ψ
/ Γ• (B)
A⊗A
ψ⊗ψ
/ Γ• (B) ⊗ Γ• (B) ,
Γ• (ηB )
ηA
A
and
∇
Γ• (B ⊗ B)
µA
Γ• (µB )
A
ψ
/B
where we consider the natural transformation ∇ : Γ• (C) ⊗ Γ• (D) → Γ• (C ⊗ D) of
Proposition 5.3.5(b), which we use to define our product on the cosimplicial module
Γ• (B).
We use the Dold-Kan equivalence for cosimplicial modules to associate this
morphism ψ : A → Γ• (B) to a morphism of cochain graded dg-modules φ : N∗ (A) →
B. We readily see that the functor N∗ (−) and the natural isomorphism N∗ Γ• (C) ' C
of the Dold-Kan equivalence carry the above diagrams to the commutative diagrams
of the definition. We just use that the natural transformation ∇ is defined as the
image of the map ∆ : N∗ (K ⊗ L) → N∗ (K) ⊗ N∗ (L) under the Dold-Kan equivalence
(see Proposition 5.3.5) when we address the preservation of the products.
In the case B = N∗] (A), the correspondence of this proposition gives a morphism
of cochain graded dg-modules χ : N∗ (A) → N∗] (A) such that χ] = id , and of which
image under the Dold-Kan equivalence represents the unit morphism η(A) : A →
Γ• N∗] (A) of our adjunction between the category of unitary commutative cochain
dg-algebras and the category of cosimplicial unitary commutative algebras. We can
moreover characterize the morphism φ] which we associate to a morphism of cochain
graded dg-modules φ in our proposition as a morphism of unitary commutative
cochain dg-algebras making the following diagram commute:
φ
N∗ (A)
χ
#
N∗] (A)
/B,
=
∃!φ]
and which is also uniquely determined by this requirement. We establish the following result:
Proposition 6.4.3. The universal morphism χ : N∗ (A) → N∗] (A) defines a
weak-equivalence of cochain graded dg-modules when the ground ring is a field of
characteristic zero and A forms a cofibrant object in the model category of cosimplicial unitary commutative algebras.
We have an analogue of this statement in the category of simplicial unitary
commutative algebras, where we consider the commutative algebra enhancement of
the simplicial Dold-Kan correspondence of Proposition 6.4.1(a), but the arguments
are different in this setting. We just give a few hints on the proof of this dual
result when we briefly examine the definition of an enhanced Dold-Kan equivalence
for simplicial unitary commutative algebras and unitary commutative chain dgalgebras (see §6.4.5).
182
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
We may also observe that the above proposition remains valid over any ring
k (not necessarily a field) such that Q ⊂ k, but we do not use this more general
statement either.
Proof. We can assume that A is a cell complex of generating cofibrations in
the category of cosimplicial unitary commutative algebras. We examine the case of
a symmetric algebra first and we prove afterwards that our claim remains true when
we perform a cell attachment in the category of cosimplicial unitary commutative
algebras.
In the case of a symmetric algebra A = S(K), where N∗] (S(K)) = S(N∗ (K)), we
can readily identify our morphism χ : N∗ (S(K)) → N∗] (S(K)) with the dg-module
morphism
(1)
∞
M
∗
N ((K
⊗n
n=0
|
∆
)Σn ) −→
{z
}
N∗ (S(K))
∞
M
(N∗ (K)⊗n )Σn
n=0
|
{z
S(N∗ (K))
}
yielded by the comonoidal transformation of Theorem 5.3.4 (check the definition of our correspondence in the proof of Proposition 6.4.2). The morphism
∆ : N∗ (K ⊗n ) → N∗ (K)⊗n is a weak-equivalence for every n ∈ N by Theorem 5.3.4,
and, in characteristic zero, this result implies that ∆ induces a weak-equivalence
on coinvariants as well. We therefore obtain that our morphism (1) defines a weakequivalence as expected.
We now consider the case of a cell attachment
(2)
S(K)
i
S(L)
/A ,
k
/B
L
L
where K = α Γ• (Bmα ), L = α Γ• (Emα ), and i is a sum
Lof generating cofibrations
•
•
•
∗
∗
Γ
(i
)
:
Γ
(B
)
→
Γ
(E
).
We
set
C
=
N
(K)
=
m
m
m
α
α
α
α Bmα and D = N (L) =
L
∗
α Emα for short. We assume that our morphism is a weak-equivalence on N (A),
and we aim to prove that this property remains true when we pass to the pushout
B = S(L) ⊗S(K) A. We have N∗] (B) = N∗] (S(L)) ⊗N∗] (S(K)) N∗] (A) = S(D) ⊗S(C) N∗] (A)
since the functor N∗] , defined as a left adjoint, preserves all colimits.
We use the bar construction of §§6.3.1-6.3.6. We readily see that our morphism
χ : N∗ (B) → N∗] (B) fits in a diagram of augmented simplicial objects
(3)
N∗ (B• (S(L), S(K), A))
/ B• (S(D), S(C), N∗ (A)) ,
]
N∗ (S(L) ⊗S(K) A)
/ S(D) ⊗S(C) N∗ (A)
]
where the upper horizontal morphism can be identified, in each dimension n ∈ N,
with the composite
(4)
∆
N∗ (S(L) ⊗ S(K)⊗n ⊗ A) −→ N∗ (S(L)) ⊗ N∗ (S(K))⊗n ⊗ N∗ (A)
→ S(N∗ (L)) ⊗ S(N∗ (K))⊗n ⊗ N∗] (A) = S(D) ⊗ S(C)⊗n ⊗ N∗] (A)
6.4. THE COMMUTATIVE ALGEBRA DOLD-KAN EQUIVALENCE
183
involving the Eilenberg-Zilber equivalence of Theorem 5.3.4 and the comparison
morphisms χ associated to the free algebras S(K), S(L), and to our algebra A. We
deduce, from our induction assumption, that this morphism is a weak-equivalence
dimension-wise.
We then consider the spectral sequence, associated to the simplicial dg-module
N∗ (B• (S(L), S(K), A)), and which has the cohomology of the cochain graded dgmodules Bm (S(L), S(K), A) (representing the vertical homology of the bicomplex
of §6.3.3) as E 1 -page. We know from Proposition 6.3.4 that this spectral sequence
converges to H∗ (B) = H∗ (S(L) ⊗S(K) A). We consider the parallel spectral sequence, associated to B• (S(D), S(C), N∗] (A)), and which has the cohomology of the
dg-modules Bm (S(D), S(C), N∗] (A)) (representing the vertical homology of the bicomplex of §6.3.5) as E 1 -page. We know from Proposition 6.3.6 that this spectral
sequence converges to H∗ (N∗] (B)) = H∗ (S(D) ⊗S(C) N∗] (A)).
We deduce from our diagram of augmented simplicial objects that these spectral sequences are connected by a morphism which defines an iso at the E 1 -stage
(because our morphism of simplicial objects is a weak-equivalence dimension-wise).
We obtain, as usual, that this spectral sequence morphism induces an iso on the
abutment, and hence, that our morphism χ : N∗ (B) → N∗] (B) defines a weakequivalence. We can therefore carry on our induction process and this verification
completes the proof that our proposition.
We can now establish the main result of this section:
Theorem 6.4.4. We still assume that the ground ring is a field of characteristic
zero (as in Proposition 6.4.3). We use that both the category of cosimplicial unitary
commutative algebras c Com + and the category of unitary commutative cochain dgalgebras dg ∗ Com + inherit a model structure in this setting. We then obtain that
the functors N∗] : c Com + dg ∗ Com + : Γ• of Proposition 6.4.1(b) define a Quillen
equivalence.
We have an analogue of this theorem for our commutative algebra upgrade
of the simplicial Dold-Kan correspondence. We just give a few hints on this case
in §6.4.5. We may still observe, after proving that the definition of our model
structures remain valid for unitary commutative algebras over a ring k such that
Q ⊂ k, that this is also the case of our theorem. We leave the verification of this
generalization to interested readers.
Proof. We first check that our functors define a Quillen adjunction. We use
the definition of the image of a symmetric algebra N∗] (S(K)) = S(N∗ (K)) under
the commutative algebra upgrade of the conormalization functor N∗] : c Com + →
dg ∗ Com + .
For a morphism of the form S(Γ• (i)) : S(Γ• (C)) → S(Γ• (D)), where we consider the inverse of the conormalization functor on cosimplicial modules, we have
N∗] (S(Γ• (i))) = S(N∗ (Γ• (i))) = S(i), and since we define the generating (acyclic)
cofibrations of unitary commutative algebras in terms of such morphisms, we readily obtain that the functor N∗] : c Com + dg ∗ Com + maps the generating (acyclic)
cofibrations of c Com + to generating (acyclic) cofibrations of dg ∗ Com + . By Proposition 4.3.1, this is enough to conclude that the functors N∗] : c Com + dg ∗ Com + :
Γ• define a Quillen adjunction.
184
6. DIFFERENTIAL GRADED ALGEBRAS AND COSIMPLICIAL ALGEBRAS
We then check that the morphism η(A) : A → Γ• (N∗] (A)), defining the unit
of our adjunction, is a weak-equivalence as soon as A is a cofibrant object in the
category of cosimplicial unitary commutative algebras. We use that the image of
this morphism under the conormalized complex functor N∗ (−) is identified with the
universal morphism χ : N∗ (A) → N∗] (A) of Proposition 6.4.3 (up to isomorphism),
and that this morphism is a weak-equivalence according to our result.
We now assume that B is a unitary commutative cochain dg-algebra, and we
∼
fix a cofibrant resolution R −
→ Γ• (B) of the cosimplicial unitary commutative
•
algebra Γ (B) associated to B. We have just checked that the adjunction unit
R → Γ• (N∗] (R)) is a weak-equivalence. We deduce from the commutative diagram
R
∼
Γ• (N∗] (R))
/ Γ• (B)
∼
/ Γ• (N∗ (Γ• (B)))
]
'
=
u
Γ• (B)
that the morphism : N∗] (R) → B, yielding the augmentation of the derived adjunction associated to our Quillen pair, induces a weak-equivalence on the inverse
Dold-Kan functor Γ• , and since we can go back to dg-modules by applying the
conormalization functor, this implies that : N∗] (R) → B is a weak-equivalence as
well.
We conclude from these verifications that both the unit η : A → Γ• (N] (A)) and
the augmentation : N∗] (R) → B of the derived adjunction associated to our Quillen
pair are weak-equivalences and this result completes the proof that our Quillen
pair is indeed a Quillen equivalence. Simply observe that N∗] (A) is automatically
fibrant as a cochain graded dg-module, and hence, as an object of the category of
unitary commutative cochain dg-algebras, so that we do not need to take a fibrant
resolution of this object when we form the unit morphism of the derived adjunction
relation.
6.4.5. Remark: Comparison with the case of simplicial algebras. We may also
establish that the adjunction Γ]• : dg ∗ Com + s Com + : N∗ in Proposition 6.4.1(a)
is a Quillen equivalence of model categories (see [180, §I.4]).
In the context of unitary commutative chain dg-algebras, the cofibrant objects
are equivalent to retracts of symmetric algebras equipped with a twisting derivation
(quasi-free unitary commutative chain dg-algebras), and we have a similar description of the cofibrant objects of the category of simplicial unitary commutative algebras. We can easily determine the value of the left adjoint Γ•] on quasi-free objects.
We then use the Eilenberg-Zilber equivalence and a spectral sequence argument to
establish that the adjunction unit η(A) : A → N∗ (Γ]• (A)) defines a weak-equivalence
when A is a quasi-free unitary commutative chain dg-algebra (see [180, §I.4] for details).
We can not adapt this argument line to the case of Theorem 6.4.4, mostly
because we can not handle the convergence of the spectral sequence associated to
quasi-free objects in the category of unitary commutative cochain dg-algebras (and
in the category of cosimplicial unitary commutative algebras similarly).