on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A model category structure on a category of dg-algebras tends to present an (∞,1)-category of ∞-algebras.
For dg-algebras bounded in negative or positive degrees, the monoidal Dold-Kan correspondence asserts that their model category structures are Quillen equivalent to the corresponding model structure on (co)simplicial algebras. This case plays a central role in rational homotopy theory.
The case of model structures on unbounded dg-algebras may be thought of as induced from this by passage to the derived geometry modeled on formal duals of the bounded dg-algebras. This is described at dg-geometry.
The category of dg-algebras is that of monoids in a category of chain complexes. Accordingly general results on a model structure on monoids in a monoidal model category apply.
Below we spell out special cases, such as restricting to commutative monoids when working over a ground field of characteristic zero, or restricting to non-negatively graded cochain dg-algebras.
We discuss the projective model structure on differential non-negatively graded-commutative algebras. This was originally introduced in Bousfield-Gugenheim 76 as a model category for Dennis Sullivan’s approach to rational homotopy theory.
For $k$ a field of characteristic zero, write
for the category of differential graded-commutative algebras over $k$ in non-negative degrees, equivalently the category of commutative monoids in the symmetric monoidal category $Ch^{\geq 0}(k)$ of cochain complexes in non-negative degrees, equipped with the tensor product of chain complexes.
(finite type)
Say that a dgc-algebra $A \in dgcAlg^{\geq 0}_k$ (def. 1) is of finite type if its underlying chain complex is in each degree of finite dimension as a $k$-vector space.
Write $(dgcAlg^{\geq 0}_k)_{proj}$ for the catgory of dgc-algebras from def. 1 equipped with the following classes of morphisms:
weak equivalences are those homomorphisms of dg-algebras whose underlying chain map is quasi-isomorphism;
fibrations are those homomorphisms which are degreewise surjections;
The category (dgcAlg^{\geq 0}_k})_{proj}
from def. 3 is a model category, to be called the projective model structure.
(Bousfield-Gugenheim 76, theorem 4.3)
(category of fibrant objects)
Evidently every object in $(dgcAlg^{\geq 0}_k}(dgcAlg^{\geq 0}_k)_{proj}$ (def. 3, prop. 1) is fibrant. Therefore these model categories structures are in particular also structures of a category of fibrant objects.
The nature of the cofibrations is discussed below.
(sphere and disk algebras)
Write $k[n]$ for the graded vector space which is the ground field $k$ in degree $n$ and 0 in all other degrees. For $n \in \mathbb{N}$, consider the semifree dgc-algebras
and for $n \geq 1$ the semifree dgc-algebras
for which the differential sends the generator of $k[n-1]$ to that of $k[n]$
Write
for the obvious morphism that takes the generator in degree $n$ to the generator in degree $n$ (and for $n = 0$ it is the unique morphism from the initial object $(0,0)$).
For $n \gt 0$ write
(generating cofibrations)
The sets
and
are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting the model category $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. 1 as a cofibrantly generated model category.
review includes (Hess 06, p. 6)
In this section we describe the cofibrations in the model structure on $(dgcalg^{\geq 0}_k)_{proj}$ (def. 3, prop. 1). Notice that it is these that are in the image of the dual monoidal Dold-Kan correspondence.
Before we characterize the cofibrations, first some notation.
For $V$ a $\mathbb{Z}$-graded vector space write $\wedge^\bullet V$ for the Grassmann algebra over it. Equipped with the trivial differential $d = 0$ this is a semifree dga $(\wedge^\bullet V, d=0)$.
With $k$ our ground field we write $(k,0)$ for the corresponding dg-algebra, the tensor unit for the standard monoidal structure on $dgAlg$. This is the Grassmann algebra on the 0-vector space $(k,0) = (\wedge^\bullet 0, 0)$.
(Sullivan algebras)
A relative Sullivan algebra is a morphism of dg-algebras that is an inclusion
for $(A,d)$ some dg-algebra and for $V$ some graded vector space, such that
there is a well ordered set $J$
indexing a basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;
such that with $V_{\lt \beta} = span(v_\alpha | \alpha \lt \beta)$ for all basis elements $v_\beta$ we have that
This is called a minimal relative Sullivan algebra if in addition the condition
holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the semifree dga $(\wedge^\bullet V,d)$ simply a Sullivan algebra. And a minimal Sullivan algebra if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.
Sullivan algebras were introduced by Dennis Sullivan in his development of rational homotopy theory. This is one of the key application areas of the model structure on dg-algebras.
($L_\infty$-algebras)
Because they are semifree dgas, Sullivan dg-algebras $(\wedge^\bullet V,d)$ are (at least for degreewise finite dimensional $V$) Chevalley-Eilenberg algebras of L-∞-algebras.
The co-commutative differential co-algebra encoding the corresponding L-∞-algebra is the free cocommutative algebra $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula
for all $\omega \in V$ and all $v_i \in V^*$.
(cofibrations are relative Sullivan algebras)
The cofibration in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the retracts of relative Sullivan algebras $(A,d) \to (A\otimes_k \wedge^\bullet V, d')$.
Accordingly, the cofibrant objects in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the Sullivan algebras $(\wedge^\bullet V, d)$
We discuss simplicial mapping spaces between dgc-algebras. These almost make the projective model structure $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. 1 into a simplicial model category, except that the tensoring/powering isomorphism holds only for finite simplicial sets or else on dgc-algebras of finite type. Still, this has useful implications, for instance it implies that the reduced suspension and loop space adjunction on [augmented algebras|augmented]] dg-algebras is a Quillen adjunction.
(simplicial mapping spaces)
For $A,B \in dgcAlg^{\geq 0}_k$ (def. 1), let
be the simplicial set whose n-simplices are the dg-algebra homomorphisms from $A$ into the tensor product of $B$ with the de Rham complex of polynomial differential forms on the n-simplex $\Omega_{poly}^\bullet(\Delta^n)$.
and whose face and degeneracy maps are the obvious ones induced from the fact that $\Omega_{poly}^\bullet \colon \Delta^{op} \to dgcAlg^{\geq 0}_k$ is canonically a simplicial object in dgc-algebras.
We also call this the simplicial mapping space from $A$ to $B$. This construction naturally extends to a functor
from the product category of the opposite category of dgc-algebras with the category itself.
Observe that
where on the right we have those dg-algebra homomorphism which in addition preserves the left dg-module structure over $\Omega^\bullet_{poly}(\Delta^n)$. This induces for any three $A,B,C \in dgcAlg^{\geq 0}_k$ a composition homomorphism of simplicial sets out of the Cartesian product of mapping spaces
The set of 0-simplices of of the mapping space $Maps(A,B)$ in def. 6 is naturally isomorphic to the ordinary hom-set of dg-algebras:
and under this identification the two notions of composition agree.
Definition 6 makes $dgcAlg^{\geq 0}_k$ an sSet-enriched category (“simplicial category”). The follows says that it is also powered, not over all of $sSet$, but over finite simplicial sets:
(powering over finite simplicial sets)
For $A, B \in dgcAlg^{\geq 0}_k$ and $S \in$ sSet, there is a natural transformation
from the hom-set of dgc-algebras into the tensor product with the polynomial differential forms on n-simplices from def. 6 to the hom-set in simplicial sets into the simplicial mapping space from def. 6.
Moreover, this morphism is an isomorphism if one of the following conditions holds:
$S$ is a finite simplicial set;
$B$ is of finite type (def. 2).
(Bousfield-Gugenheim 76, lemma 5.2)
(pullback powering axiom)
Let $i \colon V \to W$ and $p \colon X \to Y$ be two morphisms in $dgcAlg^{\geq 0}_k$. Then their pullback power with respect to the simplicial mapping space functor (def. 6)
is
a Kan fibration if $i$ is a cofibration and $p$ a fibration in the projective model category structure from prop. 1;
in addition a weak homotopy equivalence (i.e. a weak equivalence in the classical model structure on simplicial sets) if at least one of $i$ or $p$ is a weak equivalence in the projective model structure from prop. 1.
(Bousfield-Gugenheim 76, prop. 5.3)
Prop. 4 would say that $(dgcAlg^{\geq 0}_k)_{proj}$ is a simplicial model category with respect to the simplicial enrichment from def. 6 were it not for the fact that prop. 4 gives the powering only over finite simplicial sets.
this needs harmonization
from (graded-)commutative dg-algebras to dg-algebras is the right adjoint part of a Quillen adjunction
boundedness?
The forgetful functor clearly preserves fibrations and cofibrations. It has a left adjoint, the free abelianization functor $Ab$, which sends a dg-algebra $A$ to its quotient $A/[A,A]$.
Let the ground ring $k$ be a field of characteristic zero. Then every dg-algebra $A$ which has the structure of an algebra over the E-∞ operad has a dg-algebra morphism $A \to A_c$ to a commutative dg-algebra $A_c$ which is
a morphism of E-∞ algebras (where $A_c$ has the obvious E-∞ algebras structure)
a weak weak equivalence in the model structure on dg-algebras (i.e. a quasi-isomorphism of the underlying cochain complexes).
This is in (Kriz-May 95, II.1.5).
So this says that the weak equivalence classes of the commutative dg-algebras in the model category of all dg-algebras already exhaust the most general non-commutative but homotopy-commutative dg-algebras.
Discussion of a restricted kind of homotopy-faithfulness of the forgetful functor from the homotopy theory of commutative to not-necessarily commutative dg-algebras is in (Amrani 14).
We discuss now the case of unbounded dg-algebras. For these there is no longer the monoidal Dold-Kan correspondence available. Instead, these can be understood as arising naturally as function $\infty$-algebras in the derived dg-geometry over formal duals of bounded dg-algebras, see function algebras on ∞-stacks.
In derived geometry two categorical gradings interact: a cohesive $\infty$-groupoid $X$ has a space of k-morphisms $X_k$ for all non-negative $k$, and each such has itself a simplicial T-algebra of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from function algebras on ∞-stacks.
Functions on a bare $\infty$-groupoid $K$, modeled as a simplicial set, form a cosimplicial algebra $\mathcal{O}(K)$, which under the monoidal Dold-Kan correspondence identifies with a cochain dg-algebra (meaning: with positively graded differential) in non-negative degree
On the other hand, a representable $X$ has itself a simplicial T-algebra of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write
Taking this together, for $X_\bullet$ a general ∞-stack, its function algebra is generally an unbounded cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:
For $k$ a field of characteristic 0 let
be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the quasi-isomorphisms this is a
which is
The existence of the model structure follows from the general discussion at model structure on dg-algebras over an operad.
Properness and combinatoriality is discussed in (ToënVezzosi):
in lemma 2.3.1.1 they state that $cdgAlg_+$ constitutes the first two items in a triple which they call an HA context .
this implies their assumption 1.1.0.4 which asserts properness and combinatoriality
Discussion of cofibrations in $dgAlg_{proj}$ is in (Keller).
Let $cdgAg_k$ be the projective model structure on commutative unbounded dg-algebras from above.
This is a proper model category. See MO discussion here.
Let $cdgAg_k$ be the projective model structure on commutative unbounded dg-algebras from above
For cofibrant $A \in cdgAlg_k$, the functor
preserves quasi-isomorphisms.
For $A,B \in cdgAlg_k$, their derived coproduct in $k Mod$ coincides in the homotopy category with the derived tensor product in $k Mod$: the morphism
is an isomorphism in $Ho(k Mod)$.
This follows by the above with (ToënVezzosi, assumption 1.1.0.4, and page 8).
The model structure on unbounded dg-algebras is almost a simplicial model category. See the section simplicial enrichment at model structure on dg-algebras over an operad for details.
Let $k$ be a field of characteristic 0. Let $\Omega^\bullet_{poly} : sSet \to (cdgAlg_k)^{op}$ be the functor that assigns polynomial differential forms on simplices.
For $A,B \in dgcAlg_k$ define the simplicial set
This extends to a functor
The functor $cdgAlg_k(-,-)$ satisfies the dual of the pushout-product axiom: for $i : A \to B$ any cofibration in $cdgAlg_k$ and $p : X \to Y$ any fibration, the canonical morphism
is a Kan fibration, which is acyclic if $i$ or $p$ is.
This implies in particular that for $A$ cofibrant, $cdgAlg_k(A,B)$ is a Kan complex.
The proof works along the lines of (Bousfield-Gugenheim 76, prop. 5.3). See also the discussion at model structure on dg-algebras over an operad.
We give the proof for a special case. The general case is analogous.
We show that for $A$ cofibrant, and for any $B$ (automatically fibrant), $cdgAlg_k(A,B)$ is a Kan complex.
By a standard fact in rational homotopy theory (due to Bousfield-Gugenheim 76, discussed at differential forms on simplices) we have that $\Omega^\bullet_{poly} : sSet \to (cdgAlg^+_k)^{op}$ is a left Quillen functor, hence in particular sends acyclic cofibrations to acyclic cofibrations, hence acyclic monomorphisms of simplicial sets to acyclic fibrations of dg-algebras.
Specifically for each horn inclusion $\Lambda[n]_k \hookrightarrow \Delta[n]$ we have that the restriction map $\Omega^\bullet_{poly}(\Delta[n]) \to \Omega^\bullet_{poly}(\Lambda[n]_k)$ is an acyclic fibration in $cdgAlg_k^*$, hence in $cdgAlg_k$.
A $k$-horn in $cdgAlg_k(A,B)$ is a morphism $A \to B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)$. A filler for this horn is a lift $\sigma$ in
If $A$ is cofibrant, then such a lift does always exist.
For $A \in cdgAlg$ cofibrant, $cdgAlg_k(A,B)$ is the correct derived hom-space
By the assumption that $A$ is cofibrant and according to the facts discussed at derived hom-space, we need to show that
is a resolution, or simplicial frame for $B$. (Notice that every object is fibrant in $cdgAlg_k$).
Since polynomial differential forms are acyclic on simplices (discussed here) it follows that
is degreewise a weak equivalence. It remains to show that $s A$ is fibrant in the Reedy model structure $[\Delta^{op}, cdgAlg_k]_{Reedy}$.
One finds that the matching object is given by
Therefore $s B$ is Reedy fibrant if in each degree the morphism
is a fibration. But this follows from the fact that $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ is a left Quillen functor (as discussed at differential forms on simplices).
We discuss a concrete model for the $(\infty,1)$-copowering of $(cdgAlg_k)^\circ$ over ∞Grpd in terms of an operation of $cdgAlg_k$ over sSet.
First notice a basic fact about ordinary commutative algebras.
In $CAlg_k$ the coproduct is given by the tensor product over $k$:
We check the universal property of the coproduct: for $C \in CAlg_k$ and $f,g : A,B \to C$ two morphisms, we need to show that there is a unique morphism $(f,g) : A \otimes_k B \to C$ such that the diagram
commutes. For the left triangle to commute we need that $(f,g)$ sends elements of the form $(a,e_B)$ to $f(a)$. For the right triangle to commute we need that $(f,g)$ sends elements of the form $(e_A, b)$ to $g(b)$. Since every element of $A \otimes_k B$ is a product of two elements of this form
this already uniquely determines $(f,g)$ to be given on elements by the map
That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are
For these derivations it is crucial that we are working with commutative algebras.
We have that the copowering of $A$ with the map of sets from two points to the single point
is the product morphism on $A$. And that the tensoring with the map from the empty set to the point
is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring
is the morphism between tensor powers of $A$ of the cardinalities of $S$ and $T$, respectively, whose component over a copy of $A$ on the right corresponding to $t \in T$ is the iterated product $A^{\otimes_k |f^{-1}\{t\}|} \to A$ on as many tensor powers of $A$ as there are elements in the preimage of $t$ under $f$.
The analogous statements hold true with $CAlg_k$ replaced by $cdgAlg_k$: for $S \in sSet$ and $A \in cdgAlg_k$ we obtain a simplicial cdg-algebra
by the ordinary degreewise copowering over Set, using that $cdgAlg_k$ has coproducts (equal to the tensor product over $k$).
This is equivalently a commutative monoid in simplicial unbounded chain complexes
By the logic of the monoidal Dold-Kan correspondence the symmetric lax monoidal Moore complex functor (via the Eilenberg-Zilber map) sends this to a commutative monoid in non-positively graded cochain complexes in unbounded cochain complexes
Since the total complex functor $Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)$ is itself symmetric lax monoidal (…), this finally yields
Define the functor
by
We have
This appears essentially (…) as (GinotTradlerZeinalian, def 3.1.1).
The (∞,1)-copowering of $(dgcAlg_k)^\circ$ over ∞Grpd is modeled by the derived functor of $CC$.
This follows from (GinotTradlerZeinalian, theorem 4.2.7), which asserts that the derived functor of this tensoring is the unique (∞,1)-functor, up to equivalence, satisfying the axioms of $(\infty,1)$-copowering.
The functor
preserves weak equivalences in both arguments.
This is essentially due to (Pirashvili). The full statement is (GinotTradlerZeinalian, prop. 4.2.1).
This means that the assumption for the copowering models of higher order Hochschild cohomology are satsified in $cdgAlg_k$ which are described in the section Pirashvili's higher Hochschild homology is satisfied:
this means that for $A \in cdgAlg$ and $S \in sSet$, $CC(S,A)$ is a model for the function $\infty$-algebra on the free loop space object of $Spec A$. See the section Higher order Hochschild homology modeled on cdg-algebras for more details.
Let $S \in \infty Grpd$ be presented by a degreewise finite simplicial set (which we denote by the same symbol).
Then the homotopy limit in $cdgAlg_k$ over the $S$-shaped diagram constant on $k$ is given by $\Omega^\bullet_{poly}(S)$.
We show dually that for degreewise finite $S$ the assignment $(S, Spec A) \mapsto Spec (\Omega^\bullet_{poly}(S) \otimes A)$ models the $\infty$-copowering in $cdgAlg_k^{op}$.
By the discussion at (∞,1)-copowering it is sufficient to to establish an equivalence
natural in $B$. Consider a cofibrant model of $B$, which we denote by the same symbol. The we compute with 1-categorical end/coend calculus
where all steps are isomorphisms and the dot denotes the ordinary 1-categorical copowering of the 1-category $cdgAlg^{op}$ over Set. In the last step we are using that the tensor product commutes with finite limits of dg-algebras. (This is where the finiteness assumption is needed).
Now we use that $\Omega^\bullet_{poly}$ preserves products up to quasi-isomorphism (as discussed here)
This being a weak equivalence between fibrant objects and since $B$ is assumed cofibrant, we have by the above discussion of the derived hom-functor (and using the factorization lemma) a weak equivalence
Since all this is natural in $B$, this proves the claim.
For $A \in cdgAlg_k$, a path object
for $A$ is given by
This follows along the above lines. The statement appears for instance as (Behrend, lemma 1.19).
For every ring spectrum $R$ there is the notion of algebra spectra over $R$. Let $R := H \mathbb{Z}$ be the Eilenberg-MacLane spectrum for the integers. Then unbounded dg-algebras (over $\mathbb{Z}$) are one model for $H \mathbb{Z}$-algebra spectra.
There is a Quillen equivalence between the standard model category structure for $H \mathbb{Z}$-algebra spectra and the model structure on unbounded differential graded algebras.
See algebra spectrum for details.
Commutative dg-algebras over a field $k$ of characteristic 0 constitute a presentation of E-infinity algebras over $k$ ([Lurie, prop. A.7.1.4.11]).
model structure on dg-algebras over an operad
model structure on dg-algebras
The cofibrantly generated model structure on commutative dg-algebras is surveyed usefully for instance on p. 6 of
This makes use of the general discussion in section 3 of
that obtains the model structure from the model structure on chain complexes.
A standard textbook reference is section V.3 of
An original reference seems to be
For general non-commutative (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in
This is also the structure used in
where aspects of its relation to the model structure on cosimplicial rings is discussed. (See monoidal Dold-Kan correspondence for more on this).
Disucssion of the model structure on unbounded dg-algebras over a field of characteristic 0 is in
A general discussion of algebras over an operad in unbounded chain complexes is in
A survey of some useful facts with an eye towards dg-geometry is in
Discussion of cofibrations in unbounded dg-algebras are in
The derived copowering of unbounded commutative dg-algebras over $sSet$ is discussed (somewhat implicitly) in
The commutative product on the dg-algebra of the higher order Hochschild complex is discussed in
The relation to E-infinity algebras is discussed in
Igor Kriz and Peter May, Operads, algebras, modules and motives , Astérisque No 233 (1995)
Jacob Lurie, section 7.1 of Higher algebra (pdf)
The relation between commutative and non-commutative dgas is further discussed in
Ilias Amrani, Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view (arXiv:1401.7285)
Ilias Amrani, Rational homotopy theory of function spaces and Hochschild cohomology (arXiv:1406.6269)
For more see also at model structure on dg-algebras over an operad.
Discussion of homotopy limits and homotopy colimits of dg-algebras is in