and
and
nonabelian homological algebra
A dg-algebra, or differential graded algebra, is equivalently
An associative algebra $A$ which is in addition graded algebra and a differential algebra in a compatible way (with the differential derivation being of degree $\pm$ 1);
a monoid in the symmetric monoidal category of (possibly unbounded) chain complexes or cochain complexes with its standard structure of a monoidal category by the tensor product of chain complexes;
For the case of chain complexes we also speak of chain algebras.
For the case of cochain complexes we also speak of cochain algebras.
Recall
that the standard tensor product on (co)chain complexes is given by
with the differential $d_{A\otimes B} = d_A\otimes B + A\otimes d_B$.
that a monoid in a monoidal category $C$ is an object $A$ in $C$ together with a morphism
that is unital and associative in the obvious sense.
This implies that a dg-algebra is, more concretely, a graded algebra $A$ equipped with a linear map $d : A \to A$ with the property that
$d\circ d = 0$
for $a \in A$ homogeneous of degree $k$, the element $d a$ is of degree $k+1$ (for monoids in cochain complexes) or of degree $k-1$ (for monoids in chain complexes)
for all $a,b \in A$ with $a$ homogeneous of degree k the graded Leibniz rule holds
$d (a \cdot b) = (d a) \cdot b + (-1)^k a \cdot (d b)$.
A pre-graded algebra- (pre-ga) or $\mathbb{Z}$-graded algebra is a pre-gvs, $A$, together with an algebra multiplication satisfying $A_p.A_q \subseteq A_{p+q}$ for any $p,q$. The relevant morphisms are pre-gvs morphisms which respect the multiplication. This gives a category $pre GA$.
An augmentation of a pre-ga, $A$, is a homomorphism $\varepsilon : A \to k$. The augmentation ideal of $(A,\varepsilon)$ is $ker \varepsilon$ and will also be denoted $\bar{A}$. The pair $(A,\varepsilon)$ is called an augmented pre-ga.
A morphism $f:(A,\varepsilon)\to (A',\varepsilon')$ of augmented pre-gas is a homomorphism $f : A \to A'$ (thus of degree zero) such that $\varepsilon = \varepsilon ' f$. The resulting category will be written $pre \varepsilon GA$.
If $A$, $A'$ are two pre-gas, then $A\otimes A'$ is a pre-ga with
for homogeneous $a,b \in A$, $a', b' \in A'$.
If $\varepsilon, \varepsilon$ are augmentations of $A$ and $A'$ respectively, then $\varepsilon\otimes \varepsilon'$ is an augmentation of $A\otimes A'$.
Let $A$ be a pre-ga. An (algebra) derivation of degree $p\in \mathbb{Z}$ is a linear map $\theta \in Hom_p(A,A)$ such that
for homogeneous $a,b \in A$.
A derivation $\theta$ of an augmented algebra, $(A, \varepsilon)$, is an algebra derivation which, in addition, satisfies $\varepsilon \theta = 0$.
Let $Der_p(A)$ be the vector space of derivations of degree $p$ of $A$, then $Der(A) = \bigoplus_p Der_p(A)$ is a pre-gvs.
In the case of upper gradings, an element of $Der_p(A)$ sends $A^n$ into $A^{n-p}$.
A differential $\partial$ on an (augmented) pre-ga is a derivation of the (augmented) algebra of degree -1 such that $\partial\circ\partial = 0$.
The pair $(A,\partial)$ is called a pre-differential graded algebra (pre-dga). If $A$ is augmented, then $(A,\partial)$ will be called an augmented pre-dga $pre \varepsilon dga$.
If $(A,\partial)$ and $(A',\partial')$ are pre-dgas, then $(A,\partial)\otimes (A',\partial')$, with the conventions already noted, is one as well.
A morphism of pre-dgas (or pre-$\varepsilon$-dgas) is a morphism which is both of pre-gdvs and of pre-gas (with $\varepsilon$ as well if used). This gives categories $pre DGA$ and $pre \varepsilon DGA$.
A pre-ga $A$ is said to be graded commutative if $ab = (-1)^{|a||b|}ba$ for each pair, $a, b$, of elements of $A$ of homogeneous degree.
Commutativity is preserved by tensor product.
We get obvious full subcategories $pre CDGA$ and $pre \varepsilon CDGA$ corresponding to the case with differentials.
A cdga is a negatively graded pre-cdga (in upper grading), $A= \bigoplus_{p\geq 0} A^p.$
There is an augmented variant, of course. These definitions give categories $CDGA$, etc.
See at differential graded-commutative algebra.
An $\varepsilon$cdga $(A,d)$ is $n$-connected (resp. cohomologically $n$-connected if $\bar{A}^p = 0$ for $p\leq n$, (resp. $\overline{H(A,d)}^p = 0$ for $p\leq n$). This gives subcategories $CDGA^n$ and $CDGA^{c n}$.
A filtration on a pre-ga, $A$, is a filtration on $A$, so that $F_p A \subseteq F_{p+1}A$, $F_p A.F_n A \subseteq F_{p+n}A$ (and, if $A$ is differential, also $\partial F_p A \subseteq F_p A$).
Let $A$ be an augmented pre-ga and denote by
the iterated multiplication. The decreasing word length filtration, $F^p A$ is given by:
$Q(A) = \bar{A}/Im\bar{\mu}$ is the space of indecomposables of A.
If $(A,\partial)$ is an augmented pre-dga, $F^p A$ is stable under $\partial$ and we get $Q(\partial)$ is a differential on $Q(A)$ and hence we get a functor $Q: pre \varepsilon DGA\to pre DGVS.$
Given a pre-gvs, $V$, the tensor algebra generated by $V$ is given by $T(V) = \bigotimes_{n\geq 0}V^{\otimes n}$.
The augmentation sends $V$ to 0. $V^{\otimes n}$ is given the tensor product grading, and the multiplication is given by the tensor product.
If $A$ is a pre-ga and $f: V\to A$, a morphism to the underlying pre-gvs of $A$, there is a unique extension $\hat{f} :T(V)\to A$, which is a morphism of pre-gvs.
This is the tensor product of the exterior algebra on the odd elements and the symmetric algebra on the even ones:
It satisfies $\bigwedge(V \oplus W) \cong (\bigwedge V)\oplus (\bigwedge W)$.
If $A$ is a pre-cga, any morphism, $f : V\to A$, to the underlying pre-gvs of $A$, has a unique extension to a pre-cga morphism $\bar{f} :\bigwedge V \to A$.
If $(e_\alpha)_{\alpha \in I}$ is a homogeneous basis for $V$, $\bigwedge V$ and $T(V)$ may be written $\bigwedge((e_\alpha)_{\alpha \in I})$ and $T((e_\alpha)_{\alpha \in I})$ respectively.
Note:
$T(V)$ is a non-commutative polynomial algebra,
$\bigwedge V$ is a commutative polynomial algebra.
On $\bigwedge V$ (resp. $T(V)$) write
where $\bigwedge^k V$ is the subspace generated by all $v_1\wedge \ldots \wedge v_k$ with $v_1 \in V$. Then $F^p \bigwedge V = \bigwedge^{\geq p} V = \bigoplus_{k\geq p}\bigwedge^k V$, resp. $T^k (V) = V^{\otimes k}$ and $F^p T(V) = T^{\geq p}(V) = \bigoplus_{k\geq p} T^k (V)$).
If $(\bigwedge V,d)$ is a pre-cdga, which is free as a pre-cga on a fixed $V$, then $d$ is the sum of derivations $d_k$ defined by the condition $d_k (V) \subseteq \bigwedge^k V$. There is an isomorphism between $V$ and $Q(\bigwedge V)$, which identifies $d_1$ with $Q(d)$. The derivation $d_1$ (resp. $d_2$) is called the linear part (resp. quadratic part) of $d$.
If $(A,d)$ and $(A',d')$ are two cdgas, their (categorical) sum (i.e. coproduct) is their tensor product, $(A,d)\otimes(A',d' )$, whilst their product is the ‘direct sum’, $(A,d)\oplus (A',d' )$.
Given a permutation $\sigma$ of a graded object $(x_1, \ldots, x_p)$, the Koszul sign, $\varepsilon(\sigma)$ is defined by
in $\bigwedge(x_1, \ldots, x_p )$. We note that although we write $\varepsilon(\sigma)$, $\sigma$ does not suffice to define it as it depends also on the degrees of the various $x_i$.
Baues (in his book on Algebraic Homotopy) has suggested using the terminology chain algebra for positively graded differential algebras and cochain algebras for the negatively graded ones. This seems to be a very useful convention.
There is a standard model category structure on $dgAlg$.See model structure on dg-algebras.
The monoidal Dold-Kan correspondence effectively identifies non-negatively graded chain complex algebras with simplicial algebras, and non-negatively graded cochain complex algebras with cosimplicial algebras.
Since cosimplicial algebras have a fundamental interpretation dual to ∞-space, as described at ∞-quantity, this can be understood as explaining the great role differential graded algebras are playing in various context, suchh as notably in
Dually, a comonoid in chain complexes is a dg-coalgebra.
A dga $A$ is homologically smooth if as a dg-bimodule $_A A_A$ over itself it has a bounded resolution by finitely generated projective dg-bimodules.
A dg-algebra $A$ is a formal dg-algebra if there exists a morphism
to its chain (co)homology (regarded as a dg-algebra with trivial differential) that is a quasi-isomorphism.