differential graded algebra

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###Context###
#### Differential-graded objects
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[[!include differential graded objects - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
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## Definition

### Abstract definition

A __dg-algebra__, or __[[differential graded algebra]]__, is equivalently

1. 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);

1. a [[monoid]] in the [[symmetric monoidal category]] of (possibly unbounded) [[chain complexes]] or [[cochain complex]]es 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
$$
(A\otimes B)_n = \sum_{i+j=n} A_i\otimes B_j,
$$
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

  $$
    \cdot : A \otimes A \to A
  $$

  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

  $d (a \cdot b) = (d a) \cdot b + (-1)^k a \cdot (d b)$.

The dg-algebras form a category, [[dgAlg]].

### Detailed component definition

#### Pre-graded algebras

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 $pre \varepsilon GA$.

#### Tensor product

If $A$, $A'$ are two pre-gas, then $A\otimes A'$ is a pre-ga with

$$(a\otimes a')(b\otimes b') = (-1)^{|a' | |b|} a b \otimes a' b'$$

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 $A\otimes A'$.

#### Derivations

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

$$\theta(ab) = \theta(a)b + (-1)^{p|a|}a\theta(b)$$

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.

#### N.B.

In the case of upper gradings, an element of $Der_p(A)$ sends $A^n$ into $A^{n-p}$.

#### Pre-DGAs

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 $pre \varepsilon DGA$.

#### Commutative graded algebras (CGA)

A pre-ga $A$ is said to be _[[graded commutative algebra|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.

#### CDGAs

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]]_.

#### $n$-connectivity

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 $CDGA^{c n}$.

#### Filtrations

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 $\partial F_p A \subseteq F_p A$).

##### Example: Word length filtration.

Let $A$ be an augmented pre-ga and denote by

$$\bar{\mu}^p : \bigotimes^{p+1}\bar{A} \to \bar{A},$$

the iterated multiplication.  The **decreasing word length filtration**, $F^p A$ is given by:

$$F^0 A = A, \quad F^p A = Im\bar{\mu}^{(p-1)}  if p\geq 1.$$

$Q(A) = \bar{A}/Im\bar{\mu}$ 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  $Q: pre \varepsilon DGA\to pre DGVS.$

#### Free GAs: $T(V)$, the tensor algebra

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.
+-- {: .lemma }
###### Lemma (classical: freeness of $T(V)$, $T$ is a left adjoint)
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.
=--

#### Free CGAs: $\bigwedge V$

This is the tensor product of the exterior algebra on the odd elements and the symmetric algebra on the even ones:

$$\bigwedge V = E(\bigoplus V_{2p+1})\otimes S(\bigoplus V_{2p}).$$

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.

#### Word length filtrations on $\bigwedge V$ and $T(V)$.

On $\bigwedge V$ (resp.  $T(V)$) write

$$\bigwedge V = \bigoplus_{k\geq 0}\bigwedge^k V,$$

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 $d$.

#### Sum and Product of CDGAs.

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', $(A,d)\oplus (A',d' )$.

#### Koszul convention

Given a permutation $\sigma$ of a graded object $(x_1, \ldots, x_p)$, the _Koszul sign_, $\varepsilon(\sigma)$ is defined by

$$x_1\wedge \ldots \wedge x_p = \varepsilon(\sigma)x_{\sigma(1)} \wedge \ldots \wedge x_{\sigma(p)}$$

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 $x_i$.

### Terminology

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.

### Enrichment of dg-algebras over dg-coalgebras

Given a dg-coalgebra $C$ and a dg-algebra $B$,
the vector space of linear maps $C\to B$
admits a dg-algebra structure
using the coproduct on $C$ and product on $B$.
This dg-algebra is known as the __convolution algebra__ of $C$ and $B$
and is denoted by $[C,B]$.

The resulting functor
$$[-,-]: dgCoalg^op \times dgAlg \to dgAlg$$
is a part of an [[adjunction in two variables]], whose other adjoints are
$$\triangleright: dgCoalg\times dgAlg\to dgAlg$$
(the Sweedler product)
and
$$\{-,-\}:dgAlg^op\times dgAlg\to dgCoalg$$
(the Sweedler hom).

Taken together, these three functors
turn the [[symmetric monoidal category]]
of dg-algebras into an [[enriched category]]
over the [[closed symmetric monoidal category]]
of dg-coalgebras.

See [Anel and Joyal](#AnelJoyal) for more information.

## Related concepts

* [[differential algebra]]

### Model category structure

There is a standard [[model category]] structure on $dgAlg$.See [[model structure on dg-algebras]].

### Cosimplicial 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 algebra]]s.

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

* [[rational homotopy theory]].

### dg-coalgebra

Dually, a [[comonoid]] in [[chain complex]]es is a [[dg-coalgebra]].

### Homological smoothness

A dga $A$ is __[[homologically smooth dga|homologically smooth]]__ if as a dg-bimodule $_A A_A$ over itself it has a bounded resolution by finitely generated projective dg-bimodules.

### Formal dg-algebra

A dg-algebra $A$ is a [[formal dg-algebra]] if there exists a morphism

$$
  A \to H^\bullet(A)
$$

to its [[chain homology and cohomology|chain (co)homology]] (regarded as a dg-algebra with trivial differential) that is a [[quasi-isomorphism]].

### Curved dg-algebra

* [[curved dg-algebra]]

## References

* {#AnelJoyal} [[Mathieu Anel]], [[André Joyal]], _Sweedler theory of (co)algebras and the bar-cobar constructions_, [pdf](http://mathieu.anel.free.fr/mat/doc/Anel%20-%20Sweedlertheory_v1.1.pdf)

[[!redirects differential graded algebras]]
[[!redirects differential graded module]]
[[!redirects chain algebra]]
[[!redirects cochain algebra]]
[[!redirects dg-algebra]]
[[!redirects dg-algebras]]

[[!redirects differential-graded algebra]]
[[!redirects differential-graded algebras]]

[[!redirects graded differential algebra]]
[[!redirects graded differential algebras]]

[[!redirects commutative dg-algebra]]
[[!redirects commutative dg-algebras]]

[[!redirects commutative differential graded algebra]]
[[!redirects commutative differential graded algebras]]


differential graded algebras and differential graded Lie algebras-relationships

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###Context###
#### Differential graded objects
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[[!include differential graded objects - contents]]
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This entry discusses some  relationships between  [[differential graded algebra]]s and [[differential graded Lie algebra]]s (DGLA)

#Contents#
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##Idea

Here we will look at the relationships between the two topics in the title! We will explore some of the functors linking the two categories.

## The functors $C^*$ and $L_*$.

### Cochain functors $C^* : DGLA\to CDGA$

Let $(V,\partial)$ be a differential graded vector space.  We let

$$C^r(V) = \bigoplus_{p+q=r}C^{p,q} (V),$$

where $C^{p,q} (V) \subset Hom^{p+q}(\otimes^p sV, k)$ is the subspace of (graded) _symmetric_ functions:

$$f(su_1,\ldots,su_p) = \varepsilon(\sigma)f(su_{\sigma(1)},\ldots,su_{\sigma(p))}),$$

for each permutation $\sigma$, having $\varepsilon(\sigma)$ as its Koszul sign.

The product of $f\in C^{p,q}$ and $g\in C^{r,s}$ is

$$(f\wedge g)(su_1, \ldots,su_{p+r}) = \sum_{\sigma}\varepsilon(\sigma)f(su_{\sigma(1)},\ldots, su_{\sigma(p)})g(su_{\sigma(p+1)},\ldots,su_{\sigma(p+r)}),$$

where the sum is over all $(p,r)$-[[shuffles]], $\sigma$ and $\varepsilon(\sigma)$ is the Koszul sign of :

$$(f,g,su_1,\ldots, su_{p+r})\to (f,su_{\sigma(1)},\ldots, su_{\sigma(p)},g,su_{\sigma(p+1)},\ldots,su_{\sigma(p+r)}).$$

With this product, $C(V)$ is a commutative graded algebra.

Now let $(L,\partial)$ be a [[differential graded Lie algebra]]. There are two derivations with square zero, which anti-commute defined by

$$\array{
(d_1f)(su_1,\ldots,su_n) &=&(-1)^{|f|}\sum_{i=1}^p(-1)^{|su_1|+ \ldots + |su_{i-1}|}f(su_1,\ldots,s\partial u_i,\ldots,su_p) \\
(d_2f)(su_1,\ldots,su_n)&=&(-1)^{|f|}\sum_{i\lt j}\varepsilon_{ij}(-1)^{|su_i|}f(s[u_i,u_j],su_1,\ldots, s\hat{u_i}, \ldots,s\hat{u_j},\ldots,su_p),}$$

where $\varepsilon_{ij}$ is the Koszul sign of

$$(su_1, \ldots, su_p)\to (su_i,su_j,su_1,\ldots, s\hat{u_i}, \ldots,s\hat{u_j},\ldots,su_p),$$

where ''$s\hat{u_i}$'' indicates that the element $su_i$ has been omitted.  Putting $d = d_1 + d_2$, we have a commutative dga, $C^*(L,\partial)$, $d_1$ and $d_2$ being called, respectively, the linear and quadratic part of $d$.

When $L$ has finite type, this algebra can be identified with $\bigwedge s\# L$.  Before making this precise, we will develop the extension of duality to the tensor algebra.

Let $\# V$ be the dual of a gvs $V$ of finite type, the elements of the tensor algebra $T(\#V)$ can be interpreted as multilinear functions on $V$

$$\langle x_1\otimes \ldots x_p,y_1,\ldots , y_p\rangle  = \varepsilon \prod_{i=1}^{p}\langle x_i,y_i\rangle ,$$

where $\varepsilon$ is the Koszul sign of

$$(x_1,\ldots,x_p,y_1,\ldots,y_p)\to (x_1,y_1,\ldots, ,x_p,y_p).$$

The free commutative graded algebra $\bigwedge\# V$ then can be identified as the algebra $C^*(V)$ defined above, via the canonical injection $\chi : \bigwedge \# V\to T(\# V)$.

## The cochains of a dgla, $(L,\partial)$, of finite type.

Applying the preceding conventions, we get

$$\array{C^*(L,\partial) &=& (\bigwedge s^{-1}\#L, d = d_1+d_2);\\
\langle d_1s^{-1}z;sx\rangle  &=& - \langle z;\partial x\rangle ;\\
\langle d_2s^{-1}z;sx_1,sx_2\rangle &=& (-1)^{|x_1|}\langle z;[x_1,x_2]\rangle }$$
for  $ z\in \#L, x_i\in L, x\in L.$

## Remark

The differential $d$ is a sum of a linear differential $d_1$ and a quadratic differential, $d_2$.  By a word length argument, $d^2 = 0$ is equivalent to

$\left\{\array{
(a) & d_1^2 = 0\\
(b) & d_1d_2 + d_2d_1 = 0\\
(c) & d_2^2 = 0
}\right.$

If one lifts the definitions of $d_1$ and $d_2$ back to the Lie algebra $(L,\partial)$, one has

(a) is equivalent to $\partial^2 = 0$;

(b) is equivalent to '$\partial$ is a Lie algebra derivation';

(c) is equivalent to 'the bracket satisfies the Jacobi identity'.

Conversely if $(\bigwedge V,d_1 + d_2)$ is a cdga, free on $V$ as a cga, we can define a dgla structure on $s^{-1}\#V$ using the above formulae.

## Definition

$CDGA_{fg}$ will denote the subcategory of $CDGA$ with

* as objects the cdgas $(\bigwedge V, d)$ of finite type, which are free as cgas on a gvs of generators $V = \sum_{k\geq 1}V^k$ with differential given by a linear part and a quadratic part;

* as arrows, the cdga morphisms that send generators to generators.

The preceding remark translates as

+-- {: .num_prop}
######Proposition
$C^*$ is an isomorphism between $CDGA_{fg}$ and the full subcategory of $DGLA$ formed by Lie algebras of finite type.
=--

Let $\overline{C}$ be the inverse functor.

Let $(\bigwedge V, d)$ and $(\bigwedge V^\prime, d^\prime)$ be two cdgas of finite type, which are free as cgas.  Denote by $d_1$, the linear part of $d$, the differential $\overline{d}$ induced by $d$ on $H(\bigwedge V, d_1) = \bigwedge H(V,d_1)$, has zero linear part.  Its quadratic part $\overline{d_2}$ is thus a differential.  It determines, via $\overline{C}$, a gla structure on $s^{-1}\#H(V,d_1)$

If $(\bigwedge V, d) = C^*(L,\partial)$, we can easily check that the identification of $s^{-1}\#H(V,d_1)$ with $H(L,\partial)$ is a Lie algebra isomorphism.

In particular, if $d$ is decomposable, $(d(V)\subseteq \bigwedge^{\geq 2}V)$, then $s^{-1}\#V$ has a Lie algebra structure defined by $\overline{C}(\bigwedge V,d_2)$.  Applying our earlier description of this, we have that the bracket on $s^{-1}\#V$ is characterised by

$$\langle v;s[s^{-1}x_1,s^{-1}x_2]\rangle  = (-1)^{|x_2|+1}\langle d_2v;x_1,x_2\rangle ,v \in V, x_i\in \#V.$$

Now let $\Phi : (\bigwedge V, d) \to (\bigwedge V^\prime, d^\prime)$ be a morphism of cdgas, then

$$s^{-1}\#H(Q(\Phi)): s^{-1}\#H(\bigwedge V^\prime, d^\prime_1) \to  s^{-1}\#H(\bigwedge V, d_1)$$

is a Lie algebra morphism.

## The functor $L_*$

Let $(A,d)$ be a connected cdga of finite type.  Recall that $\mathbb{L}(s^{-1}\# A) \subset T(s^{-1}\#A) $ denotes the free Lie algebra on $s^{-1}\#A$ and note that $T^p(s^{-1}\#A)  = \#T^p(sA)$.  The linear and quadratic derivations $\partial_1$ and $\partial_2$ respectively are determined on $\mathbb{L}(s^{-1}\#A) $ by

$\array{
\langle sa;\partial_1s^{-1}b\rangle  &=& -\langle da;b\rangle ,\\
\langle sa_1.sa_2;\partial_2s^{-1}b\rangle & =& (-1)^{|a_1|}\langle a_1.a_2;b\rangle ; \quad a\in A, a_i\in A,b\in \#A.}$

The three conditions

(a) $d^2 = 0$,

(b) $d$ is an algebra derivative,

(c) $A$ is associative,

are equivalent, respectively, to

(a) $\partial_1^2 = 0$,

(b) $\partial_1\partial_2 + \partial_2\partial_1 = 0$,

(c) $\partial_2^2 = 0$.

## Definition

$L_* $ is the functor with domain the full subcategory $f-CDGA_0$ of $CDGA$ formed by the connected cdgas of finite type and with codomain $DGLA$.  It is defined by $L_*(A,d) = (\mathbb{L}(s^{-1}\#A,\partial = \partial_1 + \partial_2$).

Note: $L_* = P\circ \# \circ B$. [$B$ , the bar construction, $\#$ duality, and $P$, primitives.]

## Definition

Let $DGLA_{fq}$ be the subcategory of $DGLA$ having as objects the dglas $(\mathbb{L}(V),\partial)$, which, as graded Lie algebras, are free on a vector space of generators $V$ of finite type, having a differential that is the sum of a linear and a quadratic part, and for arrows the dgla morphisms that send generators to generators.

The above remarks show

+-- {: .num_prop}
######Proposition
$L_*$ is an isomorphism between $DGLA_{fq}$ and $f-CDGA$_0$.
=--

+-- {: .query}

 [[Urs Schreiber|Urs]]: eventually it would be nice if we could put these very detailed discussions in context with existing entries such as [[L-infinity-algebra]] and [[NQ-supermanifold]] which do mention closely related things. Or, in as far as these existing entries are to be regarded as not so good in the light of the present discussiion, we should point out what's amiss and eventually improve things.

[[Tim Porter|Tim]]: I agree, but before that I am hoping that others will help me to sort out the relationships here and with those other areas as I am no expert in this.
=--

We denote by $\overline{L}$, the inverse functor.

Let $(L(W), \partial)$ be a dgla, that is, as a gla, free on a finite type vector space $W$.  Denote by $\partial_1$, the linear part of $\partial$, and $\overline{\partial}_2$, the quadratic part of the differential induced by $\partial$ on $\mathbb{L}(H(W,\partial_1))$.  The space $k \oplus s^{-1}\#H(W,\partial_1)$ has then a commutative graded algebra structure induced via $\overline{L}$.  If $(\mathbb{L}(W),\partial) = L_*(A,d)$, the identification of $k \oplus s^{-1}\#H(W,\partial_1)$  with $H(A,\partial)$ is an algebra isomorphism.

In particular, if $\partial$ is decomposable, $(\partial(W) \subset \mathbb{L}^{\geq 2}(W)$), then $k \oplus s^{-1}\#W$ has a graded commutative algebra  structure defined by $\overline{L}(\mathbb{L}(W),\partial_2)$.  Applying earlier results, the multiplication on $s^{-1}\#W$ is characterised by

$\langle s(s^{-1}x_1.s^{-1}x_2;w\rangle  = (-1)^{|x_2| + 1}\langle x_1,x_2;\partial_2 w\rangle ; \quad w\in W,x_i \in\#W.$

[[!redirects differential graded algebras and differential graded Lie alg]]


differential graded categories > history

< [[differential graded categories]]
[[!redirects differential graded categories -- history]]


differential graded category > history

< [[differential graded category]]
[[!redirects differential graded category -- history]]


differential graded coalgebra

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###Context###
#### Differential-graded objects
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[[!include differential graded objects - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
=--
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#Contents#
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{:toc}

## Definition

### Abstract definition

A _dg-coalgebra_ is a [[comonoid]] in the [[category]] of [[chain complex]]es.

Equivalently, this is a graded [[coalgebra]] $C$ equipped with a [[coderivation]]

$$
  D : C \to C
$$

that is of degree -1 and squares to 0,

$$
  D^2 = 0
  \,.
$$

### Detailed component definition

#### Pre-coalgebra

 A _pre-graded coalgebra_ (pre-gc), $(C,\Delta, \varepsilon)$, is a [[graded vector space|pre-gvs]] $C$ together with linear maps of degree 0

$$\Delta: C\to C\otimes C,    \varepsilon : C\to k,$$

such that the obvious (usual) diagrams commute.  (When there is no ambiguity, we may write $C$ instead of $(C,\Delta, \varepsilon)$.)

The field $k$ is a coalgebra for the canonical isomorphism $k\to k\otimes k$ with $\varepsilon = id_k$.

A morphism $\psi : C \to C'$ of pre-gcs is a linear mapping of degree zero such that

$$(\psi \otimes \psi)\circ \Delta = \Delta' \circ \psi   and    \varepsilon = \varepsilon' \circ \psi.$$

The linear counit mapping $\varepsilon :C \to k$ is always a morphism of pre-gcs.

#### Coaugmentations

A _coaugmentation_ of a pre-gc is a morphism $\eta : k \to C$.  We will write $1$ for $\eta(1)$.

The cokernel $\bar{C}$ of $\eta$ can be identified with $Ker \varepsilon$ and so can be considered as a subspace of $C$.

The reduced diagonal $\bar{\Delta} : \bar{C} \to \bar{C}\otimes \bar{C}$, induced by $\Delta$ is defined by $\Delta x = 1\otimes x + x\otimes 1 + \bar{\Delta }x$.  The _vector space of primitives_ of $C$, denoted $P(C)$, is the kernel of the reduced diagonal.

A morphism of coaugmented pre-gcs, $\psi : (C,\eta)\to (C',\eta')$, is a morphism of the pre-gcs which satisfies $\eta' = \psi \circ \eta$. It preserves primitives.

#### The commutation morphism

Let $V$ and $V'$ be two pre-gvs.  The commutation morphism
$$\tau : V\otimes V' \to V'\otimes V$$
is defined by $\tau( v\otimes v') = (-1)^{|v||v'|} v'\otimes v$, on homogeneous elements.

#### Tensor product of pre-gcs.

Let $(C,\Delta, \varepsilon)$ and $(C',\Delta', \varepsilon')$ be two pre-graded coalgebras.  The mappings
$$C\otimes C'\stackrel{\Delta\otimes \Delta'}{\to}C\otimes C\otimes C'  \otimes C'\stackrel{C\otimes \tau \otimes C}{\to}(C\otimes C')\otimes (C\otimes C')$$

and

$$C\otimes C'\stackrel{\varepsilon\otimes \varepsilon'}{\to}k \otimes k\stackrel{\cong}{\to}k$$

give $C\otimes C'$ a pre-gc structure.

If $\eta$ and $\eta'$ are coaugmentations of $C$ and $C'$ respectively, then $\eta\otimes\eta'$ defines a coaugmentation of $C\otimes C'$.

#### Coderivations of pre-graded coalgebras

+--{: .query}
[[Tim Porter|Tim]]: These are called derivations by some sources, but I think that they are the coderivations of other workers. (to be checked)
=--

If $C$ is a pre-gc, a _coderivation_ of degree $p\in \mathbb{Z}$, is a linear map $\theta \in Hom_p(C,C)$ such that

$$\Delta \circ \theta = (\theta \otimes id_C + \tau \circ(\theta \otimes id_C)\circ \tau)\circ \Delta,  and  \varepsilon\circ \theta = 0.$$

A coderivation $\theta$ of a coaugmented pre-gc $(C,\eta)$ is a coderivation of $C$ such that $\theta\circ \eta = 0$.

#### Differential graded coalgebras

A _differential_ $\partial$ on a (coaugmented) pre-gc, $C$, is a coderivation of degree -1 such that $\partial\circ\partial = 0$.

The pair, $(C, \partial)$ is called a _differential (coaugmented) pre-graded coalgebra_ (pre-dgc).  Its homology $H(C,\partial)$ will be a pre-gc.

If $(C,\partial)$ and $(C',\partial')$ are two pre-dgcs, then their tensor product $(C,\partial)\otimes(C',\partial')$ is a pre-gdgc with the structures defined earlier.

A _morphism_ of (coaugmented) pre-dgcs is a morphism both of (coaugmented) pre-gcs and of pre-dgvs.  We denote the resulting categories by $pre DGC$ (resp. $pre \eta DGC$).

#### Cocommutative pre-dgcs

A pre-gc $C$ is _cocommutative_ if $\tau\circ\Delta = \Delta$, similarly for a pre-dgc.  The subcategories of cocommutative d.g. coalgebras will be denoted $pre CDGC$ (resp. $pre \eta CDGC$).

#### CDGC

A _cocommutative differential graded coalgebra_ is a pre-cdgc on a graded vector space of lower grading (so $C_p = 0$ for $p \lt 0$).  This gives categories $CDGC$ (resp. $\eta CDGC$).

#### $n$-connected $\eta$ cdgcs

A coaugmented cdgc $(C, \partial)$ is _$n$-connected_ if $\bar{C}_p = 0$ for $p\leq n$.

This gives a category $CDGC_n$.  Any connected (i.e. $0$-connected) cdgc is canonically coaugmented with $\bar{C}$ coinciding with $C_+$.

#### Hom-algebras and duals

Let $(C,\partial)$ be a pre-cdgc and $(A,d)$ a pre-cdga.  The pre-dgvs $(Hom(C,A),D)$ is a pre-cdga for the usual differential and the multiplication $f.g = \mu\circ (f\otimes g)\circ \Delta$,
$$ C\stackrel{\Delta}{\to}C\otimes C\stackrel{(f\otimes g)}{\to}A\otimes A\stackrel{\mu}{\to} A,$$

for $f,g \in Hom(C,A)$.

In particular $\#(C,\partial)= (Hom(C,k),D)$  defines a functor from $pre CDGC$ to $pre CDGA$, which commutes with homology and is such that $\# CDGC_n \subseteq CDGA^n$.  Conversely, if $(A,d)$ is a pre-cdga of finite type, $\#(A,d)$ is a pre-cdgc.

#### Coalgebra filtrations

Let $(C,\partial)$ be a pre-dgc. A _coalgebra filtration_ (resp. _differential coalgebra filtration_) of $(C,\partial)$ is a family of subspaces $F_p C$, $p\in \mathbb{Z}$ such that

$$F_p C\subseteq F_{p+1} C, \quad \Delta F_p C \subseteq \sum_k F_k C\otimes F_{p-k} C, \quad (resp.\quad and \quad \partial F_p C\subseteq F_p C).$$

#### Filtrations of the primitives

Let $(C,\eta)$ be a coaugmented pre-gc, $\bar{C}$ the cokernel of $\eta$, $\bar{\Delta}$, the reduced diagonal.

The iteration of $\bar{\Delta}$ is defined by

$$\bar{\Delta}^1 = \bar{\Delta}; \quad \bar{\Delta}^p = (\bar{\Delta}\otimes \bar{C} \otimes \ldots \bar{C}) \otimes \bar{\Delta}^{(p-1)}.$$

The (increasing) filtration of the primitives is $F_p C = Ker\bar{\Delta}^p$, $p\geq 1$. It is a graded coalgebra filtration.

If $(C,\partial, \eta)$ is a coaugmented pre-dgc, each $F_p C$ is stable under the differential and, in particular, $F_1 = P(C)$.  $P$ thus defines a functor from $pre \eta CDGC$ to $pre DGVS$.

Let $\mu$ be the comultiplication of the pre-ga $\# C$, the dual of $C$.  Elementary results on duality show, for finite type:  $Im\bar{\mu}^p$ is the orthogonal complement of $Ker\bar{\Delta}^p$, so, in particular, $Q(\# C) =\# P(C)$.

Let $(C,\eta)$ be a coaugmented pre-gc and $F_p C$ the filtration of its primitives.  $C$ is _conilpotent_ if $C = \bigcup_k F_k C$.
A connected coalgebra is conilpotent and conilpotency is preserved by tensor product.

#### The pre-gc structure on $T(V)$

We will denote by $T'(V)$, the gvs $T(V)$, together with the coalgebra structure in which the reduced diagonal is given by

$$\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n).$$

The counit and the coaugmentation are the natural mappings $T(V)\to k$ and $k\to T(V)$, respectively.

The coalgebra $T' (V)$ is non-commutative if $dim V\gt 1$ and has $V$ as its vector space of primitives.

If $C$ is a conilpotent pre-gc, then any morphism $f : C\to V$ of pre-gvs for which $f(1) = 0$, admits a unique lifting to a pre-gc morphism $\hat{f}:C \to T'(V)$.

#### Shuffles

A _$(p,q)$-[[shuffle]]_ $\sigma$ is a permutation of $\{1, \ldots, p+q\}$ such that

$$\sigma(i) \lt \sigma(j) \quad if \quad 1\leq i\lt j\leq p \quad or \quad p+1 \leq i \lt j\leq p+q.$$

#### The pre-cgc structure on $\bigwedge V$

We will denote $\bigwedge' V$, the gvs $\bigwedge V$ together with the coalgebra structure in which the reduced diagonal is given by

$$\bar{\Delta}(v_1\wedge \ldots \wedge v_n) = \sum_{p=1}^{n-1} \sum_\sigma\varepsilon(\sigma)(v_{\sigma(1)}\wedge \ldots \wedge v_{\sigma(p)})\otimes(v_{\sigma(p+1)}\wedge \ldots \wedge v_{\sigma(n)}),$$

in which the second sum is over all $(p,n-p)$-shuffles and $\varepsilon(\sigma)$ is the Koszul sign of $\sigma$.

The counit and coaugmentation are the natural mappings $\bigwedge V \to k$, and $k\to \bigwedge V$ respectively.

If $C$ is a conilpotent pre-cgc, any pre-gvs morphism $f:C \to V$ for which $f(1) = 0$ admits a unique lifting to a pre-cgc morphism $\hat{f} : C \to \bigwedge' V$.

There is an injective homomorphism of pre-gcs
$$\chi : \bigwedge{\!}' V \to T'(V)$$
given by
$$\chi(x_1\wedge \ldots x_n) = \sum_\nu \varepsilon(\nu)x_{\nu(1)}\otimes \ldots \otimes x_{\nu(n)},$$
where the sum is over all permutations and $\varepsilon(\sigma)$ is the corresponding Koszul sign.  It has, as image, all the symmetric tensors (in the graded sense).

On $\bigwedge' V$ and $T'(V)$, the filtration of the primitives comes from a gradation

$$F_p\bigwedge{\!}' V = \bigwedge^{\leq p}V = \bigoplus_{k\leq p}\bigwedge^k V;$$
$$F_p T'(V) = T^{\leq p}(V) = \bigoplus_{k\leq p} T^k(V).$$

## Related concepts

### Differential graded algebras

Dually, a [[dg-algebra]] is a [[monoid]] in [[chain complex]]es.

### Semifree dg-coalgebras and $L_\infty$-algebras

The notion of dg-coalgebra whose underlying [[coalgebra]] is cofree is related by duality to that of [[semifree dga]].

Semicofree dg-coalgebras concentrated in negative degree and with differential of degree -1 are  the same as [[L-∞-algebras]]

### Model category structure

There is a [[model structure on dg-coalgebras]].

## Properties

### As filtered colimits of finite-dimensional pieces
 {#AsFilteredColimits}

+-- {: .num_prop }
###### Proposition

Every dg-coalgebra is the [[filtered colimit]] of its finite-dimensional sub-dg-coalgebras.

=--

This is due to ([Getzler-Goerss 99](#GetzlerGoerss99)), in generalization to the analogous fact for plain [[coalgebras]], see at _[coalgebra -- As filtered colimits](coalgebra#AsFilteredColimits)_.

See also at _[[L-infinity algebra]]_ the section _[Ind-Conilpotency](L-infinity-algebra#IndConilpotency)_. This plays a role for instance for constructing [[model structures for L-infinity algebras]], see [there](model+structure+for+L-infinity+algebras#OnProAlg).

## References

The [[model structure on dg-coalgebras]] is due to

* {#GetzlerGoerss99} [[Ezra Getzler]], [[Paul Goerss]], _A model category structure for differential graded coalgebras_, 1999 ([ps](http://www.math.northwestern.edu/~pgoerss/papers/model.ps), [[GetzlerGoerss99.pdf:file]])

[[!redirects differential graded coalgebras]]

[[!redirects dg-coalgebra]]
[[!redirects dg-coalgebras]]


differential graded Hopf algebra

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definitions

### Graded Hopf algebras

A _$\mathbb{Z}$-graded Hopf algebra_ (pre-gha) is a $\mathbb{Z}$-graded vector space, which, for that grading, is both a $\mathbb{Z}$-graded algebra, $(A,\mu)$, with  unity, $\eta : K \to A$, and a $\mathbb{Z}$-graded coalgebra $(A, \Delta, \varepsilon)$ such that:

* $\eta : K \to A$ is a morphism of $\mathbb{Z}$-graded  coalgebras;
* $\varepsilon  : A \to K$ is a morphism of $\mathbb{Z}$-graded  algebras;
* $\mu : A \otimes A \to A$  is a morphism of $\mathbb{Z}$-graded  coalgebras

**Remark**

We can replace the third condition  by:

* $ \Delta : A \to A \otimes A$  is a morphism of $\mathbb{Z}$-graded  algebras.

Of course, wherever possible, we will abbreviate $(A,\Delta,\mu,\epsilon,\eta)$ to $A$.

A [[homomorphism]] of pre-ghas is a linear map of degree zero compatible with both the algebra and coalgebra structures.  We may write $pre GHA$ for the resulting category.

If $A$ and $A'$ are two pre-ghas, $A\otimes A'$ is a pre-gha for the algebra and coalgebra structures already defined.

### Derivations on Hopf algebras

Let $A$ be a pre-gha.  A  _Hopf algebra derivation_ of $A$ of degree $p\in \mathbb{Z}$ is a linear mapping $\theta \in Hom_p(A,A)$, defining both an algebra and a coalgebra derivation.

A differential $\partial$ of pre-ghas is a Hopf algebra derivation of degree -1 such that $\partial\circ \partial = 0$.  The pair $(A,\partial)$ is called a _differential $\mathbb{Z}$-graded Hopf algebra_ (pre-dgha).  Its homology $H(A,\partial)$ is also a pre-gha.  A morphism of pre-dghas is a morphism, at the same time, of pre-ghas and pre-dgvs.  This gives a category $pre DGHA$.

A pre-gha $(A,\Delta,\mu,\epsilon,\eta)$ is _commutative_ if $(A,\mu)$ is commutative and is _cocommutative_ if $(A,\Delta,\varepsilon)$ is cocommutative.

This gives categories $pre CDGHA$ and $pre CoDGHA$ respectively.

A _cocommutative_ (resp. _commutative_) _dgha_ is an object of $pre CoDGHA$ (resp. $pre CDGHA$, which has a lower (resp. upper) grading.

A cocommutative (resp. commutative) dgha $A$ is _$n$-connected_ if $\bar{A}_p = 0$ (resp $\bar{A}^p = 0$) for $p\leq n$.

### Shuffle product on $T(V)$

Let $V$ be a pre-gvs.  The gvs $T(V)$ is a pre-cga for the [[shuffle]] product defined by

$$(v_1\otimes \ldots \otimes v_p)\star (v_{p+1}\otimes\ldots \otimes v_n) = \sum_\sigma \varepsilon(\sigma)v_{\sigma^{-1}(1)}\otimes\ldots \otimes v_{\sigma^{-1}(n)},$$

where the sum is over all $(p,n-p)$ [[shuffles]], $\varepsilon(\sigma)$ is the Koszul sign of $\sigma$  and the elements $v_i$ of $V$ are all homogeneous.

### Graded-commutative Hopf algebra structure on $T(V)$

The underlying algebra structure is $T(V)$ with the shuffle product.  The reduced diagonal is given by

$$\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n).$$

### Graded-cocommutative Hopf algebra structure on $T(V)$

The underlying algebra structure this time is $T(V)$ with the usual product

$$(v_1\otimes \ldots \otimes v_p)\cdot(v_{p+1}\otimes \ldots \otimes v_n) = v_1\otimes \ldots \otimes v_p\otimes v_{p+1}\otimes \ldots \otimes v_n,$$

but with the reduced diagonal given by

$$\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1}\sum_\sigma \varepsilon(\sigma) (v_{\sigma(1)}\otimes \ldots \otimes v_{\sigma(p)})\otimes(v_{{\sigma(p+1)}}\otimes \ldots \otimes v_{\sigma(n)}),$$

where the sum is over all $p$ and all $(p,n-p)$-shuffles and, as usual, $\varepsilon(\sigma)$ is the Koszul sign.

The diagonal $\Delta$ is thus defined by the conditions

* $ \Delta v = v\otimes 1 + 1\otimes v$  if  $v \in V$;

* $\Delta$  is a morphism of $\mathbb{Z}$-graded algebras.

A commutative and cocommutative $\mathbb{Z}$-graded Hopf algebra structure on $\bigwedge V$ is obtained by using the algebra and coalgebra structures defined in [[differential graded algebra]] and [[differential graded coalgebra]]. respectively.

### The enveloping algebra of a Lie algebra, $U(L)$.

Let $L$ be a pre-gla, $U(L)$, is  the quotient algebra of the tensor algebra $T(L)$ by the two sided ideal generated by the elements

$$x\otimes y - (-1)^{|y||x|}y\otimes x - [x,y], \quad x,y,\in L.$$

The diagonal $\Delta : L \to L\times L$, with $\Delta(x) = (x,x)$ defines a homomorphism of pre-gas,

$$U(\Delta) : U(L)\to U(L\times L) \cong U(L)\otimes U(L),$$

which makes $U(L)$ a pre-gha which is cocommutative and conilpotent.

If $L$ is a free Lie algebra on $V$, then the enveloping algebra is the tensor algebra: $U\mathbb{L}(V) \cong T(V)$.

Let $(L,\partial)$ be a pre-dgla, the differential $\partial $ extends to an algebra differential on $T(L)$.  With the quotient differential, $U(L)$ becomes a cocommutative pre-dgha, which will be denoted $U(L,\partial)$.

The differential $\partial$ determines a differential, also denoted $\partial$, on the cocommutative pre-gca $\bigwedge' L$, (for which gca see [[differential graded coalgebra]]).  It satisfies:

$$\bigwedge' H(L,\partial) \cong H(\bigwedge' L,\partial).$$

Let $i : L \to U(L)$ be the linear mapping $L\to T(L) \to U(L)$, then define $e: \bigwedge' L \to U(L)$ by

$$e(x_1\wedge \ldots x_n) = \frac{1}{n!}\sum_\sigma \varepsilon(\sigma)i(x_{\sigma(1)})\ldots i(x_{\sigma(n)}),$$

where the sum is over all permutations and $\varepsilon(\sigma)$ is the Koszul sign.
+-- {: .num_theorem}
###### Theorem

(Poincaré-Birkhoff-Witt)(cf. **Quillen**)

The mapping $e$ is an isomorphism of pre-dgcas.
=--

+-- {: .num_corollary}
###### Corollary
$i : L \to U(L)$ defines an isomorphism between $L$ and the space of primitives of $U(L)$.
=--

+-- {: .num_corollary}
###### Corollary
The natural map $UH(L,\partial)\to H(U(L,\partial)$ is an isomorphism of cocommutative pre-ghas.
=--

### The Lie algebra of primitives, $P$

Let $(A,\partial)$ be a cocommutative pre-dgha.  The vector space $P(A)$ of primitive elements (for the coalgebra structure, cf. [[differential graded coalgebra]]), is not stable under the multiplication, however the commutator $[\alpha,\beta]$ of two elements of $P(A)$ is again in $P(A)$.  This defines a pre-gla structure on $P(A)$ and we can put the induced differential on it to obtain $P(A,\partial)$.

The inclusion $P(A)\to A$ extends to a morphism of cocommutative pre-dghas $\sigma: UP(A)\to A.$
+-- {: .num_theorem}
###### Theorem (Quillen, **Quillen**)
If $A$ is conilpotent, $\sigma$ is an isomorphism.
=--

The above theorem and earlier corollary show that $U$ and $P$ are inverse equivalences between the category, $pre DGLA$ and that of cocommutative, conilpotent pre-dghas.

**Remark**

The enveloping algebra of a free Lie algebra $\mathbb{L}(V)$ coincides with the tensor algebra, $T(V)$.  It is conilpotent from which one gets $PT(V) = \mathbb{L}(V)$.

## Related concepts

* [[Hopf algebra]]

## References

* D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Maths No. 1025, Springer, 1983.

* [[Daniel Quillen]], _Rational Homotopy Theory_, Ann. of Math., (2) 90 (1969), 205-295.

Discussion of [[bar-cobar construction]] for dg-Hopf algebras:

* [[Benoit Fresse]], _The universal Hopf operads of the bar construction_ ([arXiv:math/0701245](https://arxiv.org/abs/math/0701245))

* [[Murray Gerstenhaber]], [[Alexander Voronov]], Section 3.2 of: _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))

* Justin Young, _Brace Bar-Cobar Duality_ ([arXiv:1309.2820](https://arxiv.org/abs/1309.2820))

[[!redirects differential graded Hopf algebras]]

[[!redirects dg-Hopf algebra]]
[[!redirects dg-Hopf algebras]]


differential graded Lie algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _differential graded Lie algebra_, or _dg-Lie algebra_ for short, is equivalently

* a graded [[Lie algebra]] equipped with a [[differential]] that acts as a graded [[derivation]] with respect to the Lie bracket;

* a [[Lie algebra]] [[internalization|internal to]] the [[category of chain complexes]];

* a _strict_ [[L-∞-algebra]], i.e. an $L_\infty$-algebra in which only the unary and the binary brackets may be nontrivial.

## Definition

### Direct explicit definition
 {#DirectExplicitDefinition}

+-- {: .num_defn }
###### Definition

A _dg-Lie algebra_ $(\mathfrak{g},\partial,[-,-])$ is

1. a $\mathbb{Z}$-[[graded vector space]] $\mathfrak{g} = \bigoplus_{i} \mathfrak{g}_i$;

1. a [[linear map]] $\partial \colon \mathfrak{g} \longrightarrow \mathfrak{g}$;

1. a [[bilinear map]] $[-,-] \colon \mathfrak{g}\otimes\mathfrak{g} \longrightarrow \mathfrak{g}$, the _bracket_;

such that (all conditions are expressed for homogeneously graded elements $x_i \in \mathfrak{g}_{\vert x_i \vert}$):

1. $\partial$ is a [[differential]] that makes $(\mathfrak{g},\partial)$ into a [[chain complex]], i.e.

   1. it is of degree -1, $\partial \colon \mathfrak{g}_{i} \to \mathfrak{g}_{i-1}$;

   1. it squares to zero, $\partial \circ \partial = 0$;

1. $\partial$ is a graded [[derivation]] of the bilinear pairing, i.e.

   $$
      \partial [x_1,x_2] = [\partial x_1, x_2] + (-1)^{\vert x_1 \vert} [x_1, \partial x_2]
    \,,

   $$

1. the bilinear pairing is graded skew-symmetric, i.e.

   $$
     [x_1, x_2] = -(-1)^{\vert x_1\vert \vert x_2 \vert} [x_2,x_1]
     \,,
   $$

1. the bilinear pairing satisfies the graded [[Jacobi identity]] (saying that $[x,-]$ is a graded [[derivation]])

   $$
     [x_1,[x_2,x_3]] = [[x_1,x_2], x_3] + (-1)^{\vert x_1 \vert \vert x_2 \vert} [x_2, [x_1, x_3]]
    \,.
   $$

=--

### As graded Lie algebras with nilpotent derivations
 {#AsGradedLieAlgebrasWithDerivations}

+-- {: .num_defn}
###### Definition

A _pre-graded Lie algebra_ (pre-gla) is a [[graded vector space|pre-gvs]], $L$, together with a bilinear map of degree zero

$$[\quad,\quad ] : L\otimes L \to L,$$

such that

$$[x,y] = (-1)^{|x||y|+1}[y,x]$$

and

$$(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0$$

for every triple $(x,y,z)$ of homogeneous elements in $L$.

=--

(The first property is call _antisymmetry_, the second _the Jacobi identity_.)

A _morphism_ $f: L \to L'$ of pre-glas is a linear map of degree zero, that is compatible with the brackets,

$$f[x,y] = [f(x),f(y)].$$

+-- {: .num_example }
###### Example

To any augmented pre-ga $A$, one can associate a pre-gla, denoted $\bar{A}_L$, with underlying gvs $\bar{A}$ and with bracket, the commutator, $[x,y] = xy -(-1)^{|x||y|}yx$ for each pair $(x,y)$ of homogeneous elements.  $\bar{A}_L$ is abelian (i.e. with trivial bracket) if and only if $A$ is graded commutative.

=--

+-- {: .num_example }
###### Example

If $A$ is a pre-cga and $L$ is a pre-gla, the tensor product $A\otimes L$ has a pre-gla structure with bracket

$$[a\otimes l,a'\otimes l'] = (-1)^{|a'||l|}aa' \otimes [l,l']$$

for $a,a', l, l'$ homogeneous.

=--

+-- {: .num_defn }
###### Definition

Let $L$ be a pre-gla.  A _[[derivation]]_ of gla-s, of degree $p\in \mathbb{Z}$, is a linear mapping $\theta \in Hom_p(L,L)$ such that

$$\theta[x,y] = [\theta{x},y] + (-1)^{p|x|}[x,\theta(y)]$$

for any pair $x,y$ of homogeneous elements of $L$.  We denote by $Der_p(L)$, the vector space of degree $p$ derivations of the gla, $L$.

=--

+-- {: .num_defn }
###### Definition

A differential $\partial$ of a pre-gla is a Lie algebra derivation of degree -1 such that $\partial\circ \partial = 0$.  The pair $(L,\partial)$ is then called a _differential pre-graded Lie algebra_ (pre-dgla); its homology $H(L,\partial)$, is a pre-gla.

A morphism of pre-dglas is a morphism for both the underlying pre-gla and the pre-dgvs. We denote the corresponding category  by $pre DGLA$.

=--

This means that a  differential graded Lie algebra is an internal Lie algebra in the symmetric monoidal category of chain complexes with tensor product given as in [[differential graded vector space|differential graded vector spaces]].

+-- {: .num_example }
###### Example

If $(A,\partial)$ is an augmented pre-dga, $(\bar{A}_L,\partial)$ is a pre-dgla.

=--

+-- {: .num_example }
###### Example

If $(A,\partial)$ is a pre-cdga and $(L,\partial')$, a pre-dgla, $A \otimes L$, together with the tensor product differential, is a pre-dgla.

=--

+-- {: .num_example }
###### Example

Let $(V,\partial)$ be a pre-dgvs, then the pre-dgvs, $(Hom(V,V),D)$, constructed earlier is a pre-dga  for the multiplication law given by composition of mappings.  Its associated pre-dgla has

$$[f,g] = f\circ g - (-1)^{|f||g|}g\circ f$$

and

$$Df = [\partial,f].$$

In particular, if $(V,\partial) = (A,d)$ is a cdga (resp. $(V,\partial) = (L,\partial)$ is dgla), then $Der(A,d) = (\bigoplus_p Der_p(A),D)$, (resp. $Der(L,\partial) = (Der_p(L),D)$) is a sub-pre-dgl of $(Hom(V,V),D)$.

=--

+-- {: .num_defn }
###### Definition

A dgla is a pre-dgla with a lower grading; explicitly:

A **differential graded Lie algebra**, $(L,\partial)$, is a graded vector space $L = \bigoplus_{p\geq 0}L_p$, together with a bilinear map of degree 0
$$[\quad ,\quad] : L\otimes L \to L,$$
and a differential $\partial$ satisfying
$$\partial L_p \subseteq L_{p-1},  \quad [x,y] = (-1)^{|x||y|+1}[y,x],$$
$$(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0$$
and
$$\partial[x,y] = [\partial x,y] + (-1)^{|x|}[x,\partial y]$$
for every triple $(x,y,z)$ of homogeneous elements in $L$.

=--

Let $DGLA$ be the corresponding category.

+-- {: .num_defn }
###### Definition

A dgla is $n$-*reduced* (resp. _homologically $n$-reduced_) if $L_p = 0$ (resp. $H_p(L,\partial) = 0$) for all $p\lt n$.  Denote by $DGLA_n$ (resp.  $DGLA_{hn}$), the corresponding categories.

=--

+-- {: .num_defn }
###### Definition

If $(L,\partial)$ is a pre-dgla, a _gla-filtration_ of $L$ (resp. a _dgla-filtration_ of $( L,\partial)$ ) is a family of subgraded vector spaces $F_p L$, $p\in \mathbb{Z}$, such that $F_p L\subseteq F_{p+1}L$, $[F_p L,F_n L]\subseteq F_{p+n} L$, (resp. and $\partial F_p L\subseteq F_p L$).

=--

+-- {: .num_defn }
###### Definition

Let $L$ be a pre-gla.  Its _bracket length filtration_ is obtained from the descending central series:

$$F^1 L = L; \quad F^p L = [L,F^{p-1} L] \quad  if \quad p\geq 2.$$

It is a gla-filtration.

$Q(L) = L/F^2L$ is called the _space of indecomposables_ of $L$.

If $(L,\partial)$ is a pre-dgla, $F^p L$ is stable by $\partial$.  Letting $Q(\partial)$ be the induced differential on $Q(L)$, $Q$ then defines a functor
$$Q : pre DGLA \to pre DGVS.$$

=--

+-- {: .num_example }
###### Example
**Free Lie algebra, $\mathbb{L}(V)$**

Let $V$ be a pre-gvs, $T(V)$, the tensor algebra on $V$ with augmentation ideal $\overline{T(V)}$ (recall $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and the augmentation sends $V (= V^{\otimes 1}$ to 0).

 Let $\overline{T(V)}_L$ be $\overline{T(V)}$ with the pre-gla structure given by the commutators.  We denote by $\mathbb{L}(V)$, the Lie subalgebra of $\overline{T(V)}_L$ generated by $V$.
+--{:query}
[[Tim Porter|Tim]]: A more explicit description may help here, cf. Quillen, Rational Homotopy theory (p.281) or MacLane, Homology.
=--
If $L$ is a pre-gla, any morphism of pre-gvs $f: V\to L$ has a unique extension to a pre-gla morphism $\hat{f} :\mathbb{L}(V)\to L$.  If $(e_\alpha)_{\alpha\in I}$ is a homogeneous basis for $V$, $\mathbb{L}(V)$ may be denoted $\mathbb{L}((e_\alpha)_{\alpha\in I})$.

On the free Lie algebra $\mathbb{L}(V)$, the bracket length filtration comes from a gradation $\mathbb{L}(V) = \bigoplus_j\mathbb{L}^j(V) $, where $\mathbb{L}^j(V) $ is the subspace generated by the brackets of elements of $V$ of length $j$.  The inclusion $\mathbb{L}(V)\hookrightarrow T(V)$ identifies $\mathbb{L}^j(V)$ with $\mathbb{L}(V)\cap T^j(V)$.

If $\mathbb{L}(V), \partial)$ is a dgla, free as a gla, with $V$ fixed, $\partial$ is the sum of derivations $\partial_k$ defined by : $\partial_k \subset \mathbb{L}^k(V)$.  The isomorphism between $V$ and $Q\mathbb{L}(V)$ identifies $\partial_1V$ with $Q(\partial)$.  $\partial_1$ (resp. $\partial_2$) is called the _linear part_ (resp, the _quadratic part_) of $\partial$.

=--

+-- {: .num_defn }
###### Definition

Let $(L,\partial)$ and $(L',\partial')$ be two dglas.  Their _product_ $(L,\partial)\times(L',\partial')$ in $DGLA$ is defined by:

*  the underlying vector space is the direct sum $L\oplus L'$;

*  $(L,\partial)$ and $(L',\partial')$ are two sub differential graded Lie algebras of $(L,\partial)\times(L',\partial')$;

*  if $x\in L$ and $x' \in L'$, then $[x,x'] = 0$.

=--

Their _coproduct_ or _sum_ $(L,\partial)\star(L',\partial')$ is often called their _free product_.

+-- {: .num_example }
###### Example

$$\mathbb{L}(V)\star \mathbb{L}(V') \cong   \mathbb{L}(V\oplus V').$$

More generally if $L$ and $L'$ are given by generators and relations

$$L = \mathbb{L}(V)/I , \quad L' = \mathbb{L}(V')/I' ,$$

$$L\star L' = \mathbb{L}(V\oplus V')/{I,I'}.$$

The differential on $L\star L'$ is the unique Lie algebra derivation extending $\partial$ and $\partial'$.

=--

## Properties

### Model category structure

* [[model structure on dg-Lie algebras]]

### Relation to $L_\infty$-algebras
 {#RelationToLInfinityAlgebras}

Every dg-Lie algebra is in an evident way an
[[L-infinity algebra]]. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the _strict_ $L_\infty$-algebras: those for which the [[Jacobi identity]] holds on the nose and all its possible higher coherences are trivial.

+-- {: .num_theorem}
###### Theorem

Let $k$ be a [[field]] of [[characteristic]] 0 and write $L_\infty Alg_k$ for the [[category]] of [[L-infinity algebras]] over $k$.

Then every object of $L_\infty Alg_k$ is [[quasi-isomorphism|quasi-isomorphic]] to a [[dg-Lie algebra]].

Moreover, one can find a functorial replacement: there is a [[functor]]

$$
  W : L_\infty Alg_k \to L_\infty Alg_k
$$

such that for each $\mathfrak{g} \in L_\infty Alg_k$

1. $W(\mathfrak{k})$ is a [[dg-Lie algebra]];

1. there is a [[quasi-isomorphism]]

   $$
     \mathfrak{g} \stackrel{\simeq}{\to}
     W(\mathfrak{g})
     \,.
   $$

=--

This appears for instance as ([KrizMay, cor. 1.6](#KrizMay)).

For more see at

* _[[model structure on dg-Lie algebras]]_ the section _[Relation to L-infinity algebras](model+structure+on+dg-Lie+algebras#RectificationResolution)_.

* [[model structure for L-infinity algebras]], the section _[on dg-Lie algebras](model+structure+for+L-infinity+algebras#OndgLieAlgebras)_.

* [[relation between L-∞ algebras and dg-Lie algebras]]

### Relation to dg-coAlgebras

Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by adjunction to [[dg-coalgebras]]

$$
  dgLieAlg_k
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  dgCoAlg_k
$$

Here

* $CE$ is the [[Chevalley-Eilenberg algebra]] functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to

  $$
    CE(\mathfrak{g},\partial,[-,-])
     \;\coloneqq\;
    \left(
      \vee^\bullet \mathfrak{g}[1]
      ,\;
      D = \partial + [-,-]
    \right)
    \,,
  $$

  where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood.

* For $(X,D)$ a dg-coalgebra, then

  $$
    \mathcal{L}(X,D)
     \coloneqq
    \left(
       F(\overline{X}[-1]),\;
       \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1)
    \right)
  $$

  where

  1. $\overline{X} \coloneqq ker(\epsilon)$ is the [[kernel]] of the [[counit]], regarded as a [[chain complex]];

  1. $F$ is the [[free Lie algebra]] functor (as graded Lie algebras);

  1. on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra [[derivation]]

Moreover

$$
  Hom(\mathcal{L}(X), \mathfrak{g})
   \simeq
  Hom(X, CE(\mathfrak{g}))
    \simeq
  MC(Hom(\overline{X},\mathfrak{g}))
$$

is the [[Maurer-Cartan elements]] in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, prop 6.1, 6.2](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2](#Hinich98)).

For more see at _[[model structure on dg-Lie algebras]]_.

### Relation to simplicial Lie algebras

+-- {: .num_theorem}
###### Theorem

There is an [[adjunction]]

$$
  (N^* \dashv N) : LieAlg^\Delta \stackrel{\overset{N^*}{\leftarrow}}{\underset{N}{\to}}
  dgLieAlg
$$

between [[simplicial Lie algebras]] and [[dg-Lie algebra]]s, where $N$ acts on the underlying simplicial vector spaces as the [[Moore complex]] functor.

=--

This is ([Quillen, prop. 4.4](#Quillen)). For more see at _[[simplicial Lie algebra]]_.

+-- {: .num_theorem}
###### Theorem

This adjunction is a [[Quillen adjunction]] with respect to the projective [[model structure on dg-Lie algebras]] and the projective [[model structure on simplicial Lie algebras]] ([this prop.](model+structure+on+simplicial+Lie+algebras#QuillenAdjunctionTodgLieAlgebras)).

The corresponding [[derived functors]] constitute an [[equivalence of categories]] between the corresponding [[homotopy categories]]

$$
  (L N^* \dashv \tilde N)
   :
  Ho(LieAlg^\Delta)_1
   \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1
$$

of [[1-connected]] objects on both sides.

=--

This is in the proof of ([Quillen, theorem. 4.4](#Quillen)).

## Related concepts

* [[crossed complex]] $\Leftarrow$ [[crossed module]] $\Rightarrow$ [[2-crossed module]]

* **dg-Lie algebra** $\Leftarrow$ [[differential crossed module]] $\Rightarrow$ [[differential 2-crossed module]]

* [[L-∞ algebra]]

  * [[model structure for L-∞ algebras]]

  * [[sheaf of L-∞ algebras]]

* [[formal moduli problem]], [[deformation theory]]

## References

A standard reference in the context of [[rational homotopy theory]] is

* {#Quillen} [[Dan Quillen]], _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([JSTOR](http://www.jstor.org/stable/1970725))

For the unbounded case there is general discussion in

* {#Hinich97} [[Vladimir Hinich]], _Homological algebra of homotopy algebras_, Comm. in algebra, 25(10)(1997), 3291–3323 ([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

The relation to $L_\infty$-algebras is discussed for instance in

* {#KrizMay} [[Igor Kriz]], [[Peter May]], _Operads, algebras, modules and motives_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf))

See also the references at [[model structure on dg-Lie algebras]].

A discussion of how formal neighbourhoods of points in [[infinity-stack]]s are governed by dg-Lie algebras:

* [[Jacob Lurie]], _[[Moduli Problems and DG-Lie Algebras]]_

[[!redirects differential graded Lie algebras]]
[[!redirects dg-Lie algebra]]
[[!redirects Lie dg-algebra]]
[[!redirects dg-Lie algebras]]
[[!redirects Lie dg-algebras]]


differential graded manifold

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#### Higher geometry
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[[!include higher geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _differential graded manifold_ is a generalization of the notion of [[smooth manifold]] from ordinary [[geometry]] to [[higher geometry]], specifically to [[dg-geometry]].  Typically it is taken to be the [[Isbell duality|formal dual]] to a [[dgc-algebra]] which in degree-0 is the [[algebra of function|algebra of]] [[smooth functions]] on an ordinary [[smooth manifold]].

Hence this is a _[[graded manifold]]_ whose [[algebra of functions]] is equipped with a compatible [[differential]].

Sometimes this is called an "[[NQ-supermanifold]]".

## Examples

* An [[L-∞ algebroid]] over a smooth manifold may be thought of as a dg-manifold concentrated in non-negative degree.

* A [[derived L-∞ algebroid]] may be thought of as a dg-manifold in arbitrary degree.

## Related concepts

* [[derived smooth manifold]]

* [[dg-scheme]]

[[!redirects differential graded manifolds]]
[[!redirects dg-manifold]]
[[!redirects dg-manifolds]]


differential graded objects - contents

**[[differential object|differential]] [[graded object|graded objects]]**

and

**[[rational homotopy theory]]** ([[equivariant rational homotopy theory|equivariant]], [[rational stable homotopy theory|stable]], [[parametrized rational homotopy theory|parametrized]], [[equivariant rational stable homotopy theory|equivariant & stable]], [[parametrized rational stable homotopy theory|parametrized & stable]])

## dg-Algebra

* [[graded vector space]]

* [[differential graded vector space]]

* [[differential graded algebra]]

  * [[model structure on dgc-algebras]]

  * [[model structure on equivariant dgc-algebras]]

* [[differential graded coalgebra]]

  * [[model structure on dg-coalgebras]]

* [[differential graded Lie algebra]]

  * [[model structure on dg-Lie algebras]]

* [[L-∞ algebra]]

  * [[model structure for L-∞ algebras]]

* [[differential graded Hopf algebra]]

* [[bar and cobar construction]]

## Rational spaces

* [[nilpotent space]]

* [[finite type]]

* [[rational space]]

  * [[formal space]]

* [[rationalization]]

## PL de Rham complex

* [[differential forms on simplices]]

* [[PL de Rham complex]]

* [[Sullivan construction]]

  * [[Lie integration]]

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

## Sullivan models

* [[minimal Sullivan model|minimal]]$\,$[[Sullivan model]]

[[!include Sullivan models -- examples]]

## Related topics

* [[homotopy theory]]

* [[stable homotopy theory]]

* [[homological algebra]]

* [[∞-Lie theory]]


differential graded ring

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **differental graded ring** or **dg-ring** is a [[monoid]] in the [[category]] of [[chain complexes]] or of [[cochain complex]]es of abelian groups.

If a dg-ring is degreewise compatibly a [[vector space]] (or more generally a [[module]] over some [[ring]]) then it is called *[[dg-algebra]]*. See there for more.

## Related concepts

* [[differential graded algebra]]

* [[monoidal Dold-Kan correspondence]]

[[!redirects differential graded rings]]

[[!redirects dg-ring]]


differential graded vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential-graded objects
+-- {: .hide}
[[!include differential graded objects - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

# Contents
* automatic table of contents goes here
{:toc}

## Definition

**Differential (pre-)graded vector spaces.**
(We continue from [[graded vector space|graded vector spaces]] with the terminology of pre-gvs for the general case and gvs for the positively or negatively graded ones.)

A **differential (pre-)graded vector space**, (dgvs), is a pair $(V,\partial)$, where $V$ is a (pre-)graded vector space and $\partial \in Hom_{-1}(V,V)$ satisfies $\partial\circ\partial = 0$.

This endomorphism, $\partial$, of degree -1 is called the [[differential]] or sometimes the _boundary operator_ of the dgvs.

To link it with [[differential object]], note that using the suspension (as in [[graded vector space]]) as the translation (see [[category with translation]]) and then the differential is $\partial : V\to s(V)$. A dgvs is essentially the same as a [[chain complex]] of vector spaces. (Some questions of terminology are addressed further down this entry.)

As usual, $Ker \partial / Im \partial$ is called the [[homology]] of $(V,\partial)$, denoted $H(V,\partial)$.

Let $(V,\partial)$, $(V',\partial')$ be two pre-dgvs

$$Hom(V,V') = \bigoplus_{p\in \mathbb{Z}}Hom_p(V,V')$$

is a pre-dgvs with differential

$$Df = \partial'\circ f - (-1)^{|f|}f\circ\partial$$

for $f$ homogeneous.

A degree $r$ linear morphism $f$ is _compatible with the differentials_ if it is a cycle for this differential $D$, i.e., $Df = 0$ or $\partial' f = (-1)^{r}f\partial.$

A **morphism between pre-dgvs** is a linear morphism of degree 0 that is compatible with the differentials:

$$f: (V,\partial)\to (V',\partial').$$

This induces $H(f): H(V,\partial)\to H(V',\partial').$

We get a category $pre -  DGVS$ and $H$ is a functor $H : pre\! -\! DGVS\to pre\!-\! GVS$.

### Weak equivalences

If $f: (V,\partial)\to (V',\partial')$ in {\sf pre-DGVS}, then $f$ is a **weak equivalence** or [[quasi-isomorphism]] if $H(f)$ is an isomorphism.  In this case we write $f: (V,\partial)\stackrel{\simeq}{\to} (V',\partial')$

### Homotopies

Let  $f,f' : (V,\partial)\to (V',\partial')$ be two morphisms in $pre-DGVS$. We say $f$ and $f'$ are **homotopic** denoted $f\sim f'$ if $f-f'$ is a boundary in $(Hom(V,V'),D)$, i.e., there is some $h : V \to V'$ of degree +1 such that $f-f' = Dh = \partial' h + h\partial$.

**Contractible and Acyclic DGVSs**

 A dgvs $(V,\partial)$ is _contractible_ if the identity map on $V$ is homotopic to the zero morphism and is _acyclic_ if $H(V,\partial) = 0$.

**Remarks**

(i) A _contracting homotopy_ $h : Id_V \sim O_V$ is a degree 1 map such that $\partial h(x) + h(\partial x) = x$ for  all $x$ in $V_p$.

(ii) If $(V,\partial)$ is contractible, then it is acyclic and conversely.  The converse depends strongly on our working with vector spaces.  If we worked with modules over a commutative ring, $k$, then the correct result would be ''If $(V,\partial)$ is acyclic and projective in each dimension, then it is contractible.'' This is important when looking at, for instance, diagrams of dgvs since even if each individual object in the diagram may be contractible, it might be impossible to pick the contracting homotopies to give a map of diagrams, i.e., to be compatible with the structural maps.

### Suspension for (pre-) dgvs

If $(V,\partial)$ is a pre-dgvs, then its $r$-suspension $(s^rV,\partial)$, is the $r$-suspension, $s^rV$, of $V$ together with the differential

$$\partial s^rv = (-1)^rs^r(\partial v).$$

### Tensor product of (pre-) dgvs

If $(V,\partial)$, and $(V',\partial')$ are pre-dgvs, then we give the tensor product, $V\otimes V'$, the differential given on generators by

$$ \partial(v\otimes v') = (\partial v) \otimes v' + (-1)^{|v|}v \otimes (\partial' v' ),$$

and we denote the result by $(V,\partial)\otimes (V', \partial')$. We have ([[Kunneth theorem]])
$$H((V,\partial)\otimes (V', \partial')) \cong H(V,\partial)\otimes H(V', \partial').$$

###Duals of (pre-)dgvs

The dual $\#(V,\partial) = (\# V,\#\partial)$ of a pre-dgvs $(V,\partial)$ is given by $\# V$ with differential $(\# \partial ) = - ^t \partial$.  This satisfies

$$\langle(\# \partial)f; v\rangle + (-1)^{|f|}\langle f; \partial v\rangle = 0.$$

### Chains and cochains: terminology

If $(V,\partial)$ is a pre-dgvs with 'lower grading' that is the summands are written $V_p$, then $(V,\partial)$ may be called a [[chain complex]] and terms such as _cycle_, _boundary_, [[homology]] are used with the usual meanings.

If $(V,\partial)$ is presented with the 'upper grading', so $V^p$, then the corresponding words will have a 'co' as prefix, cochain complex, cocycle, etc.

There is no real distinction between the two cases in the abstract, but in applications there is often a fixed 'dimensional' interpretation and then the 'natural' and 'geometric' aspects determine which is more appropriate or useful.


differential graded-commutative algebra

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* table of contents
{:toc}

## Definition

### Differential graded-commutative algebra

A _differential graded-commutative algebra_ (also _DGCA_ or _dgca_, for short) is a [[differential-graded algebra]] which is [[supercommutative superalgebras|supercommutative]] in that for $v,w$ any two elements in homogeneous degree $deg(v), deg(w) \in \mathbb{Z}$, respectively, then  the product in the algebra satisfies

$$
  v w \;=\; (-1)^{deg(v) deg(w)} w v
  \,.
$$

Equivalently this is a [[commutative monoid]] in the [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes]] of [[vector spaces]] equipped with the [[tensor product of chain complexes]].

### Differential graded-commutative superalgebra
 {#DifferentialGradedCommutativeSuperalgebra}

More generally, a _differential graded commutative superalgebra_ $(A,d) \in dgcSAlg$ is a [[commutative monoid]] in the [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes of super vector spaces]].

There are (at least) two such symmetric monoidal structures $\tau_{Deligne}$ and $\tau_{Bernst}$ ([this Prop.](chain+complex+in+super+vector+spaces#SymmetricStructureOnCategoryOfChainComplexesOfSuperVectorSpaces)). While equivalent ([this Prop.](chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect)) these yield two superficially different [[signs in supergeometry|sign rules]] for differential graded-commutative superalgebras:

   1. for $a,b \in A$ two elements of homogeous degree $(n_a, \sigma_a), (n_b, \sigma_b) \in \mathbb{Z} \times \mathbb{Z}/2$, respectively, we have

1. in _Deligne's convention_

   $a b = (-1)^{n_a n_b + \sigma_a \sigma_b} \, b a$

1. in _Berstein's convention_

   $a b = (-1)^{ (n_a + \sigma_a)(n_b + \sigma_b) } \, b a$

While in both cases the [[differential]] satisfies.

$$
  d (a b) = (d a) b + (-1)^{n_1} a (d b)
  \,.
$$

[[!include sign rules in homological superalgebra -- table]]

Restricted tro bidegree $(0,-)$ both of these sign rules yield a _[[commutative superalgebra]]_, which restricted to $(-,even)$ thy yield a [[differential graded-commutative algebra]].

## Examples

* The [[de Rham algebra]] of [[differential forms]] on a [[smooth manifold]] is a differential-graded commutative algebra. The algebra of [[super differential forms]] on a [[supermanifold]] is a differential-graded commutative superalgebra.

* The dg-algebra of [[polynomial differential forms]] on an [[n-simplex]];

The following are [[semifree differential graded-commutative algebras]]:

* A [[Sullivan algebra]].

* The [[Chevalley-Eilenberg algebra]] of a [[Lie algebra]] or more generally of an [[L-infinity algebra]] or [[Lie infinity-algebroid|L-infinity algebroid]] is a differential-graded-commutative algebra, that of a [[super L-infinity algebra]] is a differential graded-commutative superalgebra.

|  |  |  |
|---|----|---|
| $\phantom{A}$[bi-degree](chain+complex+in+super+vector+spaces#ChainComplexesOfSuperVectorSpaces)$\phantom{A}$<br/> $\phantom{A}$$(n,\sigma) \in \mathbb{Z} \times \mathbb{F}_2$$\phantom{A}$ | $\phantom{A}$$n = 0$$\phantom{A}$ | $\phantom{A}$$n\;$  arbitrary$\phantom{A}$ |
| $\phantom{A}$$\sigma = even$$\phantom{A}$ | $\phantom{A}$[[commutative algebra]]$\phantom{A}$ | $\phantom{A}$[[differential graded-commutative algebra]]$\phantom{A}$ |
| $\phantom{A}$$\sigma\;$  arbitrary$\phantom{A}$ |  $\phantom{A}$e.g. [[Grassmann algebra]]$\phantom{A}$ | $\phantom{A}$[[differential graded-commutative superalgebra]]$\phantom{A}$ |

## Related concepts

* [[category of dgc-algebras]]

* [[dgc-coalgebra]]

* [[model structure on differential graded-commutative algebras]]

* [[model structure on differential graded-commutative superalgebras]]

* [[rational homotopy theory]]

* [[monoidal Dold-Kan correspondence]]

* [geometry of physics -- superalgebra -- Z-graded commutative superalgebra](https://ncatlab.org/nlab/show/geometry+of+physics+--+superalgebra#ZGradedSupercommutativeSuperalgebra)

* [[equivariant dgc-algebra]]

* [[model structure on equivariant dgc-algebras]]

* [[equivariant rational homotopy theory]]

[[!redirects differential graded-commutative algebras]]

[[!redirects DGCA]]
[[!redirects DGCAs]]

[[!redirects CDGA]]
[[!redirects CDGAs]]

[[!redirects dgca]]
[[!redirects dgca-s]]

[[!redirects dgc-algebra]]
[[!redirects dgc-algebras]]

[[!redirects cdga]]
[[!redirects cdga-s]]

[[!redirects dgc-superalgebra]]
[[!redirects dgc-superalgebras]]

[[!redirects differential graded-commutative superalgebra]]
[[!redirects differential graded-commutative superalgebras]]


differential ideal

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#Contents#
* table of contents
{:toc}

## Definition

For $(A,d)$ a [[differential algebra]], a _differential ideal_ is an [[ideal]] $I \subset A$ in the underlying [[associative algebra]] $A$ which is preserved by the [[derivation]] $d$, in that $d(I) \subset I$.

## Properties

The [[quotient algebra]] of a [[differential algebra]] by a differential ideal inherits the structure of a differential algebra.

## Related concepts

* [[germ-determined ideal]]

## References

See also

* Wikipedia, _[Differential ideal](https://en.wikipedia.org/wiki/Differential_ideal)_

[[!redirects differential ideals]]


differential K-theory

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#### Differential cohomology
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[[!include differential cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Differential K-theory_ is the refinement of the [[generalized (Eilenberg-Steenrod) cohomology]] theory [[K-theory]] to [[differential cohomology]].

In as far as we can think of cocycles in [[K-theory]] as represented by [[vector bundle]]s or [[vectorial bundle]]s, cocycles in differential K-theory may be represented by [[vector bundle]]s [[connection on a bundle|with connection]].

There are various different models that differ in the concrete realization of these cocycles and in their extra properties.

## The Simons-Sullivan model {#SimonsSullivanModel}

This section discusses the model presented in ([SimonsSullivan](#SimonsSullivan)).

More details will eventually be at

* [[Simons-Sullivan structured bundle]].

### Idea

In the Simons-Sullivan model cocycles in differential K-theory are represented by ordinary [[vector bundle]]s [[connection on a bundle|with connection]]. The crucial ingredient is that two connections on a vector bundle are taken to be the same representative of a differential K-cocycle if they are related by a [[concordance]] such that the corresponding [[Chern-Simons form]] is exact.

### Details

Let $V \to X$ be a complex [[vector bundle]] with [[connection on a bundle|connection]] $\nabla$ and [[curvature]] 2-form

$$
  F = F_\nabla \in \Omega^2(X,End(V))
  \,.
$$

**Definition**

The **[[Chern character]]** of $\nabla$ is the inhomogenous [[curvature characteristic form]]

$$
 ch(\nabla) := \sum_{j \in \mathbb{N}}
   k_j
   tr( F_\nabla \wedge \cdots \wedge F_\nabla)
  \;\;
  \in \Omega^{2 \bullet}(X)
  \,,
$$

where on the right we have $j$ wedge factors of the [[curvature]] .

**Definition**

Let $(V,\nabla)$ and $(V',\nabla')$ be two complex
vector bundles with connection.

A [[Chern-Simons form]] for this pair is a differential form

$$
  CS(\nabla,\nabla') + d \omega \in
  \Omega^{2 \bullet + 1}(X)
$$

obtained from the [[concordance]] bundle $\bar V \to X \times [0,1]$ given by [[pullback]] along $X \times [0,1] \to X$ equipped with a connection $\bar \nabla$ such that ..., by

$$
  CS(\nabla,\nabla') =
  \int_0^1 \psi_t^* (\iota_{\partial/\partial t} ch(\bar \nabla))
  + d (...)
  \,.
$$

**Proposition** This is indeed well defined in that it is independent of the chosen [[concordance]], up to an exact term.

**Definition**

A **structured bundle** in the sense of the Simons-Sullivan model is a complex vector bundle $V$ equipped with the equivalence class $[\nabla]$ of a connection under the equivalence relation that identifies two connections $\nabla$ and $\nabla'$ if their Chern-Simons form $CS(\nabla,\nabla')$ is exact.

Two structured bundles are isomorphic if there is a vector bundle isomorphism under which the two equivalence classes of connections are identified.

**Definition**

Let $Struc(X)$ be the set of [[isomorphism]] classes of structured bundles on $X$.

Under [[direct sum]] and [[tensor product]] of vector bundles, this becomes a commutatve [[rig]].

Let
$$
  \hat K(X) := K(Struct(X))
$$

be the additive [[group completion]] of this rig as usual in [[K-theory]].

So as an additive group $\hat K(X)$ is the quotient of the [[monoid]] induced by [[direct sum]] on pairs $(V,W)$ of isomorphism classes in $Struc(X)$, modulo the sub-monoid consisting of pairs $(V,V)$.

Hence the pair $(V,0)$ is the additive inverse to $(0,V)$ and $(V,W)$ may be written as $V - W$.

**Theorem**

$\hat K(X)$ is indeed a [[differential cohomology]] refinement of ordinary K-theory $K(X)$ of $X$ (i.e. of the 0th [[cohomology group]] of [[K-theory spectrum|K-cohomology]]).

Moreover...

## The Bunke-Schick model

### Idea

[[Uli Bunke]] and [[Thomas Schick]] developed in a series of articles a differential-geometric cocycle model of differential K-theory where cocycles are given by smooth families of [[Dirac operator]]s.

See the reference [below](BunkeSchickReferences).

### Properties

The restriction of the cocycles in the Bunke-Schick model to those whose "auxialiary form" $\omega$ vanishes reproduces the Simons-Sullivan model above.

## The Hopkins-Singer model

See at

* _[[differential function complex]]_

* _[differential cohomology diagram -- Hopkins-Singer differential KU](differential+cohomology+diagram#HSDifferentialKU)_

## More models in smooth spectra

See at _[Differential cohomology diagram -- Differential K-theory](differential+cohomology+diagram#DifferentialKTheory)_.

## Examples

* In [[gauge theory]] gauge fields are modeled by cocycles in [[differential cohomology]]. The field modeled by differential K-theory is the [[RR-field]]. A kind of [[integral transform]] acting on differential K-theory classes is [[T-duality]].

## Related concepts

* [[flat K-theory]]

* [[K-theory]], [[topological K-theory]]

* [[twisted K-theory]]

* **differential K-theory**

  * [[fiber integration in differential K-theory]]

  * [[algebraic K-theory of smooth manifolds]]

* [[twisted differential K-theory]]

* [[equivariant differential K-theory]], [[orbifold differential K-theory]]

* [[differential algebraic K-theory]]

## References

### General

An early sketch of a general definition, motivated by the description of [[D-brane charge]] in [[string theory]], is in

* [[Daniel Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 ([arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220))

* [[Daniel Freed]], [[Michael Hopkins]], *On Ramond-Ramond fields and K-theory*, JHEP (2000) 44, 14 [[arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027), [doi:10.1088/1126-6708/2000/05/044](https://doi.org/10.1088/1126-6708/2000/05/044)]

Then the general construction of [[differential cohomology]] theories via [[differential function complexes]] of

* {#HopkinsSinger05} [[Michael Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_,  J. Differential Geom. Volume 70, Number 3 (2005), 329-452 ([arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216))

(motivated in turn by [[7d Chern-Simons theory]] and the [[M5-brane]] [[partition function]])

provides in particular a model for differential K-theory.

For more historical remarks see section 1.6 of

* {#FreedLott10} [[Daniel Freed]], [[John Lott]], _An index theorem in differential K-theory_, Geometry and Topology 14 (2010) ([pdf](http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf))

A discussion of more models and their relation in the context of [[cohesive homotopy type theory]] and the [[differential cohomology hexagon]] then appears in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], Section 6 of: _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

Survey:

* {#BunkeSchick10} [[Ulrich Bunke]], [[Thomas Schick]], _Differential K-theory. A survey_ ([arXiv:1011.6663](http://arxiv.org/abs/1011.6663)).

Cocycle models via [[vector bundles]] [[bundle with connection]]:

* [[Max Karoubi]], *Homologie cyclique et K-théorie*, Astérisque, no. 149 (1987) ([numdam:AST_1987__149__1_0](http://www.numdam.org/item/AST_1987__149__1_0))

* [[John Lott]], *$\mathbf{R}/\mathbf{Z}$ Index theory*, Communications in Analysis and Geometry **2** 2 (1994) 279-311 ([pdf](https://math.berkeley.edu/~lott/rztheory.pdf))

* {#SimonsSullivan} [[James Simons]], [[Dennis Sullivan]], _Structured vector bundles define differential K-theory_ ([arXiv:0810.4935](http://arxiv.org/abs/0810.4935))

The basic article for the Bunke-Schick model is

* {#BunkeSchick09} [[Ulrich Bunke]], [[Thomas Schick]], _Smooth K-Theory_, Astérisque 328 (2009), 45-135 [[arXiv:0707.0046](http://arxiv.org/abs/0707.0046), [numdam:AST_2009__328__45_0](http://www.numdam.org/item/?id=AST_2009__328__45_0)]

A survey talk is

* [[Ulrich Bunke]], _Differential cohomology in geometry and analysis_ ([pdf slides](http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf))

On differential [[KO-theory]]:

* {#GradySati18} [[Daniel Grady]], [[Hisham Sati]], _Differential KO-theory: Constructions, computations, and applications_, Advances in Mathematics Volume 384, 25 June 2021, 107671 ([arXiv:1809.07059](https://arxiv.org/abs/1809.07059), [doi:10.1016/j.aim.2021.107671](https://doi.org/10.1016/j.aim.2021.107671))

* [[Kiyonori Gomi]], [[Mayuko Yamashita]], *Differential KO-theory via gradations and mass terms* [[arXiv:2111.01377](https://arxiv.org/abs/2111.01377)]

Of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]]:

* {#GradySati19} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

The equivalence of these models with the respective special case of the general construction in [Hopkins & Singer 05](#HopkinsSinger05) in terms of [[differential function complexes]] is in

* Kevin Klonoff, _An Index Theorem in Differential K-Theory_ PdD thesis (2008) ([pdf](http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf))

(assuming the existence of a universal connection, which is not strictly proven) and

* Michael L. Ortiz, _Differential Equivariant K-Theory_ ([arXiv:0905.0476](http://arxiv.org/abs/0905.0476))

(not needing that assumption).

A construction of [[differential cobordism cohomology]] theory in terms of explicit geometric cocycles is in

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup _Landweber exact formal group laws and smooth cohomology theories_ ([arXiv:0711.1134](http://arxiv.org/abs/0711.1134))

By tensoring this with the suitable ring, this also gives a model for differential K-theory, as well as for [[differential elliptic cohomology]].

A variant of this definition with the advantage that there is a natural morphism to [[Cheeger-Simons differential characters]] refining the total [[Chern class]] is (as opposed to the [[Chern character]]) is presented in

* Alain Berthomieu, _A version of smooth K-theory adapted to the
total Chern class_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.4728v1.pdf))

Discussion for the [[odd Chern character]] is in

* [[Thomas Tradler]], [[Scott Wilson]], [[Mahmoud Zeinalian]], _An Elementary Differential Extension of Odd K-theory_, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis
and Topology, ([arXiv:1211.4477](http://arxiv.org/abs/1211.4477))

* [[Scott Wilson]], _A loop group extension of the odd Chern character_ ([arXiv:1311.6393](http://arxiv.org/abs/1311.6393))

A model in terms of [[super vector bundles]] is given in

* Jae Min Lee, Byungdo Park, _A Superbundle Description of Differential K-Theory_, Axioms 2023, 12(1), 82, ([doi:10.3390/axioms12010082](https://doi.org/10.3390/axioms12010082)).

### Relation to index theory

Relation to [[index theory]]:

* Kevin Klonoff, _An Index Theorem in Differential K-Theory_ PdD thesis (2008) ([pdf](http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf))

* [[Daniel Freed]], [[John Lott]], _An index theorem in differential K-theory_, Geometry and Topology 14 (2010) ([pdf](http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf))

See also the references at _[[fiber integration in differential K-theory]]_.

### In string theory

A survey of the role of differential $K$-theory in [[quantum field theory]] and [[string theory]] is in

* [[Daniel Freed]], _K-Theory in Quantum Field Theory_, Current developments in Mathematics (2001) International Press Somerville ([arXiv:math-ph/0206031](http://arxiv.org/abs/math-ph/0206031))

The operation of [[T-duality]] on hypothetical [[twisted differential K-theory]] is discussed in

* [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]] (see also [there](D-brane#ReferencesKTheoryDescription)):

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], *Ramond-Ramond fields and twisted differential K-theory*,
Advances in Theoretical and Mathematical Physics **26** (2022) 5 [[arXiv:1903.08843](https://arxiv.org/abs/1903.08843), [doi:10.4310/ATMP.2022.v26.n5.a2](https://dx.doi.org/10.4310/ATMP.2022.v26.n5.a2)]

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

See also:

* [[Thomas Tradler]], [[Scott Wilson]], [[Mahmoud Zeinalian]], _Loop Differential K-theory_ ([arXiv:1201.4593](http://arxiv.org/abs/1201.4593))


differential linear logic

#Contents#
* table of contents
{:toc}

(work in progress)

## Idea

Differential linear logic is an extension of [[linear logic]]. New inference rules are added which allow to differentiate proofs. Proofs must be understood as morphisms and the new inference rules are here to differentiate them. It was realised than in some denotational models of linear logic, the morphisms of the [[co-Kleisli category]] are differentiable, and it was then reflected in the syntax by these new inference rules. Thus, one really needs to take the point of view of [[computational trinitarianism]] in order to understand the transition from linear logic to differential linear logic. It is more difficult to understand naively the proofs of differential linear logic than the ones of linear logic.

Differential linear logic can be presented in different guises whether we include the additives or not, and whether we include the promotion or not. We start by presenting the version without additives and with promotion. We will refer to this logic as DiLL.

## Syntax

There are two ways to present the syntax in  [[sequent calculus]]. With bilateral sequents or monolateral sequents. The first one is closer to the language of category theory but the second one allows to make the links with the syntax in [[proof nets]] and to divide by two the number of inference rules. Using the monolateral version (or proof nets) require to quotient the formulas by some equalities related to negation and [[De Morgan duality]].

### Formulas - Generalities

There are no novelties concerning formulas compared to linear logic. We will work with the formulas of Multiplicative Exponential Linear Logic (MELL). In the next subsection, the difference between MELL and DiLL will appear as additional inference rules.

Formulas are given by induction starting from atoms and applying connectives. In accordance with the tradition, formulas and atoms are denoted in the same way by upper case letters:

* Atoms are denoted by upper case letter $A,B,...$.
* If we have two formulas $A,B$, we can form $A \otimes B$.
* If we have two formulas $A,B$, we can form $A \parr B$.
* If we have a formula $A$, we can form $!{A}$.
* If we have a formula $A$, we can form $?{A}$.
* If we have a formula $A$, we can form $A^\perp$.
* We have the formula $1$.
* We have the formula $\bot$.

### Formulas - Quotienting by duality in monolateral syntax

If we want to use the monolateral sequent calculus (or proof nets), formulas must be quotiented by the following equalities:

* If we have a formula $A$, we put $A^{\perp\perp} = A$.
* If we have two formulas $A,B$, we put $(A \otimes B)^\perp = A^\perp \parr B^\perp$.
* If we have two formulas $A,B$, we put $(A \parr B)^\perp = A^\perp \otimes B^\perp$.
* If we have a formula $A$, we put $(!{A})^\perp = ?{A^\perp}$.
* If we have a formula $A$, we put $(?{A})^\perp = !{A^\perp}$.
* We have $1^\perp = \bot$.
* We have $\bot^\perp = 1$.

Notice that some equalities are implied by other ones and thus we could give a shorter list. We have preferred writting directly all the equalities of pratical use.

### Inference rules - Bilateral syntax

Inference rules will allow us to build proofs. Notice that compared to MELL, we need the ability to sum proofs. We give an explicit rule for this summation of proofs. We give also a rule for null proofs. Finally, the new inference rules are: coderiliction, cocontraction, coweakening, sum of proofs, null proof.

* Axiom: $\frac{}{A \vdash A}^{ax}$
* Left exchange: $\frac{\Gamma_{1},A,B,\Gamma_{2} \vdash \Delta}{\Gamma_{1},B,A,\Gamma_{2} \vdash \Delta}$
* Right exchange: $\frac{\Gamma \vdash \Delta_{1},A,B,\Delta_{2}}{\Gamma \vdash \Delta_{1},B,A,\Delta_{2}}$
* Left $\otimes$: $\frac{\Gamma_{1}, A, B, \Gamma_{2} \vdash \Delta}{\Gamma_{1}, A \otimes B, \Gamma_{2} \vdash \Delta}$
* Right $\parr$: $\frac{\Gamma \vdash \Delta_{1}, A, B, \Delta_{2}}{\Gamma \vdash \Delta_{1}, A \parr B, \Delta_{2}}$
* Right $\otimes$: $\frac{\Gamma_{1} \vdash \Delta_{1}, A, \Delta_{2} \qquad \Gamma_{2} \vdash \Delta_{3}, B, \Delta_{4}}{\Gamma_{1}, \Gamma_{2} \vdash \Delta_{1}, \Delta_{3}, A \otimes B, \Delta_{4}, \Delta_{2}}$
* Left $\parr$: $\frac{\Gamma_{1}, A, \Gamma_{2} \vdash \Delta_{1} \qquad \Gamma_{3}, B, \Gamma_{4} \vdash \Delta_{2}}{\Gamma_{1}, \Gamma_{3}, A \parr B, \Gamma_{4}, \Gamma_{2} \vdash \Delta_{1}, \Delta_{2}}$
* Left $1$: $\frac{\Gamma_{1}, \Gamma_{2} \vdash \Delta}{\Gamma_{1}, 1, \Gamma_{2} \vdash \Delta}$
* Right $\bot$: $\frac{\Gamma \vdash \Delta_{1},\Delta_{2}}{\Gamma \vdash \Delta_{1}, \bot, \Delta_{2}}$
* Right $1$: $\frac{}{\vdash 1}$
* Left $\bot$: $\frac{}{\bot \vdash}$
* Left negation: $\frac{\Gamma \vdash A, \Delta}{\Gamma, A^{\perp} \vdash \Delta}$
* Right negation: $\frac{\Gamma, A \vdash \Delta}{\Gamma, \vdash A^{\perp}, \Delta}$
* Left dereliction: $\frac{\Gamma_{1},A ,\Gamma_{2} \vdash \Delta}{\Gamma_{1}, !A, \Gamma_{2} \vdash \Delta}$
* Right dereliction: $\frac{\Gamma \vdash \Delta_{1},A ,\Delta_{2}}{\Gamma \vdash \Delta_{1}, ?A, \Delta_{2}}$
* Left weakening: $\frac{\Gamma_{1},\Gamma_{2} \vdash \Delta}{\Gamma_{1}, !A, \Gamma_{2} \vdash \Delta}$
* Right weakening: $\frac{\Gamma \vdash \Delta_{1}, \Delta_{2}}{\Gamma \vdash \Delta_{1}, ?A, \Delta_{2}}$
* Left contraction: $\frac{\Gamma_{1},!A ,!A, \Gamma_{2} \vdash \Delta}{\Gamma_{1}, !A, \Gamma_{2} \vdash \Delta}$
* Right contraction: $\frac{\Gamma \vdash \Delta_{1},?A , ?A, \Delta_{2}}{\Gamma \vdash \Delta_{1}, ?A, \Delta_{2}}$
* Right Promotion: $\frac{!\Gamma \vdash ?\Delta_{1}, A, ?\Delta_{2}}{!\Gamma \vdash ?\Delta_{1}, !A, ?\Delta_{2}}$
* Left promotion: $\frac{!\Gamma_{1}, A, !\Gamma_{2} \vdash \Delta}{!\Gamma_{1}, ?A, !\Gamma_{2} \vdash \Delta}$
* Right codereliction: $\frac{\Gamma \vdash \Delta_{1}, A, \Delta_{2}}{\Gamma \vdash \Delta_{1}, !A, \Delta_{2}}$
* Left codereliction: $\frac{\Gamma_{1}, A, \Gamma_{2} \vdash \Delta}{\Gamma_{1}, ?A, \Gamma_{2} \vdash \Delta}$
* Right coweakening: $\frac{}{\vdash !A}$
* Left coweakening: $\frac{}{?A \vdash}$
* Right cocontraction: $\frac{\Gamma_{1} \vdash \Delta_{1}, !A, \Delta_{2} \qquad \Gamma_{3} \vdash \Delta_{3}, !A, \Delta_{4}}{\Gamma_{1}, \Gamma_{3} \vdash \Delta_{1}, \Delta_{3}, !A, \Delta_{4}, \Delta_{2}}
$
* Left cocontraction: $\frac{\Gamma_{1}, ?A, \Gamma_{2} \vdash \Delta_{1} \qquad \Gamma_{3}, ?A, \Gamma_{4} \vdash \Delta_{2}}{\Gamma_{1}, \Gamma_{3}, ?A, \Gamma_{4}, \Gamma_{2} \vdash \Delta_{1}, \Delta_{2}}$
* Nul proof: $\frac{}{\Gamma \vdash \Delta}^{0}$
* Sum of proofs: $\frac{\Gamma \vdash \Delta \qquad \Gamma \vdash \Delta}{\Gamma \vdash \Delta}$
* Cut: $\frac{\Gamma_{1} \vdash \Delta_{1}, A, \Delta_{2} \qquad \Gamma_{2}, A, \Gamma_{3} \vdash \Delta_{3}}{\Gamma_{2}, \Gamma_{1}, \Gamma_{3} \vdash \Delta_{1}, \Delta_{3}, \Delta_{2}}$

### Inference rules - Monolateral syntax

## Isomorphisms in bilateral syntax

## Cut elimination

There is a [[rewriting system]] which, given an instance of the cut rule in a proof, reduces the proof by making the cut go higher in the proof. There is the key cases which induce important commuting diagrams in the categorical semantics and the commuting cases which are less important. The idea is that when a cut arrives at the beginning of the proof, ie. at the level of the leafs which are axioms, we can finally eliminate it by a last rewriting step. This is when the cut is made against an axiom rule, that we can eliminate it. We know that this rewriting system is [[strongly normalizing]] and that the [[normal forms]] are the cut-free proofs. It means that by using rewriting rules in the order you want, at the end you obtain a proof without cut, thus the name [[cut-elimination]]. It is important to know actually this rewriting system and not only that we can eleminate the cuts in a proof because this cut-elimination prodecure creates the commuting diagrams of the categorical semantics when we quotient the proofs by these cut-elimination steps.

### Cut elimination steps

### What is an isomorphism?

## Semantics

### Categorical semantics

We need a [[*-autonomous category]] in order to interpret the connectors and rules related to these connectors: $\otimes, \parr, 1, \bot, {}^{\perp}$.

This category must be enriched over commutative monoids in order to be able to interpret sum of proofs and null proofs.

\begin{definition}
We say that a [[*-autonomous category]] $\mathcal{C}$ is enriched over commutative monoids if it is enriched over commutative monoids as a [[symmetric monoidal category]] ie. every hom-set $\mathcal{C}[A,B]$ is a commutative monoid and $-\otimes-$ as well as $-;-$ are bilinear over morphisms ie. preserve addition and zero in every variable.
\end{definition}

The tricky part is to interpret the exponentials $!,?$. We just need to add enough to interpret $!$, the interpretation of $?$ will follow by the $*$-autonomous structure. We need:

* A natural transformation $!A \otimes !A \rightarrow !A$ to interpret the cocontraction.
* A natural transformation $!A \rightarrow !A \otimes !A$ to interpret the contraction.
* A natural transformation $I \rightarrow !A$ to interpret the coweakening.
* A natural transformation $!A \rightarrow I$ to interpret the weakening.
* A natural transformation $A \rightarrow !A$ to interpret the codereliction.
* A natural transformation $!A \rightarrow A$ to interpret the dereliction.

Thus, for every object $A$, we will have that $!A$ is a bimonoid. $!$ will be a comonad. The coderiliction is added on top of this structure.

(...)

### Concrete models

(...)

## The logic at work

### The co-Kleisli category

### Differentiating a proof

We start from a proof of $!A \vdash B$ and by using as only exponential rules cocontraction and codereliction, we obtain a proof of $!A \vdash A \multimap B$. In the co-Kleisli category we thus go from a morphism $f:A \rightarrow B$, which must be interpreted as a smooth function, to a morphism $df: A \rightarrow (A \multimap B)$. It is the differential of $f$ which associates to every point of $A$ the linear approximation of $f$ around this point.

$$
\frac{\frac{\frac{\frac{\frac{}{!A \vdash !A}^{ax} \frac{\frac{}{A \vdash A}^{ax}}{A \vdash !A}}{!A,A \vdash !A} !A \vdash B}{!A,A \vdash B}}{!A \vdash A^{\perp},B}}{!A \vdash A \multimap B}
$$

## Categorical semantics of differential linear logic and differential categories

(...)

## Additives and Seely isomorphisms

## Related concepts

* [[linear logic]]

* [[differential calculus]]

* [[differential category]]

## References

Original articles:

* {#EhrhardtRegnier09} [[Thomas Ehrhard]], [[Laurent Regnier]]: _The differential lambda-calculus_, Theor. Comput. Sci. **309** 1 (2003) 1–41  (doi:10.1016/S0304-3975(03)00392-X)

* [[Thomas Ehrhard]], [[Laurent Regnier]]: _Differential interaction nets,_ Theor. Comput. Sci. **364** 2 (2006) 166–195 ([doi:10.1016/j.tcs.2006.08.003](https://doi.org/10.1016/j.tcs.2006.08.003))

Review:

* [[Thomas Ehrhard]], *An introduction to Differential Linear Logic: proof-nets, models and antiderivatives*, Mathematical Structure in Computer Science **28** 7 (2018) 995-1060 ([arXiv:1606.01642](https://arxiv.org/abs/1606.01642), [doi:10.1017/S0960129516000372](https://doi.org/10.1017/S0960129516000372))

Recent work on categorical semantics:

* [[R. F. Blute]], [[J. R. B. Cockett]], [[Jean-Simon Lemay]], [[R. A. G. Seely]], *Differential Categories Revisited*, Appl Categor Struct **28** (2020) 171-235 ([arXiv:1806.04804](https://arxiv.org/abs/1806.04804), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y))

[[!redirects Differential linear logic]]


differential logic

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview).

[[!redirects Differential Logic]]


differential module

{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

[Roman Sikorski](http://www.ams.org/mathscinet/search/author.html?mrauthid=162010) defined the term "differential module" in 1971/72.
[Mark Mostow](http://www.ams.org/mathscinet/search/author.html?mrauthid=127450) further developed the theory in 1979.
The following definition of a differentiable space (a kind of [[generalized smooth space]]) comes from Mostow's paper where it is attributed to Sikorski.

+-- {: mynumdef #DiffSp}
###### Definition
A _differentiable space_ is a topological space $X$ together with, for each open $U$ in $X$, a collection $C^\infty(U)$ of continuous real-valued functions on $U$, satisfying the closure conditions:

1. The rule $U \to C^\infty(U)$ defines a sheaf on $X$ (denoted $C^\infty X$).
2. For any $n$, if $f_1, \dots, f_n \in C^\infty(U)$ and $g \in C^\infty(\mathbb{R}^n)$ (with the usual meaning), then $g(f_1, \dots, f_n) \in C^\infty(U)$.

The elements of $C^\infty(X)$ are called _smooth functions_ on $X$.
=--

# References #
* [Differential modules](http://www.ams.org/mathscinet-getitem?mr=482794) 1971/72
* [The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations](http://www.ams.org/mathscinet-getitem?mr=587553) 1979.