formal concept analysis

##Idea

Formal concept analysis is about extracting the relationships and hierarchies available from the common attributes shared by objects.

A _formal context_ is, simply, a dyadic [[Chu spaces, simple examples|Chu space]]. More precisely:

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

A _(dyadic or two valued) Chu space_ $\mathcal{P}$ is a triple $(P_o, \models_P, P_a)$, where $P_o$ is a set of _objects_, and $P_a$ is a set of _attributes_.  The _satisfaction_ relation $\models_P$ is a subset of $P_o\times P_a$.
=--

The terminology is that used in Formal Concept Analysis. Such an object gives an array representing a relation between a set of objects and a set of attibutes. If an object, $o\in P_o$, satisfies attribute $a\in P_a$, that is $o\models_P a$, one puts a 1 in the $(o,a)$- position in the array, otherwise one puts a 0. This is thus the usual classical representation of the relation as a subset of the product. (We will restrict attention to this simple 2-valued case. One can replace $\mathbf{2}$ but other suitably structured sets, to obtain probabilistic or fuzzy versions of a Formal Context.

The task of the analysis is to see if the data encoded in the array allows one to group objects or attributes together so as to glean more information about the given relationship between the two sets.

To quote from Zhang and Shen:
>The novel idea of FCA is the clustering of attributes based
on the algebraic principle of Galois connection, forming a partially ordered set called (a) concept lattice. The clustering determines which collection of attributes forms a coherent entity called a concept, by the philosophical criteria of unity between extension and
intension. The extension of a concept consists of all objects belonging to the concept,
while the intension of a concept consists of attributes common to all these objects. One
can then take this as the defining property of a concept: _a collection of attributes which
agrees with the intension of its extension._

### Two mappings to powersets

Given a Formal Context, $\mathcal{P}=(P_o, \models_P, P_a)$, we define two mappings:

$$\tilde{\alpha}_P:P_o\to  \mathbf{2}^{P_a}, \quad\tilde{\alpha}_P(x)=\{a\in P_a\,|\,x\models_P a\}$$

and

$$\tilde{\omega}_P:P_a\to  \mathbf{2}^{P_o}, \quad
\tilde{\omega}_P(a)=\{x\in P_o\,|\,x\models_P a\}.$$

The formal context / Chu space $\mathcal{P}$ is said to be _extensional_ if $\tilde{\omega}_P:P_a\to  \mathbf{2}^{P_o}$ is injective and is said to be _separable_ if $\tilde{\alpha}:P_o\to  \mathbf{2}^{P_a}$ is injective.

###Two mappings on powersets.

The two mappings $\tilde{\alpha}_P$ and $\tilde{\omega}_P$ extend to give mappings $\alpha_P: \mathbf{2}^{P_o} \to \mathbf{2}^{P_a}$ and $\omega_P : \mathbf{2}^{P_a}\to \mathbf{2}^{P_o}$, defined by
$$\alpha_P(X) = \{a |\forall x\in X,  x\models_P a\}$$
and
$$\omega_P(A) = \{x \,|\,\forall a \in A,  x\models_P a\}.$$

## Related concepts

* [[Chu spaces, simple examples|Chu space]]

* [[Galois connection]]

##References

* Ganter, B., & Wille, R. _Formal concept analysis: mathematical foundations_. Springer Science & Business Media, (2012). [doi](https://doi.org/10.1007/978-3-642-59830-2)
* Radim Belohlavek, Introduction to formal concept analysis [pdf](http://phoenix.inf.upol.cz/esf/ucebni/formal.pdf)
* Café discussion [Formal Concept Analysis](http://golem.ph.utexas.edu/category/2013/09/formal_concept_analysis.html#more)

* [[Guo-Qiang Zhang]], Shen, G.: [Approximable Concepts, Chu spaces, and information systems](http://www.tac.mta.ca/tac/volumes/17/5/17-05.pdf), Theory and Applications of Categories 17, 2006, no.7

* [Formal Concept Analysis Homepage](https://upriss.github.io/fca/fca.html)

[[!redirects formal concept lattice]]
[[!redirects Formal Concept Analysis]]


formal deformation quantization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of *formal deformation quantization* (often taken to be the default notion of *[[deformation quantization]]*) is one formalization of the general idea of [[quantization]] of a [[classical mechanical system]]/[[classical field theory]] to a [[quantum mechanical system]]/[[quantum field theory]], at least [[perturbative quantum field theory|perturbatively]] in small (really: [[infinitesimal]]) values of [[Planck's constant]] $\hbar$.

Formal deformation quantization focuses on the [[algebras of observables]] of a physical system (hence on the [[Heisenberg picture]]): it provides rules for how to [[deformation theory|deform]] the commutative algebra of classical observables to a non-commutative algebra of quantum observables. (This is in contrast to [[geometric quantization]], which focuses on the [[spaces of states]] and hence on the [[Schrödinger picture]].)

Usually and traditionally, _[[deformation quantization]]_ refers to (just) _formal_ deformations, in the sense that it produces [[formal power series]] expansions in a formal parameter $\hbar$ (physically: [[Planck's constant]]) of the product in the deformed algebra of observables.

| [[classical mechanics]] | [[semiclassical approximation]] |  ... | [[formal deformation quantization]] | [[quantum mechanics]] |
|--|--|--|--|--|
| $\mathcal{O}(\hbar^0)$ | $\mathcal{O}(\hbar^1)$ | $\mathcal{O}(\hbar^n)$ | $\mathcal{O}(\hbar^\infty)$  |  |

(This is related to [[perturbation theory]], which is about [[formal power series]] in [[coupling constants]], instead of in [[Planck's constant]].)

But there are refinements of this to [[C-star algebraic deformation quantization]] which studies the proper deformation to a genuine [[C-star algebra]] of observables. (This in turn is related to genuine [[geometric quantization]] via the notion of [[geometric quantization of symplectic groupoids]].)

One can, therefore, argue that only [[C-star algebraic deformation quantization|strict deformation quantization]] is genuine [[quantization]]. For instance in ([Gukov-Witten 09, section 1.4](#GukovWitten09)) it says

> Generally speaking, [[physics]] is based on $[$ strict $]$ quantization, rather than $[$ formal $]$ deformation quantization, although conventional quantization sometimes leads to problems that can be treated by deformation quantization.

As other methods of quantization, deformation quantization has as input a description of a [[classical mechanical system]], which is in this case most often a smooth [[Poisson manifold]]. The deformation quantization replaces the algebra of [[smooth functions]] on the Poisson manifold with the same [[vector space]], but equipped with new noncommutative [[associative algebra|associative unital product]] whose [[commutator]] agrees, up to order $\hbar$, with the underlying [[Poisson bracket]]. Of course the proper study of quantization of Poisson manifolds studied the appropriate notion at the level of [[sheaves]] of algebras. Gluing local solutions to the quantization problem furthermore involves [[stacks]] and specifically [[gerbes]].

## Definition

If the result of deformation quantization is an algebra over the [[power series]] [[ring]] $\mathbb{R}[ [ \hbar ] ]$ of a formal parameter $\hbar$ (thought of as [[Planck's constant]]) such that the limit $\hbar \to 0$ reproduces the starting point of the deformation, then one speaks of

* [Formal deformation quantization](#FormalDeformationQuantization)

In much of the literature this is regarded as the default meaning of "deformation quantization". But this is really the case corresponding to [[perturbation theory]] in [[quantum field theory]]. A "genuine" or "strict" deformation quantization

* [Strict deformation quantization](#StrictDeformationQuantization)

is supposed to result in a non-formal deformation, which in terms of the above formal power series at least means that one can set $\hbar = 1$ such that all expressions in $\hbar$ converge, but which in general is taken to mean something stronger, such as that there is a continuous field of [[C-star algebraic deformation quantization]].

### Formal deformation quantization
 {#FormalDeformationQuantization}

We first give the traditional

* [Explicit definition of traditional deformation quantization](#ExplicitDefinitionOfTraditionalDeformationQuantization)

of a [[Poisson manifold]]/[[Poisson algebra]]. Thought of in terms of [[physics]] this describes a [[quantization]] of a [[physical system|system]] of [[quantum mechanics]], as opposed to full [[quantum field theory]].

More abstractly, this may be formulated and generalized in terms of lifts of [[algebras over an operad]] over a [[P-n operad]] to a [[BD-n operad]] and hence an [[E-n operad]], for $n = 1$. This we discuss in

* [Formulation as lifts from P-n algebras to BD-n algebras](#FormulationAsLiftsFromPnAlgebrasToBDAlgebras)

In this formulation one sees that for genral $n$ the construction applies  to $n$-dimensional [[quantum field theory]] (with [[quantum mechanics]] for $n = 1$ be 1-dimensional quantum field theory, for instance the [[sigma-model]] "on the [[worldline]]" of a [[particle]]). A formulation of deformation quantization to _[[local quantum field theory|local]]_ quantum field theory formulated in terms of [[factorization algebras of observables]] over [[spacetime]]/[[worldvolume]] is indicated in ([Costello-Gwilliam, chapter 5](#CostelloGwilliam)).

#### Explicit definition of deformation of Poisson manifolds/Poisson algebras
 {#ExplicitDefinitionOfTraditionalDeformationQuantization}

Let $M$ be a [[Poisson manifold]] and let $A = C^\infty(M)$ be the [[Poisson algebra]] of smooth functions.

+-- {: .num_defn}
###### Definition
A **$\ast$-product** ([[star product]]) on $A$ is a product on the
[[power series]] $A [ [ t ] ]$ that is (1) bilinear over $\mathbb{R}[ [ t ] ]$, (2) associative, and (3) for $a,b \in A$ it can be written out as a [[formal power series]]
\[ a \ast b = \sum_{n=0}^\infty B_n(a,b) t^n \]
where $B_n$ are bilinear maps on $A$ such that $B_0(a,b) = ab$.
=--

+-- {: .num_defn}
###### Definition
A (formal) **deformation quantization** of $M$ is a [[star product]] on $A = C^\infty(M)$ such that the [[Poisson bracket]] $\{a,b\} = B_1(a,b) - B_1(b,a)$ for $a,b \in A$; by bilinearity over $\mathbb{R}[ [ t ] ]$, this characterizes it.
=--

#### Formulation as lifts from $P_n$-algebras to $BD_n$-algebras and $E_n$-algebras
 {#FormulationAsLiftsFromPnAlgebrasToBDAlgebras}

(...)

([Costello-Gwilliam, section 2.3, 2.4](#CostelloGwilliam))

(...)

[[!include deformation quantization - table]]

#### Deformation quantization in perturbative quantum field theory
 {#InPerturbativeQuantumFieldTheory}

Deformation quantization in [[perturbative quantum field theory]] is discussed for [[free field theory]] via [[Moyal deformation quantization]] yielding [[Wick algebras]] in ([Dütsch-Fredenhagen 00](#DuetschFredenhagen00), [Hirschfeld-Henselder 02](#HirschfeldHenselder02)),
and for interacting [[perturbative quantum field theory]] ([[perturbative AQFT]]) via [[Fedosov deformation quantization]]
in ([Collini 16](#Collini16)), see also ([Hawkins-Rejzner 16](#HawkinsRejzner16)).

### Strict deformation quantization
 {#StrictDeformationQuantization}

(...)
[[C-star algebraic deformation quantization]]
(...)

## Properties

### Existence results

[[Vladimir Drinfel'd]] has sketched a proof (and gave main ingredients) to show that every [[Poisson Lie group]] can be deformation quantized to a [[Hopf algebra]]; this proof has been completed by Etingof and Kazhdan. [[Maxim Kontsevich]] proved a certain *[[Kontsevich formality|formality theorem]]* (formality is here in the sense of _[[formal dg-algebra]]_ in [[rational homotopy theory]]) whose main corollary (and motivation) was the statement that every [[Poisson manifold]] has a deformation quantization ([Kontsevich 97](#Kontsevich97)).

For [[symplectic manifolds]] and those [[Poisson manifolds]] that have a [[regular foliation]] by [[symplectic leafs]], the theory of deformation quantization is much simpler; [[Boris Fedosov]] gave a construction of [[star products]] on symplectic manifolds using [[symplectic connections]] on [[smooth manifolds]] ([Fedosov 94](#Fedosov94)), see at _[[Fedosov's deformation quantization]]_.  An analogous argument was given by [[Roman Bezrukavnikov]] and [[Dmitry Kaledin]] in the context of an algebraic symplectic form ([BK 04](#BK)).

+-- {: .standout}
Caution: the following are rough notes from a talk by [[J.D.S. Jones]] (Cambridge, 8.1.2013); there are probably many typos and sign errors.
=--

#### Deformation of Poisson manifolds

+-- {: .num_theorem #Kontsevich}
###### Theorem
**(Kontsevich)**.  Every [[Poisson manifold]] has a (formal) deformation quantization.

=--

This was shown in ([Kontsevich 97](#Kontsevich97)). There the deformed product is constructed by a kind of [[Feynman diagram]] [[perturbation series]]. Later this was identified as the perturbation series of the [[Poisson sigma-model]] for the given Poisson manifold. See there for more details.

#### Deformation of Algebraic Varieties ####

(Not from the notes of J.D.S. Jones)

Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this:

+-- {: .num_theorem #SwYe}
###### Theorem
**(Swan, [Yekutieli](http://dx.doi.org/10.4153/CJM-2002-051-8)).**
There is a canonical isomorphism
\[ \operatorname{Ext}^i_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong
\bigoplus_q \operatorname{H}^{i - q}(X, \bigwedge^q (\mathcal{T}_X)) . \]
=--

Here $\mathcal{T}_X$ is the tangent sheaf of $X$, and $X$ is embedded diagonally in $X^2$.

This is a consequence of the following result. Let $\mathcal{C}_{cd, X}$ be the sheaf of continuous Hochschild cochains of $X$. It is a bounded below complex of quasi-coherent $\mathcal{O}_{X^2}$-modules.

+-- {: .num_theorem #Yek1}
###### Theorem
**([Yekutieli](http://dx.doi.org/10.4153/CJM-2002-051-8)).**

1. There is a canonical isomorphism
\[ \operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong
\mathcal{C}_{cd, X} \]
in the derived category of $\mathcal{O}_{X^2}$-modules.

2. There is a canonical quasi-isomorphism of complexes of $\mathcal{O}_{X^2}$-modules
\[  \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \to \mathcal{C}_{cd, X}  . \]

3. Therefore there is a  canonical isomorphism
\[ \operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong
\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \]
in the derived category of $\mathcal{O}_{X^2}$-modules.
=--

The relation to deformation quantization is this: $\mathcal{C}_{cd, X}$ is a shift by $1$ of the sheaf of $\mathcal{D}_{poly, X}$ of polydifferential operators (viewed only as a complex of quasi-coherent $\mathcal{O}_{X^2}$-modules). Similarly,
$\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$
is the shift by $1$ of the sheaf $\mathcal{T}_{poly, X}$ of polyvector fields. Thus item 2 in the theorem above says that there is a canonical $\mathcal{O}_{X^2}$-linear quasi-isomorphism
\[ \mathcal{T}_{poly, X} \to \mathcal{D}_{poly, X} . \]
Trying to replicate the global formality theorem of Kontsevich, one would like to upgrade this to an $\mathrm{L}_{\infty}$ quasi-isomorphism.
However, it seems that in general this cannot be done directly, but only after a suitable resolution.

Here is the result. (See also [Van den Bergh](http://dx.doi.org/10.1016/j.jalgebra.2007.02.012).)
Any quasi-coherent sheaf $\mathcal{M}$ on $X$ admits a canonical flasque resolution called the mixed resolution:
\[ \mathcal{M} \to \operatorname{Mix}(\mathcal{M}) . \]
This "mixes" the jet resolution with the Cech resolution (corresponding to an affine open covering of $X$ that we suppress).
In particular there are quasi-isomorphisms of sheaves of DG Lie algebras
\[ \mathcal{T}_{poly, X} \to \operatorname{Mix}(\mathcal{T}_{poly, X}) \]
and
\[ \mathcal{D}_{poly, X} \to \operatorname{Mix}(\mathcal{D}_{poly, X}) . \]

+-- {: .num_theorem #Yek2}
###### Theorem
**([Yekutieli](http://www.math.bgu.ac.il/~amyekut/publications/def-quant/def-quant.html)).**
There is an  $\mathrm{L}_{\infty}$ quasi-isomorphism
\[ \Psi : \operatorname{Mix}(\mathcal{T}_{poly, X}) \to
\operatorname{Mix}(\mathcal{D}_{poly, X}) \]
whose $1$-st order term commutes with the HKR quasi-isomorphism above. It is independent of choices up to quasi-isomorphism.
=--

A Poisson deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of Poisson
$\mathbb{k}[[\hbar]]$-algebras on $X$, with an isomorphism
$\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X$
called an augmentation. Likewise an associative deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of associative unital (but noncommutative)
$\mathbb{k}[[\hbar]]$-algebras on $X$, with an augmentation to $\mathcal{O}_X$.

[Theorem 4](#Yek2) implies:

+-- {: .num_theorem #Yek3}
###### Theorem
**(Yekutieli).**
Assume that the cohomology groups
$\operatorname{H}^{1}(X, \mathcal{O}_X)$ and
$\operatorname{H}^{2}(X, \mathcal{O}_X)$ vanish. Then there is a canonical bijection
\[ \mathrm{quant} : \quad
\frac{ \{ \text{ Poisson deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} \quad \xrightarrow{\, \simeq \,} \quad
\frac{ \{ \text{ associative deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}}  \]
called quantization. It preserves first order brackets.
=--

When $X$ is affine this is Theorem 0.1 of
[this paper](http://www.math.bgu.ac.il/~amyekut/publications/def-quant/def-quant.html).
For the full statement see Corollary 11.2 of
[this paper](http://www.math.bgu.ac.il/~amyekut/publications/twisted-defs/twisted-defs.html).

For twisted (or stacky) deformations there is a corresponding (but much more difficult to state and prove). See the
[paper](http://www.math.bgu.ac.il/~amyekut/publications/twisted-defs/twisted-defs.html)
and the
[survey](http://www.math.bgu.ac.il/~amyekut/publications/tw-defs-surv/tw-defs-surv.html).

#### Gerstenhaber's deformation theory by Hochschild (co)homology

Let $V$ be a $k$-vector space and consider $C^p(V,V) = \Hom(V^{\otimes p}, V)$.  We define a "circle operator" $\circ$ as follows: for $f \in C^p(V,V)$ and $g \in C^q(V,V)$, we define $f \circ g \in C^{p+q-1}(V,V)$ as the map
\[ (f \circ g)(v_1, \ldots v_{p+q-1}) = f(v_1, \ldots, v_{i-1}, g(v_i, \ldots, v_{i+q-1}), v_{i+q}, \ldots, v_{p+q-1}). \]

For $f \in C^\ast(V,V)$, let $A_f(g,h) = (f \circ g) \circ h - f \circ (g \circ h)$.  (This is graded symmetric.)  It follows that the commutator of $\circ$ is given by
\[ [f,g] = f \circ g - (-1)^{(|f| - 1)(|g|-1)} g \circ f \]
where $|f| = p$ when $f \in C^p(V,V)$.  This defines a _graded [[Lie bracket]]_ of degree -1.

+-- {: .num_example}
###### Example
Let $\mu \in C^2(V,V)$ ($\mu : V \otimes V \to V$).  Note that $\mu$ is associative iff $\mu \circ \mu = 0$ iff $[\mu, \mu] = 0$.  Let $d_\mu : C^{p}(V,V) \to C^{p+1}(V,V)$ be defined by $d_\mu(x) = \mu \otimes x \pm x \otimes \mu = [\mu, x]$.  We have $d_\mu \circ d_\mu = 0$ so $(C^\ast(V,V), d_\mu)$ becomes a [[differential graded algebra]].  In fact this is the [[Hochschild cochain complex]] of the [[associative algebra]] $A = (V, \mu)$.
=--

Apply this example to the construction of deformation quantization.  The star product is uniquely determined by $\theta : A \otimes A \to A[ [ t ] ]$ given by $\theta(a,b) = ab + c(a,b)$.  What we want is that
\[ (\mu + c) \otimes (\mu + c) = 0; \]
write this out and we get the equation
\[ d_\mu c + c \circ c = 0, \]
or $d_\mu c + \frac{1}{2}[c,c] = 0$; this is the [[Maurer-Cartan equation]].  Hence we are looking for solutions of the M-C equation but in the Hochschild complex $C^\ast(A,A)[ [ t ] ]$.  One should note that $d_\mu$ is actually a [[derivation]] of the [[Lie bracket]], hence we have a [[dg-Lie algebra]].

+-- {: .num_theorem #HKR}
###### Theorem
**([[HKR theorem]])**. $HH^p(A,A) = \Gamma(M, \Lambda^p TM)$.
=--

(Note that $C^p(C^\infty(M),C^\infty(M))$ should be interpreted as $\Hom_{diff}(C^\infty(M), C^\infty(M))$.)  Under this isomorphism the Poisson bracket is mapped to the [[Poisson tensor]]:
\[ \{ \cdot , \cdot \} \in HH^2(A,A) \quad \mapsto \quad P \in \Gamma(M, \Lambda^2 TM). \]
The bracket in Hochschild cohomology ([[Gerstenhaber bracket]]) goes to the [[Schouten bracket]]:
\[ [ \cdot , \cdot ]_G \quad \mapsto \quad [ \cdot, \cdot ]_S. \]

For vector fields $\xi$ and $\eta$, the [[Schouten bracket]] satisfies (1) $[\xi,\eta]_S = [\xi,\eta]$ (the Lie bracket), and (2) $[\alpha, \beta \wedge \gamma] = [\alpha,\beta] \wedge \gamma \pm [\alpha,\gamma] \wedge \beta$; note that this completely determines it (everything is locally given by wedges...).

In the Hochschild cohomology $HH^\ast(A,A)$ of $A$, $d_\mu P \mapsto 0$ and $[P,P]_S = 0$, so _we have a solution to M-C in $H^\ast(A,A)[ [ t ] ]$_.

#### In terms of differential graded Lie algebras

+-- {: .num_defn}
###### Definition
Let $L_1$ and $L_2$ be [[differential graded Lie algebras]] (dgL).  A **[[quasi-isomorphism]]** $f : L_1 \to L_2$ is a homomorphism of dgLs that induces an isomorphism on homology.  $L_1$ and $L_2$ are **quasi-isomorphic** if there exists $M$ with quasi-isomorphisms $L_1 \leftarrow M \rightarrow L_2$.  It can be verified that this is an equivalence relation.
=--

+-- {: .num_theorem #KontsevichDeformationTheorem}
###### Theorem
**([[Kontsevich]])**.  If $L_1$ is quasi-isomorphic to $L_2$ then there is a solution to the M-C equation in $L_1$ iff there is a solution to the M-C equation in $L_2$.
=--

+-- {: .num_theorem #KontsevichFormality}
###### Theorem
**([[Kontsevich formality]])**.  $C^\ast(A,A)[ [ t ] ]$ is quasi-isomorphic to $H^\ast(A,A)[ [ t ] ]$.  ($A = C^\infty(M)$)
=--

Hence there is a solution to M-C in $C^\ast(A,A)[ [ t ] ]$, and hence there is a deformation quantization (!).

#### The Deligne conjecture

We have $(C^*(A,A), d_\mu)$, the [[Gerstenhaber bracket]], and we also have a [[cup product]]
\[ (f \cup g) (a_1, \ldots, a_{p+q}) = \mu(f(a_1,\ldots,a_p), g(a_{p+1},\ldots,a_{p+q})) \]
for $f : A^{\otimes p} \to A$, $g : A^{\otimes q} \to A$; this satisfies also $d_\mu(f \cup g) = (d_\mu f) \cup g \pm f \cup d_\mu g$.
The [[Deligne conjecture]] gives a relationship between these things.

In $HH^*(A,A)$, we have:

1. $[\cdot, \cdot]$ is a graded Lie bracket of degree -1.
1. The cup product $\cup$ is graded commutative.
1. The [[Jacobi identity]] for $[\cdot,\cdot]$.
1. $[a, b \cup c] = [a,b] \cup c \pm [a,c] \cup b$.

Such a thing is called a [[Gerstenhaber algebra]].  Note that we do not have these relations in $C^*(A,A)$, they are only true modulo boundaries.

+-- {: .num_theorem #Deligne}
###### Theorem
**([[Deligne conjecture]])**.  $C^*(A,A)$ is a $G_\infty$-algebra, which is a Gerstenhaber algebra _up to coherent homotopy_.
=--

#### Deformation by universal enveloping algebras
 {#RelationToUniversalEnvelopingAlgebras}

It is a classical fact [[Gutt 1983, §4 (4.2)](#Gutt83), cf. [Gutt 2011, §2.2](#Gutt11)] that
the [[universal enveloping algebra]] of a [[Lie algebra]] provides a deformation quantization of the corresponding [[Lie-Poisson structure]] (example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below).
Remarkably, this statement generalizes to more general [[polynomial Poisson algebras]] (def. \ref{PolynomialPoissonAlgebra} below)
for a suitable generalized concept of universal enveloping algebra (def. \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below):
it is _always_ true up to third order in $\hbar$, and sometimes to higher order ([Penkava-Vanhaecke 00, theorem 3.2](#PenkavaVanhaecke00), prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3} below).
In particular it also holds true for restrictions of [[Poisson bracket Lie algebras]] to their [[Heisenberg Lie algebras]]
(example \ref{DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras} below).

+-- {: .num_defn #PolynomialPoissonAlgebra}
###### Definition
**([[polynomial]] [[Poisson algebra]])**

A [[Poisson algebra]]  $((A,\cdot), \{-,-\})$
is called a _[[polynomial Poisson algebra]]_ if the underlying [[commutative algebra]] $(A,\cdot)$
is a [[polynomial algebra]], hence a [[symmetric algebra]]

$$
  Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V)
$$

on some [[vector space]] $V$. Here

$$
  T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n}
$$

denotes the [[tensor algebra]] of $V$. We write

$$
  \mu \;\colon\; T(V) \longrightarrow Sym(V)
$$

for the canonical [[projection]] map (which is an algebra [[homomorphism]]) and

$$
  \sigma \;\colon\; Sym(V) \longrightarrow T(V)
$$

for its [[linear map|linear]] inverse (symmetrization, which is not in general an algebra [[homomorphism]]).

Notice that by its bi-[[derivation]] property the Poisson bracket
on a polynomial Poisson algebra is fixed by its restriction to linear elements

$$
  \{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V)
  \,.
$$

=--

+-- {: .num_example #PolynomialLiePoissonStructure}
###### Example
**(polynomial [[Lie-Poisson structure]])**

Let $(C^\infty(\mathbb{R}^n), \pi)$ be a [[Poisson manifold]] whose
underlying manifold is a [[Cartesian space]] $\mathbb{R}^n$. Then the restriction
of its Poisson algebra $( C^\infty(\mathbb{R}^n, \cdot), \pi^{i j} \partial_i(-) \cdot \partial_j(-)  )$
to the [[polynomial functions]] $\mathbb{R}[x^1, \cdots, x^n ] \ookrightarrow C^\infty(\mathbb{R}^n)$
is a polynomial Poisson algebra according to def. \ref{PolynomialPoissonAlgebra}.

In particular if $(\mathfrak{g}, [-,-])$ is a [[Lie algebra]] and
$(\mathfrak{g}^\ast, \{-,-\})$ the corresponding [[Lie-Poisson structure|Lie-Poisson manifold]],
then the corresponding polynomial Poisson algebra is $(Sym(\mathfrak{g}), \{-,-\})$
where the restriction of the Poisson bracket to linear polynomial elements coincides with the [[Lie bracket]]:

$$
  \{x,y\} = [x,y]
  \,.
$$

=--

+-- {: .num_defn #UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
###### Definition
**(universal enveloping algebra of polynomial Poisson algebra)**

Given a polynomial Poisson algebra $(Sym(V), \{-,-\})$ (def. \ref{PolynomialPoissonAlgebra}),
say that its _universal enveloping algebra_ $\mathcal{U}(V,\{-,-\})$ is the [[associative algebra]] which is the [[quotient]] of
the [[tensor algebra]] of $V$ with a [[power series|formal variable]] $\hbar$ adjoined
by the two-sided ideal which is generated by the the $\hbar$-Poisson bracket relation on linear elements:

$$
  \mathcal{U}(V,\{-,-\})
  \;\coloneqq\;
  T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V )
  \,.
$$

This comes with the quotient projection linear map which we denote by

$$
  \rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\})
  \,.
$$

=--

([Penkava-Vanhaecke 00, def. 3.1](#PenkavaVanhaecke00))

+-- {: .num_example #UniversalEnvelopingAlgebraOfLieAlgebra}
###### Example
**([[universal enveloping algebra]] of [[Lie algebra]])**

In the case of a polynomial [[Lie-Poisson structure]] $(Sym(\mathfrak{g}), [-,-])$ (example \ref{PolynomialLiePoissonStructure})
the universal enveloping algebra $\mathcal{U}(\mathfrak{g},[-,-])$ from def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
(for $\hbar = 1$) coincides with the standard [[universal enveloping algebra]] of the [[Lie algebra]] $(\mathfrak{g}, [-,-])$.

=--

The combined linear projection maps from def. \ref{PolynomialPoissonAlgebra} and def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra}
we denote by

$$
  \tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\})
  \,.
$$

+-- {: .num_prop #UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}
###### Proposition
**(universal enveloping algebra provides [[formal deformation quantization]] at least up to order 3)**

Let $( Sym(V), \{-,-\} )$ be a polynomial Poisson algebra (def. \ref{PolynomialPoissonAlgebra})
such that the canonical linear map to its universal enveloping algebra (def. \ref{UniversalEnvelopingAlgebraOfPolynomialPoissonAlgebra})
is [[injective function|injective]] up to order $n \in \mathbb{N}\cup \{\infty\}$

$$
  \tau/(\hbar^{n+1}) \;\colon\; Sym(V)[ [ \hbar ] ]/(\hbar^{n+1}) \hookrightarrow \mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})
  \,.
$$

Then the restriction of the product on $\mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})$ to $Sym(V)/(\hbar^{n+1})$
is a [[formal deformation quantization]] of $(Sym(V), \{-,-\})$ to order $n$ (hence a genuine deformation quantization in the case that $n = \infty$).

Moreover, this is _always_ the case for $n = 3$, hence for every polynomial Poisson algebra its universal enveloping algebra
always provides a deformation quantization of order $3$ in $\hbar$.

=--

([Penkava-Vanhaecke 00, theorem 3.2 with section 2](#PenkavaVanhaecke00))

+-- {: .num_example #DeformationQuantizationOfLiePoissonStructuresByUniversalEnvelopingAlgebras}
###### Example
**([[formal deformation quantization]] of [[Lie-Poisson structures]] by universal enveloping algebras)**

In the following cases the map $\tau$ in prop. \ref{UniversalEnvelopingAlgebraProvidesDeformationQuantizationAtLeastToOrder3}
is injective to arbitrary order, hence in these cases the [[universal enveloping algebra]] provides a genuine [[formal deformation quantization]]:

1. The case that the Poisson bracket is [[linear Poisson structure|linear]] in that it restricts as

   $$
     \{-,-\}
       \;\colon\;
     V \otimes V
       \longrightarrow
     V
       \hookrightarrow
     Sym(V)
     \,.
   $$

   This is the case of the [[Lie-Poisson structure]] from example \ref{PolynomialLiePoissonStructure} and the universal enveloping algebra that provides its deformation quantization is the standard one (example \ref{UniversalEnvelopingAlgebraOfLieAlgebra}).

1. More generally, the case that the Poisson bracket restricted to linear elements has linear and
   constant contribution in that it restricts as

   $$
     \{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V)
     \,.
   $$

   This includes notably the Poisson structures induced by [[symplectic vector spaces]], in which case the restriction

   $$
     \{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V)
   $$

   is the [[Lie bracket]] of the associated [[Heisenberg Lie algebra]].

=--

This is [Penkava & Vanhaecke 2000, p. 26](#PenkavaVanhaecke00).
The first statement in itself is a classical fact (reviewed e.g. in [Gutt 2011](#Gutt11)).

### Motivic Galois group action on the space of quantizations
 {#MotivicGaloisGroup}

In ([Kontsevich 99](Kontsevich99)) it was indicated that a [[quotient group]] of the [[motivic Galois group]] apparently equivalent to the [[Grothendieck-Teichmüller group]] naturally [[action|acts]] on the space of formal deformation quantizations of a finite dimensional manifold. See also at _[[cosmic Galois group]]_.

This has been formalized as follows. The formal deformation quantization of ([Kontsevich 97](#Kontsevich97)) is all induced by the [[Kontsevich formality]] theorem, which states that over suitable [[manifolds]]/[[varieties]] $X$ there is an [[equivalence]] of [[L-∞ algebras]]

$$
  \mathcal{X}(X) \stackrel{\simeq}{\to} C^\bullet(X)
$$

identifying the [[multivector fields]] on $X$ with the [[Hochschild cohomology]] complex (of its [[function algebra]]).  Every choice of such an equivalence induces one formal deformation quantization of a [[Poisson manifold]] $X$, and the two quantizations induced by two equivalent (homotopic) equivalences are in turn equivalent.

Therefore one may regard the [[∞-groupoid]] $Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ as the "space of formal deformation quantizations" of $X$.

In ([Dolgushev 1109, theorem 6.2](#Dolgushev1109), [Dolgushev 1111, theorem 3.1](#Dolgushev1111)) it is shown that the set $\pi_0 Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ of connected components of this space is, up to a choice of basepoint, the [[Grothendieck-Teichmüller group]], hence is a [[torsor]] over that group.

(This is based on identifications of the GRT Lie algebra with the degree-0 [[chain cohomology]] of the [[graph complex]], due to [[Thomas Willwacher]]. See at _[Grothendieck-Teichmüller group -- relation to the graph complex](Grothendieck-Teichm%C3%BCller+tower#RelationToTheGraphComplex)_.

Aspects of the generalization of this statement to more general spaces then $\mathbb{R}^n$ are discussed in [Dolgushev-Rogers-Willwacher 12](#DolgushevRogersWillwacher12).

For discussion of [[motive|motivic structures]] in [[geometric quantization]] see at _[[motivic quantization]]_.

## Examples

* [[Moyal deformation quantization]]

* [[Fedosov deformation quantization]]

## Related concepts

* [[reductions deformations resolutions in physics]]

* [[quantization]]

  * **deformation quantization**, [[geometric quantization]]

  * [[path integral]]

  * [[semiclassical approximation]]

* [[Jordan algebra]]

[[!include Isbell duality - table]]

[[!include infinitesimal and local - table]]

## References

The concept of formal deformation quantization originates with:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [<a href="https://doi.org/10.1016/0003-4916(78)90224-5">doi:10.1016/0003-4916(78)90224-5</a>]

* {#BayenFlatoFronsdalLichnerowiczSternheimer78b} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. II. Physical applications_, Annals of Physics **111** 1 (1978) 111-151 [<a href="https://doi.org/10.1016/0003-4916(78)90225-7">doi:10.1016/0003-4916(78)90225-7</a>]

Review:

* [[Alan Weinstein]], *Deformation quantization*, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [[numdam:SB_1993-1994__36__389_0](http://www.numdam.org/item/?id=SB_1993-1994__36__389_0)]

* [[Daniel Sternheimer]], _Deformation Quantization: Twenty Years After_, AIP Conf. Proc. **453** (1998) 107-145 [[arXiv:math/9809056](https://arxiv.org/abs/math/9809056)]

See also:

* {#Gutt83} [[Simone Gutt]], *An explicit $\ast$-product on the cotangent bundle of a Lie group*, Letters in Mathematical Physics **7** (1983) 249–258 [[doi:10.1007/BF00400441](https://doi.org/10.1007/BF00400441)]

* {#Gutt11} [[Simone Gutt]], _Deformation quantization of Poisson manifolds_, Geometry and Topology Monographs **17** (2011) 171-220 [[gtm:17](https://msp.org/gtm/2011/17/p003.xhtml), [pdf](http://msp.org/gtm/2011/17/gtm-2011-17-003p.pdf), [arXiv:math/0003107](https://arxiv.org/abs/math/0003107)]

Monographs:

* {#Waldmann07} [[Stefan Waldmann]], *Poisson-Geometrie und Deformationsquantisierung*, Springer (2007) [[doi:10.1007/978-3-540-72518-3](https://doi.org/10.1007/978-3-540-72518-3)]

The [[Fedosov deformation quantization]] prescription for deformation quantization of [[symplectic manifolds]] and varieties and also of [[Poisson manifolds]] that have a [[regular foliation]] by [[symplectic leaves]] is discussed in

* [[Boris Fedosov]], _Formal quantization_, Some Topics of Modern Mathematics and their Applications to Problems of Mathematical Physics (in Russian), Moscow (1985), 129-136.

* [[Boris Fedosov]], _Index theorem in the algebra of quantum observables_, Sov. Phys. Dokl. 34 (1989), 318-321.

* {#Fedosov94} [[Boris Fedosov]], _A simple geometrical construction of deformation quantization_  J. Differential Geom. Volume 40, Number 2 (1994), 213-238. ([EUCLID](http://projecteuclid.org/euclid.jdg/1214455536))

For algebraic forms this is discussed in

* {#BK09} [[Roman Bezrukavnikov]], [[Dmitry Kaledin]], _Fedosov quantization in algebraic context_ Mosc. Math. J., Volume 4, Number 3, (2004) 559–592 ([AMS pdf](http://www.ams.org/distribution/mmj/vol4-3-2004/bezrukavnikov-kaledin.pdf))

More discussion of this approach is in

* Claudio Emmrich, [[Alan Weinstein]], _The differential geometry of Fedosov's quantization_ ([arXiv:hep-th/9311094](http://arxiv.org/abs/hep-th/9311094))

A direct and general formula for the deformation quantization of any Poisson manifold was given in

* {#Kontsevich97} [[Maxim Kontsevich]], _Deformation quantization of Poisson manifolds_,  Lett. Math. Phys. __66__ (2003),  no. 3, 157--216, ([arXiv:q-alg/9709040](http://arxiv.org/abs/q-alg/9709040)).

see also

* Peter Banks, Erik Panzer, Brent Pym, _Multiple zeta values in deformation quantization_ ([arXiv:1812.11649](https://arxiv.org/abs/1812.11649))

This secretly uses the [[Poisson sigma-model]] (see there for more details) induced by the given target [[Poisson Lie algebroid]].

A popular exposition of formal deformation quantization with emphasis of the [[string theory|stringy]] [[Poisson sigma-model]]-construction:

* [[Anton Kapustin]], _Quantum geometry: The reunion of math and physics_ [[[Kapustin-QuantumGeometry.pdf:file]]]

The classification of the space of such formal deformation quantization is discussed in

* {#Dolgushev1109} [[Vasily Dolgushev]], _Stable Formality Quasi-isomorphisms for Hochschild Cochains I_ ([arXiv:1109.6031](http://arxiv.org/abs/1109.6031))

* {#Dolgushev1111} [[Vasily Dolgushev]], _Exhausting formal quantization procedures_ ([arXiv:1111.2797](http://arxiv.org/abs/1111.2797))

* {#DolgushevRogersWillwacher12} [[Vasily Dolgushev]], [[Christopher Rogers]], [[Thomas Willwacher]], _Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields_ ([arXiv:1211.4230](http://arxiv.org/abs/1211.4230))

Deformation quantization of [[polynomial Poisson algebras]] via [[universal enveloping algebra]] (generalizing that of [[Lie-Poisson structures]]) is discussed in

* {#PenkavaVanhaecke00} [[Michael Penkava]], [[Pol Vanhaecke]], *Deformation Quantization of Polynomial Poisson Algebras*, Journal of Algebra **227** 1 (2000) 365-393  [[arXiv:math/9804022](https://arxiv.org/abs/math/9804022), [doi:10.1006/jabr.1999.8239](https://doi.org/10.1006/jabr.1999.8239)]

Deformation quantization of algebraic varieties is in

* [[Amnon Yekutieli]], Deformation Quantization in Algebraic Geometry, Advances in Mathematics **198** (2005) 383-432.
Erratum: Advances in Mathematics  **217** (2008) 2897-2906.

See also

* MO discussion , _[Kontsevich's conjectures on the Grothendieck-Teichmüller group?](http://mathoverflow.net/questions/42148/kontsevichs-conjectures-on-the-grothendieck-teichmuller-group)_

Deformation quantization in [[perturbative quantum field theory]] is discussed for [[free field theory]] is due to

* {#Dito90} Joseph Dito, _Star-product approach to quantum field theory: The free scalar field_. Letters in Mathematical Physics, 20(2):125–134, 1990 ([spire](https://inspirehep.net/record/303898/))

and further expanded on in

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.1 of _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))

* {#HirschfeldHenselder02} A. C. Hirshfeld, P. Henselder, _Star Products and Perturbative Quantum Field Theory_, Annals Phys. 298 (2002) 382-393 ([arXiv:hep-th/0208194](https://arxiv.org/abs/hep-th/0208194))

and for interacting [[perturbative quantum field theory]] ([[perturbative AQFT]]) in

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

The relation to [[geometric quantization]] is discussed in

* [[Eli Hawkins]], _The Correspondence between Geometric Quantization and Formal Deformation Quantization_ ([arXiv:math/9811049](http://arxiv.org/abs/math/9811049))

* Christoph Nölle, _Geometric and deformation quantization_ ([arXiv:0903.5336](http://arxiv.org/abs/0903.5336))

and some remarks on the relation are also in section 1.4 of

* {#GukovWitten09} [[Sergei Gukov]], [[Edward Witten]], _Branes and Quantization_, Adv. Theor. Math. Phys. 13 (2009) 1–73, ([arXiv:0809.0305](http://arxiv.org/abs/0809.0305), [euclid](http://projecteuclid.org/euclid.atmp/1282054099))

which is about [[quantization via the A-model]].

The formulation of deformation quantization as lifts from [[P-n operads]] over [[BD-n operads]] to [[E-n operads]] is discussed in section 2.3 and 2.4 of

* {#CostelloGwilliam} [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory_ ([wiki](http://math.northwestern.edu/~costello/factorization_public.html), early/partial draft [pdf](http://math.northwestern.edu/~gwilliam/factorization.pdf))

See also

* [Wikipedia](http://en.wikipedia.org/wiki/Weyl_quantization#Deformation_quantization)

* F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, _Quantum mechanics as a deformation of classical mechanics_, Lett. Math. Phys. 1 (1975/77), [MR674337](http://www.ams.org/mathscinet-getitem?mr=674337), [doi](http://dx.doi.org/10.1007/BF00399745);  _Deformation theory and quantization. I. Deformations of symplectic structures_, Ann. Physics 111 (1978), no. 1, 61–110, [MR496157](http://www.ams.org/mathscinet-getitem?mr=496157); _Deformation theory and quantization. II. Physical applications_, Ann. Physics 111 (1978), no. 1, 111–151, [MR496158](http://www.ams.org/mathscinet-getitem?mr=496158)
* M. Flato, A. Lichnerowicz, D. Sternheimer, _Deformations of Poisson brackets, Dirac brackets and applications_, J. Math. Phys. __17__ (1976), no. 9, 1754–1762, [MR420723](http://www.ams.org/mathscinet-getitem?mr=420723), [doi](http://dx.doi.org/10.1063/1.523104)
* D. Arnal, J.-C. Cortet, _$\ast$-products in the method
of orbits for nilpotent groups_, J. Geom. Phys. 2 (1985), no. 2,
83–116, <a href="http://dx.doi.org/10.1016/0393-0440(85)90010-5">doi</a>

On the [[stack]] of deformation quantizations:

* [[Maxim Kontsevich]], _Deformation quantization of algebraic varieties_ ([arXiv:math/0106006](http://arxiv.org/abs/math/0106006))

* Pietro Polesello, [[Pierre Schapira]], _Stacks of quantization-deformation modules on complex symplectic manifolds_ ([arXiv:math/0305171](http://arxiv.org/abs/math/0305171))

* [[Amnon Yekutieli]], _Twisted Deformation Quantization of Algebraic Varieties_ ,
[ arxiv:0905.0488 ](http://arxiv.org/abs/0905.0488)

The action of a [[motivic Galois group]] ("[[cosmic Galois group]]") on the space of deformation quantization was observed in

* {#Kontsevich99} [[Maxim Kontsevich]], _Operads and Motives in Deformation Quantization_, Lett. Math. Phys.48:35-72,1999 ([arXiv:math/9904055](http://arxiv.org/abs/math/9904055))

See also at _[[motives in physics]]_.

[[!redirects algebraic deformation quantization]]

[[!redirects formal deformation quantization]]
[[!redirects formal deformation quantizations]]

[[!redirects formal algebraic deformation quantization]]
[[!redirects formal algebraic deformation quantizations]]


formal dg-algebra

>  This entry is about [[quasi-isomorphism]] to [[cochain cohomology]]. For [[infinitesimal neighbourhood|infinitesimal thickening]] of [[smooth manifolds]] see at _[[formal smooth manifold]]_.

***

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

# Contents
* table of contents
{:toc}

## Definition

A [[dg-algebra]] $A$ is a **formal dg-algebra** if in the [[homotopy category]] of the [[model structure on dg-algebras]] it is [[isomorphism|isomorphic]]

$$
  A \stackrel{\simeq}{\to} H^\bullet(A)
$$

to its [[chain homology and cohomology|chain (co)homology]] (regarded as a dg-algebra with trivial differential). Since all dg-algebras are fibrant in the standard model, this is equivalent to the existence of a [[span]]

$$
  A \stackrel{\simeq_w}{\leftarrow} Q A \stackrel{\simeq}{\to}
  H^\bullet(A)
$$

of [[quasi-isomorphism]]s of dg-algebras.

## Examples

In [[rational homotopy theory]] [[rational topological space]]s are encoded in their dg-algebras of Sullivan forms. A simply connected topological space $X$ whose dg-algebra of Sullivan forms $\Omega^\bullet(X)$ is formal is called a **formal topological space**.  (One can also say a __formal rational space__, to distinguish from the unrelated formal spaces in [[formal topology]].)  Such a space represents a **formal homotopy type**.

Examples are

* compact [[Kähler manifolds]] (e.g. smooth projective varieties);

* [[Lie groups]];

* [[classifying spaces]] $B G$ of [[Lie groups]] $G$;

* some [[homogeneous spaces]] $G/H$

* the unstable [[Thom spaces]] $M U_n$ and $M S O_n$

* the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$,

  whose Sullivan minimal model is the dg-algebra on a single degree-2 generator with trivial differential.

* [[the little n-disk operad is formal]]

## Related concepts

* [[H-cohomology]]

## References

For an early discussion of formal dg-algebras in the context of [[rational homotopy theory]] see section 12 of

* [[Dennis Sullivan]], _Infinitesimal computations in topology_ , Publications Mathématiques de l'IHÉS, 47 (1977), p. 269-331 (numdam: [djvu](http://archive.numdam.org/article/PMIHES_1977__47__269_0.djvu), [pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1977__47_/PMIHES_1977__47__269_0/PMIHES_1977__47__269_0.pdf))

* [[Pierre Deligne]], Phillip Griffiths, John Morgan, [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent. Math. 29 (1975), no. 3, 245--274, [doi](http://dx.doi.org/10.1007/BF01389853)
[[!redirects formal differential graded algebra]]
[[!redirects formal homotopy type]]

A survey is around definition 2.1 of

* [[Kathryn Hess]], _Rational homotopy theory: a brief introduction_ ([arXiv](http://arxiv.org/abs/math.AT/0604626))

Review with an eye towards the [[dd^c-lemma]] and [[H-cohomology]] is in

* {#Cavalcanti03} [[Gil Cavalcanti]], p. 12 of _New aspects of the $d d^c$-lemma_ ([arXiv:math/0501406](https://arxiv.org/abs/math/0501406))

The case of [[Sullivan algebras]] of [[sphere bundles]]:

* Jiawei Zhou, *Formality of Sphere Bundles* [[arXiv:2304.09594](https://arxiv.org/abs/2304.09594)]

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

[[!redirects formal rational space]]
[[!redirects formal rational spaces]]

[[!redirects formal homotopy type]]
[[!redirects formal homotopy types]]
[[!redirects formal rational homotopy type]]
[[!redirects formal rational homotopy types]]


formal disk

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a context of [[synthetic differential geometry]] or [[differential cohesion]], then given any [[space]] $X$ and a [[point]] $x \colon \ast \to X$, then the _formal disk_ in $X$ at $x$ is just the [[infinitesimally thickened point]] which is the [[formal neighbourhood]] of that point. If there is a concept of [[dimension]] for $X$, then the dimension of that formal disk is the dimension of $X$ at that point.

[[Isbell duality|Dually]] in terms of [[function algebras]], in the typical models formal disks are the [[formal spectra]] of [[formal power series]] algebras. The formal spectra of the [[Laurent series]] algebras are instead the "punctured formal disks", hence formal disks with their unique global point removed.

More concretely, if $S$ is any [[infinity-cohesive site]] of "differentiable spaces" (such as for instance [[complex analytic manifolds]]) and $S_{synthdiff}$ is the corresponding site of infinitesimal thickenings (for instance complex analytic [[smooth loci]] of the form the product of a complex manifold with an [[infinitesimally thickened point]]), then we have relative [[cohesion]] of $Sh_\infty(S_{synthdiff})$ over $Sh_\infty(InfPoints)$ (see at _[[synthetic differential ∞-groupoid]]_) and for any space $X$ then the [[flat modality]] takes it to its union $\flat X$ of all formal disks around all [[global points]] of $X$.

## Properties

### Function field analogy

## Related concepts

* [[affine line]]

* [[infinitesimally thickened point]]

* [[infinitesimal disk bundle]]

[[!include function field analogy -- table]]

## References

Discussion in [[differential cohesion]] is in

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

and formalization in [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, PhD Thesis, 2017

* {#Wellen18} [[Felix Wellen]], _[[schreiber:Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

[[!redirects formal disks]]

[[!redirects formal disc]]
[[!redirects formal discs]]


formal geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Several notions of formal geometry

Formal geometry is a highly overloaded term in mathematics, used in number of conceptually similar ways, usually meaning that we work in setup in which some crucial details of geometry or analysis are not present or satisfied, e.g.

* we work with functions on "manifolds" but the functions do not necessarily converge, the geometry is rather based on topological algebras of [[formal power series]]; this is the formal geometry of Grothendieck school and the main notion is that of a [[formal scheme]] (or more general ind-schemes). There are also noncommutative versions like [[Kapranov's noncommutative geometry]].

* Gelfand's formal geometry: study infinite dimensional manifolds of [[jet bundles]] and related objects coming from usual differential geometry, geometry of formal differential operators, study of related objects from homological algebra, including
Gelfand's [[formal manifold]] (homological vector field)

* we talk about neighborhoods,or localizations, morphisms of spaces, but not about spectra and points (a part of noncommutative geometry is done in such style) -- this is sometimes called "pseudogeometry"

* in [[algebraic geometry]]: [[formal spectrum]] of an [[adic noetherian ring]]

### Gelfand's formal geometry

See MathOverflow: [formal-geometry](http://mathoverflow.net/questions/6057/formal-geometry), [how-do-i-describe-the-gl-n-torsor-attached-to-a-smooth-morphism-of-relative-dimen](http://mathoverflow.net/questions/27531/how-do-i-describe-the-gl-n-torsor-attached-to-a-smooth-morphism-of-relative-dimen), []()

* [[Israel Gelfand]], [[David Kazhdan]], _Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields_, Soviet Math. Doklady __12__, 1367-1370 (1971)

For characteristic $p\gt 0$ case see

* Fedosov quantization in positive characteristics [arXiv](http://front.math.ucdavis.edu/0501.5247)

In a similar formal context Gelfand and collaborators introduced $(\mathfrak{A},\mathcal{D})$-systems

* I. M. Gelʹfand, Yu. L. Daletskiĭ, _Lie superalgebras and Hamiltonian operators_, Rep. No. 16 Sem. Supermanifolds, Dept. Math. Univ. Stockholm, 1987, 26 p.
* I. M. Gelʹfand, Yu. L. Daletskiĭ, B. L. Tsygan, _On a variant of noncommutative differential geometry_, Dokl. Akad. Nauk SSSR 308 (1989), no. 6, 1293--1297; translation in Soviet Math. Dokl. __40__ (1990), no. 2, 422–426 [MR91j:58015](http://www.ams.org/mathscinet-getitem?mr=1039918)

The $(\mathfrak{A},\mathcal{D})$-systems were partly motivated by the [[calculus of variations]], formalizing further the setting of works of Gelfand and Dorfman.

* И. М. Гельфанд, И. Я. Дорфман, _Гамильтоновы операторы и бесконечномерные алгебры Ли_, Функц. анализ и его прил., 15:3 (1981), 23–40, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1729&what=fullt&option_lang=rus); _Гамильтоновы операторы и классическое уравнение Янга–Бакстера_, Функц. анализ и его прил. __16__:4 (1982), 1–9 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1663&what=fullt&option_lang=rus); _Скобка Схоутена и гамильтоновы операторы_, Функц. анализ и его прил., 14:3 (1980), 71–74 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1836&what=fullt&option_lang=rus)
* Ю. Л. Далецкий, _Гамильтоновы операторы в градуированном формальном вариационном исчислении_, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1277&what=fullt&option_lang=rus), transl. Yu. L. Daletskii, _Hamiltonian operators in graded formal calculus of variations_, Funct. Anal. Appl. __20__:2, 1986, 136-138

See also

* Bernstein, Rozenfeld, Homogeneous spaces of infinite-dimensional lie algebras and characteristic classes of foliations, Uspehi Mat. Nauk, [English pdf](http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/B-Rosen-Foliations-Usp.pdf)

### Formal noncommutative symplectic geometry of Kontsevich

* [[Maxim Kontsevich]], _Formal (non)-commutative symplectic geometry_, The Gelfand Mathematical. Seminars, 1990-1992, Ed. L.Corwin, I.Gelfand, J.Lepowsky, Birkhauser 1993, 173-187, [pdf](http://www.ihes.fr/~maxim/TEXTS/Formal%20non-commutative%20symplectic%20geometry.pdf)

## Related concepts

* [[formal dg-algebra]]

* [[formal neighbourhood]], [[formal disk]]

* [[Hasse local-global principle]]

* [[formal Picard group]], [[formal Brauer group]]

* [[Kapranov's noncommutative geometry]]

[[!include infinitesimal and local - table]]


formal geometry -- contents

**[[formal geometry]]**

[[synthetic differential geometry]], [[deformation theory]]

[[local-global principle]]

[[infinitesimal space]],

* [[infinitesimal ring extension]]

[[infinitesimally thickened point]]

* [[Artin algebra]]

[[formal neighbourhood]], [[formal spectrum]]

* [[completion of a ring]]

* [[adic topology]]

* [[p-adic integers]]

[[formal group]]

[[formal deformation quantization]]


formal grammar

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

# Contents
* table of contents
{:toc}

## Idea

Formal grammars are sets of rules which describe which strings of words are considered grammatical. They can be used to characterize the syntax of both [[linguistics|natural language]] and [[programming language|programming languages]].

## Unrestricted grammar {#unrestrictedgrammar}

Fix a finite set $V$ called the vocabulary, also called the _terminal symbols_. A _language_ is a subset of the free [[monoid]] $V^*$, i.e. strings of words. Let $V^+$ denote the free [[semigroup]], i.e. non-empty strings. $X + Y$ and $X \times Y$ denote the disjoint union and the Cartesian product respectively.

An _unrestricted grammar_ is given by a tuple $G = (V, X, R, s)$ where $X$ is a finite set of _nonterminal symbols_ with a _start symbol_ $s \in X$, and $R \subseteq (V + X)^+ \times (V + X)^*$ is a finite set of _rewrite rules_. The language $L(G)$ generated by the grammar is usually defined as follows:

1. Define the "rewrite step" relation $(\to_R) \subseteq (V + X)^* \times (V + X)^*$ where $(\alpha \beta \gamma) \to_R (\alpha \delta \gamma)$ for all $(\beta, \delta) \in R$ and $\alpha, \gamma \in (V + X)^*$.
2. Take its reflexive transitive closure $(\to_R^*)$, i.e. $s \to_R^* t$ iff there is some sequence of rewrite steps $s \to_R s_1 \to_R s_2 \to_R ... \to_R t$. This sequence is called a _grammatical derivation_.
3. Let $L(G) = \{ w_1 ... w_n \in V^* | s \to_R^* w_1 ... w_n\}$ for $s \in X$ the start symbol.

The construction of $\to_R^*$ can be seen as the free [[preorder|preordered]] [[monoid]] generated by the rewrite rules $R$. Grammatical derivations can be seen as [[string diagram|string diagrams]] in the free [[monoidal category]] with the symbols $V + X$ as generating objects and the rewrite rules $R$ as generating arrows.

## Other frameworks

[[context free grammar|Context free grammars]] are defined in the same way as unrestricted grammars, but they only allow rules with a single non-terminal symbol as domain. In that particular case, the grammatical derivations are [[tree|trees]] with the start symbol as root and the words as leaves.

In [[categorial grammar|categorial grammars]], monoidal categories are replaced by [[biclosed monoidal category]]. The derivations are trees with two kinds of rules: 1) a finite language-dependent _dictionary_ of rules $w \to t$ with $w \in V$ and $t \in F(X)$, for $F(X)$ the free residuated monoid (see [Lambek (1958)](#Lambek58)) and 2) an infinite number of language-agnostic rules, for the application of each [[higher order logic|higher order type]].

Similarly, [[pregroup grammar|pregroup grammars]] are constructed from a dictionary with types coming from a free pregroup $P(X)$. The only rewrite rules are the counit maps $x^* \otimes x \to 1$ and $x \otimes {}^*x \to 1$, canceling a non-terminal symbol $x \in P(X)$ with its left and right adjoints $x^*$ and ${}^*x$. The derivations are planar [[string diagrams]] in a [[rigid monoidal category]], the counit maps are drawn as cup-shaped wires connecting the words in a sentence.

The grammatical derivations of [[dependency grammar|dependency grammars]] can also be encoded as diagrams in a rigid monoidal category.

## History

The origins of grammar go back to 4th century BC India: the Sanskrit philologist Pānini showed ancient mastery of [[context-sensitive grammar|context-sensitive grammars]] to analyse the syntax of Sanskrit.

Unrestricted grammars were introduced by [[Noam Chomsky]] as the level zero of his [[Chomsky hierarchy|hierarchy]], they are equivalent to [[Turing machine|Turing machines]] in expressive power.

The similar concept of _semi-Thue system_, also known as a string [[rewriting]] system, was introduced thirty years before Chomsky's work on grammar, see [Thue (1914)](#Thue14). It makes no distinction between terminal and non-terminal symbols, and allows rules with empty left-hand sides.

A third, equivalent, formulation was proved to be [[decidability|undecidable]] independently by [[Emil Post]] and [[Andrey Markov]]: the word problem for [[semigroup|semigroups]].

## References

* Saroja Bhate and Subhash Kak. Pānini’s Grammar and Computer Science. Annals of the Bhandarkar Oriental Research Institute (1993)
* {#Thue14} Axel Thue. Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. (1914)
* Emil L. Post. Recursive Unsolvability of a problem of Thue. Journal of Symbolic Logic (1947)
* A Markov. On certain insoluble problems concerning matrices. In Doklady Akad (1947)
* {#Lambek58} J. Lambek. The mathematics of sentence structure. American Mathematics Monthly (1958)
* J. Lambek. Type grammar revisited. Logical Aspects of Computational Linguistics (1999)


formal group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Formal geometry
+--{: .hide}
[[!include formal geometry -- contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _formal group_ is a [[group object]] [[internalization|internal to]] [[infinitesimal space]]s. More general than [[Lie algebra]]s, which are group objects in _first order infinitesimal_ spaces, formal groups may be of arbitrary infinitesimal order. They sit between [[Lie algebras]] and finite [[Lie groups]] or [[algebraic groups]].

Since [[infinitesimal spaces]] are typically modeled as [[Isbell duality|formal duals]] to [[algebras]], formal groups are typically conceived as group objects in formal duals to [[formal power series]] algebras.

Specifically, fixing a formal [[coordinate]] chart, then the product operation of a formal group is entirely expressed as a [[formal power series]] in two variables, satisfying conditions. This is called a _formal group law_, a concept that goes back to Bochner and [[Daniel Lazard|Lazard]].

Commutative formal group laws of dimension 1 notably appear in [[algebraic topology]] (originating in work by [[Sergei Novikov|Novikov]], [[Victor Buchstaber|Buchstaber]] and [[Daniel Quillen|Quillen]], see [Adams 74, part II](#Adams74)), where they express the behaviour of [[complex oriented cohomology theories]] evaluated on infinite [[complex projective space]] (i.e. on the [[classifying space]] $B U(1) \simeq \mathbb{C}P^\infty$). In particular [[complex cobordism]] cohomology theory in this context gives the universal formal group law represented by the [[Lazard ring]]. The [[height of a formal group|height of formal groups]] induces a filtering on [[complex oriented cohomology theories]] called the _[[chromatic homotopy theory|chromatic filtration]]_.

More recently [[Fabien Morel|Morel]] and [[Marc Levine]] consider the [[algebraic cobordism]] of smooth schemes in [[algebraic geometry]].  Formal groups are also useful in local [[class field theory]]; they can be used to explicitly construct the local Artin map according to Lubin and Tate.

### Formal group laws
 {#FormalGroupLaw}

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

An (commutative) [[adic ring]] is a ([[commutative ring|commutative]]) [[topological ring]] $A$ and an ideal $I \subset A$ such that

1. the [[topological space|topology]] on $A$ is the $I$-[[adic topology]];

1. the canonical morphism

   $$
     A \longrightarrow \underset{\longleftarrow}{\lim}_n (A/I^n)
   $$

   to the [[limit]] over [[quotient rings]] by powers of the ideal is an [[isomorphism]].

A [[homomorphism]] of adic rings is a ring [[homomorphism]] that is also a [[continuous function]] (hence a function that preserves the filtering $A \supset \cdots \supset A/I^2 \supset A/I $). This gives a category $AdicRing$ and a subcategory $AdicCRing$ of commutative adic rings.

The [[opposite category]] of $AdicRing$ (on [[Noetherian rings]]) is that of affine [[formal schemes]].

Similarly, for $R$ any fixed [[commutative ring]], then adic rings under $R$ are _adic $R$-algebras_. We write $Adic R Alg$ and $Adic R CAlg$ for the corresponding categories.

=--

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

For $R$ a [[commutative ring]] and $n \in \mathbb{N}$ then the [[formal power series ring]]

$$
  R[ [ x_1, x_2, \cdots, x_n ] ]
$$

in $n$ [[variables]] with [[coefficients]] in $R$ and equipped with the ideal

$$
  I = (x_1, \cdots , x_n)
$$

is an adic ring (def. \ref{AdicRing}).

=--

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

There is a [[fully faithful functor]]

$$
  AdicRing \hookrightarrow ProRing
$$

from [[adic rings]] (def. \ref{AdicRing}) to [[pro-rings]], given by

$$
  (A,I) \mapsto ( (A/I^{\bullet}))
  \,,
$$

i.e. for $A,B \in AdicRing$ two adic rings, then there is a [[natural isomorphism]]

$$
  Hom_{AdicRing}(A,B)
    \simeq
   \underset{\longleftarrow}{\lim}_{n_2}
   \underset{\longrightarrow}{\lim}_{n_1}
   Hom_{Ring}(A/I^{n_1},B/I^{n_2})
  \,.
$$

=--

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

For $R \in CRing$ a [[commutative ring]] and for $n \in \mathbb{N}$, a **formal group law** of dimension $n$ over $R$ is the structure of a [[group object]] in the category $Adic R CAlg^{op}$ from def. \ref{AdicRing} on the object $R [ [x_1, \cdots ,x_n] ]$ from example \ref{PowerSeriesAlgebra}.

Hence this is a morphism

$$
  \mu
  \;\colon\;
  R[ [ x_1, \cdots, x_n ] ]
   \longrightarrow
  R [ [ x_1, \cdots, x_n, \, y_1, \cdots, y_n ] ]
$$

in $Adic R CAlg$ satisfying unitality, associativity.

This is a **commutative formal group law** if it is an abelian group object, hence if it in addition satisfies the corresponding commutativity condition.

=--

This is equivalently a set of $n$ power series $F_i$ of $2n$ variables $x_1,\ldots,x_n,y_1,\ldots,y_n$ such that (in notation $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, $F(x,y) = (F_1(x,y),\ldots,F_n(x,y))$)

$$
  F(x,F(y,z))=F(F(x,y),z)
$$

$$
  F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms
$$

+-- {: .num_example #Commutative1DimFormalGroupLaw}
###### Example

A 1-dimensional commutative formal group law according to def. \ref{GroupObjectFormalGroupLaw} is equivalently a [[formal power series]]

$$
  \mu(x,y)
  =
  \underset{i,j \geq 0}{\sum} a_{i,j} x^i y^j
$$

(the [[image]] under $\mu$ in $R[ [ x,y ] ]$ of the element $t \in R [ [ t ] ]$) such that

1. (unitality)

   $$
     \mu(x,0) = x$ and $\mu(0,x) = x
     \,;
   $$

1. (associativity)

   $$
     \mu(x,\mu(y,z)) = \mu(\mu(x,y),z)
     \,;
   $$

1. (commutativity)

   $$
     \mu(x,y) = \mu(y,x)
     \,.
   $$

The first condition means equivalently that

$$
  a_{i,0}
    =
  \left\{
    \array{
      1 & if \; i = 1
      \\
      0 & otherwise
    }
  \right.
  \;\;\;\;\,,
  \;\;\;\;\;
  a_{0,i}
    =
  \left\{
    \array{
      1 & if \; i = 1
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

Hence $\mu$ is necessarily of the form

$$
  \mu(x,y)
   \;=\;
   x + y + \underset{i,j \geq 1}{\sum} a_{i,j} x^i y^j
  \,.
$$

The existence of inverses is no extra condition: by [[induction]] on the index $i$ one finds that there exists a unique

$$
  \iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i
$$

such that

$$
  \mu(x,\iota(x)) = 0
  \;\;\;\,,
  \;\;\;
  \mu(\iota(x),x) = 0
  \,.
$$

Hence 1-dimensional formal group laws over $R$ are equivalently [[monoids]] in $Adic R CAlg^{op}$ on $R[ [ x ] ]$.

=--

+-- {: .num_example}
###### Example
Any [[power series]] of the form $f(x) = x + a_2 x^2 + a_3 x^3 + \ldots$ in $R[ [x] ]$ has a functional or compositional inverse $f^{-1}(x)$ in the monoid $x R[ [x] ]$ under composition. Thus we may define a 1-dimensional formal group law by the formula $\mu(x, y) = f^{-1}(f(x) + f(y))$. That this is in some sense the typical way that 1-dimensional formal group laws arise is the content of [[Lazard ring|Lazard's theorem]].
=--

### Formal group schemes

Much more general are **formal group schemes** from ([Grothendieck](#Grothendieck))

__Formal group schemes__ are simply the [[group object]]s in a category of [[formal schemes]]; however usually only the case of the formal spectra of complete $k$-algebras is considered; this category is equivalent to the category of complete cocommutative $k$-[[Hopf algebras]].

### Formal groups over an operad

For a generalization over [[operads]] see ([Fresse](#Fresse)).

## Properties

### In characteristic 0

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

The quotient [[moduli stack]] $\mathcal{M}_{FG} \times Spec \mathbb{Q}$ of formal group over the [[rational numbers]] is isomorphic to $\mathbf{B}\mathbb{G}_m$, the [[delooping]] of the [[multiplicative group]] (over $Spec \mathbb{Q}$). This means that in [[characteristic]] 0 every formal group is determined, up to unique [[isomorphism]], by its [[Lie algebra]].

=--

For instance ([Lurie 10, lecture 12, corollary 3](#Lurie10)).

### Universal 1d commutative formal group law
 {#Universal1dFormalGroupLaw}

It is immediate that there exists a ring carrying a universal formal group law. For observe that for $\underset{i,j}{\sum} a_{i,j} x_1^i x_1^j$ an element in a [[formal power series]] algebra, then the condition that it defines a [[formal group law]] is equivalently a sequence of polynomial [[equations]] on the [[coefficients]] $a_k$. For instance the commutativity condition means that

$$
  a_{i,j} = a_{j,i}
$$

and the unitality constraint means that

$$
  a_{i 0}
  =
  \left\{
    \array{
      1 & if \; i = 1
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

Similarly associativity is equivalently a condition on combinations of triple products of the coefficients. It is not necessary to even write this out, the important point is only that it is some polynomial equation.

This allows to make the following definition

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

The **[[Lazard ring]]** is the [[graded commutative ring]] generated by  elements $a_{i j}$ in degree $2(i+j-1)$ with $i,j \in \mathbb{N}$

$$
  L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)
$$

quotiented by the relations

1. $a_{i j} = a_{j i}$

1. $a_{10} = a_{01} = 1$; $\forall i \neq 1: a_{i 0} = 0$

1. the obvious associativity relation

for all $i,j,k$.

The **universal 1-dimensional commutative [[formal group law]]** is the formal power series with [[coefficients]] in the Lazard ring given by

$$
  \ell(x,y) \coloneqq \sum_{i,j} a_{i j} x^i y^j \in L[ [ x , y ] ]
  \,.
$$

=--

+-- {: .num_remark #LazardRing}
###### Remark

The grading is chosen with regards to the formal group laws arising from [[complex oriented cohomology theories]] ([prop.](complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw)) where the [[variable]] $x$ naturally has degree -2. This way

$$
 deg(a_{i j} x^i y^j)
  =
  deg(a_{i j}) + i deg(x) + j deg(y)
  =
  -2
  \,.
$$

=--

The following is immediate from the definition:

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

For every [[ring]] $R$ and 1-dimensional commutative [[formal group law]] $\mu$ over $R$ (example \ref{Commutative1DimFormalGroupLaw}), there exists a unique ring [[homomorphism]]

$$
  f \;\colon\; L \longrightarrow R
$$

from the [[Lazard ring]] (def. \ref{LazardRing}) to $R$, such that it takes the universal formal group law $\ell$ to $\mu$

$$
  f_\ast \ell = \mu
  \,.
$$

=--

+-- {: .proof}
###### Proof

If the formal group law $\mu$ has coefficients $\{c_{i,j}\}$, then in order that $f_\ast \ell = \mu$, i.e. that

$$
  \underset{i,j}{\sum} f(a_{i,j}) x^i y^j
  =
  \underset{i,j}{\sum} c_{i,j} x^i y^j
$$

it must be that $f$ is given by

$$
  f(a_{i,j}) = c_{i,j}
$$

where $a_{i,j}$ are the generators of the Lazard ring. Hence it only remains to see that this indeed constitutes a ring homomorphism. But this is guaranteed by the vary choice of relations imposed in the definition of the Lazard ring.

=--

What is however highly nontrivial is this statement:

+-- {: .num_theorem #LazardTheorem}
###### Theorem
**([[Lazard's theorem]])**

The [[Lazard ring]] $L$ (def. \ref{LazardRing}) is [[isomorphism|isomorphic]] to a [[polynomial ring]]

$$
  L
  \simeq
  \mathbb{Z}[ t_1, t_2, \cdots ]
$$

in countably many generators $t_i$ in degree $2 i$.

=--

+-- {: .num_remark}
###### Remark

The [[Lazard theorem]] \ref{LazardTheorem} first of all implies, via prop. \ref{LazardRingIsUniversal}, that there exists an abundance of 1-dimensional formal group laws: given any ring $R$ then every choice of elements $\{t_i \in R\}$ defines a formal group law. (On the other hand, it is nontrivial to say which formal group law that is.)

Deeper is the fact expressed by the [[Milnor-Quillen theorem on MU]]: the Lazard ring in its polynomial incarnation of prop. \ref{LazardTheorem} is canonically identified with the [[graded commutative ring]] $\pi_\bullet(M U)$ of [[stable homotopy groups]] of the universal complex [[Thom spectrum]] [[MU]]. Moreover:

1. [[MU]] carries a [[universal complex orientation on MU|universal complex orientation]] in that for $E$ any [[homotopy commutative ring spectrum]] then homotopy classes of homotopy ring homomorphisms $M U \to E$ are in bijection to [[complex oriented cohomology theory|complex orientations]] on $E$;

1. every [[complex oriented cohomology|complex orientation]] on $E$ induced a 1-dimensional commutative [[formal group law]] ([prop.](complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw))

1. under forming stable homotopy groups every ring spectrum homomorphism $M U \to E$ induces a ring homomorphism

   $$
     L \simeq \pi_\bullet(M U) \longrightarrow \pi_\bullet(E)
   $$

   and hence, by the universality of $L$, a formal group law over $\pi_\bullet(E)$.

This is the formal group law given by the above complex orientation.

Hence the universal group law over the Lazard ring is a kind of [[decategorification]] of the [[universal complex orientation on MU]].

=--

## Examples

* [[formal multiplicative group]]

* [[Artin-Mazur formal group]] of an [[algebraic variety]],

  * [[formal Picard group]],

  * [[formal Brauer group]]

* [[Lubin-Tate formal group]]

* [[Honda formal group]]

## Related concepts

* [[height of a formal group law]]

* [[p-typical formal group law]]

* [[Morava stabilizer group]]

* [[Landweber exact functor theorem]]

* [[chromatic homotopy theory]]

* [[Lie algebra]], [[Lie group]]

* [[formal groupoid]]

Formal geometry is closely related also to the [[rigid analytic geometry]].

(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).

[[!include infinitesimal and local - table]]

## References

### General

* Shigkaki Tôgô, _Note of formal Lie groups_ , American Journal of Mathematics, Vol. 81, No. 3, Jul., 1959 ([JSTOR](http://www.jstor.org/stable/2372919))

* A. Fröhlich, _Formal group_,  Lecture Notes in Mathematics Volume 74, Springer (1968)

* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://iopscience.iop.org/article/10.1070/SM1971v013n01ABEH001030)]

* {#Adams74} [[Frank Adams]], Part II.1 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Grothendieck} [[Alexander Grothendieck]] et al. [[SGA]] III, vol. 1, Expose VIIB (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES (part B) 474-560

* {#Kochmann96} [[Stanley Kochmann]], section 4.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Fresse} [[Benoit Fresse]], _Lie theory of formal groups over an operad_, J. Alg. __202__, 455--511, 1998, [doi](http://dx.doi.org/10.1006/jabr.1997.7280)

* Michiel Hazewinkel, Formal Groups and Applications, [projecteuclid](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548600)

* Michel [[Demazure, lectures on p-divisible groups]]

* [[Jean Dieudonné]], _Introduction to the theory of formal groups_, Marcel Dekker, New York 1973.

### 1-Dimensional formal groups

A basic introduction is in

* Carl Erickson, _One-dimensional formal groups_ ([pdf](http://people.brandeis.edu/~cwe/pdfs/formal_groups.pdf))

Specifically formal group laws of [[elliptic curves]]:

* Antonia W. Bluher, _Formal groups, elliptic curves, and some theorems of Couveignes_, in: J.P. Buhler (eds.) _Algorithmic Number Theory_ ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 ([arXiv:math/9708215](https://arxiv.org/abs/math/9708215), [doi:10.1007/BFb0054887]( https://doi.org/10.1007/BFb0054887))

* {#Friedl17} Stefan Friedl, _An elementary proof of the group law for elliptic curves_ ([arXiv:1710.00214](https://arxiv.org/abs/1710.00214))

[[Quillen's theorem on MU]] is due to

* {#Quillen69} [[Daniel Quillen]], _On the formal group laws of unoriented and complex cobordism theory_, 1969, [projecteuclid](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183530915)

See also

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html)) Lecture 11 _Formal groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf))

* [[Takeshi Torii]], _One dimensional formal group laws of height $N$ and $N-1$_, PhD thesis 2001 ([pdf](http://mathnt.mat.jhu.edu/mathnew/Thesis/Torii.thesis.pdf))

* [[Takeshi Torii]], _On Degeneration of One-Dimensional Formal Group Laws and Applications to Stable Homotopy Theory_,  American Journal of Mathematics Vol. 125, No. 5 (Oct., 2003), pp. 1037-1077 ([JSTOR](http://www.jstor.org/stable/25099207))

* [[Stefan Schwede]], _Formal groups and stable homotopy of commutative rings_, Geom. Topol. 8 (2004) 335-412 ([arXiv:math/0402372](http://arxiv.org/abs/math/0402372))

The [[moduli stack]] of formal groups and its incarnation as a [[Hopf algebroid]]:

* [[Niko Naumann]], _The stack of formal groups in stable homotopy theory_, Advances in Mathematics Volume 215, Issue 2, 10 November 2007, Pages 569–600 doi:[10.1016/j.aim.2007.04.007](http://dx.doi.org/10.1016/j.aim.2007.04.007), arXiv:[math.AT/0503308](http://front.math.ucdavis.edu/math.AT/0503308)

[[!redirects formal groups]]

[[!redirects formal group scheme]]
[[!redirects formal group schemes]]

[[!redirects formal group law]]
[[!redirects formal group laws]]


formal groupoid

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## Idea

A _formal groupoid_ is a groupoid whose hom-spaces have [[infinitesimal space|infinitesimal extension]].

## Related concepts

* A formal groupoid over the point if the [[delooping]] of a [[formal group]].

* The [[∞-groupoid]]-notion of formal groupoids is discussed at  _[[formal cohesive ∞-groupoid]]_ .

## References

The notion of formal groupoid apparently goes back to

* P. Berthelot, _Cohomologie Cristalline des Schémas de Caractéristique $p \gt 0$, Springer L.N.M. 407 (1974).

and

* [[Luc Illusie]], _Complexe cotangent et déformations I_ , Springer L.N.M. 239 (1971); II, Springer L.N.M. 283

Formal groupoids in general and the [[de Rham space]] formal groupoid in particular are discussed also in section 7 of

* [[Carlos Simpson]], _The Hodge filtration on nonabelian cohomology_ ([arXiv:alg-geom/9604005](http://arxiv.org/abs/alg-geom/9604005))

Formal groupoids and their relation to Lie coalgebroids are discussed in section 1.4.15 of

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

The formal version of the notion of [[symplectic groupoid]]s is discussed in

* [[Alberto Cattaneo]], Benoit Dherin, [[Giovanni Felder]], _Formal symplectic groupoid_ ([arXiv:0312380](http://arxiv.org/abs/math/0312380))
 {#CattaneoDherinFelder}

[[!redirects formal groupoids]]


formal immersion of smooth manifolds

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## Definition

A **formal immersion** $F$ of one [[smooth manifold]], $M$, into another, $N$, is an injective [[bundle morphism]] $T M \to T N$ between their [[tangent bundles]].

That is, $F$ consists of a [[smooth function]] $f \;\colon\; M \to N$ and a [[homomorphism]] of [[vector bundles]] $F \;\colon\; T M\to T N$ covering $f$ such that the [[linear function]] $F|_{x} \;\colon\; T_x M \to T_{f(x)} N$ is  an [[injective function]] for every point $x$ in $M$.

Write $Imm^f(M,N)$ for the space of such formal immersions. There is a [[fibration]] over the space of smooth functions from $M$ to $N$, $Map^{sm}(M, N)$, forgetting the bundle homomorphism, whose [[fiber]] over $f$ is $Hom^{inj}_{Vect_M}(T M, f^{\ast}T N)$.

**Note**: Some authors define formal immersions in terms of continuous functions (e.g., [Laudenbach17, p. 6](#Laudenbach17)). However, the space $Map^{sm}(M, N)$ is homotopy equivalent to the space of all continuous functions, $Map(M, N)$, due to integrating against a smoothing kernel.

## Relation to immersions

Since an actual [[immersion of smooth manifolds]] is a formal immersion where the [[bundle morphism]] in question is specifically taken to be the pointwise [[derivative]] $d f$,
there is a natural [[continuous function]] $Imm(M,N) \to Imm^f(M,N)$, sending an actual [[immersion of smooth manifolds|immersion]] $f$ to the formal immersion with injective bundle morphism $d f$.

[[Stephen Smale]] and [[Morris Hirsch]] established that when $M$ is compact, and also either $M$ is  _open_ (in the sense that the complement of the boundary has no compact component) or $dim(M) \lt dim(N)$, then the map $Imm(M,N) \to Imm^f(M,N)$ is a [[weak  homotopy equivalence]]. This is an instance of the [[h-principle]].

When combined with the result that $Imm^f(S^k,\mathbb{R}^{n+k}) \to Map(S^k, V_k(\mathbb{R}^{n+k}))$ can be shown to be a homotopy equivalence, where $V_k(\mathbb{R}^{n+k})$ is the Stiefel manifold of $k$-frames in $\mathbb{R}^{n+k}$, the previous result establishes that isotopy classes of immersions of $S^k$ into $\mathbb{R}^{n+k}$ are in bijection with $\pi_k V_k(\mathbb{R}^{n+k})$. In the case where $k = 2$ and $n = 1$, we find that immersions of the [[2-sphere]] into $\mathbb{R}^3$ are classified by $\pi_2 V_2(\mathbb{R}^3)= \pi_2(SO(3))= 0$, in other words, all such immersions are  isotopic.  In  particular, $S^2$ can be turned inside-out ([[sphere eversion]]) inside $R^3$ by moving through a family of immersions.

##References

* [[John Francis]], _The h-principle, lectures 1 and 2: overview_, ([pdf](https://sites.math.northwestern.edu/~jnkf/classes/hprin/1overview.pdf))

* [[Konrad Voelkel]], Helene Sigloch, _Homotopy sheaves and h-principles_, ([pdf](https://www.konradvoelkel.com/wp-content/uploads/homotopy-sheaves.pdf))

* {#Laudenbach17} Francois Laudenbach, _René Thom and an anticipated h-principle_, ([arXiv:1703.08108](https://arxiv.org/abs/1703.08108))

[[!redirects formal immersions of smooth manifolds]]


formal logic

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See

* [[logic]], [[theory]]

* [[logical connective]]

* [[logical framework]]

* [[type theory]]

* [[consistency]]

## Logical operations
 {#LogicalOperations}

[[!include logic symbols -- table]]

## References

* [[Georg Kreisel]], *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

[[!redirects formal logics]]


formal loop space

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## Idea

The space of [[infinitesimal space|infinitesimal]] loops in a given [[space]] $X$ -- hence the formal neighbourhood of the constant loops $X \hookrightarrow L X$ -- is often called the _formal loop space_ (as in _[[formal geometry]]_).

Formal loop spaces have application for instance in the theory of [[elliptic genera]], since it was shown that the [[Witten genus]] is the $S^1$-equivariant [[index of a Dirac operator|index]] of a [[Dirac-Ramond operator]] which is like  [[Dirac operator on smooth loop space]], only that due to $S^1$-equivariance (orelse due to a [[large volume limit]]) the index is fully determined by the restriction of that operator to formal loops.

## Properties

### Relation to chiral differential operators
 {#RelationToChiralDifferentialOperators}

With $X$ a suitable scheme, its formal loop space $L_inf X$ in the sense of ([Kapranov-Vasserot I](#KapranovVasserotI)) has a [[Tate structure]] and hence an associated [[determinantal gerbe]] $Det_{L_{inf} X}$ with [[band]] $\mathcal{O}^\ast_{L_{inf} X}$. According to
([Kapranov-Vasserot IV](#KapranovVasserotIV)) this [[gerbe]] is essentially identified with the [[gerbe]] $CDO_X$ of [[chiral differential operators]] on $X$.

## Related concepts

* [[loop space]], [[loop space object]]

* [[smooth loop space]]

* [[free loop space]], [[free loop space object]],

* [[derived loop space]]

## References

In the context of [[algebraic geometry]] formal loop spaces have been introduced and studied in

* {#KapranovVasserotI} [[Mikhail Kapranov]], E. Vasserot, _Vertex algebras and the formal loop space_ Publications Mathématiques de l'IHÉS,
100 (2004), 209–269. ([arXiv:math/0107143](http://arxiv.org/abs/math/0107143))

* [[Mikhail Kapranov]], E. Vasserot, _Formal Loops II : the local Riemann-Roch theorem for determinantal gerbes_, Ann. Sci. ENS, ([arXiv:math/0509646](http://arxiv.org/abs/math/0509646))

* [[Mikhail Kapranov]], E. Vasserot, _Formal loops III: Factorizing functions and the Radon transform_ ([arXiv:math/0510476](http://arxiv.org/abs/math/0510476))

* {#KapranovVasserotIV} [[Mikhail Kapranov]], E. Vasserot, _Formal loops IV: Chiral differential operators_ ([arXiv:math/0612371](http://arxiv.org/abs/math/0612371))

Tentative aspects of a generalization to [[differential geometry]] are discussed in

* [[Pokman Cheung]], _Chiral differential operators: formal loop group actions and associated modules_ ([arXiv:1205.7089](http://arxiv.org/abs/1205.7089))

[[!redirects formal loop spaces]]


formal moduli problem

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## Definition

### Local definition as functors on Artinian objects

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###### Definition

Given a [[deformation context]] $(\mathcal{Y}, \{E_\alpha\}_\alpha)$, the [[(∞,1)-category]] of **formal moduli problems** over it is the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-presheaves]] over $\mathcal{Y}^{inf}$

$$
  Moduli^\mathcal{Y}
  \hookrightarrow
  [\mathcal{Y}^{inf}, \infty Grpd]
$$

on those [[(∞,1)-functors]] $X \colon \mathcal{Y}^{inf} \to \infty Grpd$ such that

1. over the [[terminal object]] they are [[contractible]]: $X(*) \simeq *$ (hence they are [[anti-reduced object|anti-reduced]]);

1. they preserve [[(∞,1)-pullbacks]] of small morphisms (are [[cohesive ∞-presheaf on E-∞ rings|infinitesimally cohesive]])

=--

([Lurie, Def. 1.1.14 with Def. 1.1.8](#Lurie))

+-- {: .num_remark }
###### Remark

This means that a "formal deformation problem" is a [[space]] in [[higher geometry]] whose geometric structure is detected by the "test spaces" in $\mathcal{Y}^{op}$ in a way that respects gluing ([[descent]]) in $\mathcal{Y}^{op}$ as given by [[(∞,1)-pullbacks]] there. The first condition requires that there is an essentially unique such probe by the point, hence that these higher geometric space has essentially a single global point. This is the condition that reflects the infinitesimal nature of the deformation problem.

=--

+-- {: .num_remark }
###### Remark

The clause about pullbacks is what makes the behaviour at arbitrary infinitesimal order be all controlled by that at first order, see [Calaque-Grivaux 18, top of p. 8](#CalaqueGrivaux18).

This ability to understand deformations order-by-order is related to the existence of a good obstruction theory. Indeed, evaluating a formal moduli problem $X$ on a pullback diagram defining an elementary morphism exhibits $X(\Omega^{\infty - n}E)$ as an obstruction space.

=--

## Properties

### Relation to $L_\infty$-algebras

For $k$ a [[field]] of [[characteristic]] 0, write
write $CAlg_k^{sm} \hookrightarrow CAlg_k$ for the [[(∞,1)-category]] of [[Artin ring|Artinian]] connective [[E-∞ algebras]] over $k$, or equivalently that of "small" commutative [[dg-algebras]] over $k$.

The smallness condition implies connectivity ([Lurie, prop. 1.1.11 (1)](#Lurie)), hence that the [[homotopy group]] of these [[E-∞ algebras]] vanish in negative degree. Notice that for the dg-algebras this means that the [[chain homology]] vanishes in negative degree _if_ the differential is taken to have degree -1 (see [Porta 13, def. 3.1.14](#Porta13) for emphasis). This is the natural condition for the function algebra in [[derived geometry]]. Here these small $E_\infty$/dg-algebras are to be thought of as function algebras on "derived [[infinitesimally thickened points]]".

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

There is an [[equivalence of (∞,1)-categories]]

$$
  L_\infty Alg_k \stackrel{\simeq}{\to} Moduli^{CAlg^{sm}_k}
$$

with that of [[L-∞ algebras]].

=--

In this form this is ([Lurie, theorem 0.0.13](#Lurie)), originally proved as ([Pridham, theorem 2.30 and proposition 4.42](#Pridham)).
See at _[[model structure for L-∞ algebras]]_ for various other incarnations of this equivalence.

### Relation to Lie differentiation

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

Given a [[deformation context]] $\mathcal{Y}$, the restricted [[(∞,1)-Yoneda embedding]] gives an [[(∞,1)-functor]]

$$
  Lie \colon \mathcal{Y} \to Moduli^{\mathcal{Y}}
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

For $Y \in \mathcal{Y}^{op}$, the object $Lie(Y)$ represents the _[[formal geometry|formal neighbourhood]]_ of the basepoint of $Y$ as seen by the infinitesimally thickened points dual to the $\{E_\alpha\}$.

Hence we may call this the operaton of **[[Lie differentiation]]** of spaces in $\mathcal{Y}^{op}$ around their given base point.

=--

## Related concepts

* [[moduli problem]]

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

* [[sheaf of L-∞ algebras]]

* [[synthetic differential geometry]]

* [[synthetic differential ∞-groupoid]]

## References

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

* {#Lurie} [[Jacob Lurie]], _[[Formal moduli problems]]_

* [[Jacob Lurie]], _Moduli problems for ring spectra_ ICM 2010 proceedings contribution [pdf](http://www.math.harvard.edu/~lurie/papers/moduli.pdf)

* {#Porta13} [[Mauro Porta]], _Derived formal moduli problems_, master thesis 2013, [pdf](http://algant.eu/documents/theses/porta.pdf).

* {#CalaqueGrivaux18} Damien Calaque, Julien Grivaux, _Formal moduli problems and formal derived stacks_ ([arXiv:1802.09556](https://arxiv.org/abs/1802.09556))

The correspondence between formal moduli problems and [[dg-Lie algebras]] is extended to [[positive characteristic]] in

* {#BrantnerMathew19} [[Lukas Brantner]], [[Akhil Mathew]], _Deformation Theory and Partition Lie Algebras_, arXiv:[1904.07352](https://arxiv.org/abs/1904.07352)

[[!redirects formal moduli problems]]


Formal Moduli Problems and DG-Lie Algebras

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This entry provides some links concerning the notes

* {#Lurie} [[Jacob Lurie]], _Formal moduli problems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf))

#Contents#
* table of contents
{:toc}

## Idea

This text establishes a statement (theorem 5.3) that characterizes [[∞-stacks]] over an [[(∞,1)-site]] of formal duals of [[Artin ring|Artin]] [[E-infinity-ring|E-∞ algebras]] over a [[field]] (hence [[infinitesimally thickened points]] in [[derived geometry]]) such that

1. they have a contractible underlying [[∞-groupoid]]

   $$
    \Gamma X \simeq *
   $$

   (in the terminology and notation for [[cohesive (∞,1)-topos]]es)

1. and are [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] in that they respects [[homotopy fiber products]] of morphisms whose [[kernels]] are [[nilpotent ideals]]

with those associated to an [[∞-Lie algebra]] $\mathfrak{g}$ in the sense of ([Hinich](#Hinich)). This is ([Lurie, theorem 0.0.13, remark 0.0.15](#Lurie)).

The construction is then generalized to noncommutative geometry.

## Keywords

* [[deformation context]]

* [[formal moduli problem]]

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

## Related entries

Am earlier (2007) [[model category]]-[[presentable (infinity,1)-category|presentation]] proof by ([Pridham](#Pridham)) exists, see the discussion at

* [[model structure for L-infinity algebras]] .

A indication of how to use the main theorem here in order to implement [[Lie differentiation]] of [[smooth ∞-groups]] is at

* _[synthetic differential infinity-groupoid -- Lie differentiation](synthetic%20differential%20infinity-groupoid#LieDifferentiation)_

## Related references

The same result was proved earlier (theorem 2.30 and corollary 4.57, cf ([Lurie, remark 0.0.14](#Lurie))), with further [[model category]] [[presentable (infinity,1)-category|presentations]] and comparisons, in

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

See at _[[model structure for L-infinity algebras]]_ for more along these lines.

The foundations of the theory appear in

* [[Vladimir Hinich]], _DG coalgebras as formal stacks_ ([arXiv:math/9812034](http://arxiv.org/abs/math/9812034))
 {#Hinich}

For more related discussion see also

* [[Jacob Lurie]], _[[Higher Algebra]]_

A survey is in

* [[Mauro Porta]], _Derived formal moduli problems_, master thesis 2013, [pdf](http://algant.eu/documents/theses/porta.pdf).

category: reference

[[!redirects Moduli Problems and DG-Lie Algebras]]
[[!redirects Formal Moduli Problems]]
[[!redirects Formal moduli problems]]


formal multiplicative group

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## Definition

The [[formal group]]  of the [[multiplicative group]] .

Its formal group law $f(x,y) \in R[ [x,y] ]$ is

$$
  f(x,y) = x + y + x y
  \,,
$$

## Examples

The formal group law of [[topological K-theory]] which is induced by its canonical [[complex oriented cohomology|complex orientation]] is the multiplicative formal group law. See at _[[topological K-theory]]_ the section _[Complex orientation and Formal group law](topological+K-theory#ComplexOrientationAndFormalGroupLaw)_

## Related concepts

* [[K-theory spectrum]]

[[!redirects formal multiplicative groups]]

[[!redirects multiplicative formal group law]]
[[!redirects multiplicative formal group laws]]


formal neighbourhood

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## Idea

A _formal neighbourhood_ is a [[neighbourhood]] in [[formal geometry]], hence an "[[infinitesimal space|infinitesimal]] neighbourhood". Also called a _[[formal completion]]_ or _[[formal disk]]_, see there for more.

## Examples

* The [[formal spectrum]] of the [[completion of a commutative ring]] in the $I$-[[adic topology]] for a [[maximal ideal]] $I$ is the formal neighbourhood of the point corresponding to that ideal in the [[spectrum of a commutative ring]].

* [[perturbative quantum field theory]] (in particular realized as [[formal deformation quantization]]) describes the infinitesimal neighbourhood of [[classical field theory|classical]] [[free field theory]] in the space of all [[quantum field theories]].

## Related concepts

[[!include infinitesimal and local - table]]

[[!redirects formal neighbourhoods]]


formal neighbourhood of the diagonal

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## Idea

Given any object $X$ in a [[cartesian monoidal category]] where [[formal completion]] is defined, then the _formal neighbourhood of the diagonal_ of $X$ is the [[formal completion]] of the [[diagonal]] map $\Delta_X \colon X \longrightarrow X \times X$.

Hence intuitively, the formal neighbourhood of the diagonal of $X$ is the space whose points are pairs of infinitesimally close points in $X$.

Specifically for $X$ a [[scheme]], then the formal neighbourhood of its diagonal is the [[formal scheme]] around $\Delta_X$. In this case the [[coequalizer]] of the two projections out of the formal neighbourhood is called the [[de Rham stack]] of $X$. Moreover, the (sheaf of modules of) sections of the formal neighbourhood, with respect to one of the two projection maps, is the [[tangent complex]] of $X$. This holds indeed more generally, at least also for $X$ a [[derived stack|derived]] [[algebraic stack]] ([Hennion 13](#Hennion13)).

More generally, see at _[[infinitesimal disk bundle]]_.

## Related concepts

* [[infinitesimal object]]

* [[infinitesimal singular simplicial complex]]

* [[infinitesimal neighborhood]]

## References

* {#Hennion13}  [[Benjamin Hennion]], _Tangent Lie algebra of derived Artin stacks_ ([arXiv:1312.3167](http://arxiv.org/abs/1312.3167))

[[!redirects formal neighbourhoods of diagonals]]