formal noncommutative symplectic geometry

Formal noncommutative symplectic geometry (generalizing [[symplectic geometry]] to the context of [[noncommutative geometry]]) has been introduced by [[Maxim Kontsevich]], motivated by several constructions in geometry and [[mathematical physics]] including the [[cohomology]] of (compactifications of) certain [[moduli space]]s, cohomology of [[foliation]]s and perturbation expansions of [[Chern-Simons theory]]. He noticed that there is a parallel between computations in such seemingly distant problems, and conjectured a generalized version of [[Lie theory]] which should apply to those, and which should be at formal level reflected in some version of duality theory. The underlying theory has been constructed by Kapranov and Ginzburg following his advice as a [[Koszul duality]] for operads.

Some of the structures are now viewed from the point of view of double derivations and double Poisson structures and are closed to derived picture.

## References

* Kontsevich gave lectures at Harvard in 1991-1992 on aspects related to the expansions of Chern-Simons theory.

* [[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://193.51.104.7/~maxim/TEXTS/Formal%20non-commutative%20symplectic%20geometry.pdf)

* Maxim Kontsevich, _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, Volume II, Progress in Mathematics 120, Birkhauser 1994, 97-121, [pdf](http://193.51.104.7/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf) [MR1247289](https://mathscinet.ams.org/mathscinet-getitem?mr=1247289)

* [[Victor Ginzburg]], [[Mikhail Kapranov]], _Koszul duality for operads_, Duke Math. J. __76__ (1994),  no. 1, 203--272; reprint [arxiv/0709.1228](https://arxiv.org/abs/0709.1228); *Erratum to: Koszul duality for operads*, Duke Math. J. __80__ (1995),  no. 1, 293.

* Victor Ginzburg, _Non-commutative symplectic geometry, quiver varieties, and operads_, Math. Res. Lett. __8__ (2001),  no. 3, 377--400, [math.QA/0005165](https://arxiv.org/abs/math/0005165); _Lectures on noncommutative geometry_, [math.AG/0506603](https://arxiv.org/abs/math/0506603)

* [[William Crawley-Boevey]], [[Pavel Etingof]], [[Victor Ginzburg]], _Noncommutative geometry and quiver algebras_,  Adv. Math. __209__:1 (2007) 274-336 [doi](https://doi.org/10.1016/j.aim.2006.05.004)


formal Picard group

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###Context###
#### Formal geometry
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[[!include formal geometry -- contents]]
=--
#### Group Theory
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[[!include group theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[algebraic variety]] with [[Picard scheme]] $Pic_X$,
if the [[connected component]] $Pic_X^0$ is a [[smooth scheme]] then the [[completion]] of $Pic_X$ at its [[neutral element|neutral]] [[global point]] is a [[formal group]]. This is called the _formal Picard group_ of $X$. ([ArtinMazur 77](#ArtinMazur77), [Liedtke 14, example 6.13](#Liedtke14))

This construction is the special case of the general construction of [[Artin-Mazur formal groups]] for $n = 1$ (see also  [this Remark](elliptic+spectrum#RoleOfFormalPicardScheme) at _[[elliptic spectrum]]_). The next case is called the _[[formal Brauer group]]_.

## Related concepts

* [[Picard stack]]

[[!include moduli of higher lines -- table]]

## References

The original account of the construction of formal Picard groups is

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)

Modern reviews include

* {#Liedtke14} Christian Liedtke, example 6.13 in _Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem_ ([arXiv.1403.2538](http://arxiv.org/abs/1403.2538))

[[!redirects formal Picard groups]]

[[!redirects formal Picard scheme]]
[[!redirects formal Picard schemes]]


formal scheme

> under construction

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _formal scheme_ is a [[formal neighbourhood]] of a [[scheme]], a scheme with infinitesimal thickening.

Generally this may be formalized via [[ind-objects]] of schemes.
Hence regarded as a [[sheaf]] on [[affine schemes]], a formal scheme is a [[filtered colimit]] of ordinary schemes (e.g. [Strickland 00, section 4](#Strickland00)).

Regarded as a [[locally ringed space]] a formal scheme has the underlying [[topological space]] of an ordinary [[scheme]], but its [[structure sheaf]] contains extra nilpotent elements.

## Motivation

Formal [[power series]] [[ring]]s $k[\![x_1,\ldots,x_n]\!]$ are limits of their truncations (e.g. in one variable $k[\![x]\!]/(x^n)$); they can be viewed as completions and they get equipped with a natural filtration and [[adic topology]].

They do not converge as a series (and make sense) in an open set or in any of the standard topologies (e.g. Zariski and complex topology over $\mathbb{Z}$), but they are rather "localized" in an *[[infinitesimal neighborhood|infinitesimal neighborhood]]* of the origin. One would like to be able to talk about functions supported only infinitesimally (in the transverse direction) to a [[closed subscheme]]. The formal schemes of [[Grothendieck]] are [[ringed spaces]] containing the information on all infinitesimal neighborhoods. Zariski's theorem on formal functions and establishing the theory of [[formal group]]s were some of the concrete motivations.

## Definition

There are roughly four equivalent definitions of a $k$-formal scheme for a field $k$ (check this):

Let $Mf_k$ denote the category of finite dimensional $k$-rings (=$k$-algebras which are rings).

1. A $k$-scheme is called a $k$_-formal scheme_ if it is is equivalent to a [[filtered colimit|directed colimit]] of finite (affine) $k$-schemes.

1. A $k$-scheme is a $k$-formal scheme if it is presented by a [[profinite group|profinite]] $k$-ring; i.e a $k$-ring which is the limit of topologically discrete quotients which are finite $k$-rings. If $A$ is such a topological $k$-ring $Spf_k(A)(R)$ denotes the set of continous morphisms from $A$ to the topologically discrete ring $R$. We have $Spf_k$ is a (contravariant) equivalence between the category of profinite $k$-rings and the category $fSch_k$ of formal $k$-schemes.

1. Instead of defining the category $fSch_k$ of formal $k$-schemes as the [[opposite category|opposite]] of $Mf_k$ define it instead covariantly on the category of finite dimensional $k$-[[coring|corings]].

1. A formal $k$-scheme is precisely a left exact (commuting with finite limits) functor $X:Mf_k\to Set$.

([Demazure, p-divisible groups, chapter I](#Demp))

The inclusion $Mf_k\hookrightarrow M_k$ induces a functor

$${}^\hat\; :Sch_k\to fSch_k$$

called _completion functor_.

### Formal spectra of Noetherian rings

If $X$ is a [[scheme]], a [[closed subscheme]] $Y\subset X$ is given by an embedding of topological spaces with the [[comorphism]] $\mathcal{O}_X\to f_*\mathcal{O}_Y$ which is a surjection; but alternatively $\mathcal{O}_Y$ can be recovered from $X$ and the defining sheaf of ideals $\mathcal{I}\subset \mathcal{O}_X$. Then one defines the [[structure sheaf]] of the completion $\mathcal{O}_{\hat{X}}$ as $lim_n \mathcal{O}_X/\mathcal{I}^n$ restricted to $Y$ and the __completion__ $\hat{X} := (Y,\mathcal{O}_{\hat{X}})$. If $X = Spec\,R$ where $R$ is a noetherian $I$-adic ring, and $Y=Spec\,R/I$ then the completion is called the **[[formal spectrum]]** of $R$ denoted

$$
  Spf\,R = \hat{X}
$$

(where $R$ is viewed as a topological ring). The formal spectrum is an [[ind-object]] in the [[category]] of [[algebraic schemes]], viewed as a formal colimit
$colim_n Spec (R/I^n)$.

A (locally) noetherian formal scheme is a formal completion of a (locally) noetherian scheme along a closed subscheme. Equivalently, a locally noetherian scheme is a locally ringed space which is locally isomorphic to the formal spectrum of a complete separated adic noetherian ring.

### More general ind-schemes

The formal spectrum can be extended to a somewhat bigger class of topological rings than the noetherian ones; Grothendieck developed the theory in the generality of [[pseudocompact ring|pseudocompact]] topological rings. However, some important rings, e.g. the ring of [[integers]] $\mathbb{Z}$, do not have a pseudocompact topology. Thus one could try to consider a more general subcategory of [[ind-schemes]] (with at least the requirement that the [[ind-object]] be represented by a diagram where the connecting morphisms are [[closed subscheme|closed immersions of schemes]]); one such approach is outlined in some detail in ([Beilinson-Drinfeld](#BeilinsonDrinfeld)).

## Examples

### Formal power series and their formal spectra
 {#FormalPowerSeries}

The [[completion of a ring]] of the [[polynomial ring]] $\mathbb{Z}[x]$ (the [[formal dual]] to the [[affine line]] $\mathbb{A}^1$) at the ideal $(x)$ is the [[limit]] -- formed in the [[category]] [[CRing]] --

$$
  \mathbb{Z}[x]^\wedge_{(x)} = \underset{\longleftarrow}{lim}_n (\mathbb{Z}[x])/(x^{n+1})
$$

of the [[quotient rings]] of $\mathbb{Z}[x]$ by the ideals generated by $x^{n+1}$ (the [[ring of dual numbers]] and its higher analogs). This yields the [[formal power series]] ring

$$
  \mathbb{Z}[x]^\wedge_{(x)}
  \simeq
   \mathbb{Z}[ [x] ]
  \,.
$$

Notice that the formal power series ring contains _no_ nilpotent elements except for zero -- even though each filtering stage $(\mathbb{Z}[x])/(x^n)$ does.

In contrast, the [[filtered colimit]] of the [[sheaves]] on [[affine varieties]] $Spec(R)$ that are [[representable functor|represented]] by this system of nilpotent rings is the formal scheme

$$
  \widehat{\mathbb{A}^1}
  =
  \underset{\longrightarrow}{lim}_n Spec(\mathbb{Z}[x]/(x^{n+1}))
  \;\;\;
  \in
  Sh(Aff) = Sh(CRing^{op})
$$

given by the sheaf which sends each ring to its [[nilradical]]

$$
  \widehat{\mathbb{A}^1} \;\colon\;  Spec(R) \mapsto Nil(R)
  \,.
$$

(see e.g. [Strickland 00, example 4.2, example 4.18](#Strickland00)).

### Formal neighbourhood of the diagonal

See at _[[formal neighbourhood of the diagonal]]_.

### Formal groups

The [[group objects]] in formal schemes are the [[formal groups]]. See there for more.

## References

Classical discussion includes

* [[Alexander Grothendieck]], _Géométrie formelle et géométrie algébrique_, FGA 2 (Séminaire Bourbaki, t. 11, 1958/59, no. 182)

* [[Luc Illusie]], _Grothendieck existence theorem in formal geometry_, chapter 8 in Fantechi, Gottsche, Illusie, Kleiman, Nitsure, Vistoli, _Fundamental algebraic geometry, Grothendieck's [[FGA explained]], Math. Surveys and Monographs_ __123__, AMS 2005 (draft version [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes))

* A. Grothendieck (avec J. Dieudonne), [[EGA]] I.10

* A. Grothendieck et al. [[SGA]] III.2 (Exp. 7a, P. Gabriel, Étude infinitésimale des schémas en groupe et groupes formels; Exp. 7b, P. Gabriel, Groupes formels)

* R. Hartshorne, _Algebraic geometry_, II.9

* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

* {#Demp}Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

* [[Daniel Murfet]], notes [Section2.9-FormalSchemes.pdf](http://therisingsea.org/notes/Section2.9-FormalSchemes.pdf)

* Leovigildo Alonso, Ana Jeremias, Marta Perez, _Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes_ ([arXiv:math/0604241](http://arxiv.org/abs/math/0604241))

More general discussion in terms of ind-schemes includes

* {#BeilinsonDrinfeld} A. Beilinson, V. Drinfel'd, _Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system_, preliminary version ([pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf))

Another approach using a certain topological extension of the [[Yoneda lemma]] on $k Alg^{op}$ has been proposed in

* B. Pareigis, R. A. Morris, _Formal groups and Hopf algebras over discrete rings_, Trans. Amer. Math. Soc.  __197__ (1974), 113--129 ([doi](http://dx.doi.org/10.2307/1996930), [[Morris-Pareigis formal scheme|nlab entry]]).

[[Nikolai Durov]] has considered a flexible bigger category (which inludes the usual schemes) of covariant functors from the category of pairs $(R,I)$ where $R$ is a commutative ring and $I$ a [[nilpotent ideal]]; the correspondence between formal groups and [[Lie algebras]] based on Hausdorff series is neatly developed and used in that language; see chapters 7--9 of

* N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, _A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra_, Journal of Algebra 309, n. 1, 318--359 (2007)
([doi:jalgebra](http://dx.doi.org/10.1016/j.jalgebra.2006.08.025))
([math.RT/0604096](http://front.math.ucdavis.edu/math.RT/0604096)).

For another generalization of formal schemes see

* T. Yasuda, _Non-adic formal schemes_, Int. Math. Research Notices 2009: 2417--2475, ([doi:imrn](http://dx.doi.org/10.1093/imrn/rnp021), [arxiv](http://arxiv.org/abs/0711.0434))

In a fundamental article in [[noncommutative algebraic geometry]],

* M. Kapranov, _Noncommutative geometry based on commutator expansions_, [math.AG/9802041](http://arxiv.org/abs/math.AG/9802041),

Kapranov introduced objects which should be interpreted as the infinitesimal neighborhoods of those commutative schemes with a closed immersion into a noncommutative scheme which is locally isomorphic to the spectrum of a free associative algebra.

And a decidedly functorial perspective is in (section 4 of)

* {#Strickland00} [[Neil Strickland]], _Formal schemes and formal groups_, [math.AT/0011121](http://arxiv.org/abs/math.AT/0011121)

For the scheme geometric picture behind the infinitesimal neighborhoods and [[D-modules]] see also

* A. Beĭlinson, J. Bernstein, J., _A proof of Jantzen conjectures_, in I. M. Gelʹfand Seminar, 1--50,
Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993 (MR1237825 (95a:22022))

Some aspects of formal completions from the point of view of the derived categories are in

* [[Dmitri Orlov|D. Orlov]], _Formal completions and idempotent completions of triangulated categories of singularities_, [arxiv/0901.1859](http://arxiv.org/abs/0901.1859)

* Alexander I. Efimov, _Formal completion of a category along a subcategory_, [arxiv/1006.4721](http://arxiv.org/abs/1006.4721)

[[!redirects formal schemes]]


formal smooth infinity-groupoid

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###Context###
#### Cohesive $\infty$-Toposes
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[[!include cohesive infinity-toposes - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _formal smooth $\infty$-groupoid_ is an [[∞-groupoid]] equipped with a [[cohesive (∞,1)-topos|cohesive structure]] that subsumes that of [[smooth ∞-groupoid]]s as well as of [[infinitesimal object|infinitesimal]] $\infty$-groupoids -- [[∞-Lie algebroids]], hence equipped with "[[differential cohesion]]".

In the [[cohesive (∞,1)-topos]] of formal smooth $\infty$-groupoids the canonical [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\mathbf{\Pi}(X)$ factors through a version relative to [[Smooth∞Grpd]]: the [[infinitesimal singular simplicial complex|infinitesimal path ∞-functor]] $\mathbf{\Pi}_{inf}(X)$. In traditional terms this is the object modeled by the [[tangent Lie algebroid]] and the [[de Rham space]] of $X$. The [[quasicoherent ∞-stack]]s on $\mathbf{\Pi}_{inf}(X)$ are [[D-module]]s on $X$.

## Definition

We consider [[(∞,1)-sheaves]] over a "twisted semidirect product" [[site]] or [[(∞,1)-site]] of

* [[Cartesian spaces]] with [[smooth functions]] between them as for [[smooth ∞-groupoids]],

* and a [[category]] or [[(∞,1)-category]] of [[infinitesimally thickened points]].

First in

* [1-localic definition](#Localic)

we consider the 1-site, then in

* [∞-localic definition](#DerivedDefinition)

we consider the $(\infty,1)$-site.

### 1-localic definition
 {#Localic}

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

Let $T := $ [[CartSp]]${}_{smooth}$ be the [[syntactic category]] of the [[Lawvere theory]] of [[smooth algebra]]s: the category of [[Cartesian space]]s $\mathbb{R}^n$ and [[smooth function]]s between them.

Write

$$
  SmoothAlg := T Alg
$$

for its category of [[algebras over a Lawvere theory|T-algebras]] -- the [[smooth algebra]]s ($C^\infty$-rings).

Let

$$
  InfPoint \hookrightarrow SmoothAlg^{op}
$$

be the [[full subcategory]] on the [[infinitesimally thickened point]]s: this smooth algebras whose underlying abelian group is a [[vector space]] of the form $\mathbb{R} \oplus V$ with $V$ a finite-[[dimension]]al real [[vector space]] and nilpotent in the algebra structure.

=--

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

Let

$$
  FormalCartSp \hookrightarrow SmoothLoc
$$

be the [[full subcategory]] of the category of [[smooth loci]] on the objects of the form

$$
  U = \mathbb{R}^n \times D
$$

that are [[product]]s of a [[Cartesian space]] $\mathbb{R}^n \in$ [[CartSp]] for $n \in \mathbb{N}$ and an [[infinitesimally thickened point]] $D \in InfPoint$. (See at _[[FormalCartSp]]_.)

Equip this category with the [[coverage]] where a family of morphisms  is [[covering]] precisely if it is of the form $\{U_i \times D \stackrel{(f_i, Id_D)}{\to} U \times D\}$ for $\{f_i : U_i \to U\}$ a covering in [[CartSp]]${}_{smooth}$ (a [[good open cover]]).

=--

This appears as ([Kock 86, (5.1)]{#Kock86}).

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

The [[sheaf topos]] over [[FormalCartSp]] is ([[equivalence of categories|equivalent]] to) the [[topos]] known as the [[Cahiers topos]], a [[smooth topos]] that constitutes a [[Models for Smooth Infinitesimal Analysis|well adapted model]] for [[synthetic differential geometry]]. See at _[[Cahiers topos]]_ for further references.

=--

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

We say the [[(∞,1)-topos]] of **formal smooth $\infty$-groupoids** is the [[(∞,1)-category of (∞,1)-sheaves]]

$$
  FormalSmooth \infty Grpd := Sh_{(\infty,1)}(FormalCartSp)
$$

on $FormalCartSp$.

=--

### $\infty$-localic
 {#DerivedDefinition}

We now generalize the 1-category $InfPoint$ of [[infinitesimally thickened points]] to the [[(∞,1)-category]] $InfPoint_\infty$ of "derived infinitesimally thickened points", the formal dual of "small commutative $\infty$-algebras" from ([Hinich](#Hinich), [Lurie](#Lurie)).

(...)

## Properties

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

$FormalSmooth \infty Grpd$ is a [[cohesive (∞,1)-topos]].

=--

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

Because [[FormalCartSp]]  is an [[∞-cohesive site]]. See there for details.

=--

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

Write $FormalSmoothMfd \hookrightarrow SmoothAlg^{op}$ for the [[full subcategory]] of [[smooth loci]] on the
 **[[formal smooth manifolds]]**: those modeled on [[FormalCartSp]] equipped with the evident [[coverage]].

=--

+-- {: .num_prop #CartSpAsDenseSubOfFSmoothMfd}
###### Observation

$FormalCartSp$ is a [[dense sub-site]] of $FormalSmoothMfd$.

=--

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

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

$$
  FormalSmooth\infty Grpd \simeq \hat Sh_{(\infty,1)}(FSmoothMfd)
$$

with the [[hypercomplete (∞,1)-topos]] over $FSmoothMfd$.

=--

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

With the [above observation](#CartSpAsDenseSubOfFSmoothMfd) this is directly analogous to the corresponding proof at [[ETop∞Grpd]].

=--

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

Write $i : CartSp_{smooth} \hookrightarrow FormalCartSp$ for the canonical [[full subcategory|embedding]].

=--

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

The functor $i^*$ given by restriction along $i$ exhibits
$FormalSmooth\infty Grpd$ as an
infinitesimal cohesive neighbourhood
of the [[(∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoids]] in that we have a quadruple of [[adjoint (∞,1)-functor]]s

$$
 ( i_! \dashv i^* \dashv i_* \dashv i^! )
  :
  Smooth \infty Grpd
  \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\to}}{\stackrel{i^!}{\leftarrow}}}}
  FormalSmooth \infty Grpd
  \,,
$$

such that $i_!$ is a [[full and faithful (∞,1)-functor]].

=--

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

Since $i : CartSp_{smooth} \hookrightarrow CartSp_{formalsmooth}$ is an [infinitesimally ∞-cohesive site](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#InfinitesimalNeighbourhoodFromInfinitesimalSite) this follows with [a proposition](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#InfinitesimalNeighbourhoodFromInfinitesimalSite) discussed at _[[cohesive (infinity,1)-topos -- infinitesimal cohesion]]_.

=--

## Structures

We discuss the realization of the general abstract structures in a cohesive (∞,1)-topos in $FormalSmooth \infty Grpd$.

Since by the [above discussion](#RelativeInftyConnectedness) $FormalSmooth\infty Grpd$ is strongly $\infty$-connected _relative_ to [[Smooth∞Grpd]] all of these structures that depend only on $\infty$-connectedness (and not on [[local (∞,1)-topos|locality]]) acquire a relative version.

### $\infty$-Lie algebroids and deformation theory
 {#StrucLieTheory}

This subsection is at

* [[∞-Lie algebroid]].

### Paths and geometric Postnikov towers
  {#StrucPaths}

We discuss the intrinsic infinitesimal path adjunction realized in $FormalSmooth\infty Grpd$.

$$
  (\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf})
   :=
  (i \circ \Pi_{inf} \dashv Disc_{inf} \Pi_{inf} \dashv Disc_{inf} \circ \Gamma_{inf})
  :
  FormalSmooth \infty Grpd
  \to
  FormalSmooth \infty Grpd
  \,.
$$

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

For $U \times D \in CartSp_{smooth} \ltimes InfinSmoothLoc = FormalCartSp \hookrightarrow FormalSmooth\infty Grpd$
we have that

$$
  \mathbf{Red}(U \times D) \simeq U
$$

is the **reduced [[smooth locus]]**: the [[Isbell duality|formal dual]] of the [[smooth algebra]] obtained by quotienting out all nilpotent elements in the [[smooth algebra]] $C^\infty(K \times D) \simeq C^\infty(K) \otimes C^\infty(D)$.

=--

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

By the [[model category]] presentation of
$\mathbf{Red} \simeq  \mathbb{L} Lan_i \circ \mathbb{R}i^*$ of the [above proof](#CohesivenessUnderSmoothInfGrpd) and using that every [[representable functor|representable]] is cofibrant and fibrant in the local projective [[model structure on simplicial presheaves]] we have

$$
  \begin{aligned}
    \mathbf{Red}(U \times D)
    &
    \simeq
     (\mathbb{L}Lan_i) (\mathbb{R}i^*) (U \times D)
    \\
     &\simeq
     (\mathbb{L}Lan_i) i^* (U \times D)
     \\
     & \simeq
     (\mathbb{L} Lan_i) U
     \\
     & \simeq
     Lan_i U
     \\
    & \simeq U
  \end{aligned}
$$

(using that $i$ is a [[full and faithful functor]]).

=--

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

For $X \in SmoothAlg^{op}  \to FormalSmooth \infty Grpd$ a [[smooth locus]], we have that $\mathbf{\Pi}_{inf}(X)$ is the corresponding [[de Rham space]], the object
in which all [[infinitesimal object|infinitesimal]] neighbour points in $X$ are equivalent, characterized by

$$
  \mathbf{\Pi}_{inf}(X) : U \times D \mapsto X(U)
  \,.
$$

=--

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

By the $(\mathbf{Red} \dashv \mathbf{\Pi}_{inf})$-adjunction relation

$$
  \begin{aligned}
    \mathbf{\Pi}_{inf}(X)(U \times D)
    & =
    FormalSmooth \infty Grpd(U \times D, \mathbf{\Pi}_{inf}(X))
    \\
    & \simeq
    FormalSmooth \infty Grpd( \mathbf{Red}(U \times D), X)
    \\
    & \simeq
     FormalSmooth \infty Grpd( U, X )
  \end{aligned}
  \,.
$$

=--

### Cohomology and principal $\infty$-bundles {#StrucCohomology}

We discuss the intrinsic cohomology in a cohesive (∞,1)-topos realized in $FormalSmooth\infty Grpd$.

#### Cohomology localization

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

The canonical [[line object]] of the [[Lawvere theory]] [[CartSp]]${}_{smooth}$ is the [[real line]], regarded as
an object of the [[Cahiers topos]], and hence of
$FormalSmooth \infty Grpd$

$$
  \mathbb{A}^1_{CartSp_{smooth}} = \mathbb{R}
  \,.
$$

=--

We shall write $\mathbb{R}$ also for the underlying
additive group

$$
  \mathbb{G}_a = \mathbb{R}
$$

regarded as an abelian [[∞-group]] object in
$FormalSmooth\infty Grpd$. For $n \in \mathbb{N}$ write
$\mathbf{B}^n \mathbb{R} \in FormalSmooth\infty Grpd$
for its $n$-fold [[delooping]].

For $n \in \mathbb{N}$ and $X \in FormalSmooth\infty Grpd$
write

$$
  H^n_{synthdiff}(X) := \pi_0 FormalSmooth\infty Grpd(X,\mathbf{B}^n \mathbb{R})
$$

for the [[cohomology group]] of $X$ with coefficients in the canonical line object in degree $n$.

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

Write

$$
  \mathbf{L}_{sdiff} \hookrightarrow FormalSmooth \infty Grpd
$$

for the [[cohomology localization]] of $FormalSmooth\infty Grpd$ at $\mathbb{R}$-[[cohomology]]: the full [[sub-(∞,1)-category]] on the $W$-[[local object]] with respect to the [[class]] $W$ of [[morphism]]s that induce [[isomorphism]]s in all $\mathbb{R}$-cohomology groups.

=--

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

Let $SmoothAlg^{\Delta}_{proj}$ be the [[model structure on cosimplicial algebras|projective model structure on cosimplicial smooth algebras]] and let $j : (SmoothAlg^{\Delta})^{op} \to [FormalCartSp, sSet]$ be the prolonged external [[Yoneda embedding]].

1. This constitutes the [[right adjoint]] of a [[Quillen adjunction]]

   $$
    (\mathcal{O} \dashv j)

    :
     (SmoothAlg^\Delta)^{op}
     \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}}
     [FSmoothMfd^{op}, sSet]_{proj,loc}
     \,.
   $$

1. Restricted to [[simplicial object|simplicial]] [[formal smooth manifold]]s along

   $$
      FSmoothMfd^{\Delta^{op}}
      \hookrightarrow
      (SmoothAlg^\Delta)^{op}
   $$

   the right [[derived functor]] of $j$ is a [[full and faithful (∞,1)-functor]] that factors through the [[cohomology localization]] and thus identifies a full [[reflective sub-(∞,1)-category]]

   $$
     (FSmoothMfd^{\Delta^{op}})^\circ
     \hookrightarrow
     \mathbf{L}_{sdiff}
     \hookrightarrow
     FormalSmooth\infty Grpd
   $$

1. The intrinsic $\mathbb{R}$-cohomology of any object $X \in FormalSmooth\infty Grpd$ is computed by the ordinary [[cochain cohomology]] of the [[Moore complex|Moore cochain complex]] underlying the cosimplicial abelian group of the image under the left [[derived functor]]$(\mathbb{L}\mathcal{O})(X)$ under the [[Dold-Kan correspondence]]:

   $$
      H_{fSmooth}^n(X)
       \simeq
      H^n_{cochain}(N^\bullet(\mathbb{L}\mathcal{O})(X))
      \,.
   $$

=--

+-- {: .proof}

###### Proof

First a remark on the sites. By the [above proposition](#EquivalenceToToposOverSoothSynthMfd)
$FormalSmooth\infty Grpd$ is equivalent to the [[hypercomplete (∞,1)-topos]] over [[formal smooth manifold]]s. This is [[presentable (∞,1)-category|presented]] by the [[nLab:Bousfield localization of model categories|left Bousfield localization]] of $[FSmoothMfd^{op}, sSet]_{proj,loc}$ at the [[∞-connected]] morphisms. But a fibrant object in $[FSmoothMfd^{op}, sSet]_{proj,loc}$ that is also [[n-truncated]] for $n \in \mathbb{N}$ is also fibrant in the hyperlocalization (only for the untruncated object there is an additional condition). Therefore the right Quillen functor claimed above indeed lands in truncated objects in $FormalSmooth \inftyGrpd$.

The proof of the above statements is given in ([Stel](#Stel)), following ([Toën](#Toen)). Some details are spelled out at
[[function algebras on ∞-stacks]].

=--

#### Cohomology of Lie groups

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

Let $G \in SmoothMfd \hookrightarrow Smooth\infty Grpd \hookrightarrow FormalSmooth\infty Grpd $ be a [[Lie group]].

Then the intrinsic group cohomology in [[Smooth∞Grpd]] and in $FormalSmooth\infty Grpd$ of $G$ with coefficients in $\mathbb{R}$ coincides with [[Graeme Segal|Segal]]'s refined [[Lie group cohomology]] ([Segal](#Segal), [Brylinski](#Brylinski)).

$$
  H^n_{smooth}(\mathbf{B}G, \mathbb{R})
  \simeq
  H^n_{fsmooth}(\mathbf{B}G, \mathbb{R})
  \simeq
  H^n_{Segal}(G,\mathbb{R})
  \,.
$$

=--

For the full proof please see [[schreiber:differential cohomology in a cohesive topos|here, section 3.4]] for the moment.

+-- {: .num_cor }
###### Corollary

For $G$ a [[compact topological space|compact]] [[Lie group]] we have for all $n \geq 1$ that

$$
  H^n_{smooth}(G,U(1))
  \simeq H_{top}^{n+1}(B G, \mathbb{Z})
  \,.
$$

=--

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

This follows from applying the [above result](#RealCohomologyOfCompactLieGroup) to the [[fiber sequence]] induced by the sequence $\mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} = U(1)$.

=--

+-- {: .num_note }
###### Note

This means that the intrinsic cohomology of [[compact topological space|compact]] [[Lie group]]s in [[Smooth∞Grpd]] and $FormalSmooth\infty Grpd$ coincides for these coefficients with the Segal-Blanc-Brylinski refined Lie group cohomology ([Brylinski](#Brylinski)).

=--

#### Cohomology of $\infty$-Lie algebroids
 {#CohomologyOfLieAlgebroids}

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

Let $\mathfrak{a} \in L_\infty \mathrm{Algd}$ be an [[L-∞ algebroid]].
Then its intrinsic real cohomology in $\mathrm{FormalSmooth}\infty \mathrm{Grpd}$

$$
  H^n(\mathfrak{a}, \mathbb{R})
  :=
  \pi_0 \mathrm{FormalSmooth}\infty \mathrm{Grpd}(\mathfrak{a}, \mathbf{B}^n \mathbb{R})
$$

coincides with its ordinary [[∞-Lie algebroid cohomology|L-∞ algebroid cohomology]]:
the cochain cohomology of its [[Chevalley-Eilenberg algebra]]

$$
  H^n(\mathfrak{a}, \mathbb{R})
   \simeq
  H^n(\mathrm{CE}(\mathfrak{a}))
  \,.
$$

=--

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

By the [above propoposition](#PresentationOfCohomologyLocalization) we have that

$$
  H^n(\mathfrak{a}, \mathbb{R}) \simeq
  H^n N^\bullet(\mathbb{L}\mathcal{O})(i(\mathfrak{a}))
  \,.
$$

By [this lemma](http://ncatlab.org/nlab/source/Lie+infinity-algebroid#CofibrantResolutionOfLinfinityAlgebroid)  this is

$$
  \cdots \simeq
  H^n N^\bullet
  \left(
    \int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathcal{O}(i(\mathfrak{a})_k)
  \right)
  \,.
$$

Observe that $\mathcal{O}(\mathfrak{a})_\bullet$ is cofibrant in the Reedy model structure
$[\Delta^{\mathrm{op}}, (\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}]_{\mathrm{Reedy}}$
relative to the opposite of the projective model structure on cosimplicial algebras:
the map from the latching object in degree $n$ in
$\mathrm{SmoothAlg}^\Delta)^{\mathrm{op}}$ is dually in
$\mathrm{SmoothAlg} \hookrightarrow \mathrm{SmoothAlg}^\Delta$ the projection
$$
  \oplus_{i = 0}^n \mathrm{CE}(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n
  \to
  \oplus_{i = 0}^{n-1} \mathrm{CE}(\mathfrak{a})_i \otimes \wedge^i \mathbb{R}^n
$$
hence is a surjection, hence a fibration in
$\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}}$ and therefore indeed a cofibration in
$(\mathrm{SmoothAlg}^\Delta_{\mathrm{proj}})^{\mathrm{op}}$.

Therefore using the Quillen bifunctor property of the coend over the tensoring in reverse to [this lemma](http://ncatlab.org/nlab/source/Lie+infinity-algebroid#CofibrantResolutionOfLinfinityAlgebroid), the above is equivalent to
$$
  \cdots
  \simeq
  H^n N^\bullet
  \left(
    \int^{[k] \in \Delta} \Delta[k] \cdot \mathcal{O}(i(\mathfrak{a})_k)
  \right)
$$
with the fat simplex replaced again by the ordinary simplex. But in brackets this is now by definition the image under the monoidal Dold-Kan correspondence of the Chevalley-Eilenberg algebra
$$
  \cdots
  \simeq
  H^n( N^\bullet \Xi \mathrm{CE}(\mathfrak{a}) )
  \,.
$$

By the [[Dold-Kan correspondence]] we have hence

$$
  \cdots \simeq
  H^n(\mathrm{CE}(\mathfrak{a}))
  \,.
$$

=--

It follows that a degree-$n$ $\mathbb{R}$-cocycle on $\mathfrak{a}$ is presented by a morphism
$$
  \mu : \mathfrak{a} \to b^n \mathbb{R}
  \,,
$$
where on the right we have the $L_\infty$-algebroid whose $\mathrm{CE}$-algebra is concentrated in
degree $n$. Notice that if $\mathfrak{a} = b \mathfrak{g}$ is the delooping of an $L_\infty$-
algebra $\mathfrak{g}$ this is equivalently a morphism of $L_\infty$-algebras
$$
  \mu : \mathfrak{g} \to b^{n-1} \mathbb{R}
  \,.
$$

#### de Rham theorem

> under construction

We consider the [[de Rham theorem]] in $FormalSmooth \infty Grpd$, with respect to the infinitesimal de Rham cohomology

$$
  H_{dR,inf}^n(X) := \pi_0 FormalSmooth \infty Grpd(X, \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R})
  \,.
$$

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

For all $n \in \mathbb{N}$, $n \gt 0$, The canonical morphism

$$
  \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}
  \to
  \mathbf{\flat} \mathbf{B}^n \mathbb{R}
$$

is an [[equivalence in an (∞,1)-category|equivalence]].

=--

This means that for all $X \in \mathbf{H}$ the infinitesimal de Rham cohomology coincides with the ordinary real cohomology of the geometric realization of $X$

$$
  H^n_{dR, inf}(X)
   \simeq
  H^n(|X|, \mathbb{R})
  \,.
$$

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

Since all representables are formally smooth, we have a colimit

$$
  U \times_{\mathbf{\Pi}_{inf}(U)} U
  \stackrel{\to}{\to}
  U
  \stackrel{}{\to}
  \mathbf{\Pi}_{inf}(U)
  \,.
$$

In the presentation over the site we have that

$$
  X \times_{\mathbf{\Pi}_{inf}(X)} X :
  K \times D \mapsto
  \{
    f,g : K \times D \to U | K \to K \times D \stackrel{\to}{\to} U
  \}
  \,.
$$

Therefore a morphism $\mathbf{\Pi}_{inf}(U) \to  \mathbb{R}$ is equivalently a morphism $\phi : U \to \mathbb{R}$ such that for all $K \times D \to U$ that coincide on $K$ we have that all the composites

$$
  K \times D \to U \stackrel{\phi}{\to} \mathbf{B}^n \mathbb{R}
$$

are equals. If $U$ is the point, then $\phi$ is necessarily constant. If $U$ is not the point, there is one nontrivial tangent vector $v$ in $U$. The composite produces the corresponding tangent vector $\phi_*(v)$ in $\mathbb{R}$. But all these tangent vectors must be equal.  Hence $\phi$ must be constant.

=--

This kind of argument is indicated in ([Simpson-Teleman, prop. 3.4](#SimpsonTeleman)).

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

Let $X \in $ [[SmoothMfd]] and write
$X^{\Delta^\bullet_{inf}} \in [CartSp_{formalsmooth}^{op}, sSet]$
for the [[tangent Lie algebroid]] regarded as a simplicial object (see [[L-infinity algebroid]] for the details).

Then there is a morphism
$X^{\Delta^\bullet_{inf}} \to \mathbf{\Pi}_{inf}(X)$ which is an equivalence in $\mathbb{R}$-cohomology.

=--

(...)

### Formally étale morphisms and cohesive étale $\infty$-groupoids
 {#StructureSheaves}

We discuss [formally étale morphisms](#http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#FormallySmoothEtaleUnramified) and [étale objects](#)
with respect to the cohesive infinitesimal neighbourhood
$i :$ [[Smooth∞Grpd]] $\hookrightarrow FormalSmooth\infty Grpd$.

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

Let $X_\bullet$ be a degreewise smooth [[paracompact topological space|paracompact]] [[simplicial manifold]],
canonically regarded as an object of [[Smooth∞Grpd]].

Then
$j_! X_\bullet$ in $FormalSmooth \infty Grpd$ is presented by the
same simplicial manifold.

=--

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

First consider an ordinary smooth paracompact manifold $X$. It admits a
[[good open cover]] $\{U_i \to X\}$ and the corresponding [[Cech nerve]]
$C(\{U_i\}) in [CartSp_{smooth}^{op}, sSet]_{proj}$ is a
[[cofibrant resolution]] of $X$. Therefore the
$\infty$-functor $j_!$ is computed on $X$ by evaluating the corresponding
simplicial functor (of which it is the [[derived functor]]) on $C(\{U_i\})$.

Since the simplicial functor

$$
  j_!
   :
  [CartSp_{smooth}^{op}, sSet]_{proj, loc}
    \to

  [CartSp_{formalsmooth}^{op}, sSet]_{proj, loc}
$$

is a [[left adjoint]] (indeed a [[left Quillen functor]]) it preserves the [[coproducts]] and [[coend]] that the Cech nerve is built from:

$$
  \begin{aligned}
    j_! C(\{U_i\})
    & =
    j_! \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n}
    U_{i_0, \cdots, i_n}
    \\
    & =
    \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n}
    j_! (U_{i_0, \cdots, i_n})
    \\
    & =
    \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_0, \cdots, i_n}
    U_{i_0, \cdots, i_n}
  \end{aligned}
  \,.
$$

Here we used that, by assumption on a [[good open cover]], all the
$U_{i_0, \cdots, i_n}$ are [[Cartesian spaces]], and that
$j_!$ coincides on [[representable functor|representables]] with the inclusion
$CartSp_{smooth} \hookrightarrow CartSp_{formalsmooth}$.

Let now $X_\bullet$ be a general simplicial manifold. Assume that in each degree there is a [[good open cover]] $\{U_{p,i} \to X_p\}$ such that these fit into a simplicial system giving a bisimplicial Cech nerve such that there is a morphism of [[bisimplicial object|bisimplicial presheaves]]

$$
  C(\mathcal{U})_{\bullet, \bullet} \to X_{\bullet}
$$

with $X_\bullet$ regarded as simplicially constant in one direction. Each
$C(\mathcal{U})_{p, \bullet} \to X_p$ is  a cofibrant resolution.

We claim that the [[coend]]

$$
  \int^{[n] \in \Delta} C(\mathcal{U})_{n, \bullet} \cdot \mathbf{\Delta}[n]
  \;\;\;
  \to
  \;\;\;
  X
$$

is a cofibrant resolution of $X$, where $\mathbf{\Delta}$ is the [[fat simplex]].
From this the proposition follows by again applying the left Quillen functor $j_!$ degreewise and pulling it through all the colimits.

This remaining claim follows from the same argument as used
above in prop. \ref{RealCohomologyOfCompactLieGroup}.

=--

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

A morphism in $FormalSmooth\infty Grpd$, is a [[formally étale morphism]] with respect to the [[infinitesimal cohesion]] $i \colon Smooth \infty Grpd \hookrightarrow FormalSmooth\infty Grpd$ precisely if for all [[infinitesimally thickened points]] $D$ the diagram

$$
  \array{
    X^D &\stackrel{f^D}{\to}& Y^D
    \\
    \downarrow && \downarrow
    \\
    Y &\stackrel{f}{\to}& Y
  }
$$

is an $\infty$-pullback under $i^*$.

=--

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

Since $i^*$ preserves $\infty$-limits, this is the case in particular if the diagram is an $\infty$-pullback already in $FormalSmooth\infty Grod$. In this form, restricted to 0-truncated objects, hence to the
[[Cahiers topos]], this characterization of formally étale morphisms appears axiomatized around p. 82 of ([Kock 81, p. 82](#Kock)).

In particular, a [[smooth function]] $f : X \to Y$ in [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] between [[smooth manifolds]] is a [[formally étale morphism]] with respect to the [[infinitesimal cohesion]] $Smooth \infty Grpd \hookrightarrow FormalSmooth\infty Grpd$ precisely if it is a [[local diffeomorphism]] in the traditional sense.

=--

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

We spell out the case for smooth manifolds.
Here we need to to show that

$$
  \array{
    i_! X &\stackrel{i_! f}{\to}& i_! Y
    \\
    \downarrow && \downarrow
    \\
    i_* X &\stackrel{i_* f}{\to}& i_* Y
  }
$$

is a [[pullback]] in $Sh(CartSp_{formalsmooth})$ precisely if $f$ is a local diffeomorphism. This is a pullback precisely if for all $U \times D \in CartSp_{smooth} \ltimes InfPoint \simeq CartSp_{formalsmooth}$ the diagram of sets of plots

$$
  \array{
    Hom(U \times D, i_! X) &\stackrel{i_! f}{\to}& Hom(U \times D, i_! Y)
    \\
    \downarrow && \downarrow
    \\
    Hom(U \times D, i_* X) &\stackrel{i_* f}{\to}& Hom(U \times D, i_* Y)
  }
$$

is a pullback. Using, by the discussion at [[∞-cohesive site]], that $i_!$ preserves colimits and restricts to $i$ on representables, and using that $i^* (U \times D ) = U$, this is equivalently the diagram

$$
  \array{
    Hom(U \times D, X) &\stackrel{f_*}{\to}& Hom(U \times D, Y)
    \\
    \downarrow && \downarrow
    \\
    Hom(U , X) &\stackrel{f_*}{\to}& Hom(U , Y)
  }
  \,,
$$

where the vertical morphisms are given by restriction along the inclusion $(id_U, *) : U \to U \times D$.

For one direction of the claim it is sufficient to consider this situation for $U = *$ the point and $D$ the first order infinitesimal interval. Then $Hom(*,X)$ is the underlying set of points of the manifold $X$ and $Hom(D,X)$ is the set of tangent vectors, the set of points of the [[tangent bundle]] $T X$. The pullback

$$
  Hom(*,X) \times_{Hom(*,Y)} Hom(D,Y)
$$

is therefore the set of pairs consisting of a point $x \in X$ and a tangent vector $v \in T_{f(x)} Y$. This set is in fiberwise bijection with $Hom(D, X) = T X$ precisely if for each $x \in X$ there is a bijection $T_x X \simeq T_{f(x)}Y $, hence precisely if $f$ is a [[local diffeomorphism]]. Therefore $f$ being a local diffeo is necessary for $f$ being formally étale with respect to the given notion of infinitesimal cohesion.

To see that this is also sufficient notice that this is evident for the case that $f$ is in fact a [[monomorphism]], and that since [[smooth function]]s are characterized locally, we can reduce the general case to that case.

=--

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

A [[Lie groupoid]] $\mathcal{G}$ is an [[étale groupoid]] in the traditional sense, precisely if regarded as an object in $i :$ [[Smooth∞Grpd]] $\hookrightarrow$ [[FormalSmooth∞Grpd]] it is an [cohesive étale ∞-groupoid](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#EtaleObjects).

=--

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

Let $\mathcal{G}_0 \to \mathcal{G}$ be the inclusion of the smooth manifold of objects. This is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]]. It remains to show that this is [formally étale](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#FormallySmoothEtaleUnramified) with respect to the given cohesive neighbourhood.

By the discussion at _[[(∞,1)-pullback]]_ we may compute the characteristic $(\infty,1)$-pullback by an ordinary pullback of a [[fibration]] of [[simplicial presheaves]] that presents $\mathcal{G}_0 \to \mathcal{G}$.

By the [[factorization lemma]] such is given by

$$
  \mathcal{G}^I \times_{\mathcal{G}} \mathcal{G}_0 \to \mathcal{G}
  \,.
$$

By inspection one see that this morphism is

* in degree 0 the target-map $t : Mor(\mathcal{G}) \to \mathcal{G}_0$;

* in degree 1 the projection $Mor(\mathcal{G}) {}_t\times_{s} Mor(\mathcal{G}) \to Mor(\mathcal{G})$.

By prop. \ref{RecoveringLocalDiffeomorphisms} both of these need to be
[[étale maps]] in the ordinary sense. By definition, this is the case
precisely if $\mathcal{G}$ is an [[étale groupoid]].

=--

### Formally smooth / formally unramified morphisms

As a direct consequence of prop. \ref{RecoveringLocalDiffeomorphisms} we have the following

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

A [[smooth function]] $f : X \to Y$ between [[smooth manifolds]],
is a [[submersion]] or [[immersion]], respectively, precisely
if, when canonically regarded as a morphism in $FormalSmooth \infty Grpd$, it is a [[formally smooth morphism]] or [[formally unramified morphism]], respectively.

=--

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

As in the proof of prop. \ref{RecoveringLocalDiffeomorphisms}
we find that the pullback $i_* X \times_{i_* Y} i_! Y$
is over the [[infinitesimal interval]] isomorphic to

$$
  X \times_Y T Y
$$

and the canonical morphism from $i_! X $ into this pullback is

$$
  T X \to X \times_Y T Y
  \,.
$$

=--

### Lie differentiation
 {#LieDifferentiation}

We indicate how to formalize [[Lie differentiation]] in the context of formal smooth $\infty$-groupoids.

Let

$$
  inf : InfPoint_\infty \hookrightarrow \mathbf{H}^{*/}
$$

be the canonical inclusion. By ([Lurie, theorem 0.0.13, remark 0.0.15](#Lurie), also [Pridham 07](#Formal+Moduli+Problems+and+DG-Lie+Algebras#Pridham)) we have a full inclusion

$$
  Lie_\infty \hookrightarrow Sh_\infty(InfPoint_\infty)
$$

on those objects whose space of global sections is contractible and which are [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] (for a somewhat different notion of "infinitesimal cohesion", beware the terminology). Consider then the $\infty$-functor

$$
  Grp(\mathbf{H})
  \simeq
  \mathbf{H}^{*/}_{\geq 1}
   \stackrel{yoneda}{\to}
  PSh_\infty( \mathbf{H}^{*/})
  \stackrel{inf^*}{\to}
  PSh_\infty(InfPoint_\infty)
$$

which sends a pointed connected formal smooth $\infty$-groupoid $\mathbf{B}G$ to the $(\infty,1)$-presheaf of pointed morphisms

$$
  \mathbf{pt} \to \mathbf{B}G
$$

for $\mathbf{pt} \in InfPoint_\infty$.

By assumption that $\mathbf{B}G$ is connected (and we need to assume that it is [[geometric infinity-stack|geometric]], which will gives infinitesimal cohesion by the [[Artin-Lurie representability theorem]]) this factors as

$$
  \mathbf{H}^{*/}_{\geq 1}
   \stackrel{Lie}{\to}
  Lie_\infty
   \hookrightarrow
  Sh_\infty(InfPoint_\infty)
  \,.
$$

The resulting $\infty$-functor

$$
  Lie : Grp(\mathbf{H}) \simeq \mathbf{H}^{*/}_{\geq 1} \to Lie_\infty
$$

is Lie differentiation.

For differentiation of smooth groupoids with [[atlas]] $U \to X$ to [[L-infinity algebroids]] this happens under $U$

$$
  \mathbf{H}^{U/}
$$

(...)

## Related concepts

* [[cohesive (∞,1)-topos]]

  * [[Euclidean-topological ∞-groupoid]]

[[!include geometries of physics -- table]]

## References

The site [[FormalCartSp]] is discussed in section 5 of

* {#Kock86} [[Anders Kock]], _Convenient vector spaces embed into the Cahiers topos_ , Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17  ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0)).

For more on this see at _[[Cahiers topos]]_.

The notion of formally étale maps as obtained here from the general abstract definition in [[differential cohesion]] coincided on 0-truncated objects with that defined on p. 82 of

* {#Kock81} [[Anders Kock]], _Synthetic Differential Geometry_ (1981) ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

The [infinitesimal path ∞-groupoid adjunction](#StrucPaths) $(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf})$ and the de Rham theorem for $\infty$-stacks is discussed at the level of [[homotopy categories]] in section 3 of

* {#SimpsonTeleman} [[Carlos Simpson]], [[Constantin Teleman]], _deRham theorem for $\infty$-stacks_ ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf))

The $(\infty,1)$-topos $SynthDiff\infty Grpd$:

* [[Urs Schreiber]], section 4.4 of: _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

* [[Hisham Sati]], [[Urs Schreiber]], Section 3.1.2 of: _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

* [[Urs Schreiber]], _[[schreiber:Introduction to Higher Supergeometry]]_, lecture at _[[Higher Structures in M-Theory 2018]]_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/)

  published in parts as: _Higher Structures in M-Theory_ (with [[Branislav Jurčo]], [[Christian Saemann]], [[Martin Wolf]]), Fortschritte der Physik (2019) ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001]( https://doi.org/10.1002/prop.201910001))

The [[cohomology localization]] of $SynthDiff\infty Grpd$ and the infinitesimal singular simplicial complex as a presentation for infinitesimal paths objects in $SynthDiff\infty Grpd$ is discussed in

* {#Stel} [[Herman Stel]], _$\infty$-Stacks and their function algebras -- with applications to $\infty$-Lie theory_ , master thesis (2010) ([[schreiber:master thesis Stel|web]])

The discussion of the cohomology localization of $SynthDiff\infty Grpd$ follows that in another context in

* {#Toen} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

The construction of the infinitesimal path object has been amplified and discussed by [[Anders Kock]] under the name _combinatorial differential forms_, for instance in

*  [[Anders Kock]], _Synthetic Geometry of Manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

The discussion that the normalized [[cosimplicial algebra]] of functions on the [[infinitesimal singular simplicial complex]] is the [[de Rham complex]] is further discussed in

* Herman Stel, _[[schreiber:Cosimplicial C-infinity rings and the de Rham complex of Euclidean space]]_ ([arXiv:arXiv:1310.7407](http://arxiv.org/abs/arXiv:1310.7407))

The results on differentiable [[Lie group cohomology]] used above is in

* {#Blanc} P. Blanc, _Cohomologie différentiable et changement de groupes_ Astérisque, vol. 124-125 (1985), pp. 113-130.

recalled in

* {#Brylinski} [[Jean-Luc Brylinski]], _Differentiable Cohomology of Gauge Groups_ ([arXiv](http://arxiv.org/abs/math/0011069))

which parallels

* {#Segal} [[Graeme Segal]], _Cohomology of topological groups_ , Symposia Mathematica, Vol IV (1970) (1986?) p. 377

The $(\infty,1)$-site of derived infinitesimal points is discussed in

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

following

* {#Hinich} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, J. Pure Appl. Algebra, 162 (2001), 209-250  ([arXiv:math/9812034](http://arxiv.org/abs/math/9812034))

[[!redirects formal smooth infinity-groupoids]]

[[!redirects formal smooth ∞-groupoid]]
[[!redirects formal smooth ∞-groupoids]]

[[!redirects synthetic differential infinity-groupoid]]
[[!redirects synthetic differential infinity-groupoids]]

[[!redirects synthetic differential ∞-groupoid]]
[[!redirects synthetic differential ∞-groupoids]]

[[!redirects synthetic-differential ∞-groupoid]]
[[!redirects synthetic-differential ∞-groupoids]]

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

[[!redirects formal smooth groupoid]]
[[!redirects formal smooth groupoids]]

[[!redirects SynthDiff∞Grpd]]

[[!redirects FormalSmooth∞Groupoid]]
[[!redirects FormalSmooth∞Groupoids]]
[[!redirects FormalSmooth∞Grpd]]


formal smooth manifold

> This entry is about [[smooth manifolds]] with [[infinitesimal neighbourhood|infinitesimal thickenings]]. For "formal spaces" in the sense that their [[de Rham complex]] is a [[formal dg-algebra]], see there.

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## Idea

A _formal smooth manifold_ is a [[smooth manifold]] equipped possibly with [[infinitesimal object|infinitesimal extension]].

In the [[differential cohesion]] of [[synthetic differential infinity-groupoids]] these are the spaces locally isomorphic to $\mathbb{R}^n \times D$, where $\mathbb{R}^n$ is a [[Cartesian space]] and $D$ is an [[infinitesimally thickened point]]. Here $\mathbb{R}^n$ is the underlying [[reduced object|reduced]] manifold.

## Related concepts

* [[jet bundle]]

## References

* {#Kock80} [[Anders Kock]], _Formal manifolds and synthetic theory of jet bundles_, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1980) Volume: 21, Issue: 3 ([Numdam](http://www.numdam.org/item?id=CTGDC_1980__21_3_227_0))

*  [[Anders Kock]], section I.17 and I.19 of _Synthetic Differential Geometry_, ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

Formal smooth manifolds of the simple [[product]] form $X \times D$ in the category of [[smooth loci]] for $X$ an ordinary [[smooth manifold]] and $D$ and [[infinitesimal space]] have been considered in section 4 of

* [[Eduardo Dubuc]], _Sur les modeles de la geometrie differentielle synthetique_ [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques, 20 no. 3 (1979), p. 231-279  ([numdam](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)).

For more on this see [[Cahiers topos]]

[[!redirects formal smooth manifolds]]

[[!redirects formal manifold]]
[[!redirects formal manifolds]]

[[!redirects infinitesimally thickened manifold]]
[[!redirects infinitesimally thickened manifolds]]

[[!redirects infinitesimally thickened smooth manifold]]
[[!redirects infinitesimally thickened smooth manifolds]]


formal smooth set

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## Idea

The concept of _formal smooth set_ is a kind of [[generalized space]] generalizing _[[smooth sets]]_ from plain [[differential geometry]] to differential geometry equipped with explicit [[infinitesimal spaces]], hence to [[synthetic differential geometry]]. Just as a [[smooth set]] is equivalently a [[sheaf]] on the [[site]] of [[Cartesian spaces]], so a _formal smooth set_ is a [[sheaf]] on the [[site]] of [[formal Cartesian spaces]], hence of [[Cartesian products]] of [[Cartesian spaces]] with [[infinitesimally thickened points]]. The resulting [[sheaf topos]] is also known as _Dubuc's [[Cahiers topos]]_.

For the moment, for more see at _[[geometry of physics]]_ the chapters _[[geometry of physics -- manifolds and orbifolds|manifolds and orbifolds]]_ and _[[geometry of physics -- supergeometry]]_.

## Related concepts

[[!include geometries of physics -- table]]

[[!redirects formal smooth sets]]

[[!redirects FormalSmoothSet]]
[[!redirects FormalSmoothSets]]


formal spectrum

> under construction -- needs attention

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## Idea

For usual ([[commutative ring|commutative]]) [[rings]], [[Grothendieck]] introduced the notion of a [[prime spectrum of a commutative ring|prime spectrum]]. In order to accommodate not only [[polynomials]] but also [[formal power series]], it is convenient to consider [[completion of a ring|completions]] and [[topological rings]]. Order $n$ [[nilpotent elements]] in an ordinary ring compare to completions as truncations of general power series and geometrically represent certain $n$-th [[infinitesimal space|infinitesimal]] neighborhood. Completions represent certain pro-objects in the category of rings. Adic completion corresponds to have all infinitesimal neighborhoods at once.

A _formal spectrum_ is a generalization of prime spectrum to adic noetherian rings, therefore containing information on all infinitesimal neighborhoods, corresponding to the ideal of completion.

## Definition

> see for instance ([Strickland 00, def. 4.13](#Strickland00))

Assume $R$ is a  [[commutative ring]] and $I \subset R$ is an ideal, such that its powers make a fundamental system of neighborhoods of zero of a complete Hausdorff topology (we say that $R$ is an separated complete ring in the $I$-[[adic topology]]).

The **formal spectrum** $Spf R$ of $(R,I)$ is the [[inductive limit]] of the [[prime spectrum of a commutative ring|prime spectra]]

$$
  Spf(R) :=colim_n Spec (R/I^n)
  \,.
$$

where the connecting morphisms are the closed nilpotent immersions $Spec(R/I^n)\hookrightarrow Spec(R/I^{n+1})$ of [[affine schemes]] and the [[colimit]] is taken in the [[category]] of [[topologically ringed space|topologically ringed spaces]].

Regarding that all affine schemes $X_n := Spec(R/I^n)$ for $n\geq 1$ have the same underlying topological space $\chi$ because nilpotents in $I^n$ do not affect the underlying reduced scheme, so does $Spf R = (\chi,\mathcal{O}_\chi)$. With our assumptions on $I$-[[adic topology]], in fact $\chi$ contains all closed points of $Spec R$ and any open subset of $Spec R$ containing $\chi$ is the whole of $Spec R$.  The structure sheaf $\mathcal{O}_\chi$ has the ring of sections $\mathcal{O}_\chi(U) = lim_n \mathcal{O}_{X_n}(U)$ where the limit is taken in the category of topological rings, and $\mathcal{O}_{X_n}(U)$ have the discrete topology. For example, the ring of global sections is $\mathcal{O}_\chi(\chi) = \hat R$.

A formal spectrum is an example of a [[formal scheme]]. Formal schemes in general form certain subcategory of the category of ind-schemes.

The formal spectrum of a separated complete topological $I$-adic ring $R$ depends just on the underlying topology on $R$ and not on a choice of the ideal $I$ generating this topology.

## Related concepts

* [[formal disk]]

## References

* standard references are [[EGA]], Hartshorne

* [[Luc Illusie]], _Grothendieck existence theorem in formal geometry_, in [[FGA explained]] (draft: [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes))

Discussion more from a [[topos theory]] perspective is in section 4.2 of

* {#Strickland00} [[Neil Strickland]], _Formal schemes and formal groups_, [math.AT/0011121](http://arxiv.org/abs/math.AT/0011121)

For the case of [[formal group laws]] see also

* _[[A Survey of Elliptic Cohomology - formal groups and cohomology]]_

[[!redirects formal spectra]]


formal topology

>This entry is about a generalized notion of [[topology]]. For the notion of _formal space_ in the sense of [[rational homotopy theory]], see [[formal dg-algebra]].

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# Formal topology
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## Idea

Formal topology is a programme for doing [[topology]] in a [[finite mathematics|finite]], [[predicative mathematics|predicative]], and [[constructive mathematics|constructive]] fashion.

It is a kind of [[pointless topology]]; in the context of [[classical mathematics]], it reproduces the theory of [[locales]] rather than [[topological spaces]] (although of course one can recover topological spaces from locales).

The basic definitions can be motivated by an attempt to study locales entirely through the [[posites]] that generate them.  However, in order to recover all basic topological notions (particularly those associated with *closed* rather than *open* features) predicatively, we need to add a 'positivity' relation to the 'coverage' relation of sites.

## Definitions

A __formal topology__ or __formal space__ is a [[set]] $S$ together with

*  an [[element]] $\top$ of $S$,
*  a [[binary operation]] $\cap$ on $S$,
*  a [[binary relation]] $\lhd$ between elements of $S$ and [[subsets]] of $S$, and
*  a [[unary relation]] $\Diamond$ on $S$,

such that

1.  $a = b$ whenever $a \lhd \{b\}$ and $b \lhd \{a\}$,
2.  $a \lhd U$ whenever $a \in U$,
3.  $a \lhd V$ whenever $a \lhd U$ and $x \lhd V$ for all $x \in U$,
4.  $a \cap b \lhd U$ whenever $a \lhd U$ or $b \lhd U$,
5.  $a \lhd \{ x \cap y \;|\; x \in U,\; y \in V \}$ whenever $a \lhd U$ and $a \lhd V$,
6.  $a \lhd \{\top\}$,
7.  $\Diamond x$ for some $x \in U$ whenever $\Diamond a$ and $a \lhd U$, and
8.  $a \lhd U$ whenever $a \lhd U$ if $\Diamond a$,

for all $a$, $b$, $U$, and $V$.

We interpret the elements of $S$ as __basic opens__ in the formal space.  We call $\top$ the __[[improper subset|entire space]]__ and $a \cap b$ the __[[intersection]]__ of $a$ and $b$.  We say that $a$ is __covered__ by $U$ or that $U$ is a __[[cover]]__ of $a$ if $a \lhd U$.  We say that $a$ is __positive__ or __[[inhabited subset|inhabited]]__ if $\Diamond a$.  (For a [[topological space]] equipped with a strict [[topological base]] $S$, taking these intepretations literally does in fact define a formal space; see the Examples.)

Some immediate points to notice:

*  If we drop (1), then the hypothesis of (1) defines an [[equivalence relation]] on $S$ which is a [[congruence]] for $\top$, $\cap$, $\lhd$, and $\Diamond$, so that we may simply pass to the [[quotient set]].  In appropriate foundations, we can even allow $S$ to be a [[preset]] originally, then use (1) as a definition of equality.
*  We can prove that $(S,\cap,\top)$ is a bounded [[semilattice]]; if (as the notation suggests) we interpret this as a [[meet]]-semilattice, then $a \leq b$ if and only if $a \lhd \{b\}$.  Conversely, we could require that $(S,\cap,\top)$ be a semilattice originally, then let (1) say that $a \leq b$ whenever $a \lhd \{b\}$.
*  We can prove that $\Diamond a$ holds iff every cover of $a$ is inhabited and that $\Diamond a$ fails iff $a \lhd \empty$.  Accordingly, this predicate is uniquely definable (in two equivalent ways, one impredicative and one nonconstructive) in a classical treatment; only in a treatment that is both predicative and constructive do we need to include it in the axioms.  See [[positivity predicate]].

## Examples

Let $X$ be a [[topological space]], and let $S$ be the collection of [[open subsets]] of $X$.  Let $\top$ be $X$ itself, and let $a \cap b$ be the literal intersection of $a$ and $b$ for $a, b \in S$.  Let $a \lhd U$ if and only $U$ is literally an [[open cover]] of $a$, and let $\Diamond a$ if and only if $a$ is literally inhabited.  Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

The above example is impredicative (since the collection of open subsets is generally large), but now let $S$ be a [[base for the topology]] of $X$ which is strict in the sense that it is closed under finitary [[intersection]].  Let the other definitions be as before.  Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

More generally, let $B$ be a [[subbase]] for the topology of $X$, and let $S$ be the [[free monoid]] on $B$, that is the set of [[finite lists]] of elements of $B$ (so this example is not strictly finitist), modulo the [[equivalence relation]] by which two lists are identified if their [[intersections]] are equal.  Let $\top$ be the [[empty list]], let $a \cap b$ be the [[concatenation]] of $a$ and $b$, let $a \lhd U$ if the intersection of $a$ is contained in the [[union]] of the intersections of the elements of $U$, and let $\Diamond a$ if the intersection of $a$ is inhabited.  Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

Let $X$ be an [[accessible locale]] generated by a [[posite]] whose underlying [[poset]] $S$ is a (meet)-[[semilattice]].  Let $\top$ and $\cap$ be as in the semilattice structure on $S$, and let $a \lhd U$ if $U$ contains a basic cover (in the posite structure on $S$) of $a$.  Then we get a formal topology, defining $\Diamond$ in the unique way.

The last example is not predicative, and this is in part *why* one studies formal topologies instead of sites, if one wishes to be strictly predicative.  (It still needs to be motivated that we want $\Diamond$ at all.)

## In dependent type theory

### Impredicative definition

In [[dependent type theory]], let $(\Omega, \mathrm{El}_\Omega)$ be the [[type of all propositions]], and let the relation $a \in_S A$ for $a:S$ and $A:S \to \Omega$ as

$$a \in_S A \coloneqq \mathrm{El}_\Omega(A(a))$$

The singleton subtype function is a function $\{-\}:S \to (S \to \Omega)$, such that for all elements $a:S$, $p(a):a \in_S \{a\}$, and for all functions $R:S \to (S \to \Omega)$ such that for all elements $a:S$, $p_R(a):a \in_S R(a)$, the type of functions $(b \in_S \{a\}) \to (b \in_S R(a))$ is contractible for all $a:A$ and $b:A$.

A **formal topology** or **formal space** is a [[type]] $S$ together with

*  an [[element]] $\top:S$,
*  a [[binary operation]] $\cap:S \times S \to S$,
*  a [[binary relation]] $a \lhd A$ between elements of $S$, $a:S$, and [[subtypes]] of $S$, $A:S \to \Omega$, and
*  a [[unary relation]] $\Diamond a$ on $S$,

such that

1. For all elements $a:S$ and $b:S$, the canonical function
$$\mathrm{idToFormalTop}(a, b):(a =_S b) \to ((a \lhd \{b\}) \times (b \lhd \{a\}))$$
is an [[equivalence of types]].
$$\prod_{a:S} \prod_{b:S} \mathrm{isEquiv}(\mathrm{idToFormalTop}(a, b))$$
2. For all elements $a:S$ and subsets $A:S \to \Omega$, if $a \in_S A$, then $a \lhd A$
$$\prod_{a:S} \prod_{A:S \to \Omega} (a \in_S A) \to (a \lhd A)$$
3.  For all elements $a:S$ and subsets $A:S \to \Omega$ and $B:S \to \Omega$, if $a \lhd A$ and for all elements $x:S$, $x \lhd B$ and $x \in_S A$, then $a \lhd B$
$$\prod_{a:S} \prod_{A:S \to \Omega} \prod_{B:S \to \Omega} \left((a \lhd A) \times \prod_{x:S} (x \in_S A) \times (x \lhd B)\right) \to (a \lhd B)$$
4.  For all elements $a:S$ and $b:S$ and subsets $A:S \to \Omega$, if $a \lhd A$ or $b \lhd A$, then $a \cap b \lhd A$.
$$\prod_{a:S} \prod_{b:S} \prod_{A:S \to \Omega} ((a \lhd A) \times (b \lhd A)) \to (a \cap b \lhd A)$$
5.  For all elements $a:S$ and subsets $A:S \to \Omega$ and $B:S \to \Omega$, if $a \lhd A$ and $a \lhd B$, then
$$a \lhd \sum_{z:S} \prod_{x:S} \prod_{y:S} (x \in_S A) \times (y \in_S B) \times (z =_S x \cap u)$$
$$\prod_{a:S} \prod_{A:S \to \Omega} \prod_{B:S \to \Omega} (a \lhd A) \times (a \lhd B) \to \left(a \lhd \sum_{z:S} \prod_{x:S} \prod_{y:S} (x \in_S A) \times (y \in_S B) \times (z =_S x \cap y)\right)$$
6.  For all elements $a:S$, $a \lhd \{\top\}$,
$$\prod_{a:S} a \lhd \{\top\}$$
7.  For all elements $a:S$ and subsets $A:S \to \Omega$, if $\Diamond a$ and $a \lhd A$, there exists an element $x:S$ such that $x \in_S A$ and $\Diamond x$.
$$\prod_{a:S} \prod_{A:S \to \Omega} (\Diamond a) \times (a \lhd A) \to \sum_{x:S} (x \in_S A) \times (\Diamond x)$$
8.  For all elements $a:S$ and subsets $A:S \to \Omega$, if $\Diamond a$ implies $a \lhd A$, then $a \lhd A$.
$$\prod_{a:S} \prod_{A:S \to \Omega} (\Diamond a \to a \lhd A) \to (a \lhd A)$$

### Predicative definition

Suppose that the [[dependent type theory]] does not have a [[type of all propositions]], nor any [[type universes]]. This means that subtypes of a type $S$ do not form a type, which is necessary for defining the cover relation, which is inherently a relation between elements of $S$ and subtypes of $S$. However, one can resolve this problem by using a definition of formal topology which does not require a cover relation in its definition, and then [[inductively define]] the cover relation on $S$ as a [[higher inductive type|higher]] [[inductive family]] of [[inductive cover relations]] for every single type in $A$.

Ayberk Tosun in [Tosun 2020](#Tosun20) defined a **formal topology** as a [[poset]] $(A, \leq)$ with families of types $x:A \vdash B(x)$, $x:A, y:B(x) \vdash C(x, y)$ and a dependent function

$$d:\prod_{x:A} \prod_{y:B(x)} C(x, y) \to A$$

such that

* for all $x:A$, $y:B(x)$, and $z:C(x, y)$, $d(x, y, z) \leq x$

$$\prod_{x:A} \prod_{y:B(x)} \prod_{z:C(x, y)} d(x, y, z) \leq x$$

* for all $x:A$ and $x':A$, $x' \leq x$ implies that for all $y:B(x)$ one can construct $y':B(x')$ such that for all $z:C(x, y)$, one can construct $z':C(x', y')$ such that $d(x', y', z') \leq d(x, y, z)$.

$$\prod_{x:A} \prod_{x:A'} (a \leq a') \to \prod_{y:B(x)}\sum_{y':B(x')} \prod_{z:C(x, y)} \sum_{z':C(x', y')} d(x', y', z') \leq d(x, y, z)$$

If one has a [[type universe]] $U$, then the locally $U$-small inductive cover relation is given by

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:\exists x:I.e(x) =_A a}{\Gamma \vdash \mathrm{dir}_U(a, I, e, p):a \lhd_U (I, e)}$$

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash b:B(a) \quad \Gamma \vdash f:\prod_{c:C(a, b)} d(a, b, c) \lhd_U (I, e)}{\Gamma \vdash \mathrm{branch}_U(a, I, e, b, f):a \lhd_U (I, e)}$$

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:a \lhd_U (I, e) \quad \Gamma \vdash q:a \lhd_U (I, e)}{\Gamma \vdash \mathrm{squash}_U(a, I, e, p, q):p =_{a \lhd_U (I, e)} q}$$

If one doesn't have type universes, then one could use the [[stack semantics]] instead. Given a type $I$, the cover relation for $I$ is a higher inductive type family $a \lhd_I e$ between elements $a:A$ and [[embeddings of types]] $e:I \hookrightarrow A$ generated by the constructors

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:\exists x:I.e(x) =_A a}{\Gamma \vdash \mathrm{dir}_I(a, e, p):a \lhd_I e}$$

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash b:B(a) \quad \Gamma \vdash f:\prod_{c:C(a, b)} d(a, b, c) \lhd_I e}{\Gamma \vdash \mathrm{branch}_I(a, e, b, f):a \lhd_I e}$$

$$\frac{\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:a \lhd_I e \quad \Gamma \vdash q:a \lhd_I e}{\Gamma \vdash \mathrm{squash}_I(a, e, p, q):p =_{a \lhd_I e} q}$$

Families of types $x:A \vdash B(x)$ are equivalently functions $B \to A$, so one can express the above definition in terms of single types. A **formal topology** is a poset $(A, \leq)$ with a set $B$, a function $f:B \to A$, a set $C$ with a function $g:C \to B$ and a function $d:C \to A$, such that

* for all $z:C$, $d(z) \leq g(f(z))$

$$\prod_{z:C} d(z) \leq g(f(z))$$

* for all $x:A$ and $x':A$, if $x \leq x'$, then for all $y:B$, $f(y) =_A x$ implies that one can construct a $y':B$ such that $f(y') =_A x'$ implies that for all $z:C$, $g(z) =_B y$ implies that one can construct a $z':C$ such that $g(z') =_B y'$ implies $d(z') \leq d(z)$.

$$\prod_{x:A} \prod_{x:A'} (a \leq a') \to \prod_{y:B} (f(y) =_A x) \to \sum_{y':B} (f(y') =_A x') \to \prod_{z:C} (g(z) =_B y) \to \sum_{z':C} (g(z') =_B y') \to (d(z') \leq d(z))$$

If one has a [[type universe]] $U$, then the locally $U$-small inductive cover relation is given by

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:\exists x:I.e(x) =_A a
\end{array}
}{\Gamma \vdash \mathrm{dir}_U(a, I, e, p):a \lhd_U (I, e)}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash b:B \quad \Gamma \vdash p:f(b) =_A a \quad \Gamma \vdash h:\prod_{c:C} (g(c) =_B b) \to (d(c) \lhd_U (I, e))
\end{array}
}{\Gamma \vdash \mathrm{branch}_U(a, I, e, b, p, h):a \lhd_U (I, e)}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I:U \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:a \lhd_U (I, e) \quad \Gamma \vdash q:a \lhd_U (I, e)
\end{array}
}{\Gamma \vdash \mathrm{squash}_U(a, I, e, p, q):p =_{a \lhd_U (I, e)} q}$$

If one doesn't have type universes, then one could use the [[stack semantics]] instead. Given a type $I$, the cover relation for $I$ is a higher inductive type family $a \lhd_I e$ between elements $a:A$ and [[embeddings of types]] $e:I \hookrightarrow A$ generated by the constructors

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:\exists x:I.e(x) =_A a
\end{array}
}{\Gamma \vdash \mathrm{dir}_I(a, e, p):a \lhd_I e}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash b:B \quad \Gamma \vdash p:f(b) =_A a \quad \Gamma \vdash h:\prod_{c:C} (g(c) =_B b) \to (d(c) \lhd_I e)
\end{array}
}{\Gamma \vdash \mathrm{branch}_I(a, e, b, p, h):a \lhd_I e}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:B \to A \quad \Gamma \vdash g:C \to B \quad \Gamma \vdash d:C \to A \\
\Gamma \vdash a:A \quad \Gamma \vdash I \; \mathrm{type} \quad \Gamma \vdash e:I \hookrightarrow A \quad \Gamma \vdash p:a \lhd_I e \quad \Gamma \vdash q:a \lhd_I e
\end{array}
}{\Gamma \vdash \mathrm{squash}_I(a, e, p, q):p =_{a \lhd_I e} q}$$

## Related concepts

* [[inductive cover]]

## References

*  [[Mike Fourman]] and Grayson (1982); _Formal Spaces_.  This is the original development, intended as an application of locale theory to logic.

*  [[Giovanni Sambin]] (1987); _Intuitionistic formal spaces_; [pdf](http://www.math.unipd.it/~sambin/txt/ifs87-97.pdf).
   *  This is the probably the main reference on the subject.
   *  Warning: you can ignore the material about [[foundations]] and [[type theory]], but if you do read any of it, the term 'category' means (roughly) [[proper class]]; this was common in type theory in the 1980s.

*  [[Giovanni Sambin]] (2001); _Some points in formal topology_; [pdf](http://www.math.unipd.it/~sambin/txt/SP.pdf).  This has newer results, alternative formulations, etc.

* {#Palmgren05} [[Erik Palmgren]], _From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory_, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 ([pdf](http://www2.math.uu.se/~palmgren/homotopy_rev2.pdf))

* {#Tosun20} Ayberk Tosun, _Formal Topology in Univalent Foundations_, ([pdf](https://odr.chalmers.se/handle/20.500.12380/301098), [slides](https://www.cs.bham.ac.uk/~axt978/talks/lab-lunch-formal-topology.pdf))

[[!redirects formal topology]]
[[!redirects formal topologies]]

[[!redirects formal space]]
[[!redirects formal spaces]]
[[!redirects formal topological space]]
[[!redirects formal topological spaces]]


FormalCartSp

[[!redirects ThCartSp]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Cohesive toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[site]] of [[formal smooth manifold|formal]] [[Cartesian spaces]].

## Definition

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

Let $FormalCartSp$ (or $FormalCartSp$) be the [[full subcategory]] of the category of [[smooth loci]] on those of the form

$$
  \mathbb{R}^n \times \ell W
  \,,
$$

consisting of a [[product]] of a [[Cartesian space]] with an [[infinitesimally thickened point]], i.e. a formal dual of a _Weil algebra_ .

Dually, the [[opposite category]] is the [[full subcategory]] $FormalCartSp^{op} \hookrightarrow SmoothAlg$ of [[smooth algebras]] on those of the form

$$
  C^\infty( \mathbb{R}^k \times \ell W)
  =
  C^\infty(\mathbb{R}^k) \otimes W
  \,.
$$

=--

This appears for instance in [Kock Reyes (1)](#KockReyes).

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

Define a structure of a [[site]] on [[FormalCartSp]] by declaring a [[covering]] family to be a family of the form

$$
  \{
    U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W
  \}
$$

where $\{U_i \stackrel{p_i}{\to} U\}$ is an [[open cover]] of the [[Cartesian space]] $U$ by Cartesian spaces $U_i$.

=--

This appears as [Kock (5.1)](#Kock).

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

The **[[Cahiers topos]]** $\mathcal{CT}$ is the [[category of sheaves]] on this site:

$$
  \mathcal{CT} := Sh(FormalCartSp)
  \,.
$$

=--

This site of definition appears in [Kock, Reyes](#KockReyes). The original definition is due to [Dubuc](#Dubuc)

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

The [[(∞,1)-topos]] of [[(∞,1)-sheaves]] over $FormalCartSp$ is that of _[[formal smooth ∞-groupoids]]_

$$
  FormSmooth\infty Grpd \coloneqq Sh_\infty(FormalCartSp)
  \,.
$$

=--

$\,$

## Related concepts

[[!include geometries of physics -- table]]

$\,$

## References

The Cahiers topos was introduced in

* {#Dubuc} [[Eduardo Dubuc]], _Sur les modèles de la géométrie différentielle synthétique_ [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques, 20 no. 3 (1979), p. 231-279  ([numdam](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)).

and got its name from this journal publication. The definition appears in theorem 4.10 there, which asserts that it is a well-adapted model for [[synthetic differential geometry]].

A review discussion is in section 5 of

* {#Kock} [[Anders Kock]], _Convenient vector spaces embed into the Cahiers topos_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17  ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0))

and with a corrected definition of the site of definition in

* {#KockReyes} [[Anders Kock]], [[Gonzalo Reyes]], _Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 ([numdam](http://www.numdam.org/item?id=CTGDC_1987__28_2_99_0))

[[!redirects formal Cartesian space]]
[[!redirects formal Cartesian spaces]]


formalism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

**Formalism** is one of the classical schools of the [[philosophy of mathematics]].

## Related entries

* [[platonism]]
* [[intuitionism]]
* [[structuralism]]
* [[David Hilbert]]
* [[free topos]]

[[!redirects formalist]]


formality

A rational homotopy type is "formal'' if its [[minimal model]] has differential zero. A [[dg-algebra]], or, more generally, an [[A-infinity algebra]] is formal if it is quasiisomorphic to its own (co)homology.

The formality for simply-connected compact [[Kähler manifold]]s implies that their real homotopy type is determined by their [[de Rham cohomology]] ring as shown using Hodge theory in the seminal work

* [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent. Math. __29__ (1975), no. 3, 245–274, [MR382702](http://www.ams.org/mathscinet-getitem?mr=382702) [doi](http://dx.doi.org/10.1007/BF01389853)

This is later improved to rational homotopy type in

* [[Dennis Sullivan]],  _Infinitesimal computations in topology,_  Publ. math. de l' IHÉS. __47__ (1977), p. 269-331, [numdam](http://www.numdam.org/item?id=PMIHES_1977__47__269_0),

A major formality result is the [[Kontsevich formality]] (see references there) and the Tamarkin formality of the [[little disks operad]] which implies it. For both see [[Kontsevich formality]].

There are many recent cases of formality in geometry, topology and algebra

* Vasiliy Dolgushev, Dmitry Tamarkin, Boris Tsygan, _Formality of the homotopy calculus algebra of Hochschild (co)chains_, [arxiv/0807.5117](http://arxiv.org/abs/0807.5117)

category: algebra


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[partial differential equation]] is said to be _formally integrable_ (e.g. [Goldschmidt 67, def. 7.2](#Goldschmidt67)) if it is [[integrable PDE|integrable]] at least over [[infinitesimal neighbourhoods]] (aka "formal neighbourhoods", whence the name).

## Related concepts

* [[integrable differential equation]]

## References

* {#Goldschmidt67} [[Hubert Goldschmidt]], _Integrability criteria for systems of nonlinear partial differential equations_, Journal of Differential Geometry 1 (1967) 269–307 ([Euclid](https://projecteuclid.org/euclid.jdg/1214428094))

* [[Maciej Zworski]], _Numerical linear algebra and solvability of partial differential equations_, Communications in Mathematical Physics 229 (2002) 293–307

* Batu Güneysu, [[Markus Pflaum]], _The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE's_ ([arXiv:1308.1005](https://arxiv.org/abs/1308.1005))

A [[synthetic mathematics|synthetic]] discussion in terms of [[differential cohesion]] is given in

* [[Igor Khavkine]], [[Urs Schreiber]], section 3.6 of _[[schreiber:Synthetic geometry of differential equations|Synthetic geometry of differential equations -- Part I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238), [v2 pdf](https://ncatlab.org/schreiber/files/SyntheticPDEs170610b.pdf), [talk slides pdf](https://ncatlab.org/schreiber/files/KhavkineTromso17.pdf), [talk video](https://www.youtube.com/watch?v=yjVXmS-sZ0w))

[[!redirects formally integrable differential equations]]

[[!redirects formally integrable partial differential equation]]
[[!redirects formally integrable partial differential equations]]

[[!redirects formally integrable PDE]]
[[!redirects formally integrable PDEs]]

[[!redirects formal integration of differential equations]]
[[!redirects formal integration of partial differential equations]]
[[!redirects formal integration of PDEs]]

[[!redirects formal integrability]]


formally real algebra

# Formally real rings
* table of contents
{: toc}

## Idea

When working with algebras (possibly [[associative algebra|associative]], possibly [[nonassociative algebra|nonassociative]]) over the [[real numbers]], we often want to express the fact that $-1$ has no [[square root]] in the algebra.  However, this by itself is not enough, since the algebra may lack square roots for other reasons, so we say that $-1$ is not a _sum_ of squares.  Even this may be too weak if the algebra has too little division, because we also want to ensure that other negative squares are not sums of squares.  So the really general statement is that $0$ is not a sum of squares, except as a sum of squares of $0$ itself.

At this point, we notice that the $\mathbb{R}$-linear structure is irrelevant, and the concept applies to more general [[rings]] (possibly [[nonassociative rings|nonassociative]]).  Even the existence of opposites is not needed, and the concept applies to (possibly nonassociative) [[rigs]].  Really, we just need multiplication, addition, and zero.

## Definitions

Let $A$ be [[ring]], or more generally a [[magma]] in the category of [[abelian monoids]].  Then $A$ is __formally real__ if, whenever
$$ \sum_i x_i^2 = 0 $$
(for a finite sum), each $x_i = 0$.

Given an [[involution]], we have a generalization: a $*$-[[star-ring|ring]] is __formally complex__ if, whenever
$$ \sum_i x_i^* x_i = 0 $$
(for a finite sum), each $x_i = 0$.  (Then a ring is formally real iff it is formally complex when equipped with the trivial involution.)

## Examples

Of course, the [[real numbers]] themselves form a formally real ring; the [[complex numbers]] do not, since
$$ \mathrm{i}^2 + 1^2 = 0 .$$
However, the complex numbers are formally complex.  (These examples are the source of the names.)

Actually, all of the [[Cayley–Dickson algebra]]s are formally complex.  (This is because $x^* x$ is always a real number and is $0$ only when $x$ is.)

The [[trivial ring]] is formally real.  Every other formally real (or formally complex) ring is [[infinite set|infinite]] (because $1 = 1^2$, $2 = 1^2 + 1^2$, $3 = 1^2 + 1^2 + 1^2$, etc are all distinct).  Even without an identity (and even in the nonassociative case), if the ring is formally real and nontrivial, then it is infinite.  (If $x \ne 0$, consider $x^2$, $x^2 + x^2$, $x^2 + x^2 + x^2$, etc.)

There are finite formally real *rigs*, however, such as the [[boolean rig]] $\{0, 1\}$.  Indeed, any [[distributive lattice]] (with a [[bottom element]]), viewed as a rig, is formally real.

Among [[Jordan algebras]], the formally real ones are especially important; it is these that (over the real numbers, in finite dimensions) have a nice [[classification theorem]].

## Properties

The [[trivial algebra]] is the only formally real algebra over any ring that is *not* formally real; more generally, if $A \to B$ is a [[monomorphism]] of rings (or of rigs, etc), then $A$ must be formally real if $B$ is.  Similarly, if $A \to B$ is a monomorphism of $*$-rings, then $A$ must be formally complex if $B$ is.

Formally real [[fields]] (like that of the real numbers) are particularly interesting; see [[formally real field]].

Every formally real (or formally complex) ring has a natural [[partial ordering]]: $a \leq b$ iff $b - a$ can be written as a sum of squares (or a sum of terms of the form $x^* x$).

[[!redirects formally real]]

[[!redirects formally real ring]]
[[!redirects formally real rings]]

[[!redirects formally real rig]]
[[!redirects formally real rig]]
[[!redirects formally real semiring]]
[[!redirects formally real semirings]]

[[!redirects formally real algebra]]
[[!redirects formally real algebras]]

[[!redirects formally complex]]
[[!redirects formally complex ring]]
[[!redirects formally complex rings]]
[[!redirects formally complex rig]]
[[!redirects formally complex rig]]
[[!redirects formally complex semiring]]
[[!redirects formally complex semirings]]
[[!redirects formally complex algebra]]
[[!redirects formally complex algebras]]


formally real field

# Formally real fields
* table of contents
{: toc}

## Definition

A [[field]] $k$ is *formally real* if one of the following equivalent conditions is satisfied:

* $k$ is a [[formally real ring]].

* $-1$ is not a sum of squares in $k$ (a special case of the previous condition).

* There exists an [[linear order]] on $k$ which makes $k$ into an [[ordered field]].  (This requires [[Zorn's Lemma]].)

* The [[Witt group]] of $k$ is not torsion.

Due to the last property, formally real fields play a special rôle in the theory of [[Witt group|Witt groups]] and in related fields of study such as [[motivic homotopy theory]].

A formally real field which is the only formally real [[algebraic extension]] of itself is a [[real closed field]]; such a field admits a unique ordering (*not* requiring Zorn\'s Lemma).

## Examples

* The field of [[real numbers]] is formally real, and even a [[real closed field]].

* The field $\mathbb{Q}$ of [[rational numbers]] is formally real but not real closed.

* [[finite field|Finite fields]] (or more generally fields of nonzero [[characteristic]]) and [[algebraically closed fields]] (such as the fields of [[complex numbers]] and [[algebraic numbers]]) are never formally real.

category: algebra

[[!redirects formally real field]]
[[!redirects formally real fields]]


formally smooth morphism

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

#Contents#
* table of contents
{:toc}

## Idea

A [[space]] $X$ is called **formally smooth** if every [[morphism]] $Y \to X$ into it has all possible [[infinitesimal object|infinitesimal]] extensions.

(If there is at most one extension per infinitesimal extension of $Y$ with no guarantee of existence it is called a [[formally unramified morphism]]. If the thickenings exist uniquely, it is called a [[formally etale morphism]]).

Traditionally this has considered in the context of [[geometry]] over formal duals of [[ring]]s and [[associative algebra]]s. This we discuss in the section ([Concrete notion](#ConcreteNotion)). But generally the notion makes sense in any context of infinitesimal cohesion. This we discuss in the section
[General abstract notion](#GeneralAbstractNotion).

## General abstract notion
 {#GeneralAbstractNotion}

### Definition

Let

$$
  \mathbf{H}
   \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}}
  \mathbf{H}_{th}
$$

be an [[adjoint triple]] of [[functors]] with $u^*$ a [[full and faithful functor]] that preserves the [[terminal object]].

We may think of this as exhibiting infinitesimal cohesion (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as [[cohesive topos|cohesive space]]s and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with [[infinitesimal object|infinitesimal extension]].

As a class of examples that is useful to keep in mind consider a [[Q-category]] $(cod \dashv \epsilon \dashv dom) : \bar A \to A$  of infinitesimal thickening of rings and let

$$
  ((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) :=
  ([dom,Set] \dashv [\epsilon, Set] \dashv [codom,Set] : [\bar A, Set] \to [A,Set])
$$

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical [[natural transformation]]

$$
  u^* \to u^!
  \,.
$$

Details of this are in the section Adjoint quadruples at [[cohesive topos]].

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

\[
  \label{MorphismIntoPullback}
  u^* X \to u^* Y \prod_{u^! Y} u^! X
  \,.
\]

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

A morphism $f : X \to Y$ in $\mathbf{H}$ is called **formally smooth** if (eq:MorphismIntoPullback) is an [[effective epimorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.1, prop. 5.3.1.1](#KontsevichRosenbergSpaces)).

The dual notion, where the above morphism is a [[monomorphism]] is that of [[formally unramified morphism]]. If both conditions hold, hence if the above morphism is an [[isomorphism]], one speaks of a [[formally  étale morphism]].

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

An object $X \in \mathbf{H}$ is called **formally smooth** if the morphism $X \to *$ to the [[terminal object]] is formally smooth.

=--

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

The object $X$ is formally smooth precisely if

$$
  u^* X \to u^! X
$$

is an [[effective epimorphism]].

=--

This appears as ([KontsevichRosenberg, def. 5.3.2](#KontsevichRosenbergSpaces)).

### Properties

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

Formally smooth morphisms are closed under composition.

=--

This appears as ([KontsevichRosenberg, prop. 5.4](#KontsevichRosenbergSpaces)).

## Concrete notion
 {#ConcreteNotion}

### Over commutative rings
 {#OverCommutativeRings}

Let $k$ be a [[field]] and let $CAlk_k$ be the [[category]] of commutative
[[associative algebra]]s over $k$. Write

$$
  \mathbf{H} = [CAlg_k, Set]
$$

for the [[presheaf topos]] over the [[opposite category]] $CAlg_k^{op}$. This is the context in which [[scheme]]s and [[algebraic space]]s over $k$ live.

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

A morphism $f :X\to Y$ in $\mathbf{H} = [CAlg_k, Set]$ is __formally smooth__ if   it satisfies the _infinitesimal lifting property_: for every algebra $A$ and nilpotent [[ideal]] $I\subset A$ and morphism $Spec(A)\to Y$ the induced map

$$ Hom_Y(Spec(A), X)\to Hom_Y(Spec(A/I),X)$$

is surjective.

=--

This is due to ([[EGAIV]]${}_4$ 17.1.1)

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

An object $X \in [CAlg_k, Set]$ is formally smooth in the concrete sense of def. \ref{CFormalSmoothness} precisely if it is so in the abstract sense of def. \ref{AbstractFormallySmoothObjectDirect}.

=--

This appears as ([KontsevichRosenbergSpaces, 4.1](#KontsevichRosenbergSpaces)).

#### Smoothness versus formal smoothness

For a morphism $f:X\to Y$ of schemes, and $x$ a point of $X$, the following are equivalent

(i) $f$ is a [[smooth morphism of schemes]] at $x$

(ii) $f$ is locally of finite presentation at $x$ and there is an open neighborhood $U\subset X$ of $x$ such that $f|_U: U\to Y$ is formally smooth

(iii) $f$ is flat at $x$, locally of finite presentation at $x$ and the sheaf of [[Kähler differential]]s $\Omega_{X/Y}$ is locally free in a neighborhood of $x$

The relative dimension of $f$ at $x$ will equal the rank of the module of Kähler differentials.

This is ([[EGAIV]]${}_4$ 17.5.2 and 17.15.15)

#### Formally smooth scheme

A [[scheme]] $S$, i.e. a scheme over the ground ring $k$, is a [[formally smooth scheme]] if the corresponding morphism $S \to Spec(k)$ is a formally smooth morphism.

There is also an interpretation of formal smoothness via the formalism of [[Q-categories]].

### Over noncommutative algebras

Let $k$ be a [[field]] and let $Alg_k$ be the category of [[associative algebra]]s over $k$ (not necessarily commutative). Let

$$
  Alg_k^{inf} : \bar A \to Alg_k
$$

be the [[Q-category]] of infinitesimal thickenings of $k$-algebras (whose objects are surjective $k$-algebra morphisms with nilpotent kernel).
Notice that the [[presheaf topos]]

$$
  \mathbf{H} := [Alg_k, Set]
$$

is the context in which [[noncommutative scheme]]s live. Let $\mathbf{H}_{th} \to \mathbf{Q}$ be the copresheaf Q-category over $Alg_k^{inf}$.

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

Let $f : R \to S$ be a morphism in $Alg_k$ such that $R$ is a [[separable algebra]]. Write $Spec f : Spec S \to Spec R$ for the corresponding morphism in $\mathbf{H} = [Alg_k, Set]$.

This $Spec f$ is formally smooth in the sense of def. \ref{AbstractFormallySmoothMorphism} precisely if the $S \otimes_k S^{op}$-[[module]]

$$
  \Omega^1_{S|R} := ker ( R \otimes_k R \stackrel{mult}{\to} R \stackrel{f}{\to} S)
$$

is a [[projective object]] in $S \otimes_k S^{op}$[[Mod]].

=--

In particular, setting $R = k$ we have that an object of the form $Spec S$ is formally smooth according to def. \ref{AbstractFormallySmoothObject} precisely if $\Omega^1(S|k)$ is projective. This is what in ([CuntzQuillen](#CuntzQuillen)) is called the condition for a [[quasi-free algebra]].

## Related concepts

**formally smooth morphism** and [[formally unramified morphism]] $\Rightarrow$ [[formally étale morphism]].

[[!include cohesion - table]]

## References

The definition over commutative rings is in

* [[EGAIV]]${}_4$, Publ. IHÉS __32__ (1967), p. 5-361, [numdam](http://www.numdam.org/item?id=PMIHES_1967__32__5_0)

The definition over noncommutative algebras is in

* [[J. Cuntz]], [[D. Quillen]], _Algebra extensions and nonsingularity_, J. Amer. Math. Soc. __8__ (1995), 251–289.
 {#CuntzQuillen}

The general abstract definition and its relation to the standard definitions is in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative spaces_, preprint MPI-2004-35 ([ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenbergSpaces}

See also

* [MO:is-formal-smoothness-a-local-property](http://mathoverflow.net/questions/22393/is-formal-smoothness-a-local-property)

* A. Ardizzoni, _Separable functors and [[formal smoothness]]_,  J. K-Theory  1  (2008),  no. 3, 535--582, [math.QA/0407095](http://arxiv.org/abs/math.QA/0407095), [doi](http://dx.doi.org/10.1017/is007011015jkt012), [MR2009k:16069](http://www.ams.org/mathscinet-getitem?mr=2009k:16069)

* T. Brzeziński, _Notes on formal smoothness_, *in*: Modules and Comodules (series _Trends in Mathematics_). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 ([doi](http://dx.doi.org/10.1007/978-3-7643-8742-6), [arXiv:0710.5527](http://arxiv.org/abs/0710.5527))

* Guillermo Cortiñas, _The structure of smooth algebras in Kapranov's framework for noncommutative geometry_, J. of Algebra __281__ (2004) 679-694, [math.RA/0002177](http://arxiv.org/abs/math/0002177)

[[!redirects formal smoothness]]
[[!redirects formally smooth]]
[[!redirects formally smooth morphisms]]


formally smooth object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]].
Here $\Im$ is called the _[[infinitesimal shape modality]]_.

An [[object]]/[[type]] $X$ is called **formally smooth** if the [[unit of an adjunction|unit]]

$$
  X \to \Im X
$$

is a [[1-epimorphism]]. This is equivalent to the essentially unique morphism $X \to *$ to the [[terminal object]] being a [[formally smooth morphism]].

## Examples

* [[formally smooth scheme]]

## Related concept

[[!include cohesion - table]]

[[!redirects formally smooth type]]

[[!redirects formally smooth type]]
[[!redirects formally smooth types]]