Introduction to Étale Cohomology

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This page collects links related to the textbook

* [[Günter Tamme]],

  _Introduction to Étale Cohomology_

  1994

on [[étale cohomology]].

#Contents#
* table of contents
{:toc}

## Chapter 0 Preliminaries

### 1. Abelian categories

* [[abelian category]]

### 2. Homological algebra in abelian categories

* [[homological algebra]]

#### 2.1 $\delta$-Functors

* [[delta-functor]]

#### 2.2 Derived functors

* [[derived functor]]

#### 2.3 Spectral sequences

* [[spectral sequence]]

* [[edge morphism]]

* (2.3.5) [[Grothendieck spectral sequence]]

### 3. Inductive limits

## Chapter I Topologies and sheaves

### 1. Topologies

* [[Grothendieck topology]]

### 2. Abelian presheaves on topologies

#### 2.1 The category of abelian presheaves

* [[group object]]

#### 2.2 Cech cohomology

#### 2.3 The functors $f^p$ and $f_p$

* [[morphism of sites]]

* [[direct image]], [[inverse image]] (here: on [[presheaves]])

### 3. Abelian sheaves on topologies

#### 3.1 The associated sheaf of a presheaf

* [[sheafification]]

#### 3.2 The category of abelian sheaves

* [[sheaf of abelian groups]]

#### 3.3 Cohomology of abelian sheaves

* [[abelian sheaf cohomology]]

#### 3.4 The spectral sequences for Cech cohomology

* [[Cech cohomology]]

#### 3.5 Flabby sheaves

* [[flabby sheaf]]

#### 3.6 The functors $f^s$ and $f_s$

* [[morphism of sites]]

* [[direct image]], [[inverse image]]

#### 3.7 The Leray spectral sequence

* [[Leray spectral sequence]]

#### 3.8 Localization

* [[localization]]

#### 3.9 The comparison lemma

#### 3.10 Noetherian topologies

#### 3.11 Commutation of the functors $H^q(U,-)$ with pseudofiltered inductive limits

## Chapter II Étale Cohomology

### 1. The Étale site of a scheme

#### 1.1 Étale morphisms

* [[étale morphism]]

#### 1.2 Étale site

* [[étale site]]

#### 1.3 Relation between Étale and Zariski cohomology

* [relation to Zariski cohomology](etale+cohomology#RelationZariskiEtaleCohomology)

#### 1.4 The functors $f_\ast$ and $f^\ast$

* [[direct image]], [[inverse image]]

* [[base change]]

#### 1.5 The restricted Étale site

### 2. The case $X = Spec(k)$

* [[Galois group]]

### 3. Examples of étale sheaves

#### 3.1 Representable sheaves

* [[sheaf]], [[descent]]

* [[étale topos]]

* [[representable presheaf]]

* [[canonical topology]]

* [[additive group]], [[multiplicative group]], [[roots of unity]]

#### 3.2 Étale sheaves of $\mathcal{O}_X$-modules

* [[sheaf of coherent modules]]

* [[constructible sheaf]]

#### 3.3 Big étale site

* [[big site]]

### 4. The theories of Artin-Schreier and of Kummer

#### 4.1 The groups $H^q(X, (\mathbb{G}_a)_X)$

#### 4.2 The Artin-Schreier sequence

* [[Frobenius morphism]]

* [[Artin-Schreier sequence]]

#### 4.3 The groups $H^q(X, (\mathbb{G}_m)_X)$

* [[Hilbert's theorem 90]]

* [[Azumaya algebra]], [[Brauer group]]

#### 4.4 The Kummer sequence

* [[Kummer sequence]]

#### 4.5 The sheaf of divisors on $X_{et}$

### 5. Stalks of étale sheaves

* [[stalk]]

### 6. Strict localizations

#### 6.1 Henselian rings and strictly local rings

* [[Henselian ring]]

* [[strictly local ring]]

### 6.2 Strict localization of a scheme

### 6.3 Étale cohomology on projective limits of schemes

### 7. The Artin spectral sequence

* [[Artin spectral sequence]]

### 8. The decomposition theorem. Relative cohomology

### 9. Torsion sheaves, locally constant sheaves, constructible sheaves

* [[torsion sheaf]]

* [[constructible sheaf]]

### 10. Étale cohomology of curves

### 11. General theorems in étale cohomology theory

* [[proper base change theorem]]

category: reference


intuitionistic logic

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# Contents
* table of contents
{:toc}

## Idea

__Intuitionistic logic__ was introduced by [[Arend Heyting]] as a [[logic]] for [[Brouwer]]\'s [[intuitionistic mathematics]].  It applies more generally to [[constructive mathematics]] and so may also be called __constructive logic__.

Beware the terminological ambiguity: Some people insist that "intuitionistic logic" refers to Brouwerian intuitionism, which includes axioms that contradict classical logic; but other people use "intuitionistic" to mean the same as what in other contexts is called "constructive", i.e. mathematics without the [[principle of excluded middle]] or the [[axiom of choice]] but nothing added that contradicts them.  Some people (particularly material set theorists) use "constructive" to mean *predicative* constructive and "intuitionistic" to mean impredicative constructive.

## Idea

### General

Intuitionistic logic is most easily described as [[classical logic]] without the [[principle of excluded middle]] ($\vdash A \vee \neg{A}$) or the [[double-negation]] rule ($\neg\neg{A} \vdash A$).  It may also be defined by starting with Gentzen\'s [[sequent calculus]] for [[classical logic]] (with $\neg$ but not $\bot$) and restricting to sequents $\Gamma \vdash \Delta$ where $\Delta$ may contain *at most one* formula, or by starting with [[sequent calculus]] with $\bot$ and restricting to such sequents where $\Delta$ must contain *exactly one* formula.

### Constructive interpretation of connectives
 {#ConstructiveInterpretationOfConnectives}

The *intuitionistic interpretation* of the [[logical connectives]] (known as the [[BHK interpretation]], due to [Kolmogorov (1932, p. 59)](#Kolmogorov32), [Heyting (1956, §7.1.1)](#Heyting56), [Troelstra (1969, §2)](#Troelstra69)) is such that the resulting [[proposition]] is regarded as [[true]] only if it is possible to [[constructive mathematics|construct]] a [[proof]] of its [[assertion]].

For instance, to assert a [[logical conjunction]] ("and") or a [[universal quantification]] ("for all") is taken to mean to provide a proof of all the instances.

Dually but more notably, to assert a [[logical disjunction]] ("or") or an [[existential quantification]] ("exists") is taken to mean to prove one of the instances, so that there is no intuitionistic existence statement without construction of an example (the "disjunction property", see [below](#DisjunctionProperty)).

This constructive interpretation of logical truth is the crux of the rejection of the [[principle of excluded middle]], for it implies that to prove $P \vee (\not P)$ (which may superficially/classically seem tautologous) one must prove $P$ or one must prove $\not P$ --- but neither proof may be known (e.g. if $P$ = [[Riemann hypothesis]]).

(Here the classical mathematician is regarded as "idealistic" in their assumption that either case must hold, even if it is impossible to tell which one.)

{#FromKolmogorov} From [Kolmogorov (1932, p. 59)](#Kolmogorov32):

\begin{imagefromfile}
    "file_name": "KolmogorovIntroducingBHK.jpg",
    "width": 550,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Heyting (1956, p. 97)](#Heyting56):

\begin{imagefromfile}
    "file_name": "HeytingIntroducingBHK.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Troelstra (1969, p. 5)](#Troelstra69):

\begin{imagefromfile}
    "file_name": "Troelstra-IntroducingBHKInterp.jpg",
    "width": 660,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

From [Troelstra (1977, p. 977)](#Troelstra77):

\begin{imagefromfile}
    "file_name": "Troelstra-AttributingBHK.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

{#FromBridges} From [Bridges (1999), p. 96](#Bridges99):

\begin{imagefromfile}
    "file_name": "Bridges-IntuitInterpOfConnectives.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

The analogous discussion for [[inference rules]] in [[intuitionistic type theory]] is then given spring in [Girard (1989, §2)](#Girard89) and with more emphasis in [Martin-Löf (1996, Lec 3)](#MartinLöf96).

## Properties

### Double negation
 {#DoubleNegation}

The [[double negation translation]] says that a [[proposition]] $P$ is provable in [[classical logic]] precisely if its [[double negation]] $\not \not P$ is provable in constructive logic.

### Disjunction property
 {#DisjunctionProperty}

Unlike classical logic, intuitionistic logic has the
_disjunction-_ and _existence_ properties (cf. [above](#ConstructiveInterpretationOfConnectives)): any [[proof]] of
$\vdash A \vee B$ must contain a proof of either $\vdash A$ or $\vdash B$, and similarly any proof of $\vdash \exists x.\,F(x)$ must
construct a term $t$ and a proof of $\vdash F(t)$.  These
properties are what justify our calling intuitionistic
logic 'constructive'.

On the other hand, (classical) [[Peano arithmetic]] is
conservative over (intuitionistic) [[Heyting arithmetic]] when
restricted to $\Pi^0_1$ formulas; that is, formulas of the
form $\forall x\colon N.\, \exists y\colon N.\, F(x,y)$.  Roughly speaking,
classical logic can be just as 'constructive' as
intuitionistic logic as far as proving the totality of
functions $\mathbb{N} \to \mathbb{N}$ is concerned.

## Classicality principles

The principle of [[excluded middle]] is not provable in intuitionistic logic, and if we assume it then the logic becomes [[classical logic]].  But there are other principles that are provable classically but not intuitionistically, but which are weaker than full PEM, such as

* various [[omniscience principles]] such as LPO and LLPO
* the [[double-negation shift]]
* [[de Morgan's law]], i.e. [[weak excluded middle]]
* the [[world's simplest axiom of choice]]

## Related concepts

* [[BHK interpretation]]

[[!include categories and logic - table]]

## References

Original articles on [[intuitionism]]:

* {#Heyting1930} [[Arend Heyting]], *Die formalen Regeln der intuitionistischen Logik. I, II, III.* Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169

  abridged reprint in:

  Karel Berka, Lothar Kreiser (eds.), *Logik-Texte*, De Gruyter (1986) 188-192 [[doi:10.1515/9783112645826](https://doi.org/10.1515/9783112645826)]

* [[Arend Heyting]], *Die intuitionistische Grundlegung der Mathematik*, Erkenntnis **2** (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)]

* {#Kolmogorov32} [[Andrey Kolmogorov]], *Zur Deutung der intuitionistischen Logik*, Math. Z. **35** (1932) 58-65 [[doi:10.1007/BF01186549](https://link.springer.com/article/10.1007/BF01186549)]

* [[Hans Freudenthal]], *Zur intuitionistischen Deutung logischer Formeln*, Comp. Math. **4** (1937) 112-116 [[numdam:CM_1937__4__112_0](http://www.numdam.org/item/?id=CM_1937__4__112_0)]

* [[Arend Heyting]], *Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln"*, Comp. Math. **4** (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)]

* [[L. E. J. Brouwer]], *Points and Spaces*, Canadian Journal of Mathematics **6** (1954) 1-17 [[doi:10.4153/CJM-1954-001-9](https://doi.org/10.4153/CJM-1954-001-9)]

Early monographs:

* {#Heyting56} [[Arend Heyting]], *Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()]

* {#Kreisel65} [[Georg Kreisel]], Section 2 of: *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

Early historical account:

* {#Troelstra90} [[Anne Sjerp Troelstra]], *On the Early History of Intuitionistic Logic*, in *Mathematical Logic*, Springer, (1990) [[doi:10.1007/978-1-4613-0609-2_1](https://doi.org/10.1007/978-1-4613-0609-2_1)]

Recognizable statemens of the [[BHK interpretation]] of [[intuitionistic logic]] appear in

* [Kolmogorov (1932, p. 59)](#Kolmogorov32)

(who however speaks not of propositions but of *Aufgaben*, i.e. "tasks", here in the sense of: "mathematical problems")

* [Heyting (1956, §7.1.1)](#Heyting56)

(who is maybe the first to speak of the "meaning of logical connectives")

and then in

* {#Troelstra69} [[Anne Sjerp Troelstra]], §2 of: *Principles of Intuitionism*, Lecture Notes in Mathematics **95** Springer Heidelberg (1969)  [[doi:10.1007/BFb0080643](https://link.springer.com/book/10.1007/BFb0080643)]

(where it is presented as the author's invention) and then in

* {#Troelstra77} [[Anne Sjerp Troelstra]], §2 of: *Aspects of Constructive Mathematics*, Studies in Logic and the Foundations of Mathematics **90** 973-1052 (1977) [doi:10.1016/S0049-237X(08)71127-3]

(where the same is now called the "Brouwer-Heyting-*Kreisel* explanation", not mentioning Kolmogorov)

and then in the context of [[intuitionistic type theory]]:

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), §1.2.2 of: *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

(where it is called *Heyting semantics*), further expanded on on:

* {#MartinLöf96} [[Per Martin-Löf]], Lecture 3 of: *On the Meanings of the Logical Constants and the Justifications of the Logical Laws*, Nordic Journal of Philosophical Logic, **1** 1 (1996) 11-60 [[pdf](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf), [[MartinLofOnTheMeaning96.pdf:file]]]

and recalled in the context of [[constructive analysis]]:

* {#Bridges99}  [[Douglas Bridges]], p. 96 of: *Constructive mathematics: a foundation for computable analysis*, Theoretical Computer Science **219** 1–2 (1999) 95-109 [doi:10.1016/S0304-3975(98)00285-0]

See also:

* Wikipedia, *[Intuitionistic logic](https://en.wikipedia.org/wiki/Intuitionistic_logic)*

* Stanford Encyclopedia, *[The Development of Intuitionistic Logic](https://plato.stanford.edu/entries/intuitionistic-logic-development/#ProoInte)*

The observation that the [[poset]] of [[open subsets]] of a [[topological space]] (the [[internal logic]] of the [[sheaf topos]]) serves as a model for [[intuitionistic logic]] is apparently originally due to

* {#Tarski} [[Alfred Tarski]], _Der Aussagenkalkül und die Topologie_, Fundamenta Mathematicae 31 (1938), pp. 103-134.

A textbook account in the context of [[programming languages]] is in section 30 of

* [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_

[[!redirects intuitionistic logic]]
[[!redirects Intuitionistic logic]]


intuitionistic mathematics

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#### Constructivism, Realizability, Computability
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# Intuitionistic mathematics
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## Idea
 {#Idea}

_Intuitionistic mathematics_ (often abbreviated INT) is the earliest full-blown variety of [[constructive mathematics]], done according to the mathematical principles developed by [[L.E.J. Brouwer]] through his philosophy of _intuitionism_.

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Beware that this terminology is not consistent across mathematics.  Not infrequently the word "intuitionistic" is used to refer simply to [[constructive mathematics]] in general, or to [[constructive logic]], or to [[predicative mathematics|impredicative]] [[set theory]] done in constructive logic.  This page is about *Brouwer's intuitionism*, which is a specific variety of constructive mathematics that (unlike [[neutral constructive mathematics]]) uses axioms that [[contradiction|contradict]] [[classical mathematics]]. For more on this see at _[Terminology](#Terminology)_ below.
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The first main philosophical idea is that mathematical truth (a true statement) can only be attained in one's mind, by carefully arranging one's concepts and constructions in such a way that there remains absolutely no doubt that every aspect of the statement is verified, unambiguously, without reliance on any 'outside' assumption, for instance about the platonic nature of reality. (The [[principle of excluded middle]] is such an assumption, and Brouwer gave mathematical counterexamples to its validity, eventually leading to the foundational crisis in mathematics (Grundlagenstreit) around 1930, and Brouwer's conflict with Hilbert).

The second main philosophical idea is that the mind works over time. One does not have everything ready and done from the start. Infinity is a (perceived) idealized property of time, but one cannot have completed any infinite process or construction on any given day. Infinity is potential, not actual.

The main mathematical idea then becomes that we can only build mathematical structures and truths starting from the natural numbers $0, 1, ...$ (where the dots indicate potential, not actual infinity). Natural numbers are definite and precise, but most real numbers, such as $\pi$, have a very different nature. We can only form something that we call '$\pi$' through a never-finished process of approximation.

Nonetheless, a substantial part of mathematics can be built up ('constructed') in this way. What mainly differentiates intuitionistic mathematics from [[constructive mathematics]] are two added axioms.

Brouwer 'deduced' two axiomatic insights, notably 'continuous choice' and a transfinite-induction based knowledge principle called `the Bar Theorem'. 'Continuous choice' conflicts with classical mathematics, the Bar Theorem is classically true (it boils down to stating that any open cover of Baire space is inductive).

Kleene & Vesley (in _Foundations of Intuitionistic Mathematics_, 1965) offered a clean axiomatic approach which is nowadays called FIM. Kleene proved that FIM is equiconsistent with classical mathematics. Kleene also proved that theorems of FIM are recursively [[realizability|realizable]], which shows the computational content of FIM.

## Terminology
 {#Terminology}

Terminological ambiguity is often present in [[constructive mathematics]] and its varieties. Intuitionistic mathematics (INT) includes [[axioms]] that contradict [[classical logic]]; but people in non-foundational disciplines often use "intuitionistic" to mean roughly the same as "[[constructive|constructive mathematics]]" (say: mathematics without the [[principle of excluded middle]], usually with computational/algorithmic content and some restriction on impredicativity, but nothing added that contradicts classical mathematics).

There are a variety of ways to use the term 'intuitionistic'.  We list them here, roughly from the most specific to the most general, and contrast (where appropriate) with the term 'constructive':

*  __Intuitionism__ is an early-20th-century [[philosophy of mathematics]] developed by [[Brouwer]], according to which [[mathematics]] is a free creation of a mind, and valid results are about what that mind creates (rather than about an external reality, as in [[platonism]], or about nothing, as in [[formalism]]).  From this controversial starting point, Brouwer drew even more controversial conclusions about both [[mathematics]] and [[logic]] (which he saw as derived from mathematics, rather than conversely as in [[logicism]]).  Intuitionism is one particular philosophy of [[constructivism]].

*  __Intuitionistic mathematics__ is the mathematics along the lines of the mathematics that Brouwer came up with.  However, it\'s not necessary to accept Brouwer\'s philosophy to practise intuitionistic mathematics; conversely, one may accept Brouwer\'s philosophical starting place but not his conclusions about the resulting mathematics.  Intuitionistic mathematics is one particular variety of [[constructive mathematics]].

   One example of intuitionistic mathematics (which nicely shows that intuitionism is not a matter of "belief" but of subject) is type II [[computable mathematics]] (see for instance [Bauer 05, section 4.3.1](#Bauer05)).

*  __[[intuitionistic set theory|Intuitionistic set theory]]__ is a [[set theory]], generally proffered as a [[foundation of mathematics]], intended to capture intuitionistic mathematics.  As the terminology is usually used (for example in the name of IZF, [[IZF|intuitionistic Zermelo-Frankel set theory]]), 'intuitionistic' means that [[excluded middle]] fails but [[power sets]] are included (making it [[impredicative mathematics|impredicative]]).  In contrast, 'constructive' set theory (such as CZF, [[CZF|constructive Zermelo-Frankel set theory]]) has [[function sets]] but not power sets (making it [[weakly predicative mathematics|weakly predicative]]).  The former is technically convenient, but the latter is better motivated.  That said, Brouwer\'s mathematics was even more predicative, making *both* of these set theories stronger than he would accept.

*  __[[intuitionistic type theory|Intuitionistic type theory]]__ is generally proffered as a [[foundation of mathematics]] that is (in most of its forms) both [[constructivism|constructive]] and [[predicativism|predicative]].  For purposes of comparing type theory to set theory, it might be nice if 'intuitionistic' and 'constructive' were distinguished for type theories as they are for set theories, but they aren\'t.  (Then again, there was never much sense in making that distinction for set theories using that terminology.)

  There is variant of the [[NuPrl]] type theory with choice sequences: [PDF](http://www.nuprl.org/html/Nuprl2Coq/continuity.pdf).

*  __[[Intuitionistic logic]]__ is the [[logic]] that intuitionistic mathematics, set theory, and type theory use, which lacks the principle of [[excluded middle]]; other forms of [[constructive mathematics]] also use intuitionistic logic, which is therefore also known as _constructive logic_.

## Mathematical principles

Brouwer did not believe in a rigourous formalization of mathematics, for various reasons (amongst which the mathematical incompleteness of formal systems, as later proved by [[Gödel's incompleteness theorem]]). He saw mathematical logic and formal systems as a correct part of mathematics, but held that this part could capture the essence nor the scope of mathematics in a meaningful way. Nonetheless he admitted that certain often-used arguments and mental constructions could be formalized `as an abbreviation'.

Brouwer's student Arend Heyting (who later succeeded Brouwer as professor in Amsterdam) formalized what is now known as [[intuitionistic logic]]. He also is largely responsible for the [[Brouwer-Heyting-Kolmogorov interpretation]] (BHK) of intuitionistic logic, BHK can be seen as a precursor to realizability.

*  In hindsight, we may say that intuitionistic mathematics is done in a [[pretopos]] identified as [[Set]].

*  We have the [[axiom of infinity]] and [[countable choice]], as in [[classical mathematics]].

*  We have the classically false principles of [[continuity principle|continuity]] and [[continuous choice]].

*  We have the [[fan theorem]] and [[bar theorem]], which are classically true but fail in [[Russian constructivism]].

*  There\'s also some stuff about [[choice sequences]] that is highly philosophical, involving mathematical principles such as the [[Kripke's schema]].

Although it\'s not enough to derive all of the above, much of the essence of intuitionistic mathematics, or at least intuitionistic [[analysis]], can be summarized in the theorem that every (set-theoretic) [[function]] from the [[unit interval]] to the [[real line]] is [[uniformly continuous map|uniformly continuous]].

Intuitionistic mathematics is often regarded as a specialization of [[Bishop's constructive mathematics]] obtained by adding the above principles, but this is somewhat questionable if it refers to what Bishop actually did (and in particular the fact that he worked with [[Bishop sets]]); see [[Bishop's constructive mathematics]] for discussion.

## Feel

In intuitionistic mathematics, already [[set theory]] behaves a lot like [[topology]], a point stressed by [[Frank Waaldijk]] ([web](http://www.fwaaldijk.nl/mathematics.html)). He uses the [[Kleene-Vesley]] system.  Fourman's [[continuous truth]] makes this remark precise using [[topos theory]].

Although intuitionistic mathematics does not accept all [[function sets]] (much less [[power sets]]), it seems to allow for both [[induction|inductive]] and [[coinduction|coinductive]] structures; see [a Café comment](http://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html#c020731).
The reluctance to add function spaces is similar to the problem of function spaces in topology; see [[nice category of spaces]].

## Related concepts

* [[intuitionistic logic]]

* [[BHK interpretation]]

* [[constructive mathematics]]

* [[intuitionistic type theory]]

* [[realizability]]

* [[choice sequence]]

## References
 {#References}

For more see also the references at _[[constructive mathematics]]_.

The roots of Brouwer's intuitionism are apparently in his PhD thesis

* [[L.E.J. Brouwer]], _Over de Grondslagen der Wiskunde, PhD thesis_, Universiteit van Amsterdam, 1907.

A reference written by Brouwer and still in print in English is

* [[L.E.J. Brouwer]], _Brouwer's Cambridge Lectures on Intuitionism_, edited by D. van Dalen

Early exposition and endorsement:

* [[Hermann Weyl]], *Über die neue Grundlagenkrise der Mathematik*, Zürich 1920 ([gdz](https://gdz.sub.uni-goettingen.de/id/PPN266833020_0010?tify={%22pages%22:[45],%22panX%22:0.525,%22panY%22:0.381,%22view%22:%22info%22,%22zoom%22:0.659}), [purl:PPN266833020_0010](http://resolver.sub.uni-goettingen.de/purl?PPN266833020_0010))

* [[Hermann Weyl]], *Die heutige Erkenntnislage in der Mathematik*, Weltkreis-Verlag, Erlangen 1926. ([GoogleBooks](https://books.google.de/books/about/Die_heutige_Erkenntnislage_in_der_Mathem.html?id=1KU_AQAAIAAJ&redir_esc=y))

Original articles on the [[formal logic]] of [[intuitionism]] ([[intuitionistic logic]]):

* {#Heyting1930} [[Arend Heyting]], *Die formalen Regeln der intuitionistischen Logik. I, II, III.* Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169

  abridged reprint in:

  Karel Berka, Lothar Kreiser (eds.), *Logik-Texte*, De Gruyter (1986) 188-192 [[doi:10.1515/9783112645826](https://doi.org/10.1515/9783112645826)]

* [[Arend Heyting]], *Die intuitionistische Grundlegung der Mathematik*, Erkenntnis **2** (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)]

* [[Andrey Kolmogorov]], *Zur Deutung der intuitionistischen Logik*, Math. Z. **35** (1932) 58-65 [[doi:10.1007/BF01186549](https://link.springer.com/article/10.1007/BF01186549)]

* [[Hans Freudenthal]], *Zur intuitionistischen Deutung logischer Formeln*, Comp. Math. **4** (1937) 112-116 [[numdam:CM_1937__4__112_0](http://www.numdam.org/item/?id=CM_1937__4__112_0)]

* [[Arend Heyting]], *Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln"*, Comp. Math. **4** (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)]

* [[L. E. J. Brouwer]], *Points and Spaces*, Canadian Journal of Mathematics **6** (1954) 1-17 [[doi:10.4153/CJM-1954-001-9](https://doi.org/10.4153/CJM-1954-001-9)]

* {#Heyting56} [[Arend Heyting]], *Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()]

* {#Kreisel65} [[Georg Kreisel]], Section 2 of: *Mathematical Logic*,  in T. Saaty et al. (ed.), *Lectures on Modern Mathematics III*, Wiley New York (1965) 95-195

Formalization in terms of [[realizability]] (the [[Kleene-Vesley topos]]), and hence proof that intuitionistic mathematics is consistent:

* [[Stephen Kleene]], R. E. Vesley, _The foundations of intuitionistic mathematics_, North-Holland (1965)

General review:

* Stanford Encyclopedia of Philosophy, _[Intuitionism in the Philosophy of Mathematics](http://plato.stanford.edu/entries/intuitionism/)

See also texts on [[constructive mathematics]], such as:

* [[Anne Sjerp Troelstra]], [[Dirk van Dalen]], *Constructivism in Mathematics -- An introduction*, Vol 1, Studies in Logic and the Foundations of Mathematics **121**, North Holland (1988) [[ISBN:9780444702661](https://www.elsevier.com/books/constructivism-in-mathematics-vol-1/troelstra/978-0-444-70266-1)]

Reviews with further developments:

* [[Frank Waaldijk]], _On the foundations of constructive mathematics -- especially in relation to the theory of continuous functions_  ([web](http://www.fwaaldijk.nl/mathematics.html#onthefoundations))

  (with a focus on [[constructive analysis]]).

Discussion of an approach to [[general topology]] in intuitionistic mathematics:

* {#Waaldijk96} [[Frank Waaldijk]], _modern intuitionistic topology_, 1996 ([pdf](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf))

[[!redirects intuitionistic mathematics]]
[[!redirects intuitionism]]


invariant

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

#Contents#
* table of contents
{:toc}

## Idea

For $A$ a [[monoid]] equipped with an [[action]] on an object $V$, an **invariant** of the action is an [[generalized element|element]] of $V$ which is taken by the action to itself, hence a [[fixed point]] for all the operations in the monoid.

## Definitions

A robust definition of invariants that generalizes to [[homotopy theory]] is via the expression of actions as [[action groupoids]] regarded as sitting over [[delooping]] [[groupoids]], as discussed at _[[infinity-action]]_ and at _[[geometry of physics -- representations and associated bundles]]_.

We describe how the ordinary concept of invariants is recovered from this perspective and then consider its immediate generalizations to [[(infinity,1)-topos theory]] and its formalization in [[homotopy type theory]].

### Via sections of action groupoid projections
 {#ViaSectionsOfActionGroupoidProjections}

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

For $G$ a [[discrete group]], $\rho$ a $G$-[[action]] on some set $S$, then the set of [[invariants]] of that action is equivalent to the groupoid of [[sections]] of the [[action groupoid]] projection of [this proposition](geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG), corresponding to the action via [this proposition](geometry+of+physics+--+representations+and+associated+bundles#EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice).

=--

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

The sections in question are diagrams in [[Grpd]] of the form

$$
  \array{
    \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{id}}\searrow
    &\swArrow_{\mathrlap{\simeq}}&
    \swarrow_{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}G
  }
  \,,
$$

hence the groupoid which they form is equivalently the [[hom-groupoid]]

$$
  Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho)
  \in Grpd
$$

in the [[slice (infinity,1)-category|slice]] of [[Grpd]] over $\mathbf{B}G$.
As in the proof of [this proposition](geometry+of+physics+--+representations+and+associated+bundles#IntertwinersOfPermutationActionAsSliceHoms), with the fibrant presentation $(p_\rho)_\bullet$ of [this proposition](geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG), this is equivalently given by strictly commuting diagrams of the form

$$
  \array{
    (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet
    \\
    & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\rho)_\bullet}}
    \\
    && (\mathbf{B}G)_\bullet
  }
  \,.
$$

These $\sigma$ now are manifestly functors that are the identity on the group labels of the morphisms

$$
  \sigma_\bullet
  \;\colon\;
  \left(
  \array{
    \ast
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \ast
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
    \sigma(\ast)
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \sigma(\ast) & = \rho(\sigma(\ast)(g))
  }
  \right)
  \,.
$$

This shows that they pick precisely those elements $\sigma(\ast) \in S$ which are fixed by the $G$-action $\rho$.

Moreover, since these functors are identity on the group labels, there are no non-trivial [[natural isomorphisms]] between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

=--

More generally, we may consider sections of these groupoid projections after pulling them back along some cocycle:

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

Given an [[associated bundle]] $P \times_G V\to X$ [[modulating morphism|modulated]], as in [this proposition](geometry+of+physics+--+representations+and+associated+bundles#Associated1BundleAsPullbackOfActionGroupoid), by a morphism of [[smooth groupoids]] of the form $g \colon X \longrightarrow \mathbf{B}G$, then its set of [[sections]] is equivalently the groupoid of diagrams

$$
  \array{
    X && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{g}}\searrow
    &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}G
  }
  \,,
$$

hence the groupoid of sections is the slice [[hom-groupoid]]

$$
  \Gamma_X(P\times_G V)
  \simeq
  Grpd_{/\mathbf{B}G}(g, p_\rho)
  \,.
$$

=--

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

By the defining [[universal property]] of the [[homotopy pullback]] in [this proposition](geometry+of+physics+--+representations+and+associated+bundles#Associated1BundleAsPullbackOfActionGroupoid).

=--

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

Taken together this means that [[invariants]] of group actions are equivalently the sections of the corresponding [[universal principal bundle|universal]] [[associated bundle]].

=--

### Invariants of $\infty$-group actions
 {#ForInfinityGroupActions}

For $\mathbf{H}$ an [[(∞,1)-topos]],  $G \in Grp(\mathbf{H})$ an [[∞-group]] and

$$
  * : \mathbf{B} G \vdash : V(*) : Type
$$

an [[∞-action]] of $G$ on $V \in \mathbf{H}$, the type of invariants is the absolute [[dependent product]]

$$
  \vdash \prod_{* : \mathbf{B}G} V(*) : Type
  \,.
$$

The connected components of this is equivalently the [[group cohomology]] of $G$ with [[coefficients]] in the [[infinity-module]] $V$.

## Properties

+-- {: .num_prop #TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero}
###### Proposition
**(in [[characteristic zero]], [[invariants]] for [[finite group]] are compatible with [[chain homology]])

Let $(V_\bullet, \partial)$ be a [[chain complex]] over a [[ground field]] of [[characteristic zero]], equipped with an [[action]] by a [[finite group]] $G$. Then taking $G$-invariants commutes with passing to [[chain homology]]:

$$
  H_\bullet((V_\bullet,\partial)^G)
  \;\simeq\;
  H_\bullet((V_\bullet,\partial))^G
  \,.
$$

=--

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

Since the [[ground field]] has [[characteristic zero]], [[group averaging]] exists and provides a [[linear map]]

$$
  \array{
    V_\bullet & \overset{p}{\longrightarrow} & V_\bullet^G
    \\
    x &\mapsto& \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} g(x)
  }
$$

onto the $G$-invariants.

Now for a [[chain homology]]-class $[x] \in H_\bullet((V_\bullet,\partial))$ being $G$-invariant means that $g[x] \coloneqq [g(x)] = [x]$ for all $g \in G$, which implies that $[x] = [p(x)]$. This means that each invariant homology class has an invariant representative, hence that the map from invariant [[cycles]] to invariant [[chain homology]]-classes

$$
  Z((V_\bullet^G,\partial)) \longrightarrow H_\bullet((V_\bullet,\partial))
$$

is an [[epimorphism]].

Next consider the [[kernel]] of this map, which a priori is $Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))$. It is now sufficient to show that this coincides with the space of $G$-invariant [[boundaries]]:

$$
  Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
  \;\simeq\;
  B((V_\bullet^G, \partial))
  \,.
$$

It is clear that there is an [[injective map|inclusion]]

$$
  B((V_\bullet^G, \partial))
    \hookrightarrow
  Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
$$

so it only remains to see that this is also a [[surjection]].

To that end, consider any

$$
  x \in Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
  \,.
$$

Since in particular $x \in B((V_\bullet,\partial))$, there is $y \in V_\bullet$ with $x = \partial y$; and since moreover $x \in V_\bullet(G)$, the above implies that

$$
  x = p(x) = p(\partial y) = \partial(p y)
$$

and hence that

$$
  x \in B((V_\bullet^G,\partial))
  \,.
$$

=--

## Related concepts

* [[group averaging]], [[norm map]]

* [[stabilizer subgroup]]

* [[invariant theory]], [[invariant polynomial]]

* [[gauge invariance]]

* [[topological invariant]]/[[homeomorphism invariant]]

* [[homotopy invariant]]

[[!include homotopy type representation theory -- table]]

[[!redirects invariants]]

[[!redirects invariance]]


invariant differential form

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

#Contents#
* table of contents
{:toc}

## Definition

### Invariant differential form

A [[differential form]] $\omega \in \Omega_{dR}^p(G)$ on a [[Lie group]] $G$ is called **left invariant** if for every $g \in G$ it is invariant under the [[pullback of differential forms]]

\[
  \label{PullbackOfDifferentialFormIsDifferentialForm}
  (L_g)^* \omega = \omega
\]

along the left multiplication action

$$
  \array{
    L_g \colon & G &\longrightarrow& G
    \\
    & x &\mapsto& g \cdot x
  }
$$

Analogously a form is **right invariant** if it is invariant under the pullback by right translations $R_g$.

More generally, given a [[differentiable function|differentiable]] (e.g. [[smooth function|smooth]]) [[group action]] of $G$ on a [[differentiable manifold|differentiable]] (e.g. [[smooth manifold|smooth]]) manifold $M$

$$
  \array{
    G \times M
    &
      \overset{\rho}{\longrightarrow}
    &
    M
    \\
    (g,x) &\mapsto& g \cdot x
  }
$$

then a [[differential form]] $\omega \in \Omega^p_{dR}(M)$ is called invariant if for all $g \in G$

$$
  \rho(g)^\ast(\omega) \;=\; \omega
  \,.
$$

This reduces to the left invariance (eq:PullbackOfDifferentialFormIsDifferentialForm) for $M = G$ and $\rho$ being the left multiplication action of $G$ on itself.

### Invariant vector field

For a vector field $X$ one instead typically defines the invariance via the pushforward $(T L_g) X = (L_g)_* X$.
Regarding that $L_g$ and $T_g$ are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field;
and the two requirements are equivalent.

## Related concepts

* [[invariant metric]]

* [[Maurer-Cartan form]]

* [[tangent Lie algebra]]

## References

* page 89 (20 of 49) at MIT course on Lie groups ([pdf 2](http://ocw.mit.edu/courses/mathematics/18-755-introduction-to-lie-groups-fall-2004/lecture-notes/chap2_topic8to14.pdf))
* [[Sigurdur Helgason]], _Differential geometry, Lie groups and symmetric spaces_
* F. Bruhat, _Lectures on Lie groups and representations of locally compact groups_, notes by S. Ramanan, TATA Bombay 1958, 1968, [pdf](http://www.math.tifr.res.in/~publ/ln/tifr14.pdf)
* MathOverflow: [Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids](https://mathoverflow.net/questions/178528/construction-of-the-lie-functor-left-vs-right-invariant-vector-fields-on-lie-g)

[[!redirects invariant differential forms]]

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[[!redirects right invariant form]]
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[[!redirects right invariant vector field]]

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invariant metric

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### For Lie groups

A [[Riemannian metric]] on a [[Lie group]] $G$ is called

* _left invariant_ if for each element $g \in G$ the left multiplication [[diffeomorphism]] $g \cdot (-) \;\colon\; G \to G$ is an [[isometry]];

* _right invariant_ if for each element $g \in G$ the right multiplication [[diffeomorphism]] $(-) \cdot g \;\colon\; G \to G$ is an [[isometry]];

* _bi-invariant_ if it is both left- and right-invariant.

More generally, a [[Riemannian metric]] on a [[differentiable manifold|differentiable]]/[[smooth manifold]] euqipped with a [[differentiable function|differentiable]]/[[smooth function|smooth]] $G$-[[action]] (a [[G-manifold]]) is called invariant if all elements of $G$ act as [[isometries]].

## Properties

### Existence on Lie groups

\begin{prop}
  Every [[compact Lie group]] admits a bi-invariant metric.
\end{prop}

([Milnor 76, Cor. 1.4](#Milnor76), reviewed in [Gallier 18, Prop. 17.6](#Gallier18))

### Existence on $G$-Manifolds

+-- {: .num_prop #ExistenceOfGInvariantRiemannianMetrics}
###### Proposition
**(existence of $G$-invariant Riemannian metric on [[G-manifolds]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[compact Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then there exists a [[Riemannian metric]] on $X$ which is [[invariant]] with respect to the $G$-[[action]], hence such that all elements of $G$ act by [[isometries]].

=--

([Bredon 72, VI Theorem 2.1](#Bredon72), see also [Ziller 13, Theorem 3.0.2](#Ziller13))

## Examples

In discrete [[metric spaces]]:

* [[Cayley distance]], [[Kendall distance]]

## Related concepts

* [[invariant differential form]]

## References

On invariant metrics on [[Lie groups]]:

* {#Milnor76} [[John Milnor]], _Curvatures of left invariant metrics on Lie groups_, Advances in Mathematics, Volume 21, Issue 3, September 1976, Pages 293-329 (doi:10.1016/S0001-8708(76)80002-3)

Review:

* {#Gallier18} [[Jean Gallier]],  Section 17.1, 17.2 in: _Metrics, Connections, and Curvatureon Lie Groups_, Chapter 17 in: _Advanced Geometric Methods in Computer Science_, Lecture notes 2018 ([pdf](https://www.cis.upenn.edu/~cis610/cis610-15-sl17.pdf), [web](https://www.cis.upenn.edu/~cis610/))

On invariant metric on [[G-manifolds]]:

* {#Bredon72} [[Glen Bredon]], _[[Introduction to compact transformation groups]]_, Academic Press 1972 ([pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf), [ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1))

Review:

* {#Ziller13} [[Wolfgang Ziller]], _Group actions_, 2013 ([pdf](https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf), [[ZillerGroupActions.pdf:file]])

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invariant polynomial

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

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

## Idea

For every [[Lie algebra]] or [[∞-Lie algebra]] or [[∞-Lie algebroid]] $\mathfrak{a}$ there is its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{a})$ and its [[Weil algebra]] $W(\mathfrak{a})$ and a canonical [[dg-algebra]] morphism

$$
  CE(\mathfrak{a}) \leftarrow W(\mathfrak{a})
  \,.
$$

Recall that a [[∞-Lie algebra cohomology|cocycle]] on $\mathfrak{a}$ is a closed element in $CE(\mathfrak{a})$. An invariant polynomial is a closed elements in $W(\mathfrak{a})$ that sits in the shifted copy $\wedge^\bullet (\mathfrak{a}^*[1])$.

This means that for $X \in \mathfrak{a}$, for $\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a})$ the contraction derivation and $ad_X := [d_W, \iota_X]$ the corresponding [[Lie derivative]], we have in particular that an invariant polynomial $\langle -\rangle \in W(\mathfrak{a})$ is invariant in the sense that

$$
  ad_X \langle -\rangle = 0
  \,.
$$

For $\mathfrak{a} = \mathfrak{g}$ an ordinary [[Lie algebra]], an invariant polynomial on $\mathfrak{g}$ is precisely a symmetric multilinear map on $\mathfrak{g}$ which is $ad$-invariant in the ordinary sense.

## Definition

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

For $\mathfrak{a}$ an [[∞-Lie algebroid]] (of finite type, i.e. degreewise of finite rank) with [[Chevalley-Eilenberg algebra]]

$$
  CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{a}^*, d_{CE(\mathfrak{a}}))
$$

and [[Weil algebra]]

$$
  W(\mathfrak{a}) = (\wedge^\bullet (\mathfrak{a}^* \oplus \mathfrak{a}^*[1]), d_{W(\mathfrak{a})})
$$

an **invariant polynomial** on $\mathfrak{a}$ is an elements $\langle - \rangle \in W(\mathfrak{a})$ with the property that

* $\langle - \rangle$ is a wedge product of generators in the shifted copy of
  $\mathfrak{a}^*$ $W(\mathfrak{a})$, i.e.

  $$
    \langle - \rangle \in \wedge^\bullet \mathfrak{a}^*[1]
  $$

  or equivalently: for all $x \in \mathfrak{a}$ and $\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a})$ the contraction [[derivation]], we have

  $$
    \iota_x \langle -\rangle = 0
    \,;
  $$

* it is closed in $W(\mathfrak{a})$ in that $d_{W(\mathfrak{a})} \langle - \rangle = 0$

   or more generally its differential is again in the shifted copy.

=--

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

This implies that for

$$
  ad_x := [d_{W(\mathfrak{a})}, \iota_X]
$$

the [[Lie derivative]] in $W(\mathfrak{a})$ along $x \in \mathfrak{a}$, which encodes the coadjoint action of $\mathfrak{a}$ on $W(\mathfrak{a})$, we have

$$
  ad_x \langle - \rangle = 0
$$

for all $x$. But the condition for an invariant polynomial is stronger than these ad-invariances. For instance there are  [[∞-Lie algebra cohomology|∞-Lie algebra cocycles]] $\mu \in CE(\mathfrak{g})$ which when regarded as elements in $W(\mathfrak{g})$ are ad-invariant. But being entirely in the un-shifted copy, $\mu \in \wedge^\bullet \mathfrak{g}^*$, these are not invariant polynomials.

=--

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

We say an invariant polynomial is _decomposable_ if it is the wedge product in $W(\mathfrak{g})$ of two invariant polynomials of non-vanishing degree.

=--

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

Two invariant polynomials $P_1, P_2 \in W(\mathfrak{g})$ are _horizontally equivalent_ if there is $\omega \in ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$ such that

$$
  P_1 = P_2 + d_W \omega
  \,.
$$

=--

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

Every decomposable invariant polynomial, def. \ref{DecomposableInvPolynomial}, is horizontally equivalent to 0.

=--

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

Let $P = P_1 \wedge P_2$ be a wedge product of two indecomposable polynomials. Then there exists a [[Chern-Simons element]] $cs_1 \in W(\mathfrak{g})$ such that $d_W cs_1 = P_1$. By the assumption that $P_2$ is in non-vanishing degree and hence in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$ it follows that

1. also $cs_1 \wedge P_2$ is in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$

1. $d_W (cs_1 \wedge P_2) = P_1 \wedge P_2$ .

Therefore $cs_1 \wedge P_1$ exhibits a horizontal equivalence $P_1 \wedge P_2 \sim 0$.

=--

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

Horizontal equivalence classes of invariant polynomials on $\mathfrak{g}$ form a [[graded vector space]] $inv(\mathfrak{g})_V$. There is a morphism of graded vector spaces

$$
  inv(\mathfrak{g})_V \hookrightarrow W(\mathfrak{g})
$$

unique up to horizontal equivalence, that sends each horizontal equivalence class to a representative.

=--

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

By prop. \ref{DecomposablePolysAreHorizontallyTrivial} it follows that
$inv(\mathfrak{g})_V$ contains only indecomposable invariant polynomials.

=--

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

We write $inv(\mathfrak{g})$ for the [[dg-algebra]] whose underlying [[graded algebra]] is the [[free construction|free]] graded algebra on the [[graded vector space]] $inv(\mathfrak{g})_V$, and whose [[differential]] is trivial.

Since invariant polynomials are closed, the inclusion of graded vector spaces from observation \ref{ClassesOfInvPolysFormGradedVectorSpace} induces an inclusion ([[monomorphism]]) of dg-algebras

$$
  inv(g) \hookrightarrow W(g)
  \,.
$$

=--

## Examples

### On Lie algebras {#OnLieAlgebras}

+-- {: .num_lemma}
###### Observation

For $\mathfrak{g}$ a [[Lie algebra]], this definition of invariant polynomials is equivalent to more traditional ones.

=--

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

To see this explicitly, let $\{t^a\}$ be a basis of $\mathfrak{g}^*$ and $\{r^a\}$ the corresponding basis of $\mathfrak{g}^*[1]$. Write $\{C^a{}_{b c}\}$ for the structure constants of the Lie bracket in this basis.

Then for $P = P_{(a_1 , \cdots , a_k)} r^{a_1} \wedge \cdots \wedge r^{a_k} \in \wedge^{r} \mathfrak{g}^*[1]$ an element in the shifted generators, the condition that it is $d_{W(\mathfrak{g})}$-closed is equivalent to

$$
  C^{b}_{c (a_1} P_{b, \cdots, a_k)} t^c \wedge r^{a_1} \wedge \cdots \wedge r^{a_k}
  \,,
$$

where the parentheses around indices denotes symmetrization, as usual, so that this is equivalent to

$$
  \sum_{i} C^{b}_{c (a_i} P_{a_1 \cdots a_{i-1} b a_{i+1} \cdots, a_k)}
  = 0
$$

for all choice of indices. This is the component-version of the familiar invariance statement

$$
  \sum_i P(t_1, \cdots, t_{i-1}, [t_c, t_i], t_{i+1}, \cdots , t_k)
  = 0
$$

for all $t_\bullet \in  \mathfrak{g}$.

=--

### On semisimple Lie algebras
 {#SemisimpLie}

See [[Killing form]], [[string Lie 2-algebra]].

### On tangent Lie algebroids

For $X$ a [[smooth manifold]], and invariant polynomial on the [[tangent Lie algebroid]] $\mathfrak{a} = T X$ is precisely a closed [[differential form]] on $X$.

### On the string Lie 2-algebra
 {#OnStringLie2Algebra}

For $\mathfrak{g}$ a [[semisimple Lie algebra]] let
$\mu_3 := \langle -,[-,-]\rangle$ be the canonical [[Lie algebra cocycle]] in degree 3, which is the one in [[transgression]] with the [[Killing form]] invariant polynomial $\langle -,-\rangle$.

Write $\mathfrak{g}_{\mu_3}$ for the corresponding [[string Lie 2-algebra]]. We have that the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g}_{\mu_3})$ is given by

$$
  d_{CE} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c
$$

$$
  d_{CE} b = \mu_3
$$

and the [[Weil algebra]] $W(\mathfrak{g}_{\mu_3})$ is given by

$$
  d_W t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c + r^a
$$

$$
  d_W b = \mu_3 - h
$$

$$
  d_W r^a = - C^a_{b c} t^b \wedge r^c
$$

$$
  d_W h = d_W \mu_3 = \sigma \mu_3
  \,,
$$

where $\sigma$ acts by degree shift isomorphism on unshifted generators.

It follows at once that every invariant polynomial

$$
  P = P_{a_1, \cdots, a_n} r^{a_1} \wedge \cdots \wedge r^{a_n}
$$

on the Lie algebra $\mathfrak{g}$ canonically identifies also with an invariant polynomial of the string Lie 2-algebra. But the differnce is that the [[Killing form]] $\langle -,- \rangle := P_{a b} r^a \wedge r^b$ is non-trivial as a polynomial on $\mathfrak{g}$, but as a polynomial on $\mathfrak{g}_{\mu_3}$ becomes
horizontally equivalent ,def. \ref{HorizontalEquivalence}), to the trivial invariant polynomial.

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

On the [[string Lie 2-algebra]] $\mathfrak{g}_{\mu_3}$ the [[Killing form]] $\langle -,-\rangle$ is horizontally equivalent to 0.

=--

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

Let $cs_3 \in W(\mathfrak{g})$ be any [[Chern-Simons element]] for $\langle -,- \rangle$, hence an element such that

1. $cs_3|_{CE(\mathfrak{g})} = \mu_3$;

1. $d_W cs_3 = \langle -,- \rangle$.

Then notice that by the above we have in $W(\mathfrak{g}_{\mu_3})$ that the differential of the new generator $h$ is equal to that of $\mu_3$:

$$
  d_W h = d_W \mu_3
  \,.
$$

We on $\mathfrak{g}_{\mu_4}$ we can replace $\mu_3$ by $h$ and still get a  [[Chern-Simons element]] for the Killing form:

$$
  \tilde cs_3 := cs_3 - \mu_3 + h
  \,.
$$

$$
  d_W \tilde cs_3 = \langle -,- \rangle
  \,.
$$

But while $\mu_3$ is not in $ker(W(\mathfrak{g}_{\mu_3}) \to CE(\mathfrak{g}_{\mu_3}))$, the element $h$ is, by definition. Therefore $\tilde cs_3$ is in that kernel, and hence exhibits a horizontal equivalence between $\langle -,- \rangle$ and $0$.

=--

This is a special case of the more general statement below, about [invariant polynomials on shifted central extensions](OnShiftedCentralExtenstions).

For illustration purposes it is useful to consider the following variant of this example:

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

Write

$$
 (b \mathbb{R} \to \mathfrak{string}) \in L_\infty Alg
$$

for the [[L-∞ algebra]] defined by the fact that its [[Chevalley-Eilenberg algebra]] is given by

$$
  d_{CE} t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
$$

$$
  d_{CE} b = \mu_3 - c
$$

$$
  d_{CE} c = 0
  \,,
$$

where $\{t^a\}$ is a dual basis in degree 1 for some [[semisimple Lie algebra]] $\mathfrak{g}$ as above, $b$ and $c$ are generators in degree 2 and 3, respectively, and $\mu_3 \propto \langle -,[-,-]\rangle$ is the canonical [[Lie algebra cohomology|Lie algebra cocycle]] in degree 3, as above.

=--

It is easily seen that

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

The canonical morphism

$$
  \mathfrak{g} \to (b \mathbb{R} \to \mathfrak{string})
$$

given dually by sending

$$
  t^a \mapsto t^a\,,\,\,\, b \mapsto 0\,,\, \,\, c \mapsto \mu_3
$$

is a weak equivalence.

=--

So the Lie 3-algebra $(b \mathbb{R} \to \mathfrak{string})$ is a kind of [[resolution]] of the ordinary Lie algebra $\mathfrak{g}$. It is for instance of use in the presentation of twisted [[differential string structure]]s, where the shifted piece $b \mathbb{R}$ in $(b \mathbb{R} \to \mathfrak{string})$ picks up the failure of $\mathfrak{so}$-valued [[connection on an ∞-bundle|connections]] to lift to $\mathfrak{string}$-[[connection on a 2-bundle|2-connections]].

The proof of the following proposition may be instructive for seeing how the definition of horizontal equivalence of invariant polynomials takes care of having the invariant polynomials of $(b\mathbb{R} \to \mathfrak{string})$ agree with those of $\mathfrak{g}$.

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

There is an [[isomorphism]]

$$
  inv(\mathfrak{g}) \simeq inv(b \mathbb{R} \to \mathfrak{string})
  \,.
$$

=--

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

Notice that the [[Weil algebra]] of $(b\mathbb{R} \to \mathfrak{string})$ is given by

$$
  d_W t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$

$$
  d_W b = \mu_3 - c - h
$$

$$
  d_W c = g
$$

for new generators $\{r^a\}$ in degree 2, $h$ in degree 3 and $g$ in degree 4, coming with their [[Bianchi identities]]

$$
  d_W r^a = - C^a{}_{b c}t^b \wedge r^c
$$

$$
  d_W h = d_W \mu_3 - g
$$

$$
  d_W g = 0
$$

For the following computations let $\{k_{a b}\}$ be the structure constants of the [[Killing form]], so that

$$
  \langle -,- \rangle = k_{a b} r^a \wedge r^b
$$

and assume that $\mu_3$ is normalized such that

$$
  \mu_3 = k_{a a'}C^{a'}_{b c} t^a \wedge t^b \wedge t^c
$$

(if another normalization is chosen, then the corresponding factor will float around the following formulas without changing anything of the end result).

Now the indecomposable invariant polynomials are those of $\mathfrak{g}$ and one additional one: $g$. This means that before deviding out horizontal equivalence on generators, the invariant polynomials of $(b \mathbb{R} \to \mathfrak{string})$ are not equal to those of $\mathfrak{g}$, due to the superfluous generator $g$.

But we do have the horizontal equivalence relation

$$
  \langle -,-\rangle = g + d_W (cs - \mu_3 + h)
 \,,
$$

where $cs$ is any [[Chern-Simons element]] for $\langle - , \rangle$, for instance

$$
  cs
   =
   \frac{1}{6} k_{a a'}C^{a'}_{b c} t^a \wedge t^b \wedge t^c
   +
   k_{a b} t^a r^b
  \,,
$$

Notice that the homotopy $cs - \mu_3 + h$ here is indeed in $ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))$: the component of $cs$ not in that kernel is precisely $\mu$. The above formula subtracts this offending summand and replaces it with the new generator $h$, which by definition _is_ in the kernel and whose image under $d_W$ is the image of $\mu$ under $d_W$, plus the superfluous new generator of invariant polynomials.

Therefore in horizontal equivalence classes of invariant polynomials on $(b \mathbb{R} \to \mathfrak{string})$ the superfluous $g$ is identified with the Killing form $\langle-,- \rangle$, and hence the claim follows.

=--

### On symplectic Lie $n$-algebroids

A [[symplectic Lie n-algebroid]] is an [[L-infinity algebroid]] that carries a binary and non-degeneraty invariant polynomial of grade $n$. This is a generalization of the notion of [[symplectic form]] to which it reduces for $n = 0$.

## Properties
 {#Properties}

### As differential forms on the moduli stack of connections
 {#AsFormsOnBGconn}

The invariant polynomials of a Lie algebra $\mathfrak{g}$, thought of as equipped with trivial differential, are the de Rham complex of differential forms on the universal moduli stack $\mathbf{B}G_{conn}$ of $G$-[[principal connections]] [Freed-Hopkins 13](#FreedHopkins13).

$$
  \mathrm{inv}(\mathfrak{g})\simeq \Omega^\bullet(\mathbf{B}G_{conn})
  \,.
$$

For more on this see also at _[Weil algebra -- Characterization in the smooth infinity-topos](Weil+algebra#CharacterizationInSmoothTopos)_.

### On reductive Lie algebras
 {#OnReductiveLieAlg}

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

Let $\mathfrak{g}$ be a [[reductive Lie algebra]]. Then the subalgebra of invariant polynomials in the [[Weil algebra]] is the [[free construction|free graded algebra]] on the [[graded vector space]] of indecomposable invariant polynomials.

This graded vector space has a vector space [[isomorphism]] of degree -1 to the graded vector space of odd generators of the [[Lie algebra cohomology]] $H^\bullet(\mathfrak{g}) = H^\bullet(CE(\mathfrak{g}))$.

=--

This appears for instance as ([GHV, vol III, page 242, theorem I](#GHV)).

## Role in $\infty$-Chern-Weil theory

In ($\infty$-)[[Chern-Weil theory]] the crucial role played by the invariant polynomials is their relation to [[∞-Lie algebra cocycle]]s. One may understand invariant invariant polynomials as extending under [[Lie integration]] $\infty$-Lie algebra cocycles from [[cohomology]] to [[differential cohomology]].

### Transgression cocycles and Chern-Simons elements {#TransgressionCocycles}

+-- {: .un_defn}
###### Definition
**(Chern-Simons elements and transgression cocycles)**

Let $\mathfrak{a} = \mathfrak{g}$ be an [[∞-Lie algebra]]. Since the [[cochain cohomology]] of the [[Weil algebra]] $W(\mathfrak{g})$ is trivial, for every invariant polynomial $\langle -\rangle \in W(\mathfrak{g})$ there is necessarily an element $cs \in W(\mathfrak{g})$ with

$$
  d_{W(\mathfrak{g})} cs = \langle -\rangle
  \,.
$$

This we call a [[Chern-Simons element]] for $\langle -\rangle$.

This element $cs$ will in general not sit entirely in the shifted copy. Its restriction

$$
  \mu := cs|_{\wedge^\bullet \mathfrak{g}^*} \in CE(\mathfrak{g})
$$

is a [[∞-Lie algebra cohomology|∞-Lie algebra cocycle]]. We say this is _in transgression_ with $\langle -\rangle$.

In total this construction yields a commuting diagram

$$
  \array{
     CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1}\mathbb{R})
     &&& cocycle
     \\
     \uparrow && \uparrow
     \\
     W(\mathfrak{g}) &\stackrel{(cs,\langle -\rangle)}{\leftarrow}&
     W(b^{n-1} \mathbb{R})
     &&&
     Chern-Simons-element
     \\
     \uparrow && \uparrow
     \\
     inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}&
     CE(b^n \mathbb{R})
     &&&
     invariant\; polynomial
  }
  \,,
$$

where $b^{n-1}\mathbb{R}$ denotes the [[∞-Lie algebra]] whose CE-algebra has a single generator in degree $n$ and vanishing differential, and where $CE(b^n \mathbb{R}) = inv(b^{n-1}\mathbb{R})$ is the algebra of invariant polynomials of $b^{n-1} \mathbb{R}$.

=--

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

The element $\mu \in CE(\mathfrak{g})$ associated to an invariant polynomial $\langle -\rangle$ by the above procedure is indeed a cocycle, and its cohomology class is independent of the choice of the element $cs$ involved.

=--

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

The procedure that assigns $\mu$ to $\langle- \rangle$ is illustarted by the following diagram

$$
  \array{
    0 && \langle-\rangle &\leftarrow & \langle-\rangle
    \\
    \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{g})}}}
    &&
    \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{g})}}}
    \\
    \mu &\leftarrow& cs
    \\
    \\
    \\
    \\
    CE(\mathfrak{a})
    &\leftarrow&
    W(\mathfrak{a})
    &\leftarrow&
    inv(\mathfrak{a})
  }
$$

From the fact that all morphisms involved respect the differential and from the fact that the image of $\langle-\rangle$ in $CE(\mathfrak{g})$ vanishes it follows that

* the element $\mu$ satisfies $d_{CE(\mathfrak{a})} \mu = 0$, hence that it is an [[∞-Lie algebra cohomology|∞-Lie algebra cocycle]];

* any two different choices of $cs$ lead to cocylces $\mu$ that are cohomologous.

=--

This construction exhibits effectively the preimage of the [[connecting homomorphism]] in the [[cochain cohomology]] sequence induced by $W(\mathfrak{g}) \to CE(\mathfrak{g})$:

The [[dg-algebra]] of invariant polynomials is a sub-dg-algebra of the kernel of the morphism $i^* : W(\mathfrak{a}) \to CE(\mathfrak{a})$ from the [[Weil algebra]] to the [[Chevalley-Eilenberg algebra]] of $\mathfrak{a}$

$$
  inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a}))
  \,.
$$

From the short [[nLab:exact sequence]]

$$
  CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to
  CE(\mathfrak{a})
$$

we obtain the long exact sequence in [[chain homology and cohomology|cohomology]]

$$
  \cdots
  \to
  H^{n+1}(CE(\mathfrak{a}))
  \stackrel{\delta}{\to} H^{n+2}(CE(\Sigma \mathfrak{a}))
  \to \cdots
  \,.
$$

We say that $\mu \in CE(\mathfrak{a})$ is in transgression with $\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a})$ if their classes map to each other under the [[connecting homomorphism]] $\delta$:

$$
  \delta : [\mu] \mapsto [\omega]
  \,.
$$

**Example.** In the case where $\mathfrak{g}$ is an ordinary semisimple [[Lie algebra]], this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with $\mathfrak{g}$-valued 1-forms. This is described in the section [On semisimple Lie algebras](#SemisimpLie).

### Chern-Simons and curvature characteristic forms

For $\mathfrak{g}$ a [[∞-Lie algebra|Lie n-alghebra]], let $\mathbf{B}G := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})$ be the [[∞-Lie group]] obtained by [[Lie integration]] from it.

For $X$ a [[paracompact space|paracompact]] [[smooth manifold]] with [[good open cover]] $\{U_i \to X\}$ whose [[Cech nerve]] we write $C(U)$, a [[cocycle]] for a $G$-[[principal ∞-bundle]] on $X$ is cocycle with coefficients in the simplicial sheaf

$$
  \mathbf{B}G =
  \mathbf{cosk}_{n+1}((U,[k]) \mapsto \{
     \Omega^\bullet_{vert}(U \times \Delta^k)
     \leftarrow
     CE(\mathfrak{g})
  \})
  \,.
$$

We say an $\infty$-connection on this is an extension to a cocycle with coefficients in the simplicial sheaf

$$
  \mathbf{B}G_{diff} =
  \mathbf{cosk}_{n+1}((U,[k]) \mapsto \left\{
    \array{
       \Omega^\bullet_{vert}(U \times \Delta^k)
       &\leftarrow&
       CE(\mathfrak{g})
       &&& underlying \; cocycle
       \\
       \uparrow && \uparrow
       \\
       \Omega^\bullet(U\times \Delta^k)
       &\stackrel{}{\leftarrow}&
       W(\mathfrak{g})
       &&&
       connection
     }
  \right\}
  \,.
$$

The diagrams on the left encode those $\mathfrak{g}$-valued forms on $U \times \Delta^k$ whose [[curvature]] vanishes on $\Delta^k$. One can show that one can always find a _genuine_ $\infty$-connection: one for which the curvatures have no leg along $\Delta^k$, in that they land in $\Omega^\bullet(U) \otimes C^\infty(\Delta^k)$. For those the above diagram extends to

$$
  \array{
     \Omega^\bullet(U \times \Delta^k)_{vert}
     &\leftarrow&
     CE(\mathfrak{g})
     &&& underlying \; cocycle
     \\
     \uparrow && \uparrow
     \\
     \Omega^\bullet(U\times \Delta^k)
     &\stackrel{}{\leftarrow}&
     W(\mathfrak{g})
     &&&
     connection
     \\
     \uparrow && \uparrow
     \\
     \Omega^\bullet(U)
     &\leftarrow&
     inv(\mathfrak{g})
     &&&
     curvature
   }
  \,.
$$

This defines the [[simplicial presheaf]] that classifies [[connections on ∞-bundles]].

By [[pasting]]-postcomposition with the above diagrams for an invariant polynomial we obtain connections with values in $b^n \mathbb{R}$

$$
  \array{
     \Omega^\bullet_{vert}(U \times \Delta^k)
     &\leftarrow&
     CE(\mathfrak{g})
     &\stackrel{\mu}{\leftarrow}&
     CE(b^{n-1}\mathbb{R})
     &&& underlying \; cocycle
     \\
     \uparrow && \uparrow && \uparrow
     \\
     \Omega^\bullet(U\times \Delta^k)
     &\stackrel{}{\leftarrow}&
     W(\mathfrak{g})
     &\stackrel{(cs,\langle- \rangle)}{\leftarrow}&
     W(b^{n-1}\mathbb{R})
     &&&
     Chern-Simons forms
     \\
     \uparrow && \uparrow && \uparrow
     \\
     \Omega^\bullet(U)
     &\leftarrow&
     inv(\mathfrak{g})
     &\stackrel{\langle -\rangle}{\leftarrow}&
     CE(b^n \mathbb{R})
     &&&
     curvature\;characteristic\;form
   }
  \,,
$$

where in the bottom row we have the [[curvature characteristic forms]] $\langle F_\nabla\rangle$ coresponding to the connection, and in the middle the corresponding [[Chern-Simons forms]].

More details for the moment at [[∞-Chern-Weil theory introduction]].

## Related concepts

* [[∞-Lie algebra cocycle]]

* [[Chern-Simons element]]

* **invariant polynomial**, [[invariant theory]]

## References

The idea of invariant polynomials as a $G$-invariant subalgebra in the [[Weil algebra]], and their use in what is now called [[Chern-Weil theory]], originates with

* [[André Weil]], _Géométrie différentielle des espaces fibres_, unpublished, item [1949e] in: _André Weil Oeuvres Scientifiques / Collected Papers_, vol. 1 (1926-1951), 422-436, Springer 2009 ([ISBN:978-3-662-45256-1](https://www.springer.com/gp/book/9783662452561))

* [[Henri Cartan]], _Cohomologie réelle d'un espace fibré principal différentiable. I : notions d'algèbre différentielle, algèbre de Weil d'un groupe de Lie _, Séminaire Henri Cartan, Volume 2  (1949-1950), Talk no. 19, May 1950  ([numdam:SHC_1949-1950__2__A18_0](http://www.numdam.org/item/?id=SHC_1949-1950__2__A18_0))

  \linebreak

  [[Henri Cartan]], _Notions d'algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie_, in: Centre Belge de Recherches Mathématiques, _Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950_, Georges Thon 1951 ([GoogleBooks](https://books.google.de/books/about/Colloque_de_topologie_espaces_fibres.html?id=sagqHQAACAAJ&redir_esc=y))

  \linebreak

  (These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed.)

* {#Chern50} [[Shiing-shen Chern]], _Differential geometry of fiber bundles_, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952) ([[Chern-DifferentialGeometryOfFiberBundles.pdf:file]], [full proceedings vol 2 pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf))

* {#KobayashiNomizu63} [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], Section XII.2 in: _Foundations of Differential Geometry, Volume 1_, Wiley 1963 ([web](https://www.zuj.edu.jo/download/foundations-of-differential-geometry-vol-1-kobayashi-nomizu-pdf/), [ISBN:9780471157335](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+Volume+1-p-9780471157335), [Wikipedia](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry))

Invariant polynomials for Lie algebras of [[simple Lie groups]] are disussed in

* [[José de Azcárraga]],  A. J. Macfarlane, A. J. Mountain, J. C. Perez Bueno, _Invariant tensors for simple groups_, Nucl. Phys. B510 (1998) 657-687 ([arXiv:physics/9706006](http://arxiv.org/abs/physics/9706006))

A standard textbook account of the traditional theory is in volume III of

* {#GHV} [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

The notion of invariant polynomials of $L_\infty$-algebras has been introduced in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-connections_ (<a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos+--+references#SSSI">web</a>).

The abstract characterization is due to

* {#FreedHopkins13} [[Daniel Freed]], [[Michael Hopkins]], _Chern-Weil forms and abstract homotopy theory_, Bull. Amer. Math. Soc. 50 (2013), 431-468 ([arXiv:1301.5959](http://arxiv.org/abs/1301.5959))

An account in the more general context of Lie theory in [[cohesive (infinity,1)-toposes]] is in section 3.3.11 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .

[[!redirects algebra of invariant polynomials]]
[[!redirects dg-algebra of invariant polynomials]]

[[!redirects invariant polynomials]]


invariant theory

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

#Contents#
* table of contents
{:toc}

## Idea

_Invariant theory_ studies _[[invariant|invariants]]_: algebraic entities -- for instance elements in a [[ring]] -- invariant under some [[group]] [[action]].

In _[[geometric invariant theory]]_ one regards the algebraic objects as [[Isbell duality|formally dual]] to a geometric space and interprets the invariants as functions on a [[quotient]] space.

## Examples
 {#Examples}

Let $V$ be a ([[graded vector space|graded]]) [[vector space]] equipped with the [[action]] $\rho$ of a [[group]] $G$. This induces an action on the symmetric [[tensor powers]] $Sym^n V$. A [[linear map]] out of sums of such symmetric powers is called a [[polynomial]] on $V$. It is an _[[invariant polynomial]]_ if it is invariant under the group action, hence if for every $g \in G$ we have (writing it for a homogeneous polynomial for convenience)

$$
  f(\rho_g(x_1), \cdots, \rho_g(x)) = f(x_1, \cdots, x_n)
  \,.
$$

For instance if $G$ is a [[Lie group]] and $V = \mathfrak{g}$ is its [[Lie algebra]], there is a canonical _[[adjoint action]]_ $\rho = Ad$ of $G$ on $Sym^n \mathfrak{g}$. The corresponding _[[invariant polynomials]]_ play a central role in [[Lie theory]], notably via [[Chern-Weil theory]].
In this case the $Ad$-invariance is often expressed in its differential form (obtained by differentiating the above equation at the neutral element), where it says that for all $y \in \mathfrak{g}$ we have

$$
  f([y,x_1], \cdots, x_n) + \cdots +
  f(x_1, \cdots, [y,x_n])
  = 0
  \,.
$$

## Related concepts

* [[invariant]], [[group averaging]], [[logicality and invariance]]

## References

* Jean Dieudonné, James B. Carrell, _Invariant theory, old and new_, Advances in Mathematics 4 (1970) 1-80. Also published as a book (1971).
* Hanspeter Kraft, [[Claudio Procesi]], _Classical invariant theory -- A primer_ ([pdf](http://jones.math.unibas.ch/~kraft/Papers/KP-Primer.pdf))
* [[Claudio Procesi]], _Lie groups, an approach through invariants and representations_, Universitext, Springer 2006, [gBooks](http://books.google.co.in/books?id=Sl8OAGYRz_AC&printsec=frontcover&hl=hr&source=gbs_atb)
* [[Igor Dolgachev]], _Lectures on invariant theory_, [ps](http://modular.math.washington.edu/people/dolgachev/invbook.ps)

* William Crawley-Boevey, _Lectures on representation theory and invariant theory_ ([pdf](www.amsta.leeds.ac.uk/~pmtwc/repinv.pdf))
* [[David Mumford]], John Fogarty, Frances Clare Kirwan, _Geometric invariant theory_, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) __34__, Springer-Verlag

* Hanspeter Kraft, _Geometrische Methoden in der Invariantentheorie_, Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp.

* Э. Б. Винберг, В. Л. Попов, _Теория инвариантов_, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 1989, том 55, с. 137–309 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=intf&paperid=158&what=fullt&option_lang=rus)
* [[B. Kostant]], S. Rallis, _Orbits and representations associated with symmetric spaces_, Amer. J. Math. __93__ (1971), 753–809 [MR0311837](http://www.ams.org/mathscinet-getitem?mr=0311837) [doi](http://dx.doi.org/10.2307/2373470)
* Edward Formanek, _Noncommutative invariant theory_, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 [doi](http://dx.doi.org/10.1090/conm/043)
* Peter Olver, _Classical Invariant Theory_, Cambridge University Press, 1999.

[[!redirects classical invariant theory]]


inverse

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# Contents
* table of contents
{: toc}

## Definition

\begin{definition}\label{InverseMorphism}
**([[inverse morphisms]])** \linebreak
An **inverse** of a [[morphism]] $f \colon X \to Y$ in a [[category]] or [[unital magmoid]] (or an element of a [[monoid]] or [[unital magma]]) is another morphism $f^{-1} \colon Y \to X$ which is both a [[left-inverse]] (a [[retraction]]) as well as a [[right-inverse]] (a [[section]]) of $f$, in that
$$
  f \circ f^{-1} \,=\, id_Y \colon Y \to Y
$$
[[equality|equals]] the [[identity morphism]] on $Y$
and
$$
  f^{-1} \circ f \,=\, id_X \colon X \to X
$$
[[equality|equals]] the [[identity morphism]] on $X$.
\end{definition}

\begin{definition}\label{Isomorphism}
A [[morphism]] that has an [[inverse morphism]] (Def. \ref{InverseMorphism}) is called an *[[isomorphism]]*.
\end{definition}

\begin{remark}
A ([[small category|small]]) [[category]] in which all morphisms have inverses is called a *[[groupoid]]*.
\end{remark}

## Properties

\begin{prop}\label{InversesAreUnique}
**([[inverse morphisms]] are unique)** \linebreak
  If $f$ is an [[isomorphism]] (Def. \ref{Isomorphism}) with [[inverse morphism]] $f^{-1}$ (Def. \ref{InverseMorphism}), then any [[left inverse]] as well as any [[right inverse]] is already an actual [[inverse morphism]] and in fact is [[equality|equal]] to $f^{-1}$.
\end{prop}
In particular, [[inverse morphisms]] are unique when they exist.
\begin{proof}
  Let $g$ be a [[left inverse]], hence such that $g \circ f \,=\, id$.
  Write $f^{-1}$ for the actual inverse, hence such that
  $f^{-1} \circ f \,=\, id$ and $f \circ f^{-1} \,=\, id$.
  Then the following sequence of equalities implies that
  $g = f^{-1}$:
  $$
  \begin{aligned}
    g
    & \;=\;
    g \circ id
    \\
    & \;=\;
    g \circ ( f \circ f^{-1} )
    \\
    & \;=\;
    (g \circ f) \circ f^{-1}
    \\
    & \;=\;
    id \circ f^{-1}
    \\
    & \;=\;
    f^{-1}
    \,.
  \end{aligned}
  $$
Here all steps use just the definitions of the various morphisms, except the third step, which uses [[associativity]] of [[composition]] in any [[category]].

An analogous argument applies to right inverses; and either argument applies to actual inverses.
\end{proof}

\begin{prop}
  Let $f$ be an [[isomorphism]] (Def. \ref{Isomorphism}).
  Then the inverse morphism $\left(f^{-1}\right)^{-1}$ of an inverse morphism $f^{-1}$ (Def. \ref{InverseMorphism}) exists and is [[equality|equal]] to the original morphism:
$$
  \left(f^{-1}\right)^{-1}
  \;=\;
  f
  \,.
$$
\end{prop}
\begin{proof}
  By the uniqueness of inverses (Prop. \ref{InversesAreUnique}) for $f^{-1}$.
\end{proof}

## Examples

\begin{example}\label{IdentityMorphismsAreTheirOwnInverses}
**([[identity morphisms]] are their own [[inverse morphisms]])** \linebreak
Any [[identity morphism]]  is its own inverse morphism (Def. \ref{InverseMorphism}):
$$
  id^{-1} \,=\, id
  \,.
$$
\end{example}

\begin{example}
In a [[balanced category]], such as in a [[topos]] (in particular in [[Sets]]) every morphism that is both a [[monomorphism]] and well as an [[epimorphism]] is actually an [[isomorphism]] and thus has an [[inverse morphism]].

To see that this is not generally the case, notice that any [[partial order]] is an (necessarily unbalanced) category where every morphism is both a monomorphism as well as an epimorphism, but only its [[identity morphisms]] have [[inverse morphisms]] (as they must, by Exp.  \ref{IdentityMorphismsAreTheirOwnInverses}).
\end{example}

## In non-unital contexts

In a [[magmoid]] or [[semicategory]] (or an element of a [[semigroup]] or [[magma]]), a morphism $f:a \to b$ has a unique __[[retraction]]__ $f^{-1}:b \to a$ if

* for every morphism $g:b \to c$, $g \circ (f^{-1} \circ f) = g$,

* for every morphism $g:c \to a$, $(f^{-1} \circ f) \circ g = g$,

and a morphism $f:a \to b$ has a unique __[[section]]__ $f^{-1}:b \to a$ if

* for every morphism $g:a \to c$, $g \circ (f \circ f^{-1}) = g$,

* for every morphism $g:a \to c$, $(f \circ f^{-1})\circ g = g$,

A morphism $f:a \to b$ has a unique __inverse__ if it has a retraction that is also a section.

## Related concepts

* [[inverse morphism]], [[inverse function]]

* [[weak inverse]]

* [[bi-invertible morphism]]

* [[inversion involution]]

* [[reversible computation]]

[[!redirects inverse]]
[[!redirects inverses]]
[[!redirects invertible]]

[[!redirects inverse element]]
[[!redirects inverse elements]]
[[!redirects inverse of an element]]
[[!redirects inverses of an element]]
[[!redirects inverses of elements]]
[[!redirects invertible element]]
[[!redirects invertible elements]]

[[!redirects inverse morphism]]
[[!redirects inverse morphisms]]
[[!redirects inverse of a morphism]]
[[!redirects inverses of a morphism]]
[[!redirects inverses of morphisms]]
[[!redirects invertible morphism]]
[[!redirects invertible morphisms]]

[[!redirects invertible map]]
[[!redirects invertible maps]]


inverse diagram model of type theory

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

New models of various type theories may be formed by taking categories of [[inverse category|inverse]] [[diagrams]] in existing models.

## Related pages

* [[categorical model of dependent types]]

## References

* [[Michael Shulman]], _Univalence for inverse diagrams and homotopy canonicity_, Math. Structures Comput. Sci.25 (2015), no. 5, 1203–1277, ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253)).

* [[Nicolai Kraus]], [[Christian Sattler]], _Space-valued diagrams, type-theoretically, extended abstract_, ([arXiv:1704.04543](https://arxiv.org/abs/1704.04543)).

* [[Chris Kapulkin]], [[Peter Lumsdaine]], _Homotopical inverse diagrams in categories with attributes_, ([arXiv:1808.01816](https://arxiv.org/abs/1808.01816)).


inverse EI (∞,1)-category

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[EI (∞,1)-category]] is an [[(∞,1)-category]] in which every [[endomorphism]] is an [[equivalence in an (∞,1)-category|equivalence]]. This induces a [[preorder|ordering]] $\prec$ on the [[equivalence classes]] of [[objects]], where x≺y means that there is a noninvertible [[morphism]] $y \to x$.

An **inverse EI (∞,1)-category** is an EI (∞,1)-category which is also an [[inverse category|inverse]] (∞,1)-category, that is, is such that $\prec$ is [[well-founded relation|well-founded]], i.e., there are no infinite chains of noninvertible morphisms $ \to \to \to \cdots$.

## Examples

* For a [[compact Lie group]] $G$ the opposite of the [[orbit category]].
* The category of [[finite sets]] and [[surjective functions]].
* An EI (∞,1)-category with [[finite number|finitely many]] objects.

## Related concepts

* [[poset-stratified space]]

## References

* [[Mike Shulman]], _Univalence for inverse EI diagrams_, ([arXiv:1508.02410](https://arxiv.org/abs/1508.02410))

[[!redirects inverse EI (∞,1)-categories]]

[[!redirects inverse EI (infinity,1)-category]]
[[!redirects inverse EI (infinity,1)-categories]]


inverse function

\tableofcontents

## Definition

### In set theory

Given [[sets]] $A$ and $B$ and a [[function]] $f\colon A \to B$, the __inverse function__ of $f$ (if it exists) is the function $f^{-1}\colon B \to A$ such that both [[composition|composite functions]] $f \circ f^{-1}$ and $f^{-1} \circ f$ are [[identity functions]].  Note that $f$ has an inverse function if and only if $f$ is a [[bijection]], in which case this inverse function is unique.

Inverse functions are [[inverse morphisms]] in the [[category]] [[Set]] of sets.

More generally, in any [[concrete category]], the inverse of any [[isomorphism]] is given by the inverse of the corresponding function between [[underlying sets]].

### In type theory

In [[type theory]], there are two different kinds of inverse functions, quasi-inverse functions, various kinds of more or less coherent quasi-inverse functions, and inverse functions.

For more on this see at *[[equivalence in type theory]]*.

#### Quasi-inverse functions

Given [[types]] $A$ and $B$ and a [[function]] $f:A \to B$, a __quasi-inverse function__ of $f$ is a function $g:B \to A$ such that $g$ is a [[retraction]] and a [[section]] of $f$.

$$\prod_{b:B} (f(g(b)) =_B b) \times \prod_{a:A} (a =_A g(f(a)))$$

Equivalently, there are functions $\epsilon_0(a, b):(a =_A g(b)) \to (f(a) =_B b)$ and $\eta_0(a, b):(f(a) =_B b) \to (a =_A g(b))$ for all $a:A$ and $b:B$.

$$\prod_{a:A} \prod_{b:B} ((a =_A g(b)) \to (f(a) =_B b)) \times ((f(a) =_B b) \to (a =_A g(b)))$$

A function $f:A \to B$ might have multiple quasi-inverse functions. The type of quasi-inverses of $f$ is given by

$$\mathrm{QuasiInv}(f) \coloneqq \sum_{g:B \to A} \prod_{b:B} (f(g(b)) =_B b) \times \prod_{a:A} (a =_A g(f(a)))$$

#### $n$-quasi inverse functions

There are also variants of quasi-inverse functions where they are the inverse functions for $n$-truncated types.

For example, every function $g:B \to A$ is a (-1)-quasi inverse function, because it is an inverse function for [[h-propositions]] $A$ and $B$.

A quasi-inverse function is a 0-quasi inverse function, because it is an inverse function for [[h-sets]] $A$ and $B$.

A 1-quasi-inverse function is a quasi-inverse function with functions $\epsilon_0(a, b):(a =_A g(b)) \to (f(a) =_B b)$ and $\eta_0(a, b):(f(a) =_B b) \to (a =_A g(b))$ for all $a:A$ and $b:B$ such that for all $c:a =_A g(b)$ and $d:f(a) =_B b$,
there are functions
$$\epsilon_1(a, b, c, d):(\epsilon_0(a, b, c) =_{f(a) =_B b} d) \to (c =_{a =_A g(b)} \eta_0(a, b, d))$$
$$\eta_1(a, b, c, d):(c =_{a =_A g(b)} \eta_0(a, b, d)) \to (\epsilon_0(a, b, c) =_{f(a) =_B b} d)$$
This is an inverse function for [[h-groupoids]] $A$ and $B$.

So on and so forth.

#### Inverse functions

Given [[types]] $A$ and $B$ and a [[function]] $f:A \to B$, an __inverse function__ of $f$ is a function $g:B \to A$ with a function $\epsilon(a, b):(f(a) =_B b) \to (a =_A g(b))$ for all $a:A$ and $b:B$, such that for all identities $p:a =_A g(b)$, the fiber of $\epsilon(a, b)$ at $p$ is contractible.

$$\mathrm{isInv}(f, g) \coloneqq \prod_{a:A} \prod_{b:B} \sum_{\epsilon(a, b):(f(a) =_B b) \to (a =_A g(b))} \prod_{p:a =_A g(b)} \mathrm{isContr}(\mathrm{fiber}(\epsilon(a, b), p))$$

#### Coherent inverse functions

Given [[types]] $A$ and $B$ and a [[function]] $f:A \to B$, a __coherent inverse function__ of $f$ is a quasi-inverse function $g:B \to A$ with [[homotopies]]

$$G:\prod_{x:B} f(g(x)) =_B x \qquad H:\prod_{x:A} g(f(x)) =_A x$$

which additionally has one of the two [[homotopies]]
$$K:\prod_{x:A} \mathrm{ap}_{B}(f, g(f(x)), x, H(x)) =_{g(f(x)) =_A x} G(f(x)) \qquad L:\prod_{x:B} H(g(x)) =_{f(g(x)) =_B x} \mathrm{ap}_{A}(g, f(g(x)), x, G(x)) $$

The type of coherent inverse functions of $f$ is given by one of the following types

$$\mathrm{CohInv}_1(f) \coloneqq \sum_{g:B \to A} \sum_{G:\prod_{x:B} f(g(x)) =_B x} \sum_{H:\prod_{x:A} g(f(x)) =_A x} \left(\prod_{x:A} \mathrm{ap}_{B}(f, g(f(x)), x, H(x)) =_{g(f(x)) =_A x} G(f(x))\right)$$

$$\mathrm{CohInv}_2(f) \coloneqq \sum_{g:B \to A} \sum_{G:\prod_{x:B} f(g(x)) =_B x} \sum_{H:\prod_{x:A} g(f(x)) =_A x} \left(\prod_{x:B} H(g(x)) =_{f(g(x)) =_B x} \mathrm{ap}_{A}(g, f(g(x)), x, G(x))\right)$$

$f$ is called a **half adjoint equivalence** if it has a coherent inverse function.

## Properties

Given functions $f:A \to B$ and $g:B \to A$ and homotopies $G:\prod_{x:B} f(g(x)) =_B x$ and $H:\prod_{x:A} g(f(x)) =_A x$, there are elements
$$\mathrm{QInvToCohInv}_1(f, g, G, H):\prod_{x:A} \mathrm{ap}_{B}(f, g(f(x)), x, H(x)) =_{g(f(x)) =_A x} G(f(x))$$
$$\mathrm{QInvToCohInv}_2(f, g, G, H):\prod_{x:B} H(g(x)) =_{f(g(x)) =_B x} \mathrm{ap}_{A}(g, f(g(x)), x, G(x))$$

which satisfies the [[commutative squares]] for all elements $a:A$ and $b:B$

$$
  \array{& f(g(f(g(b)))) & \overset{G(f(g(b))}= & f(g(b)) & \\
          \mathrm{ap}_{B}(f, g(f(b)), b, H(b)) & \Vert & =_{f(g(f(g(b)))) =_B b} & \Vert & \mathrm{QInvToCohInv}_2(f, g, G, H)(b) \\
          & f(g(b)) & \underset{G(b)}= & b & \\
}$$

$$
  \array{& g(f(g(f(a)))) & \overset{H(g(f(a))}= & g(f(a)) & \\
          \mathrm{ap}_{A}(g, f(g(a)), a, G(a)) & \Vert & =_{g(f(g(f(a)))) =_A a} & \Vert & \mathrm{QInvToCohInv}_1(f, g, G, H)(a) \\
          & g(f(a)) & \underset{H(a)}= & a & \\
}$$

Thus, there are functions
$$\lambda g:B \to A.\lambda G:\prod_{x:B} f(g(x)) =_B x.\lambda H:\prod_{x:A} g(f(x)) =_A x.\mathrm{QInvToCohInv}_1(f, g, G, H):\mathrm{QInv}(f) \to \mathrm{CohInv}_1(f)$$
and
$$\lambda g:B \to A.\lambda G:\prod_{x:B} f(g(x)) =_B x.\lambda H:\prod_{x:A} g(f(x)) =_A x.\mathrm{QInvToCohInv}_2(f, g, G, H):\mathrm{QInv}(f) \to \mathrm{CohInv}_2(f)$$

Every coherent inverse function is a quasi-inverse function by forgetting the additional coherence condition.

\begin{theorem}
If a function $f:A \to B$ has a coherent inverse function, then for all elements $b:B$ the fiber of $f$ at $b$ is contractible.
\end{theorem}

\begin{proof}
...
\end{proof}

## Related concepts

* [[inverse]], [[inverse morphisms]]

* [[inverse functor]]

* [[isomorphism]]

* [[equivalence in homotopy type theory]]

Not really related

* [[inverse image]]

## References

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

[[!redirects inverse function]]
[[!redirects inverse functions]]
[[!redirects inverse map]]
[[!redirects inverse maps]]

[[!redirects quasi-inverse]]
[[!redirects quasi-inverse function]]
[[!redirects quasi-inverse functions]]
[[!redirects quasi-inverse map]]
[[!redirects quasi-inverse maps]]

[[!redirects quasi-invertible]]
[[!redirects quasi-invertible function]]
[[!redirects quasi-invertible functions]]
[[!redirects quasi-invertible map]]
[[!redirects quasi-invertible maps]]

[[!redirects coherent inverse]]
[[!redirects coherent inverse function]]
[[!redirects coherent inverse functions]]
[[!redirects coherent inverse map]]
[[!redirects coherent inverse maps]]

[[!redirects coherently invertible]]
[[!redirects coherently invertible function]]
[[!redirects coherently invertible functions]]
[[!redirects coherently invertible map]]
[[!redirects coherently invertible maps]]

[[!redirects half adjoint equivalence]]
[[!redirects half adjoint equivalences]]
[[!redirects half-adjoint equivalence]]
[[!redirects half-adjoint equivalences]]


inverse function theorem

+-- {: .rightHandSide}
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###Context###
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

### For scalar-valued functions

The **inverse function theorem** says that given any [[sequentially Cauchy complete]] [[Archimedean field]] $\mathbb{R}$ and a [[continuously differentiable function]] $f:I \to \mathbb{R}$ from an [[open interval]] $I \subseteq \mathbb{R}$ such that for all points $a \in I$,
$$\left| \frac{d f}{d x}(a) \right| \gt 0$$
the [[inverse function]] $f^{-1}:\mathrm{im}(f) \to \mathbb{R}$ exists and is unique and is defined by the first-order nonlinear [[ordinary differential equation]]

$$\left(\frac{d f}{d x} \circ f^{-1}\right) \frac{d f^{-1}}{d x} = 1$$

with initial condition $f^{-1}(f(a)) = a$. Classically, $\mathbb{R}$ is essentially unique, but constructively, there are multiple inequivalent [[sequentially Cauchy complete]] [[Archimedean fields]].

### For vector-valued functions

To be done...

## Proof

This proof is adapted from the Wikipedia article on the inverse function theorem, in the section "A proof using the contraction mapping principle", the contraction mapping principle being another name for the [[Banach fixed-point theorem]].

\begin{lemma}
Let $(-r, r)$ denote an [[open interval]] in $\mathbb{R}$ with radius $r$ and centre $0$. If $g:(-r, r) \to \mathbb{R}$ is a map such that $g(0) = 0$ and there exists a rational constant $0 \lt c \lt 1$ such that $\vert g(y) - g(x) \vert = c \vert y - x \vert$ for all $x \in (-r, r)$ and $y \in (-r, r)$, then $f = \mathrm{id}_\mathbb{R} + g$ is injective on $(-r, r)$ and
$$(-(1 - c)r, (1 - c)r) \subset f((-r, r)) \subset (-(1 + c)r, (1 + c)r)$$
where $\mathrm{id}_\mathbb{R}$ is the identity map on the real numbers, restricted in the domain to $(-r, r)$, and given an open interval $I \subseteq \mathbb{R}$, $f(I)$ is the [[image]] of $f$ under $I$.
\end{lemma}

\begin{proof}
First, the map $f$ is injective on $(-r, r)$, since if $f(x) = f(y)$, then $g(y) - g(x) = x - y$ and so $\vert g(y) - g(x) \vert = \vert y - x \vert$,
which is a contradiction unless $y = x$. Next we show
$$(-(1 - c)r, (1 - c)r) \subset f((-r, r))$$
The idea is to note that this is equivalent to, given a point $y$ in $(-(1-c) r, (1-c) r)$, find a fixed point of the map
$$F : [-r, r] \to [-r', r']$$
defined as $F(x) = y - g(x)$ where $0 \lt r' \lt r$ such that $\vert y \vert \leq (1-c)r'$ and the bar means a closed ball. To find a fixed point, we use the [[Banach fixed-point theorem]] and checking that $F$ is a well-defined strict-contraction mapping is straightforward. Finally, we have:
$$f((-r, r)) \subset (-(1 + c)r, (1 + c)r)$$
since
$$\vert f(x) \vert = \vert x + g(x) - g(0) \vert \le (1+c) \vert x \vert$$
\end{proof}

Now that we have established the lemma, we could proved the main theorem:

\begin{proof}
It is enough to prove the special case when $a = 0, b = f(a) = 0$ and $f'(0) = \mathrm{id}_\mathbb{R}$. Let $g = f - \mathrm{id}_\mathbb{R}$. The [[mean value theorem]] applied to $t \mapsto g(x + t(y - x))$ says:
$$\vert g(y) - g(x) \vert \leq \vert y - x \vert \sup_{0 \lt t \lt 1} \vert g'(x + t(y - x)) \vert$$
Since $g'(0) = \mathrm{id}_\mathbb{R} - \mathrm{id}_\mathbb{R} = 0$ and $g'$ is continuous, we can find an $r \gt 0$ such that
$$\vert g(y) - g(x)\vert \leq 2^{-1} \vert y - x \vert$$
for all $x, y$ in $(-r, r)$. Then the earlier lemma says that $f = g + \mathrm{id}_\mathbb{R}$ is injective on $(-r, r)$ and $(-r/2, r/2) \subset (-r, r)$. Then
$$f : (-r, r) \cap f^{-1}((-r/2, r/2)) \to (-r/2, r/2)$$
is bijective and thus has the inverse, where given an open interval $I \subseteq \mathbb{R}$, $f^{-1}(I)$ is the [[inverse image]] of $f$ over $I$.

to be continued...
\end{proof}

## In constructive mathematics

The given proof of the inverse function theorem above relies on the [[mean value theorem]], which in [[constructive mathematics]] is only true for [[uniformly differentiable functions]]. There might be other proofs which might not rely on the [[mean value theorem]] and could prove the inverse function theorem for continuously differentiable functions.

## As an axiom for Archimedean ordered fields

While [[continuously differentiable functions]] and [[locally uniformly differentiable functions]] are well-defined in every [[Archimedean ordered field]], the inverse function theorem does not hold in all Archimedean ordered fields. One could instead consider adding the inverse function theorem as an axiom to an arbitrary Archimedean ordered field $R$. There are a few axioms which could be used here

* For any Archimedean ordered field $R$, the **continuously differentiable inverse function axiom** for $R$ states that given any differentiable function $f:I \to R$ on an open subinterval $I \subseteq R$ such that its derivative $\frac{d f}{d x}:I \to R$ is pointwise continuous and always apart from zero, there is a unique differentiable function $f^{-1}:\mathrm{im}(f) \to R$ called the **inverse function** with pointwise continuous derivative $\frac{d f^{-1}}{d x}:\mathrm{im}(f) \to R$ such that $f^{-1}(f(a)) = a$ and for all elements $a \in \mathrm{im}(f)$ and
$$\frac{d f}{d x}\left(f^{-1}(a)\right) \cdot \frac{d f^{-1}}{d x}\left(a\right) = 1$$

* For any Archimedean ordered field $R$, the **locally uniformly differentiable inverse function axiom** for $R$ states that given any differentiable function $f:I \to R$ on an open subinterval $I \subseteq R$ such that its derivative $\frac{d f}{d x}:I \to R$ is locally uniformly continuous and always apart from zero, there is a unique differentiable function $f^{-1}:\mathrm{im}(f) \to R$ called the **inverse function** with locally uniformly continuous derivative $\frac{d f^{-1}}{d x}:\mathrm{im}(f) \to R$ such that $f^{-1}(f(a)) = a$ and for all elements $a \in \mathrm{im}(f)$ and
$$\frac{d f}{d x}\left(f^{-1}(a)\right) \cdot \frac{d f^{-1}}{d x}\left(a\right) = 1$$

The former is stronger than the latter, although they are equivalent in the presence of [[excluded middle]]. If $R$ satisfies the locally uniformly continuous inverse function axiom, then $R$ is a [[real closed field]].

There is also a third possible axiom in [[dependent type theory]], which uses the [[shape modality]]. Given an Archimedean ordered field $R$, let $\esh \coloneqq L_R$ be the [[shape modality]], the [[localization of a type|localization]] at $R$. A type $T$ is shapewise contractible if its shape is contractible. Given an open interval $I \subseteq R$, a function $f:I \to R$ is shapewise continuous if the graph of $f$, the set of all pairs $(x, y)$ in $I \times R$ such that $y = f(x)$, is shapewise contractible:

$$\mathrm{isShapewiseContinuous}(f) \coloneqq \mathrm{isContr}\left(\esh\left(\sum_{x:I} \sum_{y:R} y =_{R} f(x)\right)\right)$$

* For any Archimedean ordered field $R$, the **shapewise continuously differentiable inverse function axiom** for $R$ states that given any differentiable function $f:I \to R$ on an open subinterval $I \subseteq R$ such that its derivative $\frac{d f}{d x}:I \to R$ is shapewise continuous and always apart from zero, there is a unique differentiable function $f^{-1}:\mathrm{im}(f) \to R$ called the **inverse function** with shapewise continuous derivative $\frac{d f^{-1}}{d x}:\mathrm{im}(f) \to R$ such that $f^{-1}(f(a)) = a$ and for all elements $a \in \mathrm{im}(f)$ and
$$\frac{d f}{d x}\left(f^{-1}(a)\right) \cdot \frac{d f^{-1}}{d x}\left(a\right) = 1$$

## Related concepts

* [[mean value theorem]]

* [[Banach fixed-point theorem]]

* [[Picard–Lindelöf theorem]]

* [[regular value]]

* [[Mather's stability theorem]]

* [[real quadratic function]]

* [[real square root]]

## References

* [[Terence Tao]], *Analysis II. Texts and Readings in Mathematics.* (1st ed. 2016 edition). New Delhi: Hindustan Book Agency. ISBN:978-9380250656

See also:

* Wikipedia, _[Inverse function theorem](https://en.wikipedia.org/wiki/Inverse_function_theorem)_

[[!redirects inverse function axiom]]
[[!redirects inverse function principle]]