De Morgan algebra

> This page is about [[distributive lattices]] equipped with a contravariant involution.  For [[Heyting algebras]] satisfying [[de Morgan's law]], which are also sometimes called "De Morgan algebras", see [[De Morgan Heyting algebra]].

# De Morgan algebras

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## Definition

A **De Morgan algebra** is a [[distributive lattice]] $D$ equipped with a [[contravariant functor|contravariant]] [[involution]], i.e. a map $\neg : D^{op}\to D$ such that $\neg\neg A = A$.  This implies that $\neg$ satisfies De Morgan's laws: $\neg(A\wedge B) = \neg A \vee \neg B$ and $\neg (A\vee B) = \neg A \wedge \neg B$.

## Examples

* Any [[Boolean algebra]] is a De Morgan algebra, with $\neg$ the logical negation.
* Any [[star-autonomous category|star-autonomous poset]] that happens to be a [[distributive lattice]] is a De Morgan algebra, with $\neg$ the star-autonomous involution $(-)^*$.
* The unit interval $[0,1]$ is a De Morgan algebra, with $\neg x = (1-x)$.
* The four-point algebra with two parallel lines: $\{0, a, b, 1\}$ with $\neg a = a$, $\neg b = b$, $a \vee b = 1$ and $a \wedge b = 0$ is a  small example of a DeMorgan algebra which is not Boolean.

## Related concepts

* [[De Morgan duality]]
* De Morgan algebras are used in one form of [[cubical type theory]]

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De Morgan duality

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# De Morgan duality
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## Idea

In [[logic]], De Morgan duality is a [[duality]] between [[intuitionistic logic]] and dual-intuitionistic [[paraconsistent logic]].  In [[classical logic]] and [[linear logic]], it is a [[self-duality]] mediated by [[negation]].  Although it goes back to [[Aristotle]] (at least), its discovery is generally attributed to [[Augustus De Morgan]].

## The dualities

More explicitly, __De Morgan duality__ is the duality between logical operators as shown in the table below:

| Intuitionistic operator                  |   | Dual-intuitionistic operator               |
| ---------------------------------------- | - | ------------------------------------------ |
| $\top$ ([[truth]])                       |   | $\bot$ ([[falsehood]])                     |
| $\wedge$ ([[conjunction]])               |   | $\vee$ ([[disjunction]])                   |
| $\Rightarrow$ ([[conditional]])          |   | $\setminus$ ([[co-Heyting algebra|subtraction]])                  |
| $\Leftrightarrow$ ([[biconditional]])    |   | $+$ ([[exclusive disjunction]])            |
| $\neg$ ($p \Rightarrow \bot$)            |   | $-$ ($\top \setminus p$)                   |
| $\forall$ ([[universal quantification]]) |   | $\exists$ ([[existential quantification]]) |
| $\Box$ ([[necessity]])                   |   | $\lozenge$ ([[possibility]])               |

The first two operators in each column exist in both intuitionistic and dual-intuitionistic [[propositional logic]] and the last two in each column exist in both forms of [[predicate logic]] and [[modal logic]] (respectively), but they are still dual as shown.  All of these exist in [[classical logic]] (although some of the paraconsistent operators are not widely used), and the two forms of negation ($\neg$ and $-$) are the same there.

In [[linear logic]], this extends to a duality between conjunctive and disjunctive operators:

| Conjunctive operator |   | Disjunctive operator |
| -------------------- | - | -------------------- |
| $\top$               |   | $0$                  |
| $1$                  |   | $\bot$               |
| $\&$                 |   | $\oplus$             |
| $\otimes$            |   | $\parr$              |
| $^\bot$              |   | $^\bot$              |
| $\bigwedge$          |   | $\bigvee$            |
| $!$                  |   | $?$                  |

As with classical negation, linear [[negation]] is self-dual. (For the [[categorical semantics]] of this see at [[dualizing object]] and at _[Wirthmüller context -- Comparison of push-forwards](Wirthmüller+context#PushforwardsIntertwinedByDuality)_.)

The first two rows of the intuitionistic/dual-intuitionistic/classical duality generalise to arbitrary [[lattices]], including [[subobject]] lattices in [[coherent category|coherent categories]], and from there to the duality between [[limits]] and [[colimits]] in category theory:

| Limit               |   | Colimit            |
| ------------------- | - | ------------------ |
| [[top]]             |   | [[bottom]]         |
| [[meet]]            |   | [[join]]           |
| [[intersection]]    |   | [[union]]          |
| [[terminal object]] |   | [[initial object]] |
| ⋮                   |   | ⋮                  |

So in a way, all [[duality]] in category theory is a generalisation of De Morgan duality.

## Related concepts

* [[De Morgan laws]]

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De Morgan Heyting algebra

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## Idea

A [[Heyting algebra]] is called a **De Morgan Heyting algebra** if it satisfies the [[De Morgan laws]], which may be considered a weak form of the [[law of excluded middle]].  The corresponding [[logic]] is an interesting intermediate logic between [[intuitionistic logic]] and [[classical logic]].

De Morgan Heyting algebras are also known as **Stone algebras** or **Stone lattices**.  They are also often called simply **de Morgan algebras**, although that clashes with a different notion; see [[De Morgan algebra]].

## Definition

A [[Heyting algebra]] $M$ that satisfies the following equivalent conditions is called a _De Morgan Heyting algebra_:

1. For all $a,b \in M$: $\not(a\wedge b) =\not a\vee\not b$. (second **[[De Morgan law]]**)

2. For all $a\in M$: $\not a\vee\not\not a=\top$. (**[[weak excluded middle]]**)

3. For all $a,b\in M$: $\not\not (a\vee b) =\not\not a\vee \not\not b$.

+-- {: .proof}
###### Proof of equivalence
(2. implies 1.) First, it is automatic that $\neg a \vee \neg b \leq \neg(a \wedge b)$. Now, in a distributive lattice such as a Heyting algebra, if $x \wedge y = \bot$ and $x \vee z = \top$, then $y \leq z$ (consider applying $- \wedge y$ to the second equation). Putting $x = \neg \neg a \wedge \neg \neg b$ and $y = \neg(a \wedge b)$ and $z = \neg a \vee \neg b$, we check
$$\neg\neg a \wedge \neg\neg b \wedge \neg(a \wedge b) = \neg\neg (a \wedge b) \wedge \neg(a \wedge b) = \bot$$
(since $\neg\neg$ preserves meets; see [[Heyting algebra]]) and
$$(\neg\neg a \wedge \neg\neg b) \vee \neg a \vee \neg b = (\neg \neg a \vee \neg a \vee \neg b) \wedge (\neg\neg b \vee \neg a \vee \neg b) = (\top \vee \neg b) \wedge (\neg a \vee \top) = \top \wedge \top = \top$$
whence $\neg(a \wedge b) \leq \neg a \vee \neg b$ follows.

(1. implies 3.) Given that $\neg(x \wedge y) = \neg x \vee \neg y$ for all $x, y$, put $x = \neg a$ and $y = \neg b$. Then
$$\neg\neg(a \vee b) = \neg(\neg a \wedge \neg b) = \neg(x \wedge y) = \neg x \vee \neg y = \neg\neg a \vee \neg\neg b$$
as desired.

(3. implies 2.) We have $\neg a = \neg\neg\neg a$, and so we may calculate
$$\neg a \vee \neg\neg a = \neg\neg\neg a \vee \neg\neg a = \neg\neg(\neg a \vee a) = \neg(\neg \neg a \wedge \neg a) = \neg(\bot) = \top$$
as desired.
=--

### Remark

The dual _first_ De Morgan law $\not (a\vee b) = \not a\wedge\not b$ is valid in every Heyting algebra.

## Examples

* Every [[Boolean algebra]] is a De Morgan Heyting algebra.
* The [[lattice of open subsets]] of the [[Sierpinski space]] is a De Morgan Heyting algebra (assuming a classical metalogic).
* The [[subobject classifier]] in any [[topos]] satisfying the [[ultrafilter principle]] is a De Morgan Heyting algebra.

## Related entries

* [[De Morgan topos]]
* [[De Morganization]]
* [[Ore condition]]
* [[Heyting algebra]]
* [[co-Heyting algebra]]
* [[Boolean algebra]]
* [[ultrafilter principle]]

## References

* [[Francis Borceux]], _Handbook of Categorical Algebra vol.3_ , Cambridge UP 1994. (sections 1.2, 7.3)

* K. B. Lee, _Equational Classes of Distributed Pseudo-Complemented Lattices_ , Can. J. Math. **22** (1970) pp.881-891. ([pdf](http://cms.math.ca/openaccess/cjm/v22/cjm1970v22.0881-0891.pdf))

* M. E. Szabo, _Categorical De Morgan Laws_ , Alg. Universalis **12** (1981) pp.93-102.

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De Morgan Heyting category

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## Definition

A **De Morgan Heyting category** is a [[Heyting category]] $C$ (such as a [[topos]] or [[pretopos]]) in which for every object $X \in \mathrm{Ob}(C)$ the pseudocomplement of the intersection of two subobjects $A \in \mathrm{Sub}(X)$ and $B \in \mathrm{Sub}(X)$ is the union of the pseudocomplements of $A$ and $B$, $((A \cap B) \implies \emptyset) = (A \implies \emptyset) \cup (B \implies \emptyset)$. Equivalently, it is a Heyting category $C$ in which for every object $X \in \mathrm{Ob}(C)$ the union of the pseudocomplement of every [[subobject]] $A \in \mathrm{Sub}(X)$ and the double pseudocomplement of $A$ is an [[improper subobject]], $A \in \mathrm{Sub}(X)$, $(A \implies \emptyset) \cup ((A \implies \emptyset) \implies \emptyset) = \Im(X)$.

Therefore, the [[subobject poset]] $Sub(X)$ of any [[object]] $X \in \mathrm{Ob}(C)$ is a [[De Morgan Heyting algebra]]. [[De Morgan's law]] and [[weak excluded middle]] hold in the [[internal logic]] of $C$.

In addition, every De Morgan Heyting category $C$ is a [[first-order hyperdoctrine|first order]] De Morgan Heyting [[hyperdoctrine]] given by the [[subobject poset]] [[functor]] $\mathrm{Sub}:\mathcal{C}^{op} \to DeMorgHeytAlg$.

## Examples

* Every [[De Morgan topos]] is a De Morgan Heyting category.

* The category [[Set]] of [[sets]] and [[functions]] in [[strongly predicative]] [[constructive mathematics]] with [[weak excluded middle]] is a De Morgan Heyting category.

## See also

* [[De Morgan Heyting algebra]]

* [[Heyting category]]

* [[Boolean category]]

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De Morgan laws

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## Idea

In [[logic]], the __De Morgan laws__ refer to the the following laws:
$$ \array {
   \neg(p \wedge q) \;\iff\; \neg{p} \vee \neg{q} ,
   \\
   \neg(p \vee q) \;\iff\; \neg{p} \wedge \neg{q} .
} $$

More generally, the __De Morgan laws__ are the statements, valid in various forms of logic, that [[De Morgan duality]] is mediated by [[negation]].

## In constructive mathematics

In the [[foundations]] of [[constructive mathematics]], __De Morgan\'s Law__ usually means the statement
$$ \neg(p \wedge q) \Rightarrow \neg{p} \vee \neg{q} ,$$
since every other aspect of the first two lines is already constructively valid, the claim that negation mediates the De Morgan self-duality already has a name (the [[double negation]] law, equivalent to the principle of [[excluded middle]]), and no other line involves only operators that appear in intuitionstic [[propositional calculus]].

### Equivalent statements

This de Morgan's law is equivalent to the following:

* [[weak excluded middle]]

* That the [[negation]] of every [[equivalence relation]] is an [[apartness relation]]

## In homotopy type theory

In the context of [[homotopy type theory]], there are two versions of the constructive De Morgan's law, depending on whether the "or" in the law is interpreted as a [[propositional truncation|propositionally truncated]] [[sum type]]
$$ \prod_{A,B} \neg(A \wedge B) \to \Vert \neg A + \neg B \Vert $$
or as an untruncated sum type:
$$ \prod_{A,B} \neg(A \wedge B) \to (\neg A + \neg B). $$

However, [[Martin Escardo]] proved that the truncated and untruncated versions of De Morgan's law are the same:

+-- {: .un_lemma}
###### Lemma
Truncated De Morgan's law implies [[weak excluded middle|weak]] [[excluded middle]]:
$$ \prod_{A} \neg A + \neg\neg A.$$
=--
Note that truncation or its absence is irrelevant in weak excluded middle, since $\neg A$ and $\neg\neg A$ are mutually exclusive so that $\neg A + \neg\neg A$ is always a proposition.
+-- {: .proof}
###### Proof
Let $B=\neg A$ in the truncated De Morgan's law, and notice that $\neg(A\wedge \neg A)$ always holds.
=--

+-- {: .un_lemma}
###### Lemma
Weak excluded middle implies that binary sums of negations have split support:
$$\prod_{A,B} \Vert \neg A + \neg B \Vert \to (\neg A + \neg B). $$
=--
+-- {: .proof}
###### Proof
By weak excluded middle, either $\neg A$ or $\neg\neg A$.  In the first case, $\neg A + \neg B$ is just true.  In the second case, either $\neg B$ or $\neg\neg B$.  In the first subcase, $\neg A + \neg B$ is again just true.  In the second subcase, we have $\neg\neg A$ and $\neg\neg B$, whence $(\neg A + \neg B) = 0$ and in particular is a proposition.
=--

+-- {: .un_theorem}
###### Theorem
Truncated De Morgan's law implies untruncated De Morgan's law,
$$ \prod_{A,B} \neg(A \wedge B) \to (\neg A + \neg B) $$
=--
+-- {: .proof}
###### Proof
Combine the two lemmas.
=--

## In the theory of pseudolattice ordered abelian groups

Peter Freyd's definition of a [[pseudolattice ordered abelian group]] in [Freyd 08](#Freyd08) is equational, and so is a [[Lawvere theory]] and could be defined in a category with finite products and a generic object $A$ where every object is equivalent to a finite product of $A$.

Freyd then showed that the de Morgan laws are satisfied in any [[pseudolattice ordered abelian group]]:

$$-(p \vee q) = (-p) \wedge (-q)$$
$$-(p \wedge q) = (-p) \vee (-q)$$

with the join and meet being the pseudolattice operations and negation being the negation of the abelian group.

In particular, the de Morgan laws are valid for the [[integers]], the [[rational numbers]], and the [[Dedekind real numbers]], with the join being the maximum function and the meet being the minimum function.

## Related concepts

* [[De Morgan Heyting algebra]]

* [[De Morgan Heyting category]]

* [[De Morgan algebra]]

* [[De Morgan topos]]

* [[excluded middle]]

* [[De Morgan duality]]

## References

Named after *[[Augustus De Morgan]]*.

* {#Freyd08} [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

See also:

* Wikipedia, *[De Morgan's laws](https://en.wikipedia.org/wiki/De_Morgan%27s_laws)*

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De Morgan topos

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## Idea

A **De Morgan topos** $\mathcal{E}$ is a middle thing between a [[Boolean topos]] with a [[classical logic|classical internal logic]] and a general [[topos]] with an [[intuitionistic logic|intuitionistic internal logic]], in that the lattices of [[subobjects]] in $\mathcal{E}$ obey a weak form of the [[law of excluded middle]] which defines a [[De Morgan Heyting algebra]].

The de Morgan property has many equivalent formulations and many facets making the class of de Morgan toposes an important one in [[topos theory]]. From the perspective on toposes as generalized spaces they correspond to _extremally disconnected spaces_ in [[topology]].

## Definition

A topos $\mathcal{E}$ is called a __De Morgan topos__ if its [[subobject classifier]] $\Omega$ is an internal [[De Morgan Heyting algebra]].

## Examples

* Every [[Boolean topos]] is a De Morgan topos.

* A simple example of a topos that is De Morgan but not Boolean is the [[Sierpinski topos]] $Set^{\to}$ the [[arrow category]] of $Set$: the diagram category $\to$ is easily seen to satify the [[Ore condition]] (cf. [below](#presheaf-deMorgan)).

* The topos $Sh(X)$ of sheaves on a topological space $X$ is De Morgan precisely iff $X$ is [[extremally disconnected topological space|extremally disconnected]], i.e. the closure of every open subset is open. (Since $Set^{\to}$ is equivalently the topos of sheaves on [[Sierpinski space]] it instantiates this case as well.)

* [[injective topos|Injective Grothendieck toposes]] are De Morgan (cf. [Johnstone 2002, p.739](#Johnstone)).

## Some equivalent formulations

The importance of the concept of a De Morgan topos stems partly from the fact that for many statements whose classical (Boolean) validity is lost in intuitionistic logic the De Morgan property delineates precisely the range of toposes where the statement is valid. An illustration of this is e.g. the implication that maximal ideals are prime in commutative rings.

Though such statements are in fact equivalent to the De Morgan property they are better viewed as higher-order properties of De Morgan toposes.

In contrast, there exists also a long list, mostly due to [Johnstone (1979)](#Johnstone79), of statements equivalent to the De Morgan property, that are logically of the same complexity and are often more convenient to work with than the original definition. Some of the most important of these appear in the following:

+-- {: .num_prop #deMorgan_equivalents}
###### Proposition
Let $\mathcal{E}$ be a topos. The following are equivalent:

* $\mathcal{E}$ is a De Morgan topos.

* For all formulae $\varphi,\psi$ : $\mathcal{E}\models \neg (\varphi\wedge\psi)\Leftrightarrow(\neg\varphi\vee\neg\psi)$ .

* For every formula $\varphi$ : $\mathcal{E}\models \neg\varphi\vee\neg\neg\varphi$.

* For every subobject $X\rightarrowtail Y$: $\neg X\vee\neg\neg X=Y$.

* $Sub(X)$ is a De Morgan Heyting algebra for every $X\in\mathcal{E}$.

* The canonical monomorphism $(\top,\bot):1\coprod 1\rightarrowtail\Omega_{\neg\neg}$ to the subobject classifier of $Sh_{\neg\neg}(\mathcal{E})$ is an isomorphism.

* $\Omega_{\neg\neg}$ is [[decidable object|decidable]].

* $1\coprod 1$ is a retract of $\Omega_{\neg\neg}$.

* $1\coprod 1$ is [[injective object|injective]].

* Every $\neg\neg$-sheaf is decidable.

* $\bot:1\to\Omega$ has a [[complement]].

* The [[real numbers object|object of Dedekind reals]] coincides with the object of [[Dedekind-MacNeille completion|Dedekind-MacNeille reals]].
=--

The first few equivalences are based mainly on the validity of the "law of excluded middle" for elements of the form $\neg a$ in any [[De Morgan Heyting algebra]]. For the rest see e.g. [Johnstone (2002, pp.999-1000)](#Johnstone). The very last point is supplied by [Johnstone (1979)](#Johnstone79).

## Properties

The following shows that 'being De Morgan' is a _local property_ i.e. stable under slicing. This is immediate from the fact that the functor $\mathcal{E}\to\mathcal{E}/A$ sending $X$ to the projection $p_A: X\times A\to A$ is [[logical functor|logical]] hence preserves the various equivalent formulations of De Morgan's law.

+-- {: .num_prop #local_property}
###### Proposition
Let $A$ be an object of the De Morgan topos $\mathcal{E}$. Then the [[slice topos]] $\mathcal{E}/A$ is De Morgan as well.
=--

+-- {: .num_prop #presheaf-deMorgan}
###### Proposition
Let $\mathcal{C}$ be a small category. The [[presheaf topos]] $Set^{\mathcal{C}^{op}}$ is De Morgan precisely if $\mathcal{C}$ satifies the [[Ore condition]].
=--

This result due to [[Peter Johnstone]] appears e.g. in [Johnstone (1979)](#Johnstone79). Compare also the generalizations in [Kock-Reyes (1994)](#KR94) and [Caramello (2012)](#Caramello12).

Since the [[dense topology]] coincides with the [[atomic site|atomic topology]] on $\mathcal{C}$ satisfying the Ore condition the preceding proposition implies that the [[double negation|double negation subtopos]] of a De Morgan presheaf topos $Set^{\mathcal{C}^{op}}$ is always [[atomic topos|atomic]].

### De Morganization

It was discovered by [[Olivia Caramello]] that every topos $\mathcal{E}$ can be associated with a De Morgan topos $Sh_m(\mathcal{E})$, its [[De Morganization]], in a universal way:

+--{: .num_prop #largest_deMorgan}
###### Proposition
Given a topos $\mathcal{E}$. There exists a unique largest [[dense subtopos]] $Sh_m(\mathcal{E})$ of $\mathcal{E}$ that is a De Morgan topos.
=--

For a proof see [Caramello (2009)](#Caramello09). The $m$ that occurs here, refers to the _De Morgan topology_ , cf. [[De Morganization]].

As $Sh_{\neg\neg}(\mathcal{E})$, the sheaf subtopos for the [[double negation|double negation topology]] $\neg\neg$ , is the _smallest_ dense De Morgan subtopos, we see that all dense De Morgan subtoposes lie in the interval between $Sh_{\neg\neg}(\mathcal{E})$ and $Sh_{m}(\mathcal{E})$.

In analogy with [[skeletal geometric morphism|skeletal geometric morphisms]] that preserve $\neg\neg$-sheaves, geometric morphisms $\mathcal{F}\to\mathcal{E}$ that map $m$-sheaves in $\mathcal{F}$ to $m$-sheaves in $\mathcal{E}$ are called $m$_-skeletal_. They occur in the following characterization of De Morgan toposes (cf. [[De Morganization]]):

+--{: .num_prop #deMorgan_m-skeletal}
###### Proposition
A topos $\mathcal{E}$ is De Morgan iff every geometric morphism $\mathcal{F}\to\mathcal{E}$ is $m$-skeletal. $\qed$
=--

### The Gleason cover

Another way to associate a De Morgan topos to an arbitrary topos $\mathcal{E}$ was proposed by [[Peter Johnstone]] in the late 70s (Johnstone [1979b](#Johnstone79b), [1980](#J80)). The so called _Gleason cover_ $\gamma\mathcal{E}$ of $\mathcal{E}$ generalizes a construction in [[topology]] that covers an arbitrary [[space]] by an extremally disconnected topological space.

......

### Relation to cohesion

Given that the [above conditions](#deMorgan_equivalents) concern $1\coprod 1$ and the contractability of $\Omega_{\neg\neg}$ it comes as no surprise that the De Morgan property interacts interestingly with [[Lawvere|Lawvere's]] axiomatic approach to [[cohesion]] and, in particular, with the part of it that concerns the connectedness of the subobject classifier in a cohesive topos of spaces. A first indication of this is the following:

+-- {: .num_prop #omega_connected}
###### Proposition
Let $Set^{\mathcal{C}^{op}}$ be a [[presheaf topos]]. Then its [[subobject classifier]] is connected iff $1\in Set^{\mathcal{C}^{op}}$ is a [[connected]] object and $Set^{\mathcal{C}^{op}}$ is not de Morgan iff there exists a connected object $X\in Set^{\mathcal{C}^{op}}$ such that $1\coprod 1\rightarrowtail X$ .
=--

This result appears in [La Palme-Reyes-Reyes-Zolfaghari (2004, p.220)](#RZZZ04) where it is attributed to [[Lawvere]].

The following result due to [Mielke (1984)](#Mielke84) shows that the De Morgan property coincides with the (local) absence of a non-trivial [[interval object|interval objects]] in a topos.

Here by an interval object we mean an internal linearly ordered object $I$ with disjoint least and greatest elements $m:1\to I$ and $M:1\to I$, respectively, i.e. roughly the sort of thing that is classified by [[sSet]].

An interval object is _trivial_ if $I\simeq I_1\coprod I_2$ and $m,M$ factor through $I_1,I_2$, respectively. A topos $\mathcal{E}$ is said to be **homotopically trivial** if every interval object in $\mathcal{E}$ is trivial, $\mathcal{E}$ is said to _locally homotopically trivial_ if the [[slice topos]] $\mathcal{E}/X$ is homotopically trivial for all $X\in\mathcal{E}$.

+-- {: .num_prop #deMorgan_interval}
###### Proposition
A topos $\mathcal{E}$ is locally homotopically trivial iff $\mathcal{E}$ is a De Morgan topos. If subobjects of $1$ generate, then $\mathcal{E}$ is homotopically trivial iff $\mathcal{E}$ is a De Morgan topos.
=--

For situations where subobjects of $1$ generate see e.g. [Johnstone (1977, sec.5.3 pp.145ff)](#Johnstone77). For a [[Grothendieck topos]] $\mathcal{E}$, this happens precisely when $\mathcal{E}$ is equivalent to $Sh(L)$ for some [[locale]] $L$. From e.g. [Borceux (1994, p.443)](#Borceux94) it follows then that $Sh(L)$ _is homotopically trivial_ precisely when $L$ is a _De Morgan locale_.

### Relation to model theory

....

+-- {: .num_prop #model_companion}
###### Proposition
Let $\mathcal{T}$ be a coherent theory. Then $\mathcal{T}$ admits a [[model companion]] and $Mod(\mathcal{T})$ has the [[amalgamation property]] iff for each $B\in\mathcal{T}$ , the lattice of subobjects of $B$ admits a negation satisfying $\neg A\vee\neg\neg A = B$.
=--

This result appears without proof in [Harun (1976, p.73)](#Harun96) where it is attributed to a preprint of [[André Joyal]] and [[Gonzalo E. Reyes]].

## Related entries

* [[De Morganization]]
* [[De Morgan Heyting algebra]]
* [[De Morgan Heyting category]]
* [[Ore condition]]
* [[double negation]]
* [[topos]]
* [[Boolean topos]]
* [[law of excluded middle]]
* [[skeletal geometric morphism]]
* [[ultrafilter principle]]

## References

* A. Bagchi, _De Morgan's law and related identities in classifying topoi_ , Proc. CT91 pp.1-32, American Mathematical Society 1992.

* A. Bagchi, _Lee Identities in Topoi I_ , JPAA **120** (1997) pp.143-159.

* [[Francis Borceux]], _Handbook of Categorical Algebra vol.3_ , Cambridge UP 1994. (sections 1.2, 7.3)

* {#Caramello09}[[Olivia Caramello]], _De Morgan classifying toposes_ , Adv. in Math. **222** (2009) pp.2117-2144. ([arXiv:0808.1519](http://arxiv.org/abs/0808.1519))

* {#Caramello12}[[Olivia Caramello]], _Topologies for intermediate logics_ , arXiv:1205.2547 (2012). ([arXiv:1205.2547](http://arxiv.org/abs/1205.2547))

* {#CJ09}[[Olivia Caramello]], [[Peter Johnstone]], _De Morgan's law and the Theory of Fields_ , Adv. in Math. **222** (2009) pp.2145-2152. ([arXiv:0808.1572](http://arxiv.org/abs/0808.1572))

* {#Harun76}R. Harun, _Applications of De Morgan toposes and the Gleason cover_ , PhD Montréal 1996. ([pdf 4.6MB](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp04/mq29711.pdf))

* {#Johnstone77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint New York 2014, exercise 5.5.3 p.162)

* {#Johnstone79}[[Peter Johnstone]], _Conditions Related to De Morgan's Law_ , pp.479-491 in LNM **753** Springer Heidelberg 1979.

* {#Johnstone79b}[[Peter Johnstone]], _Another Condition Equivalent to De Morgan's Law_ , Comm. Alg. **7** (1979) pp.1309-1312.

* {#Johnstone80}[[Peter Johnstone]], _The Gleason Cover of a Topos I_ , JPAA **19** (1980) pp.171-192.

* {#Johnstone81}[[Peter Johnstone]], _The Gleason Cover of a Topos II_ , JPAA **22** (1981) pp.229-247.

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol. II_, Oxford UP 2002. (section D4.6, pp.998ff)

* {#Johnstone13}[[Peter Johnstone]], _The Gleason Cover of a Realizability Topos_ , TAC **28** no.32 (2013) pp.1139-1152. ([abstract](http://www.tac.mta.ca/tac/volumes/28/32/28-32abs.html))

* {#KR94}[[Anders Kock]], [[Gonzalo E. Reyes]], _Relatively Boolean and De Morgan Toposes and Locales_ , Cah. Top. Géom. Diff. Cat. **35** no.3 (1994) pp.249-261. ([numdam](http://www.numdam.org/numdam-bin/item?id=CTGDC_1994__35_3_249_0))

* {#LawvereMenni15} [[William Lawvere]], [[Matías Menni]], _Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness_ , TAC **30** no.26 (2015) pp.909-932. ([abstract](http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html))

* K. B. Lee, _Equational Classes of Distributed Pseudo-Complemented Lattices_ , Can. J. Math. **22** (1970) pp.881-891. ([pdf](http://cms.math.ca/openaccess/cjm/v22/cjm1970v22.0881-0891.pdf))

* B. Loiseau, M.-M. Mawanda, _On Natural Number Objects, Finiteness and Kripke-Platek Models in Toposes_ , JPAA **61** (1989) pp.257-266.

* {#Mawanda88} M.-M. Mawanda, _Well-ordering and Choice in Toposes_ , JPAA **50** (1988) pp.171-184.

* {#Mielke84} M. V. Mielke, _Homotopically Trivial Toposes_ , Pacific J. Math. **110** no.1 (1984) pp.171-182. ([euclid](https://projecteuclid.org/euclid.pjm/1102711108))

* {#Mielke93} M. V. Mielke, _Topos based homology theory_ , Comment. Math. Univ. Carolinae **34** (1993) pp.549-565. ([pdf](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/CMUC/pdf/cmuc9303/mielke.pdf))

* [[Chris Mulvey]], _The Maximality of Filters_ , JPAA **68** (1990) pp.253-258.

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* M. E. Szabo, _Categorical De Morgan laws_ , Alg. Universalis **12** (1981) pp.93-102.

[[!redirects De Morgan topos]]
[[!redirects de Morgan topos]]
[[!redirects De Morgan topoi]]
[[!redirects de Morgan topoi]]
[[!redirects De Morgan toposes]]
[[!redirects de Morgan toposes]]

[[!redirects DeMorgan topos]]
[[!redirects demorgan topos]]
[[!redirects De Morgan Topos]]

[[!redirects De Morganization]]
[[!redirects De Morganisation]]
[[!redirects De Morganizations]]
[[!redirects De Morganisations]]


De Morganization

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

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

[[!redirects demorganization]]
[[!redirects DeMorganization]]
[[!redirects de Morganization]]
[[!redirects De Morganisation]]
[[!redirects De Morgan topology]]
[[!redirects de Morgan topology]]
[[!redirects DeMorgan topology]]

## Idea

The **De Morganization** of a [[topos]] $\mathcal{E}$ is a universal way to turn $\mathcal{E}$ into a [[de Morgan topos]] $Sh_m(\mathcal{E})$ with the use of a certain [[Lawvere-Tierney topology]] $m$, called the **De Morgan topology** on $\mathcal{E}$.

This can be viewed as an analogue to the Booleanization $Sh_{\neg\neg}(\mathcal{E})$ of $\mathcal{E}$ with the help of the [[double negation|double negation topology]] $\neg\neg$ .

## Definition

Let $\mathcal{E}$ be a [[topos]]. The _De Morgan topology_ $m$ on $\mathcal{E}$ is defined as the smallest [[Lawvere-Tierney topology]] $j$ such that the canonical monomorphism $(\top,\bot): 1\coprod 1\rightarrowtail \Omega_{\neg\neg}$ is $j$-dense. The _De Morganization_ of $\mathcal{E}$ is the associated topos $Sh_m(\mathcal{E})$ of $m$-sheaves.

### Remark

Here $\Omega_{\neg\neg}$ denotes the [[subobject classifier]] for $Sh_{\neg\neg}(\mathcal{E})$ with $\neg\neg$ the [[double negation|double negation topology]] on $\mathcal{E}$. The De Morgan topology $m$ is well-defined due to _Joyal's lemma_ (cf. [Johnstone 1977](#Johnstone77), p.99; or [Johnstone 2002](#Johnstone), p.215). Compare its definition to this [proposition](double+negation#smallest_j-dense) about $Sh_{\neg\neg}(\mathcal{E})$ .

## Example

+--{: .num_prop}
###### Proposition
The De Morganization of the [[classifying topos]] for the theory of [[field|fields]] is the classifying topos for the geometric theory of fields of finite characteristic, in which every element is algebraic over the [[prime field]].
=--

This is proposition 2.3 in [Caramello-Johnstone (2009)](#CJ09).

## Properties

+--{: .num_prop}
###### Proposition
The De Morgan topology $m$ is the [[dense subtopos|smallest dense topology]] $j$ on $\mathcal{E}$ , i.e. $j\leq \neg\neg$ , such that $Sh_j(\mathcal{E})$ is a [[De Morgan topos]].
=--

This appears as theorem 1 in [Caramello (2009)](#Caramello09). In other words, $Sh_{m}(\mathcal{E})$ is the _largest dense De Morgan subtopos_ of $\mathcal{E}$

+--{: .num_prop #DeMorgan_mono}
###### Proposition
The De Morgan topology $m$ is the smallest [[Lawvere-Tierney topology|topology]] $j$ on $\mathcal{E}$ such that all monomorphisms of the form $\neg A\vee\neg\neg A\rightarrowtail E$ for subobjects $A\rightarrowtail E$ in $\mathcal{E}$ are $j$-dense.
=--

This appears as proposition 6.2 in [Caramello (2012a)](#Caramello12a).

+--{: .num_prop #DeMorgan_prop}
###### Proposition
Let $\mathcal{E}$ be a [[topos]] and $m$ be the De Morgan topology on it.

* $Sh_m(\mathcal{E})=\mathcal{E}$ iff $\mathcal{E}$ is a [[De Morgan topos]].

* For any [[dense subtopos|dense topology]] $j$ on $\mathcal{E}$ , $Sh_m(Sh_j(\mathcal{E}))=Sh_{m\vee j}(\mathcal{E})$.
=--

[Caramello (2009)](#Caramello09), prop.1.5. In fact, in the second statement it suffices to demand that $j$ is a [[weakly open topology]] i.e. the [[associated sheaf functor]] $a_j:\mathcal{E}\to Sh_j(\mathcal{E})$ preserves the pseudo-complementation operator in the lattices of subobjects (cf. [Caramello (2012, prop.4.5)](#Caramello12)).

Notice that $Sh_{m\vee j}(\mathcal{E})=Sh_{m}(\mathcal{E})\cap Sh_{ j}(\mathcal{E})$ and, accordingly, for a dense or, more generally a weakly open subtopos De Morganization simply amounts to intersection with $Sh_m(\mathcal{E})$.

## Geometric morphisms preserving De Morganizations

Notice that in analogy to $Sh_{\neg\neg}(\mathcal{E})$ and the class of [[skeletal geometric morphism]], the universality of the De Morganization affords to define a class of **m-skeletal geometric morphisms** as those geometric morphisms $f:\mathcal{F}\to\mathcal{E}$ that restrict to geometric morphisms $f|_m:Sh_m(\mathcal{F})\to Sh_m(\mathcal{E})$ .

Due to a result in [Johnstone (2002, p.194)](#Johnstone), this is equivalent to the preservation of $m$-dense monomorphisms by $f^\ast$.

By the [above proposition](#DeMorgan_prop), $Sh_m(Sh_{j}(\mathcal{E}))\hookrightarrow Sh_{m}(\mathcal{E})$ for $j$ dense. Accordingly, _dense inclusions_ $Sh_{j}(\mathcal{E})\hookrightarrow\mathcal{E}$ are m-skeletal !

The [characterization of Boolean toposes by skeletal morphisms](http://ncatlab.org/nlab/show/skeletal+geometric+morphism#Boolean_skeletal) carries over to m-skeletal morphisms and De Morgan toposes:

+--{: .num_prop #DeMorgan_m-skeletal}
###### Proposition
A topos $\mathcal{E}$ is De Morgan iff every geometric morphism $\mathcal{F}\to\mathcal{E}$ is m-skeletal.
=--

**Proof**:
Assume $\mathcal{E}$ is De Morgan, then it coincides with $Sh_m(\mathcal{E})$ and $m$-sheaves of $\mathcal{F}$ necessarily have to land there.

Conversely, assume all $\mathcal{F}\to\mathcal{E}$ are m-skeletal. Then the surjective morphism $f:\gamma\mathcal{E}\to\mathcal{E}$ from the [[De Morgan topos|Gleason cover]] $\gamma\mathcal{E}$ is m-skeletal. But $\gamma\mathcal{E}$ is De Morgan and, therefore, so is $im(f)=\mathcal{E}$. $\qed$

## Related entries

* [[De Morgan topos]]

* [[De Morgan Heyting algebra]]

* [[double negation]]

* [[dense subtopos]]

* [[dense topology]]

* [[skeletal geometric morphism]]

## References

* {#Caramello09}[[Olivia Caramello]], _De Morgan classifying toposes_ , Adv. in Math. **222** (2009) pp.2117-2144. ([arXiv:0808.1519](http://arxiv.org/abs/0808.1519))

* {#Caramello12a}[[Olivia Caramello]], _Universal models and definability_ , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. ([arXiv:0906.3061](http://arxiv.org/abs/0906.3061))

* {#Caramello12}[[Olivia Caramello]], _Topologies for intermediate logics_ , arXiv:1205.2547 (2012). ([arXiv:1205.2547](http://arxiv.org/abs/1205.2547))

* {#CJ09}[[Olivia Caramello]], [[Peter Johnstone]], _De Morgan's law and the Theory of Fields_ , Adv. in Math. **222** (2009) pp.2145-2152. ([arXiv:0808.1572](http://arxiv.org/abs/0808.1972))

* {#Johnstone77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint New York 2014)

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002.


de re and de dicto

#Contents#
* table of contents
{:toc}

## Idea

The **de re/ de dicto** distinction has different meanings, as explained in ([SEP](#SEP)), but it typically concerns the two readings that are possible in sentences where a [[modal operator]] or propositional attitude verb is applied to a claim about some subject. For example, consider

* I believe that the person downstairs is my mother

The usual _de dicto_ reading would express my strong hunch about the person who is downstairs that she is my mother. The _de re_ reading would take the expression 'the person downstairs' to designate the person who is actually there, for example, it may in fact be my father. This would entail that I am crazy enough to believe my father to be my mother.

Similarly in the case of [[modal logic]], we could consider a statement such as

+ Someone is necessarily the monarch of this country

_De dicto_ this means that it has to be the case at this time that there be a monarch (as soon as one monarch dies, their successor becomes monarch). On the other hand, _de re_ this means that some specific individual has to be monarch. This reading leads to a false claim since that person might in fact not have been monarch. They might, say, have abdicated earlier or have been killed by their uncle when a child.

## Formalization in modal logic

In [[modal logic]] a [[proposition]] involving the consecutive application of a [[modal operator]] and an [[existential quantifier]] is called

* _de dicto_ if the modal operator is applied after the quantifier

  $\Box \underset{x \colon X}{\exists} A(x)$;

* _de re_ if it is the other way around

  $\underset{x \colon X}{\exists} \Box A(x)$;

Hence a difference between de dicto and de re statements is related to the failure of a [[modal operator]] to commute with [[base change]].

However, according to the treatment at [necessity and possibility](necessity+and+possibility#InFirstOrderLogicAndTypeTheory), these modalities operate on world dependent types:

$$
  (\underset{W}{\lozenge} \dashv \underset{W}{\Box})
  \coloneqq
  \left(
     \left(
     W^\ast \circ \underset{w\colon W}{\exists}
     \right)
     \dashv
     \left(
     W^\ast \circ \underset{w\colon W}{\forall}
     \right)
  \right)
  \;\colon\;
  \mathbf{H}_{/W}
    \longrightarrow
  \mathbf{H}_{/W}
  \,.
$$

Then, _de dicto_, I could mean by  $\Box \underset{x \colon X}{\exists} A(x)$ that we have a world-dependent type $X(w)$, and then put $\Box_W \underset{x \colon X(w)}{\exists} A(w, x)$. But this $A$ is a dependent type $w: W, x:X(w) \vdash A(w, x):Prop$.

However, _de re_, I need to have world-independent $X$, to allow the modal operator $\Box_W: \mathbf{H}/(X \times W) \to \mathbf{H}/(X \times W)$ and then $\underset{x \colon X}{\exists}: \mathbf{H}/(X \times W) \to \mathbf{H}/W$.

## Related concepts

* [[Barcan formula]]

## References

* {#SEP} Stanford Encyclopedia of Philosophy, _[The De Re/De Dicto Distinction](http://plato.stanford.edu/entries/prop-attitude-reports/dere.html)_

* Wikipedia, _[De dicto and de re](https://en.wikipedia.org/wiki/De_dicto_and_de_re)_

Detailed discussion of compatibility of [[modal operators]] with [[base change]] in a [[hyperdoctrine]] is in

* {#MakkaiReyes95} [[Michael Makkai]], [[Gonzalo Reyes]], section 4 of _Completeness results for intuitionistic and modal logic in a categorical setting_, Annals of Pure and Applied Logic 72 (1995) 25-101

Related comments are in

* {#Law91a} [[William Lawvere]], _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in A. Carboni, M. Pedicchio, G. Rosolini, _Category theory_ , [[Como|Proceedings of the International Conference held in Como 1990]], 213–236, Lecture Notes in Math. __1488__, Springer (1991)

For an attempt to represent the kinds of _opaque_ contexts considered here using the [[reader monad]], see

* Gianluca Giorgolo, Ash Asudeh, _Monads as a Solution for Generalized Opacity_, Proceedings of the EACL 2014 Workshop on Type Theory and Natural Language Semantics (TTNLS), pp. 19-27. ([pdf](http://users.ox.ac.uk/~cpgl0036/pdf/giorgolo-asudeh-eacl2014.pdf))

[[!redirects de dicto and de re]].


de Rham complex

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

#Contents#
* table of contents
{:toc}

## Idea

The _de Rham complex_  (named after [[Georges de Rham]]) $\Omega^\bullet(X)$ of a [[space]] $X$ is the [[cochain complex]] that in degree $n$ has the [[differential form]]s (which may mean: [[Kähler differential form]]s) of degree $n$, and whose [[differential]] is the **de Rham differential** or **exterior derivative**.

As $X$ varies this constitutes an [[abelian sheaf]] of complexes.

## Definition

### For smooth manifolds

The **de Rham complex** of a [[smooth manifold]] is the [[cochain complex]] which in degree $n \in \mathbb{N}$ has the [[vector space]] $\Omega^n(X)$ of degree-$n$ [[differential forms]] on $X$. The coboundary map is the deRham _[[exterior derivative]]_.

Explicitly, given a differential $k$-form $\omega$, its de Rham differential $d\omega$ can be computed as
$$d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i\lt j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),$$
where $v_i$ are [[vector fields]] on $X$, $[-,-]$ is the [[Lie bracket of vector fields]], and $\mathcal{L}_{v}$ is the [[Lie derivative]] of a smooth function with respect to a vector field $v$.

The [[chain cohomology|cohomology]] of the de Rham complex (hence the [[quotient]] of [[closed differential forms]] by [[exact differential forms]]) is __de Rham cohomology__. Under the [[wedge product]], the deRham complex becomes a [[differential graded algebra]]. This may be regarded as the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]] $T X$ of $X$.

The corresponding [[abelian sheaf]] in this case defines a [[smooth spectrum]] via the [[stable Dold-Kan correspondence]], see at _[smooth spectrum -- Examples -- De Rham spectra](smooth+spectrum#ExamplesDeRhamSpectra)_.

### For algebraic objects

For [[smooth varieties]] $X$, algebraic de Rham cohomology is defined to be the [[hypercohomology]] of the de Rham complex $\Omega_X^\bullet$.

De Rham cohomology has a rather subtle generalization for possibly singular algebraic varieties due to ([Grothendieck](#Grothendieck)).

For [[analytic spaces]]

* T. Bloom, M. Herrera, _De Rham cohomology of an analytic space_, Inv. Math. __7__ (1969), 275-296, [doi](http://dx.doi.org/10.1007/BF01425536)

### For cohesive homotopy types
 {#ForCohesiveHomotopyTypes}

In the general context of [[cohesive homotopy theory]] in a [[cohesive (∞,1)-topos]] $\mathbf{H}$, for $A \in \mathbf{H}$ a [[cohesive homotopy type]], then the [[homotopy fiber]] of the [[counit of a comonad|counit]] of the [[flat modality]]

$$
  \flat_{dR} A \coloneqq fib(\flat A \to A)
$$

may be interpreted as the de Rham complex with [[coefficients]] in $A$.

This is the [[codomain]] for the [[Maurer-Cartan form]] $\theta_{\Omega A}$ on $\Omega A$ in this generality. The [[shape modality|shape]] of $\theta_{\Omega A}$ is the general [[Chern character]] on $\Pi(\Omega A)$.

For more on this see at

* _[structures in a cohesive infinity-topos -- de Rham cohomology](cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology)_

More precisely, $\flat_{dR} \Sigma A$ and $\Pi_{dR} \Omega A$ play the role of the non-negative degree and negative degree part, respectively of the de Rham complex with coefficients in $\Pi \flat_{dR} \Sigma A$. For more on this see at

* _[differential cohomology diagram -- de Rham coefficients](differential%20cohomology%20diagram#DeRhamCoefficients)_.

## Examples

### de Rham cohomology of spheres

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

For positive $n$, the de Rham cohomology of the $n$-[[sphere]] $S^n$ is

$$
  H^p(S^n)
  =
  \left\{
    \array{
      \mathbb{R} & if\; p = 0,n
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

For $n=0$, we have

$$
  H^p(S^0)
  =
  \left\{
    \array{
      \mathbb{R} \oplus \mathbb{R} & if\; p = 0
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

=--

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

This follows from the [[Mayer-Vietoris sequence]] associated to the open cover of $S^n$ by the subset excluding just the north pole and the subset excluding just the south pole, together with the fact that the dimension of the $0^{th}$ de Rham cohomology of a smooth manifold is its number of connected components.

=--

## Properties

### Basic theorems

* [[Poincare lemma]]

* [[de Rham theorem]]

* [[nonabelian de Rham cohomology]]

### Relation to PL de Rham complex
  {#RelaionToPLDeRhamComplex}

+-- {: .num_prop #PLdeRhamComplex}
###### Proposition
**([[PL de Rham complex of smooth manifold is equivalent to de Rham complex]])**

Let $X$ be a [[smooth manifold]].

We have the following [[zig-zag]] of [[dgc-algebra]] [[quasi-isomorphisms]] between the [[PL de Rham complex]] of (the [[topological space]] underlying) $X$ and the smooth [[de Rham complex]] of $X$:

$$
  \array{
    &&
    \Omega^\bullet_{PLdR}
    \big(
      S(X)
    \big)
    &&
    &&
    \Omega^\bullet_{dR}(X)
    \\
    &
      {}^{
        \mathllap{
          i^\ast
        }
      }
      \nearrow
    & &
      \searrow^{
        \mathrlap{
          i_{poly}
        }
      }
    & &
      {}^{
        \mathllap{
          p^\ast
        }
      }
      \swarrow
    \\
    \mathllap{
      \Omega^\bullet_{PLdR}(X)
      \;=\;
    }
    \Omega^\bullet_{PLdR}
    \big(
      Sing(X)
    \big)
    && &&
    \Omega^\bullet_{PSdR}
    \big(
      S(X)
    \big)
  }
$$

Here $S(X)$ is the [[simplicial complex]] corresponding to any smooth [[triangulation]] of $X$.

=--

### Relation to Deligne complex

See at _[[Deligne complex]]_

## Related concepts

* [[pullback of differential forms]]

* [[equivariant de Rham cohomology]]

* [[de Rham-Witt complex]]

* [[twisted de Rham cohomology]]

* [[Deligne cohomology]]

* [[de Rham space]]

* [[Lie algebra valued differential forms]]

* [[L-infinity algebra valued differential forms]]

* [[absolute de Rham cohomology]]

* [[Dolbeault complex]], [[Dolbeault cohomology]]

* [[holomorphic de Rham complex]]

* [[Hodge-de Rham spectral sequence]]

* [[chiral de Rham complex]]

* [[crystalline cohomology]], [[comparison theorem (crystalline cohomology)]]

* [[Hodge cohomology]]

* [[PL de Rham complex]]

## References

### In differential geometry

Discussion in [[differential geometry]]:

* [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0))

* [[Georges de Rham]], Chapter II of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

* [[Dominic G. B. Edelen]], *Applied exterior calculus*, Wiley (1985) \[GoogleBooks\]

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 6 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

### In algebraic geometry

Discussion in [[algebraic geometry]]

A useful introduction is

* Kiran Kedlaya, _$p$-adic cohomology, from theory to practice_ ([pdf](http://swc.math.arizona.edu/aws/2007/KedlayaNotes11Mar.pdf))

A classical reference on the algebraic version is

* [[Alexander Grothendieck]], _On the De Rham cohomology of algebraic varieties_, Publications Mathématiques de l'IHÉS __29__, 351-359 (1966), [numdam](http://www.numdam.org/item?id=PMIHES_1966__29__95_0).
 {#Grothendieck}
* A. Grothendieck, _Crystals and the de Rham cohomology of schemes_, in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L. et al., Dix Exposés sur la Cohomologie des Schémas, Advanced studies in pure mathematics __3__, Amsterdam: North-Holland, pp. 306–358, [MR0269663](http://www.ams.org/mathscinet-getitem?mr=0269663), [pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf)
* [[Robin Hartshorne]], _On the de Rham cohomology of algebraic varieties_, Publ. Mathématiques de l'IHÉS __45__ (1975), p. 5-99 [MR55#5633](http://www.ams.org/mathscinet-getitem?mr=55:5633)
* P. Monsky, _Finiteness of de Rham cohomology_, Amer. J. Math. __94__ (1972), 237–245, [MR301017](http://www.ams.org/mathscinet-getitem?mr=301017), [doi](http://dx.doi.org/10.2307/2373603)

See also

* Yves André, _Comparison theorems between algebraic and
analytic De Rham cohomology_ ([pdf](http://www.emis.de/journals/JTNB/2004-2/pages335-355.pdf))

* [[Mikhail Kapranov]], _DG-Modules over the de Rham complex and the vanishing cycles functor_, Lecture Notes in Mathematics **1479**, Springer (1991) [[doi:10.1007/BFb0086264](https://doi.org/10.1007/BFb0086264)]

[[!redirects de Rham complexes]]

[[!redirects deRham complex]]
[[!redirects deRham complexes]]

[[!redirects deRham algebra]]
[[!redirects deRham algebras]]

[[!redirects de Rham algebra]]
[[!redirects de Rham algebras]]

[[!redirects deRham dga]]
[[!redirects deRham dgas]]

[[!redirects de Rham dga]]
[[!redirects de Rham dgas]]

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

[[!redirects de Rham dg-algebra]]
[[!redirects de Rham dg-algebras]]

[[!redirects de Rham cohomology]]

[[!redirects deRham cohomology]]

[[!redirects exterior calculus]]


de Rham differential

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

# The de Rham differential
* table of contents
{: toc}

## Traditional

In [[differential geometry]] the _de Rham differential_ is the [[differential]] in the [[de Rham complex]], "exterior [[derivative]]" acting on [[differential forms]].

## In cohesive homotopy theory

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] and write $T \mathbf{H}$ for its [[tangent cohesive (∞,1)-topos]].

Given a [[stable homotopy type]] $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ [[cohesion]] provides two objects

$$
  \Pi_{dR} \Omega \hat E \,,\;\;  \flat_{dR}\Sigma \hat E
  \;\;
  \in Stab(\mathbf{H})
$$

which may be interpreted as [[de Rham complexes]] with [[coefficients]] in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map

$$
  \array{
    \Pi_{dR}\Omega \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\Sigma \hat E
    \\
    & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}}
    \\
    &&
    \hat E
  }
$$

which interprets as the _de Rham differential_ $\mathbf{d}$. See at _[[differential cohomology diagram]]_ for details.

## Properties

* The de Rham differential acts as a [[graded derivation]] on the [[de Rham algebra]].

[[!redirects exterior derivative]]
[[!redirects exterior derivatives]]
[[!redirects exterior differential]]
[[!redirects exterior differentials]]
[[!redirects exterior differentiation]]
[[!redirects de Rham derivative]]
[[!redirects de Rham derivatives]]
[[!redirects deRham derivative]]
[[!redirects deRham derivatives]]
[[!redirects de Rham differential]]
[[!redirects de Rham differentials]]
[[!redirects deRham differential]]
[[!redirects deRham differentials]]
[[!redirects de Rham differentiation]]
[[!redirects deRham differentiation]]


de Rham space

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###Context###
#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Discrete and concrete objects
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[[!include discrete and concrete objects - contents]]
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=--
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#Contents#
* table of contents
{:toc}

## Idea

In a context of [[synthetic differential geometry]] or [[D-geometry]], the _de Rham space_ $dR(X)$ of a [[space]] $X$ is the [[quotient]] of $X$ that identifies [[infinitesimal object|infinitesimally close points]].

It is the [[coreduced object|coreduced reflection]] of $X$.

## Definition

### On $Rings^{op}$

Let [[CRing]] be the [[category]] of [[commutative rings]]. For $R \in CRing$, write $I \in R$ for the [[nilradical]] of $R$, the [[ideal]] consisting of the nilpotent elements. The canonical [[projection]] $R \to R/I$ to the [[quotient]] by the ideal corresponds in the [[opposite category]] $Ring^{op}$ to the inclusion

$$
  Spec (R/I) \to Spec R
$$

of the [[reduced object|reduced]] part of $Spec R$.

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

For $X \in PSh(Ring^{op})$ a [[presheaf]] on $Ring^{op}$ (for instance a [[scheme]]), its **de Rham space** $X_{dR}$ is the presheaf defined by

$$
  X_{dR} : Spec R \mapsto X\left(Spec \left(R/I\right)\right)
  \,.
$$

=--

## Properties

### As a quotient

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

If $X \in PSh(Ring^{op})$ is a [[smooth scheme]] then the canonical morphism

$$
  X \to X_{dR}
$$

is an [[epimorphism]] (hence an epimorphism over each $Spec R$) and therefore in this case $X_{dR}$ is the [[quotient]] of the relation "being infinitesimally close" between points of $X$: we have that $X_{dR}$ is the [[coequalizer]]

$$
  X_{dR} = \lim_\to
   \left(
     X^{inf} \stackrel{\longrightarrow}{\longrightarrow} X
   \right)
  \,,
$$

of the two projections out of the [[formal neighbourhood of the diagonal]].

=--

### Relation to jet bundles

For $E \to X$ a [[bundle]] over $X$, its [[direct image]] under [[base change]] along the projection map $X \longrightarrow \Pi_{inf} X$ yields its _[[jet bundle]]_. See there for more.

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

In terms of [[differential homotopy type theory]] this means that
forming "jet types" of [[dependent types]] over $X$ is the
[[dependent product]] operation along the unit of the [[infinitesimal shape modality]]

$$
  jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E
  \,.
$$

=--

### Relation to formally étale morphisms of schemes

* [[formally étale morphism of schemes]]

## Related concepts

### Crystalline site

For $X : Ring \to Set$ a [[scheme]], the [[big site]] $Ring^{op}/X_{dR}$
of $X_{dR}$, is the [[crystalline site]] of $X$.

### Grothendieck connection

Morphisms $X_{dR} \to Mod$ encode flat higher [[connection on a bundle|connections]]: [[local system]]s.

Accordingly, [[descent]] for de Rham spaces -- sometimes called **de Rham descent** encodes flat 1-connections. This is described at [[Grothendieck connection]],

### D-modules

The category of [[D-module]]s on a [[space]] is equivalent to that of [[quasicoherent sheaf|quasicoherent sheaves]] on the corresponding de Rham space
([Lurie, above theorem  0.4](#LurieCrystal)).

### Infinitesimal path $\infty$-groupoids

* [[infinitesimal path ∞-groupoid]]

## References

The term _de Rham space_ or _de Rham stack_ apparently goes back to

* {#Simpson96} [[Carlos Simpson]], _Homotopy over the complex numbers and generalized de Rham cohomology_, Toulouse preprint no. 50, April 1995, In: M. Maruyama, (ed.) _Moduli of Vector Bundles (Taniguchi symposium December 1994)_,  Lecture notes in Pure and Applied Mathematics, Issue 189, Page 229-264, Dekker (1996), 229-263 ([web](https://www.tib.eu/en/search/id/BLCP%3ACN013382849/Homotopy-Over-the-Complex-Numbers-and-Generalized/))

But actually there it just has the notation "$X_{dR}$" and then the functor it co-represents is called the "de Rham shape" of $X$.

A review of the constructions is on the first two pages of

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

The de Rham space construction on spaces ([[schemes]]) is described in [section 3, p. 7](http://math.berkeley.edu/~teleman/math/simpson.pdf#page=7)

* {#SimpsonTeleman} [[Carlos Simpson]], [[Constantin Teleman]], _de Rham theorem for $\infty$-stacks_, 1997 ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf), [[SimpsonTelemanDeRhamTheoremForStacks.pdf:file]])

which goes on to assert the existence of its [[derived functor]] on the [[homotopy category]] $Ho Sh_\infty(C)$ of [[∞-stack]]s in proposition 3.3. on the same page.

Similar discussion in a context of [[derived algebraic geometry]] is in

* [[Dennis Gaitsgory]], [[Nick Rozenblyum]], section 1 of _Crystals and D-modules_ ([pdf](http://www.math.harvard.edu/~gaitsgde/GL/Crystalstext.pdf))

The characterization of [[formally smooth scheme]] as above is also on that page.

See also online comments by [[David Ben-Zvi]]  [here](http://golem.ph.utexas.edu/category/2009/04/journal_club_geometric_infinit.html#c023287) and [here](http://golem.ph.utexas.edu/category/2009/08/question_on_synthetic_differen.html#c026218) on the $n$Café. and [here](http://mathoverflow.net/questions/10556/d-modules-derham-spaces-and-microlocalization) on MO.

[[!redirects deRham space]]
[[!redirects de Rham spaces]]

[[!redirects de Rham stack]]
[[!redirects de Rham stacks]]

[[!redirects de Rham shape]]


de Rham theorem

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### On cohomology

The **de Rham theorem** (named after [[Georges de Rham]]) asserts that the [[de Rham cohomology]] $H^n_{dR}(X)$ of a smooth [[manifold]] $X$ (without boundary) is [[isomorphism|isomorphic]] to the [[real cohomology]] $H^n(X, \mathbb{R})$, hence its [[ordinary cohomology]] with [[real number]] [[coefficients]], as computed for instance by the [[singular cohomology|singular]] or [[Čech cohomology]] with real coefficients.

The theorem has several dozens of different proofs. For example in the Čech approach one can make a double complex whose first row is the Čech complex of a covering and first column is the de Rham complex and other entries are mixed and use spectral sequence argument (see the textbook of Bott and Tu, or the geometry lectures book by Postnikov, semester III).

This is maybe best formulated, understood and proven in the context of [[abelian sheaf cohomology]]:

Write $\mathbb{R}_c$ for

* the abelian group $\mathbb{R}$

* regarded not as a [[Lie group]] with the standard [[manifold]] structure on $\mathbb{R}$ but as a topologically discrete group on the underlying set of $\mathbb{R}$

* and then regarded as a [[sheaf]] on $X$: the [[constant sheaf]] that sends connected $U \subset X$ to the set of _constant_ maps $U \to \mathbb{R}$.

Write $\mathbf{B}^n \mathbb{R}_c$ for the corresponding [[Eilenberg-MacLane object]] in [[chain complex]]es of sheaves of abelian groups: this is the complex of sheaves with $\mathbb{R}_c$ in degree $n$:

$$
  \mathbf{B}^n \mathbb{R}_c
  =
  (\cdots \to 0 \to \mathbb{R}_c \to 0 \to \cdots \to 0)
  \,.
$$

Next, write $\bar \mathbf{B}^n \mathbb{R}$ (without the subscript $c$!) for the [[Deligne cohomology|Deligne complex]] for $\mathbb{R}$

$$
  \bar \mathbf{B}^n \mathbb{R}
  =
  (C^\infty(-,\mathbb{R}) \stackrel{d_{dR}}{\to}
    \Omega^1(-) \stackrel{d_{dR}}{\to}
    \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{closed}(-))
  \,.
$$

(The notation here is borrowed from that used at [[motivation for sheaves, cohomology and higher stacks]]: we can think of $\bar \mathbf{B}^n \mathbb{R}$ as a differential refinement of the object $\mathbf{B}^n \mathbb{R}_c$).

Then we have:

* "ordinary" $\mathbb{R}$-valued cohomology of $X$ is the [[abelian sheaf cohomology]] with coefficients in $\mathbf{B}^n \mathbb{R}_c$.

* [[de Rham cohomology]] of $X$ is the [[abelian sheaf cohomology]] with coefficients in $\bar \mathbf{B}^n \mathbb{R}$ (this is semi-obvious, requires a bit more discussion).

* the [[Poincare lemma]] says that every closed [[differential form]] is locally exact, and hence there is a [[quasi-isomorphism]] of chain complexes of sheaves

  $$
    \mathbf{B}^n \mathbb{R}_c \stackrel{\simeq}{\to}
    \bar \mathbf{B}^n \mathbb{R}
  $$

  given by injecting for each $U \subset X$ the set $\mathbb{R}$ as the constant functions into $C^\infty(U,\mathbb{R})$.

It is this quasi-isomorphism of coefficient objects that induces the de Rham isomorphism of [[abelian sheaf cohomology|abelian sheaf]] [[cohomology group]]s, which is ordinarily written as

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

...

### On cochains
 {#OnCochains}

The equivalence on cohomology asserted by the de Rham theorem is but a [[decategorification]] of a more refined statement: a [[quasi-isomorphism]] of [[cochain complex]]es. This even respects the product structure:

for $X$ a [[smooth manifold]] there is an [[equivalence in an (infinity,1)-category|equivalence]] of [[A-infinity algebra]]s

$$
  (\Omega^\bullet(X), d_{dR})
  \stackrel{\simeq}{\to}
  (C(X), \cup)
$$

between the [[de Rham complex]] and the collection of [[singular cohomology|singular cochain]]s equipped with the [[cup product]].

This is due to ([Gugenheim, 1977](#Gugenheim)).

Furthermore, the [[E-infinity algebra]] structure on [[differential forms]]
(trivially induced by the [[commutative dga]] structure)
and [[singular cochains]] (as witnessed by the action of
the [[sequence operad]] of McClure and Smith on [[singular cochains]])
is also preserved.

### Synthetic version
 {#SyntheticVersion}

The de Rham theorem also holds internally in the context of suitable [[smooth topos]]es $\mathcal{T}$ modelling the axioms of [[synthetic differential geometry]].

Specifically

* the internal [[singular cohomology|singular chain complex]] in $\mathcal{T}$ is given as the $R$-linear dual of the free internal $R$-module on the [[internal hom]] objects $[\Delta^n,X]$, where $R$ is the internal incarnation of the real numbers;

* the de Rham complex is given by [[differential forms in synthetic differential geometry]].

The de Rham theorem in $\mathcal{T}$ then asserts that for $X$ a manifold regarded as an object in the well-adapted smooth topos $\mathcal{T}$ the morphism

$$
  \int : H^p(X) \to H_p(X,R)^*
$$

in $\mathcal{T}$ is an [[isomorphism]] for all $p \in \mathbb{N}$. This implies the standard (external) de Rham theorem.

This is discussed in chapter IV of

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ for the toposes $\mathcal{T} = \mathcal{G}, \mathcal{F}$ (see [models](http://ncatlab.org/nlab/show/Models+for+Smooth+Infinitesimal+Analysis#Models)).

A little bit a long these lines for [[diffeological space]]s is also in

* Patrick Iglesias-Zemmour, _De Rham calculus_ ([pdf](http://math.huji.ac.il/~piz/documents/D6.pdf))

## Related concepts

* [[PL de Rham theorem]]

* [[equivariant de Rham theorem]]

* [[equivariant PL de Rham theorem]]

* [[Poincaré lemma]]

* [[Stokes theorem]]

* **de Rham theorem**

  * [[kernel of integration is the exact differential forms]]

* [[Dolbeault theorem]]

* [[Hodge theorem]]

## References

### On cohomology

Textbook accounts:

* [[M M Postnikov]], *Differentiable manifolds*, vol. III of: *Lectures on geometry,*

* [[Raoul Bott]], [[Loring Tu]], Thm. 8.9 & Prop. 10.6 of: _[[Differential Forms in Algebraic Topology]]_, Springer (1982)

* [[Jean Gallier]], [[Jocelyn Quaintance]], Ch. 9 in: *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]

Exposition:

* Arne Lorenz, _Abstract de Rham theorem_, [pdf slides](http://wwwb.math.rwth-aachen.de/~arne/Talks/deRham.pdf)

In the context of [[rational homotopy theory]]:

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Theorem 10.15 in: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

In [[analytic geometry]]:

* M. E. Herrera, _De Rham theorems on semianalytic sets_, Bull. Amer. Math. Soc. __7__ 3 (1967) 414–418, [doi](http://dx.doi.org/10.1090/S0002-9904-1967-11772-5), [MR214094](http://www.ams.org/mathscinet-getitem?mr=214094)

In the generality of [[diffeological spaces]]:

* [[Patrick Iglesias-Zemmour]], *Une cohomologie de Čech pour les espaces differentiables et sa relation a la cohomologie de De Rham* (1988) [[pdf](https://math.huji.ac.il/~piz/documents/UCDCPLEDESRALCDDR.pdf), [[IglesiasZemmour-CechDeRham.pdf:file]]]

* [[Patrick Iglesias-Zemmour]], *Čech–De Rham bicomplex in diffeology*, Israel Journal of Mathematics (2023) 1–38 [[doi:10.1007/s11856-023-2486-8](https://doi.org/10.1007/s11856-023-2486-8)]

* [[Emilio Minichiello]], *The Diffeological Čech-de Rham Obstruction* [[arXiv:2401.09400](https://arxiv.org/abs/2401.09400)]

### On cocycles
 {#ReferencesdeRhamTheoremOnCocycles}

The refinement of the de Rham theorem from an [[isomorphism]] of [[cohomology group]]s to an [[equivalence]] of  [[A-∞ algebras]] of cochains and forms was first stated in

* [[Victor Gugenheim]], _On Chen's iterated integrals_ , Illinois J. Math. Volume 21, Issue 3 (1977), 703-715.
 {#Gugenheim}

proven using Chen's [[iterated integral]]s.

A review is in section 3 of

* [[Camilo Arias Abad]], [[Florian Schätz]], _The $A_\infty$ de Rham theorem and integration of representations up to homotopy_ ([arXiv:1011.4693](http://arxiv.org/abs/1011.4693))

[[!redirects deRham theorem]]

[[!redirects de Rham isomorphism]]
[[!redirects de Rham isomorphisms]]


de Rham-Witt complex

## Related concepts

* [[de Rham complex]], [[de Rham space]]

* [[Witt vector]], [[arithmetic jet space]]

## References

* [[Mark Behrens]] notes on the de Rham Witt complex ([pdf](http://www3.nd.edu/~mbehren1/TAGS/Davis_notes.pdf))

* [[Lars Hesselholt]], _The big de Rham-Witt complex_, [arXiv:1006.3125](http://arxiv.org/abs/1006.3125).


de Sitter gravity

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[[!include gravity contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

What is sometimes called _de Sitter gravity_ and _anti de Sitter gravity_ is the analog of [[Einstein gravity]] with the local model geometry of [[Minkowski spacetime]] acted on by the [[Poincaré group]] replaced by [[anti de Sitter spacetime]] or [[de Sitter spacetime]] acted on by their [[isometry groups]] ([[anti de Sitter group]] and [[de Sitter group]], respectively).

More in detail, in the [[first order formulation of gravity]] the [[field (physics)|field configurations]] of [[Einstein gravity]] ([[general relativity]]) are [[Cartan connections]] for the inclusion of the [[Spin group]] into the [[Poincaré group]]. Accordingly a field configuration of anti de Sitter (resp. de Sitter) gravity is a [[Cartan connection]] for a suitable [[stabilizer subgroup]] included into  $O(d-1,2)$ (resp. $O(d,1)$).

## Examples

* In the case of 3-dimensional [[spacetime]] both de Sitter and anti de Sitter gravity have been studie in much detail. See at  _[[3d quantum gravity]]_ for more.

## Related concepts

* [[AdS-CFT correspondence]]

* [[Randall-Sundrum model]]

* [[Chern-Simons gravity]]

[[!include local and global geometry - table]]

## References

* [[Paul Townsend]], _Small Scale Structure of Space-Time as the Origin of the Gravitational Constant_, Phys.Rev. D15 (1977) 2795 ([spire](https://inspirehep.net/record/5017?ln=en))

* Takeshi Fukuyama, _Gauge theory of gravity_, Zeitschrift für Physik C Particles and Fields 1981, Volume 10, Issue 1, pp 9-15 ([publisher](http://link.springer.com/article/10.1007%2FBF01545778))

* A. Achúcarro and [[Paul Townsend]], _A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories_ , Phys. Lett. B180 (1986) 89.

* {#Witten88} [[Edward Witten]], _(2+1)-Dimensional Gravity as an Exactly Soluble System_ Nucl. Phys. B311 (1988) 46. ([web](http://adsabs.harvard.edu/abs/1988NuPhB.311...46W))

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 1, chapter I.4.4 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Derek Wise]], _MacDowell-Mansouri gravity and Cartan geometry_, Class.Quant.Grav.27:155010,2010 ([arXiv:gr-qc/0611154](http://arxiv.org/abs/gr-qc/0611154))

* Theo Verwimp, _Anti-de Sitter gauge theory for gravity_ ([arXiv:1006.1614](http://xxx.tau.ac.il/abs/1006.1614))

* Wikipedia, _[de Sitter invariant special relativity](http://en.wikipedia.org/wiki/De_Sitter_invariant_special_relativity)_

* M. Gunaydin, [[Peter van Nieuwenhuizen]], N.P. Warner, _General Construction of the Unitary Representations of Anti-de Sitter Superalgebras and the Spectrum of the $S^7$ Compactification of Eleven-dimensional Supergravity_, Nucl.Phys. B255 (1985) 63 ([spire](http://inspirehep.net/record/206013?ln=en))

[[!redirects anti de Sitter gravity]]

[[!redirects AdS gravity]]
[[!redirects deSitter gravity]]


de Sitter group

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[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _de Sitter group_ is the [[isometry group]] of [[de Sitter spacetime]]. In [[spacetime]] dimension $d$ its [[connected component]] is [[isomorphism|isomorphic]] to the [[orthogonal group]] $SO(d,1)$ in Lorentzian signature in one dimension higher.

This is analgous to the [[Poincare group]], which is the [[isometry group]] of [[Minkowski spacetime]].

## Related concepts

[[!include local and global geometry - table]]

## References

The de Sitter [[Lie algebra]] is discussed for instance in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 1, chapter I.3.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

The [[representation theory]] and its [[Inönü-Wigner contraction]] to that of the [[Poincaré group]] is discussed in

* [[Jouko Mickelsson]], J. Niederle, _Contractions of Representations of de Sitter Groups_, Comm. Math. Phys. Volume 27, Number 3 (1972), 167-180. ([Euclid](http://projecteuclid.org/euclid.cmp/1103858248))

* Mauricio Ayala, Richard Haase, _Group contractions and its consequences upon representations of different spatial symmetry groups_ ([arXiv:hep-th/0206037](https://arxiv.org/abs/hep-th/0206037))

* Francisco J. Herranz, Mariano Santander, section 4 of _(Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach_, J.Phys.A41:015204,2008 ([arXiv:math-ph/0612059](https://arxiv.org/abs/math-ph/0612059))

* Thomas Basile, [[Xavier Bekaert]], Nicolas Boulanger, _Mixed-symmetry fields in de Sitter space: a group theoretical glance_ ([arXiv:1612.08166](https://arxiv.org/abs/1612.08166))

See also

* Zimo Sun, *A note on the representations of $SO(1,d+1)$* ([arXiv:2111.04591](https://arxiv.org/abs/2111.04591))


de Sitter spacetime

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#Contents#
* table of contents
{:toc}

## Definition

Full de Sitter spacetime is the [[Lorentzian manifold]] which is the [[coset space]] of [[Lorentz groups]]

$$
  dS^d \simeq O(d,1)/O(d-1,1)
$$

This models a cosmology [[FRW model]] for positive [[cosmological constant]]/[[dark energy]].

Similarly [[anti de Sitter spacetime]] is

$$
  adS^d \simeq O(d-1,2)/O(d-1,1)
$$

(e.g here)

This admits different spatial slicings of relevance:

> graphics grabbed from [this Physics.SE comment](https://physics.stackexchange.com/a/353157/5603)

## Related concepts

[[!include local and global geometry - table]]

## References
 {#References}

### In classical gravity

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 1, chapter I.3.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Matthias Blau]], chapter 38 of _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* Thomas Hartman, _Lecture Notes on Classical de Sitter Space_ ([pdf](http://www.hartmanhep.net/GR2017/desitter-lectures-v2.pdf), [[HartmanDeSitterSpacetime.pdf:file]])

* Marcus Spradlin, [[Andrew Strominger]], Anastasia Volovich, _Les Houches Lectures on De Sitter Space_ ([arXiv:hep-th/0110007](https://arxiv.org/abs/hep-th/0110007))

See also

* Wikipedia, _[de Sitter space](http://en.wikipedia.org/wiki/De_Sitter_space)_

### In perturbative quantum gravity
 {#InPerturbativeQuantumGravity}

Discussion of de Sitter spacetime in [[perturbative quantum gravity]] and its infrared instability, includes the following:

* [[Emil Mottola]], _Particle creation in de Sitter space_, Phys. Rev. D 31, 754 – Published 15 February 1985 ([doi:10.1103/PhysRevD.31.754](https://doi.org/10.1103/PhysRevD.31.754))

* {#Witten01} [[Edward Witten]], _Quantum Gravity In De Sitter Space_, talk at [Strings 2001](http://theory.tifr.res.in/strings/),  ([arXiv:hep-th/0106109](https://arxiv.org/abs/hep-th/0106109))

* Dietmar Klemm, Luciano Vanzo, _Aspects of Quantum Gravity in de Sitter Spaces_, Journal of Cosmology and Astroparticle Physics, Volume 2004, November 2004 ([arXiv:hep-th/0407255](https://arxiv.org/abs/hep-th/0407255))

* [[Richard Woodard]], _De Sitter Breaking in Field Theory_ ([arXiv:gr-qc/0408002](https://arxiv.org/abs/gr-qc/0408002))

* [[Alexander Polyakov]], _De Sitter Space and Eternity_, Nucl.Phys.B797:199-217, 2008  ([arXiv:0709.2899](https://arxiv.org/abs/0709.2899))

* [[Alexander Polyakov]], _Decay of Vacuum Energy_, Nucl.Phys.B834:316-329, 2010 ([arXiv:0912.5503](https://arxiv.org/abs/0912.5503))

* Dmitry Krotov, [[Alexander Polyakov]], _Infrared Sensitivity of Unstable Vacua_, Nucl.Phys.B 849 (2011) 410-432 ([arXiv:1012.2107](https://arxiv.org/abs/1012.2107))

* [[Alexander Polyakov]], _Infrared instability of the de Sitter space_ ([arXiv:1209.4135](https://arxiv.org/abs/1209.4135))

* Paul R. Anderson, [[Emil Mottola]], _On the Instability of Global de Sitter Space to Particle Creation_, Phys. Rev. D 89, 104038 (2014) ([arXiv:1310.0030](https://arxiv.org/abs/1310.0030))

* Paul Anderson, [[Emil Mottola]], _Quantum Vacuum Instability of 'Eternal' de Sitter Space_, Phys. Rev. D 89, 104039 (2014) ([arXiv:1310.1963](https://arxiv.org/abs/1310.1963))

* {#Rajaraman16} [[Arvind Rajaraman]], _de Sitter Space is Unstable in Quantum Gravity_, Phys. Rev. D 94, 125025 (2016) ([arXiv:1608.07237](http://arxiv.org/abs/1608.07237))

* Paul Anderson, [[Emil Mottola]], Dillon H. Sanders, _Decay of the de Sitter Vacuum_, Phys. Rev. D 97, 065016 (2018) ([arXiv:1712.04522](https://arxiv.org/abs/1712.04522))

### In string/M-theory
 {#ReferencesInStringTheory}

A [[no-go theorem]] for [[KK-compactifications]] of [[supergravity]] down to de Sitter spacetime is claimed in

* {#MaldacenaNunez01} [[Juan Maldacena]], Carlos Nunez, _Supergravity description of field theories on curved manifolds and a no go theorem_, Int.J.Mod.Phys. A16 (2001) 822-855 ([arXiv:hep-th/0007018](https://arxiv.org/abs/hep-th/0007018))

Discussion of de Sitter [[perturbative string theory vacua]] and their role in the [[landscape of string theory vacua]] or else in the [[swampland]]:

* [[David Kutasov]], Travis Maxfield, Ilarion Melnikov, [[Savdeep Sethi]], _Constraining de Sitter Space in String Theory_, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https://arxiv.org/abs/1504.00056))

* Jakob Moritz, Ander Retolaza, Alexander Westphal, _Towards de Sitter from 10D_, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https://arxiv.org/abs/1707.08678))

* {#Sethi17} [[Savdeep Sethi]], _Supersymmetry Breaking by Fluxes_ ([arXiv:1709.03554](https://arxiv.org/abs/1709.03554))

  re-amplified in:

  [[Savdeep Sethi]], _Hope or No Hope for a String Landscape?_, talk at [StringPheno2019](https://indico.cern.ch/event/782251/) ([pdf](https://indico.cern.ch/event/782251/contributions/3441886/attachments/1870138/3076971/3_Sethi.pdf) [[SethiLandscape2019.pdf:file]])

* {#DanielssonVanRiet18} [[Ulf Danielsson]], [[Thomas Van Riet]],  _What if string theory has no de Sitter vacua?_, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018)  ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070))

  (problems with [KKLT 03](landscape+of+string+theory+vacua#KKLT03))

* {#vanRiet18} [[Thomas Van Riet]], _Is dS space in the Swampland_, talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf))

* {#vanRiet18a} [[Thomas Van Riet]], _Status of KKLT_, talk at _[Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870)_ ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730))

* {#Danielsson18} [[Ulf Danielsson]], _The quantum swampland_ ([arXiv:1809.04512](https://arxiv.org/abs/1809.04512))

  (argues that the issue with [[string theory|stringy]] de Sitter [[moduli stabilization]] raised in [Sethi 17](#Sethi17) is related to the de Sitter instability seen in QFT, according to the references [above](#InPerturbativeQuantumGravity))

* Georges Obied, [[Hirosi Ooguri]], Lev Spodyneiko, [[Cumrun Vafa]], _De Sitter Space and the Swampland_ ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362))

* Jakob Moritz, Ander Retolaza, Alexander Westphal, _On uplifts by warped anti-D3-branes_ ([arXiv:1809.06618](https://arxiv.org/abs/1809.06618))

* Iosif Bena, Emilian Dudas, Mariana Graña, Severin Lüst, _Uplifting Runaways_ ([arXiv:1809.06861](https://arxiv.org/abs/1809.06861))

* [[Clay Cordova]], G. Bruno De Luca, [[Alessandro Tomasiello]], _Classical de Sitter Solutions of Ten-Dimensional Supergravity_ ([arXiv:1812.04147](https://arxiv.org/abs/1812.04147))

* [[Michael Dine]], Jamie A.P. Law-Smith, Shijun Sun, Duncan Wood, Yan Yu, _Obstacles to Constructing de Sitter Space in String Theory_ ([arXiv:2008.12399](https://arxiv.org/abs/2008.12399))

In view of [[alpha']] [[higher curvature corrections]]:

* Ivano Basile, Alessia Platania, _String Tension between de Sitter vacua and Curvature Corrections_ ([arXiv:2103.06276](https://arxiv.org/abs/2103.06276))

  > "We show that the resulting higher-derivative $\alpha'$-corrections forbid the existence of de Sitter vacua, at least in regimes where string-loop corrections can be neglected."

On  [[de Sitter spacetime]] [[cosmology]] realized in [[brane world models]] in ambient $\sim$[[anti de Sitter spacetime|AdS]]-[[bulk]] [[spacetime]]:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

Discussion in the context of [[M-theory on G2-manifolds]]:

* Beatriz de Carlos, Andre Lukas, Stephen Morris, _Non-perturbative vacua for M-theory on G2 manifolds_, JHEP 0412:018, 2004 ([arxiv:hep-th/0409255](https://arxiv.org/abs/hep-th/0409255))

  which concludes that with taking [[non-perturbative effects]] from [[membrane instantons]] into account one gets 4d vacua with vanishing and negative [[cosmological constant]] ([[Minkowski spacetime]] and [[anti-de Sitter spacetime]]) but not with positive [[cosmological constant]] ([[de Sitter spacetime]]). They close by speculating that [[M5-brane]] instantons might yield [[de Sitter spacetime]].

* Johan Blåbäck, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, _Constructing stable de Sitter in M-theory from higher curvature corrections_ ([arXiv:1902.04053](https://arxiv.org/abs/1902.04053))

  which suggests that including [[higher curvature corrections]] makes it work

* Iosif Bena, Alex Buchel, Severin Lüst, _Throat destabilization (for profit and for fun)_ ([arxiv:1910.08094](https://arxiv.org/abs/1910.08094))

For analogous discussion of embedding of the [[ekpyrotic cosmology]]-scenario into [[string theory]]/[[type IIA supergravity]]:

* {#Uzawa18} [[Kunihito Uzawa]], _No-Go theorems for ekpyrosis from ten-dimensional supergravity_, J. High Energ. Phys. (2018) 2018: 41 ([arXiv:1803.11084](https://arxiv.org/abs/1803.11084), doi:10.1007/JHEP06(2018)041)

  reviewed in:

  Kunihito Uzawa, _Ekpyrotic scenario in string theory_ ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~gc2018/slides/4th/Mar2/Uzawa.pdf))

Discussion of *asymptotic* de Sitter spacetimes from time-dependent [[KK-compactification]] of [[type II supergravity]]:

* {#MarconnetTsimpis23} Paul Marconnet, [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*, J. High Energ. Phys. **2023** 33 (2023) [[arXiv:2210.10813](https://arxiv.org/abs/2210.10813), doi:10.1007/JHEP01(2023)033]

reviewed in:

* [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [slides:[[Tsimpis-MTheoryMath2023.pdf:file]], video:video: [YT](https://www.youtube.com/watch?v=LEiqYz-G34g), [web](/nlab/show/M-Theory+and+Mathematics#Tsimpis2023)]

and further discussed in:

* David Andriot, [[Dimitrios Tsimpis]], Timm Wrase, *Accelerated expansion of an open universe, and string theory realizations* [[arXiv:2309.03938](https://arxiv.org/abs/2309.03938)]

[[!redirects de Sitter spacetimes]]

[[!redirects de Sitter space]]
[[!redirects de Sitter spaces]]


Deane Montgomery

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Deane_Montgomery)

## Selected writings

On [[transitive actions]] on [[n-spheres|$n$-spheres]] by [[compact Lie groups]]:

* {#MontogomerySamelson43} [[Deane Montgomery]], [[Hans Samelson]], _Transformation Groups of Spheres_, Annals of Mathematics Second Series **44** 3 (1943) 454-470 [[jstor:1968975](https://www.jstor.org/stable/1968975)]

Introducing the [[slice theorem]] for [[topological G-spaces]]:

* [[Deane Montgomery]], [[Chung Tao Yang]], _The existence of a slice_, Ann. of Math. 65 (1957), 108-116 ([jstor:1969667](https://www.jstor.org/stable/1969667), [doi:10.2307/1969667](https://doi.org/10.2307/1969667))

category: people


Debapriyay

I have been able to contribute a few integer sequences in Online Encyclopedia of Integer Sequences. As an avid mathematics lover thats my only identity and that is why proudly sharing my OEIS user page.
https://oeis.org/wiki/User:Debapriyay_Mukhopadhyay


Debbie W. Leung

* [personal page](https://www.math.uwaterloo.ca/~wcleung/)

## Selected writings

Introducing [[measurement-based quantum computation]]:

* [[Debbie W. Leung]], *Quantum computation by measurements*,  International Journal of Quantum Information **02** 01 (2004) 33-43 [[arXiv:quant-ph/0310189](https://arxiv.org/abs/quant-ph/0310189), [doi:10.1142/S0219749904000055](https://doi.org/10.1142/S0219749904000055)]

category: people

[[!redirects Debbie Leung ]]


Deborah Kant

* [webpage](https://deborahkant.org/)

## Selected publications

* [[Mirna Džamonja]], [[Deborah Kant]], *Interview With a Set Theorist*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_1](https://doi.org/10.1007/978-3-030-15655-8_1), [pdf](https://hal.archives-ouvertes.fr/hal-01858063/document), [pdf](https://deborahkant.org/wp-content/uploads/2018/07/PhilTalk0716.pdf)]

category: people