excluded middle

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#### Foundations
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## Idea

In [[logic]], the principle of **excluded middle** states that every [[truth value]] is either [[true]] or [[false]] ([Aristotle, MP1011b24](#AristotleMetaphysics)). (This is sometimes called the '[[axiom]]' or 'law' of excluded middle, either to emphasise that it is or is not optional; 'principle' is a relatively neutral term.) One of the many meanings of [[classical logic]] is to emphasise that this principle holds in the logic; in contrast, it fails in [[intuitionistic logic]].

The principle of excluded middle (hereafter, PEM), as a statement about truth values themselves, is accepted by nearly all mathematicians ([[classical mathematics]]); those who doubt or deny it are a distinct minority, the [[constructive mathematics|constructivists]]. However, when one [[internalization|internalises]] mathematics in [[categories]] other than [[Set|the category of sets]], there is no doubt that excluded middle often fails [[internal logic|internally]]. See the examples listed at [[internal logic]]. (Those categories in which excluded middle holds are called [[Boolean category|Boolean]]; in general, the adjective 'Boolean' is often used to indicate the applicability of PEM.)

Although the term 'excluded middle' (sometimes even _excluded third_) suggests that the principle does not apply in many-valued logics, that is not the point; many-valued logics are many-valued *externally* but may still be two-valued *internally*.  In the language of [[categorial logic]], whether a category has exactly two [[subterminal objects]] is in general independent of whether it is Boolean; instead, the category is Boolean iff the statement that it has exactly two subterminal objects holds in its [[internal logic]] (which is in general independent of whether that statement is true).

In fact, intuitionistic logic proves that there is no truth value that is neither true nor false; in this sense, the possibility of a 'middle' or 'third' truth value is still 'excluded'.  But since the relevant [[de Morgan law]] fails in intuitionistic logic, we may not conclude that every truth value is either true or false, which is the actual PEM.

## In set theory

In [[material set theory]], [Shulman 18](#Shulman18) makes the distinction between **full classical logic**, where the principle of excluded middle holds for all logical formulae in the [[material set theory]], and **$\Delta_0$-classical logic**, where the principle of excluded middle holds for only holds for $\Delta_0$-formulae. $\Delta_0$-formulae are logical formulae where every [[quantifier]] $Q$ has to be bounded by the membership relation; i.e. $Q x \in y.P(x) \coloneqq Q x.x \in y \wedge P(x)$. In [[structural set theory]], the above difference between full classical logic and $\Delta_0$-classical logic in [[material set theory]] is the difference between defining [[structural set theory]] as a [[well-pointed topos|well-pointed]] [[Boolean topos]] in [[classical logic]] and in [[intuitionistic logic]] respectively.

## In type theory

In [[dependent type theory]], there are many statements of excluded middle.

### Truncated excluded middle

The statement which directly corresponds to excluded middle in logic states that given a type $P$, if $P$ is a [[mere proposition]], then either $P$ or the negation of $P$ holds

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to (P \vee \neg P)}$$

where $\mathrm{isProp}$ is the [[isProp]] [[modality]] commonly defined as $\prod_{x:P} \prod_{y:P} \mathrm{Id}_P(x, y)$, negation $\neg P$ is simply the the function type $P \to \mathbb{0}$ into the [[empty type]] and [[disjunction]] $A \vee B$ is the [[bracket type]] of the [[sum type]] $[A + B]$.

If one additionally has a [[Russell type of all propositions]] $\Omega$ or a [[Tarski type of all propositions]] $(\Omega, \mathrm{El}_\Omega)$, the law of truncated excluded middle could be expressed as an axiom:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} P \vee \neg P}\mathrm{Russell} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} \mathrm{El}_\Omega(P) \vee \neg \mathrm{El}_\Omega(P)}\mathrm{Tarski}$$

If the type theory has a [[boolean domain]] $\mathrm{Bit}$ which is a [[univalent Tarski universe]] with [[type family]] $\mathrm{El}_\mathrm{Bit}$ and thus satisfies the extensionality principle, truncated excluded middle could be stated as given a type $P$, if $P$ is a [[mere proposition]], then there exists a boolean $Q$ such that $P$ is equivalent to the type reflection of $Q$:

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to \exists Q:\mathrm{Bit}.P \simeq \mathrm{El}_\mathrm{Bit}(Q)}$$

By definition, the only two booleans are $0$ representing false and $1$ representing true.

### Untruncated excluded middle

There is also an untruncated version of excluded middle, which uses the sum type instead of the disjunciton:

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to (P + \neg P)}$$

In the untruncated case, the requirement that $P$ be an [[h-proposition]] is necessary; the untruncated law of excluded middle for [[h-sets]] is the [[set-theoretic choice operator]], which implies the [[set-theoretic axiom of choice]] in addition to [[excluded middle]], and the untruncated law of excluded middle for general [[types]] is the [[type-theoretic choice operator]], which implies [[UIP]] in addition to the set-theoretic axiom of choice and excluded middle.

If one additionally has a [[Russell type of all propositions]] $\Omega$ or a [[Tarski type of all propositions]] $(\Omega, \mathrm{El}_\Omega)$, the law of untruncated excluded middle could be expressed as an axiom:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} P + \neg P}\mathrm{Russell} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} \mathrm{El}_\Omega(P) + \neg \mathrm{El}_\Omega(P)}\mathrm{Tarski}$$

Similarly as above, if the type theory has a [[boolean domain]] $\mathrm{Bit}$ which is a [[univalent Tarski universe]] with [[type family]] $\mathrm{El}_\mathrm{Bit}$ and thus satisfies the extensionality principle, there is also an untruncated version of excluded middle using the [[boolean domain]] and [[dependent sum types]]:

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to \sum_{Q:\mathrm{Bit}} P \simeq \mathrm{El}_\mathrm{Bit}(Q)}$$

By definition, the only two booleans are $0$ representing false and $1$ representing true.

### Contractible excluded middle

There is also a version of excluded middle, which uses [[contractible type|contractibility]] instead of [[pointed type|pointedness]] to express [[truth]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_A:\mathrm{isProp}(A) \to \mathrm{isContr}(A + \neg A)}$$

In [[dependent type theory]] as [[foundations of mathematics]], the requirement that $A$ be an [[h-proposition]] is also necessary. This alternate law of untruncated excluded middle for general types implies that every type is a [[h-proposition]] a la [[propositional logic as a dependent type theory]], because the only types $P$ such that $P + \neg P$ is equivalent to the unit type are the [[empty type]] and the [[unit type]], which are [[h-propositions]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_A:\mathrm{isContr}(A + \neg A)}$$

If one additionally has a [[Russell type of all propositions]] $\Omega$ or a [[Tarski type of all propositions]] $(\Omega, \mathrm{El}_\Omega)$, the law of excluded middle could be expressed as an axiom:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} \mathrm{isContr}(P + \neg P)}\mathrm{Russell} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} \mathrm{isContr}(\mathrm{El}_\Omega(P) + \neg \mathrm{El}_\Omega(P))}\mathrm{Tarski}$$

Similarly as above, if the type theory has a [[boolean domain]] $\mathrm{Bit}$ which is a [[univalent Tarski universe]] with [[type family]] $\mathrm{El}_\mathrm{Bit}$ and thus satisfies the extensionality principle, there is also an contractible version of excluded middle using the [[boolean domain]] and [[dependent sum types]]:

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to \mathrm{isContr}\left(\sum_{Q:\mathrm{Bit}} P \simeq \mathrm{El}_\mathrm{Bit}(Q)\right)}$$

or equivalently, using the [[boolean domain]] and the [[uniqueness quantifier]]:

$$\frac{\Gamma \vdash P \; \mathrm{type}}{\Gamma \vdash \mathrm{lem}_P:\mathrm{isProp}(P) \to \exists!Q:\mathrm{Bit}.P \simeq \mathrm{El}_\mathrm{Bit}(Q)}$$

By definition, the only two booleans are $0$ representing false and $1$ representing true.

### Relations between the notions of excluded middle

\begin{theorem}
The untruncated version of excluded middle implies the truncated version of excluded middle.
\end{theorem}

\begin{proof}
By the [[introduction rule]] of [[propositional truncations]], there exists a function $\mathrm{isInhab}_{P + \neg P}:(P + \neg P) \to [P + \neg P]$, and the truncated version of excluded middle is simply the [[composition]] of functions
$$\mathrm{isInhab}_{P + \neg P} \circ \mathrm{lem}_P:\mathrm{isProp}(P) \to (P \vee \neg P)$$
\end{proof}

\begin{theorem}
The contractible version of excluded middle implies the untruncated version of excluded middle
\end{theorem}

\begin{proof}
The type $\mathrm{isContr}(P + \neg P)$ is really the type

$$\sum_{a:P + \neg P} \prod_{b:P + \neg P} a =_{P + \neg P} b$$

Thus, by the [[introduction rules]] of [[function types]] and the elimination rules of [[negative type|negative]] [[dependent sum types]], there is a function
$$\lambda x:\mathrm{isProp}(P).\pi_1(\mathrm{lem}_P(x)):\mathrm{isProp}(P) \to (P + \neg P)$$
which is precisely the untruncated version of excluded middle
\end{proof}

## Relation to the axiom of choice
 {#RelationToTheAxiomOfChoice}

Excluded middle can be seen as a very weak form of the [[axiom of choice]] (a slightly more controversial principle, doubted or denied by a slightly larger minority, and true internally in even fewer categories).  In fact, the following are equivalent. (this is the [[Diaconescu-Goodman-Myhill theorem]] due to [Diaconescu 75](#Diaconescu75), see also [McLarty 96, theorem 19.7](#McLarty96), )

1. The principle of excluded middle.
1. [[finite set|Finitely indexed]] sets are [[projective object|projective]] (in fact, it suffices 2-indexed sets to be projective).
1. [[finite set|Finite sets]] are [[choice object|choice]] (in fact, it suffices for 2 to be choice).

(Here, a set $A$ is **finite** or **finitely-indexed** (respectively) if, for some natural number $n$, there is a bijection or surjection (respectively) $\{0, \ldots, n - 1\} \to A$.)

The proof is as follows.  If $p$ is a truth value, then divide $\{0,1\}$ by the equivalence relation where $0 \equiv 1$ iff $p$ holds.  Then we have a surjection $2\to A$, whose domain is $2$ (and in particular, finite), and whose codomain $A$ is finitely-indexed.  But this surjection splits iff $p$ is true or false, so if either $2$ is choice or $2$-indexed sets are projective, then PEM holds.

On the other hand, if PEM holds, then we can show by induction that if $A$ and $B$ are choice, so is $A\sqcup B$ (add details).  Thus, all finite sets are choice.  Now if $n\to A$ is a surjection, exhibiting $A$ as finitely indexed, it has a section $A\to n$.  Since a finite set is always projective, and any retract of a projective object is projective, this shows that $A$ is projective.

In particular, the axiom of choice implies PEM.  This argument, due originally to Diaconescu, can be internalized in any [[topos]]. However, other weak versions of choice such as [[countable choice]] (any surjection to a countable set (which for this purpose is any set isomorphic to the set of natural numbers) has a section), [[dependent choice]], or even [[COSHEP]] do not imply PEM.

While is often claimed that axiom of choice is *true* in constructive mathematics (by the BHK or [[Brouwer-Heyting-Kolmogorov interpretation]] of predicate logic), the BHK interpretation of the "axiom of choice" is simply the always true statement that indexed [[cartesian products]] distribute over indexed [[disjoint unions]]

$$\left(\prod_{x \in A} \biguplus_{y \in B(x)} \{\top \vert P(x, y)\}\right) \to \left(\biguplus_{g \in \prod_{x \in A} B(x)} \{\top \vert P(x, g(x))\}\right)$$

or the always true statement that every [[split surjection]] splits.

## Other equivalent statements

* Morgan Rogers showed [here](https://golem.ph.utexas.edu/category/2023/01/freedom.html#c062002) that the fact that the [[category of sets]] is a [[Boolean topos]] is equivalent to the fact that every [[pointed set]] is a [[free object|free]] pointed set.

* [[compact spaces equivalently have converging subnets]]

* The [[limited principle of omniscience]] holds for all sets.

* The [[boolean domain]] $2$ is a [[frame]] iff the [[category of sets]] is a [[Boolean topos]]. This implies that one can do [[topology]] (using [[topological spaces]], [[locales]], [[formal topologies]], etc) directly using the boolean domain as the frame of open truth values.

* The [[poset of truth values]] is an [[inequality space]].

## Double-negated PEM
 {#DoubleNegatedPEM}

While PEM is not valid in constructive mathematics, its double negation

$$ \neg\neg(A\vee \neg A)$$

is valid.  One way to see this is to note that $\neg (A\vee B) = \neg A \wedge \neg B$ is one of [[de Morgan's laws]] that is constructively valid, and $\neg (\neg A \wedge \neg\neg A)$ is easy to prove (it is an instance of the [[law of non-contradiction]]).

However, a more direct argument makes the structure of the proof more clear.  When [[beta-reduced]], the [[proof term]] is $\lambda x. x(inr(\lambda a. x(inl(a))))$.   This means that we first assume $\neg (A\vee \neg A)$ for a contradiction, for which it suffices (by assumption) to prove $A\vee \neg A$.  We prove that by proving $\neg A$, which we prove by assuming $A$ for a contradiction.  But now we can reach a contradiction by invoking (again) our assumption of $\neg (A\vee \neg A)$ and proving $A\vee \neg A$ this time using our new assumption of $A$.  In other words, we start out claiming that $\neg A$, but whenever that "bluff gets called" by someone supplying an $A$ and asking us to yield a contradiction, we retroactively change our minds and claim that $A$ instead, using the $A$ that we were just given as evidence.

\begin{theorem}
In [[dependent type theory]], the double negation of untruncated excluded middle holds for all types $A$
\end{theorem}

\begin{proof}
The type $\neg \neg A \to \neg \neg A$ has an element, the [[identity function]] $\mathrm{id}_{\neg \neg A}$. Given types $A$, $B$, and $C$, there are equivalences

$$\mathrm{uncurry}_{A, B, C}:(A \to (B \to C)) \simeq ((A \times B) \to C)$$

$$\mathrm{expconv}^{A, B, C}:((A \to C) \times (B \to C)) \simeq ((A + B) \to C)$$

$$\mathrm{comm}_+^{A,B}:(A + B) \simeq (B + A)$$

Thus, there is an element

$$\lambda z:\neg (A + \neg A).\lambda y:(\neg A + A).z(\mathrm{comm}_+^{\neg A,A}(\lambda x:(\neg \neg A \times \neg A).\mathrm{expconv}^{\neg A, A, \emptyset}(\mathrm{uncurry}_{\neg \neg A, \neg A, \emptyset}(\mathrm{id}_{\neg \neg A})(x))(y)):\neg \neg (A + \neg A)$$
\end{proof}

In particular, this shows how the [[double negation]] [[modality]] can be regarded computationally as a sort of [[continuation-passing]] transform.

However, there is another meaning of "double negated PEM" that is not valid.  The above argument shows that

$$ \forall A, \neg\neg(A\vee \neg A). $$

But a stronger statement is

$$ \neg\neg \forall A, (A \vee \neg A).$$

This is related to the above valid statement by a [[double-negation shift]]; and in fact, the truth of $ \neg\neg \forall A, (A \vee \neg A)$ is equivalent to the principle of double-negation shift.  In particular, it is *not* constructively provable.

## Sharp excluded middle

One could add to [[cohesive homotopy type theory]] a version of excluded middle called the **sharp law of excluded middle**, given by the following rule:

$$\frac{\Xi \vert \Gamma \vdash P \; \mathrm{type} \quad \Xi \vert \Gamma \vdash p:\mathrm{isProp}(P)}{\Xi \vert \Gamma \vdash \mathrm{lem}^\sharp:\sharp(P \vee \neg P)}$$

where $A \vee B \coloneqq \left[A + B\right]$ and $\neg A \coloneqq A \to \mathbb{0}$, and $\sharp A$ is the [[sharp modality]] of $A$, $\left[A\right]$ is the [[propositional truncation]] of $A$ and $\mathbb{0}$ is the [[empty type]].

## Related concepts

* [[axiom of choice]]

* [[law of double negation]]

* [[ex falso quodlibet]]

* [[proof by contradiction]]

* [[De Morgan law]]

* [[Cantor-Schroeder-Bernstein theorem]]

## References

### General

* {#McLarty96} [[Colin McLarty]], _Elementary Categories, Elementary Toposes_, Oxford University Press, 1996

* {#Shulman18} [[Mike Shulman]], *Comparing material and structural set theories*, Annals of Pure and Applied Logic, Volume 170, Issue 4, April 2019, Pages 465-504 ([doi:10.1016/j.apal.2018.11.002](https://doi.org/10.1016/j.apal.2018.11.002), [arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

### Relation to other axioms:

Relation to the [[axiom of choice]] ([[Diaconescu-Goodman-Myhill theorem]]):

* {#Diaconescu75} [[Radu Diaconescu]], _Axiom of choice and complementation_, Proceedings of the American Mathematical Society 51:176-178 (1975) ([doi:10.1090/S0002-9939-1975-0373893-X](https://doi.org/10.1090/S0002-9939-1975-0373893-X))

* {#GoodmanMyhill78} N. D. Goodman J. Myhill, _Choice Implies Excluded Middle_, Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik 24:461 (1978)

Relation to *[[ex falso quodlibet]]* and the *[[law of double negation]]*:

* Pedro Francisco, Valencia Vizcaíno, *Relations between ex falso, tertium non datur, and double negation elimination* [[arXiv:1304.0272](https://arxiv.org/abs/1304.0272)]

### In homotopy type theory
 {#ReferencesInHomotopyTypeTheory}

* {#AristotleMetaphysics} [[Aristotle]], _[[Metaphysics (Aristotle)|Metaphysics]]_, 1011b24: "Of any one subject, one thing must be either asserted or denied"

In [[homotopy type theory]]:

* [[Univalent Foundations Project]], section 3.5 _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

In [[cohesive homotopy type theory]]:

* Mike Shulman, *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

category: foundational axiom

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exclusive disjunction

# Exclusive disjunction
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## Idea

In [[propositional logic]], the __exclusive disjunction__ (also called __exclusive or__ or simply __xor__) of a family of statements ([[truth values]], [[propositions]], [[predicates]], velc) is a statement that is true if and only if exactly one of the statements in the family is true.  The corresponding operation on [[subsets]] (of a fixed set) or [[pure set|material sets]] is called _exclusive union_, and the binary operation agrees with _[[symmetric difference]]_ (which is generalised to higher arities in a different way).  If we think of [[quantifiers]] as infinitary operators, then exclusive disjunction becomes [[uniqueness quantification]].

## Definitions
 {#Definitions}

The __exclusive disjunction__ of $p$ and $q$, written $p ⊻ q$ (and a host of other ways), may be defined in any of these forms:
1.  $\neg(p \Leftrightarrow q)$,
2.  $(p \vee q) \wedge (\neg p \vee \neg q)$,
3.  $(p \wedge \neg{q}) \vee (\neg{p} \wedge q)$.

These are all equivalent in [[classical logic]].  In [[intuitionistic logic]], (2,3) are equivalent but (1) is weaker; (2,3) give the usual meaning in [[constructive mathematics]].

The [[false]] statement is the [[identity element|identity]] for this operation; it is the exclusive disjunction of no statements.

In classical logic (but not in intuitionistic logic), this operation is associative, but multiple applications don\'t mean what you may think.  Instead, $(p ⊻ q) ⊻ r \equiv p ⊻ (q ⊻ r)$ comes out true if $p, q, r$ are all true; in general,
$$ (p_1 ⊻ (p_2 ⊻ \cdots ⊻ p_n)\cdots) $$
is true if and only if an *odd* number of the statements $p_i$ is true.  But when we write a multiple statement with 'xor', we really mean that *exactly one* of the statements is true.  So by fiat, define the __exclusive disjunction__
$$ p_1 ⊻ p_2 ⊻ \cdots ⊻ p_n $$
(without parentheses) to mean that exactly one of the $p_i$ is true; this definition is also used in intuitionistic logic.

As the indexed version of ordinary [[disjunction]] is [[existential quantification]] $\exists$, so the indexed version of exclusive disjunction is [[uniqueness quantification]] $\exists!$; this requires a primitive notion of [[equality]] to state in [[predicate logic]].

Similarly, the __exclusive union__ of two sets $A$ and $B$, written $A \uplus B$ (and a host of other ways), may be defined using exclusive disjunction:
$$ A \uplus B = \{ x \;|\; x \in A \;⊻\; x \in B \} .$$
The __exclusive union__ of a family $(A_i)_{i\colon I}$ of sets may be defined using uniqueness quantification:
$$ \biguplus_i A_i = \{ x \;|\; \exists!{i},\; x \in A_i \} .$$

Note that the [[union]], exclusive union, and (internal, or external up to [[natural isomorphism]]) [[disjoint union]] of a family of (pairwise) [[disjoint sets]] are all the same.  For a family that is not disjoint, however, the union are exclusive union are different, the internal disjoint union does not make sense, and the external disjoint union is not isomorphic to either the union or the exclusive union (at least not naturally, and in some cases not at all).

We also have the __[[symmetric difference]]__ of two sets, which is the same as the exclusive union.  But we see this operation as the addition in a [[Boolean ring]] and so actually interpret it in the usual way as an associative operation.  So the symmetric difference of $n$ sets indeed consists of those points that belong to an odd number of sets, and there is no infinitary symmetric difference.

## As a logic gate
 {#AsALogicGate}

Exclusive disjunction as a [[logic gate]], a [[reversible computation|reversible]] gate ([[CNOT]]) and as a [[quantum logic gate]]:

\begin{imagefromfile}
    "file_name": "CNOTGates-221026c.jpg",
    "width": "750",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## In natural language

It is widely said that 'or' in English can mean either inclusive or exclusive disjunction, while Latin has two terms, respectively 'vel' and 'aut', but this is not really correct.  It\'s more correct that 'or' can sometimes mean *[[union]]* and sometimes *[[disjoint union]]* (with the latter sometimes external and sometimes internal), but never symmetric difference.  Thus, the only meaning of 'or' is for some kind of [[coproduct]], which exclusive disjunction is not.  (Similarly, 'aut' is more about disjoint union than symmetric difference.)  In some cases, exclusive disjunction may be a valid *[implicature](http://en.wikipedia.org/wiki/Implicature)* (in the sense of linguistic philosopher Paul Grice) even if the only valid *inference* (by the literal meaning of the sentence) is taken to be inclusive.

For the literature on this subject, see [the Stanford Encylopedia entry](http://plato.stanford.edu/entries/disjunction/#NatLan) and The Myth of the Exclusive 'Or' (*Mind*, 80 (317), 116--121).

## Related concepts

* [[CNOT]]

* [[logic gate]]

## References

See also

* Wikipedia, *[Exclusive or](https://en.wikipedia.org/wiki/Exclusive_or)*

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exercise

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#### Mathematics
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## Idea

In [[mathematics]], the statement of an *exercise* is typically the same as that of a [[lemma]], [[proposition]] or [[theorem]], but the attitude towards its [[proof]] is different:

An __exercise__ is for the reader to fill in a [[proof]], in private. The reader may seek help from whomever, but the idea is to practice.  A great many exercises have been publicly solved, and a great many [[theorems]] have not, thus confusing which are which, though it's generally good practice to treat them *all* as exercises and ignore published [[proofs]], within the constraints of available time and balancing of competing priorities.

When a proposition is *called* an exercise, this means at least that the omission of a proof is deliberate (or that its inclusion is accidental), but the author is confident of being able to produce one, given enough patience.

## Examples

* [[Milnor's exercise]]

## Related concepts

[[!include mathematical statements --- contents]]

[[!redirects exercises]]


exercise in groupoidification - the path integral

> under construction

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  On the formalization of the process of [[quantization]] -- by [[category theory|abstract nonsense]] -- from classical [[∞-model]] data to the corresponding [[quantum field theory]] .

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* automatic table of contents goes here
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## Background and motivation

The search is on for the abstract formalization of the process of _[[quantization]]_ -- the process that reads in a "[[classical field theory]]" -- for instance presented in form of a [[gauge theory]] or in form of a [[∞-model]] background field data -- and spits out the corresponding [[quantum field theory]].

+-- {.standout}

There is an open problem of mathematical (-[[physics]]) model building:

* what is the true formalism behind [[quantization]]?

One that is to quantization as, say, [[symplectic geometry]] is to [[Hamiltonian mechanics]]?

The formalism of [[FQFT]] clearly suggests that the fundamental description of [[quantization]] is some natural operation on higher functors.

=--

While for various aspects and facets of this question there are well-developed formalisms -- such as [[geometric quantization]] or [[deformation quantization]] or [[BV-BRST formalism]] -- a full answer is certainly still missing, not the least because the full formalization of the _question_ itself has still to be established.

Considerable progress on this formulation of the question has been achieved with the formalization and proof of the [[cobordism hypothesis]] in _[[On the Classification of Topological Field Theories]]_ by [[Jacob Lurie]]. This at least indicates what the _result_ of any full quantization procedure should be in that it clarifies what exactly a [[TQFT]] [[FQFT]] is: a morphism from the [[(∞,n)-category of cobordisms]] $Z : Bord_n \to C$.

In _[[On the Classification of Topological Field Theories]]_ Jacob Lurie indicates some first towards finding a similar formalization of "classical field theory" (in terms of his $(\infty,n)$-categories of "families") and a systematic procedure for turning the classical theory into the quantum theory. These thoughts were further developed in the article _[[Topological Quantum Field Theories from Compact Lie Groups]]_ . But for the moment, that, too, remains a bit sketchy.

For the purpose of the present entry this indication of a quantizaton proposal by Lurie et al. mainly serves as a reference for the idea itself that a formalization of something interesting is to be sought here, and of the kind of [[category theory|abstract nonsense]] answer one hopes to find. We will however discuss a somewhat different-looking approach. It may well be related to the Lurie-et al proposal in the end, but for the time being we shall not concentrate on that relation.

Rather, the approach for a formalization of the quantization procedure that shall be discussed at this entry here draws from a few different sources:

1. The observation that a classical background field that should serve as the input for a quantization of a $\sigma$-model that describes the dynamics of an object charged under this field is encoded by differential cocycles as as described at [[schreiber:differential cohomology in an (∞,1)-topos]].

1. The idea that by applying a pull-push quantization prescription a differential cocycle on a target space $X$ gives rise to a differential cocycle on a _parameter space_ $\Sigma$, which may be thought of as one of the bordisms appearing in the [[FQFT]]-description of quantum field theory.

   The pull-push operation here is akin to that in [[geometric ∞-function theory]], where a [[quantum field theory]] is obtained from a [[∞-model]] target space object $X$ by homming [[extended cobordism]] [[cospan]]s $\Sigma_in \to \Sigma \leftarrow \Sigma_{out}$ into the target object and then pull-pushing [[geometric function object]]s through the resulting [[span]]s of configuration space objects $[\Sigma_{in},X] \leftarrow [\Sigma,X] \to [\Sigma_{out},X]$.

   The main result of David Ben-Zvi et. al.'s work on this approach is that they point out that as soon as the [[geometric function object]] one uses satisfies the two [[geometric ∞-function theory|fundamental theorems of geometric infinity-function theory]], a considerable amount of rich structure that has in parts been known by itself gets unified into one coherent elegant story: the nature of partition functions (i.e. traces), of centers, of Hochschild (co)homology, Deligne-Kontsevich-statements, etc. all are understood by means of a suitable [[geometric function theory]] as induced from the underlying geometry of configuration space objects $[\Sigma,X]$ as well as the [[loop space object]]s of $X$.

   The resulting pull-push operation is an example or a generalization of what [[John Baez]] discusses under the term [[groupoidification]].

1. The observation that a differential coccycle on a [[Lorentzian manifold]] $\Sigma$ gives rise to a [[local net]] of observables, as used in the formalizaton of QFT known as [[AQFT]]. (As described [here](http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf)).

   So the procedure discussed here regards differential cocycles on target space as classicai field theories, regards their quantization as a way to obtain a differential cocycle on Lorentzian parameter space, and identifies this as a quantum field theory by associating a local net of observables to it.

   These local nets, in turn, are akin to [[factorization algebra]]s, which in the Euclidean (meaning non-Lorentzian setting) relate back to cobordism representations via the notion of [[topological chiral homology]]. However the -- physically crucial -- Lorentzian structure invoked here is not otherwise considered in these functorial axiomatization of quantum field theory.

## General ambient structure

The ambient context is the [[(∞,1)-topos]] $\mathbf{H} := Sh_{(\infty,1)}(CartSp)$ of [[Lie ∞-groupoid]]s and the $(\infty,2)$-topos of smooth $(\infty,1)$-categories, which we model, respectively, as the [[Bousfield localization of model categories|left Bousfield localization]] of the [[model structure on functors]] $[CartSp^{op}, sSet_{Quillen}]$ and $[CartSp^{op}, sSet^+]$, respectively, where

* [[CartSp]] is the site of Cartesian spaces,

* $sSet_{Quillen}$ the standard [[model structure on simplicial sets]];

* and $sSet^+$ the [[model structure for Cartesian fibrations]] over the points, hence the [[simplicial model category]] for quaasi-categories.

For the definition of the [[schreiber:path ∞-groupoid]] functor $\mathbf{\Pi}$ and the induced theory of [[schreiber:differential cohomology in an (∞,1)-topos]], we make use of the discussion at [[schreiber:differential cohomology in an (∞,1)-topos -- survey]].

When working with fibrant objects in the model, we will frequently use the constructions and notation from [[category of fibrant objects]]. Notably for $\mathbf{B}G$ a fibrant [[delooping]] object in the model we write $(\mathbf{B}G)^I$ for the [[path object]] we write $\mathbf{E}G$ for the [[pullback]]

$$
  \array{
    \mathbf{E}G &\to& *
    \\
    \downarrow && \downarrow
    \\
    (\mathbf{B}G)^I &\stackrel{d_1}{\to}& \mathbf{B}G
  }
$$

and $\mathbf{E}G \to \mathbf{B}G$ for the remaining map, induced from
$d_0 : (\mathbf{B}G)^I \to \mathbf{B}G$.

## The charged particle

We describe the general theory for the simple example of the charged
particle.

### Target space and background field

The background field for the charged particle that we want to consider is the [[electromagnetic field]]. The data involved is

* the **target space** $X$ -- a smooth [[manifold]];

* the **structure group** or **gauge group** $G = U(1)$;

* a choice of representation

  $$
    \rho : \mathbf{B}G \to Vect_{\mathbb{C}}
    \,,
  $$

  taken to be the canonical representation on $V = \mathbb{C}$;

* the **background field** given by a

  * a $U(1)$-[[principal bundle]] $P \to X$ classified by a [[cocycle]]
    $g : X \to \mathbf{B}U(1)$ in $\mathbf{H}$ which in the model is
    given by an [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}U(1)$;

  * a [[connection on a bundle|connection]] $\nabla$ on this bundle,
    which in the model is given by a diagram

    $$
      \array{
         Y &\stackrel{g}{\to}& \mathbf{B}U(1)
         \\
         \downarrow && \downarrow
         \\
         \mathbf{\Pi}(Y) &\stackrel{\nabla}{\to}&
         \mathbf{E}\mathbf{B}U(1)
      }
      \,,
    $$

    and whose [[field strength]] is given by the composite

    $$
      F : \mathbf{\Pi}(Y) \stackrel{\nabla}{\to}
      \mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1)
      \,.
    $$

### Parameter space and kinetic action

Let $\Sigma = \mathbb{R}$ be the **parameter space**, the **worldline**,
regarded as a [[Lorentzian manifold]] and write $\mathbf{\Pi}(\Sigma)$ for the [Lorentzian path category](http://ncatlab.org/nlab/show/smooth+Lorentzian+space#PathnCategory).

To define the kinetic action, we first form the $\rho$-associated background field

$$
  \array{
    X &\stackrel{g}{\to}& \mathbf{B}U(1) &\stackrel{\rho}{\to}& Vect
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\Pi}(X) &\stackrel{\nabla}{\to}& \mathbf{E} \mathbf{B}U(1)
      &\stackrel{}{\to}& \mathbf{E}\mathbf{B}U(1) \coprod_{\mathbf{B}U(1)} Vect
  }
$$

and then pull this back to the [[multisymplectic geometry|extended configuration space]] $X \times \Sigma$.

The morphisms in the [[product]] category $\mathbf{\Pi}(X) \times \mathbf{\Pi}(\Sigma)$ are paths $\gamma_X : [0,1] \to X$ in $X$ on whose base we have a (pseudo)[[Riemannian metric]], which is the pullback of the metric $\mu_\Sigma$  on $\Sigma$ along $\gamma_\Sigma : [0,1] \to \Sigma$.  We can consider the _kinetic action_ to be a differential cocycle

$$
  \array{
    \Sigma \times X
     &\stackrel{}{\to}& \mathbf{B}U(1) &\stackrel{}{\to}& Vect
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{\Pi}(\Sigma) \times
     \mathbf{\Pi}(X) &\to& \mathbf{E} \mathbf{B}U(1)
      &\stackrel{}{\to}& \mathbf{E}\mathbf{B}U(1)
      \coprod_{\mathbf{B}U(1)} Vect
  }
$$

which sends the path $\gamma_X : [0,1] \to X$ of parameter length $\gamma_\Sigma : [0,1] \to \Sigma$ to

$$
  \mathbb{C}
    \stackrel{\exp(-\frac{1}{i \hbar}\int_{0}^1 |\gamma'_X|^2 d \gamma_\Sigma}{\to}
  \mathbb{C}
  \,.
$$

Notice that $\mathbf{\Pi}(X)\times \mathbf{\Pi}(\Sigma) \simeq \mathbf{\Pi}(X \times \Sigma)$.

### The quantization

The total action is differential cocycle

$$
  \array{
    \Sigma \times X
    &\stackrel{}{\to}& Vect
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\Pi}(\Sigma) \times
     \mathbf{\Pi}(X) &\stackrel{\exp(S_{kin})tra_\nabla}{\to}& \mathbf{E}\mathbf{B}\mathbb{R}
      \coprod_{\mathbf{B}U(1)} Vect
  }
  \,.
$$

We want to consider the diagram

$$
  \array{
    \Sigma \times X & \to&  Vect
    \\
    \downarrow & \searrow && \searrow
    \\
    \Sigma && \mathbf{\Pi}(\Sigma)\times \mathbf{\Pi}(X) &\to& \mathbf{E}\mathbf{B}U(1)\coprod_{\mathbf{B}U(1)} Vect
    \\
    & \searrow & \downarrow
    \\
    && \mathbf{\Pi}(\Sigma)
  }
$$

and use it to obtain a differential cocycle on $\Sigma$, by forming something like a lax pullback ("comma object") of the point inclusion

$$
 \array{
   * &\to& *
   \\
   \downarrow && \downarrow
   \\
   Vect &\to& Vect \coprod_{\mathbf{B}U(1)} \mathbf{E}\mathbf{B}U(1)
 }
 \,,
$$

where the left vertical morphism picks the ground field $\mathbb{C}$, along this cocycle. For the underlying cocycle this is obtained as the ordinary pullback of $\mathbf{E}Vect$ defined as the ordinary pullback

$$
  \array{
    \mathbf{E}Vect &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{k}}
    \\
    Vect^I &\stackrel{d_1}{\to}& Vect
  }
  \,,
$$

where $I$ is the [[interval category]], $Vect^I$ the [[functor category]] and $k : * \to Vect$ picks the ground field vector space. Via the remaining map $d_0 : Vect^I \to Vect$ this maps to $Vect$ and then further to $Vect \coprod_{\mathbf{B}U(1)} \mathbf{E}\mathbf{B}U(1)$.

The pullback of

$$
 \array{
   \mathbf{E}Vect &\to& \mathbf{E}Vect
   \\
   \downarrow && \downarrow
   \\
   Vect &\to& Vect \coprod \mathbf{E}\mathbf{B}U(1)
 }
$$

along our differential cocycle, i. e. the pullback of the top part of the diagram

$$
  \array{
    && \mathbf{E}Vect
    \\
    && \downarrow & \searrow
    \\
    \Sigma \times X & \to&  Vect && \mathbf{E}Vect
    \\
    \downarrow & \searrow && \searrow & \downarrow
    \\
    \Sigma && \mathbf{\Pi}(\Sigma)\times \mathbf{\Pi}(X) &\to& \mathbf{E}\mathbf{B}U(1)\coprod_{\mathbf{B}U(1)} Vect
    \\
    & \searrow & \downarrow
    \\
    && \mathbf{\Pi}(\Sigma)
  }
$$

is over $X \times \Sigma$ the total space of the pullback of the underlying vector bundle $E$ on $X$ to $X \times \Sigma$, and over $\mathbf{\Pi}(X \times \Sigma)$ is a groupoid $E_{\mathbf{\Pi}}$

$$
  \array{
    E &\to& E_{\mathbf{\Pi}}
    \\
    \downarrow && \downarrow
    \\
    X\times \Sigma &\to& \mathbf{\Pi}(X \times \Sigma)
    \\
    \downarrow && \downarrow
    \\
    \Sigma &\to& \mathbf{\Pi}(\Sigma)
  }
  \,.
$$

To see what $E \to E_{\mathbf{\Pi}}$ is like, first notice that we have a [[fibration sequence]]

$$
  \array{
    &&  && V
    \\
    && && \downarrow
    \\
    V &\to& E &\to & V//U(1) &\to& *
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    {*} &\stackrel{x}{\to}
    &X &\stackrel{g}{\to}& \mathbf{B}U(1) &\stackrel{\rho}{\to}&
    Vect
  }
$$

with the bottom right square a lax pullback and everything else [[homotopy pullback]]s. Here $V$ is the vector space that $\rho : \mathbf{B}U(1) \to Vect$ is a representation on (which for the [[electromagnetic field]] will be $\mathbb{C}$ itself).

The groupoid $E_{\mathbf{\Pi}}$ has the following description:

* its objects are triples $(x,\sigma, v)$, where $v \in E_{x}$ is a vector in the fiber of $E$ over $x$;

* its morphisms are triples $(x \stackrel{\gamma}{\to} y, \sigma \to \sigma', v)$ that go from $(x,\sigma,v)$ to $(y, \sigma', \exp(\int_0^1 |...|) \rho(\nabla(\gamma)))(v)$, i.e. from a vector in the fiber over the source to the corresponding vector in the fiber over the target, obtained by evaluating the action functional on the path.

To obtain from this a cocycle on $\Sigma$, we proceed as follows: we regard an interval $\sigma := [\sigma_{in}, \sigma_{out}] \in \mathbf{\Pi}(\Sigma)_1$ as a [[cospan]]

$$
  \array{
    \sigma_{in} &\to& \sigma &\leftarrow& \sigma_{out}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \sigma_{in} &\to& \mathbf{\Pi}(\sigma) &\leftarrow& \sigma_{out}
    \,,
  }
$$

where in the top row we regard these subsets of $\Sigma$ as discrete smooth sub-categories, and in the bottom row form the path $\infty$-groupoids.

Then we take [[section]]s of $\array{E &\to& E_{\mathbf{\Pi}}\\ \downarrow && \downarrow \\ \Sigma &\to& \mathbf{\Pi}(\Sigma)}$ to produce a [[span]] of sections

$$
  \left[
    \array{
      \sigma_{in}
      \\
      \downarrow
      \\
      \sigma_{in}
    }
    \,,
    \array{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}}
    }
  \right]_\Sigma
  \leftarrow
  \left[
    \array{
      \sigma
      \\
      \downarrow
      \\
      \mathbf{\Pi}(\sigma)
    }
    \,,
    \array{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}}
    }
  \right]_\Sigma
  \to
  \left[
    \array{
      \sigma_{out}
      \\
      \downarrow
      \\
      \sigma_{out}
    }
    \,,
    \array{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}}
    }
  \right]_\Sigma
  \,.
$$

of smooth $\infty$-groupoids.

Consider an $\infty$-groupoid

$$
  \Psi \to
  \left[
    \array{
      \sigma_{in}
      \\
      \downarrow
      \\
      \sigma_{in}
    }
    \,,
    \array{
      E
      \\
      \downarrow
      \\
      E_{\mathbf{\Pi}}
    }
  \right]_\Sigma
$$

over the left foot. Under [[groupoid cardinality]], if $\Psi$ is tame, this corresponds to a collection of rational numbers over vectors in fibers of $E$. Under "degrupoidification" we may think of this as specifying a section $|\Psi| \in \Gamma(E)$. The above span is supposed to give us the propagation of this state along $\sigma$.

To determine this, consider the special case where $\Psi$ is a "delta-section", $* \mapsto (x,\sigma_{in}, v)$ supported on a single vector $v$ in a single fiber $E_x$ over $x$. Then its pull-push through this span yields the $\infty$-groupoid over $E$, which over $(y,\sigma_{out},w)$ consists of the set of paths $\gamma : x \to y$ such that $w = \exp(\int...)\rho(\gamma)(v)$, i.e. such that $w$ is the vector obtained from applying the action to $v$ along this path.

If everything were suitably finite, we could take cardinalities of the result and obtain the familiar path integral (sum)

$$
  \Psi'(y)
  =
  \int_{x \stackrel{\gamma}{\to} y}
  \exp( S_{kin}(\gamma)) tra_\nabla(\gamma) \Psi(x)
  \,.
$$

[[!redirects exercise in groupoidification -- the path integral]]
[[!redirects An Exercise in Groupoidification]]


exhaustive category

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

# Exhaustive categories

* table of contents
{: toc}

## Idea

A [[category]] is **exhaustive** if it has [[colimits]] of (transfinite) sequences of [[monomorphisms]] which interact well with [[pullbacks]].

## Definition

Given a transfinite sequence
$$ A_0 \to A_1 \to \cdots $$
(continuing through $A_\alpha$ for all $\alpha\lt\kappa$, where $\kappa$ is some limit [[ordinal]]) for which each transition map $A_\alpha \to A_\beta$ is a [[monomorphism]], if $A_\kappa$ is a colimit of this sequence (a [[transfinite composition]]) we call it a **transfinite union**.

A category $C$ is **exhaustive** if one, hence both, of the following two equivalent conditions hold.

1. $C$ has transfinite unions which are stable under pullback and for which the coprojections $A_\alpha \to A_\kappa$ are also monomorphisms.

1. $C$ has transfinite unions, and given any diagram
   $$\array{
     B_0 & \to & B_1 & \to & \cdots & \to & B_\kappa\\
     \downarrow && \downarrow &&&& \downarrow\\
     A_0 & \to & A_1 & \to & \cdots & \to & A_\kappa\\
   }$$
   in which the bottom row is a transfinite union and for each $\alpha\lt\beta\lt\kappa$, the square
   $$ \array{ B_\alpha & \to & B_\beta \\
     \downarrow && \downarrow \\
     A_\alpha & \to & A_\beta } $$
   is a [[pullback]] (hence $B_\alpha \to B_\beta$ is also monic), then the top row is a transfinite union (i.e. a colimit diagram) if and only if for all $\alpha\lt\kappa$ the square
   $$ \array{ B_\alpha & \to & B_\kappa \\
     \downarrow && \downarrow \\
     A_\alpha & \to & A_\kappa } $$
   is a pullback.  (In other words, transfinite unions are [[van Kampen colimits]].)

Note that half of the second condition is simply that the colimits in the first condition are stable under pullback.  One may obtain various weaker notions by restricting the allowable values of $\kappa$.

We will prove the equivalence of these two characterizations in Propositions \ref{CoprojectionsMonic} and \ref{AlternativeCharacterization} below.

## Examples

* The category [[Set]] is easily verified to be exhaustive.

* Since limits and colimits in [[category of presheaves|presheaf categories]] are pointwise, any presheaf category $[D^{op},Set]$ is also exhaustive.

* Since the condition involves only colimits and finite limits, it is preserved by any [[reflective subcategory|left exact localization]].  Thus, every [[Grothendieck topos]] is exhaustive.

## Properties

We first prove the equivalence of the above two definitions.

+-- {: .num_prop #CoprojectionsMonic}
###### Proposition
In a category which satisfies the second definition above, given a transfinite union
$$ A_0 \to A_1 \to \cdots \to A_\kappa$$
(so that $A_\alpha \to A_\beta$ is monic for all $\alpha\lt\beta\lt\kappa$), the coprojections $A_\alpha\to A_\kappa$ are also monic.
=--
+-- {: .proof}
###### Proof
For a fixed $\alpha$, consider the diagram
$$\array{
  A_0 & \to & A_1 & \to & \cdots & \to & A_\alpha & \to & A_\alpha & \to & \cdots & \to & A_\alpha\\
  \downarrow && \downarrow &&&& \downarrow && \downarrow &&&& \downarrow\\
  A_0 & \to & A_1 & \to & \cdots & \to & A_\alpha & \to & A_{\alpha+1} & \to & \cdots & \to & A_\kappa\\
}$$
It is easy to see that each intermediate square is a pullback, and of course the top row is a colimit.  Thus, in particular the square
$$\array{ A_\alpha & \to & A_\alpha \\
  \downarrow && \downarrow\\
  A_\alpha & \to & A_\kappa}$$
is a pullback, i.e. $A_\alpha\to A_\kappa$ is monic.
=--

+-- {: .num_prop #AlternativeCharacterization}
###### Proposition
Suppose $C$ satisfies the first condition above, i.e. it has pullback-stable transfinite unions for which the coprojections are also monic.  Then $C$ also satisfies the second condition above.
=--
+-- {: .proof}
###### Proof
It remains to show that given a diagram
$$\array{
  B_0 & \to & B_1 & \to & \cdots & \to & B_\kappa\\
  \downarrow^{f_0} && \downarrow^{f_1} &&&& \downarrow^{f_\kappa}\\
  A_0 & \to & A_1 & \to & \cdots & \to & A_\kappa\\
}$$
in which both rows are transfinite compositions of monomorphisms and the squares
$$ \array{ B_\alpha & \to & B_\beta \\
  \downarrow^{f_\alpha} && \downarrow^{f_\beta} \\
  A_\alpha & \to & A_\beta } $$
are pullbacks, also the squares
$$ \array{ B_\alpha & \to & B_\kappa \\
  \downarrow^{f_\alpha} && \downarrow^{f_\kappa} \\
  A_\alpha & \to & A_\kappa } $$
are pullbacks.

Fix an $\alpha\lt\kappa$, and suppose given $g\colon X\to B_\kappa$ such that $f_\kappa g$ factors through $A_\alpha$; we want to show that $g$ factors through $B_\alpha$.  Pulling back the $B$-colimit along $g$ and the $A$-colimit along $f_\kappa g$, we obtain another morphism of transfinite unions:
$$\array{
  g^*B_0 & \to & g^*B_1 & \to & \cdots & \to & X\\
  \downarrow && \downarrow &&&& \downarrow^{1_X}\\
  (f_\kappa g)^* A_0 & \to & (f_\kappa g)^* A_1 & \to & \cdots & \to & X\\
}$$
and it is easy to verify, by the pasting law for pullbacks, that each square
$$ \array{ g^* B_\beta & \to & g^* B_\gamma \\
  \downarrow^{f_\beta} && \downarrow^{f_\gamma} \\
  (f_\kappa g)^* A_\beta & \to & (f_\kappa g)^* A_\gamma } $$
is a pullback for $\beta\lt\gamma\lt\kappa$.  However, since $f_\kappa g$ factors through $A_\alpha$, the morphism $(f_\kappa g)^* A_\beta \to (f_\kappa g)^* A_\gamma$ is split epic whenever $\alpha\lt\beta\lt\gamma$, hence (since it is also monic) an isomorphism.  Thus, in this case so is its pullback $g^* B_\beta \to g^* B_\gamma$.  Since the coprojections into a transfinite composite of a sequence that eventually consists of isomorphisms are also eventually isomorphisms, $g^* B_\alpha \to X$ is an isomorphism; hence $g$ factors through $B_\alpha$ as desired.
=--

Exhaustiveness also interacts well with other [[exactness properties]]:

+-- {: .num_prop #FinInfExtensive}
###### Proposition
If an exhaustive category is also finitary [[extensive category|extensive]], then it is infinitary extensive.
=--
+-- {: .proof}
###### Proof
A coproduct of cardinality $\kappa$ can be constructed using transfinite compositions and finite coproducts:
$$ \sum_{\alpha\lt 0} A_\alpha \to \sum_{\alpha\lt 1} A_\alpha \to \sum_{\alpha\lt 2} A_\alpha \to \cdots \to \sum_{\alpha\lt \kappa} A_\alpha $$
Finitary extensivity implies that all the transition maps are monic, so that these transfinite composites exist in an exhaustive category.  Pullback-stability of coproducts follows easily from pullback-stability of transfinite unions and of finite coproducts.  Finally, it is known (see [[extensive category]]) that if finite coproducts exist and are disjoint, then any infinite coproducts which exist are also disjoint.
=--

Colimits in an exhaustive category preserve finite limits.

+-- {: .num_lemma #UnionsPreserveIntersections}
###### Lemma
In an exhaustive category, if $X$ is the $\kappa$-indexed transfinite union of a sequence $(A_\alpha)_{\alpha\lt\kappa}$ and also a sequence $(B_\alpha)_{\alpha\lt\kappa}$, then it is also the transfinite union of the sequence $(A_\alpha\cap B_\alpha)_{\alpha\lt\kappa}$ (the [[intersection]] being taken as [[subobjects]] of $X$).
=--
+-- {: .proof}
###### Proof
Since transfinite unions are stable under pullback, $X$ is the colimit of the $(\kappa\times\kappa)$-indexed diagram $(A_\alpha\cap B_\beta)_{\alpha\lt\kappa, \beta\lt\kappa}$.  But $\kappa$ is a [[filtered category]], so its diagonal functor is [[final functor|final]].
=--

+-- {: .num_prop #UnionsPreservePullbacks}
###### Proposition
In an exhaustive category, given a pullback diagram of $\kappa$-indexed transfinite sequences of monomorphisms:
$$ \left(\array{ A_\alpha & \to & B_\alpha \\
    \downarrow && \downarrow\\
    C_\alpha & \to & D_\alpha }\right)_{\alpha\lt\kappa} $$
the induced square of colimits:
$$ \array{ A_\kappa & \to & B_\kappa \\
    \downarrow && \downarrow\\
    C_\kappa & \to & D_\kappa } $$
is also a pullback.
=--
+-- {: .proof}
###### Proof
Define $P$ so that the following square is a pullback:
$$ \array{ P & \overset{f}{\to} & B_\kappa \\
    ^g\downarrow && \downarrow\\
    C_\kappa & \to & D_\kappa } $$
Since pullbacks preserve transfinite unions, $P$ is the transfinite union of the sequnece $(f^* B_\alpha)_{\alpha\lt\kappa}$ and also the sequence $(g^* C_\alpha)_{\alpha\lt\kappa}$.  Thus, by Lemma \ref{UnionsPreserveIntersections}, $P$ is also the transfinite union of $(f^* B_\alpha\cap g^* C_\alpha)_{\alpha\lt\kappa}$.  However, using the fact that the inclusions $D_\alpha \to D_\kappa$ are monic, it is straightforward to verify that
$$f^* B_\alpha\cap g^* C_\alpha \cong B_\alpha \times_{D_\alpha} C_\alpha \cong A_\alpha.$$
Thus, $P$ is the transfinite union of $(A_\alpha)_{\alpha\lt\kappa}$, hence $P\cong A_\kappa$.
=--

+-- {: .num_cor #UnionsPreserveMonos}
###### Corollary
In an exhaustive category, transfinite unions preserve monomorphisms.  That is, given a transformation $(A_\alpha \to B_\alpha)_{\alpha\lt\kappa}$ of transfinite sequences of monomorphisms which is pointwise monic, also the induced map on colimits $A_\kappa\to B_\kappa$ is monic.
=--

## Related pages

* [[exactness property]]

## References

* The name "exhaustive category" is a coinage; there seems to be no established name in the literature for this exactness property.  See [this MO question](http://mathoverflow.net/questions/93567/exactness-for-sequential-unions-of-monomorphisms).

[[!redirects exhaustive category]]
[[!redirects exhaustive categories]]


existential quantifier

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

# Contents
* table of contents
{: toc}

## Idea

In [[logic]], the _existential quantifier_ (or _particular quantifier_) "$\exists$" is a [[quantifier]] used to express, given a [[predicate]] $\phi$ with a [[free variable]] $x$ of [[type]] $T$, the [[proposition]]

$$
  \exists\, x\colon T, \phi x
  \,,
$$

which is intended to be [[true]] if and only if $\phi a$ is true for at least one object $a$ of type $T$.

Note that it is quite possible that $\exists\, x\colon T, \phi x$ may be [[proof|provable]] (in a given [[context]]) yet $\phi a$ cannot be proved for any [[term]] $a$ of type $T$ that can actually be constructed in that context.  Therefore, we cannot define the quantifier by taking the idea literally and applying it to terms.

## Definition

### In logic

We work in a [[logic]] in which we are concerned with which [[propositions]] entail which propositions (in a given [[context]]); in particular, two propositions which entail each other are considered equivalent.

Let $\Gamma$ be an arbitrary [[context]] and $T$ a [[type]] in $\Gamma$ so that $\Delta \coloneqq \Gamma, x\colon T$ is $\Gamma$ extended by a [[free variable]] $x$ of type $T$.  We assume that we have a [[weakening rule]] that allows us to interpret any proposition $Q$ in $\Gamma$ as a proposition $Q[\hat{x}]$ in $\Delta$.  Fix a proposition $P$ in $\Delta$, which we think of as a [[predicate]] in $\Gamma$ with the free variable $x$.  Then the __existential quantification__ of $P$ is any proposition $\exists\, x\colon T, P$ in $\Gamma$ such that, given any proposition $Q$ in $\Gamma$, we have

*  $\exists\, x\colon T, P \vdash_{\Gamma} Q$ if and only if $P \vdash_{\Gamma, x\colon T} Q[\hat{x}]$.

It is then a theorem that the existential quantification of $P$, if it exists, is unique.  The existence is typically asserted as an axiom in a particular logic, but this may be also be deduced from other principles (as in the topos-theoretic discussion below).

Often one makes the appearance of the free variable in $P$ explicit by thinking of $P$ as a [[propositional function]] and writing $P(x)$ instead; to go along with this one usually conflates $Q$ and $Q[\hat{x}]$.  Then the rule appears as follows:

*  $\exists\, x\colon T, P(x) \vdash_{\Gamma} Q$ if and only if $P(x) \vdash_{\Gamma, x\colon T} Q$.

In [[natural deduction]] the [[inference rules]] for the existential quantifier are given as

$$\frac{\Gamma, x:A \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \exists x:A.P(x) \; \mathrm{prop}}$$

$$\frac{\Gamma, x:A \vdash P(x) \; \mathrm{prop} \quad \Gamma \vdash x:A \quad \Gamma \vdash P(x) \; \mathrm{true}}{\Gamma \vdash \exists x:A.P(x) \; \mathrm{true}}$$

$$\frac{\Gamma, x:A \vdash P(x) \; \mathrm{prop} \quad \Gamma, \exists x:A.P(x) \; \mathrm{true} \vdash Q \; \mathrm{prop} \quad \Gamma, x:A, P(x) \; \mathrm{true} \vdash Q \; \mathrm{true}}{\Gamma, \exists x:A.P(x) \; \mathrm{true} \vdash Q \; \mathrm{true}}$$

### In type theory

In [[type theory]] under the identification of [[propositions as types]], the existential quantifier is given by the [[bracket type]] of the [[dependent sum type]]. Existential quantifier types in general could be regarded as a particular sort of [[higher inductive type]]. In [[Coq]] syntax:

    Inductive existquant (T:Type) (P:T->Type) : Type :=
    | exist : forall (x:T), P x -> existquant T P
    | contr0 : forall (p q : existquant T P) p == q

If the [[dependent type theory]] has a [[type of propositions]] $\mathrm{Prop}$, such as the one derived from a type universe $U$ - $\sum_{A:U} \mathrm{isProp}(A)$, then the existential quantification of the family of types $B(x)$ indexed by $x:A$ is defined as the [[dependent function type]]

$$\exists x:A.B(x) \equiv \prod_{P:\mathrm{Prop}} \left(\prod_{x:A} B(x) \to P\right) \to P$$

By [[weak function extensionality]], the disjunction of two types is a proposition.

\begin{theorem}
The two definitions above are equivalent.
\end{theorem}

\begin{proof}
The [[propositional truncation]] of a type $A$ is equivalent to the following dependent product type

$$[A] \equiv \prod_{P:\mathrm{Prop}} (A \to P) \to P$$

Substituting the dependent sum type $\sum_{x:A} B(x)$ for $A$, we have

$$\left[\sum_{x:A} B(x)\right] \simeq \prod_{P:\mathrm{Prop}} \left(\left(\sum_{x:A} B(x)\right) \to P\right) \to P$$

Given any type $C$, there is an equivalence

$$\left(\left(\sum_{x:A} B(x)\right) \to C\right) \simeq \left(\prod_{x:A} B(x) \to C\right)$$
and if $A \simeq B$, then $(A \to C) \simeq (B \to C)$. In addition, for all type families $x:A \vdash B(x)$, and $x:A \vdash C(x)$, if there is a family of equivalences $e:\prod_{x:A} B(x) \simeq C(x)$, then there is an equivalence $\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A} C(x)\right)$. All this taken together means that there are equivalences

$$\left[\sum_{x:A} B(x)\right] \simeq \left(\prod_{P:\mathrm{Prop}} \left(\left(\sum_{x:A} B(x)\right) \to P\right) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} \left(\prod_{x:A} B(x) \to C\right)\right)$$
\end{proof}

## Properties

### Categorical semantics

The [[categorical semantics]] of existential quantification is given by the [[n-truncated object of an (infinity,1)-category|(-1)-truncation]] of the [[dependent sum]]-construction along the [[projection]] morphism that projects out the [[free variable]] over which the existental quantifier quantifies.

Notice that the [[categorical semantics]] of the [[context extension]] from $Q$ to $Q[\hat{x}]$ corresponds to [[base change]]/[[pullback]] along the [[product]] [[projection]] $\sigma(T) \times A \to A$, where $\sigma(T)$ is the interpretation of the type $T$, and $A$ is the interpretation of $\Gamma$. In other words, if a statement $Q$ read in context $\Gamma$ is interpreted as a [[subobject]] of $A$, then the statement $Q$ read in context $\Delta = \Gamma, x \colon T$ is interpreted by pulling back along the projection, obtaining a subobject of $\sigma(T) \times A$.

(Often we have a class of [[display maps]] and require $f$ to be one of these.) Alternatively, any pullback functor $f^\ast\colon Set/A \to Set/B$ can be construed as pulling back along an object $X = (f \colon B \to A)$, i.e., along the unique map $!\colon X \to 1$ corresponding to an object $X$ in the slice $Set/A$, since we have the identification $Set/B \simeq (Set/A)/X$.

Therefore in terms of the [[internal logic]] of a suitable category $\mathcal{E}$ (with sufficient pullbacks)

* a [[type]] $X$ is given by an [[object]] $X \in \mathcal{E}$,

* the [[predicate]] $\phi$ is a [[(-1)-truncated]] object of the [[over-topos]] $\mathcal{E}/X$;

* a [[truth value]] is a [[(-1)-truncated]] object of $\mathcal{E}$ itself.

Existential quantification is essentially the restriction of the extra [[left adjoint]] of a canonical [[étale geometric morphism]] $X_\ast$,

$$
    (X_!\dashv X^*\dashv X_*):\mathcal{E}/X
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    \mathcal{E}
  \,,
$$

where $X^\ast$ is the functor that takes an object $A$ to the [[product]] projection $\pi \colon X \times A \to X$, where $X_! = \Sigma_X$ is the [[dependent sum]] (i.e., forgetful functor taking $f \colon A \to X$ to $A$) that is left adjoint to $X^\ast$, and where $X_\ast = \Pi_X$ is the [[dependent product]] operation that is right adjoint to $X^\ast$.

The dependent sum operation restricts to [[propositions]] by pre- and postcomposition with the [[truncated|truncation]] [[adjunctions]]

$$
  \tau_{\leq -1} \mathcal{E}
  \stackrel{\overset{\tau_{\leq -1}}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathcal{E}
$$

to give existential quantification over the domain (or context) $X$:

$$
  \array{
    \mathcal{E}/X
    &
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    &
    \mathcal{E}
    \\
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    &&
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    \\
    \tau_{\leq_{-1}}
    \mathcal{E}/X
    &
      \stackrel{\overset{\exists}{\to}}{\stackrel{\overset{}{\leftarrow}}{\underset{\forall}{\to}}}
    &
    \tau_{\leq_{-1}} \mathcal{E}
  }
  \,.
$$

In other words, we obtain the existential quantifier by applying the [[dependent sum]], then $(-1)$-[[truncated|truncating]] the result.  This is equivalent to constructing the [[image]] as a [[subobject]] of the [[codomain]].

Dually, the [[direct image functor]] $\forall$ (dependent product) expresses the [[universal quantifier]].  (In this case, it is somewhat simpler, since the dependent product automatically preserves $(-1)$-truncated objects; no additional truncation step is necessary.)

The same makes verbatim sense also in the [[(∞,1)-logic]] of any [[(∞,1)-topos]].

This interpretation of existential quantification as the left adjoint to context extension is also used in the notion of _[[hyperdoctrine]]_.

### Frobenius reciprocity

The extential quantifier and [[context extension]] is related via [[Frobenius reciprocity]]: if $\psi$ does not have $x$ as a [[free variable]] then

$$
  \exists_x (\phi \wedge \psi)
  \Leftrightarrow
  (\exists_x \phi) \wedge \psi
  \,.
$$

## Examples

Let $\mathcal{E} = $ [[Set]], let $X = \mathbb{N}$ be the set of [[natural number]]s and let $\phi \coloneqq 2\mathbb{N} \hookrightarrow \mathbb{N}$ be the [[subset]] of even natural numbers. This expresses the [[proposition]] $\phi(x) \coloneqq IsEven(x)$.

Then the [[dependent sum]]

$$
  (\phi \in Set/{\mathbb{N}})
  \mapsto
  (\sum_{x\colon X} \phi(x) \in Set)
$$

is simply the set $2 \mathbb{N} \in Set$ of even natural numbers. Since this is [[inhabited]], its [[truncated|(-1)-truncation]] is therefore the singleton set, hence the [[truth value]] _[[true]]_. Indeed, there exists an even natural number!

Notice that before the $(-1)$-truncation the operation remembers not just _that_ there is an even number, but it remembers "all proofs that there is one", namely every example of an even number.

## Related concepts

* [[possibility]], [[example]]

* [[universal quantifier]]

* [[elimination of quantifiers]]

* [[generalized quantifier]]

* [[being]]

[[!include logic symbols -- table]]

## References

The observation that [[substitution]] forms an [[adjoint pair]]/[[adjoint triple]] with the existantial/universal quantifiers is due to

* [[William Lawvere]], _[[Adjointness in Foundations]]_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

* [[William Lawvere]] _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf))

and further developed to the notion of [[hyperdoctrines]] in

* [[William Lawvere]], _Equality in hyperdoctrines and
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

The definition of the existential quantifier in dependent type theory as the propositional truncation of the dependent sum type is found in:

* {#UFP13} [[Univalent Foundations Project]], §3.7 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

And the existential quantifier defined from the type of propositions and dependent product types can be found in:

* Madeleine Birchfield, *Constructing coproduct types and boolean types from universes*, MathOverflow ([web](https://mathoverflow.net/questions/457904/constructing-coproduct-types-and-boolean-types-from-universes))

[[!redirects existential quantifier]]
[[!redirects existential quantifiers]]
[[!redirects existential quantification]]
[[!redirects existential quantifications]]
[[!redirects particular quantifier]]
[[!redirects particular quantifiers]]
[[!redirects particular quantification]]
[[!redirects particular quantifications]]

[[!redirects existence]]


existentially closed model

[[!redirects existentially closed theory]]
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### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Existential closedness is a [[property]] of [[first-order structures]] (in the same way that [[model completeness]] is a [[property]] of [[first-order theory|first-order theories]]) meant to generalize the properties of the theory [[ACF]] of [[algebraically closed fields]].

As discussed in section 8.1 of ([Hodges93](#Hodges1993)):

> Take a first-order language $\mathcal{L}$ without relation symbols and a class $\mathbf{K}$ of $\mathcal{L}$-structures. For example, $\mathcal{L}$ might be the language of rings and $\mathbf{K}$ the class of fields, or $\mathcal{L}$ might be the language of groups and $\mathbf{K}$ the class of groups.

> We say that a structure $A$ in $\mathbf{K}$ is _existentially closed_ in $\mathbf{K}$ if whenever $E$ is a finite set of equations and inequations with parameters from $A$ and $E$ has a simultaneous solution in some extension $B$ of $A$ with $B$ in $\mathbf{K}$, then $E$ has a solution already in $A$.

(c.f. [[algebraically closed field]]).

## Definition

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

1. An *existential formula* is a formula of the form $\exists y \varphi(x,y)$ where $\varphi(x,y)$ is quantifier-free.

2. A *universal formula* is a formula of the form $\forall y \varphi(x,y)$ where $\varphi(x,y)$ is quantifier-free.

=--

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

We say that a model $M$ of a first-order theory $T$ is *existentially closed* if for every $N$ also modelling $T$, every embedding $M \hookrightarrow N$, every tuple $a \in M$, and every existential formula $\exists y \varphi(x,y)$, then whenever $N \models \exists y \varphi(a, y)$, we also have that $M \models \exists y \varphi(a,y)$.

=--

("The existence of witnesses in a superstructure to existential statements about points in the existentially closed model is reflected to the existentially closed model.")

In the terminology of the passage from ([Hodges93](#Hodges1993)), we say that $M \models T$ is an *existentially closed model* of $T$ if it is existentially closed in the class $\mathbf{K} \overset{\operatorname{df}}{=} \mathbf{Mod}(T)$.

We say that a theory $T$ is an "existentially closed theory" (i.e., [[model complete]]) if all of its models are existentially closed.

## Examples

The theory [[ACF]] of [[algebraically closed field|algebraically closed fields]] is existentially closed. An existential formula $\exists y \varphi(x,y)$ in $\mathsf{ACF}$ says "for a choice of coefficients $x$, there exists a point $y$ in affine space (of fixed dimension) which is in these finitely many affine varieties depending on $x$ but not these finitely many affine varieties depending on $x$".

Checking that $\mathsf{ACF}$ is existentially closed therefore means checking that if $E \hookrightarrow K$ is an embedding of algebraically closed fields, and $x$ is a choice of coefficients from $E$, then if such a point $y$ exists in affine $K$-space, then we can find a similar $y$ in affine $E$-space.

However, we can do so very cheaply after we know that $\mathsf{ACF}$ [[quantifier elimination|eliminates quantifiers]] (and therefore so does the theory of algebraically closed fields containing $E$): $\exists y \varphi(a,y)$ is equivalent to some quantifier-free sentence involving $a$, and quantifier-free sentences are always transferred by embeddings of structures, so we are done.

## Remarks

* Theories which [[quantifier elimination|eliminate quantifiers]] are model complete: all of their models are existentially closed (again, because quantifier-free sentences are always transferred by embeddings of structures).

## Related concepts

* [[model complete theory]]

* [[quantifier elimination]]

* [[model companion]]

## References

* {#Hodges1993}Wilfrid Hodges, _Model theory_, Cambridge Univ. Press 1993

[[!redirects existentially closed]]

[[!redirects existentially-closed]]

[[!redirects existentially closed theory]]

[[!redirects existentially-closed model]]

[[!redirects existential closure]]

[[!redirects existentially-closed theory]]

[[!redirects existentially closed models]]

[[!redirects existentially closed]]

[[!redirects existential closedness]]


exotic 7-sphere

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

#Contents#
* table of contents
{:toc}

## Idea

A [[topological manifold|topological]] [[7-sphere]] equipped with an [[exotic smooth structure]] is called an _exotic 7-sphere_.

## Milnor's construction
 {#Construction}

[Milnor (1956)](#Milnor1956) gave the first examples of exotic smooth structures on the [[7-sphere]], finding at least seven.

The [[exotic 7-spheres]] constructed in [Milnor 1956](#Milnor1956) are all examples of [[fibre bundles]] over the [[4-sphere]] $S^4$ with [[fibre]] the [[3-sphere]] $S^3$, with [[structure group]] the [[special orthogonal group]] [[SO(4)]] (see also at _[[8-manifold]]_ the section _[With exotic boundary 7-spheres](8-manifold#ExoticBoundary7Spheres)_):

By the classification of bundles on spheres via the [[clutching construction]], these correspond to [[homotopy classes]] of maps $S^3 \to SO(4)$, i.e. elements of $\pi_3(SO(4))$. From the table at [orthogonal group -- Homotopy groups](orthogonal+group#HomotopyGroups), this latter group is $\mathbb{Z}\oplus\mathbb{Z}$. Thus any such bundle can be described, up to [[isomorphism]], by a [[pair]] of [[integers]] $(n,m)$. When $n+m=1$, then one can show there is a [[Morse function]] with exactly two [[critical points]] on the total space of the bundle, and hence this 7-manifold is [[homeomorphic]] to a sphere.

The  [[fractional first Pontryagin class]] $\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}$ of the bundle is given by $n-m$. Milnor constructs, using [[cobordism theory]] and [[Hirzebruch's signature theorem]] for [[8-manifolds]], a [[modulo]]-7 [[diffeomorphism]] [[invariant]] of the manifold, so that it is the standard 7-sphere precisely when $\frac{p_1}{2}^2 -1 = 0 (mod\,7)$.

By using the [[connected sum]] operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an [[abelian group]].
For the [[7-sphere]], it is the [[cyclic group]] $\mathbb{Z}/{28}$ and [Brieskorn (1966)](#Brieskorn66) found the generator $\Sigma$ so that $\underbrace{\Sigma\#\cdots\#\Sigma}_28$ is the standard sphere.

Review includes ([Kreck 10, chapter 19](#Kreck10), [McEnroe
15](#McEnroe15), [Joachim-Wraith](#JoachimWraith)).

## Examples

* [[Gromoll-Meyer sphere]]

## Properties

### As near-horizon geometries of black M2-branes
 {#AsNearHorizonGeometriesOfBlackM2Branes}

From the point of view of [[M-theory on 8-manifolds]], these [[8-manifolds]] $X$ with (exotic) [[7-sphere]] [[boundaries]] in [Milnor's construction](#Construction) correspond to [[near horizon limits]] of [[black brane|black]] [[M2 brane]] spacetimes $\mathbb{R}^{2,1} \times X$, where the [[M2-branes]] themselves would be sitting at the center of the [[7-spheres]] (if that were included in the spacetime, see also [[Dirac charge quantization]]).

([Morrison-Plesser 99, section 3.2](#MorrisonPlesser99), [[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS 19, 3.8]]))

## References

### General

* {#Milnor1956} [[John Milnor]], _On manifolds homeomorphic to the 7-sphere_, Annals of Mathematics 64 (2): 399–405 (1956) ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/exotic.pdf), [doi:10.1142/9789812836878_0001](https://doi.org/10.1142/9789812836878_0001))

* {#Brieskorn66} [[Egbert Brieskorn]], *Beispiele zur Differentialtopologie von Singularitäten*, Inventiones mathematicae **2** (1966) 1–14 $[$[doi:10.1007/BF01403388](https://doi.org/10.1007/BF01403388)$]$

* {#Kreck10} [[Matthias Kreck]], chapter 19 of _Exotic 7-spheres_ of  _Differential Algebraic Topology -- From Stratifolds to Exotic Spheres_, AMS 2010

* {#McEnroe15} Rachel McEnroe, _Milnor' construction of exotic 7-spheres_, 2015 ([pdf](http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf))

* {#JoachimWraith} [[Michael Joachim]], D. J. Wraith, _Exotic spheres and curvature_ ([pdf](https://ivv5hpp.uni-muenster.de/u/joachim/exo.pdf))

* [[Niles Johnson]], _Visualizing 7-manifolds_, 2012 ([nilesjohnson.net/seven-manifolds.html](https://nilesjohnson.net/seven-manifolds.html))

* {#CrowleyEscher03} [[Diarmuid Crowley]], [[Christine Escher]], _A classification of $S^3$-bundles over $S^4$_, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

See also

* Wikipedia, _[Exotic sphere](https://en.wikipedia.org/wiki/Exotic_sphere)_

### In M-theory

* {#MorrisonPlesser99} [[David Morrison]], [[M. Ronen Plesser]], section 3.2 of _Non-Spherical Horizons, I_, Adv. Theor. Math. Phys. 3:1-81, 1999 ([arXiv:hep-th/9810201](https://arxiv.org/abs/hep-th/9810201))

[[!redirects exotic 7-spheres]]

[[!redirects Brieskorn sphere]]
[[!redirects Brieskorn spheres]]

[[!redirects Milnor sphere]]
[[!redirects Milnor spheres]]


exotic brane

[[!redirects Exotic branes]]
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###Context###
#### String theory
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[[!include string theory - contents]]
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#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]], by "exotic branes" [[deBoer & Shigemori (2012)](#deBoerShigemori12)] one means first of all $p$-[[branes]] of small [[codimension]] $D-1-p \leq 2$ ("non-standard branes" [Bergshoeff & Riccioni (2012](#BergshoeffRiccioni12))), such as

* [[D7-branes]] (codimension 2 [[defect branes]]),

* [[D8-branes]] and maybe [[M9-branes]] (codimension 1)

* [[D9-branes]] (codimension 0).

More generally, exotic branes have been argued to be [[U-duality]]-twisted objects in [[non-geometric backgrounds]].

According to [deBoer & Shigemori 2012, p. 12](#deBoerShigemori12) all these may be identified as [[codimension]]-2 [[defect branes]] with [[U-duality]]-[[group]] [[monodromy]].

\begin{imagefromfile}
        "file_name": "exotic_branes.jpg",
        "web": "nlab",
        "width": 800,
        "height": 475,
        "unit": "px",
        "alt": "exotic branes"
\end{imagefromfile}
>Figure 1 from ([BMO19](#BMO19)). <>

[^1]: Here $\alpha$ denotes the [[exponent]] with which the brane's [[tension]] $T$ scales with the [[string coupling constant]] $g_s$ as $T \propto (g_s)^\alpha$, according to [BMO 2019](#BMO19), [p. 2](https://arxiv.org/pdf/1903.10247.pdf#page=2).

## Related concepts

* [[non-geometry]], [[T-fold]]

* [[double field theory]]

* [[defect brane]]

## References

Branes of [[codimension]] $\leq 2$ are called *non-standard branes* in:

* {#BergshoeffRiccioni12} [[Eric A. Bergshoeff]], [[Fabio Riccioni]], *Solitonic branes and wrapping rules*, Phys. Part. Nuclei **43** (2012) 557–561 [[doi:10.1134/S106377961205005X](https://doi.org/10.1134/S106377961205005X)]

  > (see also at *[[solitonic brane]]*)

The terminology "exotic branes" is due to:

* {#deBoerShigemori12} [[Jan de Boer]], [[Masaki Shigemori]], _Exotic Branes in String Theory_, Physics Reports **532** (2013) 65-118 ([arXiv:1209.6056](https://arxiv.org/abs/1209.6056), [doi:j.physrep.2013.07.003](https://doi.org/10.1016/j.physrep.2013.07.003))

  > (discussion as [[defect branes]] with [[U-duality]]-[[group]] [[monodromy]])

* [[David Berman]], Felix J. Rudolph, _Branes are Waves and Monopoles_ [[arXiv:1409.6314](https://arxiv.org/abs/1409.6314)]

* [[Jose J. Fernandez-Melgarejo]], [[Tetsuji Kimura]], [[Yuho Sakatani]], *Weaving the Exotic Web*, JHEP **2018** 72 (2018) [doi:10.1007/JHEP09(2018)072]

* {#BermanMusaevOtsuki18} [[David S. Berman]], [[Edvard T. Musaev]], [[Ray Otsuki]], *Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond*,  J. High Energ. Phys. **2018** 53 (2018) [[arXiv:1806.00430v4](https://arxiv.org/abs/1806.00430), doi:10.1007/JHEP12(2018)053]

* {#BMO19} [[David Berman]], [[Edvard Musaev]], [[Ray Otsuki]], _Exotic Branes in M-Theory_, contribution to: *[Dualities and Generalized Geometries, Corfu Summer Institute 2018](http://www.physics.ntua.gr/corfu2018/dg.html)* ([arXiv:1903.10247](https://arxiv.org/abs/1903.10247))

[[!redirects exotic branes]]
[[!redirects Exotic branes]]
[[!redirects Exotic brane]]
[[!redirects non-geometric branes]]
[[!redirects non-geometric brane]]


exotic meson

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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
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#Contents#
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## Idea

An _exotic meson_ is a [[hadron]] of [[baryon number]] [[zero]] which however, in contrast to actual [[mesons]], has [[Wigner classification]] given by an [[isospin]] and [[Lorentz group|Lorentz]]-[[spin group]]   [[linear representation|representation]] which is _not_ a [[direct summand]] of a [[tensor product of representations]] of that of a [[quark]]-[[pair]].

Hence the existence of exotic mesons is not readily explained by the [[QCD]]-based [[constituent quark model]] of [[quantum hadrodynamics]],  where [[mesons]] are [[bound states]] of two quarks (and [[baryons]] of three).

Some exotic mesons may possibly be interpreted as [[tetraquarks]]

## Examples

* [[XYZ mesons]]

## References

Review:

* Th. Gutsche, _Exotic Mesons_ ([[GutscheExoticMesons.pdf:file]])

* Luciano Maiani, Alessandro Pilloni, *GGI Lectures on Exotic Hadrons* [[arXiv:2207.05141](https://arxiv.org/abs/2207.05141)

See also

* Wikipedia, _[Exotic meson](https://en.m.wikipedia.org/wiki/Exotic_meson)

[[!redirects exotic mesons]]


exotic R^4

## Idea

There exists [[exotic smooth structure]] on the [[Euclidean space]] $\mathbb{R}^4$ of [[dimension]] 4.

(For all other dimensions there is a unique smooth structure, the standard one, see at _[[smooth structure]]_).

## References

* Wikipedia, _[Exotic R4](https://en.wikipedia.org/wiki/Exotic_R4)_

[[!redirects exotic Euclidean space]]


exotic smooth structure

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#Contents#
* table of contents
{:toc}

## Idea

An _exotic smooth structure_ is, roughly speaking, a [[smooth structure]] on a [[topological manifold]] $X$ which makes the resulting [[smooth manifold]] be [[diffeomorphism|non-diffeomorphic]] to the smooth manifold given by some evident 'standard' smooth structure on $X$.

Mostly the term is used for smooth structures on [[Euclidean space]] $\mathbb{R}^n$ and on the [[n-spheres]] $S^n$, for $n \in \mathbb{N}$. The standard smooth structure on $\mathbb{R}^n$ is exhibited by the identity [[atlas]], and the standard smooth structure on $S^n$ is that given by the atlas of the two [[hemispheres]] as given by [[stereographic projection]].

For special values of $n$ there may exist smooth structure not equivalent to these. They are the _exotic smooth structures_.

A classification of [[smooth manifold|smooth]], [[piecewise-linear manifold|PL]] and [[topological manifold|topological]] structures on manifolds in dimension 5  and higher, in terms of various groups from [[algebraic topology]] (many not known) was established by [Kirby and Siebenmann (1977)](#KirbSieb) using [[obstruction theory]].

## Existence and Examples

### No exotic smooth structure in dimensions $\leq 3$

[Rado (1925)](#Rad) proved that in dimension 2 there are no exotic differentiable structures (or the uniqueness of the standard structure). The classification of 1-dimensional manifolds and the uniqueness of the smooth structure can be found in the Appendix of [Milnor (1965b)](#Milnor1965b).

[Moise (1952)](#Moise) proved that in dimension 3 there are no exotic differentiable structures, or to put in another way, 3-dimensional differentiable manifolds which are [[homeomorphism|homeomorphic]] are [[diffeomorphism|diffeomorphic]]. In this way the 3-sphere $S^3$ inherits a unique differentiable structure, no matter which $\mathbb{R}^4$ it is considered to be embedded in.

### No exotic Euclidean space away from dimension 4

There exists a unique smooth structure on the [[Euclidean space]] $\mathbb{R}^n$ for $n\neq 4$ ([Stallings 1962](#Stallings62)).

### Exotic Euclidean 4-space
 {#ExoticEuclideal4Space}

There exists uncountably many exotic smooth structures on the [[Euclidean space]] $\mathbb{R}^4$ of dimension 4 ([Gompf 1985](#Gompf), [Freedman/Taylor 1986](#FreedTay), [Taubes 1987](#Taubes)). See also at _[[exotic R^4]]_.

There is a unique maximal exotic $\mathbb{R}^4$ into which all other 'versions' of $\mathbb{R}^4$ smoothly embed as open subsets (Freedman/Taylor 1986, [DeMichelis/Freedman 1992](#DeMFreed)).

There are two classes of exotic $\mathbb{R}^4$'s: large and small. A large exotic $\mathbb{R}^4$ cannot be embedded in the 4-sphere $S^4$ (Gompf 1985, Taubes 1987) whereas a small exotic $\mathbb{R}^4$ admits such an embedding (DeMichelis/Freedman 1992):

* A _large exotic $\mathbb{R}^4$_ is constructed by using the failure to smoothly split a smooth 4-manifold (the [[K3 surface]] for instance) as a [[connected sum]] of some factors (where a topological splitting exits).

* The _small exotic $\mathbb{R}^4$_ (or _ribbon $\mathbb{R}^4$_) is constructed by using the failure of the smooth [[h-cobordism theorem]] in dimension 4 ([Donaldson 1987](#Donaldson1), [1990](#Donaldson2)). [Bizaca and Gompf (1996)](#BizGompf) are able to present an infinite handle body of a small exotic $\mathbb{R}^4$ which serve as a coordinate representation.

### Exotic $S^2 \times S^2$

There exists an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold $S^2 \times S^2$ ([Akhmedov-Park 10](#AkhmedovPark10)).

### Exotic 4-spheres?
 {#Exotic4Spheres}

It is open whether the [[4-sphere]] admits an exotic smooth structure. See ([Freedman-Gompf-Morrison-Walker 09](#FreedmanGompfMorrisonWalker09) for review).

### Exotic 7-sphere
 {#Exotic7Sphere}

[Milnor (1956)](#Milnor1956) gave the first examples of exotic smooth structures on the [[7-sphere]], finding at least seven.

The [[exotic 7-spheres]] constructed in [Milnor 1956](#Milnor1956) are all examples of [[fibre bundles]] over the [[4-sphere]] $S^4$ with [[fibre]] the [[3-sphere]] $S^3$, with [[structure group]] the [[special orthogonal group]] [[SO(4)]] (see also at _[[8-manifold]]_ the section _[With exotic boundary 7-spheres](8-manifold#ExoticBoundary7Spheres)_):

By the classification of bundles on spheres via the [[clutching construction]], these correspond to [[homotopy classes]] of maps $S^3 \to SO(4)$, i.e. elements of $\pi_3(SO(4))$. From the table at [orthogonal group -- Homotopy groups](orthogonal+group#HomotopyGroups), this latter group is $\mathbb{Z}\oplus\mathbb{Z}$. Thus any such bundle can be described, up to [[isomorphism]], by a [[pair]] of [[integers]] $(n,m)$. When $n+m=1$, then one can show there is a [[Morse function]] with exactly two [[critical points]] on the total space of the bundle, and hence this 7-manifold is [[homeomorphic]] to a sphere.

The  [[fractional first Pontryagin class]] $\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}$ of the bundle is given by $n-m$. Milnor constructs, using [[cobordism]] theory and [[Hirzebruch's signature theorem]] for 8-manifolds, a mod-7 diffeomorphism invariant of the manifold, so that it is standard 7-sphere precisely when $\frac{p_1}{2}^2 -1 = 0 (mod 7)$.

By using the [[connected sum]] operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an [[abelian group]].
For the [[7-sphere]], it is the [[cyclic group]] $Z/{28}$ and Brieskorn (1966) found the generator $\Sigma$ so that $\underbrace{\Sigma\#\cdots\#\Sigma}_28$ is the standard sphere.

For review see for instance ([Kreck 10, chapter 19](#Kreck10), [McEnroe 15](#McEnroe15)). For more see at _[[exotic 7-sphere]]_.

From the point of view of [[M-theory on 8-manifolds]], these [[8-manifolds]] $X$ with (exotic) [[7-sphere]] [[boundaries]] correspond to [[near horizon limits]] of [[black brane|black]] [[M2 brane]] spacetimes $\mathbb{R}^{2,1} \times X$, where the [[M2-branes]] themselves would be sitting at the center of the [[7-spheres]] (if that were included in the spacetime, see also [[Dirac charge quantization]]).

([Morrison-Plesser 99, section 3.2](M-theory+on+8-manifolds#MorrisonPlesser99))

\linebreak

### Exotic 8-sphere
 {#Exotic8Sphere}

The abelian group of non-diffeomorphic structures with connected sum on the [[8-sphere]] is the cyclic group $Z/2$. The unique exotic 8-sphere corresponds to the nontrivial element of the [[cokernel]] of the [[J-homomorphism]] and is the first instance of an exotic sphere that does not bound a [[parallelizable manifold]] ([Amabel 17, Sec. 11](#Amabel17)). It admits a metric of positive [[Ricci curvature]].

### Exotic $n$-spheres for $n \geq 5$
 {#ExoticNSpheresForNGreaterThanFour}

Via the celebrated [[h cobordism theorem]] of Smale ([Smale 1962](#Smale), [Milnor 1965](#Milnor1965a)) one gets a relation between the number of smooth structures on the $n$-sphere $S^n$ (for $n \geq 5$) and the number of [[isotopy]] classes $\pi_0 (Diff(S^{n-1}))$ of the [[equator]] $S^{n-1}$.

Then [Kervaire and Milnor (1963)](#KervMil) proved that for each $n \geq 5$ there are only finitely many exotic smooth structures on the [[n-sphere]] $S^n$ (possibly none).

By using the [[connected sum]] operation, the set of smooth, non-diffeomorphic structures on the $n$-sphere has the structure of an [[abelian group]].

The only odd-dimensional spheres with _no_ exotic smooth structure are the [[circle]] $S^1$, the [[3-sphere]] $S^3$, as well as $S^5$ and $S^{61}$ ([Wang-Xu 16, corollary 1.13](#WangXu16))

In the range $5 \leq n  \leq 61$ the only $n$-spheres with _no_ exotic smooth structures are $S^5$, $S^6$, $S^{12}$, $S^{56}$ and $S^{61}$ ([Wang-Xu 16, corollary 1.15](#WangXu16)).

It is conjectured that this exhausts in fact all examples of $n$-spheres without exotic smooth structure for $n \geq 5$ ([Wang-Xu 16, conjecture 1.17](#WangXu16)).

## Related concepts

* [[differential topology]]

## References

See also

* Wikipedia, _[Exotic sphere](http://en.wikipedia.org/wiki/Exotic_sphere)_

### For the mathematical theory

The first construction of exotic smooth structures was on the 7-[[sphere]] in

* {#Milnor1956} [[John Milnor]], _On manifolds homeomorphic to the 7-sphere_, Annals of Mathematics 64 (2): 399–405 (1956) ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/exotic.pdf), [doi:10.1142/9789812836878_0001](https://doi.org/10.1142/9789812836878_0001))

(...)

* {#Smale} [[Stephen Smale]],  _On the structure of manifolds_, Amer. J. of Math. 84 : 387-399 (1962)

* {#Milnor1965a} [[John Milnor]] (1965), _Lectures on the h-cobordism theorem_ (Princeton Univ. Press, Princeton)

* {#KervMil} [[Michel Kervaire]], ; [[John Milnor]],  (1963) "Groups of homotopy spheres: I", Ann. Math. 77, pp. 504 - 537.

* {#KirbSieb} Kirby, R.; Siebenmann, L. (1977) _Foundational essays on topological manifolds, smoothings, and triangulations_, Ann. Math. Studies (Princeton University Press, Princeton).

* {#Stallings62} John R. Stallings, _The piecewise-linear structure of Euclidean space_, Proceedings of the Cambridge Philosophical Society 58: 481–488 (1962) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/stallings2.pdf))

* {#FreedTay} Freedman, Michael H.; Taylor, Laurence (1986) "A universal smoothing of four-space", J. Diff. Geom. 24, pp. 69-78

* {#DeMFreed} De Michelis, Stefano; Freedman, Michael H. (1992) "Uncountably many exotic $\mathbb{R}^4$'s in standard 4-space", J. Diff. Geom. 35, pp. 219-254.

* {#Donaldson1} [[Simon Donaldson]] (1987) "Irrationality and the h-cobordism conjecture", J. Diff. Geom. 26, pp. 141-168.

* {#Donaldson2} [[Simon Donaldson]],  (1990) "Polynomial invariants for smooth four manifolds", Topology 29, pp. 257-315.

* {#Gompf} Gompf, Robert (1985) "An infinite set of exotic $\mathbb{R}^4$'s", J. Diff. Geom. 21, pp. 283-300.

* {#Taubes} Taubes, Clifford H. (1987) "Gauge theory on asymptotically periodic 4-manifolds", J. Diff. Geom. 25, pp. 363-430

* {#BizGompf} Bizaca, Z.; Gompf, Robert (1996) "Elliptic surfaces and some simple exotic $\mathbb{R}^4$'s", J. Diff. Geom. 43, pp. 458-504.

* {#Rad} Rado, T. (1925) "Über den Begriff der Riemannschen Fläche" , Acta Litt. Scient. Univ. Szegd 2, pp. 101-121

* {#Milnor1965b} [[John Milnor]], (1965b) _Topology from the Differentiable Viewpoint_ (University Press of Virginia)

* {#WangXu16} Guozhen Wang, Zhouli Xu, _The triviality of the 61-stem in the stable homotopy groups of spheres_ ([arXiv:1601.02184](https://arxiv.org
/abs/1601.02184))

* Llohann D. Sperança, _Pulling back the Gromoll-Meyer construction and models of exotic spheres_, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 ([arXiv:1010.6039](https://arxiv.org/abs/1010.6039))

* Llohann D. Sperança, _Explicit Constructions over the Exotic 8-sphere_ ([pdf](https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf), [[SperancaExoticSpheres.pdf:file]])

* {#Amabel17} , _Exotic spheres_ ([pdf](https://math.mit.edu/~araminta/Exotic.pdf))

* {#DRL10} C. Duran, A. Rigas, Llohann D. Sperança, _Bootstrapping Ad-equivariant maps, diffeomorphisms and involutions_, Matematica Contemporanea, 35:27–39, 2010 ([pdf](http://www.ime.unicamp.br/~rigas/bootstrapping))

On the open issue of exotic [[4-spheres]]:

* {#FreedmanGompfMorrisonWalker09} [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], _Man and machine thinking about the smooth 4-dimensional Poincaré conjecture_, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 ([arXiv:0906.5177](http://arxiv.org/abs/0906.5177))

On exotic $S^2 \times S^2$

* {#AkhmedovPark10} Anar Akhmedov, B. Doug Park, _Exotic smooth structures on $S^2 \times S^2$_ ([arXiv:1005.3346](https://arxiv.org/abs/1005.3346))

Review:

* {#Kreck10} [[Matthias Kreck]], chapter 19 "Exotic 7-spheres" of  _Differential Algebraic Topology -- From Stratifolds to Exotic Spheres_, AMS 2010

* [[Rustam Sadykov]], Sections 7,8 of: _Elements of Surgery Theory_, 2013 ([pdf](https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf))

* {#McEnroe15} Rachel McEnroe, _Milnor' construction of exotic 7-spheres_, 2015 ([pdf](http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf))

See also

* Wikipedia ([spheres](http://en.wikipedia.org/wiki/Exotic_sphere#References), [$R^4$](http://en.wikipedia.org/wiki/Exotic_R4#References))

### For applications to physics
 {#ReferencesForApplicationsToPhysics}

The relevance of exotic smooth structure to [[physics]] is tantalizing but remains by and large unclear. Some of the following references probably ought to be handled with care.

#### General relativity

The argument that [[exotic spheres]] are to be regarded as [[gravitational instantons]]:

* {#Witten85} [[Edward Witten]], p. 12 of: _Global gravitational anomalies_, Comm. Math. Phys. Volume 100, Number 2 (1985), 197--229.  ([EUCLID](http://projecteuclid.org/euclid.cmp/1103943444))

* Randy A. Baadhio, _On the global gravitational instanton and soliton that are homotopy spheres_, Journal of Mathematical Physics 32, 2869 (1991) ([doi:10.1063/1.529078](https://doi.org/10.1063/1.529078))

Further discussion of exotic $4$-manifolds from the [[general relativity]] point of view is in

* [[Carl Brans]], Duane Randall, _Exotic differentiable structures and general relativity_ Gen. Rel. Grav., 25 (1993) 205--220

* [[Carl Brans]], _Exotic smoothness and physics_ J. Math. Phys. 35, (1994), 5494--5506.

The following paper contained a first proof to localize exotic smoothness in an exotic $\mathbb{R}^4$:

* [[Carl Brans]], _Localized exotic smoothness_ Class. Quant. Grav., 11, (1994), 1785--1792.

A more philosophical discussion can be found in:

* [[Carl Brans]], _Absolute spacetime: the twentieth century ether_ Gen. Rel. Grav. 31, (1999),  597--609

#### Generation of source terms (fields)

Brans conjectured in the papers above, that exotic smoothness should be a source of an additional [[gravity|gravitational field]] (Brans conjecture). This conjecture was confirmed for compact $4$-manifolds (using implicitly a mapping of basic classes):

Using the invariant of L. Taylor [arXiv](http://de.arxiv.org/abs/math/9807143), Sladkowski confirmed the conjecture for the exotic $\mathbb{R}^4$ in:

* Jan Sładkowski _Gravity on exotic R4 with few symmetries_ Int.J. Mod. Phys. D, 10, (2001) 311--313

#### Quantum (field) theory

The first real connection between exotic smoothness and [[quantum field theory]] is Witten's TQFT:

* [[Edward Witten]], _Topological quantum field theory_ Comm. Math. Phys., 117, (1988), 353--386.

and the whole work of Seiberg and Witten leading to the celebrated invariants.

The relation to particle physics by using the algebra of smooth functions can be found in

* Jan Sładkowski, _Exotic smoothness, noncommutative geometry and particle physics_ Int. J. Theor. Phys., 35, (1996), 2075–2083

* Jan Sładkowski, _Exotic smoothness and particle physics_ Acta Phys. Polon., B 27, (1996), 1649–1652

* Jan Sładkowski, _Exotic smoothness, fundamental interactions and noncommutative geometry_ [arXiv] (http://arxiv.org/abs/hep-th/9610093)

The relation between TQFT and differential-topological invariants of smooth manifolds was clarified in:

* [[Hendryk Pfeiffer]] _Quantum general relativity and the classification of smooth manifolds_ [arXiv](http://arxiv.org/abs/gr-qc/0404088)

* [[Hendryk Pfeiffer]] _Diffeomorphisms from finite triangulations and absence of 'local' degrees of freedom_ Phys.Lett. B, 591, (2004), 197-201

#### String theory
 {#ReferencesInStringTheory}

An argument for interpreting exotic smooth spheres as [[gravitational instantons]] and to cancel the gravitational [[anomalies]] of [[string theory]] is in ([Witten 85](#Witten85)).

The influence of exotic smoothness for [[Kaluza-Klein mechanism|Kaluza-Klein models]] was discussed here:

* [[Matthias Kreck]], [[Stefan Stolz]], _A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with $\mathfrak{su}(3) \times \mathfrak{su}(2) \times \mathfrak{u}(1)$-symmetry_ Ann. Math. 127, (1988), 373--388.

A discussion of topological effects (also of string theory) in relation to exotic smoothness is in

* Ryan Rohm, _Topological Defects and Differential Structures_ Annals Of Physics, 189, (1989), 223--239.

#### Cosmology

An overview can be also found in

* Jan Sładkowski, _Exotic smoothness and astrophysics_ Act. Phys. Polon. B, 40, (2009), 3157--3163

#### Quantum gravity

A first calculation of the state sum in [[quantum gravity]] by inclusion of exotic smoothness

* Kristin Schleich, Donald Witt, _Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions_ Class.Quant.Grav., 16, (1999), 2447--2469

A semi-classical approach to the functional integral is discussed here:

* Christofer Duston, _Exotic smoothness in 4 dimensions and semiclassical Euclidean quantum gravity_ [arxiv](http://arxiv.org/abs/0911.4068)

The inclusion of singularities for asymptotically flat spacetimes is discussed here (with an example of a singularity coming from exotic smoothness):

* Kristin Schleich, Donald Witt, _Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes_, [arxiv](http://arxiv.org/abs/1006.2890)

[[!redirects exotic smooth structure]]
[[!redirects exotic smooth structures]]

[[!redirects exotic sphere]]
[[!redirects exotic spheres]]


expansion

This entry is about expansion in [[shape theory]] rather than in cosmology (or elsewhere, like in theory of fluids).

Let $\mathcal{T}$ be a category, $\mathcal{P}\subset\mathcal{T}$ a subcategory and $X$ an object in $\mathcal{T}$. The canonical embedding $\mathcal{T}\to pro\mathcal{T}$ into the [[pro-category]] may be considered as an identification. A $\mathcal{T}$-expansion (or simply an expansion) of $X$ is an [[initial object]] in the [[coslice category]] $X/pro\mathcal{T}$. A $\mathcal{T}$-expansion is a $\mathcal{P}$-expansion if it is a morphism in $\mathcal{P}$.