axiom of replacement

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# Axioms of replacement and collection
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## Idea

>_Zwei äquivalente Vielheiten sind entweder beide 'Mengen' oder beide inkonsistent._ [[Georg Cantor]] (1899)[^Cant]

[^Cant]: _'Two equivalent multiplicities either are both "sets" or are both inconsistent'_, letter to Dedekind from 28th July 1899 ([Cantor 1932](#Cantor99), p.444). This is suggested as an early formulation of the axiom of replacement by van Heijenoort (1967, p.113). A categorical formalization of Cantor's idea as an extension for [[ETCS]] is given in [McLarty (2004)](#McLarty).

**Axioms of collection** and replacement are [[axiom schemata]] in [[set theory]] that permit to construct new [[sets]] from other already given sets, thereby contributing substantially to the size of the set-theoretic [[universe]], and hence are seen as 'strong' [[foundational axioms]].

The most famous of these [[axiom schemata]] is the **axiom of replacement**[^name] of [[Zermelo-Fraenkel set theory]] that was suggested by [[Abraham Fraenkel|A. Fraenkel]] and formulated by [[Thoralf Skolem|T. Skolem]] in 1922. Given a unary operation $F$ and a set $x$ it permits to collect all $F(y)$ for $y\in x$ into a new set.

[^name]: The term 'replacement', or 'Ersetzungsaxiom' in German, is apparently due to [Fraenkel (1922)](#Fraenkel22) and was intended as a provisory terminology until the final formalization of Zermelo's notion of a 'definite property' which was identified with a first-order formula in the language of set theory by Skolem in the same year (and independently earlier by [[Hermann Weyl|H. Weyl]]).

The resulting expansiveness of the set-theoretic universe is somewhat peripheral to the practice of 'ordinary' mathematics and therefore a [[structural set theory]] like [[ETCS]] can omit replacement without incurring a great loss[^etcs]. Even in the context of a ZF-equivalent material set theory the axiom of replacement can be traded in for a [[reflection principle]][^bellmach].

[^bellmach]: See [Bell-Machover 1977](#BellMach77), p.495.

[^etcs]: It is possible, however, to augment a categorical set theory with a version of replacement if necessary as shown in ([Osius 1974](#Osius74), section 9) resulting in a system with the full strength of ZF. According to [McLarty (2004)](#McLarty04), Osius' ideas go back to discussions between Lawvere and the Berkeley logicians on reflection principles in 1963. McLarty's paper proposes another equivalent way to flesh out replacement categorically!

Axioms of replacement and collection become useful, however, whenever [[recursion|recursively]] constructing a set that is 'larger' than any set known before:

>what the axiom of replacement is mainly needed for in mathematical practice is to define families of sets indexed by some set $I$ carrying some inductive structure as, typically, the set $N$ of natural numbers.[^streicher]

[^streicher]: [[Thomas Streicher]] ([2005, p.79](#Streicher05)). See there for further discussion of the role of replacement for _mathematics beyond $V_{\omega +\omega}$_ and the handling of similar iterated collection processes in toposes by universes.

There are many variations on these axiom schemata, but any given system should only need one.

## Statements

In general, these axioms apply to a [[binary relation]] that relates elements of one [[set]] $A$ to arbitrary sets.  However, we do *not* expect that the relation itself be an object in the theory; really, we have an axiom schema with one axiom for every binary [[predicate]] of the proper form.  We will write this predicate as $\phi(x,Y)$, where $x$ stands for an elment of $A$ and $Y$ stands for any set.  (Note that there may well be other free variables in the predicate.)

Generally, $\phi$ will need to be an [[entire relation]] for the axiom to apply; that is, the axiom has as a hypothesis that, for every $x \in A$, there is some $Y$ such that $\phi(x,Y)$ holds.  In versions called 'replacement' instead of 'collection', $\phi$ also needs to be [[functional relation|functional]]; that is, the axiom has the hypothesis that, for every $x \in A$, there is a *unique* $Y$ such that $\phi(x,Y)$ holds.  Thus, most of the 'replacement' versions only make sense if the language has a notion of [[equality]] of sets.

So much for the hypothesis of the axiom; the conclusion asserts the existence of a [[family of sets]] to which appropriate $Y$s belong.  In a material set theory, we can simply state the existence of set $\mathcal{F}$ such that certain $Y \in \mathcal{F}$.  In a structural set theory, we state the existence of an index set $I$, a total set $E$, and a function $f\colon E \to I$ such that each [[fibre]] $f^*(x)$ for $x \in I$ is equal to (or at least isomorphic to) certain $Y$.  (Often we can take $I$ to be $A$, but that does not come into the statement of the axioms.)

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* **Bounded replacement** or **restricted replacement** or **$\Delta_0$-replacement**: for any $\Delta_0$-formula $\phi(x, y)$, for any $a$, if for every $x \in a$ there exists a unique $y$ with $\phi(x, y)$, then the set $\{y \vert \exists x \in a.\phi(x, y)\}$ exists.

* **Full replacement**: for any formula $\phi(x, y)$, for any $a$, if for every $x \in a$ there exists a unique $y$ with $\phi(x, y)$, then the set $\{y \vert \exists x \in a.\phi(x, y)\}$ exists.

## Lawvere on replacement

>A question that has been much of a "foundational" interest, though of hardly any significance for the practice of algebra, topology, functional analysis, etc. is whether, for a given $T$, all imaginable families of sets parametrized by $T$ can be represented by $E\to T$ for some $E$ and some mapping; if "imaginable" is interpreted to mean "definable", an affirmative answer to this question is essentially equivalent (for abstract, constant sets) to the postulation of the so-called "replacement schema", whereas if $\mathcal{S}$ is considered as an object in some larger realm, an affirmative answer means that $\mathcal{S}$ itself has "inaccessible cardinality". However, in view of practice and in view of the role of $\mathcal{S}$ as a limiting case of the general notion of continuously variable sets, it seems appropriate to simply define "an internal-to-$\mathcal{S}$ $T$-parametrized family of objects of $\mathcal{S}$" to mean just a morphism of $\mathcal{S}$ with domain $T$.  [Lawvere (1976, p.121)](#Lawvere76)

See also the remarks on pages 721 and 727 of ([Lawvere 2000](#Lawvere00)).

## Related discussion

* [[Thomas Streicher]], [[William Lawvere]], [[Colin McLarty]] et al, *Categorical formulations of replacement*, March 2008. ([link](http://www.mta.ca/~cat-dist/archive/2008/08-3))

* MO-discussion: _Who needs Replacement anyway ?_  ([link](http://mathoverflow.net/questions/208711/who-needs-replacement-anyway))

* Akihiro Kanamori (2012). _In Praise of Replacement_. The Bulletin of Symbolic Logic 18:1 (2012 March).  [pdf](http://math.bu.edu/people/aki/20.pdf).

* $n$-category Cafe (2021),  [Large Sets 12](https://golem.ph.utexas.edu/category/2021/07/large_sets_12.html) ([[Tom Leinster]]) and [Large Sets 12.5](https://golem.ph.utexas.edu/category/2021/07/large_sets_125.html) ([[Mike Shulman]]) -- includes discussion of a number of different replacement axioms for [[ETCS]] and more general [[structural set theory]].

## Related entries

* [[Zermelo-Fraenkel set theory]]

* [[reflection principle]]

* [[axiom of choice]]

* [[ETCS]], [[SEAR]]

* [[large cardinals]]

* [[type theoretic axiom of replacement]]

## References

* {#BellMach77}[[John Bell|J. L. Bell]], M. Machover, *A Course in Mathematical Logic*, North-Holland Amsterdam 1977. (ch. 10, §5)

* {#Cantor99}[[Georg Cantor]], *Brief an Dedekind vom 22. Juli 1899*, pp.443-447 in Cantor, _Gesammelte Abhandlungen_, Springer Berlin 1932.  English transl. pp.113-117 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.

* {#Fraenkel22}A. Fraenkel, *Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre*, Math. Ann. **86** (1922) pp.230-237. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002268760))

* [[Harvey Friedman|H. J. Friedman]], *Higher Set Theory and Mathematical Practice*, Ann. Math. Logic **2** (1971) pp.326-357.

* [[André Joyal]], [[Ieke Moerdijk]], *A categorical theory of cumulative hierarchies of sets*, C. R. Math. Rep. Acad. Sci. Canada **13** (1991) pp.55-58.

* {#Lawvere76}[[F. William Lawvere]], *Variable Quantities and Variable Structures in Topoi*, pp.101-131 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg_ , Academic Press New York 1976.

* {#Lawvere00}[[F. William Lawvere]], *Comments on the development of topos theory*, pp.715-734 in Pier (ed.),  _Development of Mathematics 1950 - 2000_ , Birkhäuser Basel 2000. ([tac reprint](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html))

* {#McLarty04}[[Colin McLarty]], *Exploring Categorical Structuralism*, Phil. Math. **12** no.3 (2004) pp.37-53 doi:[10.1093/philmat/12.1.37](https://doi.org/10.1093/philmat/12.1.37).

* D. A. Martin, *Borel determinacy*, Ann. Math. **102** (1975) pp.363-371.

* {#Osius74}[[Gerhard Osius]], *Categorical Set Theory: A Characterization of the Category of Sets*, JPAA **4** (1974) pp.79-119.

* [[Thoralf Skolem]], *[[Some remarks on axiomatized set theory|Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre]]*, Mathematikerkongressen i Helsingfor 4-7 Juli 1922. English transl. pp.290-301 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.

* [[Thomas Streicher]], *Universes in Toposes*, pp.78-90 in Crosilla, Schuster (eds.), _From Sets and Types to Topology and Analysis_ , Oxford UP 2005. ([preprint](http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf))

* [[Paul Taylor]], *[[Practical Foundations of Mathematics]]*, Cambridge UP 1999. (ch. 9)

* George Tourlakis, *Lectures in Logic and Set Theory*, Volume 2: _Set Theory_, Cambridge University Press (2003). (section III.8)

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category: foundational axiom


axiom of separation

> This is about the [[axiom of separation]] in [[set theory]]. For the axioms in [[topology]] also called "separation", see [[separation axioms]].

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# Axiom schemes of separation
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## Idea

In [[set theory]], the [[axiom scheme]] __of separation__ aka __specification__ states that, given any [[set]] $X$ and any [[property]] $P$ of the [[elements]] of $X$, there is a set
$$ \{ X | P \} = \{ a \in X \;|\; P(a) \} $$
consisting precisely of those elements of $X$ for which $P$ holds:
$$ a \in \{ X | P \} \;\leftrightarrow\; a \in X \;\wedge\; P(a) .$$
Note that $\{X|P\}$ is a [[subset]] of $X$.

It is important to specify what language $P$ can be written in.  This connects the axiom to [[logic]] and the [[foundations of mathematics]].  Arguably, [[first-order logic]] developed in order to explain the meaning of [[Ernst Zermelo]]\'s axiom of separation (although Zermelo himself disagreed with the interpretation that this gave).  Separation is (usually) given as an axiom *[[axiom scheme|scheme]]* because there is one axiom for each way to state a property in the language.  (We also allow parameters in $P$.)

## Statements

From weaker to stronger:

*  For __bounded separation__ aka __restricted separation__ aka __$\Delta_0$-separation__, $P$ must be written in the language of first-order set theory and all [[quantifications]] must be guarded by a set: of the form $\forall x \in A$ or $\exists x \in A$ for some set $A$.

*  For __limited separation__ aka __$\Delta_0^{\mathcal{P}}$-separation__, $P$ must be written in the language of first-order set theory and all quantifications must be guarded by a set or a [[power class]]: of the form $\forall x \in A$, $\exists B \subseteq A$, etc.  Limited separation trivially implies bounded separation, while bounded separation implies limited separation if [[power sets]] exist.

*  For __full separation__ aka simply __separation__, $P$ must be written in the language of first-order set theory, but otherwise anything goes; in a [[class theory]], $P$ must be guarded by a class.  Full separation trivially implies limited separation.

*  For __large separation__, $P$ must be written in the language of first-order class theory; of course, this only makes sense *in* a class theory.  The difference in strength between the class theories $MK$ and $NBG$ is precisely that the former has large separation but the latter does not.

*  Separation is sometimes called __restricted comprehension__; for __[[full comprehension]]__, a.k.a. simply __comprehension__, no set $X$ needs to be given ahead of time.  Full comprehension was proposed by [[Gottlob Frege]], but leads to [[Russell's paradox]].  However, full comprehension can sometimes be allowed if the ambient logic is nonclassical, such as [[linear logic]] or [[paraconsistent logic]].

*  For __stratified comprehension__, no set $X$ is given, but $P$ is restricted to stratified formulas, in which each [[variable]] $x$ can be given a consistent [[natural number]] $\sigma(x)$ (its stratification) such that $x \in y$ appears in the formula only if $\sigma(y) = \sigma(x) + 1$.  This is used in [[Van Quine]]\'s [[New Foundations]].

## In structural set theory

Set theory is usually given in [[material set theory|material]] form, with a language based on a global membership relation $\in$, and we have implicitly followed this above.  However, separation makes sense also in [[structural set theory]] (although full comprehension does not, except in a structural class theory with a [[class of all sets]], where it again leads to paradox).  The conclusion of the axiom is the existence of a set
$$ \{ X | P \} = \{ a\colon X \;|\; P(a) \} $$
and an [[injection]] $i_P\colon \{X|P\} \to X$ such that
$$ \exists b\colon S,\; a = i_P(b) \;\leftrightarrow\; P(a) .$$
Note that $\{X|P\}$, equipped with $i_P$, is a [[subset]] of $X$ in the structural sense.

The structural axioms can of course be stated even in a material set theory, where they are actually weaker than the corresponding material axioms; however, the material axioms follow (as usual) from the structural axioms using [[axiom of replacement|restricted replacement]], which is quite weak (and also follows from the material form of limited separation).

If a structural set theory is given by stating axioms for the [[category of sets]], then restricted separation amounts to the property that this category is a [[Heyting category]].  If it is an [[elementary topos]], then since it satisfies the power set axiom, this implies limited separation as well.  Full separation is somewhat less natural to state category-theoretically, but the combination of full separation with the structural [[axiom of collection]] is equivalent to saying that the category of sets is [[autological category|autological]] (see [here](https://arxiv.org/abs/1004.3802) for now for a definition of autological categories).

The axiom schema of restricted separation is formally expressed by the rules

$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \{x \in S \vert P(x)\} \; \mathrm{set}}$$

$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash i:\{x \in S \vert P(x)\} \to S}$$

$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, y \in \{x \in S \vert P(x)\}, z \in \{x \in S \vert P(x)\}, i(y) = i(z) \; \mathrm{true} \vdash y = z \; \mathrm{true}}$$

$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, x \in R, \exists y \in \{x \in S \vert P(x)\}.x = i(y) \; \mathrm{true} \vdash P(x) \; \mathrm{true}}$$

$$\frac{\Gamma \vdash S \; \mathrm{set} \quad \Gamma, x \in S \vdash P(x) \; \mathrm{prop}}{\Gamma, x \in R, P(x) \; \mathrm{true} \vdash \exists y \in \{x \in S \vert P(x)\}.x = i(y) \; \mathrm{true}}$$

## In hyperdoctrines

### Lawvere's definition
 {#LawvereDefinition}

[[Lawvere]] gives a definition ([Lawvere70, p. 12-13](#Lawvere70)) of comprehension in [[hyperdoctrines]].

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###### Definition

Let $p \colon E \to B$ be a [[bifibration]] over the category $B$, and assume that each [[fibre]] $E_X$ of $E$ has a [[terminal object]] $T_X$.  For any [[morphism]] $t \colon Y \to X$ in $B$, define the _image_ $im t$ of $t$ to be the pushforward ([[dependent sum]]) $t_! T_Y$.  This gives rise to a functor $im \colon B/X \to E_X$ for each $X$.  Then $E$ is said to _have comprehension_, or to satisfy the _comprehension schema_, if each image functor has a [[right adjoint]] $\{-\} \colon E_X \to B/X$.

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###### Remark

This means that for each $P \in E_X$ there is a morphism $i_P \colon \{ P \} \to X$ in $B$ such that there is a [[bijection]] between commuting triangles $t = i_P \circ t'$ in $B$ and morphisms $im t \to P$ in $E_X$.  Lawvere calls the morphism $\{P\} \to X$ the _[[extension (semantics)|extension]]_ of $P$, so that one could say that $E$ satisfies the comprehension schema if the extensions of all predicates exist.

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### Ehrhard's reformulation

Notice that, in the above situation, morphisms $\im t \to P$ in $E_X$ are in bijection with morphisms $T_Y \to t^* P$ in $E_Y$, and that these are the same as morphisms $T_Y \to P$ in the total category $E$ that lie over $t$.  This leads to an alternative formulation of the definition, originally considered by [Ehrhard 1988](#Ehrhard88) (see also [Jacobs 1993](#Jacobs93), [Jacobs 99](#Jacobs99)), which does not depend on $p$ being a bifibration.

Suppose the fibres $E_X$ have terminal objects that are preserved by the reindexing functors.  This is equivalent to saying that $p:E\to B$ has a section $T:B\to E$ taking each object $X$ to a terminal object $T_X$ of $E_X$ and each morphism to a cartesian arrow; such a functor is then automatically [[fully faithful]] and [[right adjoint]] to $p:E\to B$.  If $B$ has a terminal object $1$, then this is additionally equivalent to saying that the entire category $E$ has a terminal object $T_1$ preserved by $p$, since then $X^*(T_1)$ will be terminal in $E_X$ by combining the universal properties of $X^*$ and $T_1$.  Note also that if $p$ is a bifibration, as in Lawvere's definition, then the reindexing functors are right adjoints and hence automatically preserve all terminal objects.

Comprehension in the sense above is equivalent to the existence of a further right adjoint to the terminal-object functor $T:B\to E$.

The implication from [Lawvere's definition](#LawvereDefinition) to Ehrhard's is clear.  Conversely, if a fibration $p \colon E \to B$ satisfies the Ehrhard definition, then there is a bijection between morphisms $T_Y \to P$ in $E$ and morphisms $Y \to \{P\}$ in $B$.  One must then show that the composite of this last with the canonical $\{P\} \to X$ (given by $p$ applied to the counit $T_{\{P\}} \to P$) is equal to $p$ applied to the morphism $Y \to \{P\}$, thus giving a morphism in $B/X$ of the right sort.  But in fact the morphism $Y \to \{P\}$ is unique with the property that applying the functor $T_{-}$ and composing with the counit gives the morphism $T_Y \to P$, by the counit's universal property, and applying $p$ to this commuting triangle in $E$ produces the required one in $B$.

### Examples

#### Subsets

The tautological example which is useful to see what the abstract [definition by Lawvere](#LawvereDefinition) axiomatizes is the following.

Let $B = $[[Set]] be the category of sets and for $X$ a [[set]] let $E_X$ be the [[poset]] of [[subsets]] of $X$, regarded as the [[propositions]] about elements in $X$. Then comprehension exists and is given by sending a subset of $X$ regarded as an object of $E_X$ (hence regarded as a proposition) to the same subset, but now regarded as a [[monomorphism]] in [[Set]] into $X$.

More generally, the same construction works for the posets of subobjects in any [[regular category]].

#### Presheaves

There is a functor $Cat_1^{op} \to CAT_1$, from the [[large category|large]] [[Cat|1-category of categories]] and functors to the '[[very large category|very large]]' 1-category of large categories and functors, that sends a category $C$ to the [[category of presheaves]] $[C^{op}, Set]$  on $C$, and a functor $F \colon C \to D$ to the pullback functor $F^* \colon [D^{op}, Set] \to [C^{op}, Set]$.  These pullback functors have left and right adjoints given by [[Kan extension]].

[Lawvere 70](#Lawvere70) shows that this fibration has comprehension, with the [[extension (semantics)|extension]] of a presheaf given by its [[category of elements]] together with the canonical projection from this to the base category.

#### In dependent linear type theory
 {#ExamplesDependentLinearTypeTheory}

If the [[hyperdoctrine]] has [[linear type theories]]/[[symmetric monoidal categories]] as fibers, then it is more natural to consider in def. \ref{LawvereianComprehension} not the [[terminal object]] in some fiber (if any) but the [[tensor unit]] (which of course happens to be the terminal object in the case of [[cartesian monoidal category|cartesian monoidal]]) fibers.

In this case then the image functor in def. \ref{LawvereianComprehension} is

$$
  \Sigma_X \;\colon\; (Y \stackrel{f}{\to} X) \mapsto \underset{f}{\sum} 1_Y
  \,.
$$

If this has a [[right adjoint]] $R_X$, hence if we have linear comprehension according to def. \ref{LawvereianComprehension}, then the induced [[comonad]]

$$
  !_X \coloneqq \Sigma_X \circ R_X
$$

is (the [[dependent type theory|dependent]] version of) the canonical [[categorical semantics|categorical interpretation]] of the [[exponential modality]] of [[linear logic]]. See at _[dependent linear type theory --  The canonical co-modality](http://ncatlab.org/nlab/show/dependent+linear+type+theory#TheCanonicalComodality)_ for more.

### The comprehensive factorization system

If the extension functors $E_X \to B/X$ are [[fully faithful]], then together they make $E$ into a 'fibrewise [[reflective subcategory|reflective]]' subfibration of the codomain fibration of $B$, which is a reflective subfibration if the image functors preserve pullbacks.  (See [[stable factorization system]].)

Call a morphism of the form $\{P\} \to X$ a _subtype inclusion_.  Every morphism $t \colon Y \to X$ of $B$ factors through one such by means of the unit $\eta_t$ of the adjunction $im \dashv \{-\}$:
$$
\array{
  Y & & \overset{\eta_t}{\to} & & \{ im t \} \\
  & \mathllap{t}\searrow & & \swarrow\mathrlap{i_t} & \\
  & & X & &
}
$$
This gives rise to an [[orthogonal factorization system]] precisely when each $\eta_t$ is [[orthogonal]] to all subtype inclusions.
It is shown by [(Carboni et. al., section 2.12)](#Carboni97) that this holds if and only if subtype inclusions are closed under composition.

For the subobject fibration of a regular category, this gives the usual ([[regular epimorphism|regular epi]], [[monomorphism|mono]]) factorization system, while for the fibration of presheaves over $Cat$ it gives the factorization of a functor into a [[final functor]] followed by a [[discrete fibration]].  (See also [[michaelshulman:comprehensive factorization]] for a description of the latter as a [[factorization system in a 2-category]].)

## Axiom or axiom scheme?

Although usually presented as an axiom scheme, in many cases, all instances of separation follow from finitely many special cases (which can then be packaged into a single axiom, using [[and|conjunction]], although this is probably pointless).  This is the case, for example, in [[ETCS]] (a structural set theory that satisfied bounded separation) and [[NBG]] (a material class theory that satisfies full separation).  In [[type theory|type-theoretic]] foundations of mathematics, separation is usually invisible, but again some form (generally only bounded) can again be proved from a few specific axioms or constructions.

## Relation to the axiom of replacement

Full separation follows from the [[axiom of replacement]] and the principle of [[excluded middle]] (along with the axiom of the [[empty set]]).  Therefore, the axiom is often left out entirely of a description of [[ZFC]] (the usually accepted foundation of mathematics).  In versions of set theory for [[constructive mathematics]], however, we often have replacement but only bounded or limited separation, and in any case separation must be listed explicitly.

## Related concepts

* [[axiom of full comprehension]]

## References

### General

(...)

### In hyperdoctrines

* {#Lawvere70} [[Lawvere]], F. W. _Equality in hyperdoctrines and comprehension schema as an adjoint functor_.  In A. Heller, ed., _Proc. New York Symp. on Applications of Categorical Algebra_, pp. 1--14.  AMS, 1970. ([[LawvereComprehension.pdf:file]])

* {#Lawvere73} [[Lawvere]], F. W. _Metric spaces, generalized logic and closed categories_ Rend. Sem. Mat. Fis. Milano, 43:135–166 (1973). Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))

* {#Ehrhard88} [[Thomas Ehrhard]]. A Categorical Semantics of Constructions. LICS 1988. ([pdf](https://www.irif.univ-paris-diderot.fr/~ehrhard/pub/categ-sem-constr.pdf))

* {#Jacobs93} [[Bart Jacobs]]. Comprehension categories and the semantics of type dependency. _Theoretical Computer Science_ 107:2 (1993), pp. 169-207.

* {#Jacobs99} [[Bart Jacobs]].  _Categorical Logic and Type Theory_.  Elsevier, 1999.

* [[Finn Lawler]], section "Tabulation and comprehension" in: _[[finnlawler:thesis outline|Fibrations of predicates and bicategories of relations]]_ (2014)

* {#Carboni97} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures **5** (1997) 1-58 [[doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444)]

On the relation between comprehension schemes and [[factorisation systems]]:

* [[Clemens Berger]], [[Ralph M. Kaufmann]], *Comprehensive Factorization Systems*, Tbilisi Math. J. 10 (2017), 255-277 ([doi:10.1515/tmj-2017-0112](https://projecteuclid.org/journals/tbilisi-mathematical-journal/volume-10/issue-3/Comprehensive-factorisation-systems/10.1515/tmj-2017-0112.short))

Discussion in [[(infinity,1)-category theory|$(\infty,1)$-category theory]]:

* [[Raffael Stenzel]], _(∞,1)-Categorical Comprehension Schemes_ [[arXiv:2010.09663](https://arxiv.org/abs/2010.09663), talk: [YT](https://www.youtube.com/watch?v=qVhRruh_1jI)]

category: foundational axiom

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axiom of set truncation

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

\tableofcontents

## Idea

In [[dependent type theory]], an **axiom of set truncation** is a [[axiom]] or unjustified rule which implies the [[uniqueness of identity proofs]] which states that every type is an [[h-set]]. Adding an axiom of set truncation to a dependent type theory results in a [[set-level type theory]].

## Examples

TODO: show that each of the axioms below implies the [[uniqueness of identity proofs]].

### Uniqueness of identity proofs

#### For identification types

In [[dependent type theory]], [[uniqueness of identity proofs]] for [[identification types]] is given by the following rule:

$$\frac{\Gamma \vdash A \; type \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash q:\mathrm{Id}_A(a, b)}{\Gamma \vdash \mathrm{UIP}_A(a, b, p, q):\mathrm{Id}_{\mathrm{Id}_A(a, b)}(p, q)}$$

#### For heterogeneous identification types

There is uniqueness of identity proofs for [[heterogeneous identification types]]:

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
      \Gamma \vdash  a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B \quad \Gamma \vdash z:B \\
     \Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{B}(a, b, p, y, z)
    \end{array}
}{\Gamma \vdash \mathrm{UIP}_{B}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, y, z)}(q, r)}$$

Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the [[unit type]] $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[mere proposition]] by uniqueness of identity proofs for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.

Similarly, there is also uniqueness of identity proofs for [[dependent heterogeneous identification types]]:

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash  a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b) \\
     \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \quad \Gamma \vdash r:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)
    \end{array}
}{\Gamma \vdash \mathrm{UIP}_{x:A.B(x)}(a, b, p, y, z, q, r):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)}(q, r)}$$

Uniqueness of identity proofs for identification types follows from uniqueness of identity proofs for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the [[unit type]] $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[mere proposition]] by uniqueness of identity proofs for dependent heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt})}(a, b)$ and thus by definition $\mathrm{Id}_{A}(a, b)$ is also always a mere proposition.

### Axiom K

#### For identification types

In [[dependent type theory]], [[axiom K (type theory)|axiom K]] is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma\vdash a:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, a)}{\Gamma\vdash K_A(a, p):\mathrm{Id}_{\mathrm{Id}_A(a, a)}(p, refl_A(a))}$$

#### For heterogeneous identification types

There is an axiom K for [[heterogeneous identification types]]:

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \\
      \Gamma \vdash f:A \to B \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\
     \Gamma \vdash q:\mathrm{hId}_{B}(a, b, p, f(a), f(b))
    \end{array}
}{\Gamma \vdash K_{B}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{B}(a, b, p, f(a), f(b))}(q, \mathrm{ap}_{B}(f, a, b, p))}$$

Axiom K for identification types follows from axiom K for heterogeneous identification types by taking canonical element $\mathrm{pt}:\mathbb{1}$ of the [[unit type]] $\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:A$ and $b:A$. Since the following two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \simeq \mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{A}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, a)$ is always a [[contractible type]] with [[center of contraction]] $\mathrm{ap}_{A}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and function $f:\mathbb{1} \to A$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.

Similarly, there is also an axiom K for [[dependent heterogeneous identification types]]:

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash  a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \\
     \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))
    \end{array}
}{\Gamma \vdash K_{x:A.B(x)}(f, a, b, p, q):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))}(q, \mathrm{apd}_{x:A.B(x)}(f, a, b, p))}$$

Axiom K for identification types follows from axiom K for dependent heterogeneous identification types by defining the type $A$ to be the dependent type $C(\mathrm{pt})$ for type family $x:\mathbb{1} \vdash C(x)$ indexed by the [[unit type]] $\mathbb{1}$, and taking the canonical element $\mathrm{pt}:\mathbb{1}$, identification $\mathrm{refl}_{\mathbb{1}}(\mathrm{pt}):\mathrm{Id}_{\mathbb{1}}(\mathrm{pt}, \mathrm{pt})$, and elements $a:C(\mathrm{pt})$ and $b:C(\mathrm{pt})$. Since the two types are equivalent:
$$\mathrm{Id}_{A}(a, b) \equiv \mathrm{Id}_{C(\mathrm{pt})}(a, b) \simeq \mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$$
and because $\mathrm{hId}_{x:\mathbb{1}.C(x)}(\mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}), a, b)$ is always a [[contractible type]] with [[center of contraction]] $\mathrm{apd}_{x:\mathbb{1}.C(x)}(f, \mathrm{pt}, \mathrm{pt}, \mathrm{refl}_{\mathbb{1}}(\mathrm{pt}))$ for element $a:A$, and dependent function $f:\prod_{x:\mathbb{1}} C(x)$, by axiom K for heterogeneous identification types, it follows that $\mathrm{Id}_{C(\mathrm{pt}}(a, a)$ and thus by definition $\mathrm{Id}_{A}(a, a)$ is also always a contractible type, with center of contraction $\mathrm{refl}_A(a)$.

### Leibniz’s identity of indiscernibles

In [[dependent type theory]] with [[power sets]], [[Leibniz's identity of indiscernibles]] is given by the following [[inference rule]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \mathrm{idOfInd}_A(x, y):\mathrm{Id}_A(x, y) \simeq \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right)}$$

This implies that all types $A$ are sets, because the type
$$\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)$$
is always a proposition, in the context of [[weak function extensionality]], and thus by the [[equivalence of types]], $\mathrm{Id}_A(x, y)$ is also a proposition for all $x:A$ and $y:A$, which means that $A$ is a set.

### Set-theoretic principle of unique choice

In [[dependent type theory]] with [[power sets]], the [[set-theoretic principle of unique choice]] is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{puc}_{A, B}:\left(\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)\right) \simeq (A \to B)}$$

Because the type $A \times B \to \mathcal{P}(\mathbb{1})$ is a [[set]], the type

$$\sum_{R:A \times B \to \mathcal{P}(\mathbb{1})} \mathrm{isTotal}(R) \times \mathrm{isFunctional}(R)$$

is also a set, and the equivalence only exists if the function type $A \to B$ is a set, which only occurs if $B$ is a set. Thus, this rule implies that all types are sets.

### Boundary separation

In [[cubical type theory]], [[boundary separation]] is given by the following rule:

$$\frac{\Gamma \vdash A \;  \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash r:I \quad \Gamma, (r =_I 0) \vee (r =_I 1) \; \mathrm{true} \vdash a \equiv b:A}{\Gamma \vdash a \equiv b:A}$$

### Axiom of existence

In [[dependent type theory]], the [[type-theoretic axiom of existence]] states that every type has a [[choice operator]], a function from its [[bracket type]] to the type itself, and is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \epsilon_A:[A] \to A}$$

Indeed, it suffices for the [[identity types]] to have [[choice operators]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \epsilon_{\mathrm{Id}_A}(x, y):[\mathrm{Id}_A(x, y)] \to \mathrm{Id}_A(x, y)}$$

### Equality reflection

In [[dependent type theory]], [[equality reflection]] is given by the following rule:

$$\frac{\Gamma \vdash A \;  \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b)}{\Gamma \vdash a \equiv b:A}$$

### Strong pattern matching

...

### $S^1$-localization

In [[dependent type theory]], the [[axiom of S1-localization|axiom of $S^1$-localization]] is given by the following rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash f:S^1 \to A}{\Gamma \vdash \mathrm{circlocal}_{A}:\mathrm{isEquiv}(\mathrm{const}_{A, S^1})}$$

### Axiom of finiteness

In [[dependent type theory]] with [[dependent product types]], [[dependent sum types]], [[identity types]], [[function extensionality]], and a [[type of all propositions]], the [[axiom of finiteness]] is given by the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{finWitn}_A:\mathrm{isFinite}(A)}$$

where

$$
\mathrm{isFinite}(A) \equiv
\begin{array}{c}
\prod_{S:(A \to \mathrm{Prop}) \to \mathrm{Prop}} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}} \prod_{Q:A \to \mathrm{Prop}} (P \in S) \\
\times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S)
\end{array}
$$

The membership relation and the subtype operations used above are defined in the nLab article on [[subtypes]].

### Circle type axioms

Any one of the following axioms on the [[circle type]] implies [[axiom K]] or [[uniqueness of identity proofs]] for all types in the dependent type theory, and thus is an axiom of set-truncation:

* That $S^1$ is a [[set]],
$$\mathrm{trunc}_0^{S^1}:\mathrm{isSet}(S^1)$$
* That $S^1$ is a [[proposition]] or a [[contractible type]], which are the same condition as $S^1$ being a [[set]] because $S^1$ is a [[pointed connected type]]
$$\mathrm{trunc}^{S^1}:\mathrm{isProp}(S^1) \qquad \mathrm{contr}_{S^1}:\mathrm{isContr}(S^1)$$
* That there is an [[identification]] $K:\mathrm{refl}_{S^1}(\mathrm{base}) =_{S^1} \mathrm{loop}$

### Kuratowski-finite implies finite

In [[dependent type theory]], in general there are [[Kuratowski-finite]] types which are not [[h-sets]], such as the [[circle type]]. However, every [[finite type]] is an [[h-set]], and hence why they are usually called [[finite sets]].

A naive translation into dependent type theory of the traditional set-theoretic theorem in classical mathematics that "Kuratowski-finite implies finite" results in the statement that every Kuratowski-finite type is finite. This implies that the circle type is a finite type, which implies that the circle type is an h-set, which implies that every type is an h-set. Thus, that every Kuratowski-finite type is finite is an axiom of set truncaiton.

The statement "Kuratowski-finite implies finite" in classical mathematics is really a statement about h-sets, similar to how the [[axiom of choice]] is really a statement about h-sets. This is because it is still true in [[constructive mathematics]] that every Kuratowski-finite type with [[decidable equality]] is a [[finite type]], and in dependent type theory, every type with decidable equality is an [[h-set]].

## See also

* [[h-set]]
* [[set-level type theory]]

[[!redirects axioms of set truncation]]

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axiom of stack completions

# Axiom of stack completion

\tableofcontents

## Definition

An [[internal category]] $\mathbb{A}$ in an [[elementary topos]] $\mathcal{E}$ is called an **(intrinsic) stack** if the [[indexed category]] that it represents, $(X\in \mathcal{E}) \mapsto \mathcal{E}(X,\mathbb{A}) \in Cat$, is a [[stack]] for the [[regular topology]] of $\mathcal{E}$.  An internal functor is called a **weak equivalence** if it is internally [[essentially surjective]] and [[fully faithful]]; this is strictly weaker than being an internal equivalence if the [[axiom of choice]] does not hold in $\mathcal{E}$.  However, if $f:\mathbb{A} \to \mathbb{B}$ is a weak equivalence and $\mathbb{C}$ is a stack, then the induced functor $\mathrm{Cat}(\mathcal{E})(\mathbb{B},\mathbb {C})\to \mathrm{Cat}(\mathcal{E})(\mathbb{A},\mathbb{C})$ is an equivalence.

We say that $\mathcal{E}$ satisfies the **axiom of stack completions (ASC)** if every internal category admits a weak equivalence to an internal category that is a stack (see [Bunge-Hermida](#BH)).

## Examples

* If $\mathcal{E}$ satisfies the [[axiom of choice]], then every internal category is a stack, so ASC holds trivially.

* If $\mathcal{E}$ is a [[Grothendieck topos]], then internal stack completions can be constructed using the [[small object argument]], as [[fibrant replacements]] in the [[model structure]] on $\mathrm{Cat}(\mathcal{E})$ constructed by [Joyal and Tierney](#JT), so it satisfies ASC.

## References

* {#BH} Bunge and Hermida, "Pseudomonadicity and 2-stack completions"

* {#JT} [[André Joyal]], [[Myles Tierney]], *Strong stacks and classifying spaces*, Lecture Notes in Mathematics **1488**, Springer (1991) [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

[[!redirects axiom of stack completion]]
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axiom of union

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### Context
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
=--
=--

# The axiom of union
* table of contents
{: toc}

## Idea

In [[material set theory]] as a [[foundation of mathematics]], the axiom of union is an important axiom needed to get the foundations off the ground (to mix metaphors).  It states that [[unions]] exist.

## Statement

The __axiom of union__ states the following:

+-- {: .un_defn}
###### Axiom (union)

If $\mathcal{X}$ is a (material) set, then there exists a set $U$ such that $a \in U$ whenever $a \in B \in \mathcal{X}$.
=--

Using the axiom of separation ([[bounded separation]] is enough), one can prove the existence of a particular set $U$ such that the members of the members of $\mathcal{X}$ are the *only* members of $U$.  Using the [[axiom of extensionality]], we can then prove that this set $U$ is unique; it is usually denoted $\bigcup\mathcal{X}$ and called the __[[union]]__ of (the elements of) $\mathcal{X}$.

A slightly different notation may be used when $\mathcal{X}$ is (Kuratowski)-[[finite set|finite]]; for example, $\bigcup\{A,B,C\}$ may be denoted $A \cup B \cup C$.  If $(B_k \;|\; k \in I)$ is a [[family of sets]], then we may write $\bigcup_{k \in I} B_k$ (and the usual variations for a [[sequence]] of sets) for $\bigcup \{B_k \;|\; k \in I\}$; however, we require the [[axiom of replacement]] to prove that the latter set (the [[range]] of the family) exists in general.

## Related notions

If $\mathcal{X}$ is given as a collection of [[subsets]] of some ambient [[set]] $S$, then the axiom of union is not necessary; $S$ itself already satisfies the conclusion of the hypothesis (and then bounded separation gives us the union that we want).  This is the only case when unions are taken in [[structural set theory]].  However, structural set theory makes use of _[[disjoint unions]]_, and [[predicative mathematics]] requires an axiom giving their existence.  (In impredicative mathematics, we can construct disjoint unions from [[power sets]] and [[cartesian products]].)

One could also include [[union structure]] in the set theory.

category: foundational axiom


axiom schema

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### Context
#### Foundations
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[[!include foundations - contents]]
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#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A family of [[axioms]] depending on parameters.

Axiom schemata are formalized in a [[deductive system]] by the use of [[inference rules]] which have [[hypotheses]]. For example, the [[axiom schema of replacement]] in [[set theory]] is given by the following [[inference rule]]:

$$\frac{\Gamma, x, y \vdash \phi(x, y) \; \mathrm{prop} \quad \Gamma \vdash a \quad \Gamma \vdash \forall x.(x \in a) \implies \exists! y.\phi(x, y) \; \mathrm{true}}{\Gamma \vdash \{y \vert \exists x.(x \in a) \implies \phi(x, y)\}}$$

Compare with the bare notion of [[axiom]], which are inference rules without any hypotheses.

## Examples

* [[axiom schema of separation]]

* [[axiom schema of replacement]]

* The [[set-theoretic axiom of choice]] in bare [[Martin-Löf type theory]] or [[cubical type theory]] without any [[type universes]] is an axiom schema, since without universes one cannot quantify over types as one could in in a [[type theory]] with [[universes]] or for sets in [[set theory]], and so the only way to express the axiom of choice is as an inference rule with a hypothesis; i.e. an axiom schema.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash P(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{ac}_{A, B, P}:(\mathrm{isSet}(A) \times \Pi x:A.(\mathrm{isSet}(B(x)) \times \Pi y:B(x).\mathrm{isProp}(P(x, y)))) \to ((\exists y:B(x).P(x, y)) \to (\exists g:(\Pi x:A.B(x)).\Pi x:A.P(x, g(x))))}$$

## Related concepts

* [[inference rule]]

## References

See also:

* Wikipedia, *[Axiom schema](https://en.wikipedia.org/wiki/Axiom_schema)*

[[!redirects axiom schemas]]
[[!redirects axiom schemata]]

[[!redirects axiom scheme]]
[[!redirects axiom schemes]]


Axiomatic field theories and their motivation from topology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

+-- {: .standout}

This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]

See there for background and context.

This entry here indicates, generally, how [[FQFT]]s may be related to [[cohomology theory|cohomology theories]].

=--

> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing

next:

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

* [[(2,1)-dimensional Euclidean field theories and tmf]]

***

+-- {: .standout}

All of the following assumes that the reader is well familiar with the basic ideas indicated at [[FQFT]].

=--

In the following $d-RB$ or $R Bord_d$ and the like denotes a category of $d$-dimensional [[Riemannian cobordism]]s that are equipped with _Riemannian structure_, i.e. with [[Riemannian metric]].
Similarly $d-E B$ or $E Bord_d$ or the like denotes a category of cobordisms with _Euclidean structure_, by which is meant a _flat_ Riemannian metric.

The definition of this is discussed in detail in

* [[bordism categories following Stolz-Teichner]]

**definition**

$d$-dimensional Riemannian field theories are symmetric monoidal functors $d-RB \to TV$ from $d$-dimensional Riemannian bordisms to [[topological vector space]]s.

A field theory is very similar to a [[representation]] of a group. Only where a representation of a [[group]] $G$ is a functor from the [[delooping]] $\mathbf{B}G = {*}//G$ of $G$ to [[Vect]], an [[FQFT]] is a representation of a more complicated domain category.

**how does [[topology]] enter?**

for $X$ some [[topological space]] there is also a [[symmetric monoidal category]]

$$
  d-RB(X)
$$

of Riemannian bordisms equipped with a continuous map to $X$.

Notice that $d RB(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(d RB(X), TV)$ is contravariant in $X$.

When special structure is around, however, we also have a push-forward of such functors along morphisms.

**Example: push-forward to the point**: for $X$ as above and $X \to {*}$ the unique map to the [[point]] heuristically we want a map

$$
  d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$

notice that this push-forward is not an [[adjoint functor]]. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of [[cohomology theory|cohomology theories]].

heuristically the pushforward

$$
    d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$

acts on field theories $E_X$ over $X$

$$
  E_X \mapsto E
$$

by the assignment

$$
  E(Y^{d-1}) \mapsto
  \Gamma\left(
  \array{
     E_X(Y)
     \\
     \downarrow
     \\
     Maps(Y,X)
  }
  regarded as a vector bundle
  \right)
$$

for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X))$.

For instance take $\Sigma$ to be the pair of pants with input $Y_0$ and output $Y_1$ and let $F : \Sigma \to X$ be a map.

Then $E(\Sigma)$ will take some section $\Psi \in E(Y_0)$ to

$$
  E(\Sigma)(\Psi) : (f_1 : Y_1 \to X)
  \mapsto
  \int_{\{F : \Sigma \to X\} | F/Y_1 = f_1} E_X(F)(\Psi(f_0))
  \frac{1}{Z}\exp(-S(F) dvol)
  \,.
$$

Here the expression $\frac{1}{Z}\exp(-S(F) dvol)$ denotes a would-be measure which is still to be defined.

Here $f_0 = F/Y_0$ is the restriction of $F$ to $Y_0$.

Contravariant functors with push-forward also arise as part of a [[cohomology theory]]

$$
  h^n : Diff^{op} \to Ab
$$

from the category [[Diff]] of smooth [[manifold]]s to the category [[Ab]] of abelian groups that satisfies the axioms of [[generalized (Eilenberg-Steenrod) cohomology]] theory.

These Eilenberg-Steenrod axioms are

1. **homotopy axiom** for $h^n$

1. **Mayer-Vietoris axiom** for $h^n$

1. **suspension isomorphism** $h^n(X) \simeq h^{n+1}_{cvs}(X \times \mathbb{R})$

here in the last axiom for [[topological space]]s we'd simply have the [[suspension]] $h^{n+1}(\Sigma X)$. Here in order to stay within [[manifold]]s we instead do as indicated, where ${}_{cvs}$ means "compact vertical support".

Concerning the **homotopy axiom**: given any [[functor]] that sends manifolds to a category of [[FQFT]]s over $X$

  $$
    d-FTs := h : Diff^{op} \to C
  $$

  in $dRFT$s we can make it a homotopy functor by defining

  $\omega_0, \omega_1 \in h(X)$ to be [[concordance|concordant]] if $\exists \omega \in h(X \times \mathbb{R})$ such that $\omega/(X\times \{i\}) \simeq \omega_i$ in $h(X)$, $i = 0,1$

then under this relation

$$
  X \mapsto  d-RFT(X)/\simeq
$$

is a homotopy invariant functor

**Homework**: for $h(X) = \Omega^n_{closed}(X)$

we have that $\omega_0$ concordant to $\omega_1$ exactly when $\omega_0 = \omega_1 + d \alpha$ for some $\alpha \in \Omega^{n-1}(X)$

Concerning the **Mayer-Vietoris axiom**: if $X = U \cup V$ then the functor should respect this gluing.

suppose $E \in 2-RFT(X)$ is a 2d Riemmanian field theory. let $\gamma : S^1 \to X$ be a loop in $X$. then $E(\gamma)$ is a [[vector space]].

If $\gamma$ sits neither entirely in $U$ or in $V$, then there is no way that the vector space $E(\gamma)$ can be reconstructed by knowing just the restriction of $R$ to $U$ and $V$.

So this is a **problem** for the definition of field theories so far.

The **proposed solution** (from [[What is an elliptic object?]]) is to use _extended_ [[FQFT]]s instead. This introduces **locality** into [[FQFT]]s, at the expense of working with [[n-category|n-categories]].

This will however not be studied here for the moment.

Concerning the **suspension isomorphism**: for that first we need for $n \in \mathbb{Z}$ the notion of a field theory over $X$ of **degree $n$**, i.e.

$$
  X \mapsto d-RFTs^n(X)
$$

such that for $n=0$ this is an ordinary Riemannian field theory, in $d-RFT(X)$.

This requires to replace [[manifold]]s by [[supermanifold]]s.

**Example** let $d = 0$ and consider 0-dimensional TFTs over $X$.

so consider

$$
  Fun^\otimes(0 Bord(X), TV)
$$

in $0 Bord(X)$ there is only a single [[object]]: the [[empty set]] $\emptyset$ which has a unique map $\emptyset \to X$

the collecton of morphisms is $\{finite set \to X\}$ in there is $Hom_{Diff}(pt,X)$.

here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.

$$
  E : 0 Bord(X) \to TV_{\mathbb{R}}
$$

$$
  \emptyset \mapsto \mathbb{R}
$$

$$
  x \in X \mapsto E(X) \in \mathb{R}
$$

so a priori we have

$$
  Fun^\otimes(0 Bord(X), TV) \simeq Maps(X, \mathbb{R})
  \,,
$$

where on the right we have _all_ maps.

This is not quite what is intended. We want to see _smooth_ maps on the right. To get that, we need to talk about _smooth functors_ on the left. This is the topic of later discussion, which will yield

$$
  SmoothFun^\otimes(0 Bord(X), TV) \simeq C^\infty(X, \mathbb{R})
  \,.
$$

But one other ting goes wrong the: the corresponding homotopy functor is $C^\infty(X)/\simeq = \{0\}$. So turning this into an Eilenberg-Steenrod theory yields the trivial theory.

The way out to that will be to go to [[supermanifold]]s.

**Punchline** of this session here:

$$
  (0|1)-TFTs(X)
  \simeq
  SmoothFun^\otimes((0|1)Bord(X), TV)
$$

and then it is a $lemma^G$ that

$$
  C^\infty(SuperDiff(\mathbb{R}^{0|1}, X))^G
  \simeq
  \Omega^\bullet(X)^G
  = (\oplus_{n = 0}^{\infty} \Omega^n(X))^G
$$

where $G$ is the [[supergroup]] $G = Diff(\mathbb{R}^{0|1}) \wim42 \mathbb{R}^{0|1} \rtimes \mathbb{R}^\times$.

The degree decomposition on forms then turns out to be the eigenvalue decompositon of the $\mathbb{R}^\times$-part, while the deRham differential is the $\mathbb{R}^{0|1}$-action (cite Kontsevich, Severa here ...)

so then from that one finds that the $G$-invariant bit is the _closed_ 0-forms

$$
  \Omega^\bullet(X)^G
  \simeq
  \Omega^0_{closed}(X)
$$

So we get $n$-**Lemma** :

$$
  (0|1)-TFT^n(X) \simeq \Omega^n_{cl}(X)
$$

and

$$
  (0|1)-TFT^n(X)_{concord} \simeq H^n_{dR}(X)
$$

**push-forward**

Now with the super-directions included, there is a notion of push-forward of the TFTs that does shift the degree, and we get the following

**Theorem (Stolz-Teichner-Kreck-Hohnhold)**

when $X$ is oriented

$$
  (0|1)TFT^n(X) \stackrel{push}{\to} (0|1)TFT^0
  \simeq \mathbb{R}
$$

$$
  \Omega^n_{cl}(X) \stackrel{\int_{X_n}}{\to}
  \Omega^0_{cl}(pt)
  \simeq
  \mathbb{R}
$$

and similarly after dividing out concordance, the push-forward becomes the push-forward in [[deRham cohomology]].

## References

* [[Stefan Stolz]] (notes by Arlo Caine), _Supersymmetric Euclidean field theories and generalized cohomology_ Lecture notes (2009) ([pdf](http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf))
 {#Stolz}


axioms of plane geometry

# Axioms of plane geometry

Under construction.

For Croatian version see [link](https://ncatlab.org/zoranskoda/show/aksiomi+planimetrije)

## Literature

* related pages include [[isometry]], [[congruence in geometry]]
* wikipedia [Euclidean geometry](https://en.wikipedia.org/wiki/Euclidean_geometry)
* Jacquelline Lelong-Ferrand, Les fondements de la géométrie, Chap. 5
* [[David Hilbert]], _Grundlagen der Geometrie_. Leipzig, Teubner (1899); _Foundations of geometry_, 1905, 1971
* Oscar Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. __5__ (1904) [jstor](https://archive.org/details/jstor-1986462)
* A. Kolmogorov
* George D. Birkhoff, A set of postulates for plane geometry, Annals of Mathematics 33: pp. 329-345 (1932)
* [[Hermann Weyl]], _Raum, Zeit, Materie_, 1918; engl. transl. Space Time Matter 1922, reprinted 1952
* Gustave Choquet, _Geometry in a modern setting_
* Friedrich Bachmann, _Aufbau der Geometrie aus dem Spiegelungsbegriff_, 1959; Russ. ed. Построение геометрии на основе понятия симметри, Nauka 1969 (transl. by Р.И.Пименов, redaction И.М.Яглом)

[[!redirects aksiomi planimetrije]]


axion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _axion_ is a hypothetical type of [[field (physics)|field]]/[[fundamental particle]] originally hypothesized ([Weinberg 77](#Weinberg77), [Wilczek 78](#Wilczek78))  as a solution to the [[strong CP problem]] in the [[standard model of particle physics]].
After the initial model for the axion was quickly ruled out by [[experiment]] (see [Wilczek 78](#Wilczek78) for the early history) a variant model was found ([Dine-Fischler-Srednicki 81](#DineFischlerSrednicki81)), called the "invisible axion", which does not violate experimental bounds.

The "invisible axion" turns out to also be a natural candidate for [[dark matter]] [Preskill-Wise-Wilczek 83](#PreskillWiseWilczek83), hence potentially also solves one of the problems with the [[standard model of cosmology]]. More recently it is argued in [Hui-Ostriker-Tremaine-Witten 16](#HOTW16) that indeed axionic dark matter ([[fuzzy dark matter]]) potentially solves the remaining problems of standard WIMP dark matter models: WIMP dark matter models work exceedingly well on [[cosmology|cosmological]] scales but has serious experimental problems on the scale of [[galaxies]]. Due to the extreme lightness of axion particles, their [[de Broglie wavelength]] may be of the scale of galaxies and hence their quantum properties may become relevant at this scale to deviate in the right way from the WIMP models.

The Pecchei-Quinn shift symmetry of the axion and the peculiar nature of the axion [[interaction term]] which are needed to make the axion model work this way have been argued to naturally arise in [[string theory]], if the axion is identified with the [[KK-reduction]] of the [[higher gauge fields]] in [[string theory]]. This we discuss [below](#AsArisingFromStringTheory).

## As a solution to the strong CP problem
 {#AsASolutionToTheStrongCPProblem}

The problem here is that the [[action functional]] for [[QCD]] ([[Yang-Mills theory]]) a priori contains an arbitrary [[theta angle]] $\theta$

$$
  S
  \;\colon\;
  \nabla \mapsto
  \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla)  \;+\;
  \theta \int_X tr(F_\nabla \wedge F_\nabla)
$$

but that -- since the term $tr(F_\nabla \wedge F_\nabla)$ causes parity violation, which is strongly constrained by [[experiment]] -- there must be some reason why $\theta$ vanishes or else is extremely small.

(Here the term $tr(F_\nabla \wedge F_\nabla)$ is the [[invariant polynomial]] for the [[second Chern class]], measuring the [[instanton number]] of the [[gauge field]] $\nabla$.)

The solution to this problem via axions is to assume that $\theta$ is not really a fundamental parameter, but instead is the [[vacuum expectation value]] of a dynamical [[field (physics)|field]] $a$ (the axion).

The idea is that a standard [[kinetic action]]

$$
  S_{kin} (a) \propto \int_X a \wedge \ast a
$$

together with the axion [[interaction]] term

$$
  S_{int} (a,\nabla) \propto  \int_X a \, tr(F_\nabla \wedge F_\nabla)
$$

makes $a$ have vanishing [[expectation value]] $\langle a\rangle$. This would give a dynamical explanation why under the identification of the [[theta angle]] with this expectation value

$$
  \theta \coloneqq \langle a\rangle
$$

the theta angle vanishes.

### The Vafa-Witten mechanism
 {#VafaWittenMechanism}

That this is indeed the case is due to [Vafa-Witten 84, around (2)](#VafaWitten84). These authors argue via the [[Wick rotation|Wick rotated]] [[path integral]] as follows:

Under [[Wick rotation]], parity-violating terms in the [[Lagrangian density]] pick up an imaginary factor $i$. Therefore the [[path integral]] expression for the Wick rotated [[vacuum energy]] is

$$
  \begin{aligned}
    E_{vac}(a)
    &=
    - log \left( \underset{\nabla,a}{\int}  \exp(- S(\nabla,a)) \, D \nabla \, D a \right)
    \\
    &=
    - log
      \left(
       \underset{\nabla}{\int} \exp\left(
        \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla)
     \right)
      \, D \nabla
      \;
     \underset{a}{\int}
     \exp\left(
        i a \int_X tr(F_\nabla \wedge F_\nabla)
     \right)
     \, D a
    \right)
  \end{aligned}
  \,.
$$

Now due to the factor of $i$ in front of $\theta$ in this expression, the real part of the argument of the logarithm necessarily becomes _smaller_ with $a$. Therefore the negative logarithm becomes _larger_ with $a$. Accordingly $E_{vac}(a)$ must have a minimum at $\theta = 0$ (according to [Vafa-Witten 84, p.2](#VafaWitten84)).

## In string theory
 {#AsArisingFromStringTheory}

In [[string theory]] axion fields

1. are naturally present as the [[KK-reduction]] of the [[higher gauge fields]] (e.g. [Svrcek-Witten 06](#SvrcekWitten06));

   reviewed below in:

   _[As KK-Reduction of higher gauge fields](#AxionsAsKKReductionOfHigherGaugeFields)_

1. are generically of positive but tiny mass ([ADDKM 09](#ADDKM09), [Acharya-Bobkov-Kumar 10](#AcharyaBobkovKumar10)) as they need to be to be a solution to the [[strong CP problem]] and be candidates for [[fuzzy dark matter|BEC/superfluid/fuzzy dark matter]];

   reviewed below in:

   _[Stringy axion phenomenology](#StringyAxionPhenomenology)_

This makes axion fields in string theory form a curious confluence point relating

* abstract concepts related to [[higher gauge theory]]

with

* fundamental question in [[standard model of particle physics|particle physics]]/[[standard model of cosmology|cosmology]] [[phenomenology]]:

$$
  \array{
    \mathbf{\text{higher gauge theory}}
    &&
    &&
    \mathbf{\text{particle physics/cosmology phenomenology}}
    \\
    \\
    \left.
      \array{
        \text{higher gauge fields}
        \\
        \text{higher characteristic classes}
        \\
        \updownarrow
        \\
        \text{non-perturbative QFT/string effects}
        \\
        \text{in HET: Green-Schwarz anomaly cancellation}
        \\
        \text{in IIA/B: higher WZW term for Green-Scharz D-branes}
      }
    \right\}
    &\longrightarrow&
    \array{
      \text{axion fields}
      \\
      \text{in the string spectrum}
    }
    &\longrightarrow&
    \left\{
      \array{
        \text{solve strong CP-problem as with P-Q robustly}
        \\
        \text{solve dark matter problem by FDM}
      }
    \right.
  }
$$

### Axions as KK-reduction of higher gauge fields
 {#AxionsAsKKReductionOfHigherGaugeFields}

In [[string theory]] axion fields _naturally_ arise as the [[KK compactification]] of the [[higher gauge fields]] ([[B-field]], [[C-field]], [[RR-fields]]). Here we review how this comes about.

Notice that this means that axions in string theory are as in the output of the original Peccei–Quinn proposal, but do not actually make use of the Peccei–Quinn mechanism.

As amplified in particular in [Svrcek-Witten 06](#SvrcekWitten06):

> An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ "symmetry" if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ "symmetries" are natural comes from string theory, which produces them without any contrivance. ... the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken. ([Svrcek-Witten 06, pages 3-4](#SvrcekWitten06))

Similarly from [ADDKM 09](#ADDKM09):

> However, at the [[effective field theory]] level  it  is  hard  to  judge  how natural it is to have such a "fake" global PQ symmetry which is explicitly broken exclusively by QCD. Note, that in order not to spoil the solution to the strong CP problem all other sources of explicit PQ symmetry breaking should be at least 10 orders of magnitude down with respect to the potential  generated  by the QCD anomaly, and one may be especially cautious about the viability of such a proposal, given the common lore that global symmetries always get broken by quantum gravitational effects [12, 13, 14]. This makes it natural to inquire whether axions arise naturally in the most developed quantum theory of gravity—string theory. ([ADDKM 09, p. 3](#ADDKM09))

The mechanism discussed in [Svrcek-Witten 06](#SvrcekWitten06), in all its incarnations in the various perspectives on string theory ([[heterotic string theory|heterotic strings]], [[type II string theory|type II strings]], [[11-dimensional supergravity]] with [[M-theory]] effects included, etc.) share the following properties:

1. the axion field itself is the [[double dimensional reduction]] of one of the [[higher gauge fields]] in string theory, the (twisted) [[Kalb-Ramond field]] for heterotic string, the [[RR-field]] for the type II string, or the [[supergravity C-field]] in 11-dimensional supergravity with [[M-brane]] effects;

1. accordingly the Peccei-Quinn periodic shift symmetry of the axion results arises as the $U(1)$-[[gauge symmetry]] that is the dimensional reduction of the $B^2 U(1)$ (heterotic [[B-field]]) or $B^3 U(1)$ (11d [[C-field]]) [[higher gauge symmetry]], where the $U(1)$-periodicity is ultimately due to the higher [[Dirac charge quantization]] for these fields;

1. the axion [[interaction term]] of the form $\propto a \langle F \wedge F\rangle $ arises as the [[double dimensional reduction]] of the self-interaction terms of these higher gauge fields:

   * in type II string theory it is a component of the [[higher WZW term]] on coincident [[D6-branes]] with a component $\propto A_{3}^{RR} \wedge \langle F \wedge F\rangle$,

   * for 11d C-fields it is the term $\propto C_{3} \wedge \langle F \wedge F\rangle$ induced by [[anomaly cancellation]] for M-theory at conical singularities (as discussed at _[[M-theory on G2-manifolds]]_)

   * for heterotic string theory it arises from integrating out the [[Lagrange multiplier]] that enforces the [[Green-Schwarz anomaly cancellation]] in 4d.

We now discuss this in more detail:

* _[In type IIA string theory](#IntypeIIA)_

* _[In heterotic string theory](#InHeteroticStringTheory)_

#### In type IIA string theory
  {#IntypeIIA}

Consider [[string phenomenology]] in [[type IIA string theory]] in the guise of [[intersecting D-brane models]] with the [[standard model of particle physics]] supported in intersecting [[D6-branes]] whose [[worldvolume]] fills all of 4d [[spacetime]].

The [[higher WZW term]] in the [[Green-Schwarz action functional]] for the [[D6-brane]] is of the form

$$
  \propto \int_{X_7} A^RR \wedge \langle \exp(F_\nabla) \rangle
$$

where $A^RR$ denotes the collective inhomogenous [[background field|background]] [[RR-field]] and $F_\nabla$ is the [[curvature]] 2-form of the [[Chan-Paton gauge field]] on the D-brane. The $\langle -\rangle$ denotes the [[Killing form]] [[invariant polynomial]], hence the [[trace]] for the [[special unitary Lie algebra]] regarded as a [[matrix Lie algebra]]. $X_7$ denotes the [[worldvolume]] of the [[D6-brane]]

Hence one of the three non-vanishing summands in this expression is

$$
  \propto \int_{X_7} A_3^{RR} \wedge \langle F_\nabla \wedge F_\nabla\rangle
  \,.
$$

Now assuming a Kaluza-Klein ansatz $X_7 = X_4 \times Y_3$ with

$$
  a \coloneqq \int_{Y_3} A_3^{RR}
$$

the effective axion field in 4d, then this term becomes the
axion [[interaction]] term

$$
  \propto \int_{X_4} a \langle F_{\nabla} \wedge F_{\nabla}\rangle
$$

([Svrcek-Witten 06, around (6.9)](#SvrcekWitten06))

#### In heterotic string theory
  {#InHeteroticStringTheory}

In [[heterotic string theory]] [[KK-compactification|KK-compactified]] to 4d, the 4d [[B-field]], dualized, serves as the axion field, called the "model independent axion" ([Svrcek-Witten 06, starting p. 15](#SvrcekWitten06)).

This works as follows: By the [[Green-Schwarz anomaly cancellation]] mechanism, then [[B-field]] in [[heterotic string theory]] is a twisted 2-form field, whose [[field strength]] $H$ locally has in addition to the exact differential $d B$ also a fundamental 3-form contribution, so that

$$
  H = d B + C
$$

(locally). Moreover, the differential $d H$ is constrained to be the Pontryagin 4-form of the gauge potential $\nabla$ (minus that of the  [[Riemann curvature]], but let's suppress this notationally for the present purpose):

$$
  d H = tr \left(F_\nabla \wedge F_\nabla\right)
  \,.
$$

Now suppose [[KK-compactification]] to 4d has been taken care of, then this constraint may be implemented in the [[equations of motion]] by adding it to the [[action functional]], multiplied with a [[Lagrange multiplier]] :

$$
  S
    =
  \underset{
      \text{kinetic action} \atop \text{for B-field}
   }{
        \underbrace{\int_X H \wedge \star H}
   }
    +
   \underset{
      \text{Green-Schwarz constraint}
   }{
    \underbrace{
      \int_X a \left( d H - tr(F_\nabla \wedge F_\nabla) \right)
    }
   }
  \,.
$$

Now by the usual argument, one says that instead of varying by $a$ and thus implementing the [[Green-Schwarz anomaly cancellation]] constraint, it is equivalent to first vary with respect to the other fields, and then insert the resulting equations in terms of $a$ into the action functional.

Now since we are dealing with a twisted [[B-field]], with free 3-form component $C$, we actually vary with respect to $C$. This yields the [[Euler-Lagrange equation]] [[equation of motion|of motion]]

$$
  d a = \star H
  \,,
$$

hence the usual relation or [[electro-magnetic duality]], expressing what used to be the [[Lagrange multiplier]] now as the magentic dual [[field (physics)|field]] to the twisted [[B-field]].

Plugging this algebraic [[equation of motion]] back into the above [[action functional]] for $H$ gives

$$
  \tilde S
    =
  \underset{\text{kinetic action} \atop \text{for axion field}}{\underbrace{\int_X d a \wedge \star d a}}
  +
  \underset{\text{axionic} \atop \text{interaction}}{\underbrace{\int_X a \, tr(F_\nabla \wedge F_\nabla)}}
  \,.
$$

This now is an action functional for an axion field $a$ of just the form required [above](#AsASolutionToTheStrongCPProblem) for the solution of the [[strong CP-problem]].

### Stringy axion phenomenology
 {#StringyAxionPhenomenology}

From [Acharya-Kane-Kumar 12, p. 3](#AcharyaKaneKumar12):

> Now consider the axions.  Due to the shift symmetries mentioned above, there are no [[perturbative QFT|perturbative]] contributions to their potential.  [[non-perturbative effect|Non-perturbative effects]] though, such as strong gauge dynamics, gauge [[instantons]], gaugino condensation and stringy instantons will generate a potential for the axions. Because any such contribution is exponentially suppressed by couplings and/or extra-dimensional volumes, in our world with perturbative gauge couplings (at high scales), the axion masses will be exponentially small.

> Furthermore, since there are large numbers of axions in general, their masses will essentially be uniformly distributed on a logarithmic scale ([5](#ADDKM09)).  See ([6](#AcharyaBobkovKumar10)) for a detailed calculation of axion masses in string/M effective theories.

> Like the moduli, the axions are also very weakly coupled to matter and therefore do not thermalize in general.  Moreover, since their masses are tiny - ranging from $m_{3/2}$ to  even  below  the  Hubble  scale  today  -  many  of them,  including  the  QCD  axion,  start  coherent  oscillations  during  the  time  that  the  moduli  are  dominating the energy density, but before BBN.

Observe that ultralight axion fields is precisely what is required in HEP [[phenomenology]] for

1. the solution to the [[strong CP-problem]];

1. the solution of the [[dark matter]] problem via [[fuzzy dark matter|BEC/superfluid/wave/fuzzy dark matter]] (e.g. [Hui-Ostriker-Tremaine-Witten 16](#HOTW16)).

## Related concepts

* [[axion inflation]]

* [[fuzzy dark matter]]

* [[strong CP problem]]

* [[G2-MSSM]]

## References

### General

The axion as such was originally proposed in

* {#Weinberg77} [[Steven Weinberg]], _New light boson_, Phys. Rev. Lett. 40:223-6 (1977)

* {#Wilczek78} [[Frank Wilczek]], _Problem of strong P and T invariance in the presence of instantons_, Phys. Rev. Lett. 40:279-82 (1978)

The experimentally viable variant as the "invisible axion" is due to

* {#DineFischlerSrednicki81} [[Michael Dine]], [[Willy Fischler]], [[Mark Srednicki]], _A simple solution to the strong CP problem with a harmless axion_, Phys. Lett. B 104:199-202 (1981)

The observation that the "invisible axion" is a candidate for [[dark matter]] is due to three groups:

* {#PreskillWiseWilczek83} [[John Preskill]], M. Wise, [[Frank Wilczek]], _Cosmology of the invisible axion_, Phys. Lett. B 120:127-32 (1983)

* {#AbbottSikivie83} L.F. Abbott, P. Sikivie, _A Cosmological Bound on the Invisible Axion_, Phys.Lett. B120:133-36 (1983)

* {#DineFischler83} [[Michael Dine]], [[Willy Fischler]], _The Not So Harmless Axion_, Phys. Lett. B120 :137-141 (1983)

A historical recollection of the development until here is in

* [[Frank Wilczek]], _Birth of axions_ in _This Week's Citation Classic_ (1991) ([pdf](http://www.garfield.library.upenn.edu/classics1991/A1991FE76900001.pdf))

The argument that the [[topological Yang-Mills theory|topological]] [[interaction]] term $\propto a \langle F \wedge F\rangle$ gives the axion field $a$ a vanishing [[vacuum expectation value]] is due to

* {#VafaWitten84} [[Cumrun Vafa]], [[Edward Witten]], _Parity Conservation in Quantum Chromodynamics_, Phys. Rev. Lett. 53, 535 (1984) ([publisher](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.535))

A reformulation of this effect in terms of [[Chern-Simons forms]] is discussed in

* [[Gia Dvali]], _Three-Form Gauging of axion Symmetries and Gravity_ ([arXiv:hep-th/0507215](https://arxiv.org/abs/hep-th/0507215))

Review:

* {#KusterRaffeltBeltran08} Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), _Axions: Theory, cosmology, and Experimental Searches_, Lect.
Notes Phys. 741 (Springer, Berlin Heidelberg 2008) (doi:10.1007/978-3-540-73518-2_2)

* Jihn E. Kim, Gianpaolo Carosi, _Axions and the Strong CP Problem_, Rev.Mod.Phys.82:557-602,2010 ([arXiv:0807.3125](https://arxiv.org/abs/0807.3125))

* Masahiro Kawasaki, Kazunori Nakayama, _Axions : Theory and Cosmological Role_, Annual Review of Nuclear and Particle Science Vol.63:1-552   ([arXiv:1301.1123](https://arxiv.org/abs/1301.1123))

* Luca Di Luzio, Maurizio Giannotti, Enrico Nardi, Luca Visinelli, _The landscape of QCD axion models_ ([arXiv:2003.01100](https://arxiv.org/abs/2003.01100))

* Igor G. Irastorza, *An introduction to axions and their detection* ([arXiv:2109.07376](https://arxiv.org/abs/2109.07376))

See also:

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

### In string theory
  {#ReferencesInStringTheory}

Discussion of the various ways that axions naturally appear in [[string theory]] is in

* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], _Axions In String Theory_, JHEP 0606:051,2006 ([arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206))

Specifically for the [[F-theory]] sector of string theory:

* {#Grimm14} Thomas Grimm, _Axion Inflation in F-theory_ ([arXiv:1404.4268](http://arxiv.org/abs/1404.4268))

Specifically open string axions

* [[Gabriele Honecker]], section 4 of _From Type II string theory towards BSM/dark sector physics_, International Journal of Modern Physics A Vol. 31 (2016) 1630050 ([arXiv:1610.00007](https://arxiv.org/abs/1610.00007))

A textbook account of axion [[string phenomenology]] is in

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 16.2 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

Discussion of stringy axion [[cosmology]] (such as [[fuzzy dark matter]]) is in

* {#ADDKM09} Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Nemanja Kaloper, John March-Russell, _String Axiverse_, Phys.Rev.D81:123530,2010 ([arXiv:0905.4720](https://arxiv.org/abs/0905.4720))

* {#AcharyaBobkovKumar10} [[Bobby Acharya]], Konstantin Bobkov, [[Piyush Kumar]], _An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse_, JHEP 1011:105, 2010 ([arXiv:1004.5138](https://arxiv.org/abs/1004.5138))

* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](https://arxiv.org/abs/1204.2795))

### In holographic QCD

Realization in the [[Witten-Sakai-SUgimoto model]] for [[holographic QCD]]:

* Francesco Bigazzi, Alessio Caddeo, Aldo L. Cotrone, Paolo Di Vecchia, Andrea Marzolla, _The Holographic QCD Axion_ ([arXiv:1906.12117](https://arxiv.org/abs/1906.12117))

### Experimental signature

Discussion of [[experiment|experimental]] signatures of and constraints on axion physics:

#### In particle physics
 {#ReferencesExperimentalSignatureInParticlePhysics}

Discussion of [[experiment|experimental]] signatures of and constraints on axion physics from [[particle physics]]:

Specifically contribution of axions to the [[anomalous magnetic moment]] of the [[electron]] and [[muon]] in [[QED]]:

* Yannis Semertzidis, _Magnetic and Electric Dipole Moments in Storage Rings_, chapter 6 of Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), _Axions: Theory, cosmology, and Experimental Searches_, Lect.
Notes Phys. 741 (Springer, Berlin Heidelberg 2008) ([Kuster-Raffelt-Beltran 08](#KusterRaffeltBeltran08), doi:10.1007/978-3-540-73518-2_2)

* Roberta Armillis, Claudio Coriano', Marco Guzzi, Simone Morelli, _Axions and Anomaly-Mediated Interactions: The Green-Schwarz and Wess-Zumino Vertices at Higher Orders and g-2 of the muon_, JHEP 0810:034,2008 ([arXiv:0808.1882](https://arxiv.org/abs/0808.1882))

* W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, _Contributions of axion-like particles to lepton dipole moments_, Phys. Rev. D 94, 115033 (2016) ([arXiv:1607.01022](https://arxiv.org/abs/1607.01022))

* Martin Bauer, Matthias Neubert, Andrea Thamm, _Collider Probes of Axion-Like Particles_, J. High Energ. Phys. (2017) 2017: 44. ([arXiv:1708.00443](https://arxiv.org/abs/1708.00443), doi:10.1007/JHEP12(2017)044)

The basic relevant [[Feynman diagrams]] are worked out here:

* [pdf](http://www-personal.umich.edu/~jbourj/peskin/6-3.pdf)

See also

* [[Frank Wilczek]], _New Ideas in Axion Searches_, talk 2017 ([pdf](http://frankwilczek.com/2017/axion_searches_01.pdf))

For the [ABRACADABRA](http://abracadabra.mit.edu/) (A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus) approach to axion detection, see

* Jonathan Ouellet, _ABRACADABRA: A Broadband Search for Axion Dark Matter_, talk 2017 ([pdf](https://indico.fnal.gov/event/13702/session/3/contribution/45/material/slides/0.pdf))

For the ADMX experiment:

* {#Avignone18} Frank T. Avignone III, _[Homing in on Axions?](https://physics.aps.org/articles/v11/34)_, APS Physics 11, 34, 2018

#### In astrophysics

Axion signatures in [[gravitational waves]] potentially visible by [[LIGO]]-type [[experiments]] are discussed in

* {#HJSSZ18} Junwu Huang, Matthew C. Johnson, Laura Sagunski, Mairi Sakellariadou, Jun Zhang, _Prospects for axion searches with Advanced LIGO through binary mergers_ ([arXiv:1807.02133](https://arxiv.org/abs/1807.02133))

#### In cosmology

Discussion of [[experiment|experimental]] signatures of and constraints on axion physics in [[cosmology]], where the axion is a ([[fuzzy dark matter|fuzzy]]) [[dark matter]]-candidate:

* Joseph P. Conlon, M.C. David Marsh, _Searching for a 0.1-1 keV Cosmic Axion Background_ ([arXiv:1305.3603](http://arxiv.org/abs/1305.3603))

  > Primordial decays of [[string theory]] [[moduli stabilization|moduli]] at $z \sim 10^{12}$ naturally generate a [[dark radiation]] Cosmic Axion Background (CAB) with $0.1 - 1 keV$ energies. This CAB can be detected through axion-[[photon]] conversion in astrophysical [[magnetic fields]] to give quasi-thermal excesses in the extreme ultraviolet and soft X-ray bands. Substantial and observable luminosities may be generated even for axion-photon couplings $\ll 10^{-11} GeV^{-1}$. We propose that axion-photon conversion may explain the observed excess emission of soft X-rays from galaxy clusters, and may also contribute to the diffuse unresolved cosmic X-ray background. We list a number of correlated predictions of the scenario.

Discussion of the axion as a candidate for [[dark matter]] ([[fuzzy dark matter]]) is in

* {#HOTW16} [[Lam Hui]], [[Jeremiah Ostriker]], [[Scott Tremaine]], [[Edward Witten]], _On the hypothesis that cosmological dark matter is composed of ultra-light bosons_, Phys. Rev. D 95, 043541 (2017) ([arXiv:1610.08297](https://arxiv.org/abs/1610.08297))

with its experimental bounds

* Nilanjan Banik, Adam J. Christopherson, Pierre Sikivie, Elisa Maria Todarello, _New astrophysical bounds on ultralight axionlike particles_, Phys. Rev. D 95, 043542 (2017) ([arXiv:1701.04573](https://arxiv.org/abs/1701.04573))

and as a candidate for [[dark energy]]:

* [[Stephon Alexander]], [[Robert Brandenberger]], [[Juerg Froehlich]], _Tracking Dark Energy from Axion-Gauge Field Couplings_ ([arXiv:1601.00057](https://arxiv.org/abs/1601.00057))

[[!redirects axions]]

[[!redirects invisible axion]]
[[!redirects invisible axions]]


axion inflation

+-- {: .rightHandSide}
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
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#Contenta#
* table of contents
{:toc}

## Idea

[[cosmic inflation]] with [[axion]] [[field (physics)|fields]] playing the role of the [[inflaton field]].

## References

General:

* Enrico Pajer, Marco Peloso, _A review of Axion Inflation in the era of Planck_ ([arXiv:1305.3557](http://arxiv.org/abs/1305.3557))

Models in [[string theory]] ([[F-theory]]):

* {#Grimm14} [[Thomas Grimm]], _Axion Inflation in F-theory_ ([arXiv:1404.4268](http://arxiv.org/abs/1404.4268))


axionic landscape

#Contents#
* table of contents
{:toc}

## Idea

The discussion of the [[landscape of string theory vacua]] is often done with rather broad stroke approximations to the structures that should really be classified. For instance the landscape of [[flux compactifications]] is often done with all [[fluxes]] modeled simply as [[differential forms]]. It is known however that more properly these flux fields are really [[cocycles]] in [[differential cohomology]] (For instance the [[B-field]] is really in [[ordinary differential cohomology]] while the [[RR-field]] is really in [[differential K-theory]]). If one does take this refinement into account, then after [[KK-compactification]] what previously looked like plane [[scalar fields]] now become _[[axion]]_ fields, which inherit periodicity from their origin in [[higher gauge fields]]. Taking this [[axion|axionic]] nature of some of the scalar fields in string theory compactifications into account is hence known as considering the _axionic landscape_ of string theory vacua.

## References

* [[Michael Dine]], Guido Festuccia, John Kehayias, Weitao Wu, _Axions in the Landscape and String Theory_, JHEP 1101:012,2011 ([arXiv:1010.4803](https://arxiv.org/abs/1010.4803))

* Gaoyuan Wang, Thorsten Battefeld, _Vacuum Selection on Axionic Landscapes_, JCAP 04 (2016) 025 ([arXiv:1512.04224](https://arxiv.org/abs/1512.04224))

[[!redirects axionic landscapes]]

[[!redirects axionic landscape of string theory vacua]]
[[!redirects axionic landscapes of string theory vacua]]