infrared divergence

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#### Algebraic Quantum Field Theory
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#Contemts#
* table of contents
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## Idea

In the formulation of ([[perturbative quantum field theory|perturbative]]) [[quantum field theory]] various naive mathematical constructions give rise to ill-defined "[[divergent series|divergent]]" expressions.

Some of these occur in the [[limit of a sequence|limit]] that [[interaction]] points are taken _close_ to each other. Since the short distances involved in this limit translate, under [[Fourier transform]], to high [[frequencies]], these are called _[[ultraviolet divergences]]_. The correct way to deal with them is called _[[renormalization]]_.

Other divergences occur in the [[limit of a sequence|limit]] that the spacetime [[support]] of the [[interaction]] (the support of the "[[adiabatic switching]]") tends without bounds to all of spacetime, also called the [[adiabatic limit]]. Since the large distances involved in this limit translate, under [[Fourier transform]], to low [[frequencies]], these are called _infrared divergences_.

In [[perturbative AQFT]] the issue is dealt with by realizing that, despite superficial appearance, the [[adiabatic switching|adiabatically switched]] [[S-matrix]] already makes physical sense, namely, for the computation of those [[quantum observables]] whose [[spacetime]] [[support]] is such that its [[causal closure]] lies inside the support of the switched S-matrix ([this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet)). This gives a well-defined meaning to [[causal perturbation theory]] _without_ having to consider the [[adiabatic limit]].

## Properties

> The [[Lee-Nauenberg theorem]] is a fundamental quantum mechanical result which provides the standard theoretical response to the problem of collinear and infrared divergences. Its argument, that the divergences due to massless charged particles can be removed by summing over degenerate states, has been successfully applied to systems with final state degeneracies such as LEP processes. If there are massless particles in both the initial and final states, as will be the case at the [[LHC]], the theorem requires the incorporation of disconnected [[Feynman diagrams|diagrams]] which produce connected interference effects at the level of the cross-section. However, this aspect of the theory has never been fully tested in the calculation of a cross-section. We show through explicit examples that in such cases the theorem introduces a divergent series of diagrams and hence fails to cancel the infrared divergences. It is also demonstrated that the widespread practice of treating soft infrared divergences by the [[Bloch-Nordsieck method]] and handling collinear divergences by the Lee-Nauenberg method is not consistent in such cases.

> ([Lavelle-McMullan 05](#LavelleMcMullan05))

## Related concepts

* [[infraparticle]]

* [[interacting vacuum]]

* [[ultraviolet divergence]]

* [[pAQFT]]

## References

Infrared divergences in [[QED]] were first observed in

* {#Mott31} N. F. Mott, _On the influence of radiative forces on the scattering of electrons_, Proc. Camb. Phil. Soc. 27, 255 (1931).

* {#BlockNordsiek37} [[Felix Bloch]] and [[Arnold Nordsieck]], _Note on the Radiation Field of the electron_, Phys. Rev. 52 , 54 (1937)

A famous approach to the problem is due to

* {#FadeevKulish70} [[Ludwig Faddeev]], P. P. Kulish, _Asymptotic conditions and infrared divergences in quantum electrodynamics_, Theoretical and Mathematical Physics June 1970, Volume 4, Issue 2, pp 745–757 ([mathnet.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=4141&option_lang=eng), [springer](https://link.springer.com/article/10.1007%2FBF01066485))

But see

* Wojciech Dybalski, _From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons_ ([arXiv:1706.09057](https://arxiv.org/abs/1706.09057))

which says:

> the [Faddeev-Kulish approach](#FadeevKulish70) is justified at best by working out some test cases in perturbation theory. The question if the infrared finite [[S-matrix]] has any [[non-perturbative quantum field theory|non-perturbative]] meaning is left completely open. Secondly, the relation between the Faddeev-Kulish approach to the more standard [[LSZ scattering theory]] has never been clarified. While a naive application of the LSZ ideas clearly fails in the presence of infrared problems, a careful LSZ description of a bare electron accompanied by real and virtual photons is in fact possible.

> In the present work we outline a bridge from the Faddeev-Kulish formalism to this LSZ descriptionin the massless Nelson model.

On the [[Lee-Nauenberg theorem]]:

* {#LavelleMcMullan05} Martin Lavelle, David McMullan, _Collinearity, convergence and cancelling infrared divergences_, JHEP 0603 (2006) 026 ([arXiv:hep-ph/0511314](https://arxiv.org/abs/hep-ph/0511314))

Discussion from the point of view of [[causal perturbation theory]] / [[perturbative AQFT]]:

* {#AAS10} Andreas Aste, Cyrill von Arx, [[Günter Scharf]], section 4 of _Regularization in quantum field theory from the causal point of view_, Prog. Part. Nucl. Phys.64:61-119, 2010 ([arXiv:0906.1952](https://arxiv.org/abs/0906.1952))

* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))

* {#Duch19} [[Paweł Duch]], _Infrared problem in perturbative quantum field theory_ ([arXiv:1906.00940](https://arxiv.org/abs/1906.00940))

Further discussion in relation to the [[soft graviton theorem]]:

* Daniel Kapec, Malcolm Perry, Ana-Maria Raclariu, [[Andrew Strominger]], _Infrared Divergences in QED, Revisited_, Phys. Rev. D 96, 085002 (2017) ([arXiv:1705.04311](https://arxiv.org/abs/1705.04311))

* {#Strominger17} [[Andrew Strominger]], _Lectures on the Infrared Structure of Gravity and Gauge Theory_ ([arXiv:1703.05448](https://arxiv.org/abs/1703.05448))

* Neelima Agarwal, Lorenzo Magnea, Chiara Signorile-Signorile, Anurag Tripathi, *The Infrared Structure of Perturbative Gauge Theories* ([arXiv:2112.07099](https://arxiv.org/abs/2112.07099))

See also:

* Holmfridur Hannesdottir, Matthew D. Schwartz, _A Finite S-Matrix_ ([arXiv:1906.03271](https://arxiv.org/abs/1906.03271))

* Wikipedia, _[Infrared divergence](https://en.wikipedia.org/wiki/Infrared_divergence)_

[[!redirects infrared divergences]]

[[!redirects infrared divergency]]
[[!redirects infrared divergencies]]

[[!redirects IR divergency]]
[[!redirects IR divergencies]]

[[!redirects Lee-Nauenberg theorem]]
[[!redirects Lee-Nauenberg method]]

[[!redirects Bloch-Nordsieck theorem]]
[[!redirects Bloch-Nordsieck method]]

[[!redirects infrared cutoff]]
[[!redirects infrared cutoffs]]


Ingemar Bengtsson

* [webpage](http://3dhouse.se/ingemar/)

## Selected writings

On [[black holes]] in [[D=3 gravity]] (generalizing [[BTZ black holes]]):

* Stefan Aminneborg, [[Ingemar Bengtsson]], Dieter Brill, Soren Holst, Peter Peldan, _Black Holes and Wormholes in 2+1 Dimensions_, Class. Quant. Grav. 15 (1998) 627-644 ([arXiv:gr-qc/9707036](https://arxiv.org/abs/gr-qc/9707036))

* Stefan Aminneborg, [[Ingemar Bengtsson]], Soren Holst, _A Spinning Anti-de Sitter Wormhole_, Class. Quant. Grav. 16 (1999) 363-382 ([arXiv:gr-qc/9805028](https://arxiv.org/abs/gr-qc/9805028))

On [[quantum mechanics]] with focus on the [[geometry]] of ([[mixed quantum state|mixed]]) [[quantum state spaces]] and their [[quantum channels]], culminating in a chapter on [[quantum entanglement]]:

* {#ŻyczkowskiBengtsson04} [[Karol Życzkowski]], [[Ingemar Bengtsson]], *On Duality between Quantum Maps and Quantum States*, Open Systems & Information Dynamics **11** 01 (2004) 3-42 [[doi:10.1023/B:OPSY.0000024753.05661.c2](https://doi.org/10.1023/B:OPSY.0000024753.05661.c2)]

* [[Ingemar Bengtsson]], [[Karol Życzkowski]], *Geometry of Quantum States --- An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048), [ResearcgGate](https://www.researchgate.net/publication/266435541_Geometry_of_Quantum_States_An_Introduction_to_Quantum_Entanglement)]

On [[mutually unbiased bases]]:

* [[Ingemar Bengtsson]], *Three ways to look at mutually unbiased bases*, AIP Conference Proceedings **889** (2007) 40-51 [[arXiv:quant-ph/0610216](https://arxiv.org/abs/quant-ph/0610216), [doi:10.1063/1.2713445](https://doi.org/10.1063/1.2713445)]

## Related entries

* [[twistor]]

category: people


Ingmar Saberi

* [website](https://www.mathi.uni-heidelberg.de/~saberi/)

### Related entries

* [[p-adic AdS-CFT]]

### Writings

On [[Chern-Simons theory]], [[D=3 gravity]], [[p-adic AdS/CFT duality]], [[BTZ black holes]], [[knot theory]] and [[3-manifolds]]:

* [[Ingmar Saberi]],  _Knots, Trees, and Fields: Common Ground Between Physics and Mathematics_ ([doi:10.7907/Z9VX0DHZ](https://thesis.library.caltech.edu/9896), [pdf](https://thesis.library.caltech.edu/9896/1/Saberi_Ingmar_2016.pdf))

On the [[p-adic AdS/CFT correspondence]]:

* [[Matthew Heydeman]], [[Matilde Marcolli]], [[Ingmar Saberi]], [[Bogdan Stoica]], _Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence_ ([arXiv:1605.07639](https://arxiv.org/abs/1605.07639))

* [[Ingmar Saberi]], _Holography and local fields_, ([arXiv:1801.04942](https://arxiv.org/abs/1801.04942))

Discussion of [[BTZ black holes]] via [[tensor networks]] in the [[p-adic AdS/CFT correspondence]]:

* Matthew Heydeman, [[Matilde Marcolli]], Sarthak Parikh, [[Ingmar Saberi]], _Nonarchimedean Holographic Entropy from Networks of Perfect Tensors_ ([arXiv:1812.04057](https://arxiv.org/abs/1812.04057))

On [[spin chains]] in [[condensed matter physics]]:

* [[Ingmar Saberi]], _An introduction to spin systems for mathematicians_, ([arXiv:1801.07270](https://arxiv.org/abs/1801.07270))

On holomorphic [[topological twist|twists]] of [[superconformal algebras]]:

* [[Ingmar Saberi]], [[Brian Williams]], *Superconformal algebras and holomorphic field theories* ([arXiv:1910.04120](https://arxiv.org/abs/1910.04120))

Relating [[supermultiplets]] to [[L-infinity algebras|$L_\infty$-algebras]] via [[pure spinor]]-formalism:

* [[Martin Cederwall]], Simon Jonsson, [[Jakob Palmkvist]], [[Ingmar Saberi]], *Canonical supermultiplets and their Koszul duals* [[arXiv:2304.01258](https://arxiv.org/abs/2304.01258)]

category: people


Ingo Battenfeld

* [webpage](http://ls10-www.cs.uni-dortmund.de/index.php?id=203)

category: people


Ingo Blechschmidt

* [webpage](https://www.ingo-blechschmidt.eu/)

## Publications

On [[synthetic algebraic geometry]]:

* [[Ingo Blechschmidt]], *Using the internal language of toposes in algebraic geometry*, PhD thesis (2017) [[pdf](https://rawgit.com/iblech/internal-methods/master/notes.pdf), [[Blechschmidt-InternalLanguage.pdf:file]]]

On [[constructivism|constructive]] [[algebra]]:

* [[Ingo Blechschmidt]], *Generalized spaces for constructive algebra*, ([arXiv:2012.13850](https://arxiv.org/abs/2012.13850))

Introduction to the [[Agda]] [[proof assistant]]:

* [[Ingo Blechschmidt]], *Agda -- a beautiful proof assistant*, course at *Teoria dei Tipi*, Padova (Apr 2023) [[webpage](https://agdapad.quasicoherent.io/~Padova)]

An online [[Agda]] interface:

* [agdapad.quasicoherent.io](https://agdapad.quasicoherent.io)

category: people

[[!redirects Ingo Blechschmidt]]
[[!redirects IngoBlechschmid]]


Ingo Kirsch

* [Institute page](http://wwwiexp.desy.de/sfb676/researchers/ikirsch/)

* [Spire page](https://inspirehep.net/authors/1020286?ui-citation-summary=true)

## Selected writings

On [[mesons]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Ingo Kirsch]], [[Ed Threlfall]], *Mesons in Gauge/Gravity Duals - A Review*, Eur. Phys. J. A **35** (2008) 81-133 [[arXiv:0711.4467](https://arxiv.org/abs/0711.4467), [doi:10.1140/epja/i2007-10540-1](https://doi.org/10.1140/epja/i2007-10540-1)]

category: people


Ingo Roth

member of the [Quantum Research Center](https://www.tii.ae/quantum) (QRC) at the [Technology Innovation Institute](https://www.tii.ae/) (TII) in Abu Dhabi

* [webpage](https://www.tii.ae/team/ingo-roth)

## Selected writings

On [[hardware verification]] for [[quantum computing]]-[[quantum systems|systems]]:

* [[Martin Kliesch]], [[Ingo Roth]], *Theory of Quantum System Certification*, PRX Quantum **2** (2021) 010201 [[doi:10.1103/PRXQuantum.2.010201](https://doi.org/10.1103/PRXQuantum.2.010201), [arXiv:2010.05925](https://arxiv.org/abs/2010.05925)]

category: people


Ingo Runkel

* [website](http://www.math.uni-hamburg.de/home/runkel/)

## Selected writings

The [[FRS-theorem on rational 2d CFT]]:

* [[Ingo Runkel]], [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Christoph Schweigert]], *Topological and conformal field theory as Frobenius algebras*, in: *Categories in Algebra, Geometry and Mathematical Physics* Contemp. Math. **431** (2007) 225-248 $[$[doi:10.1090/conm/431](http://dx.doi.org/10.1090/conm/431), [arXiv:math/0512076](https://arxiv.org/abs/math/0512076)$]$

On [[QFT with defects|defects]] and [[duality in physics|duality]] ("[[generalized global symmetry]]") in [[2d CFT]] (via the [[FRS theorem on 2d rational CFT]]):

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

Lecture notes on [[modular tensor categories]] and [[rational CFT]]:

* {#Runkel} [[Ingo Runkel]], *Algebra in Braided Tensor Categories and Conformal Field Theory* ([pdf](https://www.math.uni-hamburg.de/home/runkel/PDF/alg.pdf), [[Runkel-AlgebraInBTCAndCFT.pdf:file]])

On [[QFT with defects]]:

* [[Nils Carqueville]], [[Michele Del Zotto]], [[Ingo Runkel]], *Topological defects*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2311.02449](https://arxiv.org/abs/2311.02449)]

## Related  entries

* [[conformal field theory]]

category: people


Ingo Waschkies

* website (?)

## related $n$Lab entries

* [[locally constant ∞-stack]]

* [[homotopy group of an ∞-stack]]

* [[local system]]

category: people


inhabited object

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Category Theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[category theory]] and specifically in [[topos theory]], the notion of _inhabited object_ in a general [[topos]] is a generalization of the notion of [[inhabited set]] from the archetypal topos [[Set]]. It is the [[semantics]] of an [[inhabited type]] in [[type theory]].

## Definition

An object $X$ in a [[topos]] $\mathcal{T}$ is said to be **internally inhabited** if in the [[internal logic]] of the topos it is true that

$$
  \exists x \in X.
$$

This is equivalent to saying that the unique map $X \to 1$ is an [[epimorphism]].  More generally, we can make the same definition in any [[regular category]], in which case it is equivalent to $X\to 1$ being a [[regular epimorphism]].

The term **well-supported** is also used for this notion (in general, the [[support object]] of $X$ is the [[image]] of $X\to 1$).

In terms of [[(∞,1)-category theory]], internally inhabited means _[[(-1)-connected]]_.

On the other hand, $X$ is said to be **externally** or **globally inhabited** if there exists a morphism $1\to X$, i.e. a [[global element]].  The adjective "externally" is perhaps misleading, since this notion can also be expressed in the internal [[type theory]] of the topos (as having an [[element]]/[[term]] $x:X$), though not in its more usual internal [[higher-order logic]].  The internal assertion "every internally inhabited object is globally inhabited" is a version of the [[propositional axiom of choice]] (PAC).

Every globally inhabited object is internally inhabited, since every [[split epimorphism]] is a regular epimorphism.  The converse is true if $1$ is [[projective object|projective]], as is the case in a [[well-pointed topos]] (such as [[Set]]).  (This may seem to contradict the above, since PAC can fail even in (constructively) well-pointed toposes, but the point is that PAC also says something about *families* of inhabited objects rather than single "globally defined" objects.)

Some sources use "$X$ is inhabited" in the global sense, while others use the term "inhabited" only internally.  Regardless, a [[pointed object]] always means one _equipped with_ a global element $1\to X$ (which is the same whether interpreted internally or externally), and the terms "global element" and "well-supported" are always unambiguous.

## Examples

For [[Grothendieck topos|sheaf toposes]] with [[point of a topos|enough points]], in which epimorphisms are [[stalk]]wise epimorphisms, an object is (internally) inhabited if the [[germ]] of $X$ over every [[stalk]] is an [[inhabited set]].

One situation where this plays a role is in the study of certain [[smooth topos]]es with objects $\mathbb{I}$ of _invertible infinitesimals_ . There is an immediate definition of such a topos, the topos called $\mathcal{Z}$ at [[Models for Smooth Infinitesimal Analysis]], for which this object exists, but is not inhabited. Only the weaker internal statement $\not \not \exists x \in \mathbb{I}$ is true.

But for some useful constructions in these toposes, such as for giving an internal definition of [[distribution]]s as genuine functions (internally), it is desirable to have $\mathbb{I}$ be inhabited. In the above situation this is achieved by [[forcing]] the existence of invertible infinitesimal elements. The result is the refined topos denoted $\mathcal{B}$ at [[Models for Smooth Infinitesimal Analysis]].

## In a well-pointed topos

The category [[Set]] is a very special sort of topos.  A generally useful principle in formulating categorical properties of [[Set]] is that "external" and "internal" notions should coincide.  A general default understanding across the nLab is that $Set$ is a [[well-pointed topos]] whose terminal object $1$ is projective and [[indecomposable object|indecomposable]].  These latter properties of the terminal object hold for any well-pointed topos if the metalogic is [[classical logic|classical]]; for purposes of [[constructive mathematics]], we usually agree to add them in.

As remarked above, projectivity of $1$ easily makes internal and external inhabitedness agree.  We also present a constructive/intuitionistic proof of the following result about *emptiness*.

+-- {: .num_prop}
###### Proposition
In a well-pointed topos, an object $X$ which is _not_ internally inhabited is empty (is an initial object).
=--

Note that the negation "not" in this statement is an *external* one, which is what makes it nontrivial.  If an object is *internally* "not inhabited", then it is empty essentially by definition.

+-- {: .proof}
###### Proof
Let $X \to U \hookrightarrow 1$ be the epi-mono factorization of the unique map $X \to 1$. Here $m: U \hookrightarrow 1$ is monic but not epic if $X \to 1$ is not epic, and we use this to prove $U \cong 0$. In that case, we have a map $X \to 0$, whence $X \cong 0$ since in a topos initial objects are strict. (For given $X \to 0$, the projection $\pi_1: X \times 0 \to X$ has a section, and it follows that $X \cong 0$ since $X \times 0 \cong 0$ and retracts of initial objects are initial. This also implies that for any $V$ the map $i: 0 \to V$ is monic: for it is obviously true that for any pair of maps $h, k: A \to 0$ we have that $i h = i k$ implies $h = k$, as $A$ is initial by strictness and this makes $h = k$ automatic.)

Let $f: U \to \Omega$ be the classifying map of the mono $1_U: U \to U$, and let $g: U \to \Omega$ be the classifying map of the mono $0 \to U$. There is no map $1 \to U$, else $m: U \to 1$ would retract it and hence be epic. Hence it is vacuously true that $f x = g x$ for all $x: 1 \to U$, and so $f = g$ by well-pointedness. Hence the subobjects $1_U$ and $0$ coincide, forcing $U \cong 0$.   (In other words, the presence of a [[subobject classifier]] causes any [[generator]] to be a [[strong generator]].)
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## Related concepts

* [[inhabited set]]

* [[pointed object]]

[[!redirects inhabited objects]]

[[!redirects well-supported object]]
[[!redirects well-supported objects]]

[[!redirects inhabited]]


inhabited set

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### Context
#### Type theory
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[[!include type theory - contents]]
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# Contents
* table of contents
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## Idea

A [[set]] or [[type]] is **inhabited** if it contains an [[element]] or [[term]].

## Definition

### In set theory

In [[set theory]], an **inhabited set** is a [[set]] that contains an [[element]], i.e. a set $X$ such that $\exists x, x\in X$ is [[true]].

At least assuming [[classical logic]], this is the same thing as a set that is [[negation|not]] [[empty set|empty]]. Usually inhabited sets are simply called 'non-empty', but the positive word 'inhabited' reminds us that inhabitation is the simpler notion, which emptiness is defined as the [[negation]] of.

The term 'inhabited' comes from [[constructive mathematics]]. In constructive mathematics (such as the [[internal logic]] of some [[topos]] or generally in [[type theory]]), a set/type that is not empty is not already necessarily inhabited.  This is because [[double negation]] is nontrivial in [[intuitionistic logic]]. All the same, many constructive mathematicians use the old word 'non-empty' with the understanding that it *really* means inhabited, and write $A\neq \emptyset$ to mean that $A$ is inhabited.  The latter we can interpret literally if we regard $\neq$ as a reference to an [[inequality relation]] other than the [[denial inequality]], such as the inequality defined for subsets by $A \neq B \iff \exists x ((x\in A \wedge x\notin B)\vee(x\in B\wedge x\notin A))$.  If we prefer to reserve $\neq$ for the denial inequality, then we can write $\#$ for this stronger inequality of sets (although it is not an [[apartness relation]]), and hence $A\#\emptyset$ to mean that $A$ is inhabited.

### In type theory

In [[type theory]] there are two possible notions of **inhabited type**: a type $X$ whose [[propositional truncation]] $\Vert X \Vert$ has an element (or [[term]]), or a type $X$ that *itself* has an element (or [[term]]).  The former is what corresponds to the above notion of inhabitedness in set theory, since $\Vert X \Vert$ is the [[propositions as types]] interpretation of $\exists x:X$.  The latter is more like the notion of a [[pointed set]].

The assertion $\forall X, (\Vert X \Vert \to X)$ is a mildly nonconstructive logical principle called the [[propositional axiom of choice]].  It follows from [[excluded middle]], but in the internal logics of some toposes, it can fail, so that these two notions of "inhabited type" really are different.

### Inhabited objects

An inhabited set is the special case of an internally [[inhabited object]] in the [[topos]] [[Set]].  The two notions of inhabited type correspond to internally and externally inhabited objects (which are, respectively, those objects $X$ where $X\to 1$ is an [[epimorphism]], and those which admit a [[global element]] $1\to X$).

### Related notions

There is a distinction between 'inhabited' and '[[occupied space|occupied]]' spaces in [[Abstract Stone Duality]] (which probably corresponds to something about [[locales]], should explain that here).

## Related concepts

* [[hProp]]

* [[effective epimorphism]]

* [[inhabited object]]

* [[bracket type]], [[image]]

* [[occupied space]]

[[!redirects inhabited set]]
[[!redirects inhabited sets]]
[[!redirects inhabited subset]]
[[!redirects inhabited subsets]]
[[!redirects inhabited type]]
[[!redirects inhabited types]]
[[!redirects inhabited space]]
[[!redirects inhabited spaces]]

[[!redirects nonempty set]]
[[!redirects non-empty set]]
[[!redirects nonempty sets]]
[[!redirects non-empty sets]]
[[!redirects nonempty subset]]
[[!redirects non-empty subset]]
[[!redirects nonempty subsets]]
[[!redirects non-empty subsets]]

[[!redirects nonempty]]
[[!redirects non-empty]]


inhomogeneous cosmology

+-- {: .rightHandSide}
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### Context
#### Cosmology
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[[!include cosmology -- contents]]
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#### Gravity
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[[!include gravity contents]]
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#Contents#
* table of contents
{:toc}

## Idea

What is called _inhomogeneous cosmology_ is the study of [[cosmology]] via cosmological solutions to [[Einstein's equations]], _without_ assuming or constraining these solutions to be spatially homogeneous (in the technical sense).

This is in contrast to the [[standard model of cosmology]], based on [[FRW model]]-type solutions to [[Einstein's equations]], where [[spacetime]] _is_ assumed to be spatially homogeneous (an assumption also known as the _cosmological principle_).

Of course the [[observable universe]] is clearly not _exactly_ homogeneous (due to initial [[CMB]] fluctuation and ensuing [[structure formation]] -- see for instance the *[[KBC void]]*, [KBC 13](#KBC13), large observed an-isotropies [SHRMS22](#SHRMS22), and see [Aluri et al. 2022](#AluriEtAl22), [Peebles 2022](#Peebles22) for extensive survey of observed inhomogeneity, anisotropy and other deviations) -- but the question is whether on cosmic [[scales]] the deviation from homogeneity is small enough that it may be neglected, to good approximation, for the purpose of modelling cosmological evolution, or whether it exerts relevant "[[backreaction]]" on the global evolution of spacetime. For a clean account of the question see [Kolb-Marra-Matarrese 10](#KolbMarraMatarrese10), for review see [Buchert-Räsänen 11](#BuchertRasanen11), [Ellis 18](#Ellis18).

It has been shown that the effect of such [[backreaction]] is small or invisible if the inhomogeneity is modeled in a non-[[relativistic field theory|relativistic]] (i.e. Newtonian) limit, instead of taking [[general relativity|relativity]] into account ([Buchert 00](#Buchert00), [Buchert-Ehlers 95](#BuchertEhlers95)), which however is the standard approximation currently used in comparing the [[standard model of cosmology]] to data.

## Effective dark energy from inhomogeneity?
 {#EffectiveDarkEnergyFromInhomogeneity}

The [[standard model of cosmology]] assumes that cosmic inhomogeneities may be neglected to zeroth order, and studies [[structure formation]] as a perturbation about a spatially homogeneous [[FRW model]] [[background field|background]] [[spacetime]].

Given that the [[standard model of cosmology]] faces some issues (e.g. [BCKRW 15](standard+model+of+cosmology#BCKRW15), [Riess et. al. 16](#RiessEtAl16), [MRS 21](#MRS21)) related to _[[dark energy]]_ (a [[cosmological constant]], and possibly related issues such as [[cosmic inflation]] and the observational preference for [[phantom dark energy]]), it has been suggested that these may be but an artefact of the overly idealistic approximation of cosmic homogeneity, and that a more accurate inhomogeneous cosmology would not need to assume any [[dark energy]]/[[cosmological constant]].

References suggesting/discussing/checking this idea include the following: [Célérier 00](#Celerier00), [Buchert 00](#Buchert00), [Wetterich 01](#Wetterich01), [Schwarz 02](#Schwarz02), [Räsänen 03](#Rasanen03), [Alnes-Amarzguioui-Gron 06](#AlnesAmarzguiouiGron06), [Alnes-Amarzguioui 06](#AlnesAmarzguioui06), [Buchert-Larena-Alimi 06](#BuchertLarenaAlimi06), [Enqvist-Mattsson 06](#EnqvistMattsson06) [Buchert 07](#Buchert07), [Sarkar 08](#Sarkar08), [Notari 09](#Notari09), [Blomqvist 10](#Blomqvist10), [Buchert 11](#Buchert11), [Buchert-Räsänen 11](#BuchertRasanen11), [Scharf 13](#Scharf13), [Smoller-Temple-Vogler 14](#SmollerTempleVogler14), [Ludwick 15](#Ludwick15) [Moffat 16](#Moffat16), [Ludwick 18](#Ludwick18) [Colin-Mohayaee-Rameez-Sarkar 19](#SarkarEtAl18), [Colin-Mohayaee-Rameez-Sarkar 19b](#ColinMoymhayeeRamezSarkar19),
 [Sarkar 18](#Sarkar18), [Lombriser 19](#Lombriser19), [Deledicque 19](#Deledicque19), [Heinesen-Buchert 20](#HeinesenBuchert20).

Discussion specifically of cosmic inhomogenity as the possible cause of the [H0-tension](Hubble+constant#H0TensionReferences) is in [Bolejko 17](#Bolejko17), [Sikora-Glod 20](#SikoraGlod20), [SnowmassWhitePaper22, Section VII.H](#WhitePaper22).

From [Koksbang 19, p. 3](#Koksbang19):

> Cosmic backreaction is particularly interesting because it in principle has the potential to explain the apparent accelerated expansion of the Universe without introducing any exotic [[dark energy]] component as well as possibly being able to mimic [[dark matter]].

> Less ambitiously, cosmic backreaction might solve the $H_0$-problem through the emergence of curvature ([Bolejko 17](#Bolejko17)), or a small backreaction may bias the values obtained from analyses of
data based on FLRW models and must therefore be identified and taken into account in an era of precision cosmology. Yet another option is that cosmic backreaction is entirely negligible in the real universe.

> Whichever is the case, a theoretical quantification of cosmic backreaction is necessary for getting the foundations of cosmology onto solid ground; the
mathematics clearly shows that in principle backreaction terms affect the overall dynamics of the Universe. It is therefore an important goal of cosmologists to obtain a theoretical understanding of the size of cosmic backreaction in the real universe similarly to e.g. the
desire to theoretically understand the value of the vacuum energy density.

From [CosmoBack 18](#CosmoBack18):

> Since the early 2000's a large debate emerged in the cosmology community around the so-called «averaging problem» (Buchert 2000; Ellis & Buchert 2005), a question introduced on a more general ground in general relativity by G.F.R. Ellis in 1962. Because on the non-linearity of the Einstein equations, a non-trivial backreaction effect of the small-scale matter inhomogeneities is expected on the average large-scale dynamics of the spacetime. The occurrence of the late-time accelerated expansion of the universe at the same epoch when structure formation becomes non-linear is a tempting coincidence encouraging the backreaction conjecture against the real need of a dark energy component, which is needed instead in a Friedmann-Lemaître-Robertson-Walker setting. Besides, the same backreaction mechanism also accounts for an energy source dynamically equivalent to the dark matter. These arguments are often supported by toy models but not demonstrated in general, both from a theoretical and observational point-of-view. Simplified inhomogeneous models have been proposed to interpret the non-trivial dynamics of structures and the light propagation on cosmological scales. Only recently cosmological fully general-relativistic N-body simulations have been realised, offering a valuable ground to investigate the geometry of a realistic, "lumpy" universe.

### Qualitative mechanism

{#Qualitative} A qualitative discussion of how cosmic inhomogeneity may cause accelerated cosmic expansion is given in [Räsänen 10, section 3: "Understanding acceleration"](#Rasanen10):

> In general, underdense regions $[$"voids"$]$ are negatively curved and expand faster than the average, while overdense regions are positively curved and expand slower. ([Räsänen 03, p. 15](#Rasanen03))

> $[...]$ as the volume occupied by $[$inhomogeneous$]$ structures grows (along with the density contrast of typical structures), the expansion rate becomes dominated by voids, since their volume is large $[...]$ overdense regions slow down more as their density contrast grows, and eventually they turn around and collapse to form stable structures. Underdense regions become ever emptier, and their deceleration decreases. Regions thus become more differentiated, and the variance of the expansion rate grows. ([Räsänen 03, p. 25](#Rasanen03))

> In an inhomogeneous space, different regions expand at different rates.  Regions with faster expansion rate increase their volume more rapidly, by definition. Therefore the fraction of volume in faster expanding regions
rises, so the average expansion rate can rise ([Räsänen 10, p. 8](#Rasanen10))

> The acceleration is not due to regions speeding up locally, but due to the slower region becoming less represented in the average. First the overdense region brings down the expansion rate, but its fraction of the volume falls because of the slower expansion, so eventually the underdense region takes over and the average expansion rate rises.

> $[...]$ After the overdense region stops being important, the expansion rate will be given by the underdense region alone, and the expansion will again decelerate. Acceleration is a transient phenomenon associated with the volume becoming dominated by the underdense region.

> $[...]$ Whether the expansion accelerates depends on how rapidly the faster expanding regions catch up with the slower ones, roughly speaking by how steeply the $H t$ curve rises.  This is why the variance contributes positively to acceleration: the larger the variance, the bigger the difference between fast and slow regions, and the more rapidly the fast regions take over.

> $[...]$ So there is no ambiguity:  accelerated average expansion due to inhomogeneities is possible. The question is whether the distribution of structures in the universe is such that this mechanism is realised ([Räsänen 10, p. 10](#Rasanen10))

### Data and models

{#AbstractSmollerTempleVogler14} An analytic proof of this qualitative picture is claimed in [Smoller-Temple-Vogler 14](#SmollerTempleVogler14):

> Our analysis is based on the discovery of a closed ansatz for perturbations of the SM during the p$ = 0$ epoch of the Big Bang which triggers instabilities that create unexpectedly large regions of accelerated uniform expansion  within  Einstein’s  original  theory  without  the  cosmological constant.  We prove that these accelerated regions introduce precisely the same range of corrections to redshift vs luminosity as are produced by the cosmological constant in the theory of Dark Energy.

A survey of the field of inhomogeneous cosmology and of attitudes in the community towards open issues, as of 2016, is in [Belejko-Korzyński 16](#BelejkoKorzynski16). More recent survey of the field and the prevailing attitude is in [Sikora-Glod 20, Section 1](#SikoraGlod20).

A similar conclusion is reached in [Colin-Mohayaee-Rameez-Sarkar 19](#SarkarEtAl18), which in [Sarkar 18, slide 44](#Sarkar18) is summarized as follows:

> {#SarkarFlowQuote} There is a dipole in the recession velocities of host galaxies of supernovae $\Rightarrow$ we are in a "bulk flow" stretching out _well_ beyond the expected scale ($\sim 100 Mpc$) at which the universe is expected to become statistically homogeneous. The inference that the Hubble expansion rate is accelerating may be an artefact of the local bulk flow $[...]$ The "standard" assumptions of exact isotropy and homogeneity are _questionable $[...]$

and a potential fallacy in the widely accepted interpretation of the data is claimed, which in [Rameez 19](#Rameez19) is summarized as follows:

> [The Fitting Problem in cosmology](#EllisStoeger87) was written in 1987. In the context of this work and the significant theoretical difficulties involved in inferring fundamental physics from the real Universe, any claims of having measured a cosmological constant from directionally skewed, sparse samples of intrinsically scattered observations should have been taken with a grain of salt.  By honouring this claim with a Nobel Prize, the Swedish Academy may have induced runaway prestige bias in favour of some of the least principled analyses in science, strengthening the confirmation bias that seems prevalent in cosmology.

A conclusion along these lines, albeit weaker, is also reached independently in [Lin-Mack-Hou 19, p. 5](#LinMackHou19):

> some proposals suggest some local/environmental factors ($z \leq 0.03$) can  bias  the  local  determinations.  This would mean the locally measured $H_0$ cannot be interpreted as the global Hubble constant of the homogeneous universe. An example is a local underdense region (Lombriser 2019; Shanks et al. 2019). Recent studies have shown observational evidence supporting a  small-scale local underdense region (Boehringer et al. 2019; Pustilnik et al. 2019), though it has been argued that the likelihood for a local void to substantially affect the local measurement may be low (Kenworthyet al. 2019).   These local factors do not pose a problem tothe standard ΛCDM model at large scales, but instead point to the need for a more detailed description of our local environment to account for such a systematic effect that can shift all  local measurements in the same way. If all local measurements produce high values of $H_0$, it would favor such a local/environmental-factor explanation over systematic effects that may be unique to each observation.

A strong claim is due to [Heinesen-Buchert 20](#HeinesenBuchert20), who present a model of inhomogeneous cosmology that is claimed to be (p. 14):

> a natural and consistent explanation of

> (i) dark energy,

> (ii) the coincidence problem (here conceptually,not quantitatively),

> (iii) positive initial curvature,

> (iv) the small matter density cosmological parameterfound in local probes of the matter density,

> (v) the large angular diameter distance tothe CMB consistent with JLA supernova sample parameter constraints,

> and (vi) the local expansion rate measurements (removal of the 'Hubble tension').

> We believe that this model architecture needs convincing arguments to be rejected as a physically viable show-case, on the basis of which the model ingredients can be improved in order to build a physical cosmology in the future.

### Impact
 {#Impact}

If the apparent small positive [[cosmological constant]] ([[dark energy]]) were but an artefact of neglecting backreaction of inhomogeneities, some theoretical puzzlements regarding [[quantum gravity]] on [[de Sitter spacetimes]] would disappear (see [Rajaraman 16](de+Sitter+spacetime#Rajaraman16) for general discussion and [Danielsson-VanRiet 18, p. 27](#DanielssonVanRiet18) for discussion in terms of [[perturbative string theory vacua]]) and some theoretically attractive models could become phenomenologically viable again, such as "[[Witten's Dark Fantasy]]".

In this context, notice that even a negative cosmological constant has been argued to still be compatible with observation if suitable [[scalar fields]] are present ([Sen, Adil & Sen 2021](anti+de+Sitter+spacetime#SenAdilSen21))

## Numerical simulation
 {#NumericalSimulation}

[[computer experiment|Numerical simulations]] of inhomogeneous cosmology in the required [[relativistic field theory|relativistic]] accuracy is in its infancy (see [Belejko-Korzyński 16, p. 7](#BelejkoKorzynski16)), but includes the following results:
[ADDK 13](#ADDK13),
[Clesse-Roisin-Füzfa 17](#ClesseRoisinFufza17), [ACDDK 17](#ACDDK17), [Montanari-Räsänen 17](#MontanariRasanen17), [Adamek 18](#Adamek18) ("[[gevolution]]").

The conclusion in [Montanari-Räsänen 17, p. 20](#MontanariRasanen17) is as follows:

> $[$the model$]$ shows an increase of the expansion rate of the right order of magnitude, compared to observations, at late times. $[...]$. It is nontrivial that the right order of magnitude in the amplitude and roughly  right timescale of the change in the expansion rate follow simply from the known physics of structure formation. However, the model has shortcomings that would need to be overcome for the results to be more than suggestive.

Dependency of results on the choice of [[gauge fixing]] is highlighted in [ACDDK 17](#ACDDK17):

> We then show numerical results from the fully relativistic weak field $N$-body code _gevolution_. (p.2)

> $[...]$ The conclusion of this work is therefore that there are gauges which are relatively close to what observers measure and in these gauges backreaction is small. We used the example of Poisson gauge, but there would be others, e.g. geodesic light cone gauge [53, 54]. However, comoving synchronous gauge is not well suited to describe observations in the late time clumpy universe.  In this gauge backreaction becomes large and the gauge actually breaks down during structure formation. (p. 4)

The simulations in [Odderskov-Koksbang-Hannestad 16](#OdderskovKoksbangHannestad16), [Macpherson-Lasky-Price 18](#MacphersonLaskyPrice18) show noticeable but small effects of inhomogeneity, possibly explaining parts but not all of the measured discrepancy reported in [Riess et. al. 16](#RiessEtAl16).

Similarly [Adamek 18, slide 14](#Adamek18) on simulations obtained with [[gevolution]]:

> Backreaction is a real phenomenon that...

> - can be quantified accurately with [[computer experiment|numerical experiments]]

> - quantitatively cannot explain observed data without [[dark energy]]

> - may nevertheless be relevant for precision cosmology with
future surveys

For more see also the pointers in [Räsänen 18, slide 7](#Rasanen18).

## The "backreaction debate"
 {#BackreactionDebate}

A seminal theoretical argument that it _is_ consistent to neglect cosmic inhomogeneity was given by [Green-Wald 10](#GreenWald10), [Green-Wald 11](#GreenWald11), [Green-Wald 13](#GreenWald13), [Green-Wald 16](#GreenWald16). This was called into question in [Buchert et al. 15](#BuchertEtAl15), where it is concluded that the issue is more subtle and remains open. The reply to this criticism by [Green-Wald 15](#GreenWald15) is summarized in [Ostrowski-Roukema 15, p. 4](#OstrowskiRoukema15) as follows:

> Green and Wald state that their formalism does not apply to situations when

> $\ast$ the actual metric (e.g., at recent epochs) is far from FLRW; or

> $\ast$ one wishes to construct an effective metric (or other effective quantities) through some averaging procedure

> This, in principle, ends the debate about whether backreaction has been excluded as a dark energy candidate: the Green and Wald formalism does not apply to the main body of backreaction research; backreaction remains a viable dark energy candidate.

Later, [Ostrowski 19](#Ostrowski19) summarizes this as follows:

> Green and Wald formalism, being a special case of two-scale asymptotic homogenization is not applicable to gravitational systems with hierarchical structures

> any features of backreaction (including backreaction being trace-free) based on Green and Wald formalism are unjustified

{#OpenProblem} **Open problem**

 Accordingly, the review [Coley 18, section 3.5](#Coley18) of [[mathematical physics|mathematical]] [[general relativity]] again regards the issue as open:

> An important open question in cosmology is whether averaging of inhomogeneities can lead to significant backreaction effects on very large scales. (p. 28)

Similarly in [Huang-Gao-Xu 19, p. 2](#HuangGaoXu19):

> The lack of solid proof $[...]$ is a more serious concern.

and [CosmoBack 18](#CosmoBack18):

> Indeed, the actual amplitude of the backreaction effect, whether it requires a fully nonlinear general relativistic treatment or whether a perturbative approach is sufficient, the impact of the gauge choice, of coarse-graining, and of averaging procedures in defining the observables are still open problems.

and [Koksbang 19, p. 3](#Koksbang19):

>  Whichever is the case, a theoretical quantification of cosmic backreaction is necessary for getting the foundations of cosmology onto solid ground; the
mathematics clearly shows that in principle backreaction terms affect the overall dynamics
of the Universe.

Indeed, [Smoller-Temple-Vogler 14](#SmollerTempleVogler14) claim an analytic solution which does exhibit inhomogeneity effects mimicking dark energy (see [above](#AbstractSmollerTempleVogler14)) and a similar conclusion is claimed in [Colin-Mohayaee-Rameez-Sarkar 19](#SarkarEtAl18) (see [above](#SarkarFlowQuote)).  Also relativistic numerical simulation, albeit in their infancy, seem to exhibit noticeable backreaction (see [above](#NumericalSimulation)).

A new analytical approach is suggested by [Ginat 20](#Ginat20), where non-negligible but small effects due to ingomogeneity are found.

## Lemaitre-Tolman-Bondi models
 {#LemaitreTolmanBondi}

A particular class of exactly soluable simple examples of inhomogeneous cosmological models are Lemaitre-Tolman-Bondi models. If taken as quasi-realistic models in themselves, these require assuming that we inhabit a position close to a singled-out "center" of the universe, usually the center of an assumed cosmic "void", of low matter density. See e.g. [Moffat 05](#Moffat05), [Enkvist 07](#Enkvist07), [Moffat 16](#Moffat16). Possible observational signatures of this scenario are discussed in [Clifton-Ferreira-Land 08](#CliftonFerreiraLand08)

From [KBC 13](#KBC13):

> In particular, “minimal void” scenarios (e.g., Alexander et al. 2009; Bolejko & Sussman 2011) have shown that very simple models placing the observer near the center of an under-density that is ∼ 300 Mpc in radius and roughly half the density of its surroundings are sufficient to explain the acceleration observed via type Ia supernovae. While these models are simplistic, they make clear that an observer’s location with respect to large-scale structures could have profound implications for that observer’s measurement of cosmological observables.

It has been argued (e.g. [Moffat 16, p. 2](#Moffat16)) that the apparent unlikeliness of such a "spatial coincidence" is relativized in view of the observed  "temporal coincidence" that cosmic acceleration seems to start roughly with the onset of [[structure formation]] (the "coincidence problem" of cosmology), and the perceived fine-tuning of the [[cosmological constant]] required in the [[standard model of cosmology]].

However, this may be over-interpreting the realism of these simple models. According to [Räsänen 03, p. 15](#Rasanen03):

> In order to evaluate the importance of backreaction in the real universe, we need statistical knowledge about complex configurations of dust, not exact information about simplified models.

## Related concepts

* [[structure formation]]

## References

### General

* {#EllisStoeger87} [[George Ellis]], W. Stoeger, _The 'fitting problem' in cosmology_,  Classical and Quantum Gravity, Volume 4, Number 6, 1987 ([doi:10.1088/0264-9381/4/6/025](https://iopscience.iop.org/article/10.1088/0264-9381/4/6/025))

General review:

* {#Ellis18} [[George Ellis]], summary talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org/) ([[EllisCosmoBack18.pdf:file]])

* {#AluriEtAl22} Pavan Kumar Aluri et al.: *Is the Observable Universe Consistent with the Cosmological Principle?*, Classical and Quantum Gravity **40** 9 (2023) [[arXiv:2207.05765](https://arxiv.org/abs/2207.05765), [doi:10.1088/1361-6382/acbefc](https://doi.org/10.1088/1361-6382/acbefc)]

On [[computer experiment]]:

* {#Adamek18} Julian Adamek, _The Numerical Challenge -- Backreaction in Relativistic N-body Simulations of Cosmic Structure Formation_, talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org/) ([[AdamekCosmoBack18.pdf:file]])

  (on [[gevolution]])

See also

* Wikipedia, _[Inhomogeneous cosmology](https://en.wikipedia.org/wiki/Inhomogeneous_cosmology)_

* Wikipedia, _[Accelerating expansion of the universe -- Alternative theories](https://en.wikipedia.org/wiki/Accelerating_expansion_of_the_universe#Alternative_theories)_

Critical review of [[dark energy]] in the standard model of cosmology:

* {#MRS21} [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], *Do supernovae indicate an accelerating universe?* ([arXiv:2106.03119](https://arxiv.org/abs/2106.03119))

### Theory of backreaction

* {#BuchertEhlers95} [[Thomas Buchert]], Juergen Ehlers, _Averaging inhomogeneous Newtonian cosmologies_, Astron. Astrophys.320:1-7, 1997 ([arXiv:astro-ph/9510056](https://arxiv.org/abs/astro-ph/9510056))

* {#Buchert00} [[Thomas Buchert]], _On average properties of inhomogeneous cosmologies_, Gen.Rel.Grav.9:306-321, 2000 ([arXiv:gr-qc/0001056](https://arxiv.org/abs/gr-qc/0001056))

* {#BuchertLarenaAlimi06} [[Thomas Buchert]], Julien Larena, Jean-Michel Alimi, _Correspondence between kinematical backreaction and scalar field cosmologies - the 'morphon field'_, Class. Quant. Grav.23:6379-6408, 2006 ([arXiv:gr-qc/0606020](https://arxiv.org/abs/gr-qc/0606020))

* {#Rasanen08} [[Syksy Räsänen]], _Evaluating backreaction with the peak model of structure formation_, JCAP 0804:026,2008 ([arXiv:0801.2692](https://arxiv.org/abs/0801.2692))

* {#KolbMarraMatarrese10} Edward W. Kolb, Valerio Marra, Sabino Matarrese, _Cosmological background solutions and cosmological backreactions_, Gen.Rel.Grav.42:1399-1412, 2010 ([arXiv:0901.4566](https://arxiv.org/abs/0901.4566))

* {#Rasanen10} [[Syksy Räsänen]], _Backreaction as an alternative to dark energy and modified gravity_ ([arXiv:1012.0784](https://arxiv.org/abs/1012.0784))

* {#GreenWald10} Stephen R. Green, [[Robert Wald]], _A new framework for analyzing the effects of small scale inhomogeneities in cosmology_, Phys.Rev.D83:084020, 2011 ([arXiv:1011.4920](https://arxiv.org/abs/1011.4920))

* {#Buchert11} [[Thomas Buchert]], _Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects_, Class.Quant.Grav.28:164007, 2011 ([arXiv:1103.2016](https://arxiv.org/abs/1103.2016))

* {#GreenWald11} Stephen R. Green, [[Robert Wald]], _Newtonian and Relativistic Cosmologies_, Phys.Rev.D85:063512, 2012 ([arXiv:1111.2997](https://arxiv.org/abs/1111.2997))

* {#GreenWald13} Stephen Green, [[Robert Wald]], _Examples of backreaction of small scale inhomogeneities in cosmology_, Phys.Rev.D87:124037, 2013 ([arxiv:1304.2318](https://arxiv.org/abs/1304.2318))

* {#ADDK13} Julian Adamek, David Daverio, Ruth Durrer, Martin Kunz, _General Relativistic N-body simulations in the weak field limit_, Phys. Rev. D 88, 103527 (2013) ([arxiv:1308.6524](https://arxiv.org/abs/1308.6524))

* {#GreenWald15} Stephen R. Green, [[Robert Wald]], _Comments on Backreaction_ ([arXiv:1506.06452](https://arxiv.org/abs/1506.06452))

* {#BuchertEtAl15} [[Thomas Buchert]] et. al, _Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?_, Class. Quantum Grav. 32 215021, 2015 ([arXiv:1505.07800](https://arxiv.org/abs/1505.07800))

  exposition in _[The Universe is inhomogeneous. Does it matter?](https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/)_ CQG+, 2016

* {#OstrowskiRoukema15} Jan J. Ostrowski, Boudewijn F. Roukema, _On the Green and Wald formalism_, The Fourteenth Marcel Grossmann Meeting ([arXiv:1512.02947](https://arxiv.org/abs/1512.02947), [talk slides pdf](https://cosmoback.sciencesconf.org/data/program/Ostrowski.pdf))

* {#GreenWald16} Stephen Green, [[Robert Wald]], _A Simple, Heuristic Derivation of our "No Backreaction" Results_, Classical and Quantum Gravity, Volume 33, Number 12, 2016 ([arXiv:1601.06789](https://arxiv.org/abs/1601.06789))

* {#MontanariRasanen17} Francesco Montanari, [[Syksy Räsänen]], _Evaluating backreaction with the ellipsoidal collapse model_, JCAP12(2017)008 ([arXiv:1710.02451](https://arxiv.org/abs/1710.02451))

* {#Coley18} [[Alan Coley]], section 3.5 of _Mathematical General Relativity_ ([arXiv:1807.08628](https://arxiv.org/abs/1807.08628))

* {#HuangGaoXu19} Zhiqi Huang, Han Gao, Haoting Xu, _Revisiting Ryskin's Model of Cosmic Acceleration_ ([arXiv:1905.02441](https://arxiv.org/abs/1905.02441))

* {#Koksbang19} S. M. Koksbang, _Towards statistically homogeneous and isotropic perfect fluid universes with cosmic backreaction_, Class. Quantum Grav. 36 185004, 2019 ([arxiv:1907.08681](https://arxiv.org/abs/1907.08681))

* {#CosmoBack18} [CosmoBack 2018](https://cosmoback.sciencesconf.org/) , _From inhomogeneous gravity to cosmological backreaction. Theoretical opportunity? Observational evidence?_

* {#Ostrowski19} Jan Ostrowski, _Green and Wald formalism: The aftermath of the "backreaction debate"_, talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org)
([[OstrowskiBackreactionDebate19.pdf:file]])

* {#Rasanen18} [[Syksy Räsänen]], _Ways to settle the backreaction conjecture_, talk at [CosmoBack 2018](https://cosmoback.sciencesconf.org) ([[RasanenWaysToSettle18.pdf:file]])

* {#Macpherson19} Hayley J. Macpherson, _Inhomogeneous cosmology in an anisotropic Universe_ ([arxiv:1910.13380](https://arxiv.org/abs/1910.13380))

* {#Ginat20} Yonadav Barry Ginat, _Multiple-Scales Approach for Addressing The Averaging Problem in Cosmology_ ([arXiv:2005.03026](https://arxiv.org/abs/2005.03026))

* Martin Bojowald, Ding Ding, _Canonical description of cosmological backreaction_ ([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

In relation to [[structure formation]] during [[cosmic inflation]]:

* [[Martin Bojowald]], Ding Ding, *Canonical description of cosmological backreaction*, JCAP **03** (2021) 083
([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

### Effective dark energy from inhomogeneity
{#ReferencesOnEffectiveDarkEnergyFromInhomogeneity}

The proposal that backreaction of cosmic inhomogeneities may mimic a [[cosmological constant]]/[[dark energy]] has been discussed in the following articles:

* {#Celerier00} Marie-Noëlle Célérier, _Do we really see a cosmological constant in the supernovae data?_, Astron. Astrophys. 353:63-71, 2000 ([arxiv:astro-ph/9907206](https://arxiv.org/abs/astro-ph/9907206))

* {#Wetterich01} [[Christof Wetterich]], _Can Structure Formation Influence the Cosmological Evolution?_, Phys.Rev. D67 (2003) 043513 ([arXiv:astro-ph/0111166](https://arxiv.org/abs/astro-ph/0111166))

* {#Schwarz02} Dominik J. Schwarz, _Accelerated expansion without dark energy_ ([arXiv:astro-ph/0209584](https://arxiv.org/abs/astro-ph/0209584))

* {#Rasanen03} [[Syksy Räsänen]], _Dark energy from backreaction_, JCAP 0402:003, 2004 ([arXiv:astro-ph/0311257](https://arxiv.org/abs/astro-ph/0311257))

* {#AlnesAmarzguiouiGron06} H. Alnes, M. Amarzguioui and O. Gron, _An inhomogeneous alternative to dark energy?_, Phys. Rev. D 73, 083519 (2006) ([arXiv:astro-ph/0512006](https://arxiv.org/abs/astro-ph/0512006))

* {#EnqvistMattsson06} Kari Enqvist, Teppo Mattsson, _The effect of inhomogeneous expansion on the supernova observations_, JCAP 0702:019,2007 ([arXiv:astro-ph/0609120](https://arxiv.org/abs/astro-ph/0609120))

* {#AlnesAmarzguioui06} Havard Alnes, Morad Amarzguioui, _The supernova Hubble diagram for off-center observers in a spherically symmetric inhomogeneous universe_, Phys. Rev. D75:023506, 2007 ([arXiv:astro-ph/0610331](https://arxiv.org/abs/astro-ph/0610331))

* {#Buchert07} [[Thomas Buchert]], _Dark Energy from structure: a status report_, Gen.Rel.Grav.40:467-527, 2008 ([arXiv:0707.2153](http://xxx.lanl.gov/abs/0707.2153))

* {#Sarkar08} [[Subir Sarkar]], _Is the evidence for dark energy secure?_, Gen. Rel. Grav.40:269-284, 2008 ([arXiv:0710.5307](https://arxiv.org/abs/0710.5307))

* {#Notari09} Alessio Notari, _Can an Inhomogeneous Universe mimic Dark Energy?_, 2009 ([[NotariInhomogeneousCosmology.pdf:file]])

* {#Blomqvist10} Michael Blomqvist, _Inhomogeneous cosmologies with clustered dark energy or a local matter void_,  2010 ([web](http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A353689&dswid=2010))

* {#BuchertRasanen11} [[Thomas Buchert]], [[Syksy Räsänen]], _Backreaction in late-time cosmology_, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 ([arXiv:1112.5335](https://arxiv.org/abs/1112.5335))

* {#SmollerTempleVogler14} Joel Smoller, Blake Temple, Zeke Vogler, _An Instability of the Standard Model Creates the Anomalous Acceleration Without Dark Energy_, Proceedings of the Royal Society A, 2017 ([arXiv:1412.4001](https://arxiv.org/abs/1412.4001), [10.1098/rspa.2016.0887](http://rspa.royalsocietypublishing.org/content/473/2207/20160887), detailed talk slides: [[Temple16.pdf:file]], talk [recording I](https://www.youtube.com/watch?v=fV8KPj8vmGw), [recording II](http://cdsweb.cern.ch/record/1371553))

* {#Ludwick15} Kevin J. Ludwick, _Examining the Viability of Phantom Dark Energy_, Phys. Rev. D 92, 063019 (2015) ([arXiv:1507.06492](https://arxiv.org/abs/1507.06492))

  (on [[phantom dark energy]])

* {#OdderskovKoksbangHannestad16} I. Odderskov, S. M. Koksbang, S. Hannestad, _The Local Value of $H_0$ in an Inhomogeneous Universe_, JCAP02(2016)001 ([arXiv:1601.07356](https://arxiv.org/abs/1601.07356))

* {#RiessEtAl16} [[Adam G. Riess]] et al., _A 2.4% Determination of the Local Value of the Hubble Constant_,  The Astrophysical Journal **826** 1 (2016) [[arXiv:1604.01424](https://arxiv.org/abs/1604.01424), [doi:10.3847/0004-637X/826/1/56](https://iopscience.iop.org/article/10.3847/0004-637X/826/1/56)]

* {#BelejkoKorzynski16} Krzysztof Bolejko, Mikołaj Korzyński, _Inhomogeneous cosmology and backreaction: Current status and future prospects_, Int. J. Mod. Phys. D 26, 1730011 (2017) ([arXiv:1612.08222](https://arxiv.org/abs/1612.08222))

* {#ClesseRoisinFufza17} Sebastien Clesse, Arnaud Roisin, André Füzfa, _Mimicking Dark Energy with the backreactions of gigaparsec inhomogeneities_ ([arXiv:1702.06643](https://arxiv.org/abs/1702.06643))

* {#ACDDK17} Julian Adamek, Chris Clarkson, David Daverio, Ruth Durrer, Martin Kunz, _Safely smoothing spacetime: backreaction in relativistic cosmological simulations_ ([arXiv:1706.09309](https://arxiv.org/abs/1706.09309))

* {#Bolejko17} Krzysztof Bolejko, _Emerging spatial curvature can resolve the tension between high-redshift CMB and low-redshift distance ladder measurements of the Hubble constant_, Phys. Rev. D 97, 103529 (2018) ([arxiv:1712.02967](https://arxiv.org/abs/1712.02967))

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

* {#Ludwick18} Kevin J. Ludwick, _The Viability of Phantom Dark Energy as a Quantum Field in 1st-Order FLRW Space_, Phys. Rev. D 98, 043519 (2018) ([arXiv:1804.02987](https://arxiv.org/abs/1804.02987))

  (on [[phantom dark energy]])

* {#MacphersonLaskyPrice18} Hayley Macpherson, Paul D. Lasky, Daniel J. Price, _The trouble with Hubble: Local versus global expansion rates in inhomogeneous cosmological simulations with numerical relativity_,  ApJ Letters ([arXiv:1807.01714](https://arxiv.org/abs/1807.01714))

* {#SarkarEtAl18} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _Evidence for anisotropy of cosmic acceleration_, Astronomy & Astrophysics **631** L13 (2019) doi:[10.1051/0004-6361/201936373](https://doi.org/10.1051/0004-6361/201936373), ([arXiv:1808.04597](https://arxiv.org/abs/1808.04597))

* {#Sarkar18} [[Subir Sarkar]], _Is the universe isotropic?_, talk at _[Current Themes in High Energy Physics and Cosmology 2018](https://indico.nbi.ku.dk/event/973/)_ ([pdf](https://indico.nbi.ku.dk/event/973/contributions/8344/attachments/2648/3858/Sarkar_Copenhagen18_cosmo.pdf))

* {#Rameez19} [[Mohamed Rameez]], _On the Real Inhomogeneous Universe and the Weirdness of 'Dark Energy'_ ([Nov. 14, 2019](https://4gravitons.com/2019/11/15/guest-post-on-the-real-inhomogeneous-universe-and-the-weirdness-of-dark-energy/))

* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))

* {#Lombriser19} Lucas Lombriser, _On the cosmological constant problem_, Phys. Lett. B 797, 134804 (2019) ([arXiv:1901.08588](https://arxiv.org/abs/1901.08588), [doi:10.1016/j.physletb.2019.134804](https://doi.org/10.1016/j.physletb.2019.134804))

* {#Deledicque19} Vincent Deledicque, _Theoretical developments on the adequacy of the fitting of the FLRW metric on the universe's real metric_ ([arxiv:1907.01580](https://arxiv.org/abs/1907.01580))

* {#LinMackHou19} Weikang Lin, Katherine J. Mack, Liqiang Hou, _Investigating the Hubble Constant Tension -- Two Numbers in the Standard Cosmological Model_ ([arXiv:1910.02978](https://arxiv.org/abs/1910.02978))

* Léo Brunswic, [[Thomas Buchert]], _Gauss-Bonnet-Chern approach to the averaged Universe_, Class. Quantum Grav. 37 (2020) 215022 ([arXiv:2002.08336](https://arxiv.org/abs/2002.08336))

* {#HeinesenBuchert20} Asta Heinesen, [[Thomas Buchert]], _Solving the curvature and Hubble parameter inconsistencies through structure formation-induced curvature_, Class. Quantum Grav. 37 (2020) 164001 ([arXiv:2002.10831](https://arxiv.org/abs/2002.10831))

* {#SikoraGlod20} Szymon Sikora, Krzysztof Głód, _Construction of the cosmological model with periodically distributed inhomogeneities with growing amplitude_ ([arXiv:2005.06385](https://arxiv.org/abs/2005.06385))

### No effective dark energy from inhomogeneity
 {#ReferencesNoEffect}

In constrast, arguments that cosmic inhomogeneity can not be the cause of any sizeable amount of effective dark energy are advanced in the following articles:

* {#GeshnizjaniChungAfshordi05} Ghazal Geshnizjani, Daniel J.H. Chung, Niayesh Afshordi, _Do Large-Scale Inhomogeneities Explain Away Dark Energy?_, Phys.Rev. D72 (2005) 023517 ([arXiv:astro-ph/0503553](https://arxiv.org/abs/astro-ph/0503553))

* {#SiegelFry05} E. R. Siegel, J. N. Fry, _Effects of Inhomogeneities on Cosmic Expansion_, Astrophys.J. 628 (2005) L1-L4 ([arXiv:astro-ph/0504421](https://arxiv.org/abs/astro-ph/0504421))

* {#Flanagan05} Eanna E. Flanagan, _Can superhorizon perturbations drive the acceleration of the Universe?_, Phys.Rev. D71 (2005) 103521 ([arXiv:hep-th/0503202](https://arxiv.org/abs/hep-th/0503202))

* {#MarozziUzan12} Giovanni Marozzi, Jean-Philippe Uzan, _Late time anisotropy as an imprint of cosmological backreaction_ ([arXiv:1206.4887](https://arxiv.org/abs/1206.4887))

* {#BDGMNV13} Ido Ben-Dayan, Maurizio Gasperini, Giovanni Marozzi, Fabien Nugier, [[Gabriele Veneziano]], _Do stochastic inhomogeneities affect dark-energy precision measurements?_, Phys. Rev. Lett. 110, 021301 (2013) ([arXiv:1207.1286](https://arxiv.org/abs/1207.1286))

### Lemaitre-Tolman-Bondi models

* {#Moffat05} [[John Moffat]], _Late-time Inhomogeneity and Acceleration Without Dark Energy_, JCAP 0605 (2006) 001 ([arXiv:astro-ph/0505326](https://arxiv.org/abs/astro-ph/0505326))

* {#Enkvist07} Kari Enqvist, _Lemaitre-Tolman-Bondi model and accelerating expansion_, Gen. Rel. Grav.40:451-466, 2008 ([arXiv:0709.2044](https://arxiv.org/abs/0709.2044))

* {#CliftonFerreiraLand08} Timothy Clifton, Pedro G. Ferreira, Kate Land, _Living in a Void: Testing the Copernican Principle with Distant Supernovae_, Phys. Rev. Lett. 101 (2008) 131302 ([arXiv:0807.1443](https://arxiv.org/abs/0807.1443))

* {#Scharf13} [[Günter Scharf]], _Inhomogeneous cosmology in the cosmic rest frame without dark stuff_, chapter 6 in the latest edition of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001 ([arXiv:1312.2695](https://arxiv.org/abs/1312.2695))

* {#Moffat16} [[John Moffat]], _Inhomogeneous Cosmology Redux_ ([arXiv:1608.00534](https://arxiv.org/abs/1608.00534))

See also

* Wikipedia, _[Lemaître–Tolman metric](https://en.wikipedia.org/wiki/Lema%C3%AEtre%E2%80%93Tolman_metric)_

### Observed inhomogeneity

Introducing observation of the [[KBC void]]:

* {#KBC13} Ryan C. Keenan, Amy J. Barger, Lennox L. Cowie, _Evidence for a ~300 Mpc Scale Under-density in the Local Galaxy Distribution_, 2013, ApJ, 775, 62 ([arXiv:1304.2884](https://arxiv.org/abs/1304.2884))

Its potential relevance for the [[Hubble tension]] (apparently too small):

* Hao-Yi Wu, Dragan Huterer, _Sample variance in the local measurements of the Hubble constant_, Monthly Notices of the Royal Astronomical Society, Volume 471, Issue 4, November 2017, Pages 4946–4955 ([arXiv:1706.09723](https://arxiv.org/abs/1706.09723))

Anisotropy:

* {#SHRMS22} Nathan Secrest, Sebastian von Hausegger, [[Mohamed Rameez]], [[Roya Mohayaee]], [[Subir Sarkar]], *A Challenge to the Standard Cosmological Model*, ApJ Letters (2022) [[arXiv2206.05624](https://arxiv.org/abs/2206.05624)]

  > The joint significance of this rejection of the cosmological principle is 5.2σ.

Applying [[topological data analysis]]/[[persistent homology]] to the [[CMB]]:

* Pratyush Pranav, [[Thomas Buchert]], *Homology reveals significant anisotropy in the cosmic microwave background* ([arXiv:2308.10738](https://arxiv.org/abs/2308.10738))

Survey of deviations:

* {#Peebles22} [[Phillip James E. Peebles]], *Anomalies in Physical Cosmology* [[arXiv:2208.05018](https://arxiv.org/abs/2208.05018)]

See also:

* {#WhitePaper22} Snowmass White Paper, Section VII.H in: *Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies*, J. High En. Astrophys **34** (2022) 49-211 [[arXiv:2203.06142](https://arxiv.org/abs/2203.06142), [doi:10.1016/j.jheap.2022.04.002](https://doi.org/10.1016/j.jheap.2022.04.002)]

  > "it is possible that data has reached a requisite precision that the cosmological principle has already become obsolete"

* {#HaslbauerKroupaJerabkova23} Moritz Haslbauer, Pavel Kroupa, Tereza Jerabkova, *The cosmological star formation history from the Local Volume of galaxies and constraints on the matter homogeneity* [[arXiv:2306.16436](https://arxiv.org/abs/2306.16436)]

[[!redirects inhomogeneous cosmologies]]

[[!redirects Lemaitre-Tolman-Bondi cosmology]]
[[!redirects Lemaitre-Tolman-Bondi model]]
[[!redirects Lemaitre-Tolman-Bondi models]]

[[!redirects LTB model]]
[[!redirects LTB models]]


initial > history

< [[initial]]
[[!redirects initial -- history]]


initial algebra

# Initial algebras
* table of contents
{: toc}

## Idea

An initial algebra is an [[initial object]] in a [[category]] (or [[higher category]]) of [[algebras]].

There are many meanings of 'algebra', so many meanings of 'initial algebra'.  Most cases are fairly vacuous, but we could probably list them here.  Some cases, however, are rather important and interesting in the topic of [[induction]] and [[recursion]]:

* [[initial algebra of an endofunctor]],
* [[initial algebra of a presentable (infinity,1)-monad]].

## Examples

List them here if you like:

* (...)

[[!redirects initial algebras]]


initial algebra of a presentable (infinity,1)-monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[initial object in an (∞,1)-category|initial object]] in an [[(∞,1)-category]] of [[∞-algebras over an (∞,1)-monad]] which is [[generators and relations|presentable]].

These are thought to be the [[categorical semantics]] of [[higher inductive types]].

On [[truncated object of an (∞,1)-category|0-truncated]] objects the notion restricts to that of an [[initial algebra of an endofunctor]].

## Related concepts

* [[inductive type]], [[initial algebra of an endofunctor]]

* [[higher inductive type]]

## References

* [[Peter Lumsdaine]] and [[Mike Shulman]], Semantics of higher inductive types, 2017, [arxiv](https://arxiv.org/abs/1705.07088)

[[!redirects initial algebra of a presentable ∞-monad]]

[[!redirects initial algebras of a presentable ∞-monad]]
[[!redirects initial algebras of a presentable infinity-monad]]

[[!redirects initial algebra of a presentable infinity-monad]]

[[!redirects initial algebra of a presentable (∞,1)-monad]]
[[!redirects initial algebras of a presentable (∞,1)-monad]]

[[!redirects initial ∞-algebra of a presentable (∞,1)-monad]]
[[!redirects initial ∞-algebras of a presentable (∞,1)-monad]]

[[!redirects initial infinity-algebra of a presentable (infinity,1)-monad]]
[[!redirects initial infinity-algebras of a presentable (infinity,1)-monad]]


initial algebra of an endofunctor

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

# Contents
* table of contents
{: toc}

## Definition

An __initial [[algebra for an endofunctor|algebra]]__ for an [[endofunctor]] $F$ on a [[category]] $C$ is an [[initial object]] in the category of [[algebra over an endofunctor|algebras]] of $F$. These play a role in particular as the [[categorical semantics]] for [[inductive types]].

## Properties

### Relation to algebras over a monad

The concept of an [[algebra of an endofunctor]] is arguably somewhat odd, a more natural concept being that of an [[algebra over a monad]].  However, the former can often be reduced to the latter.

+-- {: .num_prop}
###### Proposition
If $\mathcal{C}$ is a [[complete category]], then the [[category]] of [[algebras of an endofunctor]] $F : \mathcal{C} \to \mathcal{C}$ is [[equivalence of categories|equivalent]] to the category of [[algebras over a monad]] of the [[free monad]] on $F$, if the latter exists.
=--

The proof is fairly straightforward, see for instance ([Maciej](#Maciej)) or at [[free monad]].

The existence of free monads, on the other hand, can be a tricky question.  One general technique is the [[transfinite construction of free algebras]].

### Lambek's theorem
 {#LambeksTheorem}

+-- {: .num_theorem}
###### Theorem

If $F$ has an initial algebra $\alpha: F(X) \to X$, then $X$ is [[isomorphism|isomorphic]] to $F(X)$ via $\alpha$.
=--

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

In this sense, $X$ is a [[fixed point]] of $F$.  Being initial, $X$ is the *smallest* fixed point of $F$ in that there is a map from $X$ to any other fixed point (indeed, any other algebra), and this map is an [[injection]] if $C$ is [[Set]].
=--

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

The dual concept is [[terminal coalgebra]], which is the *largest* fixed point of $F$.
=--

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

Given an initial algebra structure $\alpha: F(X) \to X$, define an algebra structure on $F(X)$ to be $F(\alpha): F(F(X)) \to F(X)$. By initiality, there exists an $F$-algebra map $i: X \to F(X)$, so that

$$\array{
F(X) & \overset{F(i)}{\to} & F(F(X)) \\
\alpha \downarrow & & \downarrow F(\alpha) \\
X & \underset{i}{\to} & F(X)
}$$

commutes. Now it is trivial, in fact tautological that $\alpha$ is itself an $F$-algebra map $F(X) \to X$. Thus $\alpha \circ i = 1_X$, since both sides of the equation are $F$-algebra maps $X \to X$ and $X$ is initial. As a result, $F(\alpha) \circ F(i) = 1_{F(X)}$, so that $i \circ \alpha = 1_{F(X)}$ according to the commutative square. Hence $\alpha$ is an isomorphism, with inverse $i$.
=--

### Adámek's theorem
 {#AdameksTheorem-sec}

In many cases, initial algebras can be constructed in a [[recursion|recursive/iterative]] fashion using [[Adámek's fixed point theorem]].

### Semantics for inductive types

Initial algebras of endofunctors provide [[categorical semantics]] for [[extensional type theory|extensional]] [[inductive types]].
The generalization to [[weak initial]] algebras captures the notion in [[intensional type theory]] and [[homotopy type theory]].

## Examples

### Natural numbers
 {#NaturalNumbers}

The archetypical example of an initial algebra is the set of [[natural numbers]] [[natural numbers object]].

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

Let $\mathcal{T}$ be  [[topos]] and let $F : \mathcal{T} \to \mathcal{T}$ the functor given by

$$
  F : X \mapsto * \coprod X
$$

(i.e. the functor underlying the "[[maybe monad]]").
Then an initial algebra over $F$ is precisely a [[natural number object]] $\mathbb{N}$ in $\mathcal{T}$.
=--

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

By definition, an $F$-algebra is an object $X$ equipped with a morphism

$$
  (0,s) :  * \coprod X \to X
  \,,
$$

hence equivalently with a [[pointed object|point]] $0 : * \to X$ and an [[endomorphism]] $s : X \to X$. Initiality means that for any other $F$-algebra $(0_Y, s_Y) : * \coprod Y \to Y$, there is a unique morphism $f : X \to Y$ such that the [[diagram]]

$$
  \array{
    * &\stackrel{0}{\to}& X &\stackrel{s}{\to}& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}}
    \\
    * &\stackrel{0}{\to}& Y &\stackrel{s}{\to}& Y
  }
$$

commutes. This is the very definition of [[natural number object]] $X = \mathbb{N}$.
=--

### Bi-pointed sets

{#BiPointedSets} Another example of an initial algebra is the [[bi-pointed set]] $\mathbf{2}$.

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

Let $\mathcal{T}$ be  [[topos]] and let $F : \mathcal{T} \to \mathcal{T}$ the functor given by

$$
  F : X \mapsto * \coprod *
$$

Then an initial algebra over $F$ is precisely a [[bi-pointed object]] $\mathbf{2}$ in $\mathcal{T}$.
=--

### More examples

Theorem \ref{AdameksTheorem} applies (in particular) to any functor $F: Set \to Set$ which is a [[colimit]] of finitely [[representable functors]] $hom(n, -): X \mapsto X^n$, as in the following examples.

\begin{example}
Let $A$ be a set, and let $F: Set \to Set$ be the functor $F(X) = 1 + A \times X$. Then the initial $F$-algebra is $A^*$, the [[free monoid]] on $A$. The $F$-algebra structure is
  $$(e, m| ): 1 + A \times A^* \to A^*$$
  where $e: 1 \to A^*$ is the identity and $m|: A \times A^* \to A^*$ is the restriction of the monoid multiplication along the evident inclusion $i \times 1: A \times A^* \to A^* \times A^*$.

  This "fixed point" of $F$ can be thought of as the result of the (slightly nonsensical) calculation
  $$1 + A \times X = X \Rightarrow X = \frac1{1 - A} = 1 + A + A^2 + \ldots = A^*$$
  which can be made rigorous by interpreting the initial equality as defining the solution $X$ by recursion, and applying the theorem above.
\end{example

\begin{example}
Let $F: Set \to Set$ be the functor $F(X) = 1 + X^2$. Then the initial $F$-algebra is the set $T$ of isomorphism classes of finite (planar, rooted) binary [[trees]]. The $F$-algebra structure is
  $$(r, j): 1 + T^2 \to T$$
  where $r: 1 \to T$ names the tree consisting of just a root vertex, and $j: T^2 \to T$ creates a tree $t \vee t'$ from two trees $t$, $t'$ by joining their roots to a new root, so that the root of $t$ becomes the left child and the root of $t'$ the right child of the new root.

  The recursive equation
  $$T = 1 + T^2$$
  would seem to imply that the structure $T$ behaves something like a structural "sixth root of unity", and indeed the structural isomorphism $T \cong F(T)$ allows one to exhibit an isomorphism
  $$T = T^7$$
  constructively, as famously explored by Andreas Blass in the paper "[Seven trees in one](#BlassSeven)".
\end{example}

\begin{example}
Let $F: Set \to Set$ be the functor $F(X) = X^*$ (the free monoid from an earlier example). Then the initial $F$-algebra is the set of isomorphism classes of finite planar rooted [[trees]] (not necessarily binary as in the previous example).
\end{example}

\begin{example}
Let $C$ be the [[coslice category]] $\mathbb{Z} \downarrow Ab$, and let $F: C \to C$ be the functor which pushes out along the multiplication-by-$p$ map $p \cdot -: \mathbb{Z} \to \mathbb{Z}$. Then the initial $F$-algebra is the [[Pruefer group]] $\mathbb{Z}[p^{-1}]/\mathbb{Z}$. See the discussion at the n-Category Café, starting [here](http://golem.ph.utexas.edu/category/2008/11/coalgebraically_thinking.html#c020660).
\end{example}

\begin{example}
Let $Ban$ be the category of complex [[Banach spaces]] and maps of [[norm]] bounded above by $1$, and let $F: \mathbb{C} \downarrow Ban \to \mathbb{C} \downarrow Ban$ be the squaring functor $X \mapsto X \times X$. Then the initial $F$-algebra is $L^1([0, 1])$ (with respect to the usual [[Lebesgue measure]]). This result is due to [[Tom Leinster]]; see this [MathOverflow discussion](http://mathoverflow.net/questions/23143/what-theorem-constructs-an-initial-object-for-this-category-formerly-integrabi/78262#78262).
\end{example}

[[!include list of notable initial algebras and terminal coalgebras]]

## Related entries

* [[list of notable initial algebras and terminal coalgebras]]

* [[inductive type]]

* [[inductive-recursive type]],  [[algebraically compact category]]

* [[higher inductive type]], [[initial algebra of a presentable ∞-monad]]

## References

### General

* {#BlassSeven} [[Andreas Blass]], _Seven Trees in One_ ([arXiv:9405205](https://arxiv.org/abs/math/9405205))

A textbook account of the basic theory is in [Chapter 10](http://www.andrew.cmu.edu/course/80-413-713/notes/chap10.pdf) of

* [[Steve Awodey]], _Category theory_ lecture notes (2011) ([web](http://www.andrew.cmu.edu/course/80-413-713/))

The relation to [[free monads]] is discussed in

* {#Maciej} [Maciej](http://maciejcs.wordpress.com/), _[Free monads and their algebras](http://maciejcs.wordpress.com/2012/04/17/free-monads-and-their-algebras/)_

Original references on initial algebras include

* {#Pohlova} Věra Pohlová. "On sums in generalized algebraic categories." Czechoslovak Mathematical Journal 23.2 (1973): 235-251. <http://eudml.org/doc/12718>.

* [[Jiří Adámek]], *Free algebras and automata realizations in the language of categories*, Commentationes Mathematicae Universitatis Carolinae 15.4 (1974): 589-602.

* [[Jiří Adámek]], Věra Trnková, *Automata and algebras in categories* Vol. 37. Springer Science & Business Media, 1990.

[[!include semantics of W-types -- references]]

[[!redirects initial algebras of an endofunctor]]
[[!redirects initial algebras of endofunctors]]

[[!redirects initial algebra for an endofunctor]]
[[!redirects initial algebras for an endofunctor]]
[[!redirects initial algebras for endofunctors]]

[[!redirects initial algebra over an endofunctor]]
[[!redirects initial algebras over an endofunctor]]
[[!redirects initial algebras over endofunctors]]

[[!redirects Lambek’s theorem]]
[[!redirects Lambek's theorem]]


initial object

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

# Contents
* table of contents
{: toc}

## Definition

\begin{definition}
An **initial object** in a [[category]] $\mathcal{C}$ is an [[object]] $\emptyset$ such that for all objects $x \,\in\, \mathcal{C}$, there is a unique [[morphism]]
$\varnothing \xrightarrow{\exists !} x$
with
[[source]] $\varnothing$.
and
[[target]] $x$.
\end{definition}

\begin{remark}
An initial object, if it exists, is unique up to unique [[isomorphism]], so that we may speak of [[the]] initial object.
\end{remark}

\begin{remark}\label{InitialObjectIsEmptyColimit}
When it exists, the initial object is the [[colimit]] over the [[empty diagram]].
\end{remark}

\begin{remark}
Initial objects are also called _coterminal_, and (rarely, though): _coterminators_, _universal initial_, _co-universal_, or simply _universal_.
\end{remark}

\begin{definition}
An initial object $\varnothing$ is called a **[[strict initial object]]** if all morphisms $x \xrightarrow{\;} \varnothing$ into it are [[isomorphisms]].
\end{definition}

\begin{remark}
Initial objects are the [[duality|dual]] concept to [[terminal objects]]: an initial object in $C$ is the same as a terminal object in the [[opposite category]] $C^{op}$.
\end{remark}

\begin{remark}
An object that is both initial and [[terminal object|terminal]] is called a [[zero object]].
\end{remark}

## Examples

* An initial object in a [[partial order|poset]] is a [[bottom element]].

* The [[empty set]] is an initial object in [[Set]].

* Likewise, the [[empty category]] is an initial object in [[Cat]], the [[empty space]] is an initial object in [[Top]], and so on.

* The [[trivial group]] is the initial object (in fact, the [[zero object]]) of [[Grp]] and [[Ab]].

* The [[ring]] of [[integers]] $\mathbb{Z}$ is the initial object of [[Ring]].

* The [[field]] of [[rational numbers]] $\mathbb{Q}$ is the initial object of $Field_0$ (category of fields with [[characteristic]] $0$) and the [[prime field]] $\mathbb{F}_p$ is the initial object of $Field_p$ (category of fields with characteristic $p$), but none are the initial object of [[Field]] (category of all fields), which actually doesn't have one at all.

* The terminal object of a [[under category|coslice category]] $x/C$ is the [[identity morphism]] $x \to x$.

* An initial object in a category of [[central extensions]] of a given algebraic object is called a _[[universal central extension]]_.

## Properties

### Left adjoints to constant functors

\begin{proposition}
\label{AdjointsToConstantFunctors}
Let $\mathcal{C}$ be a [[category]].

1. The following are equivalent:

   1. $\mathcal{C}$ has a [[terminal object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[right adjoint]]

      $$
        \ast
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \mathcal{C}
      $$

    Under this equivalence, the [[terminal object]] is identified with the image under the right adjoint of the unique object of the [[terminal category]].

1. Dually, the following are equivalent:

   1. $\mathcal{C}$ has an [[initial object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[left adjoint]]

      $$
        \mathcal{C}
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \ast
      $$

    Under this equivalence, the [[initial object]] is identified with the image under the left adjoint of the unique object of the [[terminal category]].
\end{proposition}

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

Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|the]] [[singleton]], the hom-isomorphism characterizing the [[adjoint functors]] is directly the [[universal property]] of an [[initial object]] in $\mathcal{C}$

$$
  Hom_{\mathcal{C}}( L(\ast) , X )
  \;\simeq\;
  Hom_{\ast}( \ast, R(X) )
  =
  \ast
$$

or of a [[terminal object]]

$$
  Hom_{\mathcal{C}}( X , R(\ast) )
  \;\simeq\;
  Hom_{\ast}( L(X), \ast ) = \ast
  \,,
$$

respectively.

=--

### Cones over the identity
 {#ConesOverTheIdentity}

By definition, an initial object is equipped with a universal [[cocone]] under the unique functor $\emptyset\to C$ from the [[empty category]].  On the other hand, if $I$ is initial, the unique morphisms $!: I \to x$ form a cone *over* the [[identity functor]], i.e. a natural transformation $\Delta I \to Id_C$ from the [[constant functor]] at the initial object to the identity functor.  In fact this is almost another characterization of an initial object (e.g. [MacLane, p. 229-230](#MacLane)):

+--{: .num_lemma #cone}
###### Lemma

Suppose $I\in C$ is an object equipped with a natural transformation $p:\Delta I \to Id_C$ such that $p_I = 1_I : I\to I$.  Then $I$ is an initial object of $C$.

=--

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

Obviously $I$ has at least one morphism to every other object $X\in C$, namely $p_X$, so it suffices to show that any $f:I\to X$ must be equal to $p_X$.  But the naturality of $p$ implies that $\Id_C(f) \circ p_I = p_X \circ \Delta_I(f)$, and since $p_I = 1_I$ this is to say $f \circ 1_I = p_X \circ 1_I$, i.e. $f=p_X$ as desired.

=--

+-- {: .num_theorem #LimitOverIdentityFunctorIsInitialObject}
###### Theorem

An object $I$ in a [[category]] $C$ is initial iff $I$ is the [[limit]] of the [[identity functor]] $Id_C$.

=--

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

If $I$ is initial, then there is a [[cone]] $(!_X: I \to X)_{X \in Ob(C)}$ from $I$ to $Id_C$.  If $(p_X: A \to X)_{X \in Ob(C)}$ is any cone from $A$ to $Id_C$, then $p_X = f \circ p_Y$ for any $f:Y\to X$, and so in particular $p_X = !_X \circ p_I$.   Since this is true for any $X$, $p_I: A \to I$ defines a morphism of cones, and it is the unique morphism of cones since if $q$ is any morphism of cones, then $p_I = !_I \circ q = 1_I \circ q = q$ (using that $!_I = 1_I$ by initiality). Thus $(!_X: I \to X)_{X \in Ob(C)}$ is the limit cone.

Conversely, if $(p_X: L \to X)_{X \in Ob(C)}$ is a limit cone for $Id_C$, then $f\circ p_Y = p_X$ for any $f:Y\to X$, and so in particular $p_X \circ p_L = p_X$ for all $X$. This means that both $p_L: L \to L$ and $1_L: L \to L$ define morphisms of cones; since the limit cone is the terminal cone, we infer $p_L = 1_L$. Then by Lemma \ref{cone} we conclude $L$ is initial.

=--

+-- {: .num_remark #RelevanceForAdjointFunctorTheorem}
###### Remark
**(relevance for [[adjoint functor theorem]])**

Theorem \ref{LimitOverIdentityFunctorIsInitialObject} is actually a key of entry into the [[adjoint functor theorem|general adjoint functor theorem]]. Showing that a functor $G: C \to D$ has a [[left adjoint]] is tantamount to showing that each functor $D(d, G-)$ is [[representable functor|representable]], i.e., that the [[comma category]] $d \downarrow G$ has an initial object $(c, \theta: d \to G c)$ (see at _[[adjoint functor]]_, [this prop.](#PointwiseExpressionOfLeftAdjoints)).  This is the limit of the identity functor, but typically this is the limit over a large diagram whose existence is not guaranteed. The point of a solution set condition is to replace this with a small diagram which is cofinal in the large diagram.

=--

## Related concepts

* [[terminal object]], [[bi-terminal object]]

* [[bottom type]]

* [[initial object in a 2-category]]

* [[initial object in an (∞,1)-category]]

* [[h-initial object]]

* [[adjoint functor theorem]]

## References

Textbook accounts:

* [[Francis Borceux]], Section 2.3 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

* {#MacLane} [[Saunders MacLane]], _[[Categories for the Working Mathematician]]_

[[!redirects initial object]]
[[!redirects initial objects]]
[[!redirects initial]]
[[!redirects coterminal object]]
[[!redirects coterminal objects]]
[[!redirects coterminator]]
[[!redirects coterminators]]


initial value problem

#Contents#
* table of contents
{:toc}

## Idea

The _initial value problem_ or _Cauchy problem_ for [[partial differential equation|partial]] [[differential equations]] is concerned with the solution to these [[equations]] given "initial value" data on a hypersurface.

## References

* Wikipedia, _[Initial value problem](https://en.wikipedia.org/wiki/Initial_value_problem)_
* Wikipedia, _[Cauchy problem](http://en.wikipedia.org/wiki/Cauchy_problem)_

[[!redirects initial value problems]]
[[!redirects Cauchy problem]]
[[!redirects Cauchy problems]]

[[!redirects initial value]]
[[!redirects initial values]]
[[!redirects initial condition]]
[[!redirects initial conditions]]

[[!redirects initial value data]]


initial Θ-data

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

\tableofcontents

\section{Idea}

The notion of _initial Θ-data_ or _initial theta-data_ was introduced by [[Shinichi Mochizuki]] in §3 of [Inter-Universal Teichmüller theory I](#MochizukiIUTTI) as the starting point for [[inter-universal Teichmüller theory]]. In essence, initial Θ-data consists of an [[elliptic curve]] $E$ over a [[number field]] $F$ such that the [[Galois theory]] of $F$ and the geometry of $E$ interact well.

\section{Notation}

In order to understand the definition, we recall a little notation.

1. Given a [[field]] $F$, the notation $E / F$ means that $E$ is a [[field extension]] of $F$.

1. Given a [[field]] $F$, the notation $\overline{F}$ denotes an [[algebraic closure]] of $F$.

1. Given a [[field]] $F$ and a once-punctured [[elliptic curve]] $X$ over $F$ (see below for a precise definition), the notation $F_{mod}$ denotes the [[field of moduli]] of $X$.

1. Given a field $F$, an [[elliptic curve]] $E$ over $F$, and a [[prime number|prime]] $l \geq 2$, let us write $\phi_l: Gal\left(\overline{F} / F\right) \rightarrow GL_{2}\left( \mathbb{F}_{l} \right)$ for the [[homomorphism|group homomorphism]] from the [[Galois group]] of $F$ to the [[general linear group]] of 2x2 invertible matrices with values in $\mathbb{F}_{l}$ determined by the $l$-torsion points of $E$ (see [[torsion points of an elliptic curve]] for details of this homomorphism).

1. Given a field $F$ and an [[elliptic curve]] $E$ over $F$,
 the notation $K_E$ denotes the [[field]], a [[Galois extension|finite Galois extension]] of $F$, corresponding (by [[Galois theory]]) to the kernel of $\phi_l$.

1. Given a [[number field]] $F$, the notation $\mathbb{V}(F)$ denotes the set of [[absolute value|valuations]] on $F$, both archimedean and non-archimedean.

1. Given a [[number field]] $F$ and a [[absolute value|valuation]] $v \in \mathbb{V}\left(F\right)$ on $F$, the notation $F_{v}$ denotes the [[completion of a field at an absolute value|completion]] of $F$ with respect to $v$.

\section{Definition}

The following is Definition 3.1 in [Inter-Universal Teichmüller theory I](#MochizukiIUTTI), on pg.61 (currently).

\begin{defn} _Initial Θ-data_ is a 7-tuple $(\overline{F} / F, X_{F}, l, \underline{C}_{K}, \underline{\mathbb{V}}, \mathbb{V}^{bad}_{mod}, \underline{\epsilon})$ consisting of the following data.

1. A [[number field]] $F$ such that $\sqrt{-1} \in F$. In other words, we have a field extension of the [[quotient ring]] $\mathbb{Q}[X] / (X^{2} + 1)$, which itself is a field because $X^{2} + 1$ is irreducible: see [[field extension]] for more details on this.

1. A [[scheme]] $X_{F}$ which is obtained by removing a closed point from an [[elliptic curve]] $E_{F}$ over $F$. The scheme structure on $X_{F}$ is that inherited from $E_{F}$ by virtue of the fact that $X_{F}$ is an open subset of (the underlying topological space of) $E_{F}$, as described at [[open subscheme]]. We require that $X_{F}$ satisfies certain conditions: TODO.

1. A [[prime number|prime]] $l \geq 5$ such that $\phi_l$ as defined above contains $SL_2\left( \mathbb{F}_{l} \right)$ in its [[image]].

1. The field extension $F / F_{mod}$ is [[Galois extension|Galois]].

1. A [[subset]] $\underline{\mathbb{V}}$ of $\mathbb{V}\left(K_E\right)$ which is [[bijection|isomorphic]] to $\mathbb{V}\left(F_{mod}\right)$. In other words, observing that $F_{mod}$ is a sub-field of $K_E$, a [[section]] of the map $V\left( K_E \right) \rightarrow V\left( F_{mod} \right)$ determined by the inclusion of $F_{mod}$ into $K_E$.

(TO BE CONTINUED)

\end{defn}

\section{Examples of the kind in IUTT}

Only one family of examples of initial Θ-data is given in the IUTT series of papers, pertaining directly to the [[abc conjecture]]. These examples are described in Corollary 2.2 of [IUTT IV](#IUTTIV). The language of stacks is used, but it seems possible to construct the examples without this.

We shall assume that we have defined the notion of a _compactly bounded_ subset of $\overline{\mathbb{Q}}$, the algebraic closure of $\mathbb{Q}$ (namely the [[algebraic numbers]]). This notion is due to Mochizuki.

\begin{defn} A _Mochizuki elliptic curve_ $E$ is one of the form $y^{2} = x(x-1)(x - \lambda)$ for some $\lambda \in \mathbb{Q}$ satisfying the following conditions:

1. $\lambda$ belongs to a compactly bounded subset $K_V$ of $\overline{\mathbb{Q}}$ with certain properties (omitted here for the moment);

1. $\lambda$ belongs to an [[field extension|extension]] of $\mathbb{Q}$ of degree less than or equal to a given positive integer $d$ (the choice of which comes from the statement of the [[abc conjecture]]);

1. $\lambda$ does not belong to a certain subset of $K_V$, denoted $\mathfrak{Crc}_{d}$, where $d$ is as in 2., which Mochizuki constructs (details omitted for the moment here too);

1. it is defined over a number field $F$ which is isomorphic to that obtained by beginning with the minimal field of definition of $E$ (regarded as defined over $\overline{\mathbb{Q}}$ for example) and then adjoining the fields of definition of the $30$-torsion points, that is, the $2$-torsion, $3$-torsion, and $5$-torsion points, as well as adjoining $\sqrt{-1}$;

1. there is a prime $l \geq 5$ satisfying certain properties for which $E$ does not admit a sub-group scheme which as a group is isomorphic to the [[cyclic group]] $\mathbb{Z} / l \mathbb{Z}$.

\end{defn}

\begin{rmk} The first part of Corollary 2.2 in [IUTT IV](#IUTTIV) precisely concerns the construction, given $K_{V}$ (whose construction with the required properties is standard), of $\mathfrak{Crc}_{d}$. The properties of $\mathfrak{Crc}_{d}$ are such that can one apply the theory of IUTT to deduce that the [[abc conjecture]] holds. \end{rmk}

\begin{rmk} Given a Mochizuki elliptic curve $E$, one can show that one can construct initial Θ-data whose elliptic curve is $E$. TODO: details. \end{rmk}

\section{Related concepts}

* [[Mochizuki's corollary 3.12]]

\section{References}

* {#MochizukiIUTTI} [[Shinichi Mochizuki]], _Inter-Universal Teichmüller Theory I: Construction Of Hodge Theaters_, (2017). [Link to paper](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf)

* {#IUTTIV} [[Shinichi Mochizuki]], _Inter-universal Teichmüller theory IV, Log-volume computations and set-theoretic foundations_ (2012) ([pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf))

* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_, arXiv:[2004.13228](https://arxiv.org/abs/2004.13228)

* [[Taylor Dupuy]], _Initial Theta Data_, video series.

   * [Some Prerequisites, and The Easy Part of Initial Theta Data](https://youtu.be/FTv5TnXR1HQ)
   * [Transvections](https://youtu.be/4QEntmb2xyE)
   * [Local Global l-torsion Compatibility](https://youtu.be/gcuqYsrCR5A)
   * [Existence and Examples](https://youtu.be/sUGXfC8LwwI)

[[!redirects Initial Θ-data]]
[[!redirects initial theta-data]]
[[!redirects Initial theta-data]]
[[!redirects initial theta data]]
[[!redirects Initial theta data]]