Dedekind zeta function

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## Idea

The _Dedekind zeta function_ is a generalization of the [[Riemann zeta function]] from the [[rational numbers]]/[[integers]] to [[number fields]]/their [[rings of integers]].

The analog for [[function fields]] is the [[Weil zeta function]], while the generalization to [[higher dimensional arithmetic geometry]] are the [[arithmetic zeta functions]].

## Properties

### Special values

For $K$ a [[number field]] then all [[special values of L-functions|special values]] of the Dedekind zeta function $\zeta_K(n)$ for integer $n$ happen to be [[periods]] ([MO comment](http://mathoverflow.net/a/54244/381)).

Based on this ([Deligne 79](#Deligne79)) identified _critical values_ of $L$-functions (at certain integers) and conjectured that these are all these are algebraic multiples of [[determinants]] of [[matrices]] whose entries are [[periods]]. For more on this see at _[[special values of L-functions]]_.

### Adelic integral expression

The Dedekind zeta function has an [[adelic integral]] expression in direct analogy to that of the [[Riemann zeta function]]. This is due to ([Tate 50](#Tate50)), highlighted by [[Ivan Fesenko]] in ([Goldfeld-Hundley 11, Interlude remark (1)](#GoldfeldHundley11)).

### Functional equation

The [[functional equation]] of the Dedekind zeta function follows from its [[adelic integral]] representation in direct analogy to how this works for the [[Riemann zeta function]]. This is due to ([Tate 50](#Tate50)), highlighted by [[Ivan Fesenko]] in ([Goldfeld-Hundley 11, Interlude remark (1)](#GoldfeldHundley11)).

### The pole and the class number formula

The Dedekind zeta function $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The _[[class number formula]]_ says that its [[residue]] there is proportional the the product of the [[regulator of a number field|regulator]] of $K$ with the [[class number]] of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

### Relation to other zeta-, theta-, and L-functions
 {#RelationToThetaFunctions}

* The Dedekind zeta function $\zeta_K$ is the [[Artin L-function]] $L_{K,\sigma}$ for [[trivial representation|trivial]] [[Galois representation]]

  $$
    \zeta_K = L_{K,1}
    \,.
  $$

  See at _[Artin L-function -- Relation with Dedekind zeta function](Artin+L-function#RelationToDedekindZeta)_.

* The Dedekind zeta function has an expression as an integral over a kernel given by a [[Hecke theta function]] ([Kowalski (2.3) (2.4)](#Kowalski)).

* The Dedekind zeta function of $K$ is equivalently the [[Hasse-Weil zeta function]] of $Spec(\mathcal{O}_K)$.

### Analogs over complex curves

The [[function field analogy]] in view of the discussion at _[[zeta function of an elliptic differential operator]]_ says that the Dedekind zeta function is [[analogy|analogous]] to the regulated [[functional determinant|functional trace]] of a would-be "[[Dirac operator]] on [[Spec(Z)]]".

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

Other identifications/analogies of the Riemann zeta function (and more generally the [[Dedekind zeta-function]]) with [[partition functions]] in [[physics]] have been proposed, in particular the _[[Bost-Connes system]]_.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

### Function field analogy

[[!include function field analogy -- table]]

## References

* {#Tate50} [[John Tate]], _Fourier analysis in number fields, and Hecke's zeta-functions_, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950

* Wikipedia, _[Dedekind zeta function](http://en.wikipedia.org/wiki/Dedekind_zeta_function)_

* E. Kowalski, section 1.4 of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

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Dedekind-Hasse norm

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## Idea

The notion of a *Dedekind-Hasse norm* is a generalization of the notion of the degree function in a [[Euclidean domain]]. It makes clear the links between a [[principal ideal domain]] and an [[Euclidean domain]]: a pid is like an Euclidean domain but with a weaker notion of Euclidean division.

## Definition and characterization of pids

\begin{definition}
Let $R$ be an [[integral domain]]. A *Dededekind-Hasse* norm is a function $v \colon R \rightarrow \mathbb{N}$ such that:

* $v(s)=0 \Leftrightarrow s=0$

* $\forall a \in R$, $\forall (b \neq 0) \in R$:

   * $b|a$, or

   * $\exists p,q,r \in R$ such that $pa = bq + r$ and $0 \lt v(r) \lt v(b)$

\end{definition}

\begin{proposition}
Let $R$ be an integral domain. Then it is a principal ideal domain iff it possesses a Dedekind-Hasse norm.
\end{proposition}
\begin{proof}
Suppose that $R$ possesses a Dedekind-Hasse norm. Let $I$ be a non-zero ideal. Let $b$ be a non-zero element of $I$ of minimal norm. We know that $(b) \subseteq I$. Let $a$ be an element of $I$. Suppose that $b$ doesn't divide $a$. Then, there exists $p,q,r \in R$ such that $pa = bq+r$ and $0 \lt v(r) \lt v(b)$. Thus, $r=pa-bq \in I$, $r \neq 0$ and $v(r) \lt v(b)$, absurd! Therefore $b|a$ and $a \in (b)$. Thus, $I \subseteq (b)$ and $I = (b)$. We have proved that $R$ is a pid.

Suppose that $R$ is a pid. Thus, it is a [[unique factorization domain| UFD]]. Put $v(s)=0$ if $s=0$, and $v(s)$ equal to $2^{n}$ where $n$ is the number of irreducible elements in the factorization of $n$, if $s \neq 0$. Let $a \in R$ and $(b \neq 0) \in R$. Suppose that $b$ doesn't divide $a$. We know that $(a,b) = (r)$ and thus there exists $p,q,r \in R$ such that $pa+bq = r$.  $r$ divides $b$ but $b$ doesn't divide $r$ because it would imply that $b$ divides $a$. Thus, there is strictly less irreducible elements in the factorization of $r$ than in the one of $b$ and $v(r) \lt v(b)$. Moreover $r \neq 0$ because $(a,b) = (r)$ and $b \neq 0$. Thus $0 \lt v(r) \lt v(b)$. We have proved that $v$ is a Dedekind-Hasse norm.
\end{proof}

## References

Named after [[Helmut Hasse]].

* Wikipedia, *[Dedekind-Hasse norm](https://en.wikipedia.org/wiki/Dedekind%E2%80%93Hasse_norm)*


deduction

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## Idea

In [[logic]], a __deduction__ or __derivation__ is a part of a [[proof]], drawing a conclusion from certain [[premises]].  See:

* [[natural deduction]]

* [[sequent]]

## Properties

* [[deduction theorem]]

## Related concepts

* [[deductive reasoning]]

* [[induction]]

* [[syllogism]]

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deduction and induction - contents

**[[deductive reasoning]]**, [[deduction]]

* [[sequent]]

  [[hypothesis]]/[[context]]/[[antecedent]] $\vdash$ [[conclusion]]/[[consequence]]/[[succedent]]

* [[logical framework]]

* [[deductive system]],

  * [[natural deduction]]

    * [[type theory]]

  * [[sequent calculus]]

**[[inductive reasoning]]**

* [[induction]], [[recursion]]

* [[inductive type]], [[higher inductive type]]


deduction calculus

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e.g _[[natural deduction]]_

## Related concepts

* [[sequent calculus]]

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deduction theorem

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## Idea

The _deduction theorem_ in [[formal logic]] says (when it holds) that if in some [[logical framework]] there is a [[proof]] by [[deduction]] of some [[proposition]] $B$ from the [[premise]] $A$, then there is also a proof of the [[conditional statement]] $A \to B$ (from no premises).

This hence relates [[entailment]] in the [[metalanguage]] with [[implication]] in the [[object language]].

This seems obvious, but there are formal logical systems where this fails, for instance in the original Birkhoff-vonNeumann [[quantum logic]].  On the other hand, it may be taken as an [[axiom]]; it is the [[introduction rule]] for the [[implication]]/[[function type]] in [[natural deduction]].

## References

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

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deductive reasoning

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## Idea

**Deductive** [[reasoning]] is the process of passing from [[premises]] to [[conclusions]] in such a way that the [[truth]] of the former necessitates the truth of the latter. In other words, deductive [[logic]] studies what it is for an argument to be **valid**.

## Formalizations

A formalization of the process of deductive reasoning are [[deductive systems]] in [[logical frameworks]] such as _[[natural deduction]]_ and _[[sequent calculus]]_.

## Related concepts

* [[inductive reasoning]]

* [[Bayesian reasoning]]

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deductive system

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# Deductive systems
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## Definition

In [[logic]], [[type theory]], and the [[foundations]] of [[mathematics]], a **deductive system** (or, sometimes, **inference system**) is specified by

1. A collection of [[judgments]], and
2. A collection of *steps*, each of which has a (typically finite) list of judgments as hypotheses and a single judgment as conclusion.  A step is usually written as
   $$ \frac{J_1 \quad \cdots \quad J_n}{J} $$
   If $n=0$, a step is often called an [[axiom]]. In [[set theory]], if $n \gt 0$, a step is usually called an [[axiom schema]].

Usually, one generates the steps by using **inference rules**, which are schematic ways of describing collections of steps, generally involving metavariables.

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###### Remark
This use of the terminology "deductive system" is not completely standard, but it is not uncommon, and we need *some* name by which to refer to this general notion.
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###### Example
In the concrete [[algebraic theory]] of [[groups]], the judgments are formal [[equations]] between [[terms]] built out of variables and the symbols $e$, $\cdot$, and $(-)^{-1}$.  Thus, for instance, $x\cdot e = x$ and $x = y \cdot x^{-1}$ are judgments.

The rules of inference express, among other things, that equality is a [[congruence]] relative to the "operations".  For instance, there is a rule
$$ \frac{a=a' \quad b=b'}{a\cdot b = a'\cdot b'} $$
where $a$, $b$, etc. are [[metavariable]]s.  [[substitution|Substituting]] particular terms for these metavariables produces a *step* which is an instance of this rule.
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## Proof trees and theorems

A **proof tree** in a deductive system is a rooted tree whose edges are labeled by judgments and whose nodes are labeled by steps.  We usually draw these like so:
$$
\array{\arrayopts{\rowlines{solid}}
\array{\arrayopts{\rowlines{solid}} J_1 \quad J_2 \\ J_3} \quad
\array{\arrayopts{\rowlines{solid}} J_4 \\ J_5} \\
J_6 }
$$
(To draw such trees on the nLab, see the [[HowTo]] for a hack using the `array` command.  For LaTeX papers, there is the [mathpartir](https://www.ctan.org/pkg/mathpartir) package.)

If there is a proof tree with root $J$ and no leaves (which means that every branch must terminate in an axiom), we say that $J$ is a [[theorem]] and write
$$\vdash J.$$
More generally, if there is a proof tree with root $J$ and leaves $J_1,\dots, J_n$, we write
$$ J_1, \dots, J_n \;\vdash\; J. $$
This is equivalent to saying that $J$ is a theorem in the extended deductive system obtained by adding $J_1,\dots,J_n$ as axioms.

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###### Remark
This use of $\vdash$ to express a statement *about* the deductive system should be distinguished from its use in particular deductive systems as a syntactic ingredient *in judgments*.  For instance, in [[sequent calculus]] the judgments are [[sequents]], which are sequences of statements connected by a turnstile $\vdash$.  Similarly, in [[type theory]] and [[natural deduction]] one often uses $\vdash$ inside a single judgment when that judgment is of a hypothetical sort.  However, when using a [[logical framework]], these two meanings of $\vdash$ become essentially identified.
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## Formal systems {#FormalSystems}

Depending on the strength of the metalanguage used to define the judgments and steps, simply having a deductive system does not in itself necessarily yield an *effective procedure* for enumerating valid proof trees and theorems.  Deductive systems which do yield such an enumeration are sometimes referred to as **formal systems**.  For example, Gödel's [[incompleteness theorems]] are statements about formal systems in this sense.  It is worth keeping in mind that more general deductive systems are considered in proof theory and type theory, typically because by side-stepping these coding issues one can give a simpler account of computational phenomena such as [[cut-elimination]].  A well-known example of such a so-called "semi-formal system" is [[first-order arithmetic]] with the [[∞-rule]], used by Schütte in order to simplify Gentzen's proof that the consistency of first-order arithmetic may be reduced to well-foundedness of the ordinal $\epsilon_0$.

## Examples and special cases

* [[natural deduction]]
* [[type theory]]
* [[sequent calculus]]
* [[Hilbert system]]

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defect anyons -- references

### Defect anyons
 {#ReferencesVortexAnyons}

Often the concept of anyons is introduced as if a generalization of [[perturbative quantum field theory|perturbative]] quanta like fundamental [[bosons]] and [[fermions]]. But many (concepts of) types of anyons are really *[[soliton|solitonic]]$\;$[[defects]]* such as [[vortices]].

The general concept of [[braiding]] of [[defects]] in [[solid state physics]]:

* {#Mermin79} [[N. David Mermin]], *The topological theory of defects in ordered media*, Rev. Mod. Phys. **51** (1979) 591 $[$[doi:10.1103/RevModPhys.51.591](https://doi.org/10.1103/RevModPhys.51.591)$]$

  > (including a review of [basic homotopy theory](Introduction+to+Topology+--+2))

and more specifically for [[vortices]]:

* {#LoPreskill93} [[Hoi-Kwong Lo]], [[John Preskill]], *Non-Abelian vortices and non-Abelian statistics*, Phys. Rev. D **48** (1993) 4821 $[$[doi:10.1103/PhysRevD.48.4821](https://doi.org/10.1103/PhysRevD.48.4821)$]$

Explicit discussion in terms of [[anyons]]:

* {#Kitaev06} [[Alexei Kitaev]], *Anyons in an exactly solved model and beyond*, Annals of Physics **321** 1 (2006) 2-111 $[$[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)$]$

  > Anyonic particles are best viewed as a kind of topological defects that reveal nontrivial properties of the ground state. $[$p. 4$]$

Concrete [[vortex]]$\;$[[anyons]] in [[Bose-Einstein condensates]]:

* {#PFCZ01} B. Paredes, P. Fedichev, J. I. Cirac, P. Zoller *1/2-Anyons in small atomic Bose-Einstein condensates*, Phys. Rev. Lett. **87** (2001) 010402 $[$[arXiv:cond-mat/0103251](https://arxiv.org/abs/cond-mat/0103251), [doi:10.1103/PhysRevLett.87.010402](https://doi.org/10.1103/PhysRevLett.87.010402)$]$

* Julien Garaud, Jin Dai, [[Antti J. Niemi]], *Vortex precession and exchange in a Bose-Einstein condensate*, J. High Energ. Phys. **2021** 157 (2021)
$[$[arXiv:2010.04549](https://arxiv.org/abs/2010.04549)$]$

* {#MPSS19} Thomas Mawson, Timothy Petersen, [[Joost Slingerland]], [[Tapio Simula]], *Braiding and fusion of non-Abelian vortex anyons*, Phys. Rev. Lett. **123** (2019) 140404 $[$[doi:10.1103/PhysRevLett.123.140404](https://doi.org/10.1103/PhysRevLett.123.140404)$]$

and in (other) [[superfluids]]:

* {#MMN21} Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, *Non-Abelian Half-Quantum Vortices in 3P2 Topological Superfluids* $[$[arXiv:2107.02448](https://arxiv.org/abs/2107.02448)$]$

and in condensates *of* non-defect anyons:

* {#CDLR19} Michele Correggi, Romain Duboscq, Douglas Lundholm, Nicolas Rougerie, *Vortex patterns in the almost-bosonic anyon gas*, Europhys. Lett. **126** (2019), 20005 $[$[arXiv:1901.10739](https://arxiv.org/abs/1901.10739), [doi:10.1209/0295-5075/126/20005](https://doi.org/10.1209/0295-5075/126/20005)$]$

See also [Ahn, Park & Yang 19](anyon#AhnParkYang19) who refer to the band nodes in the [[Brillouin torus]] of a  [[semi-metal]] as "vortices in momentum space".

And see at *[[defect brane]]*.


defect brane

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## Idea

A _defect brane_ ([Bershoeff-Ortin-Riccioni 11](#BershoeffOrtinRiccioni11)) is a [[brane]] of [[codimension]] 2 (hence an "[[exotic brane]]"); such as, for example

* a [[cosmic string]] in 4-dimensional [[Yang-Mills theory]],

* a [[super 1-brane in 4d]],

* a [[super 2-brane in 5d]],

* a [[super 3-brane in 6d]],

* a [[D7-brane]] in 10d [[type IIB string theory]].

In [[string theory]], defect branes with [[U-duality]]-[[group]] [[monodromy]] have been argued to be identified with "[[exotic branes]]" [deBoer & Shigemori 2012, p. 12](#deBoerShigemori12).

## Properties

### Relation to anyons
 {#RelationToAnyons}

An [[anyon]] is traditionally realized as a point-particle/defect in 2+1-dimensional spacetime, hence as a codimension-2 "0-brane" (see the references [here](braid+group+statistics#ReferencesVortexAnyons)). This suggests that general codimension-2 defect branes may serve as string-theoretic models for anyons. This general observation is briefly mentioned in [deBoer & Shigemori 2012, p. 65](#deBoerShigemori12) and further developed in [SS22](#SS22).

Concrete implementation of anyonic defect [[strings]] in [[4d TQFT]] is discussed in [Baez, Wise & Crans 2006](#BaezWiseCrans06), [Bullivant, Martins & Martin 2018](#BullivantMartinsMartin18) [Kong, Tian & Zhang 2020, Sec. 2.2](#KongTianZhang20).
More generally, circle-shaped defect strings are going to satisfy *[[loop braid group]]-statistics*, which subsumes [[braid group]]-statistics but is richer still.

## Related concepts

* [[vortex string]]

* [[black brane]]

* [[exotic brane]]

* [[domain wall]]

* [[monopole]]

## References

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

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

See also:

* [[Claudia de Rham]], *The Effective Field Theory of Codimension-two Branes*, JHEP 0801:060 (2008) [[arXiv:0707.0884](https://arxiv.org/abs/0707.0884), [doi:10.1088/1126-6708/2008/01/060](https://doi.org/10.1088/1126-6708/2008/01/060)]

The term "defect brane" was introduced in:

* {#BershoeffOrtinRiccioni11} [[Eric Bergshoeff]], [[Tomas Ortin]], [[Fabio Riccioni]], _Defect Branes_, Nuclear Physics B **856** 2 (2012) 210-227 [[arXiv:1109.4484](https://arxiv.org/abs/1109.4484), [doi:10.1016/j.nuclphysb.2011.10.037](https://doi.org/10.1016/j.nuclphysb.2011.10.037)]

Identification of codimension-2 defect branes with [[U-duality]]-group [[monodromy]] as [[exotic branes]]:

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

See also:

* Allan Bayntun, C.P. Burgess, Leo van Nierop, _Codimension-2 Brane-Bulk Matching: Examples from Six and Ten Dimensions_, New J. Phys. 12:075015, 2010 ([arXiv:0912.3039](https://arxiv.org/abs/0912.3039))

* Takashi Okada, Yuho Sakatani, _Defect branes as Alice strings_, JHEP 1503 (2015) 131 ([arXiv:1411.1043](https://arxiv.org/abs/1411.1043), doi:10.1007/JHEP03(2015)131)

* Yosuke Imamura, Hirotaka Kato, _Codimension-2 brane solutions of maximal supergravities in 9, 8, and 7 dimensions_, Prog. Theor. Exp. Phys. (2018) ([arXiv:1711.03242](https://arxiv.org/abs/1711.03242), [doi:10.1093/ptep/pty045](https://doi.org/10.1093/ptep/pty045))

Discussion of defect [[(p,q)5-branes]]:

* [[Tetsuji Kimura]], _Defect $(p,q)$ Five-branes_, Nucl. Phys. B **893** (2015) 1-20 [[arXiv:1410.8403](https://arxiv.org/abs/1410.8403), [doi:10.1016/j.nuclphysb.2015.01.023](https://doi.org/10.1016/j.nuclphysb.2015.01.023)]

* [[Tetsuji Kimura]], Shin Sasaki, Kenta Shiozawa, *On Geometries and Monodromies for Branes of Codimension Two* [[arXiv:2312.03358](https://arxiv.org/abs/2312.03358)]

Discussion of codimension-2 defects in the [[M5-brane]] [[worldvolume]], hence defect branes in [[little string theory]] (cf. [[3-brane in 6d]]):

* [[Hironori Mori]], Yuji Sugimoto, around Fig. 9 in: _Surface Operators from M-strings_, Phys. Rev. D 95, 026001 (2017) [[arXiv:1608.02849](https://arxiv.org/abs/1608.02849), [doi:10.1103/PhysRevD.95.026001](https://doi.org/10.1103/PhysRevD.95.026001)]

* [[Hironori Mori]], _M-theory Perspectives on Codimension-2 Defects_ ([spire:1519095](https://inspirehep.net/literature/1519095), [pdf](http://www2.yukawa.kyoto-u.ac.jp/~soken.editorial/shuron/DoctoralThesis_Mori.pdf))

* Nathan Haouzi, Christian Schmid, _Little String Origin of Surface Defects_,  J. High Energ. Phys. 2017, 82 (2017) ([arXiv:1608.07279](https://arxiv.org/abs/1608.07279), doi:10.1007/JHEP05(2017)082)

* Pietro Capuozzo, John Estes, Brandon Robinson, Benjamin Suzzoni, *Holographic Weyl Anomalies for 4d Defects in 6d SCFTs* [[arXiv:2310.17447](https://arxiv.org/abs/2310.17447)]

and in relation to [[quantum Seiberg-Witten curves]] of [[class S-theories]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], around Fig. 1 & Fig. 2 in:  *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

and in [[D=5 supergravity]]:

* Minkyu Park, [[Masaki Shigemori]], _Codimension-2 Solutions in Five-Dimensional Supergravity_, JHEP 1510 (2015) 011 ([arXiv:1505.05169](http://arxiv.org/abs/1505.05169))

* [[Masaki Shigemori]], *Interpolating between multi-center microstate geometries*, JHEP 09 (2021) 010 ([arXiv:2105.11639](https://arxiv.org/abs/2105.11639))

On defect strings in 4d [[TQFT]] satisfying [[braid group]]- and more generally [[loop braid group]]-statistics:

in 4d [[BF theory]]:

* {#BaezWiseCrans06} [[John Baez]], [[Derek Wise]], [[Alissa Crans]], *Exotic Statistics for Strings in 4d BF Theory*, Adv. Theor. Math. Phys. 11:707-749, 2007 ([arXiv:gr-qc/0603085](https://arxiv.org/abs/gr-qc/0603085))

* {#BullivantMartinsMartin18} Alex Bullivant, [[João Faria Martins]], Paul Martin, *Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory*, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 7 ([arXiv:1807.09551](https://arxiv.org/abs/1807.09551))

and in the [[3d toric code]]-[[4d TQFT]]:

* {#KongTianZhang20} [[Liang Kong]], Yin Tian, Zhi-Hao Zhang, Section 2.2 of: *Defects in the 3-dimensional toric code model form a braided fusion 2-category*, J. High Energ. Phys. 2020, 78 (2020) ([arXiv:2009.06564](https://arxiv.org/abs/2009.06564), doi:10.1007/JHEP12(2020)078)

Incorporation of defect branes into the [[K-theory classification of D-brane charge]] by understanding the [[hypergeometric construction of KZ solutions]] as happening in [[twisted equivariant differential K-theory]]:

* {#SS22} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Anyonic defect branes in TED-K-theory]]*, Reviews in Mathematical Physics **35** 06 (2023) 2350009 [[arXiv:2203.11838](https://arxiv.org/abs/2203.11838), [doi:10.1142/S0129055X23500095](https://doi.org/10.1142/S0129055X23500095)]

On defect branes in [[D=4 N=4 SYM]] via [[AdS/CFT]]:

* Hongliang Jiang, [[Arkady A. Tseytlin]], *On co-dimension 2 defect anomalies in $\mathcal{N}=4$ SYM and $(2,0)$ theory via brane probes in AdS/CFT* [[arXiv:2402.07881](https://arxiv.org/abs/2402.07881)]

[[!redirects defect branes]]

[[!redirects codimension 2 brane]]
[[!redirects codimension 2 branes]]

[[!redirects codimension-2 brane]]
[[!redirects codimension-2 branes]]


deferred measurement principle

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## Idea

In [[quantum physics]] and specifically in [[quantum information theory]], the *principle of deferred measurement* is a [[theorem]] which says that

* any [[quantum circuit]] involving [[quantum measurement]] of some of the [[qbits]] followed by [[quantum gates]] [[controlled quantum gate|controlled]] by the respective measurement outcomes

is equivalent ([[equality|equal]] as a [[function]] from given input to output [[data types]]) as

* a [[quantum circuit]] in which all [[quantum measurement]] happens "at the end", i.e. where no [[quantum gates]] are classically [[controlled quantum gate|controlled]] by previous measurement results, but all [[controlled quantum gate|quantumly controlled]] by coherent control qbits.

The principle can be useful in practice for optimizing [[quantum circuits]]. It also clearly relates to the issue of [[interpretations of quantum mechanics]]: Since it is the [[collapse of the wavefunction]] upon [[quantum measurement]] which makes the [[interpretation of quantum mechanics]] subtle, it is interesting to note that this collapse may be (arbitrarily) deferred, in a precise sense.

## Formalizations
 {#Formalizations}

One way to formalize the deferred measurement principle is briefly mentioned in [Staton (2015), Axiom B (p. 6 of 12)](#Staton15), there proposed as an [[axiom]] to be satisfied by [[quantum programming languages]].

Another formalization and proof has been offered in [Gurevich & Blass 2021](#GurevichBlass21).

{#InTermsOfQuantumModalLogic}
A quick *proof* of essentially the formulation of [Staton (2015)](#Staton15) exists in the [quantum modal logic](necessity+and+possibility#ModalQuantumLogic) discussed at *[[quantum circuits via dependent linear types]]*: Here the deferred measurement principle is essentially the [Kleisli equivalence](Kleisli+category#KleisliEquivalence) for the [[necessity]] [[comonad]] $\Box_B$ on [[dependent linear types]], like this:

\begin{imagefromfile}
    "file_name": "DeferredMeasurementAsKleisliEquiv-230807.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

{#ExampleCNOT} For the **example** of a [[CNOT gate]]:

\begin{imagefromfile}
    "file_name": "CNOT-DeferredMeasurement-230809.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\linebreak

## Interpretation of Deferred measurement
 {#DeferredMeasurementAndInterpretations}

The folklore of quantum physics knows [[paradox|paradoxical]]-sounding stories under the title of

* *[Schrödinger's cat](https://en.wikipedia.org/wiki/Schrödinger's_cat)* (1935)

* *[Everett's observers](#EverettObservers)* (1957)

* *[Wigner's friend](https://en.wikipedia.org/wiki/Wigner%27s_friend)* (1961)

The author of these paragraphs asserts that:

1. These are all the same story, recast with different actors: Schrödinger's cat plays the same role as Everett's observer A and the same role as Wigner's friend. The point in any case is that this first observer makes a quantum measurement and (only) ofterwards is himself observed by a second observer.

1. This is just what is formalized by the set-up of the deferred measurement principle:

   1. The first observer (called "cat" or "A" or "friend") is the controlled quantum gate denoted "$G$" above,

   1. the quantum system observed by the first observer is $\mathrm{Q}W$ above,

   1. the state space of the first observer is $\mathscr{H}$ (before) and $\mathscr{H}'$ (after the observation).

   1. The second observer inspecting the scene at the end is the right hand side of the above setup, where the measurement is made at the end of the circuit execution. Before it is made, the first observer may have been in a superposition (in $\mathscr{H}'$).

   1. But the deferred measurement principle says the outcome is indistinguishable from the situation where the first observer already collapses the original state in $\mathrm{Q}W$.

For the record, we recall Everett's version of this story:

{#EverettObservers} A key argument in the discussion of [Everett (1957)](interpretation+of+quantum+mechanics#Everett57a), which famously led its author to the *[[many-worlds interpretation of quantum mechanics]]*, is that *intermediate state collapse* caused by one observer seems [[paradox|paradoxical]], since to a later observer the analogous state collapse occurs (only) at a later time.
(This is really the same paradox as that of [Schrödinger's cat](https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat), if we grant the cat the role of the first observer).

[Everett (1957)](interpretation+of+quantum+mechanics#Everett57a) casts this into the following prose:

>> [[pp 4:](https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf#page=3)] Isolated somewhere out in space is a room containing an observer, $A$, who is about to perform a measurement upon a system $S$. After performing his measurement he will record the result in his notebook. We assume that he knows the state function of $S$ (perhaps as a result of previous measurement), and that it is not an eigenstate of the measurement he is about to perform. $A$, being an orthodox quantum theorist, then believes that the outcome of his measurement is undetermined and that the process is correctly described by Process 1 [[[collapse of the wavefunction|state collapse]]].

>> In the meantime, however, there is another observer, $B$, outside the room, who is in possession of the state function of the entire room, including $S$, the measuring apparatus, and $A$, just prior to the measurement. $B$ is only interested in what will be found in the notebook one week hence, so he computes the state function of the room for one
week in the future according to Process 2 [unitary time evolution]. One week passes, and we find $B$ still in possession of the state function of the room, which
this equally orthodox quantum theorist believes to be a complete description of the room and its contents. If $B$'s state function calculation tells beforehand exactly what is going to be in the notebook, then $A$ is incorrect in his belief about the indeterminacy of the outcome of his measurement. We therefore assume that $B$'s state function contains non-zero amplitudes over several of the notebook entries.

>> At this point, $B$ opens the door to the room and looks at the notebook (performs his observation). Having observed the notebook entry, he turns to $A$ and informs him in a patronizing manner that since his ($B$'s) wave function just prior to his entry into the room, which he knows to have been a complete description of the room and its contents, had non-zero amplitude over other than the present result of the measurement, the result must have been decided only when $B$ entered the room, so that $A$, his notebook entry, and his memory about what occurred one week ago had no independent objective existence until the intervention by $B$. In short, $B$ implies that $A$ owes his present objective existence to $B$'s generous nature which compelled him to intervene on his behalf. However, to $B$'s consternation, $A$ does not react with anything like the respect and gratitude he should exhibit towards $B$, and at the end of a somewhat heated reply, in which $A$ conveys in a colorful manner his opinion of $B$ and his beliefs, he rudely punctures $B$'s ego by observing that if $B$'s view is correct, then he has no reason to feel complacent, since the whole present situation may have no objective existence, but may depend upon the
future actions of yet another observer.

> It is now clear that the interpretation of quantum mechanics with which we began [the Copenhagen interpretation] is untenable if we are to consider a universe containing more than one observer.

## Related concepts

* [[no-cloning theorem]]

## References
 {#References}

While the principle of deferred measurement is a classical statement in [[quantum information theory]], it was not defined or proven in generality and with precision (hence has remained [[folklore]]) until [Gurevich & Blass 2021](#GurevichBlass21) (see the critical discussion of the literature provided there).

Accounts of the informal statement:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §4.4 of *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

As an [[axiom]] for [[quantum programming languages]]:

* {#Staton15} [[Sam Staton]], Axiom B (p. 6) in: *Algebraic Effects, Linearity, and Quantum Programming Languages*, POPL '15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) 395–40 [[doi:10.1145/2676726.2676999](https://doi.org/10.1145/2676726.2676999), [pdf](http://www.cs.ox.ac.uk/people/samuel.staton/papers/popl2015.pdf)]

The above discussion and graphics follows:

* {#CQTS22} [[CQTS]], *[Quantum Data Types via Linear HoTT](https://ncatlab.org/schreiber/show/QDataInLHoTT#QTML2022)* (Nov 2022)

See also:

* Wikipedia, *[Deferred Measurement Principle](https://en.wikipedia.org/wiki/Deferred_Measurement_Principle)*

General precise statement and [[proof]]:

* {#GurevichBlass21} [[Yuri Gurevich]], [[Andreas Blass]], *Quantum circuits with classical channels and the principle of deferred measurements*, Theoretical Computer Science **920** (2022) 21–32 [[arXiv:2107.08324](https://arxiv.org/abs/2107.08324), [doi:10.1016/j.tcs.2022.02.002](https://doi.org/10.1016/j.tcs.2022.02.002)]


deficit angle

See at _[[conical singularity]]_.

## Related concepts

* [[singularity]]

## References

* Wikipedia, _[Angular deficit](https://en.wikipedia.org/wiki/Angular_defect)_

[[!redirects deficit angles]]

[[!redirects angular deficit]]
[[!redirects angular deficits]]


definability

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## Related concepts

* **definability**

  * [[model theory]]

  * [[definable set]]

  * [[definable groupoid]]

  * [[Beth definability theorem]]

  * [[Craig interpolation theorem]]

  * [[definability (fibred category theory)]]

* [[decidability]]


definability (fibred category theory)

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## Idea

Suppose we have a [[fibered category]] $p \colon \mathcal{E} \to \mathcal{B}$, whose objects we think as [[categories with families|type families]] (living in the fibers of $\mathcal{E}$) indexed by [[contexts]] (living in $\mathcal{B}$, cf. the [[categorical semantics of dependent types]]).

A **definable collection** of objects of the total category $\mathcal{E}$ corresponds, intuitively, to the extension of a 'meta-property' of the types which we can use to single out subcontexts by means of a generalized '[[comprehension category|comprehension]]'.

{#ArchetypicalExample} The archetypal example is given by '[[truth]]': Suppose $p$ has [[fibred terminal objects]], i.e. over each [[context]] $B \colon \mathcal{B}$, there is a '[[unit type]]' $1_B \colon \mathcal{E}_B$ which is [[terminal object|terminal]] in the fiber.
Then the collection $\{1_B \colon \mathcal{E} \mid B \colon \mathcal{B}\}$ is an extensive definition of the meta-property of 'being true' (or better: being [[inhabited type|inhabited]]).

Once we have a definable collection $\mathcal{P}$, and given an object $X: \mathcal{E}_B$, we can ask: which members of the 'family' $X$ 'satisfy' $P$? This question is answered by a subobject $\{b:B \mid X_b : \mathcal{P}\} \rightarrowtail B : \mathcal{B}$.

In a sense, a definable collection is a very weak notion of [[coreflective subcategory|coreflective subfibration]] (i.e. a fibred coreflective subcategory).

## Definition

We follow the exposition in [(Jacobs '99)](#Jacobs).
Fix a [[Grothendieck fibration|fibration]] $p:\mathcal{E} \to \mathcal{B}$.

### Of collections of objects

\begin{definition}

Let $\mathcal{P}$ be a collection of object in $\mathcal{E}$, and denote by $\mathcal{P}_B$ the subcollection of those objects in $\mathcal{P}$ over a chosen $B:\mathcal{B}$.

1. We say $\mathcal{P}$ is **closed under substitution** if for each map $u:J \to I$ in $\mathcal{B}$ and $P : \mathcal{P}_I$, $u^*P : \mathcal{P}_J$.

2. We say $\mathcal{P}$ is **definable** if it is closed under substitution and, for each object $X: \mathcal{E}$, there is an object $\{X\}_{\mathcal{P}} : \mathcal{P}$ and  a [[cartesian morphism]] $i_X : \{X\}_{\mathcal{P}} \to X$ which is universal among cartesian morphisms into $X$ from an object in $\mathcal{P}$.

\end{definition}

The constructor $\{-\}_{\mathcal{P}}$ is the generalized comprehension we were mentioning above.
Indeed, we have the following characterization:

\begin{lemma} *(Lemma 9.6.2 in [(Jacobs '99)](#Jacobs))*

Let $\mathcal{P}$ be a collection closed under substitution.
Then the following our equivalent:

1. $\mathcal{P}$ is definable
2. For each object $X: \mathcal{E}_B$ the [[presheaf]] on $\mathcal{B}$ defined as
   \[
      J \mapsto \{u : J \to I \mid u^*X : \mathcal{P}\}
   \]
   is representable.

\end{lemma}
\begin{proof}
  The idea is that $\{X\}_P$ is the representing object for the presheaf object of (2).
\end{proof}

### Of collections of functors
\begin{definition}
Let $\mathcal{C}$ be a category, that we think of as trivially fibred $!_{\mathcal{C}} : \mathcal{C} \to 1$. A collection of fibred functors $\{(I_a, f_a) : !_{\mathcal{C}} \to p\}_{a:A}$ (where $A$ is a [[class]]) is definable iff it is definable as a collection of object of the [[exponent fibration]] $p^{!_{\mathcal{C}}}$.
\end{definition}

Recall the exponent fibration $p^{!_{\mathcal{C}}}$ is a fibration over $\mathcal{B}$ whose fiber over $I:\mathcal{B}$ is given by functors $\mathcal{C} \to \mathcal{E}_I$ and whose reindexing is given by post-composition with the reindexing functors of $p$. See [[exponent fibration]] for more details.

\begin{remark}
Clearly this definition generalizes the previous one since collections of objects of $\mathcal{E}$ are given by collections of functors out of $\mathcal{C}$.
\end{remark}

### Of collections of vertical morphisms
\begin{definition}
A collection of [[vertical morphisms]] of $\mathcal{E}$ is definable iff it is definable as collection of functors out of the arrow category $\downarrow$.
\end{definition}

### Of subfibrations
\begin{definition}
A [[subfibration]] $p': \mathcal{E}' \to \mathcal{B}$ of $p$ is definable iff its collection of objects and vertical morphisms are both definable.
\end{definition}

## Examples

\begin{example}
If $p$ has a fibred terminal objects, then their collection is definable if and only if the functor $1: \mathcal{B} \to \mathcal{E}$ has a right adjoint $\{-\}:\mathcal{E} \to \mathcal{B}$.
\end{example}

\begin{example}
For $p : \mathrm{Fam}(\mathcal{C}) \to \mathbf{Set}$ a fibration of families (a family over $I:\mathbf{Set}$ is a n $I$-indexed family of objects of $\mathcal{C}$), definable collection are those determined by a choice of objects of $\mathcal{C}$ (Lemma 9.6.4 in in [(Jacobs '99)](#Jacobs)).
\end{example}

## See also

* [[fibred category]]

* [[comprehension category]]

* [[axiom of comprehension]]

## References

Definability is introduced on p.35 ('Short annotated glossary') of

* {#Benabou85} [[Jean Bénabou]], _Fibered categories and the foundations of naive category theory_, Journal of Symbolic Logic, **50**(1) (1985) pp10-37. doi:[10.2307/2273784](https://doi.org/10.2307/2273784), ([JSTOR](http://www.jstor.org/stable/2273784))

A modern and more traditional account is given in Section 9.6 of

* {#Jacobs} [[Bart Jacobs]], *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf), [webpage](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html)]


definable and algebraic closure (in model theory)

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## Idea
In [[model theory]], the _definable closure_ of a subset of a model is a way of "generating a substructure" around that subset. The _algebraic closure_ of a subset always contains the definable closure and generalizes the usual [[algebraic closure]] of fields.

## Definition

Let $A$ be a small parameter set (i.e. a subset) of a [[monster model]] $\mathbb{M} \models T.$ The _definable closure_ $\operatorname{dcl}(A)$ of $A$ is the set of all tuples $b \in \mathbb{M}$ such that there exists a formula $\varphi(x,y)$ and a tuple $a$ from $A$ such that $b$ is the unique solution to $\varphi(a,y)$, i.e. $\varphi(\mathbb{M}_x,y) = \{b\}$.

To obtain the _algebraic closure_ $\operatorname{acl}(A)$ of $A$, we relax "unique solution" to "one of finitely many."

## Inside a monster
Inside a sufficiently saturated and homogeneous model $\mathbb{M}$, tuples $b$ are definable over $A$ if and only if they are fixed by the stabilizer $\operatorname{Aut}(\mathbb{M}/A)$ of $A$, and are algebraic over $A$ if and only if they have finite orbit under $\operatorname{Aut}(\mathbb{M}/A).$

_Proof._ To see nonobvious direction of the first claim, try the contrapositive: if every formula in the [[type (in model theory)|type]] of $b$ fails to uniquely pick out $b$, then the type of some $b'$ which is not $b$ but has the same type as $b'$ is finitely consistent, therefore realized, and by homogeneity there is an automorphism interchanging $b$ and $b'$.

To see the nonobvious direction of the second claim, try the contrapositive again: if $b$ isn't algebraic over $A$, then every formula in its type over $A$ is infinite, in particular all conjunctions of finite fragments of its type. So we can iteratively obtain infinitely many disjoint realizations of its type, and again by homogeneity these must all lie in the same orbit. $\square$

Saturation ensures that infinite definable sets become big---as big as the universe $\mathbb{M}$, while finite sets stay the same size, because the theory knows exactly how big they are.

## Examples
* In $\mathsf{ACF}_p$, the definable closure inside a monster $\mathbb{M}$ of a subset $A$ is the [[perfect field|perfect hull]] of $\mathbb{F}(A)$, where $\mathbb{F}$ is $\mathbb{Q}$ for $p = 0$ or the [[prime field]] of the characteristic $p > 0$, and algebraic closure coincides with the [[separable closure]].

* In the theory of an infinite set without structure, definable and algebraic closures coincide, and are trivial.

* In the relational structure consisting of the disjoint union of the cycle graphs of length $n$ for each $n \geq 1$, $\operatorname{dcl}(\emptyset)$ is just the cycle graph of length $1$, while $\operatorname{acl}(\emptyset)$ is the entire structure.

## Related concepts
* [[type (in model theory)]]
* [[model-theoretic Galois theory]]

## References
* Lou van den Dries, _An Introduction to Model-Theoretic Stability_.
* Dave Marker, _Model Theory: An Introduction_.

[[!redirects model-theoretic algebraic closure]]
[[!redirects definably closed]]
[[!redirects definable closure]]


definable groupoid

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## Idea
In a model $M$ of a first-order theory $T$ a definable set $G$ might have additional algebraic stucture (e.g. that of a [[group]], [[ring]], [[groupoid]], [[category]], etc.) also given by definable functions and predicates. Since this set with extra structure is given by some collection of formulas in the language of $T$, it is interpreted in every model of $T$, and is hence an invariant of the [[syntactic category]] (walking model) $\mathbf{Def}(T)$ of $T$.

## Definition
To make this precise:

* A definable group is just a [[group object]] in $\mathbf{Def}(T)$.
* A definable groupoid is just a groupoid object, i.e. an [[internal groupoid]] in $\mathbf{Def}(T)$.
* A definable ring is just a semigroup object in the category of abelian group objects of $\mathbf{Def}(T)$.
* A definable category is just a category object, i.e. an [[internal category]] in $\mathbf{Def}(T)$.

Since groups, groupoids, rings, and categories can be given by algebraic theories, a definition (modulo having [[elimination of imaginaries|EI]]) of one of these in $T$ is just an interpretation of one of those theories in $T$ is just a logical functor from the walking models of one of these to $\mathbf{Def}(T)$.

### Remarks
* Evaluating the unit of the [[Makkai duality]] adjunction at $T$ yields $\mathbf{Def}(T) \simeq \operatorname{Hom}_{\mathbf{Ult}}(\mathbf{Mod}(T), \mathbf{Set})$ the category of [[ultrafunctors]] from the category of models (logical functors $\mathbf{Def}(T) \to \mathbf{Set}$) to $\mathbf{Set}$ viewed as [[ultracategories]], so that one may identify a definable set (and definable sets with extra definable structure on them) $X \in \mathbf{Def}(T)$ with a [[functor of points]] $M \mapsto X(M)$ on the category of models.

* The fact that the [[external axiom of choice]] (all epimorphisms split) holds in $\mathbf{Set}$ if and only if every [[fully faithful]] [[essentially surjective]] [[functor]] between [[small]] categories is a true [[equivalence of categories]] can be word-for-word internalized to $\mathbf{Def}(T)$. This means: if the theory has two constants, then $T$ has definable Skolem functions if and only if every fully faithful essentially surjective definable functor between definable categories admits a [[pseudoinverse]].

* With a suitable amount of choice (definable Skolem functions, for example) Freyd's general adjoint functor theorem also carries over word-for-word to the setting of internal categories in $\mathbf{Def}(T)$.

* In particular, there is much studied case of __definable groups__, cf. e.g. ([Peterzil-Pillay](#PeterzilPillay))

## Properties

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

There is a [[bijection|bijective]] correspondence between internal imaginary sorts of $T$ and definable concrete groupoids with a single isomorphism class, up to bi-interpretability over $T$ for the internal imaginary sorts and Hrushovski-equivalence for the definable concrete groupoids.

=--

This is ([Hrushovski 2006, Th.3.2](#Hrushovski)).

## Related concepts

* [[definability]]

## References

* Y. Peterzil, A. Pillay, _Generic sets in definably compact groups_, Fundamenta Mathematicae __193__ (2007), pp. 153--170, [MR2282713](http://www.ams.org/mathscinet-getitem?mr=2282713), [doi](http://dx.doi.org/10.4064/fm193-2-4)
 {#PeterzilPillay}

* Y. Peterzil, A. Pillay, S. Starchenko, _Linear groups definable in o-minimal structures_, J. Algebra __247__ (2002), no. 1, pp. 1--23, [MR1873380](http://www.ams.org/mathscinet-getitem?mr=1873380), [doi](http://dx.doi.org/10.1006/jabr.2001.8861)

* Alessandro Berarducci, _Definable groups in o-minimal structures_, [pdf](http://www.dm.unipi.it/~dinasso/marian2004/berarducci.pdf); _Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup_, J. Symbolic Logic __74__:3 (2009), 891-900, [MR2548466](http://www.ams.org/mathscinet-getitem?mr=2548466), [euclid](http://projecteuclid.org/euclid.jsl/1245158089), [doi](http://dx.doi.org/10.2178/jsl/1245158089), _O-minimal spectra, infinitesimal subgroups and cohomology_, J. Symbolic Logic __72__ (2007), no. 4, pp. 1177--1193, [MR2371198](http://www.ams.org/mathscinet-getitem?mr=2371198), [euclid](http://projecteuclid.org/euclid.jsl/1203350779 ), [doi](http://dx.doi.org/10.2178/jsl/1203350779)

* Margarita Otero, _A survey on groups definable in o-minimal structures_, in: Model theory with applications to algebra and analysis. Vol. 2, 177–206, London Math. Soc. Lecture Note Ser. __350__, Cambridge Univ. Press 2008, [MR2010b:03042](https://www.ams.org/mathscinet-getitem?mr=2436142), [doi](https://doi.org/10.1017/CBO9780511735219.006)

* [[Ehud Hrushovski]], _Groupoids, imaginaries and internal covers_ Turk. J. Math 36 (2012) 173 – 198 [doi](https://doi.org/10.3906/mat-1001-91) [arxiv/math.LO/0603413](https://arxiv.org/abs/math/0603413) (2006) {#Hrushovski06}

* [[Ehud Hrushovski]], _On finite imaginaries_, [arxiv/0902.0842](https://arxiv.org/abs/0902.0842)

Hrushovski's correspondence between definable connected groupoids in a theory $T$ and internal generalised imaginary sorts of $T$ is extended to an equivalence of categories in

* Levon Haykazyan, Rahim Moosa, _Functoriality and uniformity in Hrushovski's groupoid-cover correspondence_, Annals of Pure and Applied Logic __169__:8 (2018) 705-- 730 [arXiv/1711.03531](https://arxiv.org/abs/1711.03531) [doi](https://doi.org/10.1016/j.apal.2018.03.005)

On higher categorical analogues of definable groupoids and internal covers:

* [[Moshe Kamensky]], _Higher internal covers_, [arXiv:2210.02699](https://arxiv.org/pdf/2210.02699)

category:

[[!redirects definable groupoids]]
[[!redirects definable category]]
[[!redirects definable categories]]
[[!redirects definable group]]
[[!redirects definable groups]]


definable set

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### Context
#### Model theory
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[[!include model theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

A [[definable set]] of an $\mathcal{L}$-theory $T$ in a [[first-order language]] is an [[equivalence class]] of [[formulas]] which evaluate the same way in every model of $T$. (In the case that $T$ is empty, $\mathbf{Mod}(T)$ is just the class of $\mathcal{L}$-structures.)

## Definition

Let $L$ be a first-order language and $T$ a theory in $L$. Any formula $\phi(x_1,\ldots,x_n)$, whose only free variables could be among $x_1,\ldots,x_n$, together with a [[model]] $M$ for $T$ evaluates to a truth value on each $a = (a_1,\ldots,a_n)\in  M^n$, hence it determines the subset $\phi(M)\subset M^n$ of all $a$ such that $\phi(a)$. Two formulas $\phi,\psi$ with the same number of free variables are equivalent if $\phi(M) = \psi(M)$ for every model $M$ of $T$. A __definable set__ $X$ in $T$ is an equivalence class of formulas in $L$ under this relation. Consider the category $\mathcal{M}_{el}(L)$ of $L$-[[structure in model theory|structures]] and [[elementary embedding|elementary embeddings]] and its full subcategory $\mathcal{M}_{el}(T)$ whose objects are the models of $T$. We can also view a __definable set__ for a theory $T$ as a [[functor]] $\mathcal{M}_{el}(T)\to Set$ which is of the form $M\mapsto\phi(M)$ for some $L$-formula $\phi$.

By $X(M)$ denote the set of all $a\in M$ such that $X(a)$ holds, i.e. $\phi(a)$ is true for any choice of representative $\phi\in X$.

Given two definable sets $X,Y$ in $T$ a __definable function__ $f: X\to Y$ is a definable subset of the product sort $X\times Y$ such that $f_M\in X(M)\times Y(M)$ is a function (we also write $f_M : X(M)\to Y(M)$) for every model $M$ of $T$. Analogously
a __definable equivalence relation__ is a definable subset $R\subset X\times X$ such that $R(M)$ is an equivalence relation for every model $M$ of $T$.

## Properties

__Proposition.__ Given a small category $\mathcal{M}$ and two functors $F,G:\mathcal{M}\to Set$ the natural transformations $\alpha : F\to G$ are in 1-1 correspondence with
functors $\tilde\alpha : \mathcal{M}\to Set$ such that $\tilde\alpha(A) \subset  F(A)\times G(A)$ and such that $\tilde\alpha(A)$ is a functional relation.

Proof. If $\alpha:F\to G$ is an $Ob(\mathcal{M})$-indexed family with components $\alpha_A: F(A)\to G(A)$ viewed as functional relations $\tilde\alpha_A\subset F(A)\times G(A)$, then the extension to a functor means
that there is a (automatically unique) dotted arrow $\tilde\alpha_f$
$$\array{
\tilde\alpha_A & \stackrel{\tilde\alpha_f}\hookrightarrow & \tilde\alpha_B\\
\downarrow && \downarrow \\
F(A)\times G(A)&\stackrel{F(f)\times G(f)}\longrightarrow & F(B)\times G(B)
}$$
making the diagram commutative (once the existence of $\tilde\alpha_f$ is known, the functoriality of $\tilde\alpha_{f\circ g} = \tilde\alpha_f\circ\tilde\alpha_g$ comes by uniqueness of $\tilde\alpha_f$ and the functoriality of $F\times G$ which it restricts from). That means that for every $x\in F(A)$
$(x,\alpha_A(x))$ should be sent to $(F(f)(x), G(f)(\alpha_A(x)))$ by $\tilde\alpha_f$, what is a restriction of $F(f)\times G(f)$, but by $\alpha_B$ being _functional_ relation this must be the same as $(F(f)(x), \alpha_B(F(f)(x)))$, i.e. that $G(f)(\alpha_A(x))=\alpha_B(F(f)(x))$. Q.E.D.

In other words, functoriality of $\tilde\alpha$ is the same as $\alpha$ being natural.

__Corollary.__ Definable functions for $L$ (for $T$) are in 1-1 correspondence with natural transformations of __definable sets__ as functors $\mathcal{M}_{el}(L)\to Set$ (resp. $\mathcal{M}_{el}(T)\to Set$).

For a fixed (multi-sorted, first-order) theory $T$, definable sets and definable functions form a category $Def(T)$. This category (or any equivalent category) is the [[syntactic category]] of the theory. Models of $T$ can be recovered as left exact
functors $Def(T)\to Set$. Notice that definable sets are functors from models to sets, and models are functors from definable sets to sets; just the latter are the "evaluation functionals" among all functors.

See also [[definable groupoid]].

An $\infty$-definable set is an intersection of definable sets.

We can also consider a _relative_ version of a definable set (definability with parameters).

Given definable sets $X,Y$, a _fixed_ model $M$ and a _fixed_ set $A\subset X(M)$, we say that an element $b\in Y(M)$ is definable over $A$ if there is $(a_1,\ldots,a_n)\in A^n$ and a $T$-definable function $f:X^n\to Y$ such that $f(a_1,\ldots,a_n)=b$. All elements $b$ definable over $A$ form the subset $Y(A)\subset Y(M)$. A subset $B\subset M$ is __definably closed__ if it is closed under definable functions. Given $A\subset M$, there is the minimal definably closed subset $B$ such that $A\subset B\subset M$, the __definable closure__ $B = dcl(A)$ of $A$.

If we extend the language by the constants from $A$ we get a new language $L_A$. The definable sets in language $L$ over $A$ are the definable sets in language $L_A$ over $\emptyset$.

## Ind- and pro-definable sets

One can also study [[ind-object]]s and [[pro-object]]s in the category of definable sets and definable functions.

+-- {: .num_defn #type-definable-set}

###### Definition

An important special case of a pro-definable set that shows up in model theory are **type-definable sets**, which are just infinite conjunctions of definable sets. (There is no restriction that the formulas representing these definable sets all sit inside a finite product of [[type|sorts]].)

=--

## Related concepts

* [[constructible set]]

* [[recursive subset]]

* [[computable set]]

* [[definable groupoid]]

## References

* Moshe Kamensky, _Ind- and Pro- definable sets_, [math.LO/0608163](http://arxiv.org/abs/math/0608163)

* Jan Holly, _Definable operations on sets and elimination of imaginaries_, Proc. Amer. Math. Soc. __117__ (1993), no. 4, 1149–1157, [MR93e:03052](http://www.ams.org/mathscinet-getitem?mr=1116261), [doi](http://dx.doi.org/10.2307/2159546), [pdf](http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116261-6/S0002-9939-1993-1116261-6.pdf)
* [[David Kazhdan]], Lecture notes in model theory, [pdf](http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf)
* D. Haskell, E. Hrushovski, H.D.Macpherson, _Definable sets in algebraically closed valued fields: elimination of imaginaries_, J. reine und angewandte Mathematik __597__ (2006), [MR2007i:03040](http://www.ams.org/mathscinet-getitem?mr=2264318), [doi](http://dx.doi.org/10.1515)/CRELLE.2006.066)

[[!redirects definable sets]]
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defining ideal of a topologizing subcategory

The notion of the [[defining sheaf of ideals]] of a closed subscheme inspires the notion of __defining ideal of a [[topologizing subcategory]]__ $S$ of an abelian category $A$ as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ where $N\in Ob(S)$. One can show that if $T\subset S$ is an inclusion of topologizing subcategories, then $\mathcal{I}_{S}\subset \mathcal{I}_{T}$.

If $R$ is an associative unital ring and $J\subset R$ a left ideal in $R$. Let $T = [J]$ be the smallest coreflective subcategory of $A$ containing $R/J$. Then $\mathcal{I}_T(R)\subset R$ is a two-sided ideal, namely the maximal 2-sided ideal in $J$, which is explicitly, $(J:R) = \{r\in R\,|\, J r \subset J\}$. Moreover, $J_{T\circ T}$ where $T\circ T$ is the square under the [[Gabriel multiplication]] agrees with $(J:R)^2$.

See also [[conormal bundle]].

* A. L.  Rosenberg, _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

* V. A. Lunts, A. L. Rosenberg, _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894)


defining sheaf

A notion of **defining sheaf of ideals** is a globalization of the notion of the [[defining ideal]] of a closed affine subvariety of an affine variety. One also says less preferrable "sheaf of definition", folowing literally the French variant.

Given a [[closed immersion of schemes]] $f= (f,f^\sharp): (Y,O_Y)\to (X,O_X)$ of (or of more general [[locally ringed spaces]], e.g. of analytic varieties), the kernel $\mathcal{I} = O_X$ of the [[comorphism]] $f^\sharp:O_Y\to f_* O_X$ is a sheaf of ideals, called the [[defining sheaf]] of the closed immersion $f$.

This can be repharsed in terms of the category of quasicoherenet sheaves. In this vein, [[Alexander Rosenberg]] defines the *defining ideal* of a [[topologizing subcategory]] $S$ of an [[abelian category]] $A$ as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of the identity $Id_A$ assigning to any $M\in A$ the intersection of [[kernels]] $Ker(f)$ of all morphisms $f: M\to N$ with $N\in Ob(S)$.

category: sheaf theory, algebraic geometry
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definite description

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### Context
#### Philosophy
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[[!include philosophy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[philosophy]], **definite description** concerns the use of the definite article, 'the', to denote an individual, as with 'the cat currently sitting in this basket' or 'the 24th President of the United States'.

A closely related use of 'the' occurs in mathematics when specifying an entity up to [[coherence|coherent]] [[isomorphism]], or more generally [[equivalence in homotopy type theory|equivalence]], such as when speaking of '*the* [[Monster group]]'. This is treated at [[generalized the]].

## Treatment in philosophy

[[Bertrand Russell]] took an important step for [[analytic philosophy]] in 1905 when he gave his famous analysis of the statement

* The present King of France is bald,

as asserting the unique existence of someone who is currently king of France and who is bald. This means that the statement uttered now is false, since there is no such person. Other philosophers, such as Peter Strawson, have taken this sentence to be meaningless due to its reliance on a false presupposition.

## Formalization

Formal treatments employ the [[uniqueness quantifier]] to specify unique existence. The use of [[homotopy type theory]] allows extension to mathematical cases of definite description ([Corfield 20, Chap. 3](#Corfield20)).

## Related entries

* [[generalized the]]
* [[uniqueness quantifier]]

## References

* Stanford Encyclopedia of Philosophy, [descriptions](https://plato.stanford.edu/entries/descriptions/)

* [[Bertrand Russell]], _On Denoting_, Mind, 14, 1905, pp. 479–493.

For a treatment of definite description using the resources of [[homotopy type theory]], which allows also for the use of [[generalized the]], see

* {#Corfield20} [[David Corfield]], Chapter 3 of [Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy](https://ncatlab.org/davidcorfield/show/Modal+Homotopy+Type+Theory).


definite form

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#Contents#
* table of contents
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## Idea

Given a [[vector space]] $V$ and an element $\phi$ in the [[exterior product]] $\wedge^p V^\ast$ (a $p$-covector), then a [[differential form|differential p-form]] $\omega$ on a [[smooth manifold]] $X$ whose [[tangent spaces]] look like $V$ is called _definite on $\phi$_ ([Bryant 05, section 3.1.1](#Bryant05)) or _stable at each point_ ([Hitchin, p. 3](#Hitchin)) if at each point $x \in X$ the restriction $\omega|_x \in \wedge^p T^\ast X \simeq \wedge^p V^\ast$ is equal to $\phi$, up to a [[general linear group|general linear transformation]].

The existence of a definite form implies a [[G-structure]] on $X$ for $G$ the [[stabilizer subgroup]] of $\phi$.

A class of examples of definite forms are the 3-forms on [[G2-manifolds]], these are definite on the "[[associative 3-form]]" on $\mathbb{R}^7$.

The [[higher prequantization]] of a definition form is a [[definite globalization of a WZW term]].

## Definition

Given a [[vector space]] $V$ and a [[stable form]] $\phi \in \wedge^p V^\ast$ (hence a form whose [[orbit]] under the [[general linear group]] $GL(V)$ is an [[open subspace]] in $\wedeg^p V$), and given a [[smooth manifold]] modeled on the [[vector space]] $V$, then a [[differential form]] $\omega \in \Omega^p(X)$ is _definite_ on $\phi$ if at each point it is in this open orbit.

## Examples

### $G_2$-manifolds

See at _[G2-manifold -- Definite forms](G2-manifold#Definite3Forms)_

## References

* {#Hitchin} [[Nigel Hitchin]], _Special holonomy and beyond_, Clay Mathematics Proceedings ([pdf](https://people.maths.ox.ac.uk/hitchin/hitchinlist/clay.pdf))

* {#Bryant05} [[Robert Bryant]], _Some remarks on $G_2$-structures_, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 [pdf](http://gokovagt.org/proceedings/2005/ggt05-bryant.pdf)

[[!redirects definite forms]]
[[!redirects definite differential form]]
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[[!redirects definite 3-form]]
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definite globalization of WZW term

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#Contents#
* table of contents
{:toc}

## Idea

Given a [[WZW term]] $\mathbf{L}$ (a [[differential cohomology|differential cocycle]]) on some $V$, and given a $V$-[[manifold]] $X$, a _definite globalization_ of $\mathbf{L}$ over $V$ is a WZW term $\mathbf{L}^X$ on $X$ which is suitably locally equivalent to $\mathbf{L}$. In particular the [[curvature]] form of $\mathbf{L}^X$ is a [[definite form]] on $X$, definite on the curvature form of the local model $\mathbf{L}$.

Hence a definite globalization of a WZW term may be thought of as a [[higher prequantization]] of a [[definite form]].

Definite gobalizations of WZW terms $\mathbf{L}$ induce [[parameterized WZW terms|definite parameterizations]], namely parameterization of the restriction $\mathbf{L}^{inf}$ of $\mathbf{L}$ to the [[infinitesimal disk]] in $V$, over the  infinitesimal disk bundle of $X$. These in turn correspond to [[G-structures]] for $G$ the [[homotopy stabilizer group]] of $\mathbf{L}^{inf}$.

## Examples

### Geometric pre-quantization

By the [[Darboux theorem]] for line bundles, every [[prequantization]] of a [[symplectic manifold]] is automatically a definite globalization of some fixed pre-quantization of $(\mathbb{R}^{2n}, \mathbf{d}p_i \wedge \mathbf{d}q^i)$.

### Super $p$-branes on curved super-spacetimes

The [[equations of motion]] of [[supergravity]] theories typically imply that the WZW curvatures of the relevant [[super p-brane sigma models]] on [[super Minkowski spacetime]] extend as [[definite forms]] over the [[super-spacetime]]. Hence the full WZW term defining the [[super p-brane sigma model]] needs to be a definite globalization over super-spacetime of the local model over super-Minkowski spacetime.

## Related concepts

* [[higher Cartan geometry]]

## References

* section "definite forms" in _[[schreiber:differential cohomology in a cohesive topos]]_

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definition

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#### Type theory
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> *There are good reasons why the theorems should all be easy and the definitions hard.* [[[Michael Spivak]], preface to "[[Calculus on Manifolds]]"]

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[type theory]] a _definition_ is the construction of a [[type]] or a [[term]] of a certain [[type]]. By "construction", we mean that the [[type]] has a [[formation rule]] and the [[term]] has an [[introduction rule]]. Then, there are in general two ways to complete the construction:

* by adding to the [[type theory]] the [[elimination rules]], [[computation rules]], and [[uniqueness rules]] for [[types]] and their associated [[terms]] which characterize the [[universal property]] of the type (see *[[natural deduction]]*),

* by adding to the type theory a rule stating that the type or term [[equality|is equal to]] an existing term or type.

Many types in type theory, such as [[function types]], [[product types]], [[coproduct types]], [[booleans type]], [[natural numbers type]], [[integers]], et cetera, are defined in both ways, by [[universal properties]] and by [[equality]] with another type.

## Rules for definitions and single assignment operators

One way that definitions of types and of terms could be formalized inside the type theory is by the use of [[equality]] with another term or type. More specifically, every definition of a symbol $A$ comes with a **formation rule** for the symbol which states that it is a type or an **introduction rule** for the symbol which states that it is a term of a type, and a **definition rule** that the term or type $A$ is equal to some existing term or type $B$. The equality used in the definition rule is called **[[definitional equality]]**.

As documented in the article on [[equality]], there are three notions of equality used in type theory: judgmental equality, propositional equality, and typal equality. All three notions of equality could be used in the definition rule. In [[Martin-Löf type theory]] and [[cubical type theory]], symbols and abbreviations are defined using judgmental equality. In [[ZFC]] and [[ETCS]], they are defined using propositional equality, and in [[objective type theories]], they are defined using typal equality.

For example, suppose that the type $B$ is already derived in some context $\Gamma$. Then, in order to define the symbol $A$ to be the type $B$ there are the following formation and definition rules for $A$:

* Formation and judgmental definition rules for $A$:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash A = B \; \mathrm{type}}$$

* Formation and propositional definition rules for $A$:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash A = B\; \mathrm{true}}$$

* Formation and typal definition rules for $A$:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \: \mathrm{ctx}}{\Gamma \vdash \delta_A:A \simeq B}$$

Similarly, suppose that the term $b:A$ is already derived in some context $\Gamma$. Then, in order to define the symbol $a$ to be the term $b:A$ there are the following introduction and definition rules for $a$:

* Introduction and judgmental definition rules for $a$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash a:A} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash a = b:A}$$

* Introduction and propositional definition rules for $a$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash a:A} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash a =_A b \; \mathrm{true}}$$

* Introduction and typal definition rules for $a$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash a:A} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \delta_a:a =_A b}$$

However, including separate rules for each new term or type is very cumbersome and would lead to an explosion of rules in the type theory. To handle that, one usually introduces the **single assignment operator**, **initialization operator**, **initialisation operator**, or **definition operator** $\coloneqq$ to the type theory, which is formally defined as a pair of judgments

* $B \coloneqq A \; \mathrm{type}$, where we judge $B$ to be defined as the type $A$, or assigned the type $A$
* $b \coloneqq a:A$, where we judge $b$ to be defined as the term $a:A$, or assigned the term $a:A$

in addition to the [[judgments]] for [[types]], [[terms]], and [[judgmental equality]]. $B \coloneqq A \; \mathrm{type}$ is called **type definition**, **type initialization**, **type initialisation** or **type single assignment**, while $b \coloneqq a:A$ is called **term definition**, **term initialization**, **term initialisation**, or **term single assignment**. Judgmental single assignment is different from judgmental equality as judgmental equality is an [[equivalence relation]], while judgmental single assignment is not an equivalence relation, but instead has a reflection rule into [[equality]].

Depending upon what notion of [[equality]] is used for [[definitional equality]], the single assignment operator has the following formation and equality reflection rules for type definitions

* Formation and judgmental equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \equiv A\; \mathrm{type}}$$

* Formation and propositional equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \equiv A\; \mathrm{true}}$$

* Formation and typal equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash P:B \simeq A}$$

There are also the following introduction and equality reflection rules for term definitions:

* Introduction and judgmental equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b \equiv a:A}$$

* Introduction and propositional equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b \equiv_A a \; \mathrm{true}}$$

* Introduction and typal equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash p:b =_A a}$$

## Copy definitions

Another way of typally defining a type $A$ to be a type $B$ is via *[[copying]]*. Copying becomes important for typally defining the [[type of equivalences]], as the usual way of typally defining types involves the [[type of equivalences]] and thus isn't available. One add rules saying that from the [[assignment]] [[judgment]] $B \coloneqq A \; \mathrm{type}$ one can derive that $B$ satisfies the universal property of a [[copy]] of $A$.

## Related concepts

* [[natural deduction]]

* [[copy]]

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

## References

The single assignment operator is defined (not by name) in Remark 2.2.1 in:

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf)) (478 pages)

See also:

* Wikipedia, _Assignment (computer science)#Single assignment_

* Wikipedia, _Assignment (computer science)#Assignment versus equality_

* Wikipedia, _Relational operator#Confusion with assignment operators_

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definition of crossed module morphisms

author: Anonymous
company:
title: definition of crossed module morphisms
subtitle:
slide_theme: default
slide_footer:
slide_subfooter:

:category: S5-slideshow

definition of crossed module morphisms
==============

My First Slide
-----------------


definition of topology

[[topological structure]]


definitional equality

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[logic]] and [[type theory]], definitional equality is the metatheoretic [[equivalence relation]] on syntactic expressions ([[types]], [[terms]], [[propositions]], [[contexts]], whatever) which states that two syntactic expressions are equivalent if they have the same meaning. Thus, definitional equality encompasses both [[syntactic equality]] and [[alpha-equivalence]], as well as equivalence of abbreviations and so forth.

According to [PML (1980), p. 31](#PML):

> Definitional equality is intensional equality, or equality of meaning (synonymy). [...] It is a relation between linguistic expressions [...] Definitional equality is the equivalence relation generated by abbreviatory definitions, changes of [[bound variables]] and the [[principle of substituting equals for equals]]. [...] Definitional equality can be used to rewrite expressions [...].

on p. 60:

> ... intensional (sameness of meaning) ...

Definitional equality is commonly written as $\equiv$, and is used to define certain symbols, abbreviations, and syntactical expressions as other syntactical expressions. For example, one might want to define the symbol $2$ as the syntactical expression $s(s(0))$ representing the successor of the successor of zero in the natural numbers; definitional equality makes it so that in the metatheory, $2 \equiv s(s(0))$.

In addition, definitions are made using the [[single assignment operator]] $\coloneqq$ in the metatheory, so one would write $2 \coloneqq s(s(0))$.

## Internalizing definitional equality

Even though definitional equality is defined on general syntactic expressions in the metatheory, for specific syntactic expressions, such as those for [[types]] and [[terms]], it is possible to internalize the notion of definitional equality inside of the object theory itself. This is formalized by the use of [[equality]] with another term or type.

More specifically, every definition of a symbol $A$ comes with a **formation rule** for the symbol which states that it is a type or an **introduction rule** for the symbol which states that it is a term of a type, and a **[[definition rule]]** that the term or type $A$ is equal to some existing term or type $B$. The equality used in the definition rule is the internal definitional equality of the theory.

As documented in the article on [[equality]], there are three notions of equality used in type theory: judgmental equality, propositional equality, and typal equality. All three notions of equality could be used as the internal definitional equality. In [[Martin-Löf type theory]] and [[cubical type theory]], [[judgmental equality]] is used for the internal definitional equality. In [[ZFC]] and [[ETCS]], [[propositional equality]] is used for the internal definitional equality, and in some [[objective type theories]], [[typal equality]] is used for the internal definitional equality.

Returning back to the initial example given in the section above, one usually defines the symbol $2$ to be the term $s(s(0))$ in the type of [[natural numbers]]. The symbol $2$ is formally defined using the internal definitional equality as follows:

* Introduction and judgmental definition rules for $2$:

$$\frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash 2:\mathbb{N}} \qquad \frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash 2 = s(s(0)):\mathbb{N}}$$

* Introduction and propositional definition rules for $2$:

$$\frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash 2:\mathbb{N}} \qquad \frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash 2 =_{\mathbb{N}} s(s(0)) \; \mathrm{true}}$$

* Introduction and typal definition rules for $2$:

$$\frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash 2:\mathbb{N}} \qquad \frac{\Gamma \vdash \mathbb{N} \; \mathrm{type}}{\Gamma \vdash \delta_2:2 =_{\mathbb{N}} s(s(0))}$$

Similarly, the symbol $\mathrm{isProp}(A)$, which is usually defined as the type
$$\prod_{x:A} \prod_{y:A} x =_A y$$
This is formally defined using the internal definitional equality as follows:

* Formation and judgmental definition rules for $\mathrm{isProp}(A)$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) = \prod_{x:A} \prod_{y:A} x =_A y \; \mathrm{type}}$$

* Formation and propositional definition rules for $\mathrm{isProp}(A)$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) = \prod_{x:A} \prod_{y:A} x =_A y\; \mathrm{true}}$$

* Formation and typal definition rules for $\mathrm{isProp}(A)$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}}{\Gamma \vdash \delta_{\mathrm{isProp}(A)}:\mathrm{isProp}(A) \simeq \prod_{x:A} \prod_{y:A} x =_A y}$$

## History

The notion of definitional equality was introduced first in [[AUTOMATH]]. The following paper presents a suggestive explanation of this notion and how proof-checking was designed in this system (especially section 10):

[On the roles of types in mathematics](http://uf-ias-2012.wikispaces.com/file/view/deB1.pdf/401388596/deB1.pdf)

The notion of definitional equality was later introduced by [[Per Martin-Löf]], first in the context of normalization proofs for higher-order logic in the paper [Hauptsatz for Intuitionistic Simple Type Theory](http://www.sciencedirect.com/science/article/pii/S0049237X09703659) and generalized in Type Theory. He discusses this notion in the paper [About Models for Intuitionistic Type Theory and The notion of Definitional Equality](http://www.sciencedirect.com/science/article/pii/S0049237X08707274).

The extension from AUTOMATH is that one adds the notion of data type (natural number), of constructors (zero and successor) and primitive recursion as definitional equality. The motivation is that one can consider the schema of primitive recursion as a definition of a function.

This was also influenced by natural deduction, where constructors correspond to introduction rules and the work of Gödel on system T.

With this extension, one obtains a programming language with dependent types and where computations correspond to unfolding of definitions (that can be primitive recursive definitions). This programming language has the feature that all computations terminate. This has been also considered in functional programming, see e.g. the discussion in [this paper](http://uf-ias-2012.wikispaces.com/file/view/turner.pdf/401400700/turner.pdf).

A description of the evaluation algorithm using techniques from functional programming
can be found in [this work of Gregoire and Leroy](http://uf-ias-2012.wikispaces.com/file/view/strong-reduction.pdf/402005168/strong-reduction.pdf).

## Note on terminology

Historically in the [[dependent type theory]] community, the term *definitional equality* was used for [[judgmental equality]]. This was because in the most common model of dependent type theory, [[Martin-Löf type theory]], judgmental equality was used for the internal definitional equality of the theory. However, we choose to make a distinction between definitional equality and judgmental equality, as not all type theories have judgmental equalities or internal definitional equalities: in particular, many [[objective type theories]] do not have judgmental equalities or an internal definitional equality, but nevertheless still have a metatheoretic definitional equality.

Even if we restrict to internal definitional equality, there is still a distinction between definitional equality and judgmental equality. First, the internal definitional equality is [[propositional equality]] in the most common [[foundations of mathematics]], such as [[ZFC]] and [[ETCS]], instead of [[judgmental equality]]. In addition, one could work in [[logic]] over [[dependent type theory]] and replace [[judgmental equality]] entirely by [[propositional equality]] throughout the type theory, resulting in a type theory which behaves almost identically to the original type theory, only that the internal definitional equality is a proposition instead of a judgment (see the second presentation of [[Martin-Löf type theory]] on that article). Finally, in the advent of [[objective type theory]] and other dependent type theories which lack [[judgmental equality]] entirely, it is [[typal equality]] which is used in the internal [[definitional equality]].

## See also

* [[equality]], [[syntactic equality]], [[alpha-equivalence]]

* [[definition]]

## References

The first paper to mention intensional equality (and the fact that it should be decidable) may be:

* [[Kurt Gödel]], _Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes_. Dialectica (1958), pp. 280–287,

The distinction between definitional equality and "book" equality:

* [[Nicolaas de Bruijn]], _[[Automath]]_,

The notion of definitional equality in the context of ([[dependent type theory|dependent]]) [[type theory]]:

* {#Thompson91} [[Simon Thompson]], §5.2.1 in: *[[Type Theory and Functional Programming]]*, Addison-Wesley (1991) [ISBN:0-201-41667-0, [webpage](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP), [pdf](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf)]

specifically in the [[Coq]] [[proof assistant]]:

* [[Adam Chlipala]], §10.1 in: _Certified programming with dependent types_, MIT Press 2013 [[ISBN:9780262026659 ](https://mitpress.mit.edu/books/certified-programming-dependent-types), [pdf](http://adam.chlipala.net/cpdt/cpdt.pdf),  [book webpage](http://adam.chlipala.net/cpdt/)]

[[!redirects Definitional equality]]
[[!redirects definitional equalities]]
[[!redirects intensional equality]]
[[!redirects intensional equalities]]
[[!redirects extensional equality]]
[[!redirects extensional equalities]]


deformation context

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Small objects
+--{: .hide}
[[!include compact object - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A **[[deformation context]]** is an [[(∞,1)-category]] $\mathcal{Y}$ such that

1. it is a [[presentable (∞,1)-category]];

1. it contains a [[terminal object in an (∞,1)-category|terminal object]]

together with a [[set]] of [[objects]] $\{E_\alpha \in Stab(\mathcal{Y})\}$ in the [[stabilization]] of $\mathcal{Y}$.

=--

This is ([Lurie, def. 1.1.3](#Lurie)) together with the assumption of a terminal object stated (and later implicialy used) on p.9.

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

Definition \ref{DeformationContext} is meant to be read as follows:

First, we think of $\mathcal{Y}$ as an [[opposite (∞,1)-category]] of [[pointed object|pointed]] spaces in some [[higher geometry]]. The point is the [[initial object]] in $\mathcal{Y}^{op}$ which is the terminal object in $\mathcal{Y}$.

Then we think of the formal duals of the objects $\{E_\alpha\}_\alpha$ as a set of generating [[infinitesimally thickened points]].

=--

The following construction generates the "[[jets]]" induced by the generating infinitesimally thickened points.

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

Given a [[deformation context]] $(\mathcal{Y}, \{E_\alpha\}_\alpha)$, we say

* a [[morphism]] in $\mathcal{Y}$ is an **elementary morphism** if it is the [[homotopy fiber]] to a map into $\Omega^{\infty -n}E_\alpha$ for some $n \in \mathbb{Z}$ and some $\alpha$;

* a morphism is a **small morphism** if it is the composite of [[finite set|finitely]] many elementary morphisms.

We write

$$
  \mathcal{Y}^{inf} \hookrightarrow \mathcal{Y}
$$

for the full [[sub-(∞,1)-category]] on those objects $A$ for which the essentially unique map $A \to *$ is small.

=--

([Lurie, def. 1.1.8](#Lurie))

## Related concepts

* [[deformation theory]]

* [[formal moduli problem]]

* [[Lie differentiation]]

* [[model structure for L-infinity algebras]]

## References

* [[Jacob Lurie]], _[[Formal moduli problems]]_
 {#Lurie}

[[!redirects deformation contexts]]


deformation functor

[[!redirects Schlessinger's criterion]]
[[!redirects schlessinger's criterion]]

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

## Idea and Motivation

One motivation for studying deformations over Artin rings is that one may wish to classify/parameterize or create a [[moduli space]] of some particular type of object. Often it is very easy to write down a [[functor of points]] for the space. For instance, curves of genus $g$, vector bundles on some fixed variety, maps between varieties, closed subschemes, quotients of a coherent sheaf, and the list goes on.

But then one must ask, is this really a space? In other words, is it [[representable functor| representable]]? This rarely happens, but even if you don't have a legitimate space, you can see what the "points" are formally locally. If everything in your original problem was noetherian, then [[completion]] of the [[local ring]] at a point gives rise to the Artin rings we consider. The deformations of the object over infinitesimal thickenings gives us a way to tell what the points near our given point look like on the original [[moduli space]].

This reduces us to now asking for a given deformation problem to be representable, but this is almost always too much to hope for as well. Schlessinger's criterion refers to two weaker notions of representability, namely that of hulls and that of [[prorepresentable functor| prorepresentability]]. The criterion gives a beautifully simple necessary and sufficient way to determine when we get each.

Once we determine these formal local points, we must then go backwards and see if these correspond to actually points in the original space. This is known as effectivization.

## Definitions

$\Lambda$ will denote a complete [[noetherian ring]].

Unless otherwise specified, everything will be in the category of local Artin $\Lambda$-algebras with residue field $k$ and will be denoted $_\Lambda Art_k$.

Define $_\Lambda Noeth_k$ to be the category of complete noetherian local $\Lambda$-algebras with residue field $k$.

A *thickening* is a surjection $A' \stackrel{f}{\to} A\to 0$ such that $I=ker(f)$ satisfies $m_{A'}I=0$.

A *small thickening* is a thickening in which $I$ is principal. Note that in our category any surjection can be factored as a sequence of small thickenings.

A *predeformation functor* is a functor $F: _\Lambda Art_k \to \text{Set}$ such that $F(k)=\{*\}$ is a singleton. This condition is the notion that we are really over a "point", and so any deformation must be the trivial one.

The functor $\widehat{F}: _\Lambda Noeth_k\to \text{Set}$ is defined to be $\displaystyle \widehat{F}(A)= \lim_n F(R/\mathfrak{m}^n)$.

A predeformation functor is [[prorepresentable functor| prorepresentable]] if $\widehat{F}$ is representable.

A hull for $F$ is a pair $(R, \eta)$ where $R\in _\Lambda Noeth_k$ and $\eta\in \widehat{F}(R)$ such that $h_R\to F$ is formally smooth and we have a bijection on tangent spaces $T_{h_R}\to T_F$.

$k[\epsilon]$ is defined to be the ring $k[\epsilon]/(\epsilon^2)$ with the trivial $\Lambda$-algebra structure.

## The Conditions

Note that whenever we are given a predeformation functor $F$, and two maps of rings: $A'\to A$ and $A''\to A$ we get an induced map $F(A'\times_A A'')\to F(A')\times_{F(A)} F(A'')$ by functoriality of $F$ and the universal property of a [[pullback]]. Call this map (\*).

Schlessinger gives four conditions in his [paper](#Schless) called (H1) - (H4).

(H1) If $A''\to A$ is a small thickening then (\*) is surjective.

(H2) If $A=k$ and $A''=k[\epsilon]$, then (\*) is bijective.

(H3) $T_F$ is finite-dimensional

(H4) If $A''=A'$ and $A'\to A$ is a small thickening, then (\*) is bijective

Geometrically the conditions can informally be thought of as follows:

One can think of (H1) as being able to "glue".

One can think of (H2) as gluing being unique over infinitesimal neighborhoods.

(H3) is having a finite dimensional tangent space

One can think of (H4) as gluing being unique on a small thickening over itself.

## Deformation Functors

We call a predeformation functor a *deformation functor* if it satisfies (H1) and (H2). Here we make precise what was meant in the motivation section. Suppose you have a moduli functor $\tilde{F}: \text{Sch}\to \text{Set}$ and you would like to know what the [[moduli space]] looks like in a neighborhood of some point say $\eta_0\in \overline{F}(\text{Spec} k)$. Then if we consider the functor of deformations of $\eta_0$, we get $F: _\Lambda Art_k \to \text{Set}$ by the following prescription $F(A)=\{ \eta \in \overline{F}(\text{Spec} A): \eta|_{\text{Spec} k}=\eta_0\}$. This $F$ can be seen to be a deformation functor.

Of course, this doesn't *always* work if we're parametrizing objects with automorphisms, but this general approach does work for most of the examples in the motivation section and explains why this class of functors is what we consider.

## Schlessinger's Criterion

The main theorem of Schlessinger's [paper](#Schless) says that  (H1), (H2), and (H3) are satisfied if and only if $F$ has a hull. The second part of the theorem is that $F$ is prorepresentable if and only if in addition (H4) is satisfied.

## Examples

Let $Def_X$ be the predeformation functor parametrizing flat deformations of $X$. Then (H1) and (H2) are always satisfied and hence it is a deformation functor. If $X$ is proper, then (H3) is satisfied and it has a hull. $Def_X$ is prorepresentable if and only if every automorphism extends over a small thickening.

Consider the node $X=\text{Spec} k[ [ x,y ] ] /(xy)$. Then the pair $(k[ [ t ] ], \text{Spec} k [ [ x,y,t ] ] /(xy - t))$ is a hull for $Def_X$. Note that this functor is not prorepresentable.

*Several more can be added.

## Tangent and Obstruction Theories

I haven't come up with a nice succinct way to do this yet. I think there is a nice streamlined gerbe/stacky way to think about it in the Brian Osserman article cited below.

## Effectivization

Given a deformation functor, $F$, that came from a moduli problem and some object $\eta\in \widehat{F}(R)$, we'd like to tell when it corresponds to a point on the original moduli space over $\text{Spec}R$, or if $(R,\eta)$ is a hull for $F$ does there exist a universal object over $\text{Spec}R$?

An example where we do have effectivization is Grothendieck's existence theorem for coherent sheaves. Let $f:X\to \text{Spec}A$ be proper and $A$ complete, local, [[noetherian ring]]. Let $A_n=A/m^n$ and $X_n=X\otimes_A A_n$ the thickenings over $A_n$. If $\mathcal{F}_n$ is a compatible collection of coherent sheaves on $X_n$, then there exists a coherent sheaf $\mathcal{F}$ on $X$ whose restriction to each $X_n$ is $\mathcal{F}_n$.

## References

See also [[deformation theory]] and references therein.

* B. Osserman, _Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces_, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, IX (2010), no. 3, 581-633.

* M. Schlessinger, _Functors of Artin rings_, Trans. AMS 130, 208-222 (1968)
{#Schless}

this was a groundbreaking article at the time, still much cited.