Edward Witten

Edward Witten is a theoretical physicist at the Institute for Advanced Study.

* [website](http://www.sns.ias.edu/~witten/)

* [wikipedia entry](http://en.wikipedia.org/wiki/Edward_Witten)

* interview by [[Hirosi Ooguri]], Notices Amer. Math. Soc, May 2015 p491 ([pdf](http://www.ams.org/notices/201505/rnoti-p491.pdf))

* _Adventures in Physics and Math_, [Kyoto Prize lecture](http://www.kyotoprize.org/en/laureates/commemorative_lectures/)  2014 ([pdf](http://www.kyotoprize.org/wp/wp-content/uploads/2016/02/30kB_lct_EN.pdf), [[WittenKyotoPrizeLecture.pdf:file]])

Witten's work originates in theoretical [[quantum field theory]] and stands out as making numerous and deep connections between that and mathematical [[geometry]] and [[cohomology]]. In the course of the 1980s Witten became the central and leading figure in [[string theory]].

Insight gained from the study of quantum field theories and specifically those relevant in [[string theory]] led Witten to mathematical results deep enough to gain him a [[Fields medal]], see [below](#FieldMedalWork). Indeed, a whole list of sub-fields in mathematics originate in aspects of Witten's work in QFT/string theory and carry his name, such as [[Chern-Simons theory]] which many people call "Chern-Simons-Witten theory", [[Wess-Zumino-Witten theory]], the [[Witten genus]], [[Gromov-Witten theory]], [[Seiberg-Witten theory]], [[Rozansky-Witten invariant]], the [[Witten conjecture]].  Other parts are still waiting to be absorbed into the mathematical literature such as [[Horava-Witten theory]], [[Diaconescu-Moore-Witten anomaly]] etc..

Despite the deeply theoretical and abstract mathematical aspects of his work, Witten has visibly always been motivated by fundamental questions in the [[phenomenology]] of the [[standard model of particle physics]] [[standard model of cosmology|and cosmology]]. (Indeed, some of his work on [[scattering amplitudes]] crucially enters into the [[experiment|experimental]] detection of the [[Higgs particle]], for more on this see at _[[string theory results applied elsewhere]]_. ) He prominently argued that specifically [[heterotic string theory]] is a plausible candidate for a fundamental [[GUT|grand unified gauge field theory]] including [[quantum gravity]].

Since about the turn of the millennium Witten has tended to more esoteric mathematical aspects of string theory, such as its relation to [[Khovanov  homology]] and [[geometric Langlands duality]] which apparently the string theory community at large is following less enthusiastically than it was the case during the excited 1990s.

#Contents#
* table of contents
{:toc}

## Quotes
 {#Quotes}

On the (lack of) understanding of [[string theory]] and [[M-theory]]:

In [Witten 87, very last sentence](#Witten87EllipticGenera):

> A properly developed theory of [[elliptic cohomology]] is likely to shed some light on what [[string theory]] really means.

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    "width": 420,
    "unit": "px",
    "margin": {
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\linebreak

{#StringTheoryAtItsFinestQuote} In an interview recorded (1988) in:

* {#DavisBrown88} P C W Davis & J Brown (eds.), *Superstrings: A theory of everything?*, Cambridge University Press 1988, 1991, Canto 1992 ([gbooks](https://books.google.us/books?id=SrSyTS4kzP8C&lpg=PP1&vq=contents&pg=PA90#v=onepage&q&f=false)):

> String theory at its finest is, or should be, a new branch of [[geometry]]. (p. 95)

> I would consider trying to elucidate this proper generalization of geometry as the central problem of physics, certainly the central problem of string theory.  (p. 96)

> What should have happened, by rights, is that the  correct mathematical structures should have been developed in the twenty-first or twenty-second century, and then finally physicists should have invented string theory as a physical theory that is made possible by these structures. (p. 102)

> we are paying the price for the fact that we didn't come by this thing in the usual way. (p. 103)

\linebreak

\begin{imagefromfile}
    "file_name": "WittenInDavisBrown_StringMathematicsGeometryII.jpg",
    "float": "right",
    "width": 420,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
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    },
    "caption": "From [Davis and Brown 1988](#DavisBrown88)"
\end{imagefromfile}

{#FurtherInDavisBrown88} Further in [Davis & Brown 1988, p. 102](#DavisBrown88):

> It's been said that string theory is part of the physics of the twenty-first century that fell by chance into the twentieth century. That's a remark that was made by a leading physicist $[$[[Daniele Amati]], see [below](#Witten21stCenturyQuote)$]$ about fifteen years ago. What he meant was that  human beings on planet Earth never had the conceptual framework that would lead them to invent string theory on purpose. String theory was invented essentially by accident in a long sequence of events, starting with the Veneziano model that was formulated in 1968. No-one invented it on purpose, it was invented by a lucky accident. By rights, twentieth century physicists shoulnd't have had the priviledge of studying this theory. By rights, string theory  shouldn't have been invented unitl our knowledge of some of the areas that are prerequisite for string theory had developed to the point that it was possible for us to have the right concept of what it was all about.

\linebreak

\linebreak

[This point](#FurtherInDavisBrown88) appears again in

* {#Witten02AstronomyIssue} [[Edward Witten]], *Can scientists' "theory of everything" really explain all the  weirdness the universe displays?*, American Scientist Astronomy Issue 2002 (pdf, [[Witten_AstronomyIssue2002.pdf:file]])

and also in [Nova interview 2003](http://www.pbs.org/wgbh/nova/elegant/view-witten.html):

> {#Witten21stCenturyQuote} Back in the early '70s, the Italian physicist, [[Daniele Amati]] reportedly said that string theory was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark.

\linebreak

In an interview with Graham Farmelo, ["The Universe Speaks in Numbers", interview 5](https://grahamfarmelo.com/the-universe-speaks-in-numbers-interview-5), 2019 (quote from 21:15 - 21:46):

> {#WhatIsMissing} I actually believe that [[string theory|string]]/[[M-theory]] is on the right track toward a deeper explanation.  But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is  missing or where to find it. The reason I am not is that in hindsight it is clear the view given in the 1980s of what is missing was too narrow. Instead of discovering what we thought was missing, we broadened the picture in the 90s, in unexpected directions.

\linebreak

## Fields medal work
 {#FieldMedalWork}

In

* [[Michael Atiyah]], _On the work of Edward Witten_, Proceedings of the International Congress of Mathematics, Kyoto 1990 ([pdf](http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf))

the following "influential papers" are listed as relevant for Edward Witten receiving the [Fields Medal in 1990](http://159.226.47.99:8080/general/prize/medal/1990.htm).

First of all

* _Supersymmetry and Morse theory_,  J. Differential Geom. Volume 17, Number 4 (1982), 661-692. ([Euclid](http://projecteuclid.org/euclid.jdg/1214437492))

  This discusses deformations of [[supersymmetric quantum mechanics]] on a [[Riemannian manifold]] and how its supersymmetric ground states are related to the [[Morse theory]] of a deformation function. The way this supersymmetric quantum mechanics appears as the point-[[particle]] limit of the [[type II superstring]] is explained at the end of

* _Global anomalies in string theory_, in W. Bardeen and A. White (eds.) _Symposium on Anomalies, Geometry_, Topology, pp. 61–99. World Scientific, 1985

which otherwise is on [[quantum anomalies]] in string theory, such as the [[Green-Schwarz anomaly]], etc.

Finally Atiyah's section 2 mentions

* (with [[Cumrun Vafa]]), _Eigenvalue inequalities for Fermions in gauge theories_, Comm. Math. Phys. 95 (1984) 257 ([Euclid](http://projecteuclid.org/euclid.cmp/1103941573))

Then

* {#Witten87EllipticGenera} [[Edward Witten]],  _Elliptic Genera And Quantum Field Theory_ , Commun.Math.Phys. 109 525 (1987) ([euclid:1104117076](http://projecteuclid.org/euclid.cmp/1104117076))

  on the [[elliptic genus]]/[[Witten genus]] as the [[partition function]] of the [[type II superstring]]/[[heterotic superstring]]

And various articles on the foundations of [[topological field theory]] such as

* _Topological quantum field theory_, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 ([Euclid](http://projecteuclid.org/euclid.cmp/1104161738))

  on [[topologically twisted D=4 super Yang-Mills theory]] and the corresponding twists of the [[superstring]] which yield the [[A-model]] and [[B-model]] [[topological string]] 2d [[TQFTs]];

* _On the structure of the topological phase of two dimensional gravity_ Nuclear Phys, B 340 (1990) 281 ([scans](http://ccdb5fs.kek.jp/cgi-bin/img_index?200033155))

  on the [[topological string]] limit of the [[worldsheet]] [[2d quantum gravity]] theory of the string, including the [[A-model]] and [[B-model]] [[topological twists]] of the [[type II superstring]];

and

* _Quantum field theory and the Jones polynomial_, Comm. Math. Phys, 121 (1989) 351

  on the [[quantization of 3d Chern-Simons theory]], its [[Jones polynomial]] [[knot invariant]] [[quantum observables]] and its [[holographic principle|holographic]] relation to the quantization of the [[WZW model]] (the [[string]] propagating on a suitable [[Lie group]] [[manifold]]).

## Related $n$Lab entries

* [[string theory]], [[M-theory]]

* [[M-theory on G2-manifolds]]

* [[Gromov-Witten invariants]], [[Rozansky-Witten invariant]]

* [[Seiberg-Witten theory]]

* [[supersymmetric quantum mechanics]]

* [[fuzzy dark matter]]

* [[Chern-Simons gravity]]

(...)

>clearly, this list deserves be further expanded...

category: people

[[!redirects E. Witten]]
[[!redirects Witten]]


Edwin E. Moise

Edwine Evariste Moise was a mathematician based at Michigan, Harvard, and CUNY.

He got his Ph.D. degree in 1947 from UT Austin, advised by [[Robert Lee Moore]].

His Ph.D. students include [[James R. Munrkes]].

* [Wikipedia entry](https://en.wikipedia.org/wiki/Edwin_E._Moise)

## Selected writings

Proving the [[triangulation theorem]] for [[3-manifolds]]:

* {#Moise52} [[Edwin E. Moise]],  *Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung*, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 ([doi:10.2307/1969769](https://doi.org/10.2307/1969769), [jstor:1969769](https://www.jstor.org/stable/1969769))

category: people

[[!redirects Edwin Moise]]


Edwin Floyd

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

## Selected writings

On [[closed covers]] in [[Čech homology]]:

* [[Edwin E. Floyd]], *Closed coverings in Čech homology theory*, Trans. Amer. Math. Soc. **84** (1957) 319-337  [[doi:10.1090/S0002-9947-1957-0087100-2](https://doi.org/10.1090/S0002-9947-1957-0087100-2), [pdf](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0087100-2/S0002-9947-1957-0087100-2.pdf)]

Introducing [[equivariant bordism homology theory]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Differentiable periodic maps_, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86 ([euclid:bams/1183524501](https://projecteuclid.org/euclid.bams/1183524501))

* [[Pierre Conner]], [[Edwin Floyd]], _Differentiable periodic maps_, Ergebnisse der Mathematik und ihrer Grenzgebiete **33**, Springer 1964 ([doi:10.1007/978-3-662-41633-4](https://link.springer.com/book/10.1007/978-3-662-41633-4))

* [[Pierre Conner]], [[Edwin Floyd]], _Maps of Odd Period_, Annals of Mathematics Second Series, Vol. 84, No. 2 (1966), pp. 132-156 ([jstor:1970515](https://www.jstor.org/stable/1970515))

Introducing [[Conner-Floyd Chern classes]] and the [[complex cobordism theory]]-interpretation of the [[e-invariant]] via the [[Todd genus]] of complex [[manifolds with boundary]] ([[MUFr]]):

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

On the [[torsion subgroup]] of the [[SU-bordism ring]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Torsion in $SU$-bordism_, Memoirs of the American Mathematical Society 1966 ([ISBN:978-1-4704-0007-1](https://bookstore.ams.org/memo-1-60))

On [[MU]]-[[bordism theory]]:

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

## Related $n$Lab entries

* [[Conner-Floyd Chern classes]]

* [[Conner-Floyd isomorphism]]

[[!redirects Edwin E. Floyd]]
[[!redirects Edwin Earl Floyd]]

category: people


Edwin Hall

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

## Selected writings

Introducing the [[Hall effect]]:

* {#Hall1879} [[Edwin Hall]], _On a New Action of the Magnet on Electric Currents_, American Journal of Mathematics Vol. 2, No. 3 (Sep., 1879), pp. 287-292 ([doi:10.2307/2369245](https://doi.org/10.2307/2369245))

category: people


Edwin Hewett

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

## Selected writings

On [[harmonic analysis]]:

* [[Edwin Hewett]], [[Kenneth Ross]], *Abstract Harmonic Analysis*

  Volume I: *Structure of Topological Groups, Integration Theory, Group Representations*,  Grundlehren der mathematischen Wissenschaften **115**, Springer 1979 ([doi:10.1007/978-1-4419-8638-2](https://link.springer.com/book/10.1007/978-1-4419-8638-2))

  Volume II: *Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups*,  Grundlehren der mathematischen Wissenschaften **152** ([doi:10.1007/978-3-662-26755-4](https://link.springer.com/book/10.1007/978-3-662-26755-4))

category: people


Edwin Hubble

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

## Selected writings

Discussion of what came to be known as *[[Hubble's law]]*:

* [[Edwin Hubble]], *A relation between distance and radial velocity among extra-galactic nebulae*, PNAS **15** (1929) 168-173 [[doi:10.1073/pnas.15.3.168](https://doi.org/10.1073/pnas.15.3.168)]

category: people


Edwin Jaynes

Edwin Jaynes (1922-1998), physicist and proponent of an objective [[Bayesian reasoning|Bayesian]] account of probability and statistics.

[Obituary](http://bayes.wustl.edu/etj/etj.html) by Larry Bretthorst.

##Works

* _Probability Theory: The Logic of Science_,  Cambridge University Press, 2003.

## related $n$Lab entries

* [[Bayesian reasoning]]

* [[statistical significance]]


Edwin Spanier

__Edwin Spanier__ was an American mathematician at the University of California at Berkeley, working in algebraic topology.  He is probably best known as the author of the influential textbook on [[algebraic topology]]

## Selected writings

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, Springer 1966, corrected reprint 1981 ([doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1))

      @misc{spanier1981algebraic,

      title={Algebraic Topology, corrected reprint of the 1966 original},

      author={Spanier, E.H.},

      year={1981},

      publisher={Springer, New York}

## Related $n$Lab entries

* [[Spanier-Whitehead category]]

* [[Spanier-Whitehead duality]]

* [[Alexander-Spanier cohomology]].

category: people

[[!redirects Spanier]]


EFC-algebra

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

#Contents#
* table of contents
{:toc}

## Definition

An __entire functional calculus algebra__ is a [[product]]-[[preserved limit|preserving]] [[functor]]
$$CartHolo \to Set,$$
where $CartHolo$ is the [[category]] of [[finite dimensional vector space|finite-dimensional]] [[complex vector spaces]]
and [[holomorphic maps]].

This is in complete analogy to [[C^∞-rings]],
and EFC-algebras are applicable in similar contexts.

## Properties

The category of globally [[finitely presented object|finitely presented]] [[Stein spaces]]
is [[contravariant functor|contravariantly]] [[equivalence of categories|equivalent]] to the category of [[finitely presented object|finitely presented]]
EFC-algebras.
The equivalence functor sends a [[Stein space]] to its EFC-algebra of [[global sections]].

The category of [[Stein spaces]] of finite embedding dimension
is contravariantly equivalent to the category of finitely generated EFC-algebras.
The equivalence functor sends a [[Stein space]] to its EFC-algebra of global sections.

These statements can thus be rightfully known as [[Stein duality]].

## Related concepts

* [[C^∞-ring]]

* [[Stein space]]

* [[Stein duality]]

* [[duality between geometry and algebra]]

## References

* [[Alexei Pirkovskii]], *Holomorphically finitely generated algebras*,  Journal of Noncommutative Geometry 9 (2015), 215–264 ([arXiv:1304.1991](https://arxiv.org/abs/1304.1991), [doi:10.4171/JNCG/192](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=9&iss=1&rank=8)).

* [[J. P. Pridham]], *A differential graded model for derived analytic geometry*,  Advances in Mathematics 360 (2020), 106922.  arXiv:1805.08538v1, doi:10.1016/j.aim.2019.106922.

[[!redirects EFC-algebras]]
[[!redirects entire functional calculus algebra]]
[[!redirects entire functional calculus algebras]]


Eff

*This page is about the programming language.  $Eff$ is also a commonly used name for the [[effective topos]].*

#Contents#
* table of contents
{:toc}

##Idea

**Eff** is a functional programming language based on algebraic effects and their [[effect handler|handlers]].

##References

* [[Andrej Bauer]], [Eff](http://math.andrej.com/eff/)

* [[Andrej Bauer]], [[Matija Pretnar]], _Programming with Algebraic Effects and Handlers_, [arXiv:1203.1539](https://arxiv.org/abs/1203.1539)


effect algebra

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

> Caveat: There is an unrelated notion of "effect of a computation"; that is rather in proximity to the entry [[monad (in computer science)]].

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum mechanics]] a [[self-adjoint operator]] $A$ on the given [[Hilbert space]] such that its [[spectrum of an operator|spectrum]] lies between 0 and 1 (hence a [[positive operator]] which is $\leq 1$) is sometimes called an _effect_ or _quantum effect_ (see e.g. ([Ludwig](#Ludwig), [Kraus](#Kraus))). These operators generalize [[projection operators]] and may be thought of as [[quantum observables]] with "unsharp" or "fuzzy" value.

The notion of _effect algebra_ (due to ([Foulis-Bennet 94](#FoulisBennet))) is an abstraction of the structure exhibited by the collection of such effect operators.

## Definition

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

A [[partial monoid|partial commutative monoid]] (PCM) consists of a set $M$ with a zero element $0 \in  M$ and a partial binary operation $\vee : M \times M \to M$ satisfying the three requirements below. They involve the notation $x \perp y$ for: $x \vee y$ is defined; in that case $x, y$ are called orthogonal.

1. Commutativity: $x\perp y$ implies $y\perp x$ and $x\vee y=y\vee x$.

1. Associativity: $y\perp z$ and $x\perp(y\vee z)$ implies $x\perp y$ and $(x\vee y)\perp z$ and $x\vee (y\vee z)=(x\vee y)\vee z$.

1. Zero: $0\perp x$ and $0\vee x=x$

([Foulis-Bennet 94](#FoulisBennet) p.22)

In a PCM, we define: $x \le y:\Leftrightarrow \exists_z. x \vee z = y$.  This is a preorder on any PCM.
=--

+-- {: .num_prop}
###### Proposition
A PCM is preordered by $\le$.
=--

+-- {: .proof}
###### Proof
Reflexivity is immediate from the Zero axiom, and transitivity follows easily from Associativity.
=--

+-- {: .num_defn}
###### Definition
A *generalized effect algebra* is a PCM $(E, 0, \vee)$ such that:

1. Cancellation Law: If $a \perp b$, $a \perp c$ and $a \vee b = a \vee c$ then $b = c$.

2. Positivity Law: If $a \perp b$ and $a \vee b = 0$ then $a = b = 0$.

In a generalized effect algebra, we define: $y\ominus x=z:\Leftrightarrow y=x\vee z$ (which exists iff $x \le y$, and is unique by the Cancellation Law).
=--

+-- {: .num_prop}
###### Proposition
A generalized effect algebra is partially ordered by $\le$.
=--

+-- {: .proof}
###### Proof
Suppose $x \le y$ and $y \le x$.  Let $x \vee a = y$ and $y \vee b = x$.  Then $x \vee (a \vee b) = x = x \vee 0$, and so $a \vee b = 0$ by the Cancellation Law.  Therefore, $a = b = 0$ and so $x = y$.
=--

+-- {: .num_defn}
###### Definition
An *effect algebra* is a PCM $(E,0,\vee)$ with an orthocomplement. The latter is a unary operation $(-)^\perp :E\to E$ satisfying:

1. Orthocomplement Law. $x^\perp\in E$ is the unique element in $E$ with $x\vee x^\perp=1$, where $1=0^\perp$.

2. Zero-One Law.  $x\perp 1\Rightarrow x=0$.

For such an effect algebra one defines: $x\wedge y:=(x^\perp\vee y^\perp)^\perp$
 ([Foulis-Bennet 94](#FoulisBennet) p. 23)
=--

+-- {: .num_prop}
###### Proposition
A structure $(E, 0, \vee)$ is an effect algebra iff it is a generalized effect algebra with a greatest element, in which case that greatest element is $1 = 0^\perp$.
=--

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

Let $(E, 0, \vee)$ be an effect algebra.  Then $E$ is a generalized effect algebra since:

1. Cancellation Law.  If $a \vee b = a \vee c$ then $a \vee b \vee (a \vee b)^\perp = a \vee c \vee (a \vee b)^\perp = 1$, and so $b = c = (a \vee (a \vee b)^\perp)^\perp$.

2. Positivity Law.  If $a \vee b = 0$ then $(a \vee b) \perp 1$, hence $a \perp 1$ and $b \perp 1$ by Associativity.  Thus, $a = b = 0$ by the Zero-One Law.

1 is the greatest elements since, for any $x$, we have $x \vee x^\perp = 1$ and so $x \leq 1$.

Conversely, let $(E, 0, \vee)$ be a generalized effect algebra with greatest element 1.  Define $x^\perp = 1 \ominus x$ for all $x$.  Then:

1. Orthocomplement Law. $x^\perp$ is the unique element such that $x \vee x^\perp = 1$ by definition.

2. Zero-One Law.  If $x \perp 1$, then $1 \leq x \vee 1$, so $x \vee 1 = 1$.  Thus, $x \vee 1 = 0 \vee 1$, and so $x = 0$ by the Cancellation Law.
=--

+-- {: .num_remark}
###### Remark
If we consider $(-)\ominus y:up(y)\to down(y^\perp)$ and $(-)\vee y:down(y^\perp)\to up$ as functors between [[posets]] we have [[adjunctions]]

$$((-\wedge y)\dashv (-\ominus y)\dashv (-\vee y)$$

Hence these functors are a [[Frobenius functor|frobenius pair]].

 ([Foulis-Bennet 94](#FoulisBennet) p.25)
=--

## Morphisms

+-- {: .num_defn}
###### Definition
Let $E$ and $F$ be effect algebras.
A *morphism of effect algebras* $f : E \rightarrow F$ is a function such that:

1. f(1) = 1

2. If $x \perp y$ then $f(x) \perp f(y)$ and $f(x \vee y) = f(x) \vee f(y)$.

We write $\mathbf{EA}$ for the category of effect algebras and morphisms of effect algebras.
=--

## Examples

(1) [[effect algebra of predicates]]

(2) The real unit inteval $[0,1]$ with $\vee$ being addition of real numbers is an effect algebra since $[0,1]$ is a pcm with zero object $0$ and commutative, associative addition of real numbers and $x\perp y$ iff $x+y\le 1$. The orthocomplement of $x\in [0,1]$ is given by $\x^\perp=1-x$.

(3) Let $D$ denote the discrete-probability-distribution [[monad]] on $Set$ which sends a set $X$ to the collection

$$D(X):=\{r_1 x_1+\dots +r_n x_n|x_i\in X, r_i\in [0,1], \Sigma_i r_i=1\}$$

of formal convex combinations of elements of $X$ and let $Kl(D)$ denote the [[Kleisli category]] of $D$ which has as objects (just) sets and a morphism $f:X\to Y$ in $Kl(D)$ is a function $f:X\to D(Y)$ which can be interpreted as a [[Markov chain]] where the [[probability]] of the transition $x\to x_i$ is the coefficient $r_i\in [0,1]$ in the the convex sum $f(x)=r_1 x_1+\dots+r_n x_n$. $Kl(D)$ has as coproducts coproducts of $Set$. A predicate on $X\in Kl(D) $is hence a function $p:X\to D(X+X)$ and $[id_X,id_X]\circ p =id_X$ means that $p(x)\in D(X+X)$ is a convex combination of elements of the form $k_1 x, k_2 x\in X+X$ such that we have $p(x)=\varphi(x)k_1 x +\psi(x)k_2 x$ with $\varphi(x),\psi(x)\in [0,1]$ such that $\varphi{x}+\psi(x)=1$. Hence $p(x)$ can be written as $p(x)=\varphi(x)k_1 x + (1-\varphi(x))k_2 x$. In particular a predicate is (uniquely determined by) a function $\varphi:X\to [0,1]$ to the [[unit interval]]. In this view the orthocomplemet of $\varphi(x)$ is the function $x\mapsto 1-\varphi(x)$ which is point-wise the orthocomplement of the unit interval in the second example.

(4) In the category $Hilb$ of [[Hilbert space|Hilbert spaces]] the coproduct coincides with the product and hence is a [[biproduct]]. In this case a predicate $p:X\to X\otimes X$ on a Hilbert space $X$ has the form $p=\lt p_1,p_2\gt$ of a pair of maps and   $[id_X,id_X]\circ p$ is equivalent to $p_1 +p_2=id_X$ where $+$ is point-wise addition. In particular $p_1$ and $p_2$ determine each other uniquely.

And now comes the eponymous feature: The category $Hilb$ is a [[dagger category]] and the dagger morphism $(-)^\dagger:Hilb^{op}\to Hilb$ is the identity on objects and complex conjugation on morphisms. An endomorphism $f:X\to X$ is called to be a *positive endomorphism* if there is a $g$ such that $f=g^\dagger\circ g$ and a predicate on $X$ is called to be an **effect** (on $X$) if $p_1$ and $p_2$ are positive. Another name for effect is "unsharp predicate"; in this terminology a "sharp predicate" is a subset of the set of projections onto $X$.

(5) In a C$^\ast$-algebra the elements between 0 and 1 form an effect algebra with $(1-a)$ as the complement of $a$.

(6) As a special case, we obtain the effect algebra of a [[von Neumann algebra]]. In general, this is not a lattice. [De Groote](#DeGroote) defines a _spectral order_ on self-adjoint operators which makes the collection of effects a [[boundedly complete lattice]]. However, this is not the canonical order on an effect algebra, as defined above.

## Relation to D-Posets

A parallel concept in the literature is that of _D-poset_ (sometimes called _D-lattice_), originally introduced for the same purpose of studying fuzzy or quantum logics. These first appeared in [Kôpka 92](#Kopka) [Chovanec-Kôpka 95](#ChovKop).

+-- {: .num_defn}
###### Definition
A partial binary operation $\ominus$ on a poset $(P, \leq)$ is called a _difference operation_ (or simply _difference_) on $P$ iff:

(1) $a \leq b \leftrightarrow b \ominus a$ is defined,

(2) $b \ominus a \leq b$,

(3) $b \ominus (b \ominus a) = a$,

(4) if $a \leq b \leq c$ implies that $c \ominus b \leq c \ominus a$ and $(c \ominus a) \ominus (c \ominus  b) = b \ominus a$.

A _D-poset_ is a poset $(P, \leq, \ominus, 1)$ with a difference operation and greatest element $1 \in P$.
=--

Any effect algebra is automatically a D-poset under the difference $c := b \ominus a \iff a \oplus c = b$, well-defined by the cancellation property of generalized effect algebras. Ultimately this determines an isomorphism of categories between D-posets and effect algebras.

## Relation to Boolean algebras

Every [[Boolean algebra]] is an effect algebra, with $a\bot b$ whenever $a \wedge b=0$. Then the partial monoid of the effect algebra is a restriction of the join of the Boolean algebra. The orthocomplement is $a^\bot=\neg a$.

Starting from a [[powerset]] Boolean algebra, for example, we have $a\bot b$ when $a$ and $b$ are disjoint, and $a\vee b$ is the disjoint union.

A function is a Boolean algebra homomorphism if and only if it is a morphism of effect algebras.

In fact the finite Boolean algebras form a [[dense subcategory]] of the category $\mathbf{EA}$ of effect algebras. This means that every effect algebra is a canonical colimit of finite Boolean algebras.

Since the powerset functor $\mathbf{FinSet}\to \mathbf{FinBool}^{\mathrm{op}}$ is an equivalence of categories ([[Stone duality#StoneSpacesAndProfiniteSets|Stone duality]]), this density property also means that we have a full and faithful functor $T:\mathbf{EA}\to [\mathbf{FinSet},\mathbf{Set}]$.
This functor can be given explicitly as finding the __tests__ of an effect algebra.
A test is a sequence of orthogonal elements that sum to $1$. Hence

$$ T(E) (n) = \{ (a_1,\dots,a_n)\ |\ a_1\vee\dots \vee a_n=1\} $$

For example, starting from the unit interval, $T([0,1])(n)$ gives the $(n-1)$-[[simplex#TopologicalSimplex|simplex]].

## Related concepts

* [[separation algebra]]

[[!include states and observables -- content]]

## References

On quantum effect operators:

* {#Ludwig} G. Ludwig, _Foundations of Quantum Mechanics I_ Springer Verlag, New York, (1983)

* {#Kraus} [[Karl Kraus]], *States, Effects, and Operations -- Fundamental Notions of Quantum Theory*, Lecture Notes in Physics **190** Springer (1983) [[doi:10.1007/3-540-12732-1](https://doi.org/10.1007/3-540-12732-1)]

* [[Teiko Heinosaari]], [[Mário Ziman]], Section 2 of: *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

The notion of effect algebra is due to

* D. J. Foulis, M. K. Bennet, _Effect algebras and unsharp quantum logics_, Found. Phys. 24 (1994), 1 331–1 352.
 {#FoulisBennet}

Discussion of effect algebras in the context of [[categorical logic]] is in

* [[Bart Jacobs]], _New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic_, (2012) ([arXiv:1205.3940 ](http://arxiv.org/abs/1205.3940))
 {#Jacobs}

Discussion in the context of [[quantum logic]] is in section 6 of

* Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto
Giuntini and Francesco Paoli, _Quantum Logic and Nonclassical Logics_, p. 127 in  Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) _Handbook of Quantum Logic and Quantum Structures: Quantum Logic_, 2009 North Holland
  {#CCGP09}

A survey of the use of effect algebras in [[quantum mechanics]] is in

* Teiko Heinosaari, Mario Ziman, _Guide to Mathematical Concepts of Quantum Theory_ ([arXiv:0810.3536](http://arxiv.org/abs/0810.3536))

  also appeared as:

  _The Mathematical Language of Quantum Theory_, Cambridge University Press (2011)

* Hans de Groote, _On a canonical lattice structure on the effect algebra of a von Neumann algebra_ [arXiv:0410018](http://arxiv.org/abs/math-ph/0410018){#DeGroote}

Discussion in relation to [[presheaves]] on [[FinSet]]${}^{op}$ (hence in the [[classifying topos]] for objects) and density of Boolean algebras is in

* Sam Staton, Sander Uijlen, _Effect algebras, presheaves, non-locality and contextuality_ ([pdf](http://www.cs.ru.nl/S.Uijlen/ICALP-camera-ready.pdf))

D-posets were first introduced in:

* F. Kôpka,_D-posets of fuzzy sets_, Tatra Mountains Mathematical Publications, vol. 1, no. 1, pp. 83-87, 1992. {#Kopka}

* F. Chovanec and F. Kôpka, _D-lattices_, International Journal of
Theoretical Physics, vol. 34, no. 8, pp. 1297–1302, 1995. {#ChovKop}

[[!redirects effect algebras]]

[[!redirects effect]]
[[!redirects effects]]

[[!redirects quantum effect]]
[[!redirects quantum effects]]
[[!redirects effect algebra]]


effect algebra of predicates

## Idea

In ([Jacobs 12](#Jacobs)) the approach to categorical logic where the substrate carrying the logical notions are [[Heyting algebras]] of [[subobjects]] in a [[topos]] is replaced by a new one where the substrate is [[effect algebra of predicates]] in [[extensive categories]].

## Predicates

Let $C$ be a category with coproducts (written $+$). A **predicate** is a map $X\stackrel{p}{\to} X+X$ such that $[id_X,id_X]\circ p=id_X$.

A predicate is hence in particular a [[nLab:coalgebra for an endofunctor|coalgebra]] for the $C$-endofunctor  $X\mapsto X+X$.

If we denote the coproduct inclusions by $k_1:X\to X+X$ and $k_2:X\to X+X$, then their coproduct $[k_2,k_1]:X+X\to X+X$ is the swap map (which is a self-inverse equivalence).

For a predicate $p$ the **orthocomplement of $p$** is defined to be the composite $p^\perp:=[k_2,k_1]\circ p$. We have $(p^\perp)^\perp\simeq p$.

## Effect algebra of predicates
+-- {: .un_lemma}
###### Lemma
In a extensive category we have the following diagrams involving coproducts and pullbacks:

(...lots of diagrams...)
=--

+-- {: .un_cor}
###### Corollary
Predicates with the operations defined by (...) assemble to an effect algebra called **the effect algebra of predicates**.
=--

## Examples

+-- {: .un_remark}
###### Examples

If $X\in Set$ is a set and $p:X\to X+X$ is a predicate on $X$, then $p(x)=k_1 x$ or $p(x)=k_2 x$.

In particular we can identify $p$ with a subset $U=\{x|p(x)=k_1 x\}\subseteq X$. Then $\neg U=\{x|p(x)=k_2 x\}=\{x|p^\perp(x)=k_1 x\}$

=--

+-- {: .proof}
###### Proof
In $Set$ we have [[nLab:disjoint coproduct|disjoint coproducts]] such that in our case we have
$$\array{
0&\to &X\\
\downarrow&&\downarrow^{k_2}\\
X&\stackrel{k_1}{\to}&X+X
}$$
is a pushout square as well as a pullback square. And in $Set$ we have [[nLab:universal colimit|universal coproducts]] such that in our case we have $X+X\simeq X$. Now if we assume $[id_X,id_X]\circ p\simeq id_X$ the couniversal property of the coproduct implies that $p$ factors through $k_1$ and $k_2$ followed by an equivalence.

A category having disjoint- and universal coproducts is called an [[nLab:extensive category]].
=--

Further examples are listed at

* [[nLab:coalgebra of the real interval]]

* some of the examples at [[nLab:coalgebra for an endofunctor]]

## References

* [[Bart Jacobs]], _New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic_, (2012) ([arXiv:1205.3940 ](http://arxiv.org/abs/1205.3940))
 {#Jacobs}


effect handler

_stub_

##Idea

An **effect handler** is an interpretation of the [[algebraic effects]] of a computation.

> Conceptually, algebraic operations and effect handlers are dual: the former could be called effect _constructors_ as they give rise to the effects; the latter could be called effect _deconstructors_ as the computations they provide proceed according to the effects already
created. ([PlotPret13](#PlotPret13))

Handlers have also been defined ([Lindley14](#Lindley14)) for more general effects, such as [[applicative functors]] and [[arrow (in computer science)|arrows]] (or idioms).

##References

* [[Matija Pretnar]], _An Introduction to Algebraic Effects and Handlers_, [tutorial](http://www.eff-lang.org/handlers-tutorial.pdf)

* {#PlotPret13} [[Gordon Plotkin]], [[Matija Pretnar]], _Handling Algebraic Effects_, [arXiv:1312.1399](https://arxiv.org/abs/1312.1399)

* Ohad Kammar, Sam Lindley, Nicolas Oury, _Handlers in Action_, [pdf](https://homepages.inf.ed.ac.uk/slindley/papers/handlers.pdf)

* Nicolas Wu, Tom Schrijvers, Ralf Hinze, _Effect Handlers in Scope_, [pdf](http://www.cs.ox.ac.uk/people/nicolas.wu/papers/Scope.pdf)

* {#Lindley14} Sam Lindley, _Algebraic Effects and Effect Handlers for Idioms and Arrows_, ([pdf](http://homepages.inf.ed.ac.uk/slindley/papers/aeia.pdf))

[[!redirects algebraic handler]]
[[!redirects algebraic handlers]]
[[!redirects effect handlers]]


effective action

* in [[group theory]]/[[representation theory]]:

  *[[effective group action]]*

* in [[quantum field theory|quantum]][[field theory]]:

  *[[effective action functional]]*

category: disambiguation


effective action functional

> this entry is about the concept of [[quantum field theory|quantum]][[field theory]], for the concept in [[group theory]]/[[representation theory]] see at [[effective group action]]; for disambiguation see [[effective action]]

+-- {: .rightHandSide}
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#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

In the context of ([[perturbative QFT|perturbative]]) [[quantum field theory]] an _effective action_ is an object obtained from a more fundamental ([[interaction]]) [[action functional]] by "averaging out" certain dependencies.

## Details

* For the plain effective action see at _[[S-matrix]]_ the section _[Effective action](S-matrix#EffectiveAction)_.

* More generally, for the _[[relative effective action]]_ see at _[[effective field theory]]_ the section _[Relative effective actions](effective+quantum+field+theory#RelativeEffectiveAction)_.

## Related concepts

* [[quasiparticle]]

## References

A discussion (with an eye towards [[supersymmetric quantum field theory]] and [[Seiberg duality]]) is in  section 2.1 of

* Flip Tanedo, _Notes on Seibergology_ ([pdf](http://www.lns.cornell.edu/~pt267/files/notes/Seibergology.pdf))

A review of effective actions in the [[background field formalism]] is in section 3.1 of

* Darren Trevor Grasso, _Higher order contributions to the effective action of $\mathcal{N} = 2$ and $4$ supersymmetric Yang-Mills theories from heat kernel techniques in superspace_ Thesis (2007) ([web](
http://repository.uwa.edu.au:80/R/-∞func=dbin-jump-full&object_id=9240&current_base=GEN01-INS01))

[[!redirects effective action functionals]]

[[!redirects effective action]]
[[!redirects effective actions]]

[[!redirects 1PI action]]
[[!redirects 1PI actions]]

[[!redirects relative effective action]]
[[!redirects relative effective actions]]


effective epimorphism

+-- {: .rightHandSide}
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#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Effective epimorphisms
* table of contents
{:toc}

## Idea

An _effective epimorphism_ is a [[morphism]] $c\to d$ in a [[category]] which behaves in the way that a [[covering]] is expected to behave, in the sense that "$d$ is the union of the parts of $c$, identified with each other in some specified way".

## Definition

\begin{definition}
An **effective epimorphism** in a [[category]] $C$ is a [[morphism]] $f \colon c \to d$ that has a [[kernel pair]] $c \times_d c$ and is the [[quotient object]] of this kernel pair, in that

$$
  c \times_d c \;\rightrightarrows\; c \overset{f}{\to} d
$$

is a [[colimit]] diagram (a [[coequalizer]]).
\end{definition}

In other words, this says that $f : c \to d$ is effective if $d$ is the [[coimage]] of $f$.

Sometimes we say that such morphism $f$ is an [[effective quotient]].

The [[formal duality|dual]] concept is that of [[effective monomorphism]].

\begin{remark}
A morphism with a [[kernel pair]] (such as any morphism in a category with [[pullbacks]]) is an effective epimorphism if and only if it is a [[regular epimorphism]] (see [there](regular+epimorphism#MorphismsWithKernelPairsAreRegularIffEffective)) and a [[strict epimorphism]].  For morphisms without kernel pairs, the notion of effective epimorphism is of questionable usefulness.
\end{remark}

## Properties

### Relation to other notions of epimorphism

Every effective epimorphism is, of course, a [[regular epimorphism]] and hence a [[strict epimorphism]].  Conversely, a strict epimorphism which *has* a kernel pair is necessarily an effective epimorphism.  (This is a special case of the theory of [[generalized kernels]].)  For this reason, some writers use "effective epimorphism" in general to mean what is here called a [[strict epimorphism]].

## Examples

* In [[Set|the category of sets]], every [[epimorphism]] is effective.  Thus, it can be hard to know, when generalising concepts from $\Set$ to other categories, what kind of epimorphism to use.  In particular, one may define a [[projective object]] (and hence the [[axiom of choice]]) using effective epimorphisms.

* More generally, in any [[pretopos]], hence in particular in every [[topos]], every [[epimorphism]] is an effective epimorphism.  See, for instance, ([MacLane & Moerdijk 1992, Thm IV.7.8](#MacLaneMoerdijk), [Borceux 1994III, Prop. 3.4.13, 3.4.15](#Borceux94III)).

* In an [[(∞,1)-topos]] the bare notion of epimorphism disappears, and [[effective epimorphism in an (∞,1)-category]] becomes the default notion of epiness. A morphism in an $(\infty,1)$-topos is effective epi precisely if its [[n-truncated object in an (∞,1)-category|0-truncation]] is an epimorphism (hence an effective epimorphism) in the underlying 1-topos. This is Proposition 7.2.1.14 in Higher Topos Theory.

## Related concepts

* [[epimorphism]], [[regular epimorphism]], **effective epimorphism**

* [[effective epimorphism in an (∞,1)-category]]

## References

Original articles:

* {#Grothendieck61} [[Alexander Grothendieck]], p. 101 (4 of 21) in: _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf))

Textbook accounts:

* [[Francis Borceux]], Def. 2.5.3 in: *[[Handbook of Categorical Algebra]]*, Vol. 2: *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

Exposition and examples:

* [[Qiaochu Yuan]], *[Regular and effective monomorphisms and epimorphisms](https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/)*, 2012

Discussion in [[toposes]]:

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], section IV.7  of _[[Sheaves in Geometry and Logic]]_

* {#Borceux94III} [[Francis Borceux]], Section 3.3.4 of: *[[Handbook of Categorical Algebra]]*. Vol. 3. *Categories of Sheaves*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525872](https://doi.org/10.1017/CBO9780511525872)]

Discussion in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 7.6 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

[[!redirects effective epi]]
[[!redirects effective epimorphism]]
[[!redirects effective epimorphisms]]


effective epimorphism in an (infinity,1)-category

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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of [[effective epimorphism]] from [[category theory]] to [[(∞,1)-category theory]].

See also at _[[1-epimorphism]]_. However, beware of the [[red herring principle]]: effective epimorphisms in an $(\infty, 1)$-category need _not_ be [[epimorphism in an (infinity,1)-category|epimorphisms]].

## Definition

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

A [[morphism]] $f : Y \to X$ in an [[(∞,1)-category]] is an **effective epimorphism** (alias __quotient morphism__) if it has a [[Cech nerve]], of which it is the [[(∞,1)-colimit]]; in other words the [[augmented simplicial set|augmented simplicial]] [[diagram]]

$$
  \cdots
  Y \times_X Y \times_X Y
  \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
 Y \times_X Y \stackrel{\longrightarrow}{\longrightarrow}  Y \stackrel{f}{\longrightarrow} X
$$

is an [[(∞,1)-colimit|colimiting]] diagram.

=--

This appears below [[Higher Topos Theory|HTT, cor. 6.2.3.5]] for $C$ a [[(∞,1)-semitopos]], but seems to be a good definition more generally.

## Properties

### Factorization

In an [[(∞,1)-topos]] the effective epis are the [[n-epimorphisms]] for $n = 1$ sitting in the [[(n-epi, n-mono) factorization system]] for $n = 1$ with the [[monomorphism in an (∞,1)-category]], factoring every morphism through its [[1-image]].

### Stability

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

In an [[(∞,1)-semitopos]], effective epimorphisms are stable under [[(∞,1)-pullback]].

=--

This appears as ([Lurie, prop. 6.2.3.15](#Lurie)).

### Characterization
 {#Characterization}

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

For $C$ an [[(∞,1)-semitopos]]  we have that $f : X \to Y$ is an effective epimorphism precisely if its (-1)-[[truncated|truncation]] is a [[terminal object in an (∞,1)-category|terminal object]] in the [[over-(∞,1)-category]] $C/Y$.

=--

This is [[Higher Topos Theory|HTT, cor. 6.2.3.5]].

More generally,

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

The effective epimorphisms in any [[(∞,1)-topos]] are precisely the [[n-connected object of an (infinity,1)-topos|(-1)-connected morphisms]], and form a [[orthogonal factorization system in an (∞,1)-category|factorization system]] together with the [[monomorphism in an (∞,1)-category|monomorphisms]] (the (-1)-truncated morphisms).

=--

See [[n-connected/n-truncated factorization system]] for more on this.

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

For $C$ an [[(∞,1)-topos]], a morphism $f : X \to Y$ in $C$ is effective epi precisely if the induced morphism on [[subobjects]] ([[monomorphism in an (∞,1)-category|(∞,1)-monos]], they form actually a [[small set]]) by [[(∞,1)-pullback]]

$$
  f^* : Sub(Y) \to Sub(X)
$$

is [[injective function|injective]].

=--

This appears as ([Rezk, lemma 7.9](#Rezk)) and ([Lurie, prop. 6.2.3.10](#Lurie)).

Useful is also the following characterization:

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

A morphism in an [[(∞,1)-topos]] is an effective epimorphism precisely if its [[0-truncated|0-truncation]] is an [[effective epimorphism]] in the underlying 1-[[topos]].

=--

This is ([Lurie, prop. 7.2.1.14](#Lurie)).

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

In words this means that a map is an effective epimorphism if it induces an epimorphism on connected components.

=--

This is true generally in the [[internal logic]] of the $(\infty,1)$-topos (i.e. in [[homotopy type theory]], see at [[1-epimorphism]] for more on this), but in [[∞Grpd]] $\simeq L_{whe}$ [[sSet]] it is also true externally (prop. \ref{EffectiveEpisOfInfinityGroupoids} below).

### In a sheaf $\infty$-topos

In the [[infinity-topos]] of [[infinity-sheaves]] on an $\infty$-site (i.e. in a [[topological localization]]), one has the following characterization of morphisms which become effective epimorphisms after applying the [[associated sheaf functor]].

+-- {: .num_prop #EffectiveEpiInSheafTopos}
###### Proposition
Let $(C,t)$ be an $\infty$-site and $f : F \to G$ a morphism of [[presheaves]] in $P(C)$.
The morphism $a_t(f)$ is an effective epimorphism in $Shv_t(C)$ if and only if $f$ is a [[local epimorphism]], i.e.
  $$ \colim \check{C}(f) \longrightarrow G $$
is $t$-covering, or in other words
  $$ \colim \check{C}(f) \times_G h(X) \longrightarrow h(X) $$
is a $t$-[[covering sieve]] for all morphisms $h(X) \to G$, where $h$ is the [[Yoneda embedding]].
(Here $a_t$ denotes the [[associated sheaf functor]].)
=--

This is clear.  See [MO/177325/2503](http://mathoverflow.net/a/177325/2503) by [[David Carchedi]] for the argument.

## Examples

### In $\infty Grpd$

As a corollary of prop. \ref{CharacterizationAsEpiOnZeroTruncation} we have:

+-- {: .num_prop #EffectiveEpisOfInfinityGroupoids}
###### Proposition
**(effective epis of $\infty$-groupoids**)

In $C = $ [[∞Grpd]] a morphism $f : Y \to X$ is an effective epimorphism precisely if it induces an [[epimorphism]] $\pi_0 f : \pi_0 Y \to \pi_0 X$ in [[Set]] (a [[surjection]]) on [[connected components]].

=--

This appears as [[Higher Topos Theory|HTT, cor. 7.2.1.15]].

+-- {: .num_example #AnEffectiveEpiNotBeingEpiInInfinityGroupoids}
###### Example

If $S^1 = \ast \underset{\ast \coprod \ast}{\coprod} \ast$ denotes the [[homotopy type]] of the [[circle]], then the unique morphism $S^1 \to \Delta^0$ is an effective epimorphism, by prop. \ref{EffectiveEpisOfInfinityGroupoids}, but it not an [[epimorphism in an (infinity,1)-category|epimorphism]], because the [[suspension]] of $S^1$ is the [[sphere]] $S^2$, which is not [[contractible]].

=--

## Related concepts

* [[effective epimorphism]]

* [[effective epimorphism in homotopy type theory]]

* [[groupoid objects in an (∞,1)-topos are effective]]

* [[regular epimorphism in an (∞,1)-category]]

* [[n-connected object of an (∞,1)-category]]

* [[n-types cover]]

## References

* {#Rezk} [[Charles Rezk]], Section 7.7 of: _Toposes and homotopy toposes_ ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

* {#Lurie} [[Jacob Lurie]], Section 6.2.3 of: _[[Higher Topos Theory]]_

[[!redirects effective epimorphism in an (∞,1)-category]]
[[!redirects effective epimorphisms in an (∞,1)-category]]
[[!redirects effective epimorphisms in an (infinity,1)-category]]
[[!redirects effective epimorphism in an infinity1-category]]
[[!redirects (∞,1)-quotient morphism]]
[[!redirects (∞,1)-quotient morphisms]]


effective epimorphism in homotopy type theory

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## Definition

In [[homotopy type theory]], given [[types]] $A$ and $B$, a [[function]] $f:A \to B$ is an __effective epimorphism__ if for all [[terms]] $b:B$ the [[fiber]] of $f$ over $b$ is [[inhabited]].

$$isEffectiveEpic(f) \coloneqq \prod_{b:B} \Vert fiber(f, b) \Vert$$

The type of effective epimorphisms between $A$ and $B$ is defined as

$$EffectiveEpis(A, B) \coloneqq \sum_{f:A \to B} isEffectiveEpic(f)$$

## Examples ##

* The [[surjections]] $f:A \to B$ between two [[set]]s $A$ and $B$ are effective epimorphisms.

## See also

* [[n-monomorphism]]
* [[effective epimorphism in an (infinity,1)-category]]


effective field theories of nuclear physics -- contents

**[[effective field theories]] of [[nuclear physics]], hence for [[confinement|confined]]-[[phase of matter|phase]] [[quantum chromodynamics]]**:

* with effective [[light meson|ligh]] [[meson]] [[field (physics)|fields]]:

  * [[chiral perturbation theory]]

    * [[vector meson dominance]]

    * [[hidden local symmetry]]

  * with emergent ([[soliton|solitonic]])  [[baryon]] [[field (physics)|fields]]:

    * [[quark bag model]]

    * [[Skyrme model]]

    * [[Witten-Sakai-Sugimoto model]]

  * with explicit [[effective field theory|effective]] [[baryon]] [[field (physics)|fields]]:

    * [[baryon chiral perturbation theory]]

    * [[quantum hadrodynamics]]

      * [[Walecka model]]

  * with explicit [[quark]] [[field (physics)|fields]]:

    * [[quark-meson coupling model]]


effective group action

> this entry is about the concept in [[group theory]]; for the concept in [[quantum field theory|quantum]][[field theory]] see at _[[effective action functional]]_; for disambiguation see [[effective action]]

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## Idea

A [[group action]] is _effective_ if no group element other than the [[neutral element]] acts [[trivial action|trivially]] on _all_ elements of the space, hence if no other element acts the way the neutral element does.

## Definition

A [[group action]] of a [[group]] ([[group object]]) $G$ on a  [[set]] ([[object]]) $X$ is _effective_ if $\underset{x \in X}{\forall} g x = x$ implies that $g = e$ is the [[neutral element]].

Beware the similarity to and difference with [[free action]]: a free action is effective, but an effective action need not be free.

## Related concepts

* [[transitive action]], [[free action]], [[semi-free action]], [[regular action]],

* [[proper action]], [[properly discontinuous action]]

* [[effective orbifold]]


effective homology

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#### Cohomology
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#### Constructivism, Realizability, Computability
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## Idea

What is called _effective homology_ is the study of (effective) [[algorithms]] for computing invariants in [[algebraic topology]], such as [[homology groups]] but also [[homotopy groups]].

From the first part of "Effective homology, a survey", we can see why effective homology is useful.

It's much better to consider only the simply-connected spaces. We have two methods

1. Sullivan method for rational homotopy (based on minimal model
   theory) [fn:1]
2. Edgar Brown method for integer homotopy. [fn:2]

Sullivan's method is more restrictive (to Q) but much more
efficient than Brown's method. Brown's method is not practical at
all, even with the most powerful computer at hand. Nevertheless,
even Sullivan's method is at least #P-hard.

That's how effective method kicks in. It adapts Hirsch's
method [fn:3]. Using functional programming, this becomes a real
computing tool for homology and homotopy groups.

## Related concepts

* [[persistent homology]]

## References

* [[Francis Sergeraert]], _Effective homology, a survey_ ([pdf](http://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Survey.pdf))

* [fn:1] [20] Dennis Sullivan. Infinitesimal calculations in
topology.

* [fn:2] [3] Edgar H. Brown Jr.. Finite computability of Postnikov
complexes. Annals of Mathematics, 1957, vol. 65, pp 1-20.

* [fn:3] [11] G. Hirsch. Sur les groupes d’homologie des espaces
fibr´es.

[[!redirects effective homology theory]]


effective Lie groupoid

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## Idea

A [[Lie groupoid]] is said to be _effective_ if its morphisms act locally freely on [[germs]], in a sense.

(Beware that this use of the term is entirely independent of "effective" in the sense of [[Giraud theorem|Giraud's axioms]], as discussed are [[groupoid object in an (infinity,1)-category]].)

## Definition

Let $X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} Y)$ be a [[Lie groupoid]]. Equivalently, let $X$ be a [[differentiable stack]] equipped with an [[atlas]] $X_0 \to X$.

Then given any element $f$ in $X_1$, hence given a [[morphism]] $f \colon x \to y$, it induces a [[germ]] of a local [[diffeomorphism]] $\tilde f \colon (X_0,x) \to (X_0,y)$ as follows:

choose $U_x \subset X_0$ to be any [[neighbourhood]] of $x$ small enough such that the restricted [[source]] and [[target]] maps

$$
  \array{
    X_1 \times_{X_0} U_x &\to& X_1
    \\
    {}^{\mathllap{s|_U, t|_U}}\downarrow && \downarrow^{\mathrlap{s,t}}
    \\
    U_x &\stackrel{}{\hookrightarrow} & X_0
  }
$$

are [[diffeomorphisms]]. Then define $\tilde f$ to be the germ $t \circ (s|_U)^{-1}$.

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###### Definition

The Lie groupoid $X_\bullet$ is called _effective_ if this assignment of morphisms to germs of local diffeomorphisms is [[injective]].

Similarly a [[differentiable stack]] is called an _effective étale stack_ if it is represented by an effective étale Lie groupoid.

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This means that the [[action]] of the [[automorphism group]] at any point $x$ on the germ at $x$ is [[faithful action|faithful]].

## Properties

### Equivalent characterizations
 {#EquivalentCharacterizations}

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###### Proposition

The following are equivalent

1. $X_\bullet$ is an effective Lie groupoid, def. \ref{EffectiveLieGroupoid};

1. the canonical smooth [[functor]] $X_\bullet \to \mathbb{H}(X_0)$ (see [here](etale+groupoid#RelationToHaefligerGroupoids)) to the [[Haefliger groupoid]] of the manifold of objects is [[faithful functor|faithful]], i.e. gives a [[representable morphism of stacks]];

1. under the equivalence ([here](etale+groupoid#CharacterizationBySiteOfManifolds)) between smooth étale stacks and stacks on the [[site]] $SmthMfd^{et}$ of smooth manifolds with [[local diffeomorphisms]] between them, $X$ corresponds to a [[sheaf]] (i.e. to a [[0-truncated]] stack) on $SmthMfd^{et}$.

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([Carchedi 12, theorem 4.1](#Carchedi12)).

## Related concepts

* [[proper Lie groupoid]], [[étale groupoid]]

## References

A standard textbook aacount is in

* [[Ieke Moerdijk]], Janez Mrčun _Introduction to Foliations and Lie Groupoids_ ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)

Brief survey is in

* [[David Carchedi]], _A quick note on étale stacks_ ([pdf](http://people.mpim-bonn.mpg.de/carchedi/Etale_Stacks.pdf))

Discussion in a more general context of [[étale stacks]] is in

* {#Carchedi12} [[David Carchedi]], _Étale Stacks as Prolongations_ ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))


effective monomorphism

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# Effective monomorphisms
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## Idea

* [[effective epimorphism]] $\Rightarrow$ [[regular epimorphism]] $\Leftrightarrow$ [[covering]]

* [[effective monomorphism]] $\Rightarrow$ [[regular monomorphism]] $\Leftrightarrow$ [[embedding]] .

## Definition

A [[morphism]] $f : X \to Y$ in a [[category]] $C$ is an **effective monomorphism** if

1. it has a cokernel pair, i.e. if the [[pushout]] $Y \coprod_X Y$ exists;

1. it is [[generalized the|the]] [[equalizer]] of the canonical pair of morphisms $Y \stackrel{\to}{\to}Y \coprod_X Y$.

Such a morphism, $f$, may probably also be called an [[embedding]].

The dual concept is that of [[effective epimorphism]]. See there for more discussion.

## References

Exposition and examples:

* [[Qiaochu Yuan]], *[Regular and effective monomorphisms and epimorphisms](https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/)*, 2012

[[!redirects effective mono]]
[[!redirects effective monomorphism]]
[[!redirects effective monomorphisms]]