entwined module

## Idea

Given an [[entwining structure]] between a $k$-[[algebra]] $A$ and a $k$-[[coalgebra]] $C$ one defines the corresponding analogue of [[Hopf module]]s: they are $A$-modules with structure of $C$-comodules with a compatibility dictated by the entwining structure. If an algebra and a coalgebra are generalized to a monad and a comonad, then entwining structure generalizes to a mixed distributive law. Every entwining structure defines the composed coring (as observed by Takeuchi); the entwined modules are then precisely the objects in the category of comodules ([[Eilenberg-Moore category]]) of the composed coring.

## Literature

* [[T. Brzeziński]], [[S. Majid]], _Coalgebra bundles_, Comm. Math. Phys.  191  (1998),  no. 2, 467--492 ([arXiv version](http://arxiv.org/abs/q-alg/9602022)).
* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.
* [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal __16__ (2009), No. 1, 183–202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770) [MR2011b:14004](http://www.ams.org/mathscinet-getitem?mr=2527623)

Historically, the entwined modules are first introduced under the name "bialgebras" by van Osdol for the mixed distributive laws between monads and comonads instead of $k$-algebras and $k$-coalgebras. The same terminology is used by Power and Watanabe.

* {#vanOsdol73} [[Donovan van Osdol]], *Bicohomology Theory*, Transactions of the American Mathematical Society **183** (1973) 449-476 [[jstor:1996479](https://www.jstor.org/stable/1996479)]

* [[John Power]], [[Hiroshi Watanabe]], *Combining a monad and a comonad*, Theoretical Computer Science **280** 1--2 (2002) 137--162 [doi:10.1016/S0304-3975(01)00024-X]

[[!redirects entwined modules]]


entwining structure

## Idea

In addition to the usual [[distributive laws]] between a [[monad]] and a monad (in a bicategory) there are many other combinations between monad and [[comonad]], comonad and [[endofunctor]], [[action of a monoidal category]] and a monad and so on.

Suppose $V$ is any [[symmetric monoidal category]] with [[coequalizers]] preserved by the tensor product, for example $V$ is the category of $k$-[[modules]] where $k$ is a [[commutative ring]]. Then for any $k$-algebra $A$, endofunctors $A\otimes -$ and $-\otimes A$ are monads on $V$; similarly if $C$ is a $k$-[[comodule|coalgebra]], then endofunctors $C\otimes -$ and $-\otimes C$ are comonads on $V$.

## Definition
A **left entwining structure** is a morphism
$\phi:A\otimes C\to C\otimes A$ in $V$ satisfying four coherence axioms so that $\phi\otimes -$ is (after postcomposing by the [[associator]]) a mixed [[distributive law]] $A\otimes(C\otimes -)\to C\otimes(A\otimes -)$.

This amounts to ensuring $C\otimes -$ lifts to a comonad on $A\otimes -$-modules (which are the same as _left_ $A$-modules) or, equivalently, that $A\otimes -$ lifts to a monad on $C\otimes -$-comodules.

A **right entwining structure** is a morphism $\psi:C\otimes A\to A\otimes C$ such that $-\otimes\psi$ defines a [[distributive law]] ensuring a lift of the comonad $-\otimes C$ to $-\otimes A$-modules, that is _right_ $A$-modules.
Entwining structures in the monoidal category of endofunctors are the usual distributive laws.

In category of vector spaces (and $k$-modules for $k$ being a commutative ring) (right) entwining structure was introduced by [Brzeziński and Majid 1998](#BrzezinskiMajid1998) in the context of the study of [[noncommutative principal bundles]], not being aware at the time of the mixed distributive laws in category theory ([Van Osdol, 1973](#vanOsdol73)). Namely, they noticed that some examples of noncommutative principal bundles do not have a bialgebra as a structure "group", but a coalgebra together with an entwining.  Entwinings organize into a [[bicategory]]. Every entwining
induces a coring (as observed by Takeuchi; in [Škoda 2008](#Škoda2008) this correspondence is extended to a 2-functor).
To every entwining structure one associates the corresponding category of [[entwined module]]s.

## Literature

* {#BrzezinskiMajid1998} [[T. Brzeziński]], [[S. Majid]], _Coalgebra bundles_, Comm. Math. Phys.  191  (1998),  no. 2, 467--492 [arXiv version](https://arxiv.org/abs/q-alg/9602022)
*  [[Gabriella Böhm]], _Internal bialgebroids, entwining structures and corings_, Algebraic structures and their representations, 207-226, Contemp. Math., 376, Amer. Math. Soc., Providence, RI, 2005, arXiv:[math/0311244](https://arxiv.org/abs/math/0311244)
* {#vanOsdol73} [[Donovan van Osdol]], *Bicohomology Theory*, Trans. Amer. Math. Soc. **183** (1973) 449--476 [jstor:1996479](https://www.jstor.org/stable/1996479)]
* {#Škoda2008} [[Zoran Škoda]], _Bicategory of entwinings_, [arXiv:0805.4611](https://arxiv.org/abs/0805.4611)
* B. Mesablishvili, _Entwining structures in monoidal categories_, J. Algebra __319__:6 (2008) 2496--2517 [doi](http://dx.doi.org/10.1016/j.jalgebra.2007.08.030)
* Bachuki Mesablishvili, Robert Wisbauer, _Galois functors and entwining structures_, J. Algebra __324__:3 (2010) 464--506 [doi](https://doi.org/10.1016/j.jalgebra.2010.04.004)

[[!redirects entwining]]
[[!redirects entwinings]]
[[!redirects entwining structures]]


enumerated type

# Contents
* table of contents
{: toc}

## Idea ##

An enumerated type is any [[type]] given by a finite [[sum type | disjoint sum]] of [[unit type|unit types]].

## Definition ##

...

## Properties ##

Every enumerated type is a [[subtype]] of the [[natural numbers type]].

## Examples ##

...

## References ##

* Wikipedia, _[Enumerated type](https://en.wikipedia.org/wiki/Enumerated_type)_


enumerative combinatorics

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### Context
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[combinatorics]] with emphasis on counting [[natural numbers|numbers]] of possibilities.

## References

* {#Stanley11} [[Richard Stanley]], *[Enumerative combinatorics](http://www-math.mit.edu/~rstan/ec/)* -- [Volume 1](http://www-math.mit.edu/~rstan/ec/ec1/), Wadsworth & Brooks/Cole Mathematics Series book series 1986, 1997 ([doi:10.1007/978-1-4615-9763-6](https://link.springer.com/book/10.1007/978-1-4615-9763-6)) reprinted in: Cambridge Studies in Advanced Mathematics, Cambridge University Press 2011 ([ISBN:9781107602625](https://www.cambridge.org/us/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/enumerative-combinatorics-volume-1-2nd-edition?format=PB&isbn=9781107602625), [pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf))

* Federico Ardila, _Algebraic and geometric methods in enumerative combinatorics_, survey for an upcoming handbook on enumeration, ([arxiv/1409.2562](http://arxiv.org/abs/1409.2562))

See also:

* Wikipedia, *[Enumerative combinatorics](https://en.wikipedia.org/wiki/Enumerative_combinatorics)*

category: combinatorics


enumerative geometry

#Contents#
* table of contents
{:toc}

## Overview

__Enumerative geometry__ is a branch of [[algebraic geometry]] which counts a number of solutions to various geometric problems, usually stated for data in _general position_. Usually the solution is a nonnegative integer, but some vaguely defined intuitive enumerative problems have rigorous reformulations in terms of integration on moduli spaces and can yield answers which are, say, rational numbers. Most studied enumerative problems are problems of intersection theory of algebraic subvarieties and the solution is often found by cohomological methods.

## Examples

How many plane conics are tangent simultaneously to 5 given plane conics in general position ? The answer is 3264, see Griffiths-Harris, _Principles of algebraic geometry_ for a detailed treatment. A further question is how many of them are real ? see Frank Sottile, [3264 real curves](http://www.math.tamu.edu/~sottile/research/stories/3264/index.html).

## Related entries

* [[intersection theory]]

* [[Gromov-Witten invariant]]

* [[Schubert calculus]]

## References

* Wikipedia, *[Enumerative geometry](http://en.wikipedia.org/wiki/Enumerative_geometry)*

* [[William Fulton]], _Introduction to intersection theory in algebraic geometry_, CBMS 54, AMS, 1996, second edition

* [[M. Kontsevich]], _Enumeration of rational curves via torus actions_, [arXiv:hep-th/9405035](https://arxiv.org/abs/hep-th/9405035).

* G. Ellingsrud, S. A. Strømme], _Bott's formula and enumerative geometry_, JAMS vol. 9, num. 1, 1996, [arXiv:alg-geom/9411005](https://arxiv.org/abs/alg-geom/9411005).

In relation to [[string theory]] (mostly [[Gromov-Witten theory]]):

* {#Katz06} [[Sheldon Katz]], *Enumerative Geometry and String Theory*, Student Mathematical Library **32**, AMS 2006 ([ISBN:978-1-4704-2143-4](https://bookstore.ams.org/stml-32), [spire:739788](https://inspirehep.net/literature/739788))


envelope of an adjunction

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

#Contents#
* table of contents
{:toc}

## Definition

Let $L \dashv R$ be a pair of [[adjoint functors]] (an [[adjunction]] in [[Cat]]).
\begin{tikzcd}
	C & D
	\arrow[""{name=0, anchor=center, inner sep=0}, "L", bend left, from=1-1, to=1-2]
	\arrow[""{name=1, anchor=center, inner sep=0}, "R", bend left, from=1-2, to=1-1]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=1]
\end{tikzcd}

The **envelope** of the adjunction, denoted $Env(L \dashv R)$, is the [[category]] whose [[objects]] are [[quadruples]]

$$
(c \in C, d \in D, f : L c \to d, g \colon c \to R d)
$$

such that $f$ and $g$ are each other's [[mate]], and whose [[morphisms]] are [[pairs]]

$$
(p \colon c \to c', q \colon d \to d')
$$

such that $(R q) \circ f = f' \circ p$ or equivalently $q \circ g = g' \circ (L p)$.

## Properties

* {#EnvelopeOfAdjunctionCOmmaCategory} A [[functor]] $L \colon C \to D$ is [[left adjoint]] to a functor $R \colon D \to C$ if and only if there is an isomorphism of [[comma categories]] $L \downarrow D \cong C \downarrow R$ and this isomorphism commutes with the [[forgetful functors]] to the [[product category]] $C \times D$ (see [there](adjoint+functor#InTermsOfCommaCategories)). In this case, the envelope $Env(L \dashv R)$ is also isomorphic to these comma categories over the evident [[forgetful functor]] to $C \times D$.

* Every adjunction factors through its envelope via a [[reflection|reflective subcategory]] and a [[coreflection|coreflective subcategory]].
\begin{tikzcd}
	C & {\mathrm{Env}(L \dashv R)} & D
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{cod}}", shift left=3, from=1-2, to=1-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, shift left=3, hook, from=1-3, to=1-2]
	\arrow[""{name=2, anchor=center, inner sep=0}, shift left=3, hook, from=1-1, to=1-2]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{\mathrm{dom}}", shift left=2, from=1-2, to=1-1]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=0, to=1]
	\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=2, to=3]
\end{tikzcd}
See Lemma 4.1 of [Pavlovic and Hughes](#PavlovicHughes).

* Denoting by $Inv(F \dashv G)$ the [[fixed point of an adjunction|fixed point]] of $F \dashv G$, the following square of [[fully faithful]] functors forms a [[2-pullback]] in [[Cat]].
\begin{tikzcd}
	{Inv(F \dashv G)} & C \\
	D & {Env(F \dashv G)}
	\arrow[hook, from=1-1, to=1-2]
	\arrow[hook, from=1-1, to=2-1]
	\arrow[hook, from=1-2, to=2-2]
	\arrow[hook, from=2-1, to=2-2]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2]
\end{tikzcd}
In particular, $Inv(F \dashv G)$ is the full subcategory of $Env(F \dashv G)$ whose objects $(c, d, f, g)$ have $f$ and $g$ isomorphisms.
(See Lemma 12.1 of [Avery and Leinster](#AveryLeinster).)

* The envelope construction (respectively the [[fixed point of an adjunction|fixed point]] construction) can be seen as [[2-adjunctions]] between [[Cat]], and the 2-category of adjunctions (respectively the [[wide subcategory|wide]] sub-2-category of adjunctions whose morphisms $(P, Q, \alpha, \beta)$ have $\alpha$ and $\beta$ invertible)).
\begin{tikzcd}
	& {\mathrm{ADJ}} \\
	{\mathrm{CAT}} \\
	& {\mathrm{ADJ}_{\cong}}
	\arrow[""{name=0, anchor=center, inner sep=0}, shift left=3, hook, from=2-1, to=3-2]
	\arrow[""{name=1, anchor=center, inner sep=0}, shift left=3, hook, from=2-1, to=1-2]
	\arrow[hook, from=3-2, to=1-2]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{\mathrm{Env}}", shift left=3, from=1-2, to=2-1]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{\mathrm{Inv}}", shift left=3, from=3-2, to=2-1]
	\arrow["\dashv"{anchor=center, rotate=-48}, draw=none, from=1, to=2]
	\arrow["\dashv"{anchor=center, rotate=-132}, draw=none, from=0, to=3]
\end{tikzcd}
(See Remark 12.2 of [Avery and Leinster](#AveryLeinster).)

## Examples

* The envelope of the [[Isbell duality]] adjunction associated to a category $A$ is the [[Isbell envelope]] of $A$ (called the _category of gaps_ in $A$ by [Pavlovic and Hughes](#PavlovicHughes)).

* The envelope of the [[nuclear adjunction]] associated to the [[Isbell duality]] adjunction for a small category $A$ is what [Pavlovic and Hughes](#PavlovicHughes) call the _category of intervals_ in $A$.

## Related concepts

* [[fixed point of an adjunction]]

## References

* {#AveryLeinster} [[Tom Avery]], [[Tom Leinster]]. _Isbell conjugacy and the reflexive completion_. Theory and Applications of Categories, **36** 12 (2021) 306-347 [[tac:36-12](http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf)]

* {#PavlovicHughes} [[Dusko Pavlovic]], and Dominic JD Hughes. _Tight limits and completions from Dedekind-MacNeille to Lambek-Isbell_. ([arXiv](https://arxiv.org/abs/2204.09285))


enveloping algebra

> There is also a distinct notion of an [[universal enveloping algebra|enveloping algebra of a Lie algebra]].

***

# Contents
* table of contents
{: toc}

## Definition

Given a [[monoid]] $A$ with multiplication $\mu$ in a [[symmetric monoidal category]] $C$ with symmetry $\tau$ its opposite $A^{op}$ is the same underlying object $A$ with multiplication $\mu\circ\tau$. The __enveloping monoid__ $A^e$ of $A$ is the monoid whose underlying object is $A\otimes A^{op}$ and for which the multiplication is given by

$$
(\mu\otimes (\mu\circ\tau))\circ(id\otimes\tau\otimes id): (A\otimes A^{op})\otimes (A\otimes A^{op})\to A\otimes A^{op}
$$

where $\tau=\tau_{A,A}$ is the symmetry $A\otimes A\to A\otimes A$.

The enveloping monoid is sometimes called the __enveloping algebra__, especially if the monoidal category is a category of vector spaces. The left $A^e$-modules in $C$ are in 1-1 correspondence with the $A$-$A$-bimodules in $C$.

[[!redirects enveloping algebra]]
[[!redirects enveloping algebras]]


enveloping von Neumann algebra

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### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Enveloping von Neumann algebras
* table of contents
{: toc}

## Definitions

Let $A$ be a $C^*$-[[C-star algebra|algebra]].  We may define its _enveloping [[von Neumann algebra]]_ in a few different but equivalent ways.

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

The __enveloping von Neumann algebra__ $E(A)$ of $A$ is the [[free object|free]] von Neumann algebra on $A$.  That is, we have an [[adjoint functors|adjunction]]
$$ W^* Alg \overset{E}\underset{U}\leftrightharpoons C^* Alg ,$$
where $W^* Alg$ is the category of von Neumann algebras (which are $C^*$-algebras with [[preduals]]) and von Neumann algebra [[homomorphisms]] (which are $C^*$-algebra homomorphisms with preduals), $C^* Alg$ is the category of $C^*$-algebras and $C^*$-algebra homomorphisms, $U$ is the [[forgetful functor]] or [[inclusion functor]], and $E$ is the [[functor]] that we wish to define.
=--

Definition \ref{adjunction} defines [[the]] functor $E$ up to unique [[natural isomorphism]], if it exists.  We may prove that it exists by the [[adjoint functor theorem]] or by proving that one of the explicit constructions below satisfies the relevant [[universal property]].

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

Consider the [[direct sum]] of the the [[GNS representation]]s of the [[positive functional|positive]] linear functionals on $A$; this is a [[Hilbert space]] $H$ and [[representation]] $\pi : A \to B(H)$, the a [[universal representation of a C-star algebra|universal representation]] of $A$.  The [[image]] $\pi(A)$ is a [[linear subspace|subspace]] of $B(H)$; consider its [[double commutant]] (or equivalently its [[topological closure|closure]] in the [[weak operator topology]]) $A''$.  Ignoring the representation of $A''$ on $H$, $A''$ is a von Neumann algebra, the __enveloping von Neumann algebra__ of $A$.
=--

To obtain an [[adjoint functors|adjunction]] from Definition \ref{concrete}, we need also the [[unit of the adjunction]], which is the map
$$ A \to \pi(A) \to Cl_{wk^*}(\pi(A)) = A'' .$$

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

Think of $A$ as a [[Banach space]], and consider its [[double dual]] $A^{**}$.  We have (as with any Banach space) a [[short linear map]] $i\colon A \to A^{**}$, so that $i(A)$ has the structure of a $C^*$-algebra.  Since $i(A)$ is weak-$*$-[[dense subspace|dense]] in $A^{**}$ and the $C^*$-algebraic operations are [[continuous map|continuous]], they extend to $A^{**}$. These extensions turn $A^{**}$ into a [[Banach algebra]]; the $C^*$ identity also extends, making $A^{**}$ into $C^*$-algebra.  Since $A^{**}$ has $A^*$ as a [[predual]], it is a von Neumann algebra, the __enveloping von Neumann algebra__ of $A$.
=--

Here, the [[unit of the adjunction]] is simply $i$.

The claim that the definitions above are all equivalent is the __Sherman–Takeda theorem__, due (naturally enough) to [Sherman (1950)](#Sherman1950) and [Takeda (1954)](#Takeda1954).

## Properties

A $C^*$-algebra and its enveloping von Neumann algebra have the same [[spectrum]].  The [[functional calculus]] on a $C^*$-algebra (which treats [[continuous functions]]) extends to the functional calculus on its enveloping von Neumann algebra (which treats [[measurable functions]]).  In particular, we can apply a measurable function to an element of a $C^*$-algebra to obtain an element of its enveloping von Neumann algebra.

## Applications

Some astract treatments of [[quantum mechanics]] use $C^*$-algebras, while others use von Neumann algebras.  If a [[physical system]] is described by a $C^*$-algebra in the first case, then it may described by its enveloping von Neumann algebra in the second case.

## References

* S. Sherman (1950). The second adjoint of a C* algebra. _Proceedings of the International Congress of Mathematicians_ 1950 (1): 470. American Mathematical Society.
  {#Sherman1950}

* Zirô Takeda (1954). Conjugate spaces of operator algebras. _Proceedings of the Japan Academy_ 30 (2): 90–95.
  {#Takeda1954}

* Wikipedia (English): [Enveloping von Neumann algebra](http://en.wikipedia.org/wiki/Enveloping_von_Neumann_algebra), [Sherman–Takeda theorem](http://en.wikipedia.org/wiki/Sherman–Takeda_theorem)

category: operator algebras

[[!redirects enveloping von Neumann algebra]]
[[!redirects enveloping von Neumann algebras]]
[[!redirects enveloping W-star-algebra]]
[[!redirects enveloping W-star-algebras]]
[[!redirects enveloping W-*-algebra]]
[[!redirects enveloping W-*-algebras]]
[[!redirects enveloping W*-algebra]]
[[!redirects enveloping W*-algebras]]

[[!redirects Sherman-Takeda theorem]]
[[!redirects Sherman–Takeda theorem]]
[[!redirects Sherman--Takeda theorem]]


environment rep of quantum measurement channel - section

### Environmental representation of measurement channels

By the general theorem about [environmental representations](quantum+channel#QuantumChannelsAndDecoherence) of quantum channels, every [[quantum channel]] on a [[quantum system]] $\mathscr{H}$ may be decomposed as

1. coupling of $\mathscr{H}$ to an environment/bath system $\mathscr{B}$,

1. [[unitary quantum channel|unitary evolution]] of the [[composite system]] $\mathscr{H} \otimes \mathscr{B}$,

1. [[averaging quantum channel|averaging]] the result over the environment states.

The way this works specifically for [[quantum measurement channels]] has precursor discussion [von Neumann 1932 §VI.3 ](quantum+measurement#vonNeumann32) and received much attention in discussion of *[[quantum decoherence]]* following [Zurek 1981](quantum+decoherence#Zurek81) and [Joos & Zeh 1985](quantum+decoherence#JoosZeh85) (independently and apparently unkowingly of the general discussion of [environmental representations](quantum+channel#QuantumChannelsAndDecoherence) in [Lindblad 1975](quantum+channel#Lindblad75)).

Concretely,

> (we shall restrict attention to [[finite-dimensional Hilbert spaces]] not to get distracted by technicalities that are irrelevant to the point we are after)

if $\left\vert b_{\mathrm{ini}} \right\rangle \,\colon\, \mathscr{B}$ (we use [[bra-ket notation]]) denotes the initial state of a "device" [[quantum system]] then any notion of this device measuring the given [[quantum system]] $\mathscr{H}$ (in its measurement basis $W$, $\mathscr{H} \simeq \underset{W}{\oplus}\mathbb{C}$) under their joint [[unitary quantum channel|unitary quantum evolution]]  should be reflected in a [[unitary operator]] under $[$[Zurek 1981 (1.1)](quantum+decoherence#Zurek81), [Joos & Zeh 1985 (1.1.)](quantum+decoherence#JoosZeh85), following [von Neumann 1932 §VI.3 ](quantum+measurement#vonNeumann32), review includes [Schlosshauer 2007 (2.51)](quantum+decoherence#Schlosshauer07)$]$:

1. the system $\mathscr{H}$ remains invariant *if* it is purely in any eigenstate $\left\vert w \right\rangle$ of the measurement basis,

1. while in this case the measuring system evolves to a corresponding "pointer state" $\left\vert b_w \right\rangle$:

\[
  \label{UnitaryMeasurementProcess}
  \array{
    &\mathclap{
      \color{green}
      \array{
        \text{unitary}
        \\
        \text{measurement interaction}
      }
   }&
    \\
    \mathllap{
      U_W
      \;\colon\;\;
    }
    \mathscr{H} \otimes \mathscr{B}
    &\longrightarrow&
    \mathscr{H} \otimes \mathscr{B}
    \\
    \left\vert w, b_{ini} \right\rangle
    &\mapsto&
    \left\vert w, b_w \right\rangle
  }
\]

for $b_{\mathrm{ini}}$ and $b_w$ distinct elements of an (in practice: approximately-)[[orthonormal basis]] for $\mathscr{B}$. (There is always a unitary operator with this mapping property (eq:UnitaryMeasurementProcess), for instance the one which moreover maps $\left\vert w, b_{w}\right\rangle \mapsto \left\vert w, b_{\mathrm{ini}}\right\rangle$ and is the [[identity map|identity]] on all remaining basis elements.)

But then the composition of the corresponding [[unitary quantum channel]] with the [[averaging quantum channel|averaging channel]] over $\mathscr{B}$ is indeed equal to the $W$-[[measurement quantum channel]] on $\mathscr{H}$ (cf. eg. [Schlosshauer 2007 (2.117)](quantum+decoherence#Schlosshauer07), going back to [Zeh 1970 (7)](quantum+decoherence#Zeh70)), as follows:

\begin{imagefromfile}
    "file_name": "EnvRepOfQuantumMeasurementChannel-230924b.jpg",
    "width": 940,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}


Enxin Wu

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=167873)

## Selected writings

On the [[D-topology]] on [[diffeological spaces]]:

* {#CSW13} [[J. Daniel Christensen]], [[Gord Sinnamon]], [[Enxin Wu]], _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

On the [[homotopy theory]] of [[diffeological spaces]] via their [[cohesion|cohesive]] [[shape modality|shape]]:

* [[J. Daniel Christensen]],  [[Enxin Wu]], _The homotopy theory of diffeological spaces_, New York J. Math. 20 (2014), 1269-1303 ([arXiv:1311.6394](http://arxiv.org/abs/1311.6394))

category: people


EO(n)

#Contents#
* table of contents
{:toc}

## Idea

A real form of [[Morava E-theory]].

## Properties

### Orientation

The [[orientation in generalized cohomology|orientation]] in $EO(2)$-cohomology of a manifold with [[spin structure]] ($w_1 = 0$ and $w_2 = 0$) is essentially obstructed by the 4th [[Stiefel-Whitney class]] $w_4$ ([Kriz-Sati 04, section 5.2](SatiKriz04)).

## References

The construction and general properties are discussed in

* Po Hu, [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence_, Topology 40 (2001) ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

The [[orientation in generalized cohomology|orientation]] in $EO(2)$-theory is discussed in section 5.2 of

* [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor.
Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))
 {#SatiKriz04}


Eon-Kyung Lee

* [institute page](https://sejong.elsevierpure.com/en/persons/eon-kyung-lee)

## Selected writings

On [[finitely presented group|finite presentations]] of the [[pure braid group]]:

* [[Eon-Kyung Lee]], *A positive presentation of the pure braid group*, Journal of the Chungcheong Mathematical Society **23** 3 (2010) 555-561 [[JAKO201007648745187](https://koreascience.kr/article/JAKO201007648745187.view?orgId=anpor), [pdf](http://www.ccms.or.kr/data/pdfpaper/jcms23_3/23_3_555.pdf)]

category: people


EOS

__EOS__ is a [[high performance distributed ledger]] using [[WebAssembly]] [[virtual machine]] for [[smart contract]]s. [eos.io](https://eos.io) claims speeds up to 300000 TPS.

* EOS technical white paper v2 (March 2018) is at github, [md](https://github.com/EOSIO/Documentation/blob/master/TechnicalWhitePaper.md)
* Shankar Nathan, _How to deploy and run a smart contract on the EOS blockchain_, [web](https://hackernoon.com/how-to-deploy-and-run-a-smart-contract-on-the-eos-blockchain-from-zero-to-hero-72ca592803ba)
* gurghet, [Write EOS contracts in Rust instead of C++](https://steemit.com/crypto/@gurghet/write-eos-contracts-in-rust-instead-of-c)
* How does WASM get interpreted by the EOS virtual machine? [eosio.stackexchange](https://eosio.stackexchange.com/questions/167/how-does-wasm-get-interpreted-by-the-eos-virtual-machine)

See also [[blockchain]].

category: computer science, applications


epi-pullback

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

#Contents#
* table of contents
{:toc}

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

In [[category theory]], a [[commutative square]]

$$
  \array{
    q
    & \longrightarrow &
    a
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    b
    &
    \longrightarrow
    &
    c
  }
$$

in a [[category]] with [[finite limits]] is (sometimes) called an **epi-pullback** (or **quasi-pullback** or **epi cartesian-square**) if the canonical morphism $q\to a\times_c b$ to the [[fiber product]]  (induced by its [[universal property]]) is an [[epimorphism]].

Spcifically, the object $q$ is then called an **epi-pullback** or **quasi-pullback** of the span $b\to c\leftarrow a$.

=--

## Examples

\begin{proposition}
For a [[topos]] $T$ and $T^I$ its [[arrow category]] which is a topos, epi-pullback squares (in $T$) form a [[class]] of [[open morphisms]] in $T^I$.
\end{proposition}

\begin{proposition}
Lax [[double functors]] $Rel(A) \to Rel(B)$ are equivalent to [[functors]] $A \to B$ preserving quasi-pullbacks.
\end{proposition}

## Related pages

* [[pullback]]

## References

- [[Robert Paré]], _Some things about double categories_, Talk at _Virtual Double Categories Workshop_ 2022, [pdf](https://bryceclarke.github.io/virtual-double-categories-workshop/slides/robert-pare.pdf)


epic map in a dagger 2-poset

[[!redirects injective map in a dagger 2-poset]]

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

## Definition ##

A morphism $f \in Hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is an **epic map** if it is an [[onto morphism in a dagger 2-poset|onto]] [[map in a dagger 2-poset|map]].

The type of all epic maps in $Hom_A(a,b)$ is defined as

$$EpiMap(a, b) \coloneqq \sum_{f:Hom_A(a,b)} isOnto(f) \times isMap(f)$$

## See also ##

* [[onto morphism in a dagger 2-poset]]
* [[map in a dagger 2-poset]]


epicyclic space

An __epicyclic category__ is certain small category containing the [[cyclic category]]; it has been introduced by [[Thomas Goodwillie]] (in an unpublished letter to Waldhausen, August 10, 1987).
__Epicyclic sets__ ( __epicyclic spaces__) are presheaves of sets (topological spaces, respectively) on the epicyclic category.

* D. Burghelea, Z. Fiedorowicz, W. Gajda, _Power maps and epicyclic spaces_, J. of Pure and Appl. Algebra __96__:1,
pp. 1–14 (Sep 1994) doi
* [[Alain Connes]], [[Caterina Consani]], _Projective geometry in characteristic one and the epicyclic category_, [arxiv/1309.0406](http://arxiv.org/abs/1309.0406)

[[!redirects epicyclic set]]
[[!redirects epicyclic space]]


epimorphism

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

#Contents#
* table of contents
{:toc}

## Idea

In [[category theory]], the concept of _epimorphism_ is a generalization or strengthening of the concept of _[[surjective functions]]_ between [[sets]] (example \ref{EpimorphismsOfSets} below).

The [[formal duality|formally dual concept]] is that of _[[monomorphism]]_, similarly related to the concept of [[injective function]].

Common jargon includes "is a mono" or "is monic" for "is a monomorphism", and "is an epi" or "is epic" for "is an epimorphism", and "is an iso" for "is an isomorphism".

## Definition

A [[morphism]]  $f \colon X \to Y$ in some [[category]] is called an _epimorphism_ (sometimes abbrieviated to _epi_, or described as being _epic_), if for every other [[object]] $Z$ and every [[pair of parallel morphisms]] $g_1,g_2 \colon Y \to Z$ then

$$
  \left(
    g_1\circ f
    \,=\,
    g_2 \circ f
  \right)
  \;\Rightarrow\;
  \left(
    g_1 = g_2
  \right)
  \,.
$$

Stated more abstractly, this says that $f$ is an epimorphism precisely if for every object $Z$ the [[hom-functor]] $Hom(-,Z)$ sends it to an [[injective function]]

$$
  Hom(Y,Z)
   \xhookrightarrow{\;\; f^* \;\;}
  Hom(X,Z)
$$

between [[hom-sets]]. Since injective functions are the [[monomorphisms]] in [[Set]] (example \ref{EpimorphismsOfSets} below) this means that $f$ is an epimorphism precisely if $Hom(f,Z)$ is a monomorphism for all $Z$.

Finally, this means that $f$ is an epimorphism in a [[category]] $\mathcal{C}$ precisely if it is a [[monomorphism]] in the [[opposite category]] $\mathcal{C}^{op}$.

## Examples

### General

+-- {: .num_example #EpimorphismsOfSets}
###### Example
**(epimorphisms of sets)**

The epimorphisms in the category [[Set]] of [[sets]] are precisely the [[surjective functions]].

=--

Thus the concept of epimorphism may be thought of as a category-theoretic generalization of the concept of surjection.

But beware that in categories of sets with [[extra structure]], epimorphisms need not be surjective (in contrast to [[monomorphisms]], which are usually [[injective]]).

+-- {: .num_example #EpimorphismsOfSheaves}
###### Example
**(epimorphisms of sheaves)**

A morphism in the category of sheaves of sets over a topological space $X$ is epimorphism iff the induced map on the level of each stalk is surjective. See also math.stackexchange [here](https://math.stackexchange.com/questions/205658/an-easy-way-to-prove-that-epimorphism-of-sheaves-implies-surjectivity-on-stalk).

=--

+-- {: .num_example #EpimorphismsOfRings}
###### Example
**(epimorphisms of rings)**

In the categories [[Ring]] or [[CRing]] of ([[commutative ring|commutative]]) [[rings]] and [[ring homomorphisms]] between them, a morphism $f: R\to S$ is epi iff the canonical inclusion of the right module $S_R$ into its extension of scalars along $f$, $S_R\to S_R\otimes_R S$ is
a bijection (hence isomorphism), or equivalently, the multiplication map $S\otimes S\to S$ factors to an isomorphism $S\otimes_R S\to S$ of $R$-bimodules.

It follows that the multiplication induces natural isomorphism of endofunctors with components $M\mapsto M\otimes_R S\otimes_R S\cong M\otimes_R S$. While the extension of scalars followed by restriction of scalars is a monad (by adjunction), this implies that it is an idempotent monad, hence the extension of scalars is a localization functor.

In both categories, every surjective ring homomorphism is an epimorphism, however there are many epimorphisms which are not surjective maps. For example, the inclusion $\mathbb{Z} \to \mathbb{Q}$ is an epimorphism of rings, and of commutative rings.  More generally, every flat localization of rings (e.g. [[Ore localization]]) is a flat epimorphism, but (nontrivial) localizations are not surjective.

Any faithfully flat epimorphism of rings is an isomorphism.

An epimorphism of commutative rings is surjective iff it is a finite morphism. For epimorphisms of commutative rings see [[Stacks Project]], _[10.106 Epimorphisms of rings](http://stacks.math.columbia.edu/tag/04VM)_ and MathOverflow [what-do-epimorphisms-of-commutative-rings-look-like](https://mathoverflow.net/questions/109/what-do-epimorphisms-of-commutative-rings-look-like).

=--

Often, though, the surjections correspond to a stronger notion of epimorphism.

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

Every [[isomorphism]] is both an epimorphism and a [[monomorphism]].

=--

But beware that the converse fails:

### Examples of monos that are epi but not iso
 {#ExamplesOfMonosThatAreEpiButNotIso}

The following lists some examples of [[morphisms]] that are both [[monomorphisms]] and [[epimorphisms]], but not necessarily [[isomorphisms]].

\begin{example}

In the [[category]] of [[Hausdorff topological spaces]], the inclusion $A \hookrightarrow X$ of a [[dense subspace]] is an [[epimorphism]].

\end{example}

See [this Prop.](Hausdorff+space#DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces) for proof.

\begin{example}
 \label{InclusionOfIntegersIntoRationalsIsEpimorphismOfRings}

In [[unital ring|unital]] [[Rings]], the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the [[integers]] into the [[rational numbers]] is an [[epimorphism]].

\end{example}

See [this Prop.](Ring#InclusionOfIntegersIntoRationalsIsEpimorphismOfRings) for proof.

## Properties

\begin{prop}\label{BasicCharacterizationOfEpimorphisms}
The following are equivalent:

* $f : x \to y$ is an epimorphism in $C$;

* $f$ is a [[monomorphism]] in the [[opposite category]] $C^{op}$;

* precomposition with $f$ is a monomorphism in [[Set]]: that is, for all $c \in C$, $- \circ f : Hom(y,c) \to Hom(x,c)$ is an [[injection]];

* the [[commuting diagram]]
  $$
    \array{
      x &\stackrel{f}{\to}& y
      \\
      {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Id}}
     \\
     y &\underset{Id}{\to}& y
    }
  $$

  is a [[pushout]] diagram.

\end{prop}

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

If $f \colon x \to y$ and $g \colon y \to z$ are epimorphisms, so is their composite $g f$.  If $g f$ is an epimorphism, so is $g$.

=--

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

Every [[coequalizer]] $x \to y$

$$
  z \stackrel{\to}{\to} x \to y
$$

is an epimorphism.

=--

The converse of the above proposition fails, and an epimorphism is called a [[regular epimorphism]] if it is the coequalizer of some pair of morphisms.

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

Epimorphisms are preserved by [[pushout]]: if $f \colon x \to y$ is an epimorphism and

$$
  \array{
    x &\to& a
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{g}
    \\
    y &\to& b
  }
$$

is a [[pushout]] diagram, then also $g$ is an epimorphism.

=--

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

Let $h_1,h_2 : b \to c$ be two morphisms such that $\stackrel{g}{\to} \stackrel{h_1}{\to} = \stackrel{g}{\to} \stackrel{h_2}{\to} $. Then by the commutativity of the diagram also $x \to y \to b \stackrel{h_1}{\to} c$ equals $x \to y \to b \stackrel{h_2}{\to} c$. Since $x \to y$ is assumed to be epi, it follows that $y \to b \stackrel{h_1}{\to} c$ equals $y \to b \stackrel{h_2}{\to} c$. But this means that $h_1$ and $h_2$ define the same [[cocone]]. By the universality of the pushout $b$ there is a unique map of cocones from $b$ to $c$. Hence $h_1$ must equal $h_2$. Therefore $g$ is epi.

=--

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

Epimorphisms are preserved by any [[left adjoint]] [[functor]], or more generally any functor that preserves [[pushouts]]: if $F \colon C \to D$ is a [[functor]] that preserves pushouts and $f \in Mor(C)$ an epimorphism then $F(f) \in Mor(D)$ is an epimorphism.

=--

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

If $F : C \to D$ is a left adjoint we can argue this way: by the [[adjunction]] [[natural isomorphism]] we have for all $d \in Obj(D)$

$$
  Hom_D(L(f),d) \simeq Hom_C(f,R(d))
  \,.
$$

The right hand is a monomorphism by assumption, hence so is the left hand, hence $L(f)$ is epi.

More generally, if $F$ preserves pushouts we can use the fact that $f$ is epic iff

$$
    \array{
      x &\stackrel{f}{\to}& y
      \\
      {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Id}}
     \\
     y &\underset{Id}{\to}& y
    }
  $$

is a [[pushout]] diagram.

=--

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

Epimorphisms are [[reflected limit|reflected]] by [[faithful functors]].

=--

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

Let $F \colon \mathcal{C}\longrightarrow \mathcal{D}$ be a [[faithful functor]]. Consider $f \colon x \longrightarrow y$ a [[morphism]] in $\mathcal{C}$ such that $F(f) \colon F(x)\longrightarrow F(y)$ is an epimorphism in $\mathcal{D}$. We need to show that then $f$ itself is an epimorphism.

So consider morphisms $g,h \colon y \longrightarrow z$ such that $g \circ f = h \circ f$. We need to show that this implies that already $g = h$ (injectivity of $Hom(f,z)$). But functoriality implies that $F(g)\circ F(f) = F(h) \circ F(f)$, and since $F(f)$ is epic this implies that $F(g) = F(h)$. Now the statement follows with the assumption that $F$ is faithful, hence [[injection|injective]] on morphisms.

=--

Epimorphisms get along with [[colimits]] in a number of ways, some of which we have seen above.   Here is another:

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

1. Any [[morphism]] to an [[initial object]] is an epimorphism.

1. The [[coproduct]] of epimorphisms is an epimorphism.

=--

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

For the first suppose $0 \in C$ is initial and $f : x \to 0$.  Given morphisms $g,h: 0 \to y$ with $g \circ f = h \circ f$ we have $g = h$ simply because $0$ is initial.

For the second suppose $f_1 : x_1 \to y_1$ and $f_2 : x_2 \to y_2$ are epimorphisms; we wish to show that $f_1 + f_2 : x_1+x_2 \to y_1 + y_2$ is an epimorphism.   Suppose we have morphisms $g, h: y_1+y_2 \to z$ with $g \circ (f_1+f_2) = h \circ (f_1 + f_2)$.   Then $g \circ i_1 \circ f_1 = h \circ i_1 \circ f_1$ where $i_1 : y_1 \to y_1 + y_2$ is the canonical map into the coproduct.  Since $f_1$ is epic we conclude $g \circ i_1 = h \circ i_1$.  Similarly we have $g \circ i_2 = h \circ i_2$.  It follows that $g = h$.
=--

{#EpimorphismsDoNotGetAlongWithLimits} Epimorphisms do not get along quite as well with [[limits]].  For example, the [[projections]] from a [[Cartesian product]] onto its factors, e.g. $p_1 \colon x_1 \times x_2 \to x_1$, are not always epimorphisms (even in $Set$: take $x_2$ to be empty).

## Variations {#Variations}

There are a sequence of variations on the concept of epimorphism, which conveniently arrange themselves in a [[total order]].  In order from strongest to weakest, we have:

* [[split epimorphism]] = morphism having a [[section]]
* [[effective descent morphism]]
* [[descent morphism]] = [[pullback-stability|stable]] [[regular epimorphism]]
* [[effective epimorphism]] = [[coequalizer]] of its [[kernel pair]]
* [[regular epimorphism]] = [[coequalizer]] of some [[parallel pair]] of morphisms
* [[strict epimorphism]] = joint coequalizer of all pairs which it coequalizes
* [[strong epimorphism]] = an epimorphism [[orthogonality|left orthogonal]] to [[monomorphisms]]
* [[extremal epimorphism]] = an epimorphism not factoring through any nontrivial monomorphism.
* epimorphism.

In [[Set|the category of sets]], every epimorphism is effective descent (and even split if you believe the [[axiom of choice]]).  Thus, it can be hard to know, when generalising concepts from [[Set]] to other categories, what kind of epimorphism to use.  The following discussion may be helpful in this regard.

First we note:

* [[descent|Descent]] and effective descent morphisms are only defined in a category with [[pullbacks]].  The other notions can be defined in any category, although of course for an effective epimorphism one must in general assert the existence of the kernel pair.

Moreover, if the category has [[finite limits]], then the picture becomes much simpler:

* If a [[strict epimorphism]] has a [[kernel pair]], then it is [[effective epimorphism|effective]] and hence also [[regular epimorphism|regular]].  Thus, in a category with [[pullbacks]], effective = regular = strict.  Probably for this reason, there is substantial variation among authors in their use of these words; some use "effective epi" or "regular epi" to mean what we have called a "strict epi".

* Likewise, in a category with pullbacks, every extremal epimorphism is strong, since monomorphisms are always pullback-stable.

* Moreover, in a category with [[equalizers]], strong and extremal epimorphisms do not need to explicitly be asserted to be epic; that follows from the other condition in their definition.

Also worth noting are:

* In a [[regular category]], every extremal epimorphism is a descent morphism (i.e. a pullback-stable regular epimorphism).  Thus in this case there remain only four types of epimorphism: split, effective descent, regular, and plain.

* In an [[exact category]], or a category that has pullback-stable [[reflexive coequalizers]] (which implies that it is regular), any regular epimorphism is effective descent.  Thus in this case we have only three types: split, regular, and plain.

* In a [[pretopos]] (hence also in a [[topos]]), every epimorphism is regular, leaving only two types: split and plain.  The collapsing of these two types into one is called the [[axiom of choice]] for that category.

Thus, in general, the two serious distinctions come

* Between split epimorphisms and regular ones: in very few categories are all regular epimorphisms split.  Splitting of even regular epimorphisms is a form of the axiom of choice, which may be valid in [[Set]] (if you believe it) but very often fails [[internalization|internally]].

* Between extremal epimorphisms and "plain" epimorphisms: in many categories, the plain epimorphisms are oddly behaved, but the extremal ones are what we would expect.  For instance, the inclusion $\mathbb{Z}\hookrightarrow\mathbb{Q}$ is an epimorphism of [[ring|rings]], but the extremal epimorphisms of rings are just the surjective ring homomorphisms.  More generally, in all [[algebraic category|algebraic categories]] (categories of algebra for a [[Lawvere theory]]), which are regular, the regular epimorphisms are the morphisms whose underlying function is surjective.

Moreover, even in non-regular categories, there seems to be a strong tendency for strong/extremal epimorphisms to coincide with regular/strict ones.  For example, this is the case in [[Top]], where both are the class of quotient maps.  (The plain epimorphisms are the surjective continuous functions.)

However, the distinction is real.  For instance, in the category generated by the following graph:
$$
\array{ &&&& C\\
  &&& ^f\nearrow\\
  A& \underoverset{h}{k}{\rightrightarrows} & B \\
  &&& _g\searrow\\
  &&&& D}
$$
subject to the equations $f h = f k$ and $g h = g k$, both $f$ and $g$ are strong, but not strict, epimorphisms.

## Related concepts

* [[monomorphism]]

* [[image]], [[coimage]]

* [[epimorphism in an (infinity,1)-category|epimorphism in an $(\infty,1)$-category]]

* [[n-epimorphism]]

## References

Textbook accounts:

* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.8 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

See also:

* [[G. M. Kelly]], _Monomorphisms, Epimorphisms, and Pullbacks_, Journal of the Australian Mathematical Society 9.1-2 (1969), pp. 124–142.  [doi:10.1017/s1446788700005693](https://doi.org/10.1017/s1446788700005693).

[[!redirects epimorphisms]]
[[!redirects epic]]
[[!redirects epics]]
[[!redirects epi]]
[[!redirects epis]]
[[!redirects Epimorphism]]
[[!redirects Epimorphisms]]
[[!redirects Epic]]
[[!redirects Epics]]
[[!redirects Epi]]
[[!redirects Epis]]


epimorphism in an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One analog in [[(∞,1)-category theory]] of _[[epimorphism]]_ in [[category theory]]. Beware that there are other notions such as _[[effective epimorphism in an (infinity,1)-category|effective epimorphism in an $(\infty,1)$-category]]_ and, more generally, the concept of _[[n-epimorphism]]_.

## Definition

\begin{definition}
 \label{EpimorphismInAnInfinityCategory}
For $\mathcal{C}$ an [[(∞,1)-category]], a [[1-morphism]] $f \colon X \to Y$ in $\mathcal{C}$ is an **epimorphism** if for all [[objects]] $A \in \mathcal{C}$ the induced morphism on [[hom infinity-groupoids|hom $\infty$-groupoids]]

$$
  \mathcal{C}(f,A)
  \,\colon\,
  \mathcal{C}(Y,A)
    \xhookrightarrow{\;\;}
  \mathcal{C}(X,A)
$$

is a [[monomorphism in an (∞,1)-category|monomorphism in]] [[∞Grpd]].
\end{definition}

## Examples
 {#Examples}

\begin{example}\label{TerminalEpimorphismsInInfinityGroupoids}
**(terminal epimorphisms of $\infty$-groupoids)**
\linebreak
  For $\mathcal{X} \,\in\,$ [[Infinity-Grpd|$Grpd_\infty$]], if the [[terminal object in an (infinity,1)-category|terminal map]] $\mathcal{X} \to \ast$ is an epimorphism in the sense of Def. \ref{EpimorphismInAnInfinityCategory} then

1. $\mathcal{X}$ is [[connected homotopy type|connected]];

1. [[fundamental group|$\pi_1(\mathcal{X})$]] is [[perfect group|perfect]].

([Hoyois 2019, Lem. 3](#Hoyois19))

In particular, this means that in order for the [[delooping groupoid]] $\mathbf{B}G$ of a [[discrete group]] $G$ (i.e. the [[Eilenberg-MacLane space]] $K(G,1)$) to be such that $\mathbf{B}G \to \ast$ is epi, the group $G$ must be [[perfect group|perfect]].
\end{example}

\begin{remark}
**(terminal epimorphisms of 1-groupoids)**
  In contrast to Ex. \ref{TerminalEpimorphismsInInfinityGroupoids},
  in the [[(2,1)-category]] [[Grpd|$Grpd_1$]] of [[1-groupoid|1-]][[groupoids]] the maps

$$
  \mathcal{X}
  \xrightarrow{Map(B G \to \ast, \mathcal{X})}
  Map(B G, \mathcal{X})
$$

are always [[fully faithful functor|fully faithful]], for all $\mathcal{X} \in Grpd_1$, without further conditions on $G$, notably so for non-trivial [[abelian groups]] $G$. Explicitly, for the case $\mathcal{X} \simeq B K$ (to which the general situation reduces by [[extensive category|extensivity]]), we have the coproduct decomposition

$$
  Map(B G, \, B K)
  \;\simeq\;
  \underset{
    c \in H^1_{Grp}(G,K)
  }{\coprod}
  B Stab_K(c)
  \;\;\;\simeq\;\;\;
  B K
  \,
  \sqcup
  \,
  \underset{
    {
      c \in H^1_{Grp}(G,K)
    }
    \atop
    {
      c \neq 1
    }
  }{\coprod}
  B Stab_K(c)
$$

which plays a central role in discussion such as of [[inertia orbifolds]], [[equivariant principal bundles]], [[equivariant K-theory]] and other aspects of [[equivariant cohomology]].

The point is that a [[natural transformation]] out of $B G$ into a [[1-groupoid]] has only a single component, corresponding to the point $\ast \to B G$, but a [[pseudonatural transformation]] out of $B G$ into a [[2-groupoid]] (and higher) has in addition a component for each element of $G$, which are not reflected on $\ast \to B G$.
\end{remark}

In generalization of Ex. \ref{TerminalEpimorphismsInInfinityGroupoids}:
\begin{example}
A morphism $X\to Y$ between [[connected spaces]] is an epimorphism iff $Y$ is formed via a [[Quillen plus construction]] from a [[perfect group|perfect]] [[normal subgroup]] of the [[fundamental group]] $\pi_1(X)$.

More generally, a map of spaces is an epi iff all its fibers are acyclic spaces in the sense that their [[suspensions]] are [[contractible]].
\end{example}
This is discussed in [Raptis 2017](#Raptis17).

A generalization of this to epimorphisms and acyclic spaces in [[(infinity,1)-toposes|$(\infty,1)$-toposes]] is discussed in [Hoyois 19](#Hoyois19).

\begin{example}
A morphism $A \to B$ of [[E-infinity rings|$E_\infty$-rings]] is an epimorphism iff $B$ is [[smashing localization|smashing]] over $A$, i.e., if $ B \wedge_A B \approx B$.
\end{example}

## Related concepts

* [[Quillen plus construction]]

## References

* {#Raptis17} [[George Raptis]], *Some characterizations of acyclic maps*, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 ([arxiv:1711.08898](https://arxiv.org/abs/1711.08898), [doi:10.1007/s40062-019-00231-6](https://doi.org/10.1007/s40062-019-00231-6))

* {#Hoyois19} [[Marc Hoyois]], *On Quillen's plus construction*, 2019 ([pdf](http://www.mathematik.ur.de/hoyois/papers/acyclic.pdf), [[Hoyois_PlusConstruction.pdf:file]])

* [[Oren Ben-Bassat]], D. Mukherjee, _Analytification, localization and homotopy epimorphisms_, Bulletin des Sciences Mathématiques __176__ (2022) 103129 [arXiv:2111.04184](https://arxiv.org/pdf/2111.04184) [doi](https://doi.org/10.1016/j.bulsci.2022.103129)

[[!redirects epimorphisms in an (infinity,1)-category]]

[[!redirects epimorphism in an (∞,1)-category]]
[[!redirects epimorphisms in an (∞,1)-category]]

[[!redirects epimorphism in an infinity1-category]]


epimorphisms > history

< [[epimorphisms]]
[[!redirects epimorphisms -- history]]


epimorphisms of groups are surjective

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#Contents#
* table of contents
{:toc}

## Statement

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

Every [[epimorphism]] in the [[Grp|category of groups]] is [[surjective function|surjective]] (a [[regular epimorphism]]). Equivalently, the category of groups is [[balanced category|balanced]] (every [[monomorphism|monic]] epimorphism is an [[isomorphism]]).

=--

There are probably many [[proofs]] of this statement. Some (for example, I think the one in [[Saunders Mac Lane|Mac Lane's]] _[[Categories for the Working Mathematician]]_) involve some mild case analysis. The following (originally written up [here](http://ncatlab.org/toddtrimble/published/epimorphisms+in+the+category+of+groups)) is more uniform (and [[constructive mathematics|constructive]]).

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

Suppose $i: H \hookrightarrow G$ is a monic epi. Let $A$ be a nontrivial [[abelian group]], say $\mathbb{Z}/(2)$. There is a left [[action]] of $G$ on $A^G = \prod_{g \in G} A$, the group of *[[functions]]* $G \to A$, where $(g \cdot \phi)(g') \coloneqq \phi(g'g^{-1})$; this gives rise to a left $G$-[[module]]. It suffices to prove that any element $m \in A^G$ of the form

$$
  G \stackrel{proj}{\to} G/H \to A
$$

is a [[constant function]], i.e., is [[fixed point|fixed]] under the action of $G$, or in other words that the map $\phi_m: G \to A^G$ mapping $g \in G$ to $g \cdot m - m$ is identically $0$. As $\phi_m$ is an $A^G$-valued [[derivation on a group|derivation]], it yields a [[splitting]] of the [[exact sequence]] of groups

$$0 \to A^G \to G \ltimes A^G \to G \to 1$$

given by $k: G \to G \ltimes A^G$, with $k(g) \coloneqq (g, \phi_m(g))$. Of course we also have the trivial splitting $j(g) \coloneqq (g, 0)$. But note that the [[restrictions]] along $i: H \hookrightarrow G$ coincide: $j \circ i = k \circ i$ since $\phi_m(h) = 0$ for all $h \in H$ (as $H$ acts trivially on the right on [[cosets]] $g H$). Since $i$ is epic, we conclude $j = k$, or that $\phi_m \equiv 0$, as was to be shown.

=--


epipresheaf

#Contents#
* table of contents
{:toc}

## Idea

The notion of _epipresheaf_ is formally dual to the more standard notion of  [[separated presheaf]]: where the latter has a [[monomorphism]], the former has an [[epimorphism]].

From this perspective a [[sheaf]] is a [[presheaf]] satisfying two properties: the epipresheaf condition and the "[[monopresheaf]]" (or [[separated presheaf]]) condition. Thus there are epipresheaves, monopresheaves and sheaves.

## Definition

A presheaf $A$ is called _epipresheaf_ if for any [[local isomorphism]] $f:X\to Y$ the map $A(Y)\to A(X)$ is an [[epimorphism]]

## References

The notion is introduced in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative spaces_, preprint MPI-2004-35 ([ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenbergSpaces}, [arxiv 9812158](http://arxiv.org/abs/math/9812158)

category: sheaf theory
[[!redirects epipresheaves]]


epistemic logic > history

see at *[[epistemic modal logic]]*


epistemic logics > history

< [[epistemic logics]]
[[!redirects epistemic logics>history]]


epistemic modal logic

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#### Modality
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[[!include modalities - contents]]
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# Epistemic logic
* table of contents
{:toc}

## Idea

*Epistemic logic* is the branch of [[modal logic]] which is concerned with [[epistemology]], hence with notions of knowledge and sometimes also notions of belief, although these may be treated separately by [[doxastic logic]]. In its applied form it has found considerable use in [[computer science]] and Artificial Intelligence.

The key [[modal operators]] for epistemic logic are forms of *[[necessity and possibility]]* corresponding to "It is known to be the case that" and "It may be the case that (so far as is known)". As a form of modal logic, the [[semantics]] typically used is *[[possible worlds semantics]]* where the worlds correspond to maximally specific ways the world could have been. Philosophers disagree as to whether to distinguish the space of *metaphysically* possible worlds from the space of *epistemically* possible worlds ([Kment 21](#Kment21)).

## Introduction

Epistemic modalities are captured in [[epistemic logic|epistemic modal logic]] by interpreting "*necessarily $\phi$*" as saying _"It is known that proposition $\phi$ is true"_. It is common also to indicate the epistemic agent, so that corresponding to agent $x$ we have "$x$ knows that proposition $\phi$ is true".

In the closely related 'provability logic', the basic modal operator interprets  as _"it is provable that $\phi$"_.

Notice that the notions of _[[possibility]]_ and _[[necessity]]_ have different senses in ordinary language. For example, if we say '$P$ is possible', we may mean that $P$ is: *epistemically* possible, not ruled out by anything I know; *physically* possible, not ruled out by the laws of physics; *logically* possible, not ruled out by the laws of logic. Some suggest that there is a further type of possibility, *metaphysical* possibility intermediate between logical and physical possibility. Metaphysical possibility would allow that different laws of physics might apply.

## Epistemic formulae

These are variants of the formulae of the basic [[modal logic|modal language]].  The basic modal operators are, here, labelled $K_i$ since they relate to 'knowledge' and indicate the epistemic agent, the knower, $i$. These correspond to the $\Box$ operators in  the standard form, and are used in preference to the dual $\lozenge$ forms since they are more immediately relevant to applications.

More formally, we have $P$ or $Prop$, is a set of countably (finite or infinite) many atomic formulae.  there is also a set $A$, often called the set of 'agents' and taken to be $A = \{1,\ldots,m\}$.  The set of epistemic formulae (= basic $m$-agent epistemic language) will be denoted  $\mathcal{L}^m_K(P)$ is given by the rules

$$\phi ::= p \mid \bot \mid \neg \phi \mid \phi_1 \wedge \phi_2 \mid K_i\phi for i\in A.$$

We read $K_i \phi$ as ''agent $i$ knows that $\phi$''.

The converse or dual operators, denoted $M_i$ (so that $M_i \phi = \neg K_i\neg \phi$) reads as ''agent $i$ considers $\phi$ is possible''.

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

The 'agent' terminology is extremely useful, but in pure modal logic texts is not used so much.  It does provide an 'intuition' and an interpretation however.
=--

## Axioms for epistemic logic {#Axioms}

The question then arises as to which axioms of modal logic are appropriate to the epistemic case. Answers will depend on what interpretation is given to 'know' in specifying how the operator $K$ behaves. E.g., there is a difference between the knowledge of a realistic, cognitively-limited agent and that of some idealized agent with boundless resources.

It is generally admitted that axiom (T) should hold, here

$$
\mathbf{T}: K \phi \to \phi,
$$

which states that if $\phi$ is known then $\phi$ is true. Truth is generally taken to be a precondition of knowledge.

A much more contentious issue in the field is whether to admit an epistemic version of axiom (4). Known as the _[[KK principle]]_ or _KK thesis_, this corresponds to

$$
\mathbf{4}: K \phi \to K K \phi,
$$

which states that when $\phi$ is known to be true then it is also known that it is known. T

{#EvenPossible} It is even possible to challenge the admission of two of the most fundamental modal axioms, axioms (N) and (K). These correspond to

$$
  \mathbf{N}
  \phantom{,}\colon\phantom{,}
  \text{If}\phantom{,}P\phantom{,}\text{is a theorem, then so is}\phantom{,}K P
$$
and

$$
  \mathbf{K}: K(P \to Q) \to (K P \to K Q).
$$
An understanding of **N** that it states that all theorems are known is evidently problematic. Similarly for **K**, it is not clear that we know the deductive consequences of the collection of propositions that we know, such as the application of _modus ponens_ to every instance of known $P \to Q$ and $P$.

On the other hand, we might consider epistemic logic to represent the reasoning of an ideal, logically omniscient, agent, and so admit **N** and **K**.

## Models for epistemic logics

The geometric or combinatorial semantics of epistemic models follows the same techniques of [[Kripke frames]] as at [[geometric models for modal logics]], whilst the [[algebraic models for modal logics|algebraic models]] are BAOs that is [[algebraic models for modal logics|Boolean algebras with operators]].  As usual the [[Kripke frames]] semantics is an example of [[coalgebraic semantics]].

## Related entries

* [[epistemology]]

* [[necessity and possibility]]

* [[modal logic]]

* [[epistemology of mathematics]]

* [[the logic K(m)| K(m)]];

* [[the logic T(m)| T(m)]];

* [[the logic S4(m)|S4(m)]];

* [[the logic S5(m)|S5(m)]].

## References

### General

Early discussion of epistemic modal logic:

* [[Georg H. von Wright]], Section IV of: *An Essay in Modal Logic*, North-Holland Publishing (1951)

and introducing the formalization as [[K modal logic]]:

* [[K. Jaakko J. Hintikka]], *Knowledge and belief: An introduction to the logic of the two notions*, Cornell University Press (1962) [[ark:/13960/t9k437s65](https://archive.org/details/knowledgebeliefi00hint_0)]

A fairly recent book on epistemic logics and their applications (which was used for some of the material above):

* J.- J. Ch. Meyer and W. Van der Hoek, _Epistemic logic for AI and Computer Science_, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995,

General books on [[modal logic]] with discussion of epistemic logic:

* [[Patrick Blackburn]], M. de Rijke and [[Yde Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

* [[Marcus Kracht]], _Tools and Techniques in Modal Logic,_ Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999,

* Johan van Benthem, _Dynamic logic for belief revision_ ([pdf](http://www.illc.uva.nl/Research/Reports/PP-2006-11.text.pdf))

* {#Kment21} Boris Kment, _Varieties of Modality_, [SEP](https://plato.stanford.edu/entries/modality-varieties/#Dua)

See also:

* Wikipedia, *[Epistemic modal logic](https://en.wikipedia.org/wiki/Epistemic_modal_logic)*

[[!include S5 modal logic as epistemic logic -- references]]

[[!redirects epistemic modal logics]]

[[!redirects epistemic logic]]
[[!redirects epistemic logics]]

[[!redirects epistemic modality]]
[[!redirects epistemic modalities]]


epistemology

> This entry is about the notion in [[philosophy]] and [[modal logic]]. For Trimble's approach to [[formal category theory]], see [this page](https://ncatlab.org/toddtrimble/published/Epistemologies).

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\tableofcontents

## Idea

In [[philosophy]], *epistemology* refers to the theory of knowledge.

The [[mathematics|mathematical]] [[formal logic|formalization]] of epistemology is the subject of [[epistemic modal logic]] (and possibly of [[modal logic]] and [[modal type theory]] more generally).

## Related concepts

* [[epistemology of mathematics]]

* [[epistemic modal logic]]

## Literature

### General

Textbook accounts:

* Robert Audi , *Epistemology -- A Contemporary Introduction to the Theory of Knowledge*, Routledge (2011) [[ISBN:9780415879231](https://www.routledge.com/Epistemology-A-Contemporary-Introduction-to-the-Theory-of-Knowledge/Audi/p/book/9780415879231)]

*  Duncan Pritchard, *Epistemology*, Springer (2016) [[doi:10.1007/978-1-137-52692-2](https://doi.org/10.1007/978-1-137-52692-2)]

See also:

* Wikipedia, *[Epistemology](https://en.wikipedia.org/wiki/Epistemology)*

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

### In quantum physics
 {#InQuantumPhysics}

Discussion of epistemological issues in [[quantum physics]] and its [[interpretation of quantum mechanics|interpretation]] (see there for more):

* {#Bohr49} [[Nils Bohr]], *Discussion with Einstein on Epistemological Problems in Atomic Physics*, in: P. A. Schilpp (ed.), *Albert Einstein, Philosopher-Scientist*, The Library of Living Philosophers **VII** Evanston (1949) 201-241, and *Niels Bohr Collected Works* **7** (1996) 339-381 [doi:10.1016/S1876-0503(08)70379-7]

* [[Jagdish Mehra]], *The quantum principle: Its interpretation and epistemology*, Dialectica **27** 2 (1973) 75-157 [[jstor:42968519](https://www.jstor.org/stable/42968519)]

* [[Eugene P. Wigner]], *Epistemology of Quantum Mechanics*, Part I in: [[Jagdish Mehra]] (ed.), *Philosophical Reflections and Syntheses* Vol B.VI in: *The Collected Works of Eugene Wigner*, Springer (1973) [[doi:10.1007/978-3-642-78374-6_5](https://doi.org/10.1007/978-3-642-78374-6_5)]

* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]

* Florian J. Boge, *Quantum Mechanics Between Ontology and Epistemology*,  European Studies in Philosophy of Science (ESPS) **10**, Springer (2008) [[doi:10.1007/978-3-319-95765-4](https://doi.org/10.1007/978-3-319-95765-4)]


epistemology of mathematics

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\tableofcontents

## Overview

The [[epistemology]] of [[mathematics]] is the study of mathematical knowledge.

There are a few branches of [[epistemology]] which could be applied to mathematics and the [[foundations of mathematics]].

### Rationalism

Rationalism states that reason is what determines what knowledge is. As formal mathematics entirely relies on deduction, such as [[natural deduction]], to derive all of mathematics from some collection of basic primitives and axioms in the [[foundations of mathematics]], it relies on reason to derive mathematical knowledge.

### Empiricism

Empiricism states that experience is what determines what knowledge is.

add something here about Quine...

From the point of view of the [[history of mathematics]], many mathematical concepts are first developed from physical intuition, and then only later are they formalized in logic. For example, Newton and Leibniz developed [[calculus]] on the [[derivative]] and the [[Riemann integral]] from the experience of [[geometry]] and [[physics]] in real life, and only in the 19th century was calculus (in particular the [[derivative]] and the [[Riemann integral]]) put on a formal rationalist basis with [[epsilontic analysis]].

The pipeline from [[physics]] to pure mathematics abounds with examples which demonstrate empiricism. Similar things have occurred with [[Minkowski geometry]] and [[Riemannian geometry]] in the early 20th century with the advent of [[special relativity]] and [[general relativity]], and with [[differential cohomology]] in the mid 20th century with the advent of [[quantum field theory]].

### Pragmatism

Pragmatism states that all mathematical knowledge, including the words and language and symbols and notation associated with mathematics, are tools for mathematicians to use to solve problems in mathematics and to communicate to each other.

[Clarke-Duane 2022](#JCD22) argued that [[pluralism]] implies Carnap's pragmatism, as the relevant questions are not whether certain axioms are [[true]] or rules are derivable, but rather normative statements of which collection of axioms or rules to use in the [[foundations of mathematics]] in the broader mathematics community.

### Epistemic relativism

Epistemic relativism states that what is true or justified for one person is not necessarily true or justified for another person.

Looking at the mathematical community as a whole, epistemic relativism seems to be true in formal [[mathematics]] and in the [[foundations of mathematics]]. Take [[excluded middle]] as an example. For classical mathematicians working in [[ZFC]], [[excluded middle]] is simply true. However, for constructive mathematicians working in Brouwer's intuitionism, or in the [[sheaf topos]] of the [[Dedekind real numbers]], or in a [[smooth topos]], [[excluded middle]] is provably false. In [[neutral constructive mathematics]], excluded middle can neither be proven true nor false (it is independent), and neither it nor its negation is justified. Thus, the differing status of excluded middle for different mathematicians demonstrates epistemic relativism in mathematics.

### Epistemic idealism

For now see [[idealism]], [[absolute idealism]], [[subjective idealism]], [[objective idealism]].

## Related entries

* [[epistemology]]

* [[philosophy of mathematics]]

* [[epistemic modal logic]]

## References

* {#JCD22} Justin Clarke-Doane (2022). *Mathematics and Metaphilosophy (Elements in the Philosophy of Mathematics)*. Cambridge: Cambridge University Press. [[doi:10.1017/9781108993937](https://doi.org/10.1017/9781108993937), [philsci-archive:20728](https://philsci-archive.pitt.edu/20728/)]

See also:

* Wikipedia, *[Epistemology](https://en.wikipedia.org/wiki/Epistemology)*


EPR paradox

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## Idea

The "EPR paradox" is an argument due to [Einstein, Podoldsky & Rosen (1935)](#EinsteinPodoldskyRosen35) (historically perceived of as leading to a "[[paradox]]") about fundamental properties of [[quantum mechanics]] related to [[quantum entanglement]].

The argument begins by giving a necessary condition for a [[theory]] to be complete, that every element of reality features in the theory.

It goes on to note that for noncommuting operators/[[quantum observables]], the [[wave function]] cannot simultaneously be an [[eigenstate]] for both. Hence, it is not the case that both quantities are represented in the theory. From this they conclude that either [[quantum mechanics]] is incomplete or else not both of the quantities are real.

The second part of the argument sees the famous example of an [[entanglement|entangled pair]] of [[particles]]. The argument here now relies on what they take to be a sufficient condition for reality, that a quantity be predictable without disturbing the system. However, if I allow the two particles to travel far from each other, it appears that by making [[quantum measurements]] on one particle I can predict both of two noncommuting quantities of the other system (admittedly, not simultaneously) without disturbing it. Both quantities then are real.

Thus they conclude that quantum mechanics is not complete.

...  Need to talk about separability and locality. Then link to [[Bell inequality|Bell's inequalities]].

## Related entries

* [[ER = EPR]]

* [[parameterized quantum systems]]

## References

The original article:

* {#EinsteinPodoldskyRosen35} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review **47** 10 (1935) 777-780  [[doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777)]

Review and discussion:

* [[Franco Selleri]] (ed.), *Quantum Mechanics Versus Local Realism -- The Einstein-Podolsky-Rosen Paradox*, Physics of Atoms and Molecules (PAMO), Springer (1988) [[doi:10.1007/978-1-4684-8774-9](https://doi.org/10.1007/978-1-4684-8774-9)]

* [[Roland Omnès]], §9 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

* Arthur Fine, *The Einstein-Podolsky-Rosen Argument in Quantum Theory*, Stanford Encyclopedia of Philosophy [[web](http://plato.stanford.edu/entries/qt-epr/)]

* [[Anthony Sudbery]], §5.3 of: *Quantum mechanics and the particles of nature: an outline for mathematicians*, Cambridge University Press (1986) [[pdf](http://users.uoa.gr/~navidcon/sudbery.pdf), [spire:240835](https://inspirehep.net/literature/240835)]

* [[Jeffrey Bub]], §2.1 in: *Interpreting the quantum world*, Cambridge University Press (1999) [[ISBN:9780521653862](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/interpreting-quantum-world?format=PB&isbn=9780521653862)]

Background discussion with emphasis on [[quantum logic]] and [[Bell's inequalities]]:

* Laura Molenaar, *Quantum logic and the EPR paradox*, Delft (2014) [[uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b](http://resolver.tudelft.nl/uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b)]

See also:

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

Realization in [[experiment]]:

with massive bodies:

* Roman Schnabel, *Einstein-Podolsky-Rosen -- entangled motion of two massive objects*,  Phys. Rev. A **92** 012126 (2015) [[arXiv;1508.06462](https://arxiv.org/abs/1508.06462), [doi:10.1103/PhysRevA.92.012126](https://doi.org/10.1103/PhysRevA.92.012126)]

with [[Bose-Einstein condensates]]:

* Paolo Colciaghi, Yifan Li, Philipp Treutlein, Tilman Zibold, *Einstein-Podolsky-Rosen experiment with two Bose-Einstein condensates* [[arXiv:2211.05101](https://arxiv.org/abs/2211.05101)]

[[!redirects EPR paradox]]
[[!redirects Einstein-Podolsky-Rosen paradox]]
[[!redirects Einstein–Podolsky–Rosen paradox]]
[[!redirects Einstein--Podolsky--Rosen paradox]]

[[!redirects EPR argument]]
[[!redirects Einstein-Podolsky-Rosen argument]]
[[!redirects Einstein–Podolsky–Rosen argument]]
[[!redirects Einstein--Podolsky--Rosen argument]]

[[!redirects EPR phenomenon]]
[[!redirects EPR phenomena]]


epsilon-number

# $\varepsilon$-numbers

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## Definition

An **$\varepsilon$-number** is an [[ordinal number]], or more generally a [[surreal number]], $\alpha$, such that $\alpha = \omega^\alpha$, where $\omega$ is the first infinite ordinal.  In other words, it is a [[fixed point]] of the [[exponential map]] $\lambda x. \omega^x$.

## Properties

The ordinal $\varepsilon$-numbers are unbounded: for every ordinal $\beta$ there is an $\varepsilon$-number greater than $\beta$, namely the limit of the sequence $\beta, \omega^\beta ,\omega^{\omega^\beta},\dots$.  Thus, there are a proper class of ordinal $\varepsilon$-numbers, and their [[order type]] is the same as that of the ordinals themselves.  Hence, we can label them by writing $\varepsilon_n$ for the $n^{th}$ $\varepsilon$-number, where $n$ is any ordinal.

In particular, $\varepsilon_0$ is the first ordinal $\varepsilon$-number, namely the limit of the sequence $1,\omega, \omega^\omega ,\omega^{\omega^\omega},\dots$.  It may be tempting to write $\varepsilon_0$ as
$$ \omega^{\omega^{\omega^{\dots}}} $$
but in fact *every* $\varepsilon$-number could be written in this way, so this notation does not specify a unique number.

Similarly, the surreal $\varepsilon$-numbers have the same order type as the surreals, and can be notated as $\varepsilon_n$ where $n$ is a surreal.

[[!redirects epsilon-number]]
[[!redirects epsilon-numbers]]
[[!redirects epsilon number]]
[[!redirects epsilon numbers]]
[[!redirects ∞-number]]
[[!redirects ∞-numbers]]
[[!redirects ε number]]
[[!redirects ε-numbers]]
[[!redirects ∞-number]]
[[!redirects ∞-numbers]]
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[[!redirects ε numbers]]
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epsilontic analysis

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# Epsilontic analysis
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## Idea

Epsilontic analysis is the now standard rigorous approach to [[analysis]], developed by [[Bernard Bolzano]], [[Augustin Cauchy]], [[Karl Weierstraß]], [[Richard Dedekind]], and others in the mid-to-late 19th century.  It contrasts with [[infinitesimal analysis]], both the less-than-rigorous analysis that preceded it and the rigorous [[nonstandard analysis]] (and other approaches to infinitesimals, such as [[synthetic differential geometry]]) that followed it.

Ironically, [[classical mathematics|classical]] epsilontic analysis is not rigorous enough for [[constructive mathematics]], which requires it to be applied using [[intuitionistic logic]].  This was developed thoroughly by [[Errett Bishop]]; see [[constructive analysis]].

## History

Epsilontic analysis developed out of the practical problems of finding errors of [[approximation]].  Taking the basic principles of calculus for granted, 18th-century mathematicians discovered various methods to approximate the values of [[limit of a sequence|limits]], [[derivatives]], [[integrals]], and [[series]] with an error guaranteed to be bounded above by whatever [[positive number]] $\epsilon$ (for 'error') one wished.

Epsilontic analysis turns these results around into definitions: the value of a limit, derivative, integral, or series is *defined* to be the (necessarily unique) [[real number]] $L$ such that, given any positive number $\epsilon$, one can produce an approximation within $\epsilon$ of $L$.

[[Augustin Cauchy]] is often credited with promoting this approach, in his 1821 _[[Cours d'Analyse]]_.  However, Cauchy also used [[infinitesimal number|infinitesimals]], and he did not actually *define* concepts with epsilontics.  Cauchy did, however, use epsilontic arguments in many proofs.

Other mathematicians used epsilontics to clarify Cauchy\'s work.  [[Bernard Bolzano]] gave epsilontic definitions and proofs even earlier than Cauchy, but his work was largely ignored at the time.  This later came to the attention of [[Karl Weierstraß]], who set out a systematic treatment of epsilontic analysis, the basis of the standard analysis of today.

## Topological vs uniform properties

The precise post-Cauchy development of epsilontics was necessary to clarify the difference between (as we now see them) [[topological property|topological]] and [[uniform property|uniform properties]], which show up in the order of [[quantifiers]] in the definitions.

Typically, topological properties apply at a point, although one can then require them at all points.  Doing so, one has a definition of the form
$$ \forall x,\; \forall \epsilon,\; \exists M,\; \phi(M,x,\epsilon) ,$$
where $M$ is some parameter for a method of approximation and $\phi(M,x,\epsilon)$ states that the method, applied with the parameter $M$, approximates a value of interest at $x$ to within an error $\epsilon$.  Equivalently, one may write
$$ \forall \epsilon,\; \forall x,\; \exists M,\; \phi(M,x,\epsilon) ,$$
but we cannot shift the quantifiers further.

In contrast, a uniform property necessarily refers to a range of points and typically takes the form
$$ \forall \epsilon,\; \exists M,\; \forall x,\; \phi(M,x,\epsilon) ,$$
which is stronger.

It is difficult to make this distinction without the $\epsilon$.  It is not impossible to do this otherwise, as [[Abraham Robinson]] showed with infinitesimals, but it\'s telling that nobody did so before epsilontics.  Even Cauchy\'s definitions are difficult to understand in this respect, since he used epsilontics only for proofs (and not always even then) and not for definitions.  (In this regard, see the [[Cauchy sum theorem]].)

## Examples

* for an epsilontic proof that [[polynomials are continuous]] see for instance ([Miller 14](#Miller14))

## References

The extent to which Cauchy anticipated modern epsilontics is debated.  For arguments pro and contra (respectively), see:

* {#Grabiner1983} Judith Grabiner (1983). Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus. The American Mathematical Monthly 90:3 (1983). [pdf](http://www.maa.org/pubs/Calc_articles/ma002.pdf).

* {#BKS2012} Piotr Błaszczyk, Mikhail G. Katz, David Sherry (2012). Ten Misconceptions from the History of Analysis and Their Debunking. [arXiv](http://arxiv.org/abs/1202.4153). Foundations of Science 18:1 (2013), 43--74. [doi](http://dx.doi.org/10.1007/s10699-012-9285-8).

On Bolzano, this looks promising, but I have not read it:

*  {#Rusnock2000} Paul Rusnock (2000).  Bolzano\'s Philosophy and the Emergence of Modern Mathematics.  Rodopi (2000).

Examples of epsilontic proofs include

* {#Miller14} Kyle Miller, _Polynomials are continuous functions_, 2014 ([pdf](https://math.berkeley.edu/~kmill/math1afa2014/poly.pdf))

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Epstein zeta function

### Definition

Let $\tau = \tau_1 + i\tau_2\in\mathbb{C}$ with $\tau_2\gt 0$. Then, the corresponding __Epstein Zeta Function__ is a [[meromorphic function]] which appears by [[analytic continuation]] to the complex plane of the function $\zeta_\tau:s\to\zeta_\tau(s)$ which is for $Re(s)\gt 0$ defined by the formula
$$
\zeta_\tau(s) = \sum_{\overset{n,m\in\mathbb{Z}}{(n,m)\neq(0,0)}}\left(\frac{\tau_2}{|n+m\tau|}\right)^s
$$

### Bibliography

* for related notions see [[zeta function]] and the links therein
* L. Kronecker, _Zur Theorie der Elliptischen Functionen I, IV_ (1883) in Leopold
Kronecker's Werke IV
* C. L. Siegel, _Lectures on advanced analytic number theory_, Tata Institute
(1961)