Ismail Zahed

* [Mathematics genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=121088)

## Selected writings

On [[nuclear physics]] via [[chiral perturbation theory]]/[[quantum hadrodynamics]]:

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

On [[skyrmions]]:

* [[Ismail Zahed]], [[Gerald E. Brown]] *The Skyrme model*, Physics Reports **142** 1–2 (1986) 1-102 [doi:10.1016/0370-1573(86)90142-0]

* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _The Multifaceted Skyrmion_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

On the [[Cheshire cat principle]] of the [[quark bag model]]:

* {#NNZ85} S. Nadkarni, [[Holger Bech Nielsen]], [[Ismail Zahed]], _Bosonization Relations As Bag Boundary Conditions_, Nucl. Phys. B 253 (1985) 308 (doi:10.1016/0550-3213(85)90533-4)

On [[nucleons]] as [[Skyrmions]] in [[quantum hadrodynamics]] with [[pions]], [[omega-mesons]] and [[rho-mesons]] ($\pi$-$\rho$-$\omega$-model):

* {#MeissnerZahed86} [[Ulf-G. Meissner]], [[Ismail Zahed]], _Skyrmions in the Presence of Vector Mesons_, Phys. Rev. Lett. 56, 1035 (1986) ([doi:10.1103/PhysRevLett.56.1035](https://doi.org/10.1103/PhysRevLett.56.1035))

  (includes also the [[A1-meson]])

Its derivation from the [[AdS/QCD correspondence]]:

* [[Holger Bech Nielsen]], [[Ismail Zahed]], _The Cheshire Cat Principle from Holography_ ([doi:10.1142/9789814280709_0017](https://doi.org/10.1142/9789814280709_0017), [arXiv:0904.0429](https://arxiv.org/abs/0904.0429)), Chapter 17 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

On [[heavy flavor]] via [[holographic QCD]]:

* Yizhuang Liu, [[Ismail Zahed]], _Heavy and Strange Holographic Baryons_, Phys. Rev. D 96, 056027 (2017) ([arXiv:1705.01397](https://arxiv.org/abs/1705.01397))

On [[nucleon]] [[form factors]] via [[holographic QCD]]:

* Kiminad A. Mamo, [[Ismail Zahed]],  _Nucleon mass radii and distribution: Holographic QCD, Lattice QCD and GlueX data_ ([arXiv:2103.03186](https://arxiv.org/abs/2103.03186))

On [[quantum hadrodynamics]] via [[light-front QCD]]:

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials* ([arXiv:2110.15927](https://arxiv.org/abs/2110.15927))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials* ([arXiv:2111.01775](https://arxiv.org/abs/2111.01775))

category: people


Ismar Volic > history

see at _[[Ismar Volić]]_


Ismar Volić

* [webpage](https://www.wellesley.edu/math/faculty/volici)

## Selected writings

On [[graph complex]]-models for [[configuration spaces of points]] and [[formality of the little n-disk operad]]:

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

On higher order [[Vassiliev invariants]] as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]:

* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))

On [[cubical objects]] in [[homotopy theory]], such as in [[n-excisive functors]] and [[Goodwillie calculus]]:

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volić]], _Cubical homotopy theory_, Cambridge University Press, 2015  ([pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf), [doi:10.1017/CBO9781139343329](https://doi.org/10.1017/CBO9781139343329))

## Related $n$Lab entries

* [[graph complex]]

* [[little n-disk operad]]

* [[formality of the little n-disk operad]]

* [[configuration space of points]]

* [[Fulton-MacPherson compactification]]

* [[semi-algebraic manifold]]

* [[Blakers-Massey theorem]]

* [[Goodwillie calculus]]

category: people

[[!redirects Ismar Volic]]


isobar

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### Context
#### Physics
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[function]] which is thought of as assigning to each [[element]] of its [[domain]] the _[[pressure]]_ [[observable|observed]] at that locus, then the [[preimage]] of any element in its [[codomain]] (a value of the function)  is called the _isobar_ for that value.

## Related concepts

* [[isosurface]]

## References

See also:

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

[[!redirects isobars]]


isocofibration

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{:toc}

## Idea

A morphism $f:A\to B$ in a [[strict 2-category]] $K$ is an **isocofibration** if the corresponding morphism $B\to A$ in $K^{op}$ is an [[isofibration]].

## Examples

In [[Cat]] (and [[Grpd]]), the isocofibrations are the [[functors]] which are [[injective-on-objects functor|injective on objects]].  There they are the [[cofibrations]] in the [[canonical model structure on Cat]] ([[canonical model structure on groupoids|on Grpd]]).

However, a general 2-category has two candidates for a canonical model structure, one involving the [[isofibrations]] and one the isocofibrations, and in general the two may not be the same.

## Related concepts

* [[isofibration]]

[[!redirects isocofibrations]]
[[!redirects iso-cofibration]]
[[!redirects iso-cofibrations]]
[[!redirects co-isofibration]]
[[!redirects co-isofibrations]]
[[!redirects co-iso-fibration]]
[[!redirects co-iso-fibrations]]


isofibration

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\tableofcontents

\section{Idea}

An _isofibration_ is, roughly speaking, a [[functor]] $F: E \rightarrow B$ between [[category|categories]] such that every [[isomorphism]] $f$ in $B$ can be 'lifted' to an isomorphism in $E$. Here 'lifted' means that there is an isomorphism $g$ in $E$ such that $F(g) = f$.

Alternatively, an isofibration is the analogue of a [[Hurewicz fibration]] for the [[interval object]] in $\mathsf{Cat}$, the [[Cat|category of categories]], given by the [[free-standing isomorphism]]. That this definition is equivalent to the former one will be established below.

It is the second of these definitions that often generalises better to higher categories.

\section{Definition}

\begin{defn} An **isofibration** is a [[functor]] $p:E\to B$ such that for any [[object]] $e$ of $E$ and any [[isomorphism]] $\phi:p(e) \cong b$, there exists an isomorphism $\psi:e \cong e'$ such that $p(\psi)=\phi$. \end{defn}

\begin{rmk} Equivalently: for any commutative diagram

\begin{tikzcd}
1 \ar[r, "e"] \ar[d, swap, "0"] & E \ar[d, "p"] \\
I \ar[r, swap, "\phi"] & B
\end{tikzcd}

in $\mathsf{Cat}$, where $I$ is the [[free-standing isomorphism]], there is a functor $l: I \rightarrow E$ such that the following diagram in $\mathsf{Cat}$ commutes.

\begin{tikzcd}
1 \ar[r, "e"] \ar[d, swap, "0"] & E \ar[d, "p"] \\
I \ar[r, swap, "\phi"] \ar[ur, "l"] & B
\end{tikzcd}

\end{rmk}

\begin{rmk} We can picture this as follows.

$$
  \array{
    e &\stackrel{\exists \psi \in p^{-1}(\phi)}{\to} & \exists e'&&& E
    \\
     &&&&&  \downarrow^p
    \\
    p(e) &\stackrel{\phi}{\to} & b &&& B
  }
  \,.
$$
\end{rmk}

\begin{rmk} If $p$ is a [[forgetful functor]], then being an isofibration says that whatever stuff $p$ forgets can be "transported along isomorphisms." \end{rmk}

\begin{rmk} {#NotInvariantUnderEquivalence} Notice that this definition of isofibration violates the 1-categorical [[principle of equivalence]] where it demands that $p(\psi)=\phi$ (which includes the requirement that $p(e') = b$): this condition is not invariant under [[equivalence of categories]]. If one changed the definition to involve just an [[isomorphism]] $p(\psi)\cong\phi$, then of course, any functor would qualify. But the point of isofibrations is rather to help present the [[(2,1)-category]] of categories/groupoids in terms of 1-categorical data. For more on this see below at _[As fibrations in canonical model structures](#AsFibrationsInCanonicalModelStructures)_. \end{rmk}

\section{Equivalent definition}

The following may at first seem a little surprising. It says that isofibrations have in fact a stronger lifting property, namely the analogue of that of a [[Hurewicz fibration]] with respect to the [[interval object]] in $\mathsf{Cat}$ given by the [[interval groupoid]]. This stronger lifting property is more conceptually fundamental with regard to finding the correct generalisation of isofibrations to higher categories, where 'correct' refers for instance to defining the fibrations of a [[model structure]].

\begin{prpn} Let $p: E \rightarrow B$ be a [[functor]] between categories. Then $p$ is an isofibration if and only if for every commutative diagram

\begin{tikzcd}
X \ar[r, "f"] \ar[d, swap, "id \times 0"] & E \ar[d, "p"] \\
X \times I \ar[r, swap, "g"] & B
\end{tikzcd}

in $\mathsf{Cat}$, where $I$ is the [[free-standing isomorphism]], there is a  functor $l: X \times I \rightarrow E$ such that the following diagram in $\mathsf{Cat}$ commutes.

\begin{tikzcd}
X \ar[r, "f"] \ar[d, swap, "id \times 0"] & E \ar[d, "p"] \\
X \times I \ar[r, swap, "g"] \ar[ur, "l"] & B
\end{tikzcd}

\end{prpn}

\begin{proof} The "only if" direction is immediate. Let us demonstrate that the "if" direction holds by constructing $l$. To this end, for any object $x$ of $X$, let $l_{x}$ be the unique functor such that the following diagram in $\mathsf{Cat}$ commutes, which exists since $p$ is an isofibration.

\begin{tikzcd}
1 \ar[r, "x"] \ar[d, swap, "0"] & E \ar[d, "p"] \\
I \ar[r, swap, "\phi_{x}"] \ar[ur, "l_{x}"] & B
\end{tikzcd}

Here $\phi_{x}$ is the functor representing the isomorphism $g(x, i)$ of $B$, where $i$ is the arrow $0 \rightarrow 1$ of $I$.

* For any object $x$ of $X$, we define $l(0)$ to be $f(x)$, define $l(x, i)$ to be $l_{x}(i)$, define $l\left(x,i^{-1}\right)$ to be $l_{x}\left(x^{-1}\right)$, and define $l(1)$ to be $l_{x}(1)$.
* For any arrow $r: x \rightarrow x'$ of $X$, we define $l(r,0)$ to be $f(r)$, and define $l(r, 1)$ to be $l_{x'}(i) \circ f(r) \circ l_{x}\left(i^{-1}\right)$.

It is immediately checked that $l$ is well-defined, is indeed a functor, and fits into the required commutative diagram.

\end{proof}

## Properties

### General

Isofibrations have a number of good properties.  For example, any [[strict 2-limit|strict pullback]] of an isofibration is also a [[2-limit|weak pullback]]. (This is also explained by the role of isofibrations as the fibrations in the [[canonical model structures]], see [below](#AsFibrationsInCanonicalModelStructures).)

\begin{example}\label{GrothendieckFibrationsAreIsofibrations}
Any [[Grothendieck fibration]] or opfibration is an isofibration (take the lifts to be the Cartesian lifts), but not in general conversely, unless $B$ is a [[groupoid]].
\end{example}

\begin{example}\label{GrothIsIsoFibBetweenGroupoids}
  For [[groupoids]] $\mathcal{E}, \mathcal{B} \,\in\,$ [[Grpd]], a functor $P \,\colon\, \mathcal{E} \to \mathcal{B}$ the following are equivalent:

1. $P$ is a [[Grothendieck fibration]],

1. $P$ is an [[isofibration]].

\end{example}
\begin{proof}
  The implication $IsoFib \Rightarrow GrothFib$ is Ex. \ref{GrothendieckFibrationsAreIsofibrations}.

Conversely, if $P$ is an isofibration, it is sufficient to show that any of the lifts $f$ it provides is a [[Cartesian morphism]]. But by assumption the lift is again an [[isomorphism]], and all isomorphisms are Cartesian.

More explicitly (in the notation [there](Grothendieck+fibration#CartesianMorphism))
\begin{tikzcd}[
  sep=20pt
]
  &
  z
  \ar[
    drrr,
    bend left=20,
   "{ \forall \, g }"{description}
  ]
  \ar[
    dr,
    dashed,
    "{
       \exists! \, \widehat{w}
    }"{description}
  ]
  \\
  \mathcal{E}
  \ar[dd, "{ P }"]
  &&
  x
  \ar[
     rr,
     "{ f }"{description},
     "{ \mathrm{cartesian} }"{swap, yshift=-1pt}
  ]
  &&
  y
  \\
  &
  P(z)
  \ar[drrr, bend left=20, "{ P(g) }"{description}]
  \ar[
    dr,
    "{ \forall \, w }"{description}
  ]
  \\
  \mathcal{B}
  &&
  P(x)
  \ar[rr, "{ P(f) }"{description}]
  &&
  P(y)
\end{tikzcd}
the [[invertible morphism|invertibility]] of all morphisms implies that both $w$ and $w'$ are uniquely determined by $f$ and $g$, and then [[functor|functoriality]] of $P$ implies that indeed $P(\widehat{w}) = w$.
\end{proof}

### As fibrations in canonical model structures
 {#AsFibrationsInCanonicalModelStructures}

The isofibrations are the _[[fibrations]]_ in the [[canonical model structure on categories]] and the [[canonical model structure on groupoids]].  More generally, the fibrations in [[canonical model structures]] on various types of higher categories are usually some generalization of isofibrations.  For example, the fibrations in the Lack model structure on 2-Cat, known as [[equifibrations]], have "equivalence lifting" and "local isomorphism lifting," and the fibrations in the Joyal model structure for [[quasi-category|quasicategories]] have "equivalence lifting" at all levels.

Generalizing in another direction, internalized isofibrations are the fibrations in the [[2-trivial model structure]] on any finitely complete and cocomplete [[strict 2-category]].

## Related concepts

* [[amnestic isofibration]]

* [[isocofibration]]

* [[equifibration]]

## References

For groupoids the definition appears (called "star surjectivity" there) on p. 105 (3 of 30) in

* {#Brown70} [[Ronnie Brown]], _Fibrations of groupoids_, Journal of Algebra Volume 15, Issue 1, May 1970, Pages 103-132 doi:[10.1016/0021-8693(70)90089-X](https://doi.org/10.1016/0021-8693%2870%2990089-X), [author's pdf](http://groupoids.org.uk/pdffiles/fibrationsgpds.pdf)

[[!redirects isofibrations]]


isogeny

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## Idea

An **isogeny** is a [[surjection]] of [[algebraic groups]] that has a [[finite set|finite]] [[kernel]]. For [[abelian varieties]], this is equivalently a [[rational map]] between groups of the same dimension that preserves the neutral element (which implies that it is a [[group homomorphism]]).

## References

* This Stack Exchange discussion, _[What is isogeny?](https://math.stackexchange.com/questions/36724/what-is-isogeny)_

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

Examples of sporadic (exceptional) isogenies from [[spin groups]] onto [[orthogonal groups]] are discussed in

* {#Garrett13} [[Paul Garrett]], _Sporadic isogenies to orthogonal groups_ (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), [[Garrett-SporadicIsogenies.pdf:file]] ]

[[!redirects isogenies]]


isoinserter

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## Idea

An _iso-inserter_ is a particular kind of [[2-limit]] in a [[2-category]], which universally inserts an [[isomorphism|invertible]] [[2-morphism]] between a pair of parallel 1-morphisms.

## Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category.  The **iso-inserter** of $f$ and $g$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ and an invertible 2-morphism $\alpha\colon f v \xrightarrow{\cong} g v$.

More precisely, universality means that for any object $X$, the induced functor
$$Hom(X,V) \to IsoIns(Hom(X,f),Hom(X,g)) $$
is an equivalence, where $IsoIns(Hom(X,f),Hom(X,g))$ denotes the category whose objects are pairs $(u,\beta)$ where $u\colon X\to A$ is a morphism and $\beta\colon f u \xrightarrow{\cong} g u$ is an invertible 2-morphism.  If this functor is an *isomorphism* of categories, then we say that $V\xrightarrow{v} A$ is a **strict iso-inserter**.

Iso-inserters and strict iso-inserters can be described as a certain sort of [[weighted limit|weighted]] 2-limit, where the diagram shape is the [[walking structure|walking]] parallel pair $P = (\cdot \rightrightarrows\cdot)$ and the weight $P\to Cat$ is the diagram
$$1 \;\rightrightarrows\; Iso$$
where $1$ is the [[terminal category]] and $Iso$ is the walking [[isomorphism]].

An iso-inserter in $K^{op}$ (see [[opposite 2-category]]) is called a **co-iso-inserter** in $K$.

## Properties

* Any iso-inserter is a [[faithful morphism]], and also a [[conservative morphism]], but not in general a [[fully faithful morphism]].

* Any strict iso-inserter is, in particular, an iso-inserter.  (This is not true for all strict 2-limits.)

* Since $Iso$ is [[equivalence of categories|equivalent]] to $1$, iso-inserters (but not strict iso-inserters) can equivalently be described as the [[conical limit]] of a diagram of shape $P$, which might be called a (non-strict) *pseudo-equalizer*.  A strict pseudo-equalizer---that is, a [[strict pseudolimit]] of a diagram of shape $P$---is not the same as a strict isoinserter, but if it exists, a strict pseudo-equalizer is also, in particular, a (non-strict) iso-inserter.  The relationship between isoinserters and pseudoequalizers is analogous to the relationship between [[iso-comma-objects]] and [[pseudo-pullbacks]].

* Iso-inserters can be constructed as an [[inserter]] followed by an [[inverter]] (for both the strict and non-strict versions).  In particular, it follows that strict iso-inserters are a type of [[PIE-limit]].

[[!redirects isoinserters]]
[[!redirects strict isoinserter]]
[[!redirects strict isoinserters]]
[[!redirects co-isoinserter]]
[[!redirects co-isoinserters]]
[[!redirects strict co-isoinserter]]
[[!redirects strict co-isoinserters]]
[[!redirects iso-inserter]]
[[!redirects iso-inserters]]
[[!redirects strict iso-inserter]]
[[!redirects strict iso-inserters]]
[[!redirects co-iso-inserter]]
[[!redirects co-iso-inserters]]
[[!redirects strict co-iso-inserter]]
[[!redirects strict co-iso-inserters]]
[[!redirects pseudo-equalizer]]
[[!redirects strict pseudo-equalizer]]
[[!redirects biequalizer]]


isolated point

# Isolated points
* table of contents
{: toc}

## Definitions

A [[concrete point|point]] $P$ of a [[topological space]] $X$ is __isolated__ if it is a [[neighbourhood]] of itself, in other words if the [[singleton subset]] $\{P\}$ is [[open subset|open]].

More generally, a point $P$ of a [[subset]] $A$ of a space $X$ is __isolated__ in $A$ if it is isolated when viewed as a point in the [[subspace]] $A$ with the [[subspace topology]].  More explicitly, for some [[neighbourhood]] $U$ of $P$ (in $X$), $U \cap A = \{P\}$.

The antithetical concept is that of an [[accumulation point]].

## Properties

Every [[function]] on $X$ is [[continuous function|continuous]] at $P$ if $P$ is isolated.

[[!redirects isolated point]]
[[!redirects isolated points]]


isometry

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# Contents
* table of contents
{: toc}

## Idea

An _isometry_ is a [[function]] that preserves a [[metric]], either in the sense of a [[metric space]] or in the sense of a [[Riemannian manifold]].

## Metric spaces

An __isometry__ $f\colon (X,d) \to (X',d')$ between metric spaces is a function $f\colon X \to X'$ between the underyling sets that respects the metrics in that $d = f^* d'$.  More explicitly, $d'(f(a),f(b)) = d(a,b)$ for any points $a,b$ in $X$.

The same idea holds for extended quasi-pseudo-generalisations of metric spaces.

## Manifolds

An __isometry__ $f\colon (X,g) \to (X',g')$ between Riemannian manifolds is a morphism $f\colon X \to X'$ between the underlying [[manifolds]] that respects the metrics in that $g = f^* g'$.  More explicitly, $g'(f_*v,f_*w) = g(v,w)$ for any [[tangent vectors]] $v,w$ on $X$.

## Global isometries

Global isometries are the [[isomorphisms]] of metric spaces or Riemannian manifolds.  An isometry is __global__ if it is a [[bijection]] whose inverse is also an isometry.  Between metric spaces, isometries are necessarily [[injections]] and bijective isometries necessarily have isometries as inverses, so global isometries between metric spaces are also called _[[surjection|surjective]] isometries_; this does not work for Riemannian manifolds (where the inverse of an isometry need not be a morphism of manifolds), nor does it work for [[pseudometric spaces]] (where an isometry need not be injective).

## Infinitesimal isometries

see [[Killing vector field]]

## Isometries on normed vector spaces

In practice, isometries $E \to F$ between [[normed vector space|normed vector spaces]] tend to be affine maps. The following theorem gives a precise meaning to this.

A norm on a vector space is **strictly convex** if, whenever ${\|u\|} = 1 = {\|v\|}$, we have ${\|t u + (1-t)v\|} \lt 1$ for some (hence all!) $t$ in the range $0 \lt t \lt 1$. In brief, no sphere contains a line segment. Examples of strictly convex spaces include spaces of type $L^p$ for $1 \lt p \lt \infty$.

+-- {: .un_thm}
###### Theorem
Let $f \colon E \to F$ an isometry between normed vector spaces, and suppose $F$ is strictly convex. Then $f$ is affine.
=--

+-- {: .proof}
###### Proof
To say that $f \colon E \to F$ is affine means that $f$ preserves linear combinations of the form $t x + (1-t)y$. It suffices to consider only the case where $0 \lt t \lt 1$ and, by continuity considerations, only the case of dyadic rationals between $0$ and $1$. Continuing this train of thought, it suffices to prove that $f(\frac1{2}(x + y)) = \frac1{2}(f(x) + f(y))$ for all $x, y$.

In the case of strict convexity, midpoints $\frac1{2}(u+v)$ are determined in terms of the norm, as the unique point $w$ such that

$${\|w - u\|} = \frac1{2}{\|u-v\|} = {\|w-v\|}.$$

The midpoint satisfies these equations for any normed vector space, but the uniqueness is a consequence of strict convexity. For if there were two such points $w, w'$, then for some point $w''$ on the line segment between them, we would have ${\|w'' - u\|} \lt \frac1{2}{\|u-v\|}$, and ${\|w''-v\|} \leq \frac1{2}{\|u-v\|}$ by ordinary convexity. But these two inequalities taken together would violate the triangle inequality.

As a result, since $f$ is an isometry, $w = f(\frac1{2}(x+y))$ is forced to be the midpoint between $f(x)$ and $f(y)$ if $F$ is strictly convex. This completes the proof.
=--

If $F$ is not strictly convex, then isometries need not be affine. For example, consider $E = \mathbb{R}$, and $F = \mathbb{R}^2$ equipped with the $l_\infty$ (max) norm. For any contractive map $\phi \colon \mathbb{R} \to \mathbb{R}$, e.g., any smooth function with ${|\phi'|} \leq 1$, the map $E \to F$ sending $x$ to $(x, \phi(x))$ is easily seen to be an isometry.

If however $f$ is a _surjective_ isometry between normed vector spaces, then $f$ is affine, by the [[Mazur-Ulam theorem]].

## Related concepts

* [[isometry group]]

* [[linear isometry]]

* [[superisometry]]

[[!redirects isometry]]
[[!redirects isometries]]
[[!redirects global isometry]]
[[!redirects global isometries]]

[[!redirects isometric isomorphism]]
[[!redirects isometric isomorphisms]]

[[!redirects orthogonal map]]
[[!redirects orthogonal maps]]

[[!redirects orthogonal linear map]]
[[!redirects orthogonal linear maps]]


isometry group

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###Context###
#### Riemannian geometry
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#### Differential geometry
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#Contents#
* table of contents
{:toc}

## Definition

For $(X,g)$ a [[Riemannian manifold]] or [[pseudo-Riemannian manifold]] its **isometry group**

$$
  Iso(X,g) \subset Diff(X)
$$

is that subgroup of the group of all [[diffeomorphism]]s $\phi : X \to X$ that are [[isometries]]: which preserve the [[metric]] $g$ in that

$$
  \phi^* g = g
  \,.
$$

## Properties

### Killing vectors

The [[Lie algebra]] of $Iso(X,g)$ is spanned by the [[Killing vector]]s of $(X,g)$.

## Examples

* The isometry group of [[Minkowski spacetime]] is the [[Poincaré group]].

* The isometry group of [[anti de Sitter spacetime]] is the [[anti de Sitter group]].

* The isometry group of [[de Sitter spacetime]] is the [[de Sitter group]].

Special [[subgroups]] of isometry groups:

* [[finite rotation group]]

* [[wallpaper group]]

## Related concepts

* [[isometry]]

* [[superisometry group]]

[[!redirects isometry groups]]


isomonodromic deformation

* Claude Sabbah, _Isomonodromic deformations and Frobenius manifolds_, Springer 2007, [doi](http://dx.doi.org/10.1007/978-1-84800-054-4)
* [[Nigel Hitchin]], _Twistor spaces, Einstein metrics and isomonodromic deformations_, J. Diff. Geom. __3__ (1995) 52-134


isomorphic functors > history

< [[isomorphic functors]]


isomorphism

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#### Equality and Equivalence
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#Contents#
* table of contents
{:toc}

## Idea

The concept of _isomorphism_ generalizes the concept of [[bijection]] from the category [[Set]] of [[sets]] to general [[categories]].

An _isomorphism_ is an invertible [[morphism]], hence a morphism with an [[inverse morphism]].

Two [[objects]] of a [[category]] are said to be _isomorphic_ if there exists an isomorphism between them. This means that they "are the same for all practical purposes" as long as one does not violate the [[principle of equivalence]].

But beware that two objects may be isomorphic by more than one isomorphism. In particular a single object may be isomorphic to itself by nontrivial isomorphisms other than the [[identity morphism]]. Frequently the particular choice of isomorphism matters.

Every isomorphism is in particular an [[epimorphism]] and a [[monomorphism]], but the converse need not hold.

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".

[[Henri Poincaré]] wrote in [Poincaré 1908](#Poincaré1908) about isomorphisms:

> "It is scarcely credible, as Mach said, how much a well-chosen word can economize thought. I do not know whether or not I have said somewhere that mathematics is the art of giving the same name to different things. We must so understand it. It is appropriate that things different in substance, but alike in form, should be put into the same mold, so to speak. When our language is well chosen, it is astonishing to see how all the demonstrations made upon some known fact immediately become applicable to many new facts. Nothing has to be changed, not even the words, since the names are the same in the new cases. There is an example, which comes at once to my mind; it is quaternions, upon which, however, I will not dwell.

> A word well chosen very often causes the disappearance of exceptions to rules as announced in their former forms; it was for this purpose that the terms 'negative quantities', 'imaginary quantities', 'infinite points', have been invented. And let us not forget that these exceptions are pernicious, for they conceal laws. Very well then, one of those marks by which we recognize the pregnancy of a result is in that it permits a happy innovation in our language. The mere fact is oftentimes without interest; it has been noted many times, but has rendered no service to science; it becomes of value only on that day when some happily advised thinker perceives a relationship, which he indicates and symbolizes by a word.

> The physicists also do it just the same way. They invented the term 'energy', a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form.

> Among the words which have had this happy result, I will mention the 'group' and the 'invariant'. They make us perceive the gist of many mathematical demonstrations; they make us realize how often mathematicians of the past must have run across groups without recognizing them and how, believing these groups to be isolated things, they have found them to be in close relationship without knowing why. Today we would say that they were looking right in the face of isomorphic groups. We feel now that in a group the substance interests us but very little; it is the form alone which matters, and so, when we once know well a single group, then we know through it all the isomorphic groups; thanks to the words 'groups' and 'isomorphism', which sum in a few syllables this subtle law and make it at once familiar to us all, we take our step at once and in so doing economize all effort of thought."

##Definitions

An __isomorphism__, or __iso__ for short, is an invertible [[morphism]], i.e. a [[morphism]] with a 2-sided [[inverse]].

A morphism could be called __isic__ (following the more common 'monic' and 'epic') if it is an isomorphism, but it\'s more common to simply call it _invertible_.  Two [[objects]] $x$ and $y$ are __isomorphic__ if there exists an isomorphism from $x$ to $y$ (or equivalently, from $y$ to $x$).  An __[[automorphism]]__ is an isomorphism from one object to itself.

## Properties

It is immediate that isomorphisms satisfy the [[two-out-of-three]] property. But they also satisfy [[two-out-of-six property]] satisfied by the [[weak equivalences]] in any [[homotopical category]].

Note that the [[inverse morphism]] of an isomorphism is an isomorphism, as is any [[identity morphism]] or [[composite]] of isomorphisms.  Thus, being isomorphic is an [[equivalence relation]] on objects.  The equivalence classes form the [[fundamental 0-groupoid]] of the category in question.

Every isomorphism is both a [[split monomorphism]] (and thus about any other kind of [[monomorphism]]) and a [[split epimorphism]] (and thus about any other kind of [[epimorphism]]).  In a [[balanced category]], every morphism that is both a [[monomorphism]] and an [[epimorphism]] is invertible, but this does not hold in general.  However, any monic [[regular epimorphism]] (or dually, any epic [[regular monomorphism]]) must be an isomorphism.

A [[groupoid]] is precisely a [[category]] in which every morphism is an isomorphism.  More generally, the [[core]] of any category $C$ is the [[subcategory]] consisting of all objects but only the isomorphisms; it is a kind of underlying groupoid of $C$.  In a similar way, the automorphisms of any given object $x$ form a [[group]], the [[automorphism group]] of $x$.

In [[n-category|higher categories]], isomorphisms generalise to [[equivalence]]s, which we expect to have only [[weak inverse]]s.

In the context of [[homotopy type theory]], for every [[morphism]] $f : hom_A(a,b)$ the [[type]] "f is an isomorphism" is a [[proposition]]. Therefore, for any $a,b:A$ the type $a \cong b$ is a set.
\begin{proof}
Suppose given $g:hom_A(b,a)$ and $\eta:1_a = g \circ f$ and $\epsilon : f \circ g = 1_b$, and similarly $g',\eta'$, and $\epsilon'$. We must show $(g,\eta,\epsilon)=(g',\eta', \epsilon')$. But since all hom-sets are sets, their identity types are mere propositions, so it suffices to show $g=g'$. For this we have $ g' = 1_{a} \circ g'= (g \circ f) \circ g' = g \circ (f \circ g') = g \circ 1_{b}= g$
\end{proof}

## Examples

*  A __[[bijection]]__ is an isomorphism in [[Set]].

*  A __[[homeomorphism]]__ is an isomorphism in [[Top]].

*  A __[[diffeomorphism]]__ is an isomorphism in [[Diff]].

* Every morphism in a [[groupoid]] is an isomorphism.
  By definition of groupoid.

## Related concepts

* [[equality]]

* **isomorphism**

  * [[isomorphism class]]

  * [[exceptional isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homotopy equivalence]], [[weak homotopy equivalence]]

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

* [[equivalence of (∞,1)-categories]]

* [[EI-category]]

## References

* {#Poincaré1908} [[Henri Poincaré]], *The future of mathematics*, Revue generale des Sciences pures et appliquees, Paris, 19th year, No. 23, December, 1908 ([pdf](https://sites.math.rutgers.edu/courses/535/535-f08/Poincare.pdf))

* [[Francis Borceux]], Section 1.9 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

[[!redirects isomorphic]]
[[!redirects isomorphisms]]
[[!redirects iso]]
[[!redirects isic]]

[[!redirects isomorphic]]


isomorphism class

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## Idea

Given a [[category]], the *isomorphism class* of one of its [[objects]] is its [[equivalence class]] under the [[equivalence relation]] which regards two objects as [[relation|related]] when they are connected by an [[isomorphism]].

The set of isomorphism classes of a category is also sometimes thought of as the category's *[[decategorification]]*.

## Examples

* The isomorphism class of a [[finite set]] is a [[natural number]].

* The isomorphism class of a [[set]] is its [[cardinality]].

* The isomorphism class of a [[vector space]] is (labeled by) its [[dimension of a vector space|dimension]].

* The isomorphism classes of an [[action groupoid]] are the [[orbits]] of the [[group action]].

## Related concepts

* [[K-theory]]

* [[Grothendieck group]]

* [[homotopy class]]

[[!redirects isomorphism classes]]


isomorphism classes of Banach spaces

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This page is inspired by the following question, which appeared [on MathOverflow](http://mathoverflow.net/questions/79713/lp-mathbbr-vs-lq-mathbbr).

> Let $p,q \in (1,\infty)$ with $p\neq q$.  Are the [[Banach spaces]] $L^p(\mathbb{R})$, $L^q(\mathbb{R})$  [[norm isomorphism|isomorphic]]?

More generally, one can ask:

> Given two [[Banach spaces]], $X$ and $Y$, when are they [[norm isomorphism|isomorphic]]?

The following started out as an adapted version of Bill Johnson's answer to the MathOverflow question.

One way to prove that a [[Banach space]] $X$ is not [[norm isomorphism|isomorphic]] to a Banach space $Y$ is to exhibit a property which is preserved under isomorphisms that $X$ has but $Y$ does not.  For example, among the spaces $L_p(\mathbb{R})$ for $p \in [1,\infty]$, $L_\infty$ is the only nonseparable space, and $L_1$ is the only separable space with a nonseparable dual.  Thus $L_1$ and $L_\infty$ are not isomorphic to each other or to any $L_p$ with $p \in (1,\infty)$.

To distinguish among the $L_p$ with $p \in (1,\infty)$ finer properties are needed.
[[type (functional analysis)|Type]] and [[cotype (functional analysis)|cotype]] are examples of such properties.  The [[type (functional analysis)|(best) type]] and [[cotype (functional analysis)|cotype]] of $L_p$ are standard calculations: if $p \in [1,2]$ then $L_p$ has type $p$ and cotype $2$ (and no better), and if $p \in [2,\infty)$ then $L_p$ has type $2$ and cotype $p$ (and no better). See for example in Theorem 6.2.14 of [AK06](#AK06).  From that, one can see that if $p \ne q$, then $L_p$ and $L_q$ either have different (best) type or different (best) cotype.

Type and cotype depend only on the collection of finite dimensional subspaces of a space (we call such a property a [[local property]]).  So neither can be used to prove, e.g., that for $p \ne 2$, $L_p$ is not isomorphic to $\ell_p$.  One way of proving this is to show that for $p \ne 2$, $\ell_2$ embeds isomorphically into $L_p$ but not into $\ell_p$ (see also [AK](#AK)).

### References ###

* {: #AK06 } **AK06** Albiac, Fernando and Kalton, Nigel.
Topics in Banach space theory.
Graduate Texts in Mathematics, 233. Springer, New York, 2006. xii+373 pp. ISBN: 978-0387-28141-4; [MR2192298](http://www.ams.org/mathscinet-getitem?mr=2192298)


isomorphism in a Segal type

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\tableofcontents

## Definition

In [[simplicial type theory]], given a [[Segal type]] $A$ and [[terms]] $x \colon A$ and $y \colon A$, a "morphism" ([[term]] of [[hom type]]) $f \colon \mathrm{hom}_A(x, y)$ is an **isomorphism** if there exist morphisms $g:\mathrm{hom}_A(y, x)$ and $h:\mathrm{hom}_A(y, x)$ such that $g \circ f = \mathrm{id}_x$ and $f \circ h = \mathrm{id}_y$
$$\mathrm{isIso}(f) \coloneqq \left(\sum_{g:\mathrm{hom}_A(y, x)} g \circ f = \mathrm{id}_x\right) \times \left(\sum_{h:\mathrm{hom}_A(y, x)} f \circ h = \mathrm{id}_y\right)$$

The type of all isomorphisms between $x$ and $y$ in $A$ is defined as
$$
  \big(x \cong_A y\big)
   \;\;\coloneqq\;\;
  \sum_{
    \mathclap{
      f \colon \mathrm{hom}_A(x, y)
    }
  }
  \mathrm{isIso}(f)
$$

## Related concepts

* [[hom type]]

* [[Segal type]]

* [[Rezk type]]

* [[isomorphism]]

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

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$

[[!redirects isomorphism in a Segal type]]
[[!redirects isomorphisms in a Segal type]]

[[!redirects type of isomorphisms in a Segal type]]


isospin

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Properties

### Spin-isospin transmutation

* [[spin-isospin transmutation]]

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

Textbook accounts:

* [[Howard Georgi]], §5 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

* Wiukipedia, _[Isospin](https://en.wikipedia.org/wiki/Isospin)_


isosurface

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] and [[data analysis]], where an [[observable]] is typically a [[function]] $X \overset{f}{\longrightarrow} \mathbb{R}^n$ from a given data set $X$ to the set of [[real numbers]] $\mathbb{R}^1$, or to an [[n-tuple]] of such (varying in a [[Cartesian space]] $\mathbb{R}^n$), an _iso-hypersurface_ or _level-hypersuface_ is a [[level set]] of this observable, hence the [[subset]] of $X$ consisting all those points $x \in X$ on which this observable $f$ takes the same given value $f(x) = c$.

For example, if $X$ is a model for the Earth's atmosphere, and if $f$ assigns to each point $x \in X$ the atmospheric [[pressure]] $p(x)$ at this point -- in some suitable [[physical units]] and to some suitable approximation --, then an iso-surface for $p$ is an _[[isobar]]_: a [[surface]] of [[constant function|constant]] [[pressure]].

For such a [[level set]] to actually be a [[hypersurface]], hence a [[differentiable manifold|differentiable]]/[[smooth manifold|smooth]] [[submanifold]], some regularity conditions on $f$ need to be satisfied, such as that $f$ is a [[differentiable function]] to suitable degree, and, crucially, that its value $c$ (whose [[pre-image]] is formed) is a _[[regular value]]_.

Phrased this way, the construction of iso-hypersurfaces turns out to be a central topic also in areas of pure mathematics, such as in [[differential topology]] and [[cobordism theory]], where the formation of [[pre-images]] of [[regular values]] inside $\mathbb{R}^n$ is known as part of the [[Pontryagin construction]].

Curiously, this means that[[Pontryagin's theorem]] applies to iso-hypersurfaces, saying here that, as the value $c$ of the [[observable]] $f$ varies, the shape ([[topology]]) of the corresponding iso-hypersurfaces changes (at most) by a _[[cobordism]]_, and that the resulting ([[normally framed submanifold|normally framed]]) [[cobordism class]] of all these hypersurfaces corresponds to the class of $f$ in the $n$-[[Cohomotopy theory]] of the data set $X$.

While Pontryagin's theorem is ancient, the idea that it thus implies the possibility of [[topological data analysis]] via [[Cohomotopy theory]] of [[observables]] and [[cobordism theory]] of their iso-hypersurfaces appears only recently ([Franek-Krčál 16](#FranekKrcal16), [Franek-Krčál 17](#FranekKrcal17)).

## Examples

* [[isobar]]

## References

### General

See also

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

[[!include cohomotopy in topological data analysis -- references]]

[[!redirects isosurfaces]]

[[!redirects iso-surface]]
[[!redirects iso-surfaces]]

[[!redirects iso-hypersurface]]
[[!redirects iso-hypersurfaces]]


isotope

For isotope in algebra see unrelated entry [[isotope (algebra)]].

## Definition

Two atoms (in chemistry and nuclear physics) or nuclei are isotopes if they have the same number of protons, but different number of neutrons.

category: physics
[[!redirects isotopes]]


isotope (algebra)

This entry is about the isotope and isotopy in algebra, for isotopy in geometry see [[isotopy]]; homotopy of quasigroups is also covered, do not mix with the notion from [[homotopy theory]].
For [[isotope]] in chemistry and nuclear physics see wikipedia; along with [[isotopic spin]] might be once covered in $n$Lab.

## Idea

Homotopy and isotopy are relaxed notions of homomorphism and isomorphism suitable for nonassociative binary algebraic structures.

## Definition

Let $(A,\cdot), (B,\star)$ be [[binary algebraic structure]]s (magmas)
and $\xi,\eta,\zeta : A\to B$ set maps. The triple $(\xi,\eta,\zeta)$ is a homotopy of binary structures if for all $x,y\in A$

$$
\xi(a)\star \eta(b) = \zeta (a\cdot b).
$$

A homotopy is an __isotopy__ if $\xi,\eta,\zeta$ are
bijections of sets; if there is an isotopy $(A,\cdot)\to (B,\star)$, then we say that $(A,\cdot)$ is isotopic to $(B,\star)$, or that they are __isotopes__. Autotopy is an isotopy from $(A,\cdot)$ to itself.

## Properties

Being isotopic is a relation of equivalence. Magmas and their isotopies from a groupoid.

Every [[loop (algebra)|loop]] isotopic to a group is isomorphic to a group. This is why isotopy is a non-interesting notion for groups.
There exist a [[quasigroup]] which is isotopic to a group but not isomorphic to a group.

## Literature

* R. Artzy, _Isotopy an parastrophy of quasigroups_, Proc. Amer. Math. Soc. __14__ (1963)

category: algebra
[[!redirects isotopy (algebra)]]
[[!redirects homotopy (algebra)]]
[[!redirects homotopy of quasigroups]]


isotopy

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# Contents
* table of contents
{: toc}

## Idea ##

A [[homotopy]] between two maps $f,g \;\colon\; X \to Y$ may be thought of as a [[continuous map|continuous]] [[path]] in the [[mapping space]] $Map(X,Y)$.  Similarly, a path in the  [[topological subspace|subspace]] $Emb(X,Y) \hookrightarrow Map(X,Y)$ of [[embeddings]] of $X$ into $Y$, is an **isotopy**.

Since, for arbitrary [[topological spaces]], the mapping space $Map(X,Y)$ is not always well-behaved, defining a homotopy as a _left homotopy_ using a [[cylinder object]], hence as a map $H \colon X \times I \to Y$ is the rigorous way of saying "$H$ is a path from $f$ to $g$ in the mapping space.".  Similarly, the proper definition of an isotopy as a map $H \colon X \times I \to Y$ with the property that $H(-,t) \colon X \to Y$ is always an embedding, is the rigorous way of saying "$H$ is a path from $f$ to $g$ in the embedding space.".

In some settings, even this is not strong enough.  In the notion of an **ambient isotopy**, one requires the path to extend to a path in the space of homeomorphisms of the ambient space.

## Definition ##

+-- {: .num_defn #isotopy}
###### Definition
Let $X$ and $Y$ be two [[topological spaces]].  Let $f,g \colon X \to Y$ be two [[embeddings]] of $X$ in to $Y$.  An **isotopy** from $f$ to $g$ is a [[continuous map]] $H \colon X \times [0,1] \to Y$ with the following properties:

1. $H(-,0) = f$
2. $H(-,1) = g$
3. $H(-,t)$ is an embedding of $X$ in to $Y$ for each $t \in [0,1]$.

Two maps for which there exists an isotopy are said to be **isotopic**.
=--

+-- {: .num_defn #amiso}
###### Definition
Let $X$ and $Y$ be two [[topological spaces]].  Let $f,g \colon X \to Y$ be two [[embeddings]] of $X$ in to $Y$.  An **ambient isotopy** from $f$ to $g$ is a [[continuous map]] $H \colon Y \times [0,1] \to Y$ with the following properties:

1. $H(-,0)$ is the identity on $Y$
2. $H(-,1) \circ f = g$
3. $H(-,t)$ is a self-homeomorphism of $Y$ for each $t \in [0,1]$.
=--

## Properties ##

The first two conditions on an isotopy $H$ imply that $H$ is a homotopy from $f$ to $g$.  Therefore the condition of being isotopic is stronger than that of being homotopic.  As the third condition applies level-wise, the proof that homotopy is an equivalence relation carries over to show that the same is true of isotopy.

## Examples ##

Isotopy is used where one wishes to study deformations of an object inside some ambient space that do not change the object itself.  An extremely important example of this is the theory of [[knots]] and [[links]] where, to prevent unknottings and unlinkings, there have to be some restrictions on the allowed movements.  These are usually encoded in terms of isotopies. It is unfortunately true, however, that a naive use of isotopy leads to strange results. If you use *continuous* isotopies then any two continuous embeddings of the circle into $S^3$ are isotopic, (basically since you just pull the knot tighter and tighter, (so the knotted bit gets smaller and smaller) and at the end you just put the 'unknot').  One way to handle this is to demand the isotopy to be [[piecewise linear]] (or [[smooth map|smooth]]), another is to work explicitly with ambient isotopy.

One of the beauties of isotopy of knots is that it can be realised very simply at the level of [[knot diagrams]]. Two knots are isotopic if their respective knot diagrams can be related using [[Reidemeister moves]]. (This is a formal theorem, but will be given elsewhere after some more development.)

## Related concepts

* [[smooth isotopy]]

## References ##

General discussion includes

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

For isotopy in [[knot theory]] see

 * R. H Crowell and R.H. Fox, Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.

 * N.D. Gilbert and [[Tim Porter]], _Knots and Surfaces_, Oxford U.P., 1994.

Specifically for [[smooth isotopy]]:

* {#Greene13} Josh Greene, _Combinatorial methods in knot theory_, 2013 ([pdf](https://www2.bc.edu/joshua-e-greene/MT885S13/Lecture%201.pdf))

[[!redirects isotopies]]
[[!redirects Isotopy]]
[[!redirects Isotopies]]
[[!redirects isotopic]]
[[!redirects Isotopic]]
[[!redirects ambient isotopy]]
[[!redirects Ambient Isotopy]]
[[!redirects Ambient isotopy]]
[[!redirects ambient isotopies]]
[[!redirects Ambient Isotopies]]
[[!redirects Ambient isotopies]]

[[!redirects isotopy class]]
[[!redirects isotopy classes]]


isotropic Grassmannian

#Contents#
* table of contents
{:toc}

## Definition

Given an [[inner product space]], its **isotropic Grassmannian** is the space of all its [[isotropic subspaces]].

## Related concepts

* [[Grassmannian]]

  * **isotropic Grassmannian**

  * [[Lagrangian Grassmannian]]

[[!redirects isotropic Grassmannians]]


isotropic submanifold

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## Definition

### Traditional definition

A [[submanifold]] of a [[symplectic manifold]] each [[tangent space]] of which is an [[isotropic subspace]] with respect to the ambient symplectic structure is an **isotropic submanifold**.

## Related concepts

* [[coisotropic submanifold]]

* [[Lagrangian submanifold]]

[[!include (co)isotropic subspaces - table]]

[[!redirects isotropic submanifolds]]


isotropic subspace

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#Contents#
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 {:toc}

## Definition

A **totally isotropic subspace** of an [[inner product space]] is a sub-[[vector space]] on which the [[bilinear form]] vanishes.

An **isotropic subspace** of a [[symplectic vector space]] is a [[vector space|vector subspace]] on which the [[symplectic form]] vanishes.

A maximal isotropic subspace is called a **[[lagrangian subspace]]**.

The space of all isotropic subspaces of a given inner product space is called its **[[isotropic Grassmannian]]**.

A [[submanifold]] of a [[symplectic manifold]] each [[tangent space]] of which is isotropic with respect to the ambient symplectic structure is an **[[isotropic submanifold]]**.

## Related concepts

[[!include (co)isotropic subspaces - table]]

[[!redirects isotropic subspaces]]


isotropy group of a topos

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# Isotropy group of a topos

* table of contents
{: toc}

## Definition

The **isotropy group of a [[topos]]** $T$ is its [[free loop space object]] $I_T$ in [[Topos]], regarded as a group object over $T$.  It is therefore a [[group topos]] in the world of $T$-toposes.

Specifically, this means it is the [[pullback]] of the [[diagonal]] $T \to T\times T$ against itself, in the sense appropriate to $Topos$ as a higher category.

This definition makes sense for (Grothendieck) 1-toposes and also for [[(infinity,1)-topos|higher toposes]].  But in the 1-topos case, the map $I_T \to T$ is automatically [[localic geometric morphism|localic]], since it is a pullback of the diagonal $T\to T\times T$ which is localic, and therefore corresponds to a [[localic group]] internal to $T$.  (In general, if $T$ is an $n$-topos, then $I_T\to T$ will be $(n-1)$-[[n-localic geometric morphism|localic]].)  The group of points of this localic group is then a [[group object]] $Z_T$ in $T$, the **etale isotropy group** of $T$.  The latter captures some but not all of the information about $I_T$.

## Isotropy quotient

Like a free loop space object in any category, $I_T$ acts on all toposes over $T$.  In particular, it acts on [[etale geometric morphisms]] over $T$, and hence on objects of $T$.  The **isotropy quotient** of $T$ is the [[full subcategory]] of objects for which this action is trivial.

The isotropy quotient is contained in the **etale isotropy quotient**, namely the full subcategory of objects for which the action of the etale isotropy group $Z_T$ is trivial, but it may be strictly smaller.

## References

The etale isotropy group of a 1-topos (then called just the "isotropy group") was defined in

* Jonathon Funk, Pieter Hofstra, and Benjamin Steinberg, _Isotropy and crossed toposes_, [TAC](http://www.tac.mta.ca/tac/volumes/26/24/26-24abs.html)

It was shown to be the group of points of the localic isotropy group in

* [[Simon Henry]], _The localic isotropy group of a topos_, [arxiv](https://arxiv.org/abs/1706.04835)


Israel Gelfand

__Israel Moiseevich Gel'fand__ (Израиль Гельфанд, 1913--2009) played a preeminent role in Russian mathematics for many years.  His  legendary seminar in Moscow ran for 50 years, and set the pattern for many others, including Sullivan's seminar at CUNY and Drinfeld's at Chicago.    His name is attached to many mathematical achievements, including

* the [[Gelfand representation]] of a [[Banach algebra]],

* an important special case, the [[Gelfand-Naimark theorem]] describing commutative [[C*-algebra]]s,

* the [[Gelfand-Naimark-Segal construction]] describing general C*-algebras,

* the [[Gelfand-Fuks cohomology]] of a [[foliation]],

* the [[Gelfand-Kirillov dimension]] of an [[associative algebra]],

* the [[Bernstein-Gelfand-Gelfand resolution]] for [[representations]] of simple [[Lie groups]].

* [[Gel'fand triple]]

He is the author of a 6-volume treatise on [[distribution]] theory (various volumes in collaboration with Shilov, Vilenkin, Graev).

## Selected writings

The characterization of [[state on a star-algebra|states]] on [[group algebras]] and what came to be known as the [[Gelfand-Raikov theorem]]:

* {#GelfandRaikov43} И. М. Гельфанд, Д. А. Райков, *Неприводимые унитарные представления локально бикомпактных групп*, Матем. сб., 13(55):2–3 (1943)  301–316 [[mathnet pdf](https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf)]

  \linebreak

  [[Israel Gelfand]], [[Dmitri Raikov]], *Irreducible unitary representations of locally bicompact groups*, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [[mathnet:eng/sm6181](https://www.mathnet.ru/eng/sm6181)]

Introducing the [[GNS construction]]:

* {#GelfandNaimark43} [[Israel Gelfand]], [[Mark Naimark]], *On the imbedding of normed rings into the ring of operators on a Hilbert space*,  Matematicheskii Sbornik. 12 (2): 197–217 (1943)

  reprinted in:

  Robert Doran (ed.), *$C^\ast$-Algebras: 1943–1993*, Contemporary Mathematics **167**, AMS 1994 ([doi:10.1090/conm/167](http://dx.doi.org/10.1090/conm/167))

One of the first comprehensive series of texts on [[generalized functions]] (also known as a [[distributions]]):

* [[Israel M. Gelfand]], [[Georgiy E. Shilov]], *Generalized Functions* Vol. 1-6 (1958), Acad. Press  (1966-1968), AMS (2016) [[ISBN:978-1-4704-2885-3](https://bookstore.ams.org/view?ProductCode=CHELGELFSET)]

category: people

[[!redirects Израиль Моисеевич Гельфанд]]
[[!redirects Израиль Гельфанд]]
[[!redirects Гельфанд]]
[[!redirects И. М. Гельфанд]]
[[!redirects И. Гельфанд]]
[[!redirects Israel Gel'fand]]
[[!redirects I. M. Gelfand]]
[[!redirects I. M. Gel'fand]]
[[!redirects Israel M. Gelfand]]


Israel Vainsencher

* [institute page](https://www.mat.ufmg.br/~israel/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=38671)

## Selected writings

On [[quantum cohomology]]:

* [[Joachim Kock]], [[Israel Vainsencher]], *An Invitation to Quantum Cohomology -- Kontsevich's Formula for Rational Plane Curves*, Birkhäser (2007) [[doi:10.1007/978-0-8176-4495-6](https://doi.org/10.1007/978-0-8176-4495-6)]

category: people


Issac Newton

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

category: people


Issai Schur

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

## Selected writings

Introducing [[Schur functions]]:

* [[Issai Schur]], *Über eine Klasse von Matrizen die sich einer gegeben Matrix zuordnen lassen*, Inaugural-Dissertation, Berlin (1901)

## Related $n$Lab entries

* [[Schur's lemma]]

* [[Schur function]], [[Schur polynomial]], [[Schur functor]]

* [[Schur decomposition]]

category: people


ItaCa

The name _ItaCa_ stands for **Ita**lian **Ca**tegory theorists, but it also reminds the isle where Odysseus strived so hard to come back.

[website](https://progetto-itaca.github.io).


Itamar Pitowsky

* [Institute page](http://edelstein.huji.ac.il/people/past-members/itamar-pitowsky)

A brief rememberance:

* [[Gil Kalai]], *[Itamar Pitowsky: Probability in Physics, Where does it Come From?](https://gilkalai.wordpress.com/2010/02/15/itamar-pitowski-probability-in-physics-where-does-it-come-from/)* (Feb 2010)

See also:

*  Meir Hemmo, Orly Shenker (eds.): *Quantum, Probability, Logic -- The Work and Influence of Itamar Pitowsky*, Springer (2020) &lbrack[doi:10.1007/978-3-030-34316-3](https://doi.org/10.1007/978-3-030-34316-3)]

## Selected writings

On how [[Bell's inequalities]] in [[quantum probability theory]] have been preconceived by [[George Boole]] (in *[[Boole -- The Laws of Thought|The Laws of Thought]]*) in classical  [[probability theory]]:

* {#Pitowsky98} [[Itamar Pitowsky]], *From George Boole To John Bell — The Origins of Bell’s Inequality*, in: *Bell’s Theorem, Quantum Theory and Conceptions of the Universe*, Fundamental Theories of Physics **37** Springer (1989) [[doi:10.1007/978-94-017-0849-4_6](https://doi.org/10.1007/978-94-017-0849-4_6)]

On [[quantum probability]] and [[quantum logic]] with emphasis on this perspective on [[Bell's inequalities]]:

* [[Itamar Pitowsky]], *Quantum Probability -- Quantum Logic*, Lecture Notes in Physics **321**, Springer (1989) [[doi:10.1007/BFb0021186](https://doi.org/10.1007/BFb0021186)]

category: people


Itaï Ben Yaacov

* [website](http://math.univ-lyon1.fr/~begnac/)

### Related nLab entries

* [[continuous logic]]


iterated algebraic K-theory

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## Idea

### General

The construction of [[algebraic K-theory]] $K(R)$, originally defined for [[rings]] $R$, generalizes to [[A-∞ ring|$A_\infty$]]-[[ring spectra]].
When $R$ happens to be a [[connective spectrum|connective]] [[E-infinity ring|$E_\infty$]]-[[ring spectrum]], then also the [[Brown representability theorem|representing spectrum]] $K(R)$ of its algebraic K-theory is a [[connective spectrum|connective]] [[E-infinity ring|$E_\infty$]]-[[ring spectrum]]
([Schwänzl & Vogt 1994, Thm. 1](#SchwaenzlVogt94), [EKMM 1997, Thm. 6.1](#EKMM97)) so that this construction may then be iterated ([Rognes 2014](#Rognes14)) to yield _iterated algebraic K-theories_ $K(K(R))$, $K(K(K(R)))$, etc.

The [[red-shift conjecture]] says that this iteration plays a special role in [[chromatic homotopy theory]].

### On topological K-theory

The construction of iterated algebraic K-theory has received particular attention for the case that $R = $ [[ku]] is the [[connective spectrum|connective]] [[ring spectrum]] [[Brown representability theorem|representing]] [[complex vector bundle|complex]] [[topological K-theory]].

Here the first iterated stage $K(ku)$ is related to [[BDR 2-vector bundles]] essentially like [[ku]] is related to ordinary [[complex vector bundles]].

The tower $K^{2r}(ku)$ of higher iterated algebraic K-theories of topological K-theory has been shown to accommodate a generalization of the [[Fourier-Mukai transform|Fourier-Mukai]]-type transform on [[twisted K-theory]] that is given by [[topological T-duality]], generalizing it to [[spherical T-duality]] ([Lind-Sati-Westerland 16](#LindSatiWesterland16)).

## Properties

### Red-shift conjecture

See at *[[red-shift conjecture]]*.

## References

### Algebraic K-theory of ring spectra

On the [[algebraic K-theory]] of ([[connective spectrum|connective]]) [[ring spectra]]:

* {#SchwaenzlVogt94} [[Roland Schwänzl]], [[Rainer Vogt]], *Basic Constructions in the K-Theory of Homotopy Ring Spaces*, Transactions of the American Mathematical Society, Vol. 341, No. 2 (Feb., 1994), pp. 549-584 ([jstor:2154572](https://www.jstor.org/stable/2154572), [doi:10.2307/2154572](https://doi.org/10.2307/2154572))

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VI of _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], Section 9.5 of: _A universal characterization of higher algebraic K-theory_, Geom. Topol. 17 (2013) 733-838 ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282), [doi:10.2140/gt.2013.17.733](http://dx.doi.org/10.2140/gt.2013.17.733))

* [[Jacob Lurie]], _Algebraic K-Theory of Ring Spectra_, Lecture 19 of _[Algebraic K-Theory and Manifold Topology](https://www.math.ias.edu/~lurie/281.html)_, 2014 ([pdf](http://people.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf))

The  algebraic K-theory specifically of [[suspension spectra]] of [[loop spaces]] (Waldhausen's _[[A-theory]]_) is originally due to

* [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_, In: A. Ranicki N.,  Levitt, F. Quinn (eds.), Algebraic and Geometric Topology, Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg (1985) ([doi:10.1007/BFb0074449](https://doi.org/10.1007/BFb0074449))

On the iteration of the construction and the [[red-shift conjecture]]:

* {#Rognes14} [[John Rognes]], _Chromatic redshift_, lecture notes, MSRI 2014 ([arXiv:1403.4838](https://arxiv.org/abs/1403.4838))

On the [[algebraic K-theory]] $K(R)$ of a [[ring spectrum]] $R$ as the [[Grothendieck group]] of [[(∞,1)-module bundles]] over $R$:

* [[John Lind]], _Bundles of spectra and algebraic K-theory_, Pacific J. Math. 285 (2016) 427-452 ([arXiv:1304.5676](https://arxiv.org/abs/1304.5676), [doi:10.2140/pjm.2016.285.427](https://doi.org/10.2140/pjm.2016.285.427))

### Algebraic K-theory of topological K-theory

On the first [[algebraic K-theory]] $K(ku)$ of [[connective spectrum|connective]] [[topological K-theory]]:

* [[Christian Ausoni]], _On the algebraic K-theory of the complex K-theory spectrum_, Inventiones mathematicae volume 180, pages 611–668 (2010) ([arXiv:0902.2334](http://arxiv.org/abs/0902.2334), [doi:10.1007/s00222-010-0239-x](https://doi.org/10.1007/s00222-010-0239-x))

* [[Christian Ausoni]], [[John Rognes]], _Algebraic K-theory of topological K-theory_, Acta Math. Volume 188, Number 1 (2002), 1-39 ([euclid:acta/1485891473](https://projecteuclid.org/euclid.acta/1485891473))

* [[Christian Ausoni]], [[John Rognes]], _Rational algebraic K-theory of topological K-theory_, Geom. Topol. 16 (2012) 2037-2065 ([arXiv:0708.2160](https://arxiv.org/abs/0708.2160), [doi:10.2140/gt.2012.16.2037](http://dx.doi.org/10.2140/gt.2012.16.2037))

Interpretation of $K(ku)$ as the [[K-theory]] of [[BDR 2-vector bundles]]:

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[John Rognes]], _Two-vector bundles and forms of elliptic cohomology_, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004 ([arXiv:math/0306027](https://arxiv.org/abs/math/0306027),  [doi:10.1017/CBO9780511526398.005](https://doi.org/10.1017/CBO9780511526398.005))

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Stable bundles over rig categories_, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 ([arXiv:0909.1742](https://arxiv.org/abs/0909.1742), [doi:10.1112/jtopol/jtr016](https://doi.org/10.1112/jtopol/jtr016))

### Algebraic K-theory of algebraic K-theory

On the [[algebraic K-theory]] of algebraic K-theory of [[finite fields]] $K(K(\mathbb{F}))$:

* Gabe Angelini-Knoll, _Detecting the $\beta$-family in iterated algebraic K-theory of finite fields_, Wayne State University 2017 ([arXiv:1810.10088](https://arxiv.org/abs/1810.10088), [oa_dissertations:1778](https://digitalcommons.wayne.edu/oa_dissertations/1778))

### Higher iterated algebraic K-theory of topological K-theory

Discussion of higher and of [[twisted cohomology|twisted]] iterated K-theory on $ku$, and realization of the [[spherical T-duality]] on twisted $K^{2r}(ku)$:

* {#LindSatiWesterland16}  [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], _A higher categorical analogue of topological T-duality for sphere bundles_, Annals of K-Theory, Vol. 5 (2020), No. 1, 1–42 ([arXiv:1601.06285](http://arxiv.org/abs/1601.06285), [doi:doi:10.2140/akt.2020.5.1](https://msp.org/akt/2020/5-1/p01.xhtml))

[[!redirects iterated algebraic K-theories]]

[[!redirects iterated K-theory]]
[[!redirects iterated K-theories]]


iterated distributive law

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## Idea

Iteration of [[distributive laws]] on [[monads]].

## References

* [[Eugenia Cheng]], *Iterated distributive laws*, Mathematical Proceedings of the Cambridge Philosophical Society **150** 3 (2011) 459-487  [[arxiv:0710.1120](https://arxiv.org/abs/0710.1120), [doi:10.1017/S0305004110000599](https://doi.org/10.1017/S0305004110000599)]

[[!redirects iterated distributive laws]]


iterated inductive definitions

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## Idea

Systems of iterated inductive definitions are important formal systems introduced by Feferman (1970), especially of use in proof-theoretic investigations. They are obviously related as well to [[inductive types]] in [[type theory]].

Below we record the main definitions and proof-theoretic results about these systems. See also the table at [[ordinal analysis]].

## Definition

### Non-iterated case

Let us begin by the case of a non-iterated inductive definition, as incarnated in the system ID$_1$, which axiomatizes the least fixed point of an arithmetically definable positive inductive definition.

A subset $I \subseteq \mathbb{N}$ is called inductively defined if it is the least fixed point of a monotone operator $\Gamma : \mathcal{P}(\mathbb{N}) \to \mathcal{P}(\mathbb{N})$. In order to formalize this situation relative to first-order number theory, we consider operators $\Gamma_A$ defined by positive operator forms.

The language of $ID_1$, $\mathcal{L}_{ID_1}$, is obtained from that of first-order number theory, $\mathcal{L}_{\mathbb{N}}$, by the addition of a set (or predicate) constant $I_A$ for every $X$-positive formula $A(X,x)$ in $\mathcal{L}_{\mathbb{N}}[X]$ that only contains $X$ (a new set variable) and $x$ (a number variable) as free variables. The term _$X$-positive_ means that $X$ only occurs positively in $A$. If we take the background logic to be constructive, it is common to require that $X$ only occurs _strictly_ positively (never to the left of an implication).

We allow ourselves a bit of set-theoretic notation, and we consider formulas $F(x)$ in a distinguished free variable $x$ to represent the subclass $\{ x\in\mathbb{N} \mid F(x) \}$. We write $s\in F$ to mean $F(s)$, and if $F$ and $G$ are two such formulas, we write $F \subseteq G$ to mean $\forall x. F(x) \to G(x)$.

Then $ID_1$ contains the axioms of first-order number theory with the induction scheme extended to the new language as well as the axiom
\[
  (ID_1)^1\qquad
  A(I_A) \subseteq I_A
\]
and the scheme
\[
  (ID_1)^2\qquad
  A(F) \subseteq F \to I_A \subseteq F
\]
where $F(x)$ ranges over all $\mathcal{L}_{ID_1}$-formulas.

Note that $(ID_1)^1$ expresses that $I_A$ is closed under the arithmetically definable set operator $\Gamma_A(S) = \{ x\in\mathbb{N} \mid \mathbb{N} \vDash A(S,x) \}$, while $(ID_1)^2$ expresses that $I_A$ is the least such (at least among sets definable in $\mathcal{L}_{ID_1}$).

Thus, $I_A$ is meant to be the least pre-fixed-point, and hence the least fixed point of the operator $\Gamma_A$.

### Iterated inductive definitions

It is possible to define systems of finitely iterated inductive definitions by simply repeating the process of adding a level of inductive definitions and extending the existing schemas to the new language. It's important that the constants $I_A$ for the previous levels may occur negatively in the operators for the later levels. So these systems can be defined in stages by (metamathematical) induction on $\mathbb{N}$. However, to define the transfinitely iterated variants another approach is needed.

To define the system of $\nu$-times iterated inductive definitions, where $\nu$ is an [[ordinal]], let ${\prec}$ be a primitive recursive well-ordering of order type $\nu$. We use Greek letters to denote elements of the field of ${\prec}$.

The language of $ID_\nu$, $\mathcal{L}_{ID_\nu}$ is obtained from $\mathcal{L}_{\mathbb{N}}$ by the addition of a binary predicate constant $J_A$ for every $X$-positive $\mathcal{L}_{\mathbb{N}}$[X,Y] formula $A(X,Y,\mu,x)$ that contains at most the shown free variables, where $X$ is again a unary (set) variable, and $Y$ is a fresh binary predicate variable. We write $x\in J_A^\mu$ instead of $J_A(\mu,x)$, thinking of $x$ as a distinguished variable in the latter formula.

The system $ID_\nu$ is now obtained from the system of first-order number theory by expanding the induction scheme to the new language and adding
the scheme
\[
  (TI_\nu)\qquad
  TI({\prec},F)
\]
expressing transfinite induction along ${\prec}$ for an arbitrary $\mathcal{L}_{ID_\nu}$ formula $F$ as well as the axiom
\[
  (ID_\nu)^1\qquad
  \forall\mu\prec\nu. A^\mu(J_A^\mu) \subseteq J_A^\mu
\]
and the scheme
\[
  (ID_\nu)^2\qquad
  \forall\mu\prec\nu.
  A^\mu(F) \subseteq F \to J_A^\mu \subseteq F
\]
where $F(x)$ is an arbitary $\mathcal{L}_{ID_\nu}$ formula. In $(ID_\nu)^1$ and $(ID_\nu)^2$ we used the abbreviation $A^\mu(F)$ for the formula $A(F, (\lambda\gamma y. \gamma\prec\mu \wedge y\in J_A^\gamma), \mu, x)$, where $x$ is the distinguished variable. We see that these express that each $J_A^\mu$, for $\mu\prec\nu$, is the least fixed point (among definable sets) for the operator $\Gamma_A^\mu(S) = \{ n\in\mathbb{N} \mid (\mathbb{N},(J_A^\gamma)_{\gamma\prec\mu}) \vDash A(S,(J_A^\gamma)_{\gamma\prec\mu},\mu,n) \}$. Note how all the previous sets $J_A^\gamma$, for $\gamma\prec\mu$, are used as parameters.

We define $ID_{\prec\nu} = \bigcup_{\xi\prec\nu} ID_\xi$.

## Examples

It is for instance possible to define the constructive ordinal number classes $\mathcal{O}_\mu$ for $\mu\le\nu$ in $ID_\nu$. Here we describe the simpler example of the constructive tree classes.

The first tree class $\mathcal{T}_0$ is a non-iterated inductive definition, which informally states that

* $0 \in \mathcal{T}_0$, and
* if $e$ is (the code of) a total recursive function enumerating elements of $\mathcal{T}_0$, then $e+1 \in \mathcal{T}_0$.

Elements of $\mathcal{T}_0$ represent $\mathbb{N}$-branching trees. Formally, $\mathcal{T}_0$ is represented by $I_A$ for the operator $A(X,x)$ defined by
\[
  x = 0 \vee (x \ne 0 \wedge \forall n. \{x-1\}(n) \in X).
\]

The higher tree classes $\mathcal{T}_\mu$, for $\mu\prec\nu$, are given by an iterated inductive definition, which informally states that

* $0 \in \mathcal{T}_\mu$, and
* if $e$ is (the code of) a total recursive function enumerating elements of $\mathcal{T}_\mu$, then $\langle 0,e\rangle \in \mathcal{T}_\mu$, and
* if $\gamma\prec\mu$ and $e$ is (the code of) a partial recursive function whose domain includes $\mathcal{T}_\gamma$ and whose values thereon are in in $\mathcal{T}_\mu$, then $\langle 1+\gamma,e\rangle \in \mathcal{T}_\mu$.

Now the elements represent infinite trees branching at each internal node over either $\mathbb{N}$ or one of the previous tree classes $\mathcal{T}_\gamma$, for $\gamma\prec\mu$.

To define these higher tree classes, consider the operator form $A(X,Y,\mu,x)$ defined by the disjunction of the corresponding three clauses:

* $x = 0$,
* $\exists e. x = \langle 0,e\rangle \wedge \forall n. \{e\}(n) \in X$,
* $\exists \gamma, e. \gamma \prec \mu \wedge x = \langle 1+\gamma,e\rangle \wedge \forall y \in Y^\gamma. \{e\}(y) \in X$.

In the last clause we used the class term $Y^\gamma$, thinking of $y$ in $Y(\gamma,y)$ as the distinguished variable.

## Proof-theoretic results

The book by Buchholz, Feferman, Pohlers and Sieg (1981) contains most proof-theoretic results concerning these systems. For example, the [[proof-theoretic ordinal]] of $ID_\nu$ is $\psi_\Omega(\varepsilon_{\Omega_\nu+1})$ for $\nu$ a limit ordinal, and that of $ID_{\prec\nu}$ is $\psi_\Omega(\Omega_\nu)$ when $\nu=\omega^\rho$ for $\rho$ a limit ordinal.

Girard (1982) gave another analysis of the $ID_\nu$ systems using what he calls $\Pi^1_2$-logic, making heavy use of [[dilators]].

See [[ordinal analysis]] for the relations between some of the $ID_\nu$ and $ID_{\prec\nu}$ systems and other systems.

## References

* Buchholz, Feferman, Pohlers and Sieg: Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics 897 (1981).

* Feferman: Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. In: Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 1968). 1970.

* Feferman: The proof theory of classical and constructive inductive definitions, a 40 year saga, 1968-2008. In: Ways of Proof Theory. 2013. [pdf](https://math.stanford.edu/~feferman/papers/id-saga.pdf)

* Girard: Proof-theoretic investigations of inductive definitions, I. In: Logic and algorithmic (Proc. Conf., Zurich, 1980). 1982.

* Pohlers: Subsystems of Set Theory and Second Order Number Theory. In: Handbook of Proof Theory. 1998.


iterated integral

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# Contents
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## Idea

An _iterated integral_ is a expression involving nested [[integrals]], such as
$$ \int_{x=a}^b \bigg(\int_{y=g(x)}^{h(x)} f(x,y) \,\mathrm{d}y\bigg) \,\mathrm{d}x ,$$
where the grouping parentheses are usually left out.  Iterated integrals are the subject of [[Fubini theorem]]s.

Iterated integrals include nested [[integration of differential forms]] of the kind as it appears in the formulation of [[parallel transport]] for nonabelian-value [[connection on a bundle|connection]] form (known as the _[[Dyson formula]]_ in [[physics]]).

Applied to higher degree forms iterated integrals serve to express generalized [[transgression]] of differential forms to [[loop spaces]] and other [[mapping spaces]] relaized as [[diffeological spaces]].

## Related concepts

* [[Dyson formula]]

* [[Chen connection]]

* [[higher holonomy]]

* [ordinary homology -- description in terms of higher linear algebra](ordinary%20homology#InTermsOfHigherLinearAlgebra)

## References

The notion of iterated integration of differential forms originates in informal observation like the [[Dyson formula]] for [[parallel transport]]. It was formalized in the context of [[differential geometry]] on [[diffeological spaces]] ([[Chen spaces]]) (especially [[loop spaces]]) in

* [[Kuo Tsai Chen]], _Iterated integrals of differential forms and loop space homology_, Ann. Math. 97 (1973), 217–246. [JSTOR](http://www.jstor.org/stable/1970846)

* [[Kuo Tsai Chen]], Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83–98. [doi:10.1090/S0002-9947-1975-0377960-0](http://dx.doi.org/10.1090/S0002-9947-1975-0377960-0)

* [[Kuo Tsai Chen]], _Iterated path integrals_. Bull. Amer. Math. Soc., 83(5):831-879, 1977 [doi:10.1090/S0002-9904-1977-14320-6](http://dx.doi.org/10.1090/S0002-9904-1977-14320-6)

Essentially these formulas are used in

* [[Kiyoshi Igusa]], _Iterated integrals of superconnections_ ([arXiv:0912.0249](arxiv.org/abs/0912.0249))

for expressing [[higher holonomy]] of certain [[flat infinity-connections]] given by [[Lie infinity-algebroid representation|infinity-representations]] of [[tangent Lie algebroids]].

A review of this and more discussion in the context of a higher [[Riemann-Hilbert correspondence]] is in

* [[Jonathan Block]], Aaron Smith, _A Riemann--Hilbert correspondence for infinity local systems_ ([arXiv:0908.2843](http://arxiv.org/abs/0908.2843))

A relation to [[algebraic cycles]] is discussed in

* [[Richard Hain]], _Iterated Integrals and Algebraic Cycles: Examples and Prospects_ ([arXiv:math/0109204](http://arxiv.org/abs/math/0109204))

[[!redirects iterated integral]]
[[!redirects iterated integrals]]


iterated localization

See CatLab:[[joyalscatlab:Factorisation systems]].


iterated loop space

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

#Contents#
* table of contents
{:toc}

## Idea

A multiple [[loop space]].

A grouplike [[E-k algebra]] in [[Top]].

An [[iterated loop space object]] in [[Top]].

## Properties

### Higher group structure

See at _[[May recognition theorem]]_.

### Homotopy
 {#Homotopy}

\begin{prop}\label{IteratedLoopSpaceAsHomotopyFiberOfMappingSpace}
 For $A$ a [[pointed homotopy type]], hence an [[∞-groupoid]] equipped with a base point $\ast \xrightarrow{ pt_A } A$,
then for $n \,\in\, \mathbb{N}$, the [[iterated loop space|n-fold loop space]] of $A$ is the [[homotopy fiber]] of the basepoint-[[evaluation map]] on the [[mapping space]] from the [[homotopy type]] of the [[n-sphere]]:
$$
  \Omega^n A
  \xrightarrow{ hofib(ev_{\ast}) }
  Maps
  \big(
    ʃ S^n
    ,\,
    A
  \big)
  \xrightarrow{ ev_\ast }
  A
$$
\end{prop}
\begin{proof}
  We may [[presentable (infinity,1)-category|present]] the sequence in the [[classical model structure on topological spaces]] or the [[classical model structure on simplicial sets]], in the latter case we may assume that $A$  is presented by a [[Kan complex]], so that, in either case, it is a [[fibrant object]].

In either case, the canonical model for the [[iterated loop space]] is evidently the ordinary [[1-category]]-theoretic [[fiber]] of the [[evaluation map]] out of the [[internal hom]]:
$$
  \Omega^n A
  \xrightarrow{ ev_\ast }
  Maps( ʃ S^d ,\, A )
  \xrightarrow{ ev_\ast }
  A
  \,.
$$
Moreover, the evaluation map is equivalently the image of the point inclusion under the [[internal hom]]-functor
$$
  ev_\ast
  \;=\;
  Maps( \ast \to S^n ,\, A )
  \,.
$$
Since either model category is a [[cartesian closed category|cartesian closed]] [[monoidal model category]], hence an [[enriched model category]] over itself ([this Exp.](enriched+model+category#TheClassicalModelCategoriesOfTopSpacesAndSimpSets))
and since the canonical model for $\ast \to ʃ  S^n$ is a [[cofibration]], in either case, the [[pullback-power axiom]] implies that $ev_\ast$ is a [[fibration]]. Therefore its ordinary fiber above models the [[homotopy fiber]], and the claim follows.
\end{proof}
\begin{corollary}\label{LESOfHomotopyGroupswithMappingSpace}
The [[homotopy groups]] of the [[mapping space]] $Maps(ʃ  S^n ,\, A)$ out of an [[n-sphere]] form a [[long exact sequence]] with those of $A$, of the following form:
\begin{tikzcd}[column sep=10pt]
    \cdots
    \ar[r]
    &
    \pi_{\bullet + 1}(A)
    \ar[rr]
    &&
    \pi_{\bullet+d}(A)
    \ar[
      rr
    ]
    &&
    \pi_{ \bullet }
    \big(
      \mathrm{Maps}
      (
        S^n
        \,
         A
      )
    \big)
    \ar[
      rr
    ]
    &&
    \pi_{\bullet}(A)
    \ar[
      rr
    ]
    &&
    \pi_{\bullet + d-1}(A)
    \ar[r]
    &
    \cdots
\end{tikzcd}
\end{corollary}
\begin{proof}
  This is the [[long exact sequence of homotopy groups]]
  applied to the [[homotopy fiber sequence]]
  from Prop. \ref{IteratedLoopSpaceAsHomotopyFiberOfMappingSpace}.
\end{proof}

\begin{example}
  If $A \,\in\, Grp_\infty$ is [[n-truncated object of an (infinity,1)-category|n-truncated]], then the [[evaluation map]] out of the [[mapping space]] from the [[n-sphere|(n+2)-sphere]] into it is a [[weak homotopy equivalence]]:

$$
  \tau_{n}(A) \,\simeq\, A
  {\phantom{AAAAA}}
  \Rightarrow
  {\phantom{AAAAA}}
  Maps
  \big(
     ʃ S^{n+2}
     ,\,
     A
  \big)
  \underoverset
    {\in \mathrm{W}_{wh}}
    {ev_\ast}
    {\longrightarrow}
  A
  \,.
$$
\end{example}
\begin{proof}
  By assumption, the [[long exact sequence]] from Cor. \ref{LESOfHomotopyGroupswithMappingSpace} collapses to exact segments of the form
$$
  \ast
  \to
  \pi_{\bullet}
  \big(
     Maps(ʃ S^{n+2},\, A)
  \big)
  \xrightarrow{
    \;\;\;
    \pi_\bullet(ev_\ast)
    \;\;\;
  }
  \pi_\bullet(A)
  \to
  \ast
  \,.
$$
\end{proof}

### Cohomology

+-- {: .num_prop #RationalCohomologyOfIteratedLoopSpaceOf2kSphere}
###### Proposition
**([[rational cohomology]] of [[iterated loop space]] of the [[n-sphere|2k-sphere]])**

Let

$$
  1 \leq D \lt n = 2k \in \mathbb{N}

$$

(hence two [[positive number|positive]] [[natural numbers]], one of them required to be [[even number|even]] and the other required to be smaller than the first) and consider the [[iterated loop space|D-fold loop space]] $\Omega^D S^n$ of the [[n-sphere]].

Its [[rational cohomology|rational]] [[cohomology ring]] is the [[free construction|free]] [[graded-commutative algebra]] over $\mathbb{Q}$ on one [[generators and relations|generator]] $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:

$$
  H^\bullet
  \big(
    \Omega^D S^n
    ,
    \mathbb{Q}
  \big)
  \;\simeq\;
  \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big]
  \,.
$$

=--

([Kallel-Sjerve 99, Prop. 4.10](#KallelSjerve99))

\linebreak

### Homology

See at _[[homology of iterated loop spaces]]_.

\linebreak

### Relation to configuration spaces of points

+-- {: .num_prop #ScanningMapEquivalenceOverCartesianSpace}
###### Proposition
**([[iterated loop spaces]] equivalent to [[configuration spaces of points]])**

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

the electric field map/[[scanning map]] constitutes a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^d, Y
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d \Sigma^d (Y/\partial Y)
$$

between

1. the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. \ref{ConfigurationSpacesOfnPoints})

1. the [[iterated loop space|d-fold loop space]] of the $d$-fold [[reduced suspension]] of the [[quotient space]] $Y / \partial Y$ (regarded as a [[pointed topological space]] with basepoint $[\partial Y]$).

In particular when $Y = \mathbb{D}^k$ is the [[closed ball]] of [[dimension]] $k \geq 1$ this gives a [[homotopy equivalence]]

$$
  Conf\left(
    \mathbb{R}^d, \mathbb{D}^k
  \right)
  \overset{scan}{\longrightarrow}
  \Omega^d S^{ d + k }
$$

with the [[iterated loop space|d-fold loop space]] of the [[n-sphere|(d+k)-sphere]].

=--

([May 72, Theorem 2.7](#May72), [Segal 73, Theorem 3](#Segal73), see [Bödigheimer 87, Example 13](#Boedigheimer87))

+-- {: .num_prop #StableSplittingOfMappingSpacesOutOfEuclideanSpace}
###### Proposition
**([[stable splitting of mapping spaces]] out of [[Euclidean space]]/[[n-spheres]])**

For

1. $d \in \mathbb{N}$, $d \geq 1$ a [[natural number]] with $\mathbb{R}^d$ denoting the [[Cartesian space]]/[[Euclidean space]] of that [[dimension]],

1. $Y$ a [[manifold]], with [[inhabited set|non-empty]] [[manifold with boundary|boundary]] so that  $Y / \partial Y$ is [[connected topological space|connected]],

there is a [[stable weak homotopy equivalence]]

$$
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
$$

between

1. the [[suspension spectrum]] of the [[configuration space of points|configuration space]] of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})

1. the [[direct sum]] (hence: [[wedge sum]]) of [[suspension spectra]] of the [[configuration space of points|configuration spaces]] of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).

Combined with the [[stabilization]] of the electric field map/[[scanning map]] [[homotopy equivalence]] from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a [[stable weak homotopy equivalence]]

$$
  Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  =
  \Omega^d \Sigma^d (Y/\partial Y)
  \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow}
  \Sigma^\infty Conf(\mathbb{R}^d, Y)
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)
$$

between the latter direct sum and the [[suspension spectrum]] of the [[mapping space]] of pointed [[continuous functions]] from the [[n-sphere|d-sphere]] to the $d$-fold [[reduced suspension]] of $Y / \partial Y$.

=--

([Snaith 74, theorem 1.1](#Snaith74), [Bödigheimer 87, Example 2](#Boedigheimer87))

In fact by [Bödigheimer 87, Example 5](#Boedigheimer87) this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

$$
  Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y))
  =
  Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y))
  \overset{\simeq}{\longrightarrow}
  \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)
$$

## Related concepts

* [[loop space]], [[free loop space]]

* [[infinite loop space]]

* [[relative loop space]]

[[!include k-monoidal table]]

## References

### General

* [[Peter May]], _Infinite loop space theory_, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. ([euclid:bams/1183538891](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538891))

  _Infinite loop space theory revisited_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/28.pdf))

* {#Adams} [[John Frank Adams]], _Infinite loop spaces_, Hermann Weyl lectures at IAS, Princeton University Press (1978) ([ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90), [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259))

* [[Peter May]], _The uniqueness of infinite loop space machines_, Topology, vol 17, pp. 205-224 (1978) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/22.pdf))

* {#Lurie} [[Jacob Lurie]], Section 5.2.6 of _[[Higher Algebra]]_

On the [[Morava K-theory]] of iterated loop spaces of [[n-spheres]]:

* [[Douglas Ravenel]], *What we still don't know about loop spaces of spheres*, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), *Homotopy Theory via Algebraic Geometry and Group Representations*, Contemporary Mathematics **220**, AMS 1998  ([pdf](https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf), [[Ravenel_LoopSpacesOfSpheres.pdf:file]], [doi:10.1090/conm/220](http://dx.doi.org/10.1090/conm/220))

### Relation to configuration spaces of points

In relation to [[configuration spaces of points]]:

* {#May72} [[Peter May]], _The geometry of iterated loop spaces_, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf), [doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491))

* {#Segal73} [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. **21** (1973) 213-221. MR 0331377 [[pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf), [doi:10.1007/BF01390197](https://doi.org/10.1007/BF01390197)]

* {#Snaith74} [[Victor Snaith]],  _A stable decomposition of $\Omega^n S^n X$_, Journal of the London Mathematical Society 7 (1974), 577 - 583 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf))

* {#McDuff75} [[Dusa McDuff]], _Configuration spaces of positive and negative particles_, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], _Stable splittings of mapping spaces_, Algebraic topology. Springer 1987. 174-187 ([pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]])

### Rational cohomology

On [[ordinary cohomology]] of iterated loop spaces in relation to [[configuration spaces of points]] (see also at _[[graph complex]]_):

* [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, _On the homology of configuration spaces_, Topology Vol. 28 No. 1, p. 111-123 1989 ([pdf](https://core.ac.uk/download/pdf/82124359.pdf))

On the [[rational cohomology]]:

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structur eof Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

[[!redirects iterated loop spaces]]

[[!redirects iterated based loop space]]
[[!redirects iterated based loop spaces]]

[[!redirects n-fold loop space]]
[[!redirects n-fold loop spaces]]


iterated loop space object

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of [[iterated loop space]] in [[Top]] has direct generalization to any [[(∞,1)-category]] with [[(∞,1)-pullbacks]].

## Related concepts

[[!include k-monoidal table]]

[[!redirects iterated loop space object]].


iterated monoidal category

# Iterated monoidal categories
* table of contents
{: toc}

## Idea

An _iterated monoidal category_, or _$n$-fold monoidal category_ for varying $n$, is an algebraic analogue of the concept of [[n-fold loop space]]. By means of a suitable [[bar construction]], the [[geometric realization]] of an $n$-fold monoidal category, or rather its [[group completion]], bears a structure of $n$-fold loop space.

Roughly speaking, the iterative idea is that an $(n+1)$-fold monoidal category is a [[pseudomonoid|(pseudo-)monoid]] in the monoidal $2$-category of $n$-fold monoidal categories and ([[normal lax monoidal functor|normal lax]]) $n$-fold monoidal functors. Were the laxity to be strengthened so that the relevant structure constraints become isomorphisms ([[strong monoidal functor|strong]] $n$-fold monoidal functors), we would get [[braided monoidal categories]] in the case $n = 2$, and [[symmetric monoidal categories]] at $n = 3$ and beyond (in other words, the concept stabilizes at $n = 3$). Without that strengthening, however, we get a new type of structure for each $n$, without [[stabilization]].

## Definition

Let $C$ be a [[2-category]] with [[2-products]]. Form a new 2-category with 2-products $Mon_{lax,norm}(C)$ whose

* $0$-cells are [[pseudomonoids]] in $C$.

* $1$-cells are _[[normal lax morphism|normal]]_ lax morphisms of pseudomonoids.

* $2$-cells are [[monoidal transformations]] between normal lax morphisms of pseudomonoids.

The $2$-product structure on $Mon(C)$ is inherited from the $2$-product structure of $C$.

+-- {: .num_def}
###### Definition
A __$0$-fold monoidal category__ is an ordinary [[category]]; the $2$-category of $0$-fold monoidal categories is [[Cat]]. By [[recursion]], the $2$-category of $n$-fold monoidal categories is $Mon_{norm}(C)$ where $C$ is the $2$-category of $(n-1)$-fold monoidal categories; objects of $Mon_{norm}(C)$ are of course called __$n$-fold monoidal categories__.
=--

## Related pages

* 2-fold monoidal categories are a special sort of [[duoidal category]].

* [[distributivity for monoidal structures]]

## References

* Balteanu, Fiedorowicz, Schwänzl, and Vogt, _Iterated monoidal categories II_, ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.290&rep=rep1&type=pdf)).

[[!redirects iterated monoidal category]]
[[!redirects iterated monoidal categories]]
[[!redirects n-fold monoidal category]]
[[!redirects n-fold monoidal categories]]
[[!redirects 2-fold monoidal category]]
[[!redirects 2-fold monoidal categories]]


iteration theory

# Iteration theory

* table of contents
{: toc}

## Idea

An __iteration theory__ is an [[algebraic theory]] that supports iteration on its operators.

In particular, in an iteration theory, any equation

$$ x = t $$

has a well behaved solution, $t^\dagger$, such that $t[t^\dagger/x]=t^\dagger$.
More generally, any finite sequence of equations has a solution,

$$ x_1 = t_1, x_2 = t_2, \dots x_n = t_n$$

In general, $t_i$'s may have other free variables too.

Keeping it simple for a moment, suppose for a second that $n=1$ and $t$ has just one free variable, $x$. For any algebra $A$, the  term is interpreted as a function $\llbracket t\rrbracket:A\to A$, and the solution $t^\dagger$ is interpreted as a constant in $\llbracket t^\dagger\rrbracket\in A$, which is a fixed point for the function $\llbracket t\rrbracket$. Thus an iteration theory provides canonical fixed points for equations, in every algebra.

## Definition

Let $T$ be an abstract [[clone]]. It is a _Conway theory_ when it is equipped with a family of operations

$$ \dagger_{m,n} : T(m+n)^m \to T(n)^m $$

such that, for $f\in T(m+n)^m$, (omitting subscripts):

$$ f^\dagger(i) \quad = \quad f(i) \rhd j. \begin{cases} f^\dagger(j) & j\in m \\ \eta(j) & j\in n\end{cases}$$

and satisfying axioms:

* parameter identity: $((1_n \oplus g) \circ f)^\dagger = g \circ f ^\dagger$ where $f:n \to n+p, \ g:p \to q$

* composition identity: $\langle (\langle f , \ 0_m \oplus 1_p \rangle \circ g)^\dagger , 1_p \rangle \circ f = (\langle g , \ 0_n \oplus 1_p \rangle \circ f)^\dagger$ where $f:n \to m+p, \ g:m \to n+p$

* double dagger identity: $(\langle 1_n , \ 1_n \rangle \oplus 1_p) \circ f)^\dagger = (f^\dagger)^\dagger$ where $f: n \to n+n+p$

in its [[Kleisli category]].

It is an _iteration theory_ if the operations also satisfy:

* commutative identity: $\langle (\rho_1 \oplus 1_p) \circ (f \circ \rho)_1 , \ ... \ , \ (\rho_m \oplus 1_p) \circ (f \circ \rho)_m) \rangle ^\dagger = ((\rho \oplus 1_p) \circ f)^\dagger \circ \rho$

where $f: n \to m+p, \ \rho: m \to n$ is a surjective base morphism and $\rho_i: m \to m$ are base with $\rho \circ \rho_i = \rho$ for every i.

This connects with the basic idea, because $f\in T(m+n)^m$ can be thought of as a sequence of $m$ terms with $m+n$ variables, and the solution $f^\dagger$ comprises $m$ terms just having the $n$ free variables.

## Properties

To equip an abstract clone $T$ with the structure of a Conway theory is to equip the corresponding [[Lawvere theory]] with the structure of a [[traced monoidal category]] for the categorical product structure.

## Continuous theories

An abstract clone $T$ is _continuous_ if each set $T(n)$ is equipped with the structure of a [[cpo]], and the clone operations are continuous.

If $T$ is also pointed, i.e. each $T(n)$ has a least element $\bot$, then $T$ has a canonical structure of a Conway theory.

For any $f\in T(m+n)^m$, we define an endofunction
$F_f : T(n)^m \to T(n)^m$ by

$$ F_f (g) \quad = \quad f(i) \rhd j. \begin{cases} g(j) & j\in m \\ \eta(j) & j\in n\end{cases}$$

and then use [[Tarski]]'s [[fixed point]] theorem to define

$$ f^\dagger \quad = \quad \bigvee_{i=1}^\infty (F_f)^n(\bot) .$$

## References

The ideas originated in the work by Calvin C. Elgot in the 1970s. A textbook reference is:

* Stephen L. Bloom & Zoltán Ésik , _Iteration theories_, Springer 1993. ([[Iteration Theories.pdf:file]])

The connection with traced monoidal categories is due to [[Masahito Hasegawa]] and [[Martin Hyland]], see Theorem 3.1 of:

* {#Hasegawa97} [[Masahito Hasegawa]], _Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi_, Proc. 3rd International Conference on Typed Lambda Calculi and Applications (TLCA 1997). Springer LNCS1210, 1997 ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.31))

This setting is further studied by [[Alex Simpson]] and [[Gordon Plotkin]] in

* [[Alex Simpson]] and [[Gordon Plotkin]]. _Complete Axioms for Categorical Fixed-point Operators_. Proc. LICS 2000. [pdf](https://homepages.inf.ed.ac.uk/gdp/publications/fixpoints.pdf)

[[!redirects Conway theory]]


Ittay Weiss

* [website](https://www.port.ac.uk/about-us/structure-and-governance/our-people/our-staff/ittay-weiss)

#related $n$Lab entries#

* [[(∞,1)-operad]]

  * [[dendroidal set]]

  * [[model structure on dendroidal sets]]

categories: people


Itzhak Bars

* [webpage](http://physics.usc.edu/~bars/)

## Selected writings

On [[1/N corrections]] in [[2d QCD]]:

* [[Itzhak Bars]], _QCD and Strings in 2D_ ([arXiv:hep-th/9312018](https://arxiv.org/abs/hep-th/9312018))

On [[D=12 supergravity]]:

* [[Itzhak Bars]], _Supersymmetry, p-brane duality and hidden space-time dimensions_, Phys. Rev. D54, 5203 (1996) ([arXiv: hep-th/9604139](https://arxiv.org/abs/hep-th/9604139)).

* [[Itzhak Bars]], _S-Theory_, Phys.Rev. D55 (1997) 2373-2381 ([arXiv:hep-th/9607112](https://arxiv.org/abs/hep-th/9607112))

On [[D=14 supersymmetry]]:

* [[Itzhak Bars]], _A case for 14 dimensions_, Phys. Lett. B403 (1997) 257-264 ([arXiv:hep-th/9704054](https://arxiv.org/abs/hep-th/9704054))

* [[Itzhak Bars]], _Duality Covariant Type IIB Supersymmetry and Nonperturbative Consequences_, Phys. Rev. D 56, 7954 (1997) ([arXiv:hep-th/9706185](https://arxiv.org/abs/hep-th/9706185))

On [[string field theory]] with [[orthosymplectic super Lie algebra|$OSp(d\vert 2)$]]-symmetry:

* [[Itzhak Bars]], Dmitry Rychkov. *Background Independent String Field Theory* [[arXiv:1407.4699](https://arxiv.org/abs/1407.4699)]

## Related entries

* [[supersymmetry and division algebras]]

category: people


IulianUdrea


Ivan Cherednik

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

* [Institute page](https://cas-wp1.oasis.unc.edu/faculty-member/cherednik-ivan/)

## Selected writings

On the [[Knizhnik-Zamolodchikov equation]] and [[Hecke algebras]]:

* [[Ivan Cherednik]], *Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras*, Mathematical Society of Japan Memoirs **1998** (1998) 1-96 $[$[doi:10.2969/msjmemoirs/00101C010](https://doi.org/10.2969/msjmemoirs/00101C010)$]$

category: people