irreducible closed subspace

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### Context
#### Topology
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[[!include topology - contents]]
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=--
=--

# Contents
* table of contents
{: toc}

## Definition

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

Given a [[topological space]] $X$, a [[closed subspace]] $F$ of $X$ is __irreducible__ if it is [[inhabited]] and not the [[union]] of two closed proper (i.e. smaller) subspaces. In other words, $F$ is irreducible if whenever $F_1$ and $F_2$ are closed subsets of $X$ such that

$$
  F = F_1 \cup F_2
$$

then $F_1 = F$ or $F_2 = F$.

=--

Equivalently this may be expressed in terms of [[open subsets]]:

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

Let $(X, \tau)$ be a [[topological space]], and let $P \in \tau \subset P(X)$ be a proper [[open subset]],
so that the [[complement]] $F \coloneqq X\backslash P$ is an [[inhabited]] [[closed subspace]]. Then $F$
is [[irreducible closed subspace|irreducible]] in the sense of def. \ref{ClosedIrreducible} precisely if whenever $U_1,U_2 \in \tau$ are open subsets
with $U_1 \cap U_2 \subset P$ then $U_1 \subset P$ or $U_2 \subset P$:

$$
  X \backslash P \,\text{irreducible}
  \;\Leftrightarrow\;
  \left(
    \underset{U_1, U_2 \in \tau}{\forall }
    \left(
      \left(
        U_1 \cap U_2 \subset P
      \right)
      \;\Rightarrow\;
      \left(U_1 \subset P \;\text{or}\; U_2 \subset P\right)
    \right)
  \right)
$$

=--

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

Every [[closed subset]] $F_i \subset F$ may be exhibited as the [[complement]]

$$
  F_i = F \backslash U_i
$$

for some open subset $U_i \in \tau$. Observe that under this identification the condition
that $U_1 \cap U_2 \subset P$ is equivalent to the condition that $F_1 \cup F_2 = F$,
because it is equivalent to the equation labeled $(\star)$ in the following sequence of equations:

$$
\begin{aligned}
  F_1 \cup F_2
  & =
  (F \backslash U_1) \cup (F \backslash U_2)
  \\
  &
  = \left( X \backslash (P \cup U_1) \right) \cup \left( X \backslash P \cup U_2 \right)
  \\
  & = X \backslash ( P \cup (U_1 \cap U_2)  )
  \\
  & \stackrel{(\star)}{=}   X \backslash P
  \\
  & = F
\end{aligned}
\,.
$$

Similarly, the condition that $U_i \subset P$ is equivalent to the condition that $F_i = F$,
because it is equivalent to the equality $(\star)$ in the following sequence of equalities:

$$
  \begin{aligned}
    F_i &= F \backslash U_i
    \\
    & = X \backslash ( P \cup U_i )
    \\
    & \stackrel{(\star)}{=} X \backslash P
    \\
    & =
    F
  \end{aligned}
  \,.
$$

Under these equivalences, the two conditions are manifestly the same.

=--

Yet another equivalent characterization is in terms of frame homomorphisms:

In the following we write

$$
  \ast \coloneqq (\{1\}, \tau_\ast = \left\{ \emptyset, \{1\}\right\})
$$

for the [[point]], regarded, uniquely, as a topological space, the [[point space]].

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

For $(X,\tau)$ a [[topological space]], then there is a [[bijection]] between
the [[irreducible closed subspaces]] of $(X,\tau)$ and the
[[frame]] [[homomorphisms]] from $\tau_X$ to $\tau_\ast$, given by

$$
  \array{
    Hom_{Frame}(\tau_X, \tau_\ast)
      &\underoverset{\simeq}{}{\longrightarrow}&
    IrrClSub(X)
    \\
    \phi
      &\mapsto&
    X \backslash U_\emptyset(\phi)
  }
$$

where $U_\emptyset(\phi)$ is the [[union]] of all elements $U \in \tau_x$ such that $\phi(U) = \emptyset$:

$$
  U_{\emptyset}(\phi)
    \coloneqq
  \underset{{U \in \tau_X} \atop {\phi(U) = \emptyset} }{\cup}
   U
  \,.
$$

=--

See also ([[Stone Spaces|Johnstone 82, II 1.3]]).

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

First we need to show that the function is well defined in that given
a frame homomorphism $\phi \colon \tau_X \to \tau_\ast$ then  $X \backslash U_\emptyset(\phi)$
is indeed an irreducible closed subspace.

To that end observe that:

$(\ast)$ _If there are two elements $U_1, U_2 \in \tau_X$ with $U_1 \cap U_2 \subset U_{\emptyset}(\phi)$
then $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$._

This is because

$$
  \begin{aligned}
    \phi(U_1) \cap \phi(U_2)
    & =
    \phi(U_1 \cap U_2)
    \\
    & \subset \phi(U_{\emptyset}(\emptyset))
    \\
    & =
    \emptyset
  \end{aligned}
  \,,
$$

(where the first equality holds because $\phi$ preserves finite intersections, the inclusion holds because $\phi$ respects
inclusions, and the second equality holds because $\phi$ preserves arbitrary unions).
But in $\tau_\ast = \{\emptyset, \{1\}\}$ the intersection of two open subsets is empty precisely if at least one of them is empty,
hence $\phi(U_1) = \emptyset$ or $\phi(U_2) = \emptyset$. But this means that $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$, as claimed.

Now according to prop. \ref{OpenSubsetVersionOfClosedIrreducible}
the condition $(\ast)$ identifies the [[complement]]
$X \backslash U_{\emptyset}(\phi)$ as an [[irreducible closed subspace]] of $(X,\tau)$.

Conversely, given an irreducible closed subset $X \backslash U_0$, define $\phi$ by

$$
  \phi
    \;\colon\;
  U
    \mapsto
  \left\{
    \array{
      \emptyset & \vert \, \text{if} \, U \subset U_0
      \\
      \{1\} & \vert \, \text{otherwise}
    }
  \right.
  \,.
$$

This does preserve

1. arbitrary unions

   because $\phi(\underset{i}{\cup} U_i) = \emptyset$ precisely if $\underset{i}{\cup}U_i \subset U_0$ which is the
   case precisely if all $U_i \subset U_0$, which means that all $\phi(U_i) = \emptyset$ and because $\underset{i}{\cup}\emptyset = \emptyset$;

   while $\phi(\underset{i}{\cup}U_1) = \{1\}$ as soon as one of the $U_i$ is not contained in $U_0$, which means that
   one of the $\phi(U_i) = \{1\}$ which means that $\underset{i}{\cup} \phi(U_i) = \{1\}$;

1. finite intersections

   because if $U_1 \cap U_2 \subset U_0$, then by $(\ast)$ $U_1 \in U_0$ or $U_2 \in U_0$, whence $\phi(U_1) = \emptyset$
   or $\phi(U_2) = \emptyset$, whence with $\phi(U_1 \cap U_2) = \emptyset$ also $\phi(U_1) \cap \phi(U_2) = \emptyset$;

   while if $U_1 \cap U_2$ is not contained in $U_0$ then neither $U_1$ nor $U_2$ is contained in $U_0$ and hence with
   $\phi(U_1 \cap U_2) = \{1\}$ also $\phi(U_1) \cap \phi(U_2) = \{1\} \cap \{1\} = \{1\}$.

Hence this is indeed a frame homomorphism $\tau_X \to \tau_\ast$.

Finally, it is clear that these two operations are inverse to each other.

=--

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

Note that frame homomorphisms from $\tau_X$ to $\tau_\ast$ are the same thing as [[continuous maps]] from the [[locale]] $\mathcal{O}_\ast$ to the locale $\mathcal{O}_X$ (that is, in the other direction, regarding the frames of opens as instead being locales).  Thus, the irreducible closed subspaces correspond to what *should* be the points of the space, if we regard it as much as possible as a locale, that is the [[points of the locale]] $\mathcal{O}_X$.  At a more elementary level, these are also the same thing as the [[completely prime filters]] in the frame $\tau_X$.
=--

## Properties

Note that the [[closure]] of (the [[singleton]] set on) any [[point]]/[[element]] of $X$ is an irreducible closed subspace.  $X$ is [[sober space|sober]] if and only if every irreducible closed subspace is the closure of a unique point of $X$.  In general, the irreducible closed subspaces of $X$ correspond to the [[point of a locale|points]] of the [[topological locale]] $\Omega(X)$, which are (by definition) the [[completely prime filters]] on the [[frame of open subspaces]] of $X$.  Specifically, given an irreducibly closed subspace, the filter of open subspaces that contain it is completely prime; conversely, given a completely prime filter of open subspaces, the closure of its intersection is irreducible.

The theory of irreducible closed subspaces is not useful in [[constructive mathematics]]; instead, one must use the [[completely prime filters]] directly.  While one might hope that the irreducibly open subsets (that is those that satisfy the conditions of Proposition \ref{OpenSubsetVersionOfClosedIrreducible}) might be more tractible constructively, they are in fact no better.  (We should probably put in the classical proof and see where it goes wrong.)

## Related concepts

* [[closed point]]

## References

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], around definition IX.3.2 in _[[Sheaves in Geometry and Logic]]_

[[!redirects irreducible closed subspace]]
[[!redirects irreducible closed subspaces]]
[[!redirects irreducible closed subset]]
[[!redirects irreducible closed subsets]]
[[!redirects irreducible closed set]]
[[!redirects irreducible closed sets]]


irreducible component

Let $X$ be a [[topological space]] (usually a [[scheme]]).  An **irreducible component** of $X$ is a maximal [[irreducible space|irreducible subspace]] of $X$.

[[!redirects irreducible components]]


irreducible element

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#### Algebra
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[[!include higher algebra - contents]]
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#### Monoid theory
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[[!include monoid theory - contents]]
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# Contents
* table of contents
{: toc}

## Definition

In a [[monoid]], an element $x$ is __irreducible__ if it is neither [[invertible element|invertible]] nor the product of two non-invertible elements.  Without [[bias]], we can say that $x$ is irreducible if, whenever it is written as a product of a [[finite list]] of elements, all but one elements in the list are invertible.

In a [[commutative ring]], an element is **irreducible** if it is neither [[invertible element|invertible]] nor the product of two non-invertible elements, with respect to the multiplication operation on the commutative ring.

## Examples

* Every [[prime number]] is an irreducible element in the integers.

* Given a field $K$, every [[monic polynomial]] of [[degree of a polynomial|degree]] one is an irreducible element in the univariate [[polynomial ring]] $K[x]$.

## Related concepts

* [[unique factorization monoid]]
* [[unique factorization domain]]

[[!redirects irreducible element]]
[[!redirects irreducible elements]]


irreducible ideal

# Irreducible ideals
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{: toc}

## Idea

_Irreducible ideals_ generalize [[prime ideals]] by replacing [[ideal product|multiplication]] of [[ideals]] with [[intersection]].  (Strictly speaking, this gives us the notion of _strongly_ irreducible ideals; the irreducible ideals are found by replacing a containment with an equality.  However, in many common situations, these are equivalent.)

In [[lattices]] (or [[prosets]] more broadly), where multiplication of ideals *is* intersection, the (strongly) irreducible ideals are the same as the prime ideals; this is true in some other cases, such as the [[ring of integers]].  (For this reason, the term 'strongly irreducible' is not needed in [[order theory]].)

There is also an infinitary version, the _completely_ irreducible ideals, which also have a strong version.  (The completely strongly irreducible ideals are usually called 'completely prime' in order theory, although they are different from the [[completely prime ideals]] of noncommutative ring theory).

Irreducible ideals are related to [[irreducible elements]] but don\'t line up perfectly.

## Definitions

Let $R$ be a [[ring]], or more generally a [[rig]], possibly even [[nonassociative ring|nonassociative]], or even just a [[monoid]], possibly just a [[magma]]; or on the other hand a [[lattice]], or even just a [[proset]].  In other words, anything with a concept of _[[ideals]]_.  In the noncommutative case, we could be talking about left ideals, right ideals, or two-sided ideals; pick a meaning and stick with it.  We need the concept of [[intersection]] of ideals (which all of these have), and that\'s pretty much it; that is, we use the structure of the [[meet-semilattice]] $Idl(R)$ of ideals of $R$.

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

An ideal $I$ of $R$ is __irreducible__ if $I$ is [[proper ideal|proper]] and, whenever $I$ equals the intersection of two ideals of $R$, $I$ equals at least one of those two ideals:

$$ \forall\, J \in Idl(R),\; \forall\, K \in Idl(R),\; I = J \cap K \;\Rightarrow\; I = J \;\vee\; I = K .$$

Also, $I$ is __completely irreducible__ if, whenever $I$ contains the intersection of any collection of ideals of $R$, $I$ contains at least one of the ideals in that collection:

$$ \forall\, \mathcal{J} \subseteq Idl(R),\; I = \bigcap \mathcal{J} \;\Rightarrow\; \exists\, J \in Idl(R),\; J \in \mathcal{J} \;\wedge\; I = J .$$

Alternatively, $I$ is __strongly irreducible__ if $I$ is [[proper ideal|proper]] and, whenever $I$ contains the intersection of two ideals of $R$, $I$ contains at least one of those two ideals:

$$ \forall\, J \in Idl(R),\; \forall\, K \in Idl(R),\; I \supseteq J \cap K \;\Rightarrow\; I \supseteq J \;\vee\; I \supseteq K .$$

Finally, $I$ is __strongly completely irreducible__ if, whenever $I$ contains the intersection of any collection of ideals of $R$, $I$ contains at least one of the ideals in that collection:

$$ \forall\, \mathcal{J} \subseteq Idl(R),\; I \supseteq \bigcap \mathcal{J} \;\Rightarrow\; \exists\, J \in Idl(R),\; J \in \mathcal{J} \;\wedge\; I \supseteq J .$$
=--

Every strongly irreducible ideal is irreducible, every completely irreducible ideal is irreducible, and every strongly completely irreducible ideal is both strongly irreducible and completely irreducible.  (The converses of these hold in various situations that we should describe below.)

The definition of 'completely irreducible' doesn\'t need to explicitly require $I$ to be proper; that is covered by considering the [[empty collection]] $\mathcal{J}$ of ideals.  The definition of 'irreducible' can take care of that automatically too if it is made [[unbiased definition|unbiased]]:

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

An ideal $I$ of $R$ is __irreducible__ if, whenever $I$ equals the intersection of a [[finite list]] of ideals of $R$, $I$ equals at least one of the ideals in the list:

$$ \forall\, n \in \mathbb{N},\; \forall\, J \in Idl(R)^n,\; I = \bigcap_{k\in[n]} J_k \;\Rightarrow\; \exists\, k \in [n],\; I = J_k .$$

Also, $I$ of $R$ is __strongly irreducible__ if, whenever $I$ contains the intersection of a [[finite list]] of ideals of $R$, $I$ contains at least one of the ideals in the list:

$$ \forall\, n \in \mathbb{N},\; \forall\, J \in Idl(R)^n,\; I \supseteq \bigcap_{k\in[n]} J_k \;\Rightarrow\; \exists\, k \in [n],\; I \supseteq J_k .$$
=--

As is typical, the case for $n \gt 2$ follows from the case for $n = 2$ by [[mathematical induction|induction]], the case for $n = 2$ is the significant requirement in the unbiased definition, the case for $n = 1$ is trivial, and the case for $n = 0$ is the extra requirement (in this case, being proper) that rules out ideals that are [[too simple to be simple|too irreducible to be irreducible]].

As with [[prime ideals]], it may be helpful to focus on the [[complements]] of the ideals involved, which are known (especially in [[constructive mathematics]]) as [[anti-ideals]].  Of course, we could just repeat the definitions involving the [[join-semilattice]] $AIdl(R)$ of antiideals of $R$ (which classically is equivalent to $Idl(R)^op$); I won\'t write that out here, but it does make the term 'irreducible' have more of the flavour of its usual meaning; a proper anti-ideal is irreducible if it cannot be _reduced_ as the union of two smaller anti-ideals.

## Properties

If $Idl(R)$ is a [[distributive lattice]], then irreducible ideals are the same as strongly irreducible ideals.  If $R$ is a distributive lattice, then not only are these the same, but the [[prime ideals]] are also the same as them.

An ideal generated by an [[irreducible element]] is irreducible, but not always conversely (not even if we ignore $0$).

## References

*  Kiyoshi Iséki. 1956. Ideal Theory of Semiring. _Proceedings of the Japan Academy_ A 32:8, 554--559. [Project Euclid](https://projecteuclid.org/euclid.pja/1195525272).

[[!redirects irreducible ideal]]
[[!redirects irreducible ideals]]

[[!redirects completely irreducible ideal]]
[[!redirects completely irreducible ideals]]

[[!redirects strongly irreducible ideal]]
[[!redirects strongly irreducible ideals]]

[[!redirects completely strongly irreducible ideal]]
[[!redirects completely strongly irreducible ideals]]

[[!redirects strongly completely irreducible ideal]]
[[!redirects strongly completely irreducible ideals]]


irreducible manifold

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###Context###
#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[manifold]] of [[dimension]] $n$ is _irreducible_ if any [[embedding|embedded]] $(n-1)$-[[sphere]] [[boundary|bounds]] an embedded $n$-[[ball]].

## Related concepts

* [[virtually fibered conjecture]]

[[!redirects irreducible manifolds]]


irreducible polynomial

A non-zero [[polynomial]] $f$ with coefficients in a [[field]] $k$ is **irreducible** if when written as the product $g h$ of two polynomials, one of $g$ or $h$ is a constant (and necessarily non-zero) polynomial. Equivalently, a polynomial $f$ is __irreducible__ if the [[ideal]] it generates is a [[maximal ideal]] of the polynomial ring $k[x]$.

In other words, a polynomial $f$ is irreducible if it is an [[irreducible element]] of $k[x]$ as an [[integral domain]].

Notice that under this definition, the zero polynomial is not considered to be irreducible. An alternative definition, which applies to the case of coefficients in a [[commutative ring]] $R$, is that a polynomial $f$ is irreducible if, whenever $f$ divides $g h$, either $f$ divides $g$ or $f$ divides $h$. Under this definition, a polynomial is irreducible if it generates a [[prime ideal]] in $R[x]$, and the zero polynomial is irreducible if $R$ is an integral domain.

If $L$ is a [[field extension]] of $k$ and $\alpha \in L$ is algebraic over $k$ (i.e., the smallest subextension containing $\alpha$, $k(\alpha)$, is finite-dimensional over $k$), then the $k$-algebra map $k[x] \to k(\alpha): x \mapsto \alpha$ has a non-trivial kernel which is a [[maximal ideal]], which being [[principal ideal|principal]] is generated by an irreducible polynomial $f$. The unique [[monic polynomial]] generator is called the *irreducible polynomial of $\alpha$*.

Over a [[unique factorization domain]] [[eisenstein's criterion]] determines irreducibility

[[!redirects irreducible polynomials]]


irreducible representation

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

#Contents#
* table of contents
{:toc}

## Idea ##

An _irreducible representation_ --  often abbreviated _irrep_ -- is a [[representation]] that has no smaller non-trivial representations "sitting inside it".

Similarly for irreducible [[modules]].

## Definition ##

Given some algebraic structure, such as a [[group]], equipped with a notion of ([[linear representation|linear]]) [[representation]], an _irreducible representation_ is a [[representation]] that has no nontrivial proper [[subobject]] in the [[category of representations|category of all representations]] in question and yet which is not itself trivial either.  In other words, an irrep is a [[simple object]] in the [[category of representations]].

Notice that there is also the closely related but in general different notion of an [[indecomposable representation]].  Every irrep is indecomposable, but the converse may fail.

A representation that has proper nontrivial subrepresentations but can not be decomposed into a [[direct sum]] of such representations is an **indecomposable representation** but still reducible.

In good cases for [[finite dimensional vector space|finite dimensional]] representations, the two notions (irreducible, indecomposable) coincide.

## Examples

+-- {: .num_example #IrreducibleRealRepresentationsOfCyclicGroups}
###### Example
**([[irreducible representation|irreducible]] [[real numbers|real]] [[linear representations]] of [[cyclic groups]])**

For $n \in \mathbb{N}$, $n \geq 2$,
the [[isomorphism classes]] of [[irreducible representation|irreducible]] [[real numbers|real]] [[linear representations]] of the [[cyclic group]] $\mathbb{Z}/n$ are given by precisely the following:

1. the 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1}$;

1. the 1-[[dimension|dimensional]] [[sign representation]] $\mathbf{1}_{sgn}$;

1. the 2-[[dimension|dimensional]] standard representations $\mathbf{2}_k$ of [[rotations]] in the [[Euclidean plane]] by [[angles]] that are [[integer]] multiples of $2 \pi k/n$, for $k \in \mathbb{N}$, $0 \lt k \lt n/2$;

   hence the [[restricted representations]] of the defining real rep of [[SO(2)]] under the [[subgroup]] inclusions $\mathbb{Z}/n \hookrightarrow SO(2)$, hence the representations generated by [[real number|real]] $2 \times 2$ [[trigonometric function|trigonometric]] [[matrices]] of the form

   $$
    \rho_{\mathbf{2}_k}(1)
    \;=\;
     \left(
       \array{
         cos(\theta) & -sin(\theta)
         \\
         sin(\theta) & \phantom{-}cos(\theta)
       }
     \right)
     \phantom{AA}
     \text{with}
     \;
     \theta = 2 \pi \tfrac{k}{n}
     \,,
   $$

  (For $k = n/2$ the corresponding 2d representation is the [[direct sum]] of two copies of the [[sign representation]]: $\mathbf{2}_{n/2} \simeq \mathbf{1}_{sgn} \oplus \mathbf{1}_{sgn}$, and hence not [[irrep|irreducible]]. Moreover, for $k \gt n/2$ we have that $\mathbf{2}_{k}$ is irreducible, but [[isomorphism|isomorphic]] to $\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}$).

In summary:

$$
  Rep^{irr}_{\mathbb{R}}
  \big(
    \mathbb{Z}/n
  \big)_{/\sim}
  \;=\;
  \big\{
    \mathbf{1},
    \mathbf{1}_{sgn},
    \mathbf{2}_k
    \;\vert\;
    0 \lt k \lt n/2
  \big\}
$$

=--

(e.g. [tom Dieck 09 (1.1.6), (1.1.8)](#tomDieck09))

***

## Related concepts

* [[reducible representation]]

* [[irreducible character]]

* [[simple object]]

* [[fundamental representation]]

* In [[quantum physics]] an irreducible representation of an [[algebra of observables]] is called a _[[superselection sector]]_.

  In [[Wigner classification]] the irreps of the [[spacetime]] symmetry groups are identified with [[elementary particle]] species.

## References

See any text on [[representation theory]], for instance

* {#tomDieck09} [[Tammo tom Dieck]], _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

See also

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

[[!redirects irreducible representations]]

[[!redirects indecomposable representation]]
[[!redirects indecomposable representations]]

[[!redirects irrep]]
[[!redirects irreps]]

[[!redirects irreducible module]]
[[!redirects irreducible modules]]


irreducible topological space

#Contents#
* table of contents
{:toc}

## Definition

A [[topological space]] $X$ is called __irreducible__ if it cannot be expressed as [[union]] of two proper [[closed subsets]], or equivalently if any two [[inhabited set|inhabited]] [[open subsets]] have inhabited intersection.

A subset $S$ of a topological space $X$ is an __irreducible subset__ if $S$ is an irreducible topological space with the [[subspace topology]].

An [[algebraic variety]] is irreducible if its underlying topological space (in the [[Zariski topology]]) is irreducible.

A _[[sober topological space]]_, is one whose only irreducible closed subsets are the [[closures]] of single points.

## Related concepts

* [[separation axiom]]

[[!redirects irreducible topological spaces]]
[[!redirects irreducible space]]


irreflexive comparison

# Irreflexive comparison
* table of contents
{: toc}

## Idea

Just as [[preorders]] generalise [[equivalence relations]] and [[total orders]], _irreflexive comparisons_ should generalise [[apartness relations]] and [[strict total orders]]

## Definitions

An **irreflexive comparison** on a set $S$ is a (binary) [[relation]] $\lt$ on $S$ that is both [[irreflexive relation|irreflexive]] and a [[comparison]].  That is:

* $x \lt x$ is always false;
* If $x \lt z$, then $x \lt y$ or $y \lt z$

## Properties

A [[set]] $S$ equipped with an irreflexive comparison is a [[category]] (with $S$ as the set of [[objects]]) [[enriched category|enriched]] over the [[cartesian monoidal category]] $TV^\op$, that is the [[opposite category|opposite]] of the [[partial order|poset]] of [[truth value|truth values]], made into a [[monoidal category]] using [[disjunction]]. $TV^\op$ is a [[co-Heyting algebra]].

### Preorder of an irreflexive comparison

An important part of an irreflexive comparison is that it is a preorder.

\begin{theorem}
The negation of an irreflexive comparison is transitive.
\end{theorem}

\begin{proof}
The [[contrapositive]] of comparison says that

> for all $x$, $y$, and $z$, if $x \lt y$ or $y \lt z$ is false, then $x \lt z$ is false.

By one of [[de Morgan's laws]], that $x \lt y$ or $y \lt z$ is false is logically to equivalent to that $\neg(x \lt y)$ and $\neg(y \lt z)$, and substituting this into the original statement results in

> if $\neg(x \lt y)$ and $\neg(y \lt z)$, then $\neg(x \lt z)$

which is precisely transitivity for the negation of the irreflexive comparison.
\end{proof}

\begin{theorem}
The negation of an irreflexive comparison is reflexive.
\end{theorem}

\begin{proof}
Irreflexivity states that $\neg (x \lt x)$ is true, which is precisely reflexivity for the negation of the strict weak order.
\end{proof}

\begin{theorem}
The negation of an irreflexive comparison is a preorder.
\end{theorem}

\begin{corollary}
The incomparability relation of a strict weak order, $\neg (x \lt y) \wedge \neg (y \lt x)$, is an [[equivalence relation]]
\end{corollary}

\begin{proof}
For every preorder, $(x \leq y) \wedge (y \leq x)$ is an [[equivalence relation]]. Since $\neg (x \lt y)$ is a [[preorder]], $\neg (x \lt y) \wedge \neg (y \lt x)$ is an equivalence relation.
\end{proof}

\begin{theorem}
If the irreflexive comparison is a [[connected relation]], then its negation is a [[partial order]].
\end{theorem}

\begin{proof}
The connectedness condition states that $\neg (x \lt y) \wedge \neg (y \lt x)$ implies equality, which is precisely the antisymmetry condition for the negation of the strict weak order, implying that its negation is a [[partial order]].
\end{proof}

\begin{theorem}
If the irreflexive comparison is an [[apartness relation]], then its negation is an [[equivalence relation]].
\end{theorem}

\begin{proof}
The symmetry condition of the apartness relation states that $x \lt y$ implies $y \lt x$ for all $x$ and $y$. It's contrapositive states that $\neg(y \lt x)$ implies $\neg(x \lt y)$ for all $x$ and $y$, which is the symmetry condition for the negation of $\lt$. A [[symmetric preorder]] is the same thing as an equivalence relation, which means that the negation of $\lt$ is an equivalence relation.
\end{proof}

\begin{theorem}
If the irreflexive comparison is a [[tight apartness relation]], then its negation is the [[equality relation]].
\end{theorem}

\begin{proof}
This follows from the previous two theorems.
\end{proof}

## Examples

* If an irreflexive comparison satisfies symmetry (if $x \lt y$ then $y \lt x$ then it is an [[apartness relation]].

* If an irreflexive comparison is asymmetric (if $x \lt y$, then $\neg(y \lt x)$) or transitive (if $x \lt y$ and $y \lt z$, then $x \lt z$), then it is a [[strict weak order]].

* Connected versions of the above result in [[tight apartness relations]] and [[strict total orders]].

### Connected irreflexive comparisons

* An irreflexive comparison that is also a [[connected relation]] (if $x \lt y$ is false and $y \lt x$ is false, then $x = y$) is a __connected irreflexive comparison__.

* If the set is an [[inequality space]], then an irreflexive comparison is __strongly connected__ if $x \neq y$ implies $x \lt y$ or $y \lt x$.

## Related concepts

* [[apartness relation]]

* [[tight apartness relation]]

* [[strict total order]]

* [[strict weak order]]

* [[preorder]]

* [[partial order]]

* [[strict preorder]]

[[!redirects irreflexive comparisons]]
[[!redirects connected irreflexive comparison]]
[[!redirects connected irreflexive comparisons]]


irreflexive relation

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

#Contents#
* table of contents
{:toc}

## Definition

A (binary) [[relation]] $\sim$ on a set $A$ is __irreflexive__ if no element of $A$ is related to itself:
$$ \forall (x: A),\; x \nsim x .$$
(where $x\nsim x$ means that $x\sim x$ is false, i.e. that $\neg (x\sim x)$).

In the language of the $2$-poset [[Rel]] of sets and relations, a relation $R: A \to A$ is __irreflexive__ if it is [[disjoint]] from the [[identity relation]] on $A$:
$$ \id_A \cap R \subseteq \empty .$$
Of course, this containment is in fact an equality.

In [[constructive mathematics]], if $A$ is equipped with a [[tight apartness]] $\#$, we say that $\sim$ is __strongly irrelexive__ if only distinct elements of $A$ are related:
$$ \forall (x: A),\; \forall (y: A),\; x \sim y \;\Rightarrow x \# y .$$
Since $\#$ is irrelexive itself, any strongly irrelexive relation must be irrelexive.  The converse holds using [[excluded middle]], through which every set has a unique tight apartness.

## Examples

* A [[digraph]] is a graph in which the edge relation is irreflexive.

## Related concepts

* [[reflexive relation]]

[[!redirects irreflexive relation]]
[[!redirects irreflexive relations]]
[[!redirects irreflexivity]]
[[!redirects irreflexive]]

[[!redirects strongly irreflexive relation]]
[[!redirects strongly irreflexive relations]]
[[!redirects strong irreflexivity]]
[[!redirects strongly irreflexive]]


irreflexive relations > history

< [[irreflexive relations]]
[[!redirects irreflexive relations -- history]]


irreflexive symmetric relation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

\tableofcontents

## Definition

An **irreflexive symmetric relation** is a [[binary relation]] that is [[irreflexive relation|irreflexive]] and [[symmetric relation|symmetric]].

## Examples

* Every [[apartness relation]] is an irreflexive symmetric relation which is also a [[comparison]].

* The negation of a [[reflexive symmetric relation]] is an irreflexive symmetric relation.

* Every [[denial inequality]] is a [[weakly tight relation|weakly tight]] irreflexive symmetric relation.

## In constructive mathematics
 {#ConstructiveMathematics}

In constructive mathematics, not ever tight irreflexive symmetric relation is a denial inequality, and not every denial inequality is tight. Instead, every [[denial inequality]] is only [[weakly tight]]. Indeed, since excluded middle cannot be proven, we could distinguish between the following irreflexive symmetric relations in constructive mathematics:

* [[denial inequality]], an irreflexive symmetric relation $\neq$ which additionally satisfies $\neg(a = b)$ implies $(a \neq b)$

* [[tight]] irreflexive symmetric relation, an irreflexive symmetric relation $\neq$ which additionally satisfies $\neg(a \neq b)$ implies $(a = b)$

* [[weakly tight relation|weakly tight]] irreflexive symmetric relation, an irreflexive symmetric relation $\neq$ which additionally satisfies $\neg(a \neq b)$ implies $\neg \neg (a = b)$

* [[stable relation|stable]] irreflexive symmetric relation, an irreflexive symmetric relation $\neq$ which additionally satisfies $\neg \neg (a \neq b)$ implies $(a \neq b)$

* [[decidable relation|decidable]] irreflexive symmetric relation, an irreflexive symmetric relation $\neq$ which additionally satisfies $(a \neq b)$ or $\neg (a \neq b)$

* [[apartness relation]], an irreflexive symmetric relation $\neq$ which additionally satisfies $a \neq b$ implies that $a \neq c$ or $c \neq b$

or any combination of the above, such as [[tight apartness relation]].

The notion of "[[inequality relation]]" in constructive mathematics usually refers to either the [[denial inequality]] or a [[tight apartness relation]]. On the other hand, some authors have used "inequality relation" to refere to general irreflexive symmetric relations.

## See also

* [[apartness relation]]
* [[inequality relation]]
* [[reflexive symmetric relation]]

[[!redirects irreflexive symmetric relation]]
[[!redirects irreflexive symmetric relations]]

[[!redirects tight irreflexive symmetric relation]]
[[!redirects tight irreflexive symmetric relations]]

[[!redirects weakly tight irreflexive symmetric relation]]
[[!redirects weakly tight irreflexive symmetric relations]]

[[!redirects stable irreflexive symmetric relation]]
[[!redirects stable irreflexive symmetric relations]]

[[!redirects decidable irreflexive symmetric relation]]
[[!redirects decidable irreflexive symmetric relations]]


Irving Kaplansky

Irving Kaplansky (1917 - 2006) was a mathematician who made major contributions to [[ring theory]], [[group theory]] and the theory of [[fields]].

## Reference

* [MacTutor biography](https://mathshistory.st-andrews.ac.uk/Biographies/Kaplansky/)

## Selected writings

On [[Jordan superalgebras]]:

* [[Irving Kaplansky]], _Graded Jordan algebras I_ ([preprint](https://www1.osu.cz/~zusmanovich/links/files/kaplansky/gja.pdf))

category: people


Irving Segal

* [Wikipedia page](http://en.wikipedia.org/wiki/Irving_Segal)

## Selected writings

Introducing the [[GNS construction]]:

* {#Segal47} [[Irving Segal]], _Irreducible representations of operator algebras_, Bull. Am. Math. Soc. 53: 73–88, (1947)  ([pdf](http://www.ams.org/journals/bull/1947-53-02/S0002-9904-1947-08742-5/S0002-9904-1947-08742-5.pdf), [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-2/Irreducible-representations-of-operator-algebras/bams/1183510397.full))

category: people

[[!redirects Irving Ezra Segal]]
[[!redirects Irving E. Segal]]
[[!redirects I. E. Segal]]
[[!redirects I. Segal]]


Is probability theory a branch of mathematics?

+-- {: .query}

[[Urs Schreiber]]: This page contains what turned out to be the beginning of a discussion that was further had on the nForum [here](https://nforum.ncatlab.org/discussion/1232/is-probability-theory-a-branch-of-mathematics/#Item_0).

Eventually, I suggest, we should have a page [[probability theory]] following the usual [[template page]] that contains a remark on the fact that there is a certain philosophical discussion to be had about the nature of probability theory. That remark should then point to this page here, or to a page like it, which would contain this philosophical material.

This page here should eventually be more structured, so that the reader gets a better idea of what it is about. Roughly following the [[template page]], what we need here is

1. A brief indication of what the issue is.

1. A brief summary of what the proposals for answering it are.

1. A structured section list reflecting what the different pieces of the material given here are about.

=--

Andrei N. Kolmogorov is often held to have shown that probability theory is [[measure theory]] of spaces of measure 1.  Sometimes it is asserted that the difference between probability and measure theory is that in the former one is concerned with the concepts of independence of events and independence of random variables.  By these accounts, one who does probability theory is doing mathematics.

A different view is that probability theory is a science that, like physics, must rely very heavily upon mathematics, but some questions in probability theory are not questions of mathematics.  That seems at least consonant with views of the physicist Edwin Jaynes in his book ''Probability Theory: The Logic of Science.''  Perhaps probability theory may be viewed as a branch of epistemology.

Some reasons for holding that view follow.

Consider an "unfair" die whose outcome 2/7 of the time is "1", and for which the other outcomes---"2" through "6"---occur with equal frequencies, thus each in 1 trial out of 7.

A question that must be considered vague or, at best, ambiguous asks: "What is the probability that a '1' appeared the first time the die was cast?"

All probabilities are conditional.  If one follows Kolmogorov's conventions, whereby a probability space consists of an underlying set $\Omega$ and a countably additive probability measure $P$ on a $\sigma$-algebra of subsets of $\Omega$, then one could say that the probability of a measurable subset $A$ of $\Omega$ is the conditional probability $P(A | \Omega)$.

* If one asks for the conditional probability that a "1" appeared on the first toss, given the whole record of outcomes, then the answer is either 0 or 1.

* If one asks for the conditional probability that a "1" appeared on the first toss, given only the information about relative frequencies above, then the answer is 2/7.

* But what if one asks for the conditional probability that a "1" appeared on the first toss given even less information: one knows there are six possible outcomes but one is wholly ignorant of the relative frequencies above?

The answer proposed to the question after the second bullet above may appear to be uncontroversial.  But it is not necessitated by Kolmogorov's theory unless one construes the question in a way that makes it trivial.  If it is construed as meaning "How sure should we be, given the state of our knowledge", then Kolmogorov's theory neither answers it nor understands it.

The question after the third bullet does not express a well-formed mathematical problem about measure spaces.  It is anything but clear which measure space it asks about.  Nonetheless, it seems like a perfectly sensible question if such a thing as epistemic probabilities can exist.  And it does not seem unreasonable to say that the conditional probability that a "1" appeared on the first trial, given that particular state of knowledge, is 1/6.

Within Kolmogorov's theory, one can ask the following sort of question: Suppose the 6-tuple $(p_1,\dots,p_6)$ is uniformly distributed in the space $p_1+\cdots+p_6 = 1,\quad p_1,\dots,p_6 \ge 0$.  Suppose that given the value of $(p_1,\dots,p_6)$, the probability that the outcome of any throw of the die is $i$ is $p_i$.  Finally, suppose that the outcomes on different trials are conditionally independent given the value of $(p_1,\dots,p_6)$.  Then what is the "unconditional" probability that the outcome of the first throw was "1"?  The answer is 1/6.  What is the conditional probability that the outcome of the 1st throw was "1", given that a "1" appeared on each of the 2nd through 5th trials?  The answer is substantially more than 1/6, since the given information about the record makes it probable that $p_1$ is large.  (The trials, although conditionally independent given $(p_1,\dots,p_6)$, are thus seen not to be "unconditionally" independent.)

The reasonable lay person unschooled in the mathematics of probability theory, observing a long string of consecutive "1"s, or even a less extreme case of unusually high frequency of "1"s, will begin to suspect bias. If one were to attempt to model such a layperson's behavior as some probability distribution assigned to the aforementioned vector $(p_1,\dots,p_6)$, it is not immediately clear how, or even whether, one could do so. However, one aspect of the answer that appeals to common sense, is that, given symmetry of information that treats the six outcomes equally, the conditional epistemic probability distribution of the vector $(p_1,\dots,p_6)$ would be symmetric under permutations of the six indices. Among the consequences of this observation are:

* The probability of a "1" on the first toss is 1/6; and
* The probability that a "1" appeared on the first toss, given the record of outcomes a long sequence of subsequent tosses, will approach the observed relative frequency with which a "1" has appeared.

But how fast will it approach that observed relative frequency?

From Kolmogorov's point of view, that depends on what probability distribution has been assigned to the vector $(p_1,\dots,p_6)$.  Only when that has been specified is there a well-defined mathematical problem.  But one does not approach the question of how to model the reasonable layperson's inferences with such a distribution already assigned. We might nonetheless reasonably approach it with the constraint of symmetry in the six indices. That is enough to draw the conclusion stated in the bullet point above that says the probability of a "1" on the first toss is 1/6. It is not enough to answer the question: how fast?

Richard von Mises said that it makes no more sense to say the conditional probability that the outcome of the first trial was "1", given the state of complete ignorance of relative frequencies, is 1/6, than it does to say that the heights of all members of a group of people are estimated to be equal because one is ignorant of their heights.  von Mises missed the point.  The problem was not to estimate the relative frequencies, but to say how epistemically probable that outcome on the first trial is, given a particular state of knowledge.

From the point of view of probability-as-measure-theory put forth by Kolmogorov, it makes sense to speak of the probability of ["1" on the first trial], but it does not make sense to speak of the die's probability of landing on "1" divorced from any particular trial and thought of as a relative frequency or propensity.  One assigns a probability only to a measurable subset of a probability space.

But Kolmogorov took a philosophical position on how his mathematical theory should be applied to the world: he wanted to assign probabilities only when they could be viewed as relative frequencies or proportions.  That is called "frequentism".

When one applies frequentism to statistical inference, one finds oneself making subjective economic decisions about the frequencies of false positives and false negatives one is willing to tolerate.

On the frequentist view, one must not assign probability 1/3, or any number, to the proposition that there was life on Mars a billion years ago, because one cannot say that that happened in 1/3 of all cases.

An alternative view is called [[Bayesianism]], after the 18th-century English Presbyterian minister and mathematician Thomas Bayes.  On the Bayesian view, probabilities are degrees of belief in uncertain propositions, not relative frequencies or proportions.

When one applies Bayesianism to statistical inference, one faces the problem of what probability distribution to assign to the vector $(p_1,\dots,p_6)$.  It is often asserted by both Bayesians and frequentists that that is subjective.

Thus Bayesians and frequentists put their subjectivity in different places.

It can be argued that it is a principle of logic that the conditional probability, given an epistemic state of complete ignorance of the relative frequencies of the die's outcomes, that a "1" appeared on the first trial, is 1/6.  If that is viewed as eliminating the subjectivity from that particular epistemic probability, it nonetheless falls short of solving the problem of logically rather than subjectively determining the distribution of $(p_1,\dots,p_6)$.

+-- {: .query}
[[Michael Hardy]]: Much of this material appeared in a subpage of my Wikipedia user page.
=--


Isaac H. Kim

* [personal page](https://sites.google.com/view/isaackimquantum)

## Selected writings

On [classically controlled quantum computation](quantum+computation#ClassicalControlQuantumData) on [[NISQ]]-machines in terms of [[holographic entanglement entropy|holographic]] [[tensor network]] states:

* {#KimSwingle17} [[Isaac H. Kim]], [[Brian Swingle]], *Robust entanglement renormalization on a noisy quantum computer* ([arXiv:1711.07500](https://arxiv.org/abs/1711.07500))

category: people

[[!redirects Issac H. Kim]]


Isaac L. Chuang

* [personal page](http://feynman.mit.edu/ike/homepage/index.html)

* [institute page](https://physics.mit.edu/faculty/isaac-chuang/)

## Selected writings

Introducing [[measurement-based quantum computation]]:

* [[Daniel Gottesman]], [[Isaac L. Chuang]], *Quantum Teleportation is a Universal Computational Primitive*, Nature **402** (1999) 390-393 [[doi:10.1038/46503](https://doi.org/10.1038/46503), [arXiv:quant-ph/9908010](https://arxiv.org/abs/quant-ph/9908010)]

On [[quantum computation]] and [[quantum information]]:

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

On [[quantum error correction]]:

* [[Isaac L. Chuang]], *Quantum error correction*, Chapter 7 in: *Quantum Machines: Measurement and Control of Engineered Quantum Systems* Lecture Notes of the Les Houches Summer School **96** (2011) 273–320  [[doi:10.1093/acprof:oso/9780199681181.003.0007](https://doi.org/10.1093/acprof:oso/9780199681181.003.0007)]

category: people

[[!redirects Isaac Chuang]]


Isaac Newton

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

The founder of the science of _[[physics]]_ ([[mechanics]]) as a science rooted in [[mathematics]], and of _[[differential calculus]]_.

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

## Selected writings

Laying the foundations of modern [[physics]] (in contrast to the previous *[[natural philosophy]]*) by what came to be known as *[[Newton's laws of motion]]* and *[[Newtonian mechanics]]*:

* [[Isaac Newton]], *[[Philosophiæ naturalis principia mathematica]]* (1687)

## Quotes

On [[Nature conformable to herself]]:

> "For Nature is very consonant and conformable to her self... For we must learn from the phaenomena of nature what bodies attract one another, and what are the laws and properties of the attraction, before we enquire the cause by which the attraction is perform'd, the attractions of gravity, magnetism and electricity, react to very sensible distances, and so have been observed by vulgar eyes, and there may be others which reach to so small distances as hitherto escape observation; and perhaps electrical attraction may react to such small distances, even without being excited by friction." [Opticks]

## Related entries

* [[Newton's laws of motion]]

* [[Newtonian mechanics]] ([[classical mechanics]])

* [[physics]]

category: people

[[!redirects I. Newton]]

[[!redirects Newton]]

[[!redirects Newtonian]]


Isabelle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Isabelle__ is a [[proof assistant]]. Its main application is _[[HOL]]_. Related proof assistants are HOL4, [[HOL Light]], HOL Zero, and ProofPower. These are programmed, as Isabelle itself, in ML.
In contrast to other proof assistants, Isabelle is based on classical [[set theory|set theoretic]] [[foundation of mathematics|foundations]].
The [[Archive of Formal Proofs]] is an online library of proofs formalized  in Isabelle.

## Related entries

[[!include proof assistants and formalization projects -- list]]

## Links

* Wikipedia, [Isabelle (proof assistant)](http://en.wikipedia.org/wiki/Isabelle_%28proof_assistant%29)

* Isabelle [homepage](http://isabelle.in.tum.de)

* a tutorial [webpage](http://www.inf.kcl.ac.uk/staff/urbanc/Cookbook)

* the [[Archive of Formal Proofs]] [webpage](https://www.isa-afp.org/index.html)

* _[The Isabelle2014 Library](http://isabelle.in.tum.de/library/)_

## References

* T. Nipkow, L. Paulson, M. Wenzel. _Isabelle/HOL — A Proof Assistant for Higher-Order
Logic_, Springer (2002)

* T. Nipkow, G. Klein. _Concrete Semantics with Isabelle/HOL_, Springer (2014)  ([web](http://www.concrete-semantics.org))

* Florian Rabe and Mihnea Iancu, _A Formalized Set-Theoretical Semantics of
Isabelle/HOL_ ([pdf](https://kwarc.info/people/frabe/Research/RI_isabelle_10.pdf))

Implementation of [[homotopy type theory]] in Isabelle:

* {#Chen19} Joshua Chen, _An Implementation of Homotopy Type Theory in Isabelle/Pure_ ([arXiv:1911.00399](https://arxiv.org/abs/1911.00399), [github:jaycech3n/Isabelle-HoTT](https://github.com/jaycech3n/Isabelle-HoTT))

category: software


Isadore Singer

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

## Selected writings

Introducing the [[Kadison-Singer problem]]:

* [[Richard V. Kadison]], [[Isadore Singer]], _Extensions of pure states_, American Journal of Mathematics, 81(2):383–400, 1959 ([jstor:2372748](https://www.jstor.org/stable/2372748))

Introducing what came to be known as [[Karoubi K-theory]]:

* [[Michael Atiyah]], [[Isadore Singer]], *Index theory for skew-adjoint Fredholm operators*, Publications Mathématiques de l'IHÉS, Tome **37** (1969) 5-26 $[$[numdam:PMIHES_1969__37__5_0](http://www.numdam.org/item/PMIHES_1969__37__5_0)$]$

On [[quantum anomalies]] via [[index theory]]:

* [[Michael Atiyah]], [[Isadore Singer]], _Dirac operators coupled to vector potentials_, Proc. Nat. Acad. Sci. USA __81__, 2597-2600 (1984) ([doi:10.1073/pnas.81.8.2597]( https://doi.org/10.1073/pnas.81.8.2597), [jstor:23378](https://www.jstor.org/stable/23378))

On [[topological Yang-Mills theory]]:

* L. Baulieu, [[Isadore Singer]], _Topological Yang-Mills symmetry_, Nuclear Physics B - Proceedings Supplements Volume 5, Issue 2, December 1988, Pages 12-19 (doi:10.1016/0920-5632(88)90366-0)

On [[differential cohomology]] with motivation from [[M5-branes]]-phyiscs

* _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_

category: people


Isaiah Kantor

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

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=198055)

## Selected writings

Introducing the [[Kantor-Koecher-Tits construction]]:

* {#Kantor64} [[Isaiah Kantor]], _Classification of irreducible transitive differential groups_, Dokl. Akad. Nauk SSSR, 158:6 (1964), 1271–1274 ([mathnet:dan30260](http://mi.mathnet.ru/eng/dan30260))

On [[super Lie algebras]]:

* {#Kantor70} [[Isaiah Kantor]], _Graded Lie algebras_, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.

## Related $n$Lab entries

* [[Kantor-Koecher-Tits construction]]

category: people


Isamu Iwanari

* [Home page](https://sites.google.com/site/isamuiwanarishomepage/)

## Related $n$Lab entries

* [[Tannaka duality]]

category: people


Isar Stubbe

* [homepage](http://www-lmpa.univ-littoral.fr/~stubbe/)

category:people


Isbell conjugation > history

< [[Isbell conjugation]]


Isbell duality

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

[[higher geometry]] $\leftarrow$ **Isbell duality**  $\rightarrow$ [[higher algebra]]

***

#Contents#
* tabel of contents
{:toc}

## Idea

A general abstract [[adjunction]]

$$
  (\mathcal{O} \dashv Spec)
  :
  CoPresheaves
  \underoverset{Spec}{\mathcal{O}}{\leftrightarrows}
  Presheaves
$$

relates (higher) [[presheaves]] with (higher) [[copresheaves]] on a given ([[higher category theory|higher]]) [[category]] $C$: this is called **Isbell conjugation** or **Isbell duality** (after [[John Isbell]]).

To the extent that this adjunction descends to presheaves that are ([[higher topos theory|higher]]) [[sheaves]] and copresheaves that are ([[(infinity,1)-algebraic theory|higher]]) [[algebra over an algebraic theory|algebras]] this duality relates [[higher geometry]] with [[higher algebra]].

Objects preserved by the [[monad]] of this adjunction are called **Isbell self-dual**.

Under the interpretation of [[presheaves]] as generalized [[spaces]] and [[copresheaves]] as generalized [[quantities]] modeled on $C$ ([Lawvere 86](#Lawvere86), see at _[[space and quantity]]_), Isbell duality is the archetype of the [[duality]] between [[geometry]] and [[algebra]] that permeates mathematics (such as [[Gelfand duality]], [[Stone duality]], or the [[embedding of smooth manifolds into formal duals of R-algebras]]).

## Definition

Let $\mathcal{V}$ be a good enriching category (a [[cosmos]], i.e. a [[complete category|complete]] and [[cocomplete category|cocomplete]] [[closed monoidal category|closed]] [[symmetric monoidal category]]).

Let $\mathcal{C}$ be a [[small category|small]] $\mathcal{V}$-[[enriched category]].

Write $[\mathcal{C}^{op}, \mathcal{V}]$ and $[\mathcal{C}, \mathcal{V}]$ for the [[enriched functor categories]].

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

There is a $V$-[[adjunction]]

$$
  (\mathcal{O} \dashv Spec)
  \colon
  [C, \mathcal{V}]^{op}
  \underoverset{Spec}{\mathcal{O}}{\leftrightarrows}
  [C^{op}, \mathcal{V}]
$$

where

$$
  \mathcal{O}(X) \colon c \mapsto [C^{op}, \mathcal{V}](X, C(-,c))
  \,,
$$

and

$$
  Spec(A) \colon c \mapsto [C, \mathcal{V}]^{op}(C(c,-),A)
  \,.
$$

=--

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

This is also called [[Isbell duality]]. Objects which are preserved by $\mathcal{O} \circ Spec$ or $Spec \mathcal{O}$ are called __Isbell self-dual__.

=--

The proof is mostly a tautology after the notation is unwound. The mechanism of the proof may still be of interest and be relevant for generalizations and for less tautological variations of the setup. We therefore spell out several proofs.

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

Use the [[end]]-expression for the [[hom-object]]s of the [[enriched functor categories]] to compute

$$
  \begin{aligned}
     [C,\mathcal{V}]^{op}(\mathcal{O}(X), A)
     & :=
     \int_{c \in C} \mathcal{V}(A(c), \mathcal{O}(X)(c))
     \\
     & :=
     \int_{c \in C} \mathcal{V}(A(c), [C^{op}, \mathcal{V}](X, C(-,c)))
     \\
     & :=
     \int_{c \in C} \int_{d \in C} \mathcal{V}(A(c), \mathcal{V}(X(d), C(d,c)))
     \\
     & \simeq
     \int_{d \in C} \int_{c \in C} \mathcal{V}(X(d), \mathcal{V}(A(c), C(d,c)))
    \\
     & =:
     \int_{d \in C}  \mathcal{V}(X(d), [C,\mathcal{V}]^{op}(C(d,-),A))
    \\
     & =:
     \int_{d \in C}  \mathcal{V}(X(d), Spec(A)(d))
    \\
     & =:
     [C^{op}, \mathcal{V}](X, Spec(A))
  \end{aligned}
  \,.
$$

=--

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

Here apart from writing out or hiding the ends, the only step that is not a definition is precisely the middle one, where we used that $\mathcal{V}$ is a [[symmetric monoidal category|symmetric]] [[closed monoidal category]].

=--

The following proof does not use ends and needs instead slightly more preparation, but has then the advantage that its structure goes through also in great generality in [[higher category theory]].
{#ProofB}

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

Notice that

**Lemma 1:** $Spec(C(c,-)) \simeq C(-,c)$

because we have a natural isomorphism

$$
  \begin{aligned}
    Spec(C(c,-))(d)
    & :=
    [C,\mathcal{V}](C(c,-), C(d,-))
    \\
    & \simeq
    C(d,c)
  \end{aligned}
$$

by the [[Yoneda lemma]].

From this we get

**Lemma 2:** $[C^{op}, \mathcal{V}](Spec C(c,-), Spec A) \simeq
    [C,\mathcal{V}](A, C(c,-))$

by the sequence of natural isomorphisms

$$
  \begin{aligned}
     [C^{op}, \mathcal{V}](Spec C(c,-), Spec A)
     & \simeq
     [C^{op}, \mathcal{V}](C(-,c), Spec A)
     \\
     & \simeq (Spec A)(c)
     \\
     & := [C, \mathcal{V}](A, C(c,-))
  \end{aligned}
  \,,
$$

where the first is Lemma 1 and the second the [[Yoneda lemma]].

Since (by what is sometimes called the [[co-Yoneda lemma]]) every object $X \in [C^{op}, \mathcal{V}]$ may be written as a [[colimit]]

$$
  X \simeq {\lim_\to}_i C(-,c_i)
$$

over [[representable functor|representables]] $C(-,c_i)$ we have

$$
  X \simeq {\lim_\to}_i Spec(C(c_i,-))
  \,.
$$

In terms of the same diagram of representables it then follows that

**Lemma 3:**

$$
  \mathcal{O}(X) \simeq {\lim_{\leftarrow}}_i C(c_i,-)
$$

because using the above colimit representation and the Yoneda lemma we have natural isomorphisms

$$
  \begin{aligned}
     \mathcal{O}(X)(d)
     &=
     [C^{op}, \mathcal{V}](X, C(-,c))
     \\
     & \simeq
     [C^{op}, \mathcal{V}]({\lim_\to}_i C(-,c_i), C(-,c))
     \\
     & \simeq
     {\lim_\leftarrow}_i [C^{op}, \mathcal{V}](C(-,c_i), C(-,c))
     \\
     & \simeq
     {\lim_\leftarrow}_i C(c_i,c)
  \end{aligned}
  \,.
$$

Using all this we can finally obtain the adjunction in
question by the following sequence of natural isomorphisms

$$
  \begin{aligned}
     [C,\mathcal{V}]^{op}(\mathcal{O}(X), A)
     &
     \simeq
     [C,\mathcal{V}](A, {\lim_\leftarrow}_i C(c_i,-))
     \\
     & \simeq
     {\lim_{\leftarrow}}_i [C, \mathcal{V}](A, C(c_i,-))
     \\
     & \simeq
     {\lim_{\leftarrow}}_i [C^{op}, \mathcal{V}](Spec C(c_i,-), Spec A)
     \\
     & \simeq
     [C^{op}, \mathcal{V}]({\lim_{\to}}_i  Spec C(c_i,-), Spec A)
     \\
     & \simeq
     [C^{op}, \mathcal{V}](X, Spec A)
  \end{aligned}
  \,.
$$

=--

The pattern of this proof has the advantage that it goes through in great generality also on [[higher category theory]] without reference to a higher notion of enriched category theory.

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

Under certain circumstances, Isbell duality can be extended to large $\mathcal{V}$-enriched categories $C$. For example, if $C$ has a small generating subcategory $S$ and a small cogenerating subcategory $T$, then for each $F: C^{op} \to \mathcal{V}$ and $G: C \to \mathcal{V}$, one may construct $\mathcal{O}(F)$ and $Spec(G)$ objectwise as appropriate subobjects in $\mathcal{V}$:

$$\mathcal{O}(F)(c) = [C^{op}, \mathcal{V}](F, C(-, c)) \hookrightarrow \int_{s: S} \mathcal{V}(F s, \hom(s, c))$$

$$Spec(G)(c) = [C, \mathcal{V}](G, C(c, -)) \hookrightarrow \int_{t: T} \mathcal{V}(G t, \hom(c, t))$$

=--

## Example

In the simplest case, namely for an ordinary category $\mathcal{C}$, the adjunction between presheaves and copresheaves arises as follows.

The category of [[presheaves]] $[\mathcal{C}^{op}, \mathrm{Set}]$ is the [[free cocompletion]] of $\mathcal{C}$.  This means that any functor

$$f \colon \mathcal{C} \to \mathcal{D}$$

to a [[cocomplete category]] $\mathcal{D}$ extends along the [[Yoneda embedding]] $y \colon \mathcal{C} \to [\mathcal{C}^{op}, \mathrm{Set}]$ to a [[cocontinuous functor]]

$$F \colon [\mathcal{C}^{op}, \mathrm{Set}] \to \mathcal{D}$$

in a manner unique up to natural isomorphism.

Dually, the category of [[copresheaves]]  $[\mathcal{C}, \mathrm{Set}]^{op}$ is the [[free completion]] of $\mathcal{C}$.  This means that any functor

$$g \colon \mathcal{C} \to \mathcal{D}$$

to a [[complete category]] $\mathcal{D}$ extends along the [[co-Yoneda lemma|co-Yoneda embedding]] $z \colon \mathcal{C} \to [\mathcal{C}, \mathrm{Set}]^{op}$ to a [[continuous functor]]

$$G \colon [\mathcal{C}, \mathrm{Set}]^{op} \to \mathcal{D}$$

in a manner unique up to natural isomorphism.

We can apply these ideas to get the functors involved in Isbell duality.   The presheaf category $[\mathcal{C}^{op}, \mathrm{Set}]$ has all limits, so we can extend the Yoneda embedding to a continuous functor

$$ Y \colon [\mathcal{C}, \mathrm{Set}]^{op} \to [\mathcal{C}^{op}, \mathrm{Set}] $$

from copresheaves to presheaves.   Dually, the copresheaf category $[\mathcal{C}, \mathrm{Set}]^{op}$ has all colimits, so we can extend the co-Yoneda embedding to a cocontinuous functor

$$ Z \colon [\mathcal{C}^{op}, \mathrm{Set}] \to [\mathcal{C}, \mathrm{Set}]^{op} $$

from presheaves to copresheaves.

Isbell duality says that these are adjoint functors: $Y$ is right adjoint to $Z$.

## Properties

###Relation to Yoneda embedding

$Spec$ is the [[left Kan extension]] of the [[Yoneda embedding]] along the contravariant Yoneda embedding, while $\mathcal{O}$ is the left Kan extension of the contravariant Yoneda embedding along the Yoneda embedding.

The [[codensity monad]] of the [[Yoneda embedding]] is isomorphic to the monad induced by the Isbell adjunction, $Spec \mathcal{O}$ ([Kock 66, Theorem 4.1](#Kock66) and [Di Liberti 19, Theorem 2.7](#DiLiberti)).

### Respect for limits

Choose any [[class]] $L$ of [[limit]]s in $C$ and write $[C,\mathcal{V}]_\times \subset [C,\mathcal{V}]$ for the [[full subcategory]] consisting of those functors preserving these limits.

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

The $(\mathcal{O} \dashv Spec)$-adjunction does descend to this inclusion, in that we have an adjunction

$$
  (\mathcal{O} \dashv Spec)
  :
  [C, \mathcal{V}]_{\times}^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op}, \mathcal{V}]
$$

=--

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

Because the [[hom-functor]]s preserves all [[limit]]s:

$$
  \begin{aligned}
    \mathcal{O}(X)({\lim_{\leftarrow}}_j c_j)
    & := [C^{op}, \mathcal{V}](X,C(-,{\lim_{\leftarrow}}_j c_j))
    \\
    & \simeq [C^{op}, \mathcal{V}](X,{\lim_{\leftarrow}}_j  C(-,c_j))
    \\
    & \simeq {\lim_{\leftarrow}}_j  [C^{op}, \mathcal{V}](X,C(-,c_j))
    \\
    & =: {\lim_{\leftarrow}}_j \mathcal{O}(X)(c_j)
  \end{aligned}
  \,.
$$

=--

### Isbell self-dual objects

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

An object $X$ or $A$ is **Isbell-self-dual** if

* $A \stackrel{}{\to} \mathcal{O} Spec(A)$ is an [[isomorphism]] in $[C,\mathcal{V}]$;

* $X \to Spec \mathcal{O} X$ is an [[isomorphism]] in $[C^{op}, \mathcal{V}]$, respectively.

=--

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

All [[representable functor|representables]] are Isbell self-dual.

=--

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

By [Proof B , lemma 1](#ProofB) we have a [[natural isomorphism]]s in $c \in C$

$$
  Spec(C(c,-)) \simeq C(-,c)
  \,.
$$

Therefore we have also the natural isomorphism

$$
  \begin{aligned}
     \mathcal{O} Spec C(c,-)(d)
     & \simeq
     \mathcal{O} C(-,c) (d)
     \\
     & :=
      [C^{op}, \mathcal{V}](C(-,c), C(-,d))
     \\
     & \simeq
     C(c,d)
  \end{aligned}
  \,,
$$

where the second step is the [[Yoneda lemma]]. Similarly the other way round.

=--

### Isbell envelope

See [[Isbell envelope]].

### Reflexive completion

See [[reflexive completion]].

## Examples and similar dualities

Isbell duality is a template for many other [[space]]/[[algebra]]-[[dualities]] in [[mathematics]].

### Function $T$-Algebras on presheaves {#FunctionAlgebrasOnPresheaves}

Let $\mathcal{V}$ be any [[cartesian closed category]].

Let $C := T$ be the [[syntactic category]] of a $\mathcal{V}$-enriched [[Lawvere theory]], that is a $\mathcal{V}$-category with finite [[product]]s such that all objects are generated under products from a single object $1$.

Then write $T Alg := [C,\mathcal{V}]_\times$ for category of product-preserving functors: the category of $T$-algebras. This comes with the canonical forgetful functor

$$
  U_T : T Alg \to \mathcal{V} :   A \mapsto A(1)
$$

Write

$$
  F_T : T^{op} \hookrightarrow T Alg
$$

for the [[Yoneda embedding]].

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

Call

$$
  \mathbb{A}_T := Spec(F_T(1))  \in [C^{op}, \mathcal{V}]
$$

the **$T$-line object**.

=--

+-- {: .num_lemma}
###### Observation

For all $X \in [C^{op}, \mathcal{V}]$ we have

$$
  \mathcal{O}(X) \simeq
  [C^{op}, \mathcal{V}](X, Spec(F_T(-)))
  \,.
$$

In particular

$$
  U_T(\mathcal{O}(X)) \simeq [C^{op}, \mathcal{V}](X,\mathbb{A}_T)
  \,.
$$

=--

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

We have isomorphisms natural in $k \in T$

$$
\begin{aligned}
  [C^{op}, \mathcal{V}](X, Spec(F_T(k)))
  & \simeq
  T Alg(F_T(k), \mathcal{O}(X))
  \\
  & \simeq
  \mathcal{O}(X)(k)
\end{aligned}
$$

by the above adjunction and then by the [[Yoneda lemma]].

=--

All this generalizes to the following case:

instead of setting $C := T$ let more generally

$$
  T \subset C \subset T Alg^{op}
$$

be a [[small category|small]] [[full subcategory]] of $T$-algebras, containing all the free $T$-algebras.

Then the original construction of $\mathcal{O} \dashv Spec$ no longer makes sense, but that in terms of the line object still does

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

Set

$$
 Spec A : B \mapsto T Alg(A,B)
$$

and

$$
  \mathcal{O}(X) : k \mapsto
  [C^{op}, \mathcal{V}](X, Spec(F_T(k)))
  \,.
$$

Then we still have an adjunction

$$
  (\mathcal{O} \dashv Spec)
  :
  T Alg^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op}, \mathcal{V}]
  \,.
$$

=--

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

$$
  \begin{aligned}
    T Alg^{op}(\mathcal{O}(X), A)
    & :=
    \int_{k \in T} \mathcal{V}( A(k), \mathcal{O}(X)(k) )
    \\
    & :=
    \int_{k \in T} \mathcal{V}( A(k), [C^{op}, \mathcal{V}](X, Spec(F_T(k))) )
    \\
    & := \int_{k \in T} \int_{B \in C}
    \mathcal{V}(A(k), \mathcal{V}(X(B), T Alg(F_T(k), B) ))
    \\
    & \simeq \int_{k \in T} \int_{B \in C}
    \mathcal{V}(A(k), \mathcal{V}(X(B), B(k) ))
    \\
    & \simeq \int_{k \in T} \int_{B \in C}
    \mathcal{V}(X(B), \mathcal{V}(A(k), B(k) ))
    \\
    & =: \int_{B \in C} \mathcal{V}(X(B), T Alg(A,B))
    \\
    & =:
    \int_{B \in C} \mathcal{V}(X(B), Spec(A)(B))
    \\
    & =:
    [C^{op}, Set](X,Spec(A))
  \end{aligned}
  \,.
$$

The first step that is not a definition is the [[Yoneda lemma]]. The step after that is the symmetric-closed-monoidal structure of $\mathcal{V}$.

=--

### Function $k$-algebras on derived $\infty$-stacks {#FuncCompDerStacks}

The structure of our [Proof B](#ProofB) above goes through in higher category theory.

Formulated in terms of [[derived stack]]s over the [[(∞,1)-category]] of [[dg-algebra]]s, this is essentially the argument appearing on [page 23](http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.3636v1.pdf#page=23) of ([Ben-ZviNadler](#Ben-ZviNadler)).

### Function $T$-algebras on $\infty$-stacks

for the moment see at _[[function algebras on ∞-stacks]]_.

### Function 2-algebras on algebraic stacks

see [[Tannaka duality for geometric stacks]]

### Gelfand duality
 {#GelfandDuality}

[[Gelfand duality]] is the [[equivalence of categories]] between (nonunital) commutative [[C-star algebra|C*-algebras]] and ([[locally compact space|locally]]) [[compact topological spaces]]. See there for more details.

### Serre-Swan theorem

The [[Serre-Swan theorem]] says that suitable [[modules]] over an commutative [[C-star algebra|C*-algebra]] are equivalently modules of [[sections]] of [[vector bundles]] over the [Gelfand-dual](#GelfandDuality) topological space.

## Related concepts

* [[duality between algebra and geometry]]

* [[function algebra]]

* [[function algebras on ∞-stacks]]

* [[nucleus of a profunctor]]

[[!include Isbell duality - table]]

## References

The original articles on Isbell duality and the [[Isbell envelope]] are

* {#Isbell66} [[John Isbell]], *Structure of categories*, Bulletin of the American Mathematical Society **72** (1966), 619-655. [[euclid:1183528163](http://projecteuclid.org/euclid.bams/1183528163), [ams:1966-72-04/S0002-9904-1966-11541-0](https://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11541-0/)]

* {#Isbell67} [[John Isbell]], _Normal completions of categories_, Reports of the Midwest Category Seminar, **47**, Springer (1967) 110-155 [[doi:10.1007/BFb0074302](https://doi.org/10.1007/BFb0074302)]

* {#Kock66} [[Anders Kock]]. _Continuous Yoneda representation of a small category_. University of Aarhus, Denmark, 1966. ([pdf](https://tildeweb.au.dk/au76680/CYRSC.pdf))

More recent discussion:

* {#Lawvere86} [[William Lawvere]], p. 17 of _Taking categories seriously_, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted as: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 ([web](http://tac.mta.ca/tac/reprints/articles/8/tr8abs.html))

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell Duality* Theory and Applications of Categories **20** 15 (2008) 504-542 [[tac:20-15](http://www.tac.mta.ca/tac/volumes/20/15/20-15abs.html)]

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell duality for for modules*, Theory and Applications of Categories **22** 17 (2009) 401-419 [[tac:22-17](http://www.tac.mta.ca/tac/volumes/22/17/22-17abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/rmod.pdf)]

* [[Richard Garner]], _The Isbell monad_, Advances in Mathematics **274** (2015) pp.516-537. ([draft](http://comp.mq.edu.au/~rgarner/Papers/Isbell.pdf))

* [[Vaughan Pratt]], _Communes via Yoneda, from an elementary perspective_, Fundamenta Informaticae 103 (2010), 203–218.

* {#DL19} [[Ivan Di Liberti]], [[Fosco Loregian]], _On the Unicity of Formal Category Theories_, arXiv:1901.01594 (2019). ([abstract](https://arxiv.org/abs/1901.01594))

* {#DiLiberti} [[Ivan Di Liberti]], _Codensity: Isbell duality, pro-objects, compactness and accessibility_, ([arXiv:1910.01014](https://arxiv.org/abs/1910.01014))

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

  > (relation to [[reflexive completion]])

* {#Baez22} [[John Baez]], Isbell duality, *Notices Amer. Math. Soc.* **70** (2022) 140-141 [[doi:10.1090/noti2602](https://doi.org/10.1090/noti2602), [pdf](https://www.ams.org/journals/notices/202301/rnoti-p140.pdf)]

Isbell conjugacy for [[(∞,1)-presheaves]] over the [[(∞,1)-category]] of duals of [[dg-algebra]]s is discussed around page 32 of

* [[David Ben-Zvi]], [[David Nadler]], _Loop spaces and connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))
{#Ben-ZviNadler}

in

* {#Toen} [[nLab:Bertrand Toën]], _Champs affines_ ([arXiv:math/0012219](http://arxiv.org/abs/math/0012219))

Isbell self-dual [[∞-stack]]s over duals of commutative [[associative algebra]]s are called _affine stacks_. They are characterized as those objects that are _small_ in a sense and local with respect to the [[cohomology]] with coefficients in the canonical [[line object]].

A generalization of this latter to $\infty$-stacks over duals of [[algebra over a Lawvere theory|algebras over arbitrary abelian Lawvere theories]] is the content of

* {#Stel} [[Herman Stel]], _$\infty$-Stacks and their function algebras -- with applications to $\infty$-Lie theory_, master thesis (2010) ([[schreiber:master thesis Stel|web]])

See also

* MathOverflow: [theme-of-isbell-duality](http://mathoverflow.net/questions/84641/theme-of-isbell-duality)

* R.J. Wood, _Some remarks on total categories_, J. Algebra __75_:2, 1982, 538–545 doi

[[!redirects Isbell dualities]]

[[!redirects Isbell conjugation]]
[[!redirects Isbell conjugations]]

[[!redirects Isbell adjunction]]
[[!redirects Isbell adjunctions]]


Isbell duality - table

**[[duality between algebra and geometry|duality between]]
$\;$[[algebra]] and [[geometry]]**

| $\phantom{A}$[[nLab:geometry]]$\phantom{A}$  | $\phantom{A}$[[nLab:category]]$\phantom{A}$ | $\phantom{A}$[[nLab:dual category]]$\phantom{A}$ |  $\phantom{A}$[[nLab:algebra]]$\phantom{A}$ |
|--------------------|--|------------------|---|
| $\phantom{A}$[[nLab:topology]]$\phantom{A}$ | $\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$ | $\phantom{A}$[[nLab:commutative algebra]]$\phantom{A}$ |
| $\phantom{A}$[[nLab:topology]]$\phantom{A}$ | $\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$ | $\phantom{A}$[[nLab:commutative C*-algebra|comm. C-star-algebra]]$\phantom{A}$ |
| $\phantom{A}$[[nLab:noncommutative topology|noncomm. topology]]$\phantom{A}$ | $\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$ | $\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$ | $\phantom{A}$general [[nLab:C-star-algebra]]$\phantom{A}$ |
| $\phantom{A}$[[nLab:algebraic geometry]]$\phantom{A}$ |  $\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$ | $\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} $$\phantom{A}$ |$\phantom{A}$[[nLab:finitely generated algebra|fin. gen.]]$\phantom{A}$
  $\phantom{A}$[[nLab:commutative algebra]]$\phantom{A}$ |
| $\phantom{A}$[[nLab:noncommutative algebraic geometry|noncomm. algebraic]]$\phantom{A}$
 $\phantom{A}$[[nLab:noncommutative algebraic geometry|geometry]]$\phantom{A}$ |  $\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$ | $\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$ | $\phantom{A}$[[nLab:finitely generated algebra|fin. gen.]]
$\phantom{A}$[[nLab:associative algebra]]$\phantom{A}$$\phantom{A}$|
| $\phantom{A}$[[nLab:differential geometry]]$\phantom{A}$ |  $\phantom{A}$$SmoothManifolds$$\phantom{A}$ | $\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$ | $\phantom{A}$[[nLab:commutative algebra]]$\phantom{A}$ |
| $\phantom{A}$[[nLab:supergeometry]]$\phantom{A}$ | $\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$ | $\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}}  \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$ | $\phantom{A}$[[nLab:supercommutative superalgebra|supercommutative]]$\phantom{A}$
$\phantom{A}$[[nLab:superalgebra]]$\phantom{A}$  |
| $\phantom{A}$[[nLab:formal moduli problem|formal]] [[nLab:higher differential geometry|higher]]$\phantom{A}$
 $\phantom{A}$[[nLab:supergeometry]]$\phantom{A}$
 $\phantom{A}$([[nLab:super L-∞ algebra|super Lie theory]])$\phantom{A}$ | $\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$ | $\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A}   }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$ | $\phantom{A}$[[nLab:differential graded-commutative superalgebra|differential graded-commutative]]$\phantom{A}$
 $\phantom{A}$[[nLab:superalgebra]]
 $\phantom{A}$ ("[[nLab:D'Auria-Fre formulation of supergravity|FDAs]]")  |
{: style='margin:auto'}

**in [[physics]]**:

| $\phantom{A}$[[algebra]]$\phantom{A}$ | $\phantom{A}$[[geometry]]$\phantom{A}$ |
|--|--|
| $\phantom{A}$[[Poisson algebra]]$\phantom{A}$ | $\phantom{A}$[[Poisson manifold]]$\phantom{A}$ |
| $\phantom{A}$[[deformation quantization]]$\phantom{A}$ | $\phantom{A}$[[geometric quantization]]$\phantom{A}$ |
| $\phantom{A}$[[algebra of observables]] | $\phantom{A}$[[space of states]]$\phantom{A}$ |
| $\phantom{A}$[[Heisenberg picture]] | $\phantom{A}$[[Schrödinger picture]]$\phantom{A}$ |
| $\phantom{A}$[[AQFT]]$\phantom{A}$ | $\phantom{A}$[[FQFT]]$\phantom{A}$ |
| | |
| $\phantom{A}$**[[higher algebra]]**$\phantom{A}$ | $\phantom{A}$**[[higher geometry]]**$\phantom{A}$ |
| $\phantom{A}$[[Poisson n-algebra]]$\phantom{A}$ | $\phantom{A}$[[n-plectic manifold]]$\phantom{A}$ |
| $\phantom{A}$[[En-algebras]]$\phantom{A}$ | $\phantom{A}$[[higher symplectic geometry]]$\phantom{A}$ |
| $\phantom{A}$[[Beilinson-Drinfeld algebra|BD]]-[[BV quantization]]$\phantom{A}$ | $\phantom{A}$[[higher geometric quantization]]$\phantom{A}$ |
| $\phantom{A}$[[factorization algebra of observables]]$\phantom{A}$ | $\phantom{A}$[[extended quantum field theory]]$\phantom{A}$ |
| $\phantom{A}$[[factorization homology]]$\phantom{A}$ | $\phantom{A}$[[cobordism hypothesis|cobordism representation]]$\phantom{A}$ |
{: style='margin:auto'}


Isbell envelope

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

# The Isbell Envelope of a Category
* table of contents
{: toc}

## Idea

Given a [[category]] $\mathcal{T}$ of "test objects", one can consider "things" which $\mathcal{T}$ can detect.  If one already has a class of [[object]]s in mind, one can consider $\mathcal{T}$-structures on those objects by looking at [[morphism]]s to and from objects of $\mathcal{T}$.  Then one considers questions only in so far as they can be distinguished after mapping into $\mathcal{T}$.  A simple example is that of smooth [[manifold]]s where many questions are solved by transporting the problem to Euclidean spaces via charts.  Another example is the use of weak topologies on locally convex topological spaces.  Without a class of objects in mind, the natural definition of an object probeable by $\mathcal{T}$-objects involves presheaves and copresheaves and is as follows.

+-- {: .num_defn #IsbellEnvelope}
###### Definition
Let $\mathcal{T}$ be an [[small category|essentially small]] category.  Then the _Isbell envelope_ of $\mathcal{T}$, written $E(\mathcal{T})$, is defined as follows.
An object is a triple $X = (P,F,c)$ where

1. $P$ is a [[contravariant functor]] $\mathcal{T} \to Set$ (a [[presheaf]]),
2. $F$ is a [[covariant functor]] $\mathcal{T} \to Set$ (a [[copresheaf]]),
3. $c : P \times F \to \mathcal{T}(-,-)$ is a [[natural transformation]] of bifunctors $\mathcal{T}^{op} \times \mathcal{T} \to $ [[Set]].

(We conventionally write $X = (P_X, F_X, c_X)$.)
A morphism $X \to Y$ is a pair of natural transformations $(\alpha,\beta)$ with $\alpha : P_X \to P_Y$ $\beta : F_Y \to F_X$ which satisfy the relation $c_X(-,\beta -) = c_Y(\alpha -,-)$.

Abstractly, this is (a reformulation of) the [[envelope of an adjunction|envelope]] of the [[Isbell duality]] adjunction associated to $\mathcal{T}$.
=--

The requirement that $\mathcal{T}$ be essentially small implies that the collections of natural transformations form sets, and thus that this is a [[locally small category]].

Certain elementary properties are easy to prove.

+-- {: .num_prop #ElemProp}
###### Proposition
There is a ("double/twosided Yoneda") embedding of $\mathcal{T}$ as a full subcategory of $E(\mathcal{T})$ via

\[
T \mapsto (\mathcal{T}(-,T), \mathcal{T}(T,-), \circ)
\]

Identifying $\mathcal{T}$ with its image, there are [[natural isomorphism]]s

\[
\mathcal{T}^g(T,X) \cong P_X(T), \qquad \mathcal{T}^g(X,T) \cong F_X(T)
\]

=--

_Other elementary properties to follow_

## Profunctors

The Isbell envelope of a category can be viewed as a category of [[profunctors]].
In short, the Isbell envelope of $\mathcal{T}$ consists of the _lax factorisations of $Hom$ through $1$_.

Let us spell this out.
Recall that a profunctor $\mathcal{A} \to \mathcal{B}$ is a functor $\mathcal{A} \times \mathcal{B}^{op} \to Set$.
Both covariant and contravariant functors to $Set$ are special examples of profunctors: a covariant functor $\mathfrak{F} : \mathcal{A} \to Set$ is a profunctor $\mathcal{A} \to 1$, where $1$ is the terminal category, and a contravariant functor $\mathfrak{G} : \mathcal{B} \to Set$ is a profunctor $1 \to \mathcal{B}$.
The composition of these as profunctors produces the obvious  profunctor $\mathfrak{F} \times \mathfrak{G} : \mathcal{A} \times \mathcal{B}^{op} \to Set$.
Thus extracting the functor part of the definition of an object in the Isbell envelope of $\mathcal{T}$ produces a profunctor $\mathcal{T} \to \mathcal{T}$ which factors through $1$.

There is an obvious profunctor $\mathcal{T} \to \mathcal{T}$ given by the $Hom$-bifunctor.
This is the identity for profunctor composition.
The natural transformation from the definition of an object in the Isbell envelope of $\mathcal{T}$ defines a morphism of profunctors from the corresponding profunctor to $Hom$.
Thus an object in the Isbell envelope of $\mathcal{T}$ corresponds to an object in the subcategory of the slice category of profunctors over $Hom$ of those objects which factor through $1$.

In other words, a lax factorisation of $Hom$ through $1$.

This characterization relates directly to a definition of [[Cauchy completion]]. One definition of a point of the Cauchy completion is an adjoint pair of a presheaf and copresheaf, and these define a subcategory of the Isbell envelope where $c_X$ is the counit of an adjunction. This exhibits the Cauchy completion as a subcategory of the Isbell envelope, that factorizes through both the free completion and free cocompletion:

$$\mathcal{T} \hookrightarrow \bar{\mathcal{T}} \hookrightarrow E(\mathcal{T})$$

## Concrete Envelopes

A variant of the above involves a background category, say $\mathcal{U}$.  The test objects should be viewable also as objects of $\mathcal{U}$, usually via a faithful functor.  Any object of $\mathcal{U}$ defines a profunctor $\mathcal{T} \to \mathcal{T}$ (which factors through $1$) via

\[
\begin{aligned}
\mathcal{T} \times \mathcal{T}^{op} &\to \mathcal{U} \times \mathcal{U}^{op} &&\to Set \\
(T_1, T_2) &\mapsto (|T_1|, |T_2|) &&\mapsto \mathcal{U}(U,|T_1|) \times \mathcal{U}(|T_2|,U)
\end{aligned}
\]

Then one can consider those objects of the Isbell envelope of $\mathcal{T}$ that are sub-profunctors of one of this type.
In addition one should restrict the morphisms to those that are induced by morphism of objects of $\mathcal{U}$.
Translating this back to the language of functors yields the following definition.

+-- {: .num_defn #ConcIsbellEnvelope}
###### Definition
Let $\mathcal{T}$ and $\mathcal{U}$ be categories with a [[faithful functor]] $\mathcal{T} \to \mathcal{U}$ which we shall write as $T \mapsto |T|$.  The _$\mathcal{U}$-concrete  Isbell envelope_ of $\mathcal{T}$, which we shall write $E_{\mathcal{U}}(\mathcal{T})$, is the category whose objects are triples $(U,P,F)$ where

1. $U$ is an object of $\mathcal{U}$,
2. $P : \mathcal{T} \to Set$ is a subfunctor of the (contravariant) functor $T \mapsto \mathcal{U}(|T|,U)$,
3. $F : \mathcal{T} \to Set$ is a subfunctor of the (covariant) functor $T \mapsto \mathcal{U}(U,|T|)$,

such that the image of the natural transformation
\[
P \times F \to \mathcal{U}(|-|,|-|),
\]
which comes from composition in $\mathcal{U}$, lies in the image of the natural transformation $\mathcal{T}(-,-) \to \mathcal{U}(|-|,|-|).$

A morphism in $E_{\mathcal{U}}(\mathcal{T})$ is a $\mathcal{U}$-morphism on the underlying $\mathcal{U}$-objects that defines natural transformations $P_X \to P_Y$ and $F_Y \to F_X$.
=--

Having a background category ensures that the size issues with natural transformations do not occur and so we can drop the requirement that $\mathcal{T}$ be essentially small.

## Enrichment

These notions can be considered in the setting of [[enriched category theory]] by replacing [[Set]] wherever it occurs (explicitly or implicitly) by the enriching category.

## Isbell Duality

Within the Isbell envelope of $\mathcal{T}$ one can consider various [[subcategories]] where the objects satisfy extra conditions.  An obvious condition is that the presheaf is actually a sheaf.  Another useful condition is that of [[Isbell duality]].

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

An object $X$ of $E(\mathcal{T})$ is said to be _$P$-saturated_ if the obvious natural transformations

\[
P_X(T) \to NatTrans(F_X,\mathcal{T}(T,-))
\]

are isomorphisms.  It is said to be _$F$-saturated_ if the obvious natural transformations

\[
F_X(T) \to NatTrans(P_X,\mathcal{T}(-,T))
\]

are isomorphisms.  It is said to satisfy _[[Isbell duality]]_ if it is both $P$- and $F$-saturated.
=--

Within $E(\mathcal{T})$ one can consider the [[full subcategories]] of $P$-saturated objects, of $F$-saturated objects, and those satisfying Isbell duality.  Clearly, the last is the intersection of the first two.  There are idempotent functors onto the first two categories given by replacing one of $P_X$ or $F_X$ by the natural transformations of the other.  An interesting question is to ask whether or not the obvious iteration stabilises after a finite number of steps (which would result in an object satisfying Isbell duality).

If the test category has, or "morally has", a [[representable functor]] to $Set$ then there is a strong relationship between saturation and concreteness.

+-- {: .num_defn #ConstSep}
###### Definition
A _constant separator_ in a category is an object, say $S$, with the property that if $f \ne g : A \to B$ then there is a [[constant morphism]] $c : S \to A$ such that $f c \ne g c$.
=--

As the constant morphisms form a two-sided ideal, any object $C_0$ in a category $\mathcal{C}$ defines a covariant functor $|-|_{C_0} : \mathcal{C} \to Set$ which on objects sends $C$ to the set of constant morphisms from $C_0$ to $C$.
A morphism $C_0 \to C_0'$ defines a natural transformation (in the opposite direction) between the corresponding functors.
From the properties of constant morphisms one can easily deduce that two different morphisms between the same objects induce the same natural transformations between the functors.
Thus if there are morphisms between two objects in both directions the two functors are naturally isomorphic.

The property of being a constant separator is clearly equivalent to the condition that this constant functor be faithful.
Moreover, it is easy to show that in a non-trivial category, any two constant separators have morphisms between them in both directions and so induce naturally isomorphic functors.
Indeed, not just naturally isomorphic but naturally naturally isomorphic in that there is a canonical choice of natural isomorphism.

Now we transfer this to the Isbell envelope of $\mathcal{T}$.

+-- {: .num_defn #GenSpaceConst}
###### Definition
Let $X=(P,F,c)$ be an object of $E(\mathcal{T})$.
An element $\alpha \in P(T)$, for $T$ an object of $\mathcal{T}$, is said to be _constant_ if $\phi \circ \alpha \in \mathcal{T}(T,T')$ is constant for all $T'$ objects of $\mathcal{T}$ and $\phi \in F(T')$.

Let us write $|X|_T$ for the set of constant elements in $P(T)$.
=--

+-- {: .num_lemma #GenSpConsFun}
###### Lemma
Let $T_0$ be an object in $\mathcal{T}$.
The assignment $X \to |X|_{T_0}$ is functorial and extends the assignment $T \to |T|_{T_0}$.
A $\mathcal{T}$--morphism $T_0 \to T_1$ defines a natural transformation of functors.
=--

+-- {.proof}
###### Proof
Let $X_i = (P_i,F_i,c_i)$, $i = 1,2$, be two objects in $E(\mathcal{T})$.
Let $(\alpha,\beta)$ be a morphism from the first to the second.
Then $\alpha$ is a natural transformation $P_1 \to P_2$ so in particular defines a morphism of sets $P_1(T_0) \to P_2(T_0)$.
Let $\gamma \in P_1(U_0)$ be a constant element.
Let $T$ be an object in $\mathcal{T}$ and $\phi \in F_2(T)$.
We need to show that $\phi \circ (\alpha \circ \gamma)$ is a constant morphism.
By the definition of a morphism in $E(\mathcal{T})$, $\phi \circ (\alpha \circ \gamma) = (\phi \circ \beta) \circ \gamma$ and this latter is a constant morphism since $\gamma \in P_1(T_0)$ is a constant element.

For the extension, we observe that for $\gamma \in P_T(T_0)$ then if $\gamma$ is a constant morphism, $c_T(\phi,\gamma) = \phi \circ \gamma$ is constant for all $\phi$; whilst if $c_T(\phi,\gamma)$ is constant for all $\phi$ then in particular $\gamma = c_T(1_T,\gamma)$ is constant.
That the morphisms correspond is obvious.

A $\mathcal{T}$--morphism $\psi : T_0 \to T_1$ defines a map $P(T_1) \to P(T_0)$ as $P$ is a contravariant functor.
Let $\gamma \in P(T_1)$ be a constant element.
Then for $T'$ an object in $\mathcal{T}$ and $\phi \in F(U')$, $\phi \circ (\gamma \circ \psi) = (\phi \circ \gamma) \circ \psi$ and $\phi \circ \gamma$ is a constant morphism in $\mathcal{T}(T_1,T')$ so $\phi \circ (\gamma \circ \psi)$ is a constant morphism in $\mathcal{T}(T_0,T')$.
Hence $\gamma \circ \psi$ is a constant element in $P(T_0)$.
=--

The result that the natural transformations depend only on the existence of a morphism does not carry over to this extended setting.

+-- {: .query}
[[Andrew Stacey|Andrew]]: Useful to have an example here, of course.
=--

Let us fix a $\mathcal{T}$--object $T_0$.
We have two bifunctors $\mathcal{T} \times E(\mathcal{T}) \to Set$ given by
$$
(T,X) \mapsto P_X(T)
$$
and
$$
(T,X) \mapsto Set(|T|_{T_0},|X|_{T_0})
$$
Let us, to simplify notation, identify $T$ with its image in $E(\mathcal{T})$.
Then $P_X(T)$ is naturally isomorphic to $E(\mathcal{T})(T,X)$ so the fact that $X \mapsto |X|_{T_0}$ is a functor defines a natural transformation from the first to the second via
$$
P_X(T) \cong E(\mathcal{T})(T,X) \to Set(|T|_{T_0}, |X|_{T_0})
$$
In a similar fashion, we obtain a natural transformation
$$
F_X(T) \cong E(\mathcal{T})(X,T) \to Set(|X|_{T_0},|T|_{T_0}).
$$
We therefore obtain a functorial choice of underlying concrete object (with the underlying category being $Set$).
We can refine the notion of concreteness slightly.

+-- {: .num_defn #ConcGenSp}
###### Definition
Let $\mathcal{T}$ be an essentially small category with a concrete separator, say $T_0$.
Let $X = (P_X,F_X,c_X)$ be an object of $E(\mathcal{T})$.
We say that $X$ is _$P$--concrete_ if the map of sets
$$
P_X(T) \to Set(|T|_{T_0},|X|_{T_0})
$$
is injective for all $\mathcal{T}$--objects, $T$.

Similarly, we say that $X$ is _$F$--concrete_ if the map of sets
$$
F_X(T) \to Set(|X|_{T_0}, |T|_{T_0})
$$
is injective for all $\mathcal{T}$--objects, $T$.
=--

Clearly, if it is both $P$-concrete and $F$-concrete then it is concrete.  Changing the concrete separator does not alter concreteness.

+-- {: .num_theorem #ConcreteIsbell}
###### Theorem
Let $\mathcal{T}$ be an essentially small category admitting a constant separator.
Then for an object $X$ of $E(\mathcal{T})$, the following hold:

1. If $X$ is $P$-saturated then it is $P$-concrete,
2. If $X$ is $F$-saturated then it is $F$-concrete,
3. If $X$ satisfies Isbell duality then it is concrete.
=--

+-- {: .proof}
###### Proof
Clearly the first and second statements imply the third.

Let us consider the first.

Let $X$ be an object of $E(\mathcal{T})$ such that $P_X(T) = NatTrans(F_X,F_T)$ for all $\mathcal{T}$--objects $T$.
Let $S$ be a concrete separator in $\mathcal{T}$.  We shall write $|-|$ for $|-|_S$.
Let $T$ be an object of $\mathcal{T}$.
Let $\alpha \ne \beta \in P_X(T)$.
As $P_X(T) = NatTrans(F_X,F_T)$, their inequality means that they differ as natural transformations.
Thus there is some $T'$ for which $\alpha$ and $\beta$ induce different morphisms $F_X(T') \to F_T(T')$.
Thus there is some $\phi \in F_X(T')$ such that $\alpha_{T'}(\phi) \ne \beta_{T'}(\phi) \in \mathcal{T}(T,T')$.

As $S$ is a constant separator, there is thus a constant morphism $\delta : S \to T$ such that $\alpha_{T'}(\phi) \circ \delta \ne \beta_{T'}(\phi) \circ \delta$.
Using composition notation, we rewrite this as $\phi \circ \alpha \circ \delta \ne \phi \circ \beta \circ \delta$.
As $\delta$ is a constant morphism, $\alpha \circ \delta$ and $\beta \circ \delta$ are constant elements of $P_X(S)$.
Since $\phi \circ \alpha \circ \delta \ne \phi \circ \beta \circ \delta$, they must be different constant elements.
Thus the maps $\alpha,\beta : |T| \to |X|$ are different and so $X$ is $P$--concrete.

The second statement is very similar.

Let $X$ be an object of $E(\mathcal{T})$ such that $F_X(U) = NatTrans(P_X,P_T)$ for all $\mathcal{T}$--objects $T$.
Let $S$ be a concrete separator in $\mathcal{T}$.
Let $T$ be an object of $\mathcal{T}$.
Let $\phi \ne \psi \in F_X(T)$.
As $F_X(T) = NatTrans(P_X,P_T)$, their inequality means that they differ as natural transformations.
Thus there is some $T'$ for which $\phi$ and $\psi$ induce different morphisms $P_X(T') \to P_T(T')$.
Thus there is some $\alpha \in P_X(T')$ such that $\phi_{T'}(\alpha) \ne \psi_{T'}(\alpha) \in \mathcal{T}(T',T)$.

As $S$ is a constant separator, there is thus a constant morphism $\delta : S \to T'$ such that $\phi_{T'}(\alpha) \circ \delta \ne \psi_{T'}(\alpha) \circ \delta$.
Using the composition notation, we rewrite this as $\phi \circ \alpha \circ \delta \ne \psi \circ \alpha \circ \delta$.
As $\delta$ is a constant morphism, $\alpha \circ \delta \in P_X(S)$ is a constant element.
We therefore have an element of $|X|$ which distinguishes between the induced maps from $\phi$ and $\psi$.
Hence $X$ is $F$--concrete.
=--

### Examples

When $\mathcal{T}$ is a [[poset]], the category of saturated objects in its Isbell envelope coincides with its [[MacNeille completion]].

## Pseudomonadicity and factorisation systems

The assignment of a category to its Isbell envelope extends to a pseudomonad on [[CAT]], whose algebras are categories equipped with [[cylinder factorisation systems]]. See [Garner](#Garner).

## References

The original reference, in which the Isbell envelope was called the **couple category**:

* [[John Isbell]], _Structure of categories_, Bulletin of the American Mathematical Society 72 (1966), 619– 655.

* [[John Isbell]], _Normal completions of categories_, Reports of the Midwest Category Seminar, vol. 47, Springer, 1967, 110–155.

* {#Garner} [[Richard Garner]], _The Isbell monad_, Advances in Mathematics **274** (2015) 516–537.  [arxiv:1410.7108](https://arxiv.org/abs/1410.7108)

* [[Vaughan Pratt]], _Communes via Yoneda, from an elementary perspective_, Fundamenta Informaticae 103 (2010), 203–218.

For more see the [references](Isbell+duality#references) at _[[Isbell duality]]_.

[[!redirects Isbell envelope]]
[[!redirects Isbell envelopes]]
[[!redirects generalized object]]
[[!redirects generalized objects]]


Ising model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functorial Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Ising model_ is a simple [[lattice model|lattice]] [[model (in theoretical physics)]] of [[physical systems]] roughly similar to ferromagnets. Its configurations are functions on a lattice with values in $\{-1,+1\}$, roughly to be thought of as the magnetic polarizations of elementary magnets in a crystal lattice.

The Ising model gained importance as a toy model in theoretical [[physics]]. It is about the simplest possible model that allows methods of [[Euclidean field theory|Euclidean]] [[quantum field theory]] in [[statistical physics]] and the study of [[critical phenomena]]. In fact at a critical [[temperature]] and in [[dimension]] 2 the model exibits the behaviour of a [[2d CFT|2d]][[conformal field theory]], one of the examples of [[rational conformal field theory]].

Real-world [[physical systems]] that show behaviour described by the Ising model were only found later ([Wolf](#Wolf)).

## Properties

* [[2d CFT]]

* [[Kramers-Wannier duality]]

## Related concepts

* [[Euclidean field theory]]

* [[Potts model]]

* [[Gepner model]]

## References

On historical work by [[Yoichiro Nambu]]:

* [[Lars Brink]], [[Pierre Ramond]], *Nambu and the Ising Model* [[arXiv:2209.01122](https://arxiv.org/abs/2209.01122)]

Original articles:

* {#Wegner71} [[Franz Wegner]], _Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters_, Journal of Mathematical Physics 12, 2259 (1971) ([doi:10.1063/1.1665530](https://doi.org/10.1063/1.1665530))

  > (introducing the idea of [[lattice QCD]])

(...)

Review:

* {#PeetersZamaklar09} [[Kasper Peeters]], [[Marija Zamaklar]], section 2.1 of _Euclidean Field Theory_, Lecture notes 2009-2011 ([web](http://maths.dur.ac.uk/users/kasper.peeters/eft.html), [pdf](http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf))

* {#Wolf} W. P. Wolf, _The Ising model and real magnetic materials_, Braz. J. Phys. vol.30 no.4 São Paulo Dec. 2000 ([web](http://www.scielo.br/scielo.php?script=sci_arttext&pid=s0103-97332000000400030))

See also

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

Discussion of the [[Ising model]] [[2d CFT]] as a boundary theory to a 3d [[TQFT]] based on the [[Turaev-Viro model]], similar to the [[CS-WZW correspondence]], and the phenomenon of [[Kramers-Wannier duality]], is in:

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _Kramers-Wannier duality from conformal defects_ ([arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051))

(via [[FRS-formalism]]) and also in

* {#FreedTeleman18} [[Daniel Freed]], [[Constantin Teleman]], _Topological dualities in the Ising model_ ([arXiv:1806.00008](https://arxiv.org/abs/1806.00008))

3d version:

* {#Behan18} Connor Behan, _Bootstrapping the long-range Ising model in three dimensions_,  Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 7 ([arXiv:1810.07199](https://arxiv.org/abs/1810.07199), [doi:10.1088/1751-8121/aafd1b](https://iopscience.iop.org/article/10.1088/1751-8121/aafd1b))

[[!redirects Ising models]]


Isle of Thorns

The *Isle of Thorns* was a conference centre owned and run by the University of Sussex.  It was situated in the Ashdown Forest not far from Gatwick Airport. During the 1970s it was the venue for regular informal meetings on [[category theory]] organised by [[Gavin Wraith]] and [[Chris Mulvey]].

As a very non-categorical aside, if anyone wants to get an impression of the countryside there, they should look at the pictures of E. H. Shepard in the original editions of Winnie-the-Pooh as those are based on parts of the Ashdown Forest only a short distance away from the Isle of Thorns. A lovely place and a lot of good category theory went on there!

(Ironically, or appropriately, when the university sold the centre in 2002 it was bought by the Cats Protection League!)

## Links

- [Abstracts of the 1976 meeting](https://web.archive.org/web/20170210175423/https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/Isle-of-Thorns-1976.pdf)

category: reference