inverse functor

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### Context
#### Category theory
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[[!include category theory - contents]]
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## Idea

The concept of _inverse functor_ is the generalization of that of _[[inverse function]]_ from [[sets]] to [[categories]]. Where the existence of an inverse function exhibits a [[bijection]] of sets, the existence of an inverse functor exhibits an [[equivalence of categories]].

## Definition

Given a [[functor]]

$$
  F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
$$

then an _inverse_ to $F$ is a [[functor]] going the other way around

$$
  G \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}
$$

together with [[natural isomorphisms]] relating their [[composition|composites]] to the respective [[identity functors]]:

$$
  G\circ F \simeq id_{\mathcal{C}}
  \phantom{AAAA}
  F \circ G  \simeq id_{\mathcal{D}}
  \,.
$$

## Related concepts

* [[inverse]], [[inverse morphisms]]

[[!redirects inverse functors]]


inverse Galois problem

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## Idea

In [classical Galois theory for fields](http://ncatlab.org/nlab/show/Galois+theory#ClassicalGaloisTheory) the **inverse Galois problem** is the question whether or not every [[finite group]] appears as the [[Galois group]] of some [[Galois extension]] of the [[rational number|rational numbers]].

This problem is ''inverse'' to the problem of computing the Galois group of some [[algebraic equation]]. It was first posed in the 19th century and is still unsolved. However there are several results dealing with special and related cases.

## References

* Helmut Völklein, Groups as Galois Groups, an Introduction, Cambridge University Press, 1996.

* ''inverse Galois problem'' in wikipedia, [web](http://en.wikipedia.org/wiki/Inverse_Galois_problem)


inverse image

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### Context
#### Topos Theory
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> This page is about **inverse images of sheaves** and related subjects.  For the set-theoretic operation, see [[preimage]].

***

# Inverse images
* tic
{: toc}

## Idea

An _inverse image_ operation is the [[left adjoint]] part $f^*$ of a [[geometric morphism]] $(f^* \dashv f_*) \colon E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} F$ between [[topoi]].

For the case of [[sheaf topoi]] the inverse image

$$
  f^{-1} : Sh(Y) \to Sh(X)
$$

may be interpreted as encoding the idea of *[[pullback bundle|pulling back]]* along $f$ the "[[bundle]] of which the sheaf is the sheaf of sections".

In the case that $X$ and $Y$ are (the [[sites]] corresponding to) [[topological spaces]], this interpretation becomes literally true: the inverse image of a sheaf on topological spaces is the pullback operation on the corresponding [[etale spaces]].

## Definition

Consider a [[functor]] $f^t \colon Y \to X$ between the [[underlying]] categories of [[sites]].

### on presheaves

The [[direct image]] operation $f_* \colon PSh(X) \to PSh(Y)$ on [[presheaves]] is just precomposition with $f^t$

$$
  \array{
    Y^{op}
    &\overset{f_* F}{\longrightarrow}&
    Set
    \\
    \mathllap{
     {}^{f^t}
    }\big\downarrow & \nearrow\mathrlap{_{F}}
    \\
    X^{op}
  }
  \,.
$$

The **inverse image** operation

$$
  f^{-1} \colon PSh(Y) \to PSh(X)
$$

on [[presheaves]] is the [[left adjoint]] to the [[direct image]] operation on presheaves, hence the left [[Kan extension]] along $f^t$:

$$
  f^{-1} F \;\coloneqq\;  Lan_{f^t} F
  \,.
$$

.

### on sheaves

The inverse image operation on the [[category of sheaves]] $Sh(Y) \subset PSh(Y)$ inside the category of presheaves involves [[Kan extension]] followed by [[sheafification]].

First notice that

+-- {: .un_lemma}
###### Lemma

The [[direct image]] operation $f_* \colon PSh(X) \to PSh(Y)$ restricts to a functor $f_* \colon Sh(X) \to Sh(Y)$ that sends sheaves to sheaves.

=--

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

The direct image $f_* \colon PSh(X) \to PSh(Y)$ is more generally characterized by
$$
  Hom_{PSh(Y)}(A, f_* F)
  \simeq
  Hom_{PSh(X)}(\hat {f^t} A, F)
$$
where $\hat f^t$ is the [[Yoneda extension]] of $Y \circ f^t \colon Y \to PSh(X)$ to a functor $\hat {f^t} : PSh(Y) \to PSh(X)$, because using the [[co-Yoneda lemma]] and the colim expression for the [[Yoneda extension]] we have

$$
 \begin{aligned}
   Hom(A, f_* F) & \simeq
   Hom(colim_{Y(U) \to A}) U, f_* F)
   \\
   & \simeq
    \lim_{Y(U) \to A} Hom(U, f_* F)
   \\
   & \simeq
    \lim_{Y(U) \to A} F(f^t(U))
   \\
   & \simeq
    Hom( colim_{Y(U) \to A} f^t(U), F )
   \\
   & \simeq Hom(\hat {f^t}(A), F)
   \,.
 \end{aligned}
$$

Let now $\pi \colon B \to A$ be a [[local isomorphism]] in $PSh(Y)$.  By definition of morphism of [[sites]] we have that
$$
  \hat {f^t}(\pi) : \hat{f^t}(B) \to \hat{f^t}(A)
$$
is a [[local isomorphism]] in $X$. From this and the above we obtain the isomorphism

$$
  Hom(B, f_* F)
  \simeq
  Hom(\hat {f^t}(B), F)
  \stackrel{\simeq}{\to}
  Hom(\hat {f^t}(A), F)
  \simeq
  Hom(A, f_* F)
  \,,
$$
where the isomorphism in the middle is due to the fact that
$F$ is a sheaf on $X$. Since this holds for all local isomorphism $\pi : B \to A$ in $PSh(Y)$, $f_* F$ is a sheaf on $Y$.

=--

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

For $f : X \to Y$ a morphism of [[site]]s, the
**inverse image of sheaves** is the functor
$$
  f^{-1} : Sh(Y) \to Sh(X)
$$
defined as the inverse image on presheaves followed
by [[sheafification]]
$$
  f^{-1} : Sh(Y) \hookrightarrow PSh(Y) \stackrel{Lan_{f^t}}{\to} PSh(X) \stackrel{\bar{-}}{\to}
  Sh(X)
  \,.
$$

=--

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

The inverse image $f^{-1} : Sh(Y) \to Sh(X)$
of sheaves has the following properties:

* it is [[left adjoint]] to the [[direct image]]
  $(f^{-1} \dashv f_*)$;

* it therefore commutes with small [[colimit]]s but is in addition left [[exact functor|exact]] in that it commutes with finite [[limit]]s.

=--

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

The left-adjointness is obtained by the following computation, for any
two $F \in Sh(X)$ and $G \in Sh(Y)$ and using the above facts as well as the fact that [[sheafification]] $\bar {(-)} : PSh(X) \to Sh(X)$ is [[left adjoint]] to the inclusion $Sh(X) \hookrightarrow PSh(X)$:

$$
  \begin{aligned}
    Hom_{Sh(Y)}(G, f_*F)
     & \simeq
    Hom_{PSh(Y)}(G, f_* F)
     \\
     & \simeq
       Hom_{PSh(X)}(Lan_{f^t} G, F)
     \\
     & \simeq
      Hom_{Sh(X)}( \bar{(Lan_{f^t} G)}, F)
     \\
     & =:
      Hom_{Sh(X)}(f^{-1}G, F)
  \end{aligned}
  \,.
$$

The proof of left-exactness requires more technology and work.

=--

### on sheaves on topological spaces

In the case where the [[site]]s $X$ and $Y$ in question are given by [[category of open subsets|categories of open subsets]] of [[topological space]]s denoted, by an abuse of symbols, also by $X$ and $Y$, one can identify sheaves with their corresponding [[etale space]]s over $X$ and $Y$. In that case the inverse image is simply obtained by the pullback along the continuous map $f : X \to Y$ of the corresponding [[etale space]]s.

## Properties

* See also [[restriction and extension of sheaves]].

* It follows that direct image and inverse image of
sheaves define a [[geometric morphism]]
$f : Sh(X) \to Sh(Y)$
of [[sheaf]] [[topos|topoi]]

* Generally, therefore, the left adjoint partner in
the adjoint pair defining a [[geometric morphism]]
of [[topos|topoi]] (or abelian categories of quasicoherent sheaves) is
called the **inverse image functor**. In fact more general in geometry,
including [[noncommutative geometry|noncommutative]] morphisms often induce or are defined via pairs of adjoint functors among some associated categories of objects over a geometric space; then the left adjoint part is called the inverse image part. Geometers also often say inverse image for an arbitrary functor of the form $f^*$ in a [[fibered category]]. For abelian categories of sheaf-like objects, the corresponding higher derived functors of inverse image functors are sometimes called higher (derived) inverse image functors.

* The other adjoint to the [[direct image]], the [[adjoint functor|right adjoint]], is (if it exists) the [[restriction and extension of sheaves|extension]] of sheaves.

## Examples

The standard example is that where $X$ and $Y$ are [[topological space]]s and $S_X = Op(X)$, $S_Y = Op(Y)$ are their [[category of open subsets|categories of open subsets.]]

A [[continuous map]] $f : X \to Y$ induces the obvious functor $f^{-1} : Op(Y) \to Op(X)$, since [[preimages]] of open subsets under continuous maps are open.

Hence presheaves canonically push forward

$$
  f_* : PSh(X) \to PSh(Y)
$$

They do not in the same simple way pull back, since images of open subsets need not be open. The Kan extension  computes the best possible approximation:

The inverse image $(f^{-1})^\dagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to

$$
  f^\dagger F(U) = \colim_{U \subseteq f^{-1}(V)} F(V)
  \,.
$$

What's this really doing is using $V$ to approximate $f(U)$ from the outside. Thus a section of $f^\dagger F$ on $U$ will be an equivalence class of sections of $F$ on open neighbourhoods of $f(U)$, under the equivalence given by agreement on a restriction to a smaller neighbourhood:
$$
  (s, V_1) \sim (t, V_2) \quad\underline{iff}\quad \exists W \in Op(Y), V_1 \cap V_2 \supseteq W \supseteq f(U), \; s\vert_W = t\vert_W.
$$
Compare this with the definition of [[germ|germs]] at a [[stalk]].

On the other hand, the extension $(f^{-1})^\ddagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to

$$
  f^\ddagger F(U) = \lim_{f^{-1}(V) \subseteq U} F(V)
  \,.
$$

This approximates the possibly non-open subset $f(U)$ by all open subsets $V$  _contained_ in it.
This corresponds to taking the [[Kan extension|right Kan extension]] instead of the left one, and when it exists it's called [[restriction and extension of sheaves|extension of presheaves]].

## Related concepts

* [[derived inverse image]]

* [[exceptional inverse image]]

## References

For the general description in terms of Kan extension and sheafification see section 17.5 of

* Kashiwara-Schapira, [[Categories and Sheaves]]

For the description in terms of pullback of étale spaces, see section VII.1 of

* MacLane-Moerdijk, [[Sheaves in Geometry and Logic]]

[[!redirects inverse image]]
[[!redirects inverse images]]
[[!redirects inverse image functor]]


inverse limit

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### Context
#### Category theory
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#### Limits and colimits
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An __inverse limit__ is the same thing as a [[limit]].  (Similarly, a [[direct limit]] is the same thing as a [[colimit]].)  In this context, an __inverse system__ is the same thing as a [[diagram]], and an __inverse cone__ is the same thing as a [[cone]].

Many authors restrict this terminology to limits over [[codirected sets]] (or [[cofiltered categories]]); see __[[codirected limit]]__ (or __[[cofiltered limit]]__) for discussion of this case if you think that it may be what you want, or see __[[limit]]__ for the more general notion. Especially common is the codirected set $(\mathbb{N},\geq)$ of [[natural numbers]], in which case the inverse system may be called an __inverse sequence__.

## Related concepts

* [[projective limit]], another terminology for limit

[[!redirects inverse limit]]
[[!redirects inverse limits]]

[[!redirects inverse system]]
[[!redirects inverse systems]]

[[!redirects inverse sequence]]
[[!redirects inverse sequences]]

[[!redirects inverse cone]]
[[!redirects inverse cones]]


inverse matrix

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### Context
#### Linear algebra
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## Properties

+-- {: .num_prop #FundamentalTheoremOfLinearAlgebra}
###### Proposition
**(fundamental theorem of invertible matrices)**

For $n \in \mathbb{N}$ and $A \in Mat_{n \times n}(k)$
a [[square matrix]], the following are equivalent:

1. $A$ is an invertible matrix;

1. $A$ is the [[matrix product]] of [[elementary matrices]].

=--

## Related concepts

* [[rank of a matrix]]

* [[matrix group]]

* [[general linear group]]

* [[matrix equivalence]]

## References

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

* WolframMathWorld, _[Invertible Matrix Theorem](http://mathworld.wolfram.com/InvertibleMatrixTheorem.html)_

[[!redirects inverse matrices]]

[[!redirects invertible matrix]]
[[!redirects invertible matrices]]

[[!redirects matrix inverse]]

[[!redirects fundamental theorem of invertible matrices]]


inverse semigroup

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###Context###
#### Algebra
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> This entry is about [[semigroups]] with a unary operator $i$ such that $s \cdot i(s) \cdot s = s$ and $i(s) \cdot s \cdot i(s) = i(s)$. For the semigroup with two-sided [[inverses]], see instead at *[[invertible semigroup]]*.

***

# Contents #
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{:toc}

## Definition

An **inverse semigroup** is a [[semigroup]] $S$ (ie. a [[set]] equipped with an [[associativity|associative]] [[binary operation]]) such that for every [[element]] $s\in S$, there exists a *unique* "[[inverse]]" $s^*\in S$ such that $s s^* s = s$ and $s^* s s^* = s^*$. It is evident from this that $s^{\ast\ast} = s$.

## Examples

Needless to say, a [[group]] is an inverse semigroup. More to the point however:

* The fundamental example is the following: for any set $X$, let $I(X)$ be the set of all partial [[bijections]] on $X$, i.e. bijections between subsets of $X$.  The composite of partial bijections is their composite as [[relations]] (or as [[partial functions]]).  In fact, any inverse semigroup is isomorphic to a sub-inverse-semigroup of $I(X)$ by the "Cayley's theorem" for inverse semigroups (see below).

This inverse semigroup plays a role in the theory similar to that of [[permutation]] groups in the theory of [[groups]].  It is also paradigmatic of the general philosophy that

> Groups describe global symmetries, while inverse semigroups describe local symmetries.

Other examples include:

* If $X$ is a topological space, let $\Gamma(X)\subseteq I(X)$ consist of the homeomorphisms between open subsets of $X$.  Then $\Gamma(X)$ is a [[pseudogroup of transformations]] on $X$ (a general pseudogroup of transformations is a sub-inverse-semigroup of $\Gamma(X)$).

* If $L$ is a [[meet-semilattice]], then $L$ is an inverse semigroup under the meet operation.

## Properties

+-- {: .query}
Lots to say here: the meet-[[semilattice]] of [[idempotents]], the connection with [[ordered groupoid]]s, various representation theorems.
=--

### Idempotents form a subsemigroup

+-- {: .num_prop #triv}
###### Proposition
For any $x$ in an inverse semigroup, $x x^\ast$ and $x^\ast x$ are idempotent. If $e$ is idempotent, then $e^\ast = e$.
=--

The proof is trivial.

+-- {: .num_lemma #comm}
###### Lemma
In an inverse semigroup, the product of any two idempotents $e, f$ is idempotent, and any two idempotents commute.
=--

+-- {: .proof}
###### Proof
One easily checks that $(e f)^\ast = f(e f)^\ast e$, and that $f(e f)^\ast e$ is an idempotent. So $(e f)^\ast$ is idempotent; as a result, $(e f)^\ast = e f$. Thus $e f$ and similarly $f e$ are idempotent. Next we have

$$e f (f e) e f = e f^2 e^2 f = e f e f = e f, \qquad f e (e f) f e = f e^2 f^2 e = f e f e = f e$$

since $e, f, e f, f e$ are all idempotent, and so $f e = (e f)^\ast = e f$, which completes the proof.
=--

Thus the idempotents in an inverse semigroup form a subsemigroup which is commutative and idempotent. Such a structure is the same as a meet-semilattice except for the fact that there might not have an empty meet or top element; that is, we define an order $\leq$ on idempotents by $e \leq f$ if and only if $e = e f$, whence multiplication of idempotents becomes the binary meet.

### $\ast$ is an anti-involution

+-- {: .num_lemma #invol}
###### Lemma
For any two elements $x, y$ in an inverse semigroup, $(x y)^\ast = y^\ast x^\ast$.
=--

+-- {: .proof}
###### Proof
Since the idempotents $x^\ast x, y y^\ast$ commute, we have

$$x y (y^\ast x^\ast) x y = x (y y^\ast)(x^\ast x) y = x x^\ast x y y^\ast y = x y$$

and similarly $y^\ast x^\ast (x y)y^\ast x^\ast = y^\ast y y^\ast x^\ast x x^\ast = y^\ast x^\ast$, which is all we need.
=--

### Order structure

+-- {: .num_prop #ord}
###### Proposition
For elements $x, y$ in an inverse semigroup, the following are equivalent:

1. There exists an idempotent $e$ such that $x = e y$,
1. $x = x x^\ast y$,
1. There exists an idempotent $f$ such that $x = y f$,
1. $x = y x^\ast x$.
=--

+-- {: .proof}
###### Proof
We show $3. \Rightarrow 2.$; a similar proof shows $1. \Rightarrow 4.$ Clearly then we have $3. \Rightarrow 2. \Rightarrow 1. \Rightarrow 4. \Rightarrow 3.$

Given an idempotent $f$ such that $x = y f$, we have

$$\array{
x & = & y f & \\
 & = & y y^\ast y f & \\
 & = & y f y^\ast y & since \; idempotents \; commute \\
 & = & y f f y^\ast y & \\
 & = & y f f^\ast y^\ast y & \text{Proposition 1} \\
 & = & y f (y f)^\ast y & \text{Lemma 2} \\
 & = & x x^\ast y &
}$$

which gives $3. \Rightarrow 2.$
=--

A [[preorder]] $\leq$ is defined on an inverse semigroup by saying $x \leq y$ if any of the four conditions of Proposition \ref{ord} is satisfied; transitivity follows by equivalence to 1. and closure of idempotents under multiplication. When restricted to idempotents, this preorder coincides with the meet-semilattice order.

+-- {: .num_prop #order}
###### Proposition
If $a \leq b$ and $x \leq y$ in an inverse semigroup, then $a x \leq b y$ and $x^\ast \leq y^\ast$.
=--

+-- {: .proof}
###### Proof
Writing $a = e b$ for some idempotent $e$, we have $a x = e (b x)$ and so $a x \leq b x$. Similarly $b x \leq b y$, so $a x \leq b y$ by transitivity.
This gives $a x \leq b y$. If $x = e y$ for an idempotent $e$, then $x^\ast = (e y)^\ast = y^\ast e^\ast$; this gives $x^\ast \leq y^\ast$,
=--

+-- {: .num_cor}
###### Corollary
The preorder $\leq$ on an inverse semigroup is a [[partial order]], i.e., if $x \leq y$ and $y \leq x$, then $x = y$.
=--

+-- {: .proof}
###### Proof
From $x \leq y$ we derive $x^\ast \leq y^\ast$ and $x x^\ast \leq y y^\ast$, and similarly from $y \leq x$ we derive $y y^\ast \leq x x^\ast$. Thus $x x^\ast = y y^\ast$ since the preorder on idempotents is a meet-semilattice, which is a partial order. Then from $x \leq y$ we derive $x = x x^\ast y = y y^\ast y = y$.
=--

Thus an inverse semigroup is naturally regarded as an internal semigroup in the category of posets (equivalently, a finite-product preserving functor from the [[Lawvere theory]] of semigroups to [[Pos]]).

### Connection with ordered groupoids

In this section, an *ordered groupoid* means an [[internal category|internal]] [[groupoid]] in the [[finitely complete category]] of posets [[Pos]]. For any finitely complete category $C$, we observe that the forgetful functor $Gpd(C) \to SemiCat(C)$, taking an internal groupoid in $C$ to the underlying [[semicategory]] (remembering only composition of morphisms, forgetting presence of inverses and identity morphisms), has a right adjoint which takes a semicategory to the [[core]] groupoid of the category of idempotents attached to a semicategory (see [here](/nlab/show/semicategory#idempotents) for details). (This observation is formulated in [[internal logic|finite limit logic]], and thus by a Yoneda lemma argument, its validity reduces to that of the observation in the special case $C = Set$.)

In particular, this construction may be applied to an inverse semigroup seen as a semigroup in $Pos$:

+-- {: .num_defn}
###### Definition
The groupoid $Ind(S)$ attached to an inverse semigroup $S$ is the core of the category of idempotents $Idem(S)$ of $S$, which as a semigroup in $Pos$ is viewed as a one-object semicategory $B S$ in $Pos$.
=--

In more detail: an arrow $e \to e'$ in $Idem(S)$ is a triple $(e, x, e')$ of elements in $S$, where $e, e'$ are idempotent elements and $x$ is an element such that $x e = x = e' x$. Such an arrow is invertible precisely when $e = x^\ast x$ and $e' = x x^\ast$, with inverse $x^\ast$. Thus the core consists of such arrows $x: x^\ast x \to x x^\ast$.

A key example to keep in mind is the inverse semigroup of partial bijections $\phi$ on a set, where the arrows of the corresponding groupoid are actual invertible maps $\dom(\phi) \to range(\phi)$ between subsets. In general, the object part of the associated groupoid is not just a poset, but a poset with binary meets.

The reason for the notation $Ind(S)$ is that this ordered groupoid is a so-called *inductive groupoid*, defined as follows:

+-- {: .num_defn}
###### Definition
An **inductive groupoid** is an internal groupoid $G$ in $Pos$ with the following additional properties:

* The object part $G_0$ admits binary meets;

* Given $x: e \to f$ in $G_1$ and $e' \leq e$ in $G_0$, there exists a unique $x': e' \to f$ in $G_1$ with $x' \leq x$, called the *restriction* $[x|_\ast e']$.

* Given $x: e \to f$ in $G_1$ and $f \geq f'$ in $G_0$, there exists a unique $x': e \to f'$ in $G_1$ with $x \geq x'$, called the *corestriction* $[f' _\ast| x]$.
=--

In fact conditions 2. and 3. in this definition are equivalent. A *morphism* of inductive groupoids $G \to G'$ is an internal functor from $G$ to $G'$ in $Pos$.

For $G$ an inductive groupoid, a tensor product $\otimes: G_1 \times G_1 \to G_1$ may be defined by the rule

$$x \otimes y = [x|_\ast dom(x) \wedge cod(y)] \cdot [dom(x) \wedge cod(y) _\ast|y]$$

where $\cdot$ indicates composition in $G$. It may be shown that $(G_1, \otimes)$ is an inverse semigroup $Inv(G)$, and the two notions are equivalent:

+-- {: .num_theorem}
###### Theorem
**(Ehresmann-Schein-Nampooripad)**
There are canonical isomorphisms $S \to Inv(Ind(S))$ and $G \to Ind(Inv(G))$, providing an equivalence of categories $InvSemiGrp \simeq IndGpd$.
=--

### Warning on the definition

With only a subtle change in definition, the result is that one gets only groups:

+-- {: .num_prop}
###### Proposition
Let $S$ be an [[inhabited set|inhabited]] semigroup with the property that for every $a \in S$ there exists a unique $x \in S$ such that $a x a = a$. Then $S$ is a group.
=--

+-- {: .proof}
###### Proof
Since $S$ is inhabited, say by an element $b$, it has an idempotent $e$, for example $b b^\ast$. We will show that $x e = x$ for any $x$; by a similar argument $e x = x$, so that any idempotent $e$ is an identity (*the* identity $1$), whence the idempotents $a a^\ast$ and $a^\ast a$ equal $1$ for any $a$ and $S$ is a group.

If $a y a = a$ for unique $y$, then from $(a y a) y a = a y a = a$ it follows $y a y = y$ and hence $S$ is an inverse semigroup. The same observation means it is enough to show $(x e) x^\ast (x e) = x e$, since then also $x^\ast (x e) x^\ast = x^\ast$, which by uniqueness implies $x e = x$.

The above results on inverse semigroups apply and we derive

$$\array{
x e x^\ast x e & = & x e e x^\ast x e & \\
 & = & x e e^\ast x^\ast x e & Proposition \; 1 \\
 & = & x e (x e)^\ast x e & Lemma 2 \\
 & = & x e
}$$

as was to be shown.
=--

### [Wagner-Preston Theorem](https://en.wikipedia.org/wiki/Inverse_semigroup#Homomorphisms_and_representations_of_inverse_semigroups): "Cayley's Theorem" for inverse semigroups

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

Every inverse semigroup $S$ can be realized as a semigroup of partial bijections on a set.
=--

+-- {: .proof}
###### Proof
First use the conventional Cayley's theorem to embed $S$ in $\text{End}(S)$ through the map sending $s\in S$ to the map $x\mapsto sx$.  We can now consider $S$ a subset of $\text{End}(S)$.

For each map $s\in S$, $s^\ast s s^\ast=s^\ast$, so $s|_{s^\ast(S)}:s^\ast(S)\rightarrow s(S)$ has left inverse $s^\ast|_{s(S)}:s(S)\rightarrow s^\ast(S)$.  Similarly, as $s s^\ast s=s$, this map is a right inverse, so $s|_{s^\ast(S)}$ is a bijection.

Consider the map $S\rightarrow I(S)$ defined by $s\mapsto (s|_{s^\ast(S)}: s^\ast(S)\rightarrow s(S))$.  We now check that this map is a homomorphism.  For $s,t\in S$, we want to compose the partial bijections $s|_{s^\ast(S)}$ and $t|_{t^\ast(S)}$.  To do this, we first compute the overlap between the codomain of the former and the domain of the latter, which is $s(S)\cap t^\ast (S)$.  The domain of $t|_{t^\ast(S)}\circ s|_{s^\ast(S)}$ is then $s^\ast (s(S)\cap t^\ast (S))=s^\ast t^\ast(S)=(ts)^\ast (S)$ as required.  Finally, we check injectivity.  If $s,t\in S$ and $s_{s^\ast(S)}=t_{t^\ast(S)}$, then $s^\ast(S)=t^\ast(S)$ so $ss^\ast=ts^\ast$.  But then $s^\ast s s^\ast = s^\ast t s^\ast$, so $s=t$.
=--

##Links

* [[homotopy theory of inverse semigroups]]

* [[cohomology of inverse semi-groups]]

## Related concepts

* [[invertible semigroup]] (semigroups with a stronger notion of inverse)

## References

* [[Mark V. Lawson]], tutorial lectures on semigroups in Ottawa, notes available [here](http://www.ma.hw.ac.uk/~markl/ottawa.html).

*  [[Mark V. Lawson]], _Constructing ordered groupoids_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123--138, [URL stable](http://www.numdam.org/item?id=CTGDC_2005__46_2_123_0)

* [[Mark V. Lawson]], Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.

* [[Mark V. Lawson]], G. Kurdyavtseva, _The classifying space of an inverse semigroup_,  Period. Math. Hungar., to appear, [as preprint: ArXiv 1210.4421](http://arxiv.org/pdf/1210.4421.pdf)

* Alan L. T. Paterson, _Groupoids, inverse semigroups, and their operator algebras_, Progress in Mathematics __170__, Birkhäuser 1999, [MR 1724106](http://www.ams.org/mathscinet-getitem?mr=1724106)

* Alcides Buss, Ruy Exel, [[Ralf Meyer]], _Inverse semigroup actions as groupoid actions_, Semigroup Forum __85__ (2012), 227--243, [arxiv/1104.0811](http://arxiv.org/abs/1104.0811)

* Ruy Exel, _Inverse semigroups and combinatorial $C^\ast$-algebras_, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313, [doi](http://dx.doi.org/10.1007/s00574-008-0080-7) MR 2419901

* Alcides Buss, Ruy Exel, _Fell bundles over inverse semigroups and twisted étale groupoids_, J. Oper. Theory __67__, No. 1, 153-205 (2012) [MR2821242](http://www.ams.org/mathscinet-getitem?mr=2821242) [Zbl 1249.46053](http://zbmath.org/?q=an:1249.46053) [arxiv/0903.3388](http://arxiv.org/abs/0903.3388)[journal](http://www.mathjournals.org/jot/2012-067-001/0000-000-000-000.html); _Twisted actions and regular Fell bundles over inverse semigroups_, [arxiv/1003.0613](http://arxiv.org/abs/1003.0613)

* [[Pedro Resende]], _Lectures on étale groupoids, inverse semigroups and quantales_, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4-18 Sep 2006, 115 pp. [pdf](http://www.math.ist.utl.pt/%7Epmr/poci55958/gncg51gamap-version2.pdf); _Étale groupoids and their quantales_, Adv. Math. 208 (2007) 147-209; also published electronically: [doi](http://dx.doi.org/10.1016/j.aim.2006.02.004) [math/0412478](http://arxiv.org/abs/math/0412478); _A note on infinitely distributive inverse semigroups_, Semigroup Forum 73 (2006) 156-158; [doi](http://dx.doi.org/10.1007/s00233-005-0547-4) [math/0506454](http://arxiv.org/abs/math.RA/0506454)

* [[B. Steinberg]], _Strong Morita equivalence of inverse semigroups_, Houston J. Math. 37 (2011) 895-927

* Darien DeWolf and Dorette Pronk, _The Ehresmann-Schein-Nampooripad Theorem for Inverse Categories_, [arXiv](https://arxiv.org/abs/1507.08615).

* Igor Dolinka, Sergey V. Gusev, Mikhail V. Volkov, _Semiring and involution identities of powers of inverse semigroups_, [arXiv:2309.11432](https://arxiv.org/abs/2309.11432)

> The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

[[!redirects inverse semigroups]]
[[!redirects ordered groupoids]]
[[!redirects ordered groupoid]]


inverse temperature

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## Idea

In formulas in [[thermodynamics]] it is often the (multiplicative) inverse of the [[temperature]] $T$ that is the natural [[variable]], then traditionally abbreviated

$$
  \beta
  \;\coloneqq\;
  \frac{1}{k T}
  \,,
$$

where $k$ denotes the [[Boltzmann constant]], a [[physical unit]] which for purposes of pure mathematics may be identified with unity and hence omitted from the formula.

This inverse temperature appears, in particular, in the formula for the [[Boltzmann distribution]], which is the [[probability distribution]] such that the [[probability density]] for a [[physical system]] to be in a [[state]] with [[energy]] $E$ is proportional to the [[exponential]] of $E$ weighted by minus the inverse temperature:

$$
  d p \;=\; e^{- \beta E} \, d E
  \,.
$$

Essentially a special case of this is the formula for the [[heat kernel]] on a [[Riemannian manifold]], which is an [[integral kernel]] proportial to the [[exponential]] of the Riemannian [[distance]] squared and again weighted by minus the inverse temperature.

$$
  K_{Heat}(x,y) \;\propto\; e^{  - \beta \cdot d(x,y)^2 }
  \,.
$$

Extrapolating from this case, the scale of the exponent in various [[kernel method|kernels]] may often be understood as akin to inverse temperature.

Finally, when understanding [[thermal field theory|thermal]]/[[Euclidean field theory]] as the result of [[Wick rotation]] of [[relativistic field theory]] (when possible), the inverse temperature $\beta$ is proportional to the [[radius]] of the "thermal circle" on which the [[Euclidean field theory]] must be understood to be [[Kaluza-Klein compactification|KK-compactified]], see [there](Euclidean+field+theory#TemporalCompactificationToThermalRelativisticFieldTheory) for more.

## Related concepts

* [[temperature]], [[absolute zero]]

* [[thermal field theory]], [[infinite-temperature thermal field theory]]

* [[kernel methods]]

[[!redirects inverse temperatures]]


inverse trigonometric function

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## Idea

An inverse trigonometric function is a [[right inverse]] to one of the six standard [[trigonometric functions]], or more precisely a [[continuous function|continuous]] right inverse to the [[epimorphism]] part of the [[(epi, mono) factorization]] of a trigonometric function.

Inverse functions are often named using the prefix "arc", so that for example "[[arcsine]]" is another name for the inverse [[sine]]. This naming arises because the values of an inverse trigonometric function are considered as [[arc lengths]], just as trigonometric functions themselves are defined in terms of arc length parametrizations. Another possible reason for using the *arc-* prefix is to avoid a notational confusion, where for example $\cos^{-1}$ might easily be confused with the [[reciprocal]] of the [[cosine]] function, inasmuch as the notation $\cos^n$ is commonly used to denote an $n^{th}$ power of the cosine.

Since many such right inverses are possible, the choice of right inverse is a matter of convention. For the [[arcsine]] and [[arctangent]], the choice is dictated by having $0$ as a [[fixed point]]. This leads to convenient [[power series]] representations in [[neighborhoods]] of the origin, for example

$$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots$$

For other inverse trigonometric functions, the choice is dictated by convenient formulas such as

$$\arccos(x) = \frac{\pi}{2} - \arcsin(x), \qquad \arccot(x) = \frac{\pi}{2} - \arctan(x)$$

while we also have

$$\arcsec(x) = \arccos(\frac1{x}), \qquad \arccsc(x) = \arcsin(\frac1{x}).$$

## Example

* [[arccos]]

* [[arcsin]]

* [[arctan]]

[[!redirects inverse trigonometric functions]]


inversion involution

#Contents#
* table of contents
{:toc}

## Idea

For $A$ an [[abelian group]] there is a group [[automorphism]]

$$
  - \;\colon \; A \stackrel{\simeq}{\longrightarrow} A
$$

given by sending any element $a$ to its [[inverse]] $-a$.

This is an [[involution]].

## Examples

* The inversion involutuion on the abelian groups underlying [[elliptic curves]] (over the [[complex numbers]]) extends to an inversion action on the universal elliptic curve $\mathcal{E} \to \mathcal{M}_{ell}(\mathbb{C})$ over the compactified [[moduli stack of elliptic curves]] over the [[complex numbers]] ([Hain 08,lemma 5.6](#Hain08)). It extends along the inclusion $\mathcal{M}_{ell}\to \mathcal{M}_{\overline{ell}}$ to the [[Deligne-Mumford compactification|compactified]] moduli stack to give the inversion involution on the [[nodal cubic curve]] ([Hain 08, prop. 5.7](#Hain08)).

* The inversion involution on the [[multiplicative group]] induces, via the [[Goerss-Hopkins-Miller theorem]] the [[real oriented cohomology theory]] structure on [[complex K-theory]] [[KU]] (namely [[KR-theory]]), see at _[[moduli stack of tori]]_ for details.

## References

* {#Hain08} [[Richard Hain]], _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

[[!redirects inversion involutions]]


inverter

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## Idea

An _inverter_ is a particular kind of [[2-limit]] in a [[2-category]], which universally renders a [[2-morphism]] invertible.

## Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category and $\alpha\colon f\to g$ a 2-morphism.  The **inverter** of $\alpha$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ such that $\alpha v$ is invertible.

More precisely, universality means that for any object $X$, the induced functor
$$Hom(X,V) \to Hom(X,A)$$
is fully faithful, and its [[replete image]] consists precisely of those morphisms $u\colon X\to A$ such that $\alpha u$ is invertible.  If the above functor is additionally an *isomorphism* of categories onto the exact subcategory of such $u$, then we say that $V\xrightarrow{v} A$ is a **strict inverter**.

Inverters and strict inverters can be described as a certain sort of [[weighted limit|weighted]] 2-limit, where the diagram shape is the [[walking structure|walking]] 2-morphism

$T=$ [[!include 2-morphism - SVG]]

and the weight $T\to Cat$ is the diagram
$$\array{ & \to \\ 1 & \Downarrow & Inv\\ & \to } $$
where $1$ is the [[terminal category]] and $Inv$ is the walking [[isomorphism]].  Since $Inv$ is equivalent to $1$, this weight is equivalent to the terminal weight, and thus an inverter (but not a strict inverter) can also be defined simply as a [[conical limit|conical]] 2-limit over a diagram of shape $T$.

An inverter in $K^{op}$ (see [[opposite 2-category]]) is called a **coinverter** in $K$.  Coinverters in [[Cat]] are also called [[localizations]].

## Properties

* The above description makes it clear that any inverter is in particular a [[fully faithful morphism]].

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

* Inverters can be constructed in a straightforward way from [[inserters]] and [[equifiers]]: first we insert a 2-morphism $\beta$ going in the opposite direction from $\alpha$, then we equify $\beta\alpha$ and $\alpha\beta$ with identities.  In particular, it follows that strict inverters are [[PIE-limits]].

## References

In

* [[Peter Johnstone]], _[[Elephant]]_

inverters appear in B1.1.4. The inverter of a transformation between [[geometric morphism]]s of [[topos]]es is constructed in the proof of corollary 4.1.7 in section B4.1. Coinverters are discussed in section B4.5 there.

[[!redirects inverters]]
[[!redirects strict inverter]]
[[!redirects strict inverters]]
[[!redirects coinverter]]
[[!redirects coinverters]]
[[!redirects strict coinverter]]
[[!redirects strict coinverters]]


invertible field theory

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*Stub*

## Idea

An invertible ([[extended functorial field theory|extended]]) [[functorial field theory]] is one which is an [[invertible object]] under the [[tensor product]] on TQFTs [[Freed & Moore (2006), Def. 5.7](#FreedMoore06)], which essentially means that it is one whose target category is a rigid symmetric monoidal groupoid, a [[Picard groupoid]] ([DGP 19, p.20](#DGP19)).

## References

As [[quantum anomalies]] of lower-dimensional non-invertible (and possibly non-topological) [[quantum field theories]]:

* {#FreedMoore06} [[Daniel S. Freed]], [[Gregory W. Moore]], Def. 5.7 in: *Setting the quantum integrand of M-theory*, Commun. Math. Phys. **263** (2006) 89-132 [[arXiv:hep-th/0409135](https://arxiv.org/abs/hep-th/0409135), [doi:10.1007/s00220-005-1482-7](https://doi.org/10.1007/s00220-005-1482-7)]

* {#Freed13} [[Daniel Freed]], _Anomalies and Invertible Field Theories_, talk at [StringMath2013](http://scgp.stonybrook.edu/events/event-pages/string-math-2013) [[arXiv.1404.7224](https://arxiv.org/abs/1404.7224)]

As formalizing [[short-range entanglement]] in [[topological phases of matter]]:

* {#Freed14} [[Daniel S. Freed]], *Short-range entanglement and invertible field theories* [[arXiv:1406.7278](https://arxiv.org/abs/1406.7278)]

On classification (via [[reflection positivity]]):

* {#FreedHopkins21} [[Daniel Freed]], [[Michael Hopkins]], *Reflection positivity and invertible topological phases*, Geometry & Topology **25** (2021) 1165–1330  [[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)]

Review:

* {#DGP19} [[Arun Debray]], [[Soren Galatius]], [[Martin Palmer]], _Lectures on invertible field theories_, [[arXiv:1912.08706](https://arxiv.org/abs/1912.08706)]

Relation to [[cutting and pasting of manifolds]]:

* [[Carmen Rovi]], Matthew Schoenbauer, *Relating Cut and Paste Invariants and TQFTs*, The Quarterly Journal of Mathematics **73** 2 (2022) 579–607 [[arXiv:1803.02939](https://arxiv.org/abs/1803.02939), [doi:10.1093/qmath/haab044](https://doi.org/10.1093/qmath/haab044)]

* [[Carmen Rovi]], *Relating cut and paste invariants and TQFTS*, [talk](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#RoviArp2023) at [[CQTS]] (Apr. 2023) [video: [YT](https://www.youtube.com/watch?v=lD8wQhwyrdg)]

* Mayuko Yamashita, *Invertible QFTs and differential Anderson duals* [[arXiv:2304.08833](https://arxiv.org/abs/2304.08833)]

[[!redirects invertible field theories]]

[[!redirects invertible quantum field theory]]
[[!redirects invertible quantum field theories]]

[[!redirects invertible topological field theory]]
[[!redirects invertible topological field theories]]

[[!redirects invertible TQFT]]
[[!redirects invertible TQFTs]]


invertible magma

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## Idea

It is possible to define inverses in a magma without defining an [[identity element]] first, yielding a notion of _invertible magma_

## Definition

A __left invertible magma__ is a [[magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __left inverse__ or __retraction__ such that

* $a \cdot (b^{-1} \cdot b) = a$
* $(b^{-1} \cdot b) \cdot a = a$

for all $a,b \in G$.

A __right invertible magma__ is a [[magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __right inverse__ or __section__ such that

* $a \cdot (b \cdot b^{-1}) = a$
* $(b \cdot b^{-1}) \cdot a = a$

for all $a,b \in G$.

An __invertible magma__ is a [[magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __[[inverse]]__ such that

* $a \cdot (b^{-1} \cdot b) = a$
* $(b^{-1} \cdot b) \cdot a = a$
* $a \cdot (b \cdot b^{-1}) = a$
* $(b \cdot b^{-1}) \cdot a = a$

for all $a,b \in G$.

## Properties

Every invertible magma is a [[cancellative magma]].

The submagma of every [[power-associative]] invertible magma $M$ [[generator|generated]] by an element $a \in M$ is a [[cyclic group]]. This means in particular there is a $\mathbb{Z}$-[[action]] on $M$ $(-)^{(-)}:M\times\mathbb{Z}\to M$ called the __[[power]]__.

## Examples

* Every group is an invertible magma.

* Every [[invertible semigroup]] and [[nonassociative group]] is an invertible magma.

## Related concepts

* [[magma]] (noninvertible version)

* invertible [[unital magma]] (unital version)

* [[commutative invertible magma]] (commutative version)

* [[invertible semigroup]] (associative version)

* [[invertible quasigroup]] (divisible version)

* [[reciprocal algebra]]

* [[centipede mathematics]]

[[!redirects invertible magmas]]


invertible object

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## Definition

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

In a [[monoidal category]], a [[dualizable object]] $A$ for which the structure unit (and counit) maps between $A \otimes A^\ast$ (and $A^\ast \otimes A$) and the [[unit object]] are [[isomorphisms]] is called an __invertible object__.
=--

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

A [[monoidal category]] in which all objects are invertible is called a _[[2-group]]_.
=--

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

In terms of [[linear type theory]] one might speak of _invertible [[types]]_.
=--

## Related concepts

* [[Picard group]]

* [[line]], [[line object]]

* [[dualizable object]]

* [[invertible field theory]]

[[!redirects invertible object]]
[[!redirects invertible objects]]

[[!redirects invertible type]]
[[!redirects invertible types]]


invertible quasigroup

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## Idea

A quasigroup with a two-sided [[inverse]]

## Definition
An __invertible quasigroup__ is a [[quasigroup]] $(G,\cdot,\backslash,/)$ with a unary operation $(-)^{-1}:G \to G$ called the __[[inverse]]__ such that

* $a \cdot (b^{-1} \cdot b) = a$
* $(b^{-1} \cdot b) \cdot a = a$
* $a \cdot (b \cdot b^{-1}) = a$
* $(b \cdot b^{-1}) \cdot a = a$

for all $a,b \in G$.

### Without division

An __invertible quasigroup__ is a [[magma]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the __[[inverse]]__ such that

* $a \cdot (b^{-1} \cdot b) = a$
* $(b^{-1} \cdot b) \cdot a = a$
* $a \cdot (b \cdot b^{-1}) = a$
* $(b \cdot b^{-1}) \cdot a = a$

and

* $b \cdot (b^{-1} \cdot a) = a$
* $b^{-1} \cdot (b \cdot a) = a$
* $(a \cdot b) \cdot b^{-1} = a$
* $(a \cdot b^{-1}) \cdot b = a$

for all $a,b \in G$.

## Examples

* Every group is an invertible quasigroup.

* Every [[associative quasigroup]] and [[nonassociative group]] is an invertible quasigroup.

## Related concepts

* [[quasigroup]] (noninvertible version)

* [[nonassociative group]] (unital version)

* [[commutative invertible quasigroup]] (commutative version)

* [[associative quasigroup]] (associative version)

* [[invertible magma]] (nondivisible version)

* [[Malcev variety]]

* [[centipede mathematics]]

[[!redirects invertible magmas]]


invertible semigroup

> This entry is about [[semigroups]] with two-sided [[inverses]]. For semigroups with a unary operator $i$ such that $s \cdot i(s) \cdot s = s$ and $i(s) \cdot s \cdot i(s) = i(s)$, see instead at *[[inverse semigroup]]*.

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## Idea

A [[semigroup]] that is also an [[invertible magma]].

## Definition

### With only multiplication

An __invertible semigroup__ is a [[semigroup]] $(G,(-)\cdot(-):G\times G\to G)$ such that for every element $a \in A$, left multiplication and right multiplication by $a$ are both [[bijections]].

### With multiplication and inverses

An __invertible semigroup__ is a [[semigroup]] $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation called the __inverse__ $(-)^{-1}:G \to G$ such that

* $a \cdot b^{-1} \cdot b = a$
* $b^{-1} \cdot b \cdot a = a$
* $b \cdot b^{-1} \cdot a = a$
* $a \cdot b \cdot b^{-1} = a$

for all $a,b \in G$.

### Torsor-like definition

There is an alternate definition of an invertible semigroup that looks like the usual definition of a [[torsor]] or [[heap]]:

An __invertible semigroup__ is a set $S$ with a binary operation $(-)\cdot(-):S\times S\to S$ called __multiplication__ and a unary operation $(-)^{-1}:S\to S$ called __inverse__ satisfying the following laws:

* [[associativity]]: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a,b,c\in S$
* left [[Malcev operation|Malcev identity]]: $b \cdot b^{-1} \cdot a = a$ for all $a,b\in S$
* right [[Malcev operation|Malcev identity]]: $a \cdot b^{-1} \cdot b = a$ for all $a,b\in S$
* [[commutative|commutativity]] with [[inverse elements]]: $a \cdot a^{-1} = a^{-1} \cdot a$ for all $a\in S$

## Pseudo-torsor

Every invertible semigroup $G$ has a [[pseudo-torsor]], or associative [[Malcev operations|Malcev algebras]], $t:G^3\to G$ defined as $t(x,y,z)=x\cdot y^{-1}\cdot z$. If the invertible semigroup is inhabited, then those pseudo-torsors are actually [[torsors]] or [[heaps]].

## Properties

* Every invertible semigroup is either a [[group]] or the empty semigroup.

## Related concepts

* [[invertible magma]] (nonassociative version)

* [[semigroup]] (non-invertible version)

* [[group]] (unital version)

* [[commutative invertible semigroup]] (commutative version)

* [possibly empty heap](heap#empty) (forgetting the unit element)

* [[pseudo-torsor]], [[torsor]]

[[!redirects invertible semigroups]]


involution

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## Definition
 {#Definition}

\begin{definition}
An **involution** is an [[endomorphism]] $\sigma$ whose [[composition]] with itself is the [[identity morphism]]:

\[
  \label{InvolutiveProperty}
  \sigma \circ \sigma \;=\; id
  \,.
\]
\end{definition}
Such an endomorphism is necessarily an [[automorphism]], being its own [[inverse morphism|inverse]].

\begin{remark}
\label{InvolutionsAreZModTwoActions}
**(involutions are equivalently $\mathbb{Z}/2$-actions)**
\linebreak
Involutions are equivalently the value on the single non-trivial element $\sigma$ in [[cyclic group of order 2|$\mathbb{Z}/2$]] of a [[group action]] by $\mathbb{Z}/2$.

In this guise, involutions appear throughout [[representation theory]], [[transformation groups]], [[equivariant homotopy theory]], [[equivariant cohomology]], etc.

In particular, involutions, are the defining ingredients of *Real* (with capital "R"!) [[Whitehead-generalized cohomology theories]] such as [[KR-theory]], [[MR-theory]], [[BPR-theory]] and [[ER-theory]].
\end{remark}

\begin{remark}
**(terminology in algebra)**
\linebreak
An [[associative algebra]] equipped with an anti-involution is called a *[[star-algebra]]*.

On [[algebras]] and other [[mathematical structures]] where this makes sense, an __anti-involution__ is an [[anti-homomorphism]] satisfying (eq:InvolutiveProperty), instead of a [[homomorphism]] (hence an anti-[[endomorphism]] and necessarily an anti-[[automorphism]]).
\end{remark}

\begin{remark}
**(terminology in topology)**
\linebreak
A [[topological space]] equipped with an involutive [[homeomorphism]] is sometimes called a *[[real space]]* (at least in the context of [[KR-theory]]).

In view of Rem. \ref{InvolutionsAreZModTwoActions}, involutions on topological spaces are equivalently known as *[[topological G-spaces]]* for $G = $[[cyclic group of order 2|$\mathbb{Z}/2$]]. The case of [[n-spheres]] with involution is discussed [here](group+actions+on+spheres#Involutions).
\end{remark}

### In dependent type theory

In [[dependent type theory]], there are two different notions of involution.

\begin{definition}
A **non-coherent involution** on a type $A$ is an element of the type
$$\sum_{i:A \to A} \prod_{x:A} i(i(x)) =_{A} x$$
\end{definition}

\begin{definition}
A **coherent involution** on a type $A$ is an element of the type
$$\sum_{i:A \to A} \sum_{p:\prod_{x:A}i(i(x)) =_{A} x} \prod_{x:A} p(i(x)) =_{_{i(i(i(x))) =_{A} i(x)}} \mathrm{ap}_i(i(i(x)), x, p(x))$$
\end{definition}

## Properties

### Commuting involutions

Two involutions $f, g \colon X \to X$ [[commutative diagram|commute]] if and only if their [[composition]] $f g$ is also an involution, as shown by the following manipulations:

$$
  \begin{aligned}
  f g
   \;=\;
  g f
  &\;\;\;\;\;\implies\;\;\;\;\;
  (f g) (f g)
    \;=\;
  (f g) (g f)
    \;=\;
  f (g g) f
    \;=\;
  f f
   \;=\;
  id
  \\
  (f g) (f g)
    \;=\;
  id
  &\;\;\;\;\;\implies\;\;\;\;\;
  f g
    \;=\;
  f
  \big(
     (f g) (f g)
  \big)
  g
    \;=\;
  (f f) (g f) (g g)
    \;=\;
  g f
  \,.
  \end{aligned}
$$

### Fixed point free involutions

In [[combinatorics]], an important class of involutions are the [[fixed point]] free ones: an arbitrary involution on a [[finite set]] of cardinality $n$ may be specified by the choice of $k$ elements which are fixed together with a fixed point free involution on the remaining $(n-k)$.  The number of fixed point free involutions on a set of $2n$ labelled elements is counted by the double factorial $(2n-1)!! = (2n-1)\cdot (2n-3)\cdot\dots\cdot 3\cdot 1 = \frac{(2n)!}{2^n n!}$, while arbitrary involutions on a set of $n$ labelled elements are counted by OEIS sequence [A000085](https://oeis.org/A000085), which also counts the number of [[Young tableaux]] with $n$ cells.

### Monad of involutions
 {#MonadOfInvolutions}

An involution on a set $X$ is the same thing as an [[action]] of $\mathbb{Z}/2\mathbb{Z}$ on $X$.

More generally, let $(C,\otimes,1)$ be a [[monoidal category]] with [[distributive monoidal category|distributive]] finite [[coproducts]]. The object $2 = 1 + 1$ is equipped with an involution
$$
not : 2 \to 2
$$
defined as the [[copairing]] $not = [inr,inl]$ of the right and left injections.
Moreover, 2 can be given the structure of a [[monoid]] in $C$, with unit and multiplication
$$false : 1 \to 2 \qquad xor : 2 \otimes 2 \to 2$$
defined by $false = inl$ and $xor = [id,not]$ (here we make use of the isomorphism $2 \otimes 2 \cong 2 + 2$ to define $xor$ by copairing). The mapping
$$
X \mapsto 2 \otimes X \cong X + X
$$
thus extends to a [[monad]] on $C$, sending any object $X$ to the free object equipped with an involution over $X$. Explicitly, the unit $\eta_X : X \to 2\otimes X$ and the multiplication $\mu_X : 2\otimes 2\otimes X \to 2\otimes X$ of the monad are defined by tensoring the unit and the multiplication of the monoid with the identity on $X$, while the involution on $2 \otimes X$ is likewise defined by tensoring the involution on 2 with the identity on $X$.

We then have that involutions in $C$ are precisely the [[module over a monad|algebras]] of the monad $(2\otimes-,false\otimes-,xor\otimes-)$. In the forward direction, given an involution $f : X \to X$, we define a monad algebra structure $\alpha : 2\otimes X \to X$ on $X$ by $\alpha = [id,f]$ (again using the isomorphism $2\otimes X \cong X+X$). Conversely, given a monad algebra $\alpha : 2\otimes X \to X$, we can define an endomorphism $f : X \to X$ by $f = \alpha \circ inr$. The monad algebra laws imply that
$$\alpha \circ inr \circ \alpha \circ inr = \alpha \circ (2\otimes \alpha) \circ (2\otimes inr) \circ inr = \alpha \circ (xor\otimes id) \circ (2\otimes inr) \circ inr$$
and since $xor$ is defined such that $(xor\otimes id) \circ (2\otimes inr) \circ inr = id$, we derive that $\alpha \circ inr$ is an involution.

## Related concepts

* [[star-algebra]]

* [[dagger category]]

* [[complex conjugation]]

* [[anti-linear map]], [[anti-dual linear space]]

* [[duality]]

* [[involutive Hopf algebra]]

* [[idempotent]]

## References

Discussion in [[combinatorics]]:

* Philippe Flajolet and Robert Sedgewick, _Analytic Combinatorics_, CUP, 2009. ([author pdf](http://algo.inria.fr/flajolet/Publications/book.pdf))

Discussion in [[differential topology]]:

* [[Santiago López de Medrano]], *Involutions on Manifolds*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **59**, Springer 1971 ([doi:10.1007/978-3-642-65012-3](https://link.springer.com/book/10.1007/978-3-642-65012-3))

[[!redirects involution]]
[[!redirects involutions]]

[[!redirects non-coherent involution]]
[[!redirects non-coherent involutions]]

[[!redirects coherent involution]]
[[!redirects coherent involutions]]

[[!redirects antiinvolution]]
[[!redirects antiinvolutions]]
[[!redirects anti-involution]]
[[!redirects anti-involutions]]


involutive Hopf algebra

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

#Contents#
* table of contents
{:toc}

## Definition

A [[Hopf algebra]] is called *involutive* (or *involutory*, e.g. [Kuperberg 91](#Kuperberg91), [Virelizier 02, Sec. 1.7](#Virelizier02)) if its [[antipode]] is an [[involution]].

Since the antipode of a Hopf algebra is necessarily an [[anti-homomorphism]] of the underlying [[associative algebra]]-structure (by [this Prop.](Hopf+algebra#AntipodeIsAnAntihomomorphism)), an involutive Hopf algebra is in particular a [[star algebra]].

## Examples

* [[group algebra]], [[groupoid algebra]], [[category algebra]] of a [[dagger-category]]

* [[homology of loop spaces]] under [[Pontrjagin product]]

## References

* {#Kuperberg91} [[Greg Kuperberg]], *Involutory Hopf algebras and 3-manifold invariants*, Internat. J. Math. 2 (1991), no. 1, 41--66 ([arXiv:math/9201301](https://arxiv.org/abs/math/9201301))

  > (in relation to [[3-manifolds]])

* {#Virelizier02} Alexis Virelizier, *Involutory Hopf group-coalgebras and flat bundles over 3-manifolds* ([arXiv:math/0206254](https://arxiv.org/abs/math/0206254))

[[!redirects involutive Hopf algebras]]

[[!redirects involutory Hopf algebra]]
[[!redirects involutory Hopf algebras]]


Inönü-Wigner group contraction > history

see _[[group contraction]]_


IO-monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In purely [[functional programming languages]] (such as [[Haskell]]) all would-be side effects are [[data type|data typed]] in terms of *[[monads in computer science]]* which make the side effects look like and hence be formally treated like [[type checking|verifiable]] deterministic pure functions. If the actual side effect operations of reading input from or writing output to actual physical devices is [[data type|typed]] in this [[monad in computer science|monadic]] way, one speaks of an **I/O-monad**.

So to the programming language an IO-monad looks just like a [[state monad]] or similar, but when run on an actual computer the bind operation does cause actual physical effects.

## Related concepts

* [[monad in computer science]]

* [[quantum IO monad]]

## References

Original discussion:

* [[Simon L. Peyton Jones]], [[Philip Wadler]], *Imperative functional programming*, Principles of programming languages (1993) 71–84 [[doi:10.1145/158511.158524](https://doi.org/10.1145/158511.158524), [pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/1993/01/imperative.pdf)]

Review and exposition:

* {#BentonHughesMoggi02} [[Nick Benton]], [[John Hughes]], [[Eugenio Moggi]], §5.1 in: *Monads and Effects*, in: *Applied Semantics*, Lecture Notes in Computer Science **2395**, Springer (2002) 42-122 [[doi:10.1007/3-540-45699-6_2](https://doi.org/10.1007/3-540-45699-6_2)]

* {#Milewski19} [[Bartosz Milewski]], §21.2.8 & §21.29 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

In [[Haskell]]:

* [A Gentle Introduction to Haskell](https://www.haskell.org/tutorial/), Section 7: *[Input/Output](https://www.haskell.org/tutorial/io.html)*

* Wikipedia, *IO monad (Haskell)*

(...)

[[!redirects IO-monads]]

[[!redirects IO monad]]
[[!redirects IO monads]]

[[!redirects I/O-monad]]
[[!redirects I/O-monads]]

[[!redirects I/O monad]]
[[!redirects I/O monads]]

[[!redirects IO]]

[[!redirects I/O]]

[[!redirects input]]
[[!redirects inputs]]

[[!redirects output]]
[[!redirects outputs]]

[[!redirects input and output]]
[[!redirects inputs and soutput]]

[[!redirects output and input]]
[[!redirects outputs and inputs]]


Ioan James

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

## Selected writings

On [[general topology]] and [[homotopy theory]]:

* [[Ioan Mackenzie James]], *General Topology and Homotopy Theory*, Springer 1984
([doi:10.1007/978-1-4613-8283-6](https://doi.org/10.1007/978-1-4613-8283-6))

On [[algebraic topology]]:

* [[Handbook of Algebraic Topology]]

On [[parameterized topology]] and [[parameterized homotopy theory]]:

* [[Ioan Mackenzie James]]: *Fibrewise topology*, Cambridge Tracts in Mathematics, Cambridge University Press (1989) $[$ISBN:9780521360906$]$

On [[parametrized homotopy theory]] and [[parameterized stable homotopy theory]]:

* [[Ioan Mackenzie James]], _[[Introduction to fibrewise homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ (1995)

* [[Michael C. Crabb]], [[Ioan Mackenzie James]]: *Fiberwise homotopy theory*, Springer Monographs in Mathematics, Springer (1998) $[$[doi:10.1007/978-1-4471-1265-5](https://doi.org/10.1007/978-1-4471-1265-5), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabbjames.pdf) ,[pdf](https://www.gbv.de/dms/ilmenau/toc/243540361.PDF)$]$

On the [[Samelson product]] and [[Whitehead product]] for [[homotopy groups of spheres]]:

* {#James71} [[Ioan Mackenzie James]], _Products between homotopy groups_,  Compositio Mathematica, Volume 23 (1971) no. 3, p. 329-345  ([numdam](http://www.numdam.org/item/?id=CM_1971__23_3_329_0))

* {#James56} [[Ioan Mackenzie James]], _On the Suspension Triad_, Annals of Mathematics Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 191-247 ([arXiv:1969607](https://www.jstor.org/stable/1969607))

* {#James57} [[Ioan Mackenzie James]], _On the Suspension Sequence_, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 ([arXiv:1969666](https://www.jstor.org/stable/1969666))

Proving the [[equivariant Whitehead theorem]]:

* {#JamesSegal78}  [[Ioan James]], [[Graeme Segal]], Theorem 1.1 in: _On equivariant homotopy type_, Topology Volume 17, Issue 3, 1978, Pages 267-272 (doi:10.1016/0040-9383(78)90030-7)

## Related $n$Lab entries

* [[Stiefel manifold]]

* [[EHP spectral sequence]]

category: people

[[!redirects Ioan Mackenzie James]]

[[!redirects Ioan M. James]]


Ioanid Rosu

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=32966)

## Selected writings

On [[elliptic genera]], [[equivariant elliptic cohomology]] and the [[rigidity theorem for elliptic genera]]:

* [[Ioanid Rosu]], _Equivariant Elliptic Cohomology and Rigidity_, American Journal of Mathematics 123 (2001), 647-677 ([arXiv:math/9912089](https://arxiv.org/abs/math/9912089))

category: people


Ioannis Dalianis

* [publications](http://pubviz.icapsid.net/author/Dalianis)

## related $n$Lab entries

* [[Starobinsky model of cosmic inflation]]

category: people


Ioannis Markakis

* [personal page](https://jmarkakis.github.io/)

* [Institute page](https://www.cst.cam.ac.uk/people/im496)

## Selected writings

On [[weak omega-category|weak $\omega$-categories]] via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

category: people


ionad

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

# Contents #
* table of contents
{:toc}

## Idea ##

As a [[Grothendieck topos]] is a [[categorification|categorified]] [[locale]], so an ionad is a categorified [[topological space]].  While the opens are primary in toposes and locales, the points are primary in ionads and topological spaces.

[[Richard Garner]] developed the theory of ionads, in which the [[topos]] $Set^X$ plays an role analogous to that of the [[lattice]] $\mathbb{2}^X$ (the [[power set]] of $X$) in the theory of topological spaces.  Intuitively, we are [[categorification|categorifying]] the [[subobject classifier]] $\mathbb{2}\in Set$ to the "categorified subobject classifier" $Set\in Cat$ (which classifies [[discrete opfibrations]].

The word 'ionad' is Irish for a location, place, or site; 'Ionad' often translates 'Centre' in titles of institutions.  It is pronounced /ˈɪnəd/ (roughly 'INN-ad' or 'UNN-ad', not 'i-NAD' or 'yonad'; 'ЫН-ад' in a North Slavic language), or more precisely [ˈɨ̞n̪ˠəd̪ˠ] (at least in Munster), following [Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Irish_phonology).  The plural (which you can use if you like to use 'topoi' too) is 'ionaid' (/ˈɪnəɟ/, 'INN-adge' or 'UNN-adge', 'ЫН-адь', [ˈɨ̞n̪ˠədʲ]).  We could go on to decline it out of the nominative case, but now it\'s getting silly.

## Definition

Recall that a [[topological space|topology]] on a set $X$ can be defined by giving its interior operator, an operation $Int\colon \mathbb{2}^X \to \mathbb{2}^X$ (where $\mathbb{2}$ is the poset of [[truth values]]) such that

*  $A \supseteq Int(A)$,
*  $Int(Int(A)) \supseteq Int(A)$,
*  $Int(X) = X$, and
*  $Int(A \cap B) = Int(A) \cap Int(B)$

for all [[subsets]] $A, B$ of $X$.

In more sophisticated language, $Int$ is a finite [[meet]]-preserving [[comonad]] on the [[power set]] of $X$.

[[categorification|Categorifying]] (mostly), we have this:

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

An __ionad__ is a set $X$ together with a [[left exact functor|finite limit-preserving]] [[comonad]] $Int_X$ on the category $Set^X$.  An ionad is __bounded__ if the comonad is [[accessible monad|accessible]].
=--

Although Garner does not require an ionad to be bounded, the nicest results hold for them, and all of his applications involve only bounded ionads.  In fact, Garner writes, 'Indeed, the existence of unbounded ionads is a problem that seems to be independent of the axioms of [[Zermelo-Fraenkel set theory]].' (Section 3.8).

+-- {: .un_defn}
###### Remark (Garner Remark 3.9)
Bounded ionads give rise to exactly the Grothendieck toposes with [[point of a topos|enough points]] as their topos of coalgebras. (See below)
=--

### Sets, groupoids, or categories?

It may seem odd to take $X$ to be a *set* rather than something else such as a [[groupoid]] or a [[category]].  An analogous definition can be given where $X$ is a groupoid or a category, of course, but the reason for taking it to be a set is that it makes the analogy to classical topological spaces closer.  Consider the following three notions:

1. A set $X$ together with a finite-limit-preserving comonad on its powerset $\mathbb{2}^X$.
2. A set $X$ equipped with an [[equivalence relation]] (which we can regard as a [[preorder]] that happens to be symmetric) together with a finite-limit-preserving comonad on the hom-preorder $\mathbb{2}^X$.
3. A [[preorder]] $X$ together with a finite-limit-preserving comonad on $\mathbb{2}^X$.

All three of these *induce* a topology on the underlying set of $X$.  But it is (1) that is *exactly* a topological space: (2) and (3) include the extra data of a (perhaps symmetric) preorder on $X$ that maps bijectively-on-objects into the [[specialization preorder]] of that topology.

However, as in other cases such as [[Segal categories]]/[[complete Segal spaces]] and [[generalized multicategories]], another way to "get rid of extra data" is to  force it to duplicate data that's already present (a "completeness" condition).  Thus we could consider instead (still in the uncategorified case):

* A structure as in (2) above, but such that the given equivalence relation coincides with the relation "$x$ and $y$ are in all the same open sets" (which it automatically *implies*).
* A structure as in (3) above, but such that the given preorder coincides with the specialization preorder.

These would give equivalent definitions to (1), but may be better-behaved in some ways.  In [[homotopy type theory]] without [[sets cover]], they would no longer be equivalent, but the groupoidal/preorder versions might be better.  For ionads the corresponding definitions would be

* A groupoid $X$ with a finite-limit-preserving comonad on $Set^X$ such that the induced functor from $X$ to the category of points of the resulting topos is [[pseudomonic functor|pseudomonic]].
* A category $X$ with a finite-limit-preserving comonad on $Set^X$ such that the induced functor from $X$ to the category of points of the resulting topos is [[fully faithful]].

(Asking that these functors also be surjective on objects would be a [[sober space|sobriety]] condition on an ionad.)

When categorifying further to "$n$-ionads" and "$\infty$-ionads", the possible options multiply further; but that is probably a topic for another page.  For discussion of these questions, see the [nForum thread](https://nforum.ncatlab.org/discussion/5857).

## Morphisms of ionads ##

Recall that, given two topological spaces $X$ and $Y$, a [[continuous map]] from $X$ to $Y$ is a [[function]] $f\colon X \to Y$ (on the [[underlying set]]s) such that

*  $f^*(Int(A)) \subseteq Int(f^*(A))$

for every subset $A$ of $Y$, where $f^*$ takes the [[preimage]].

Categorifying (and adding a coherence law), we have this:

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

Given ionads $X$ and $Y$, a __continuous map__ from $X$ to $Y$ consists of a [[function]] $f\colon X \to Y$ (on the underlying sets of points) together with a [[natural transformation]]
$$ Int_f\colon f^* \circ Int_Y \to Int_X \circ f^* ,$$
where $f^*\colon Set^Y \to Set^X$ is $Set^f$, such that $Int_f$ 'respects the comonad structures'.
=--

+-- {: .query}
I need to figure out exactly what this last clause means.
=--

Note that $Int_f$ is part of the [[extra structure|structure]] here; it is not merely a property.  (In other words, the [[forgetful functor]] from ionads to topological spaces is not [[faithful functor|faithful]].)

There is an obvious notion of [[2-morphism]], which turns out to be trivial (but probably would not be if $X$ were allowed to be a groupoid).  However, the category of ionads is presumably (like [[Top]]) still a [[locally prosetal 2-category]] under the [[specialisation order]].

## The topos of opens

As every topological space has a [[Heyting algebra]] (in fact a [[frame]], or dually a [[locale]]) of open subsets, so every ionad has a [[topos]] (in fact a [[Grothendieck topos]], if it is bounded) of opens.

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

Given an ionad $X$, an __open__ in $X$ is simply a [[algebra of a monad|coalgebra]] of the comonad $Int_X$.
=--

The opens of $X$ form a topos $\Omega(X)$, and we have a [[surjective geometric morphism|surjective]] [[geometric morphism]] from $Set^X$ to $\Omega(X)$.  In fact, an ionad may be defined as a set $X$ together with a surjective geometric morphism from $Set^X$ to some topos, much as a topological space may be defined as a set $X$ together with a surjective locale morphism from $\mathbb{2}^X$ to some [[locale]].  The topos $\Omega(X)$ is essentially unique by surjectivity, and this also shows that $\Omega(X)$ has [[enough points]].

Just as $\Omega(X)$ is a [[frame]] whenever $X$ is a topological space, so $\Omega(X)$ should be a [[Grothendieck topos]] when $X$ is an ionad.  In fact, this holds only for *bounded* ionads; an ionad may be defined to be bounded if and only if its topos of opens is [[cocomplete category|cocomplete]] (or equivalently a Grothendieck topos).

A continous map between topological spaces may be given by a function $f\colon X \to Y$ and a commuting square

$$ \array {
   \Omega(Y)    & \to              & \Omega(X) \\
   \downarrow   &                  & \downarrow \\
   \mathbb{2}^Y & \overset{f^*}\to & \mathbb{2}^X
} $$

Similarly, a continuous map between ionads may be given by a function $f\colon X \to Y$ and a commuting square

$$ \array {
   \Omega(Y)    & \to              & \Omega(X) \\
   \downarrow   &                  & \downarrow \\
   Set^Y        & \overset{f^*}\to & Set^X
} $$

Without loss of generality, we may require this square to commute on the nose; this is related to the triviality of $2$-morphisms in the category of ionads.

Note that the map $\Omega(Y) \to \Omega(X)$ must be the preimage half of a [[geometric morphism]] from $\Omega(X)$ to $\Omega(Y)$; we may also define a continous map $X$ to $Y$ to be such a geometric morphism together with a compatible map between the generating points of the toposes.

## Bases of ionad structures

...

## References ##

Ionads have been introduced and studied in

*  [[Richard Garner]], _Ionads_,  J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747. ([arXiv](http://arxiv.org/abs/0912.1415)) ([doi](http://dx.doi.org/10.1016/j.jpaa.2012.02.013)) ([author-archived version of published copy](http://comp.mq.edu.au/~rgarner/Papers/Ionads.pdf))

Some basic aspects of the theory are developed there, and applications to [[topology]], [[logic]] and [[geometry]] are discussed.

Further aspects of the theory of ionads, with a focus on their logical importance were studied in

*  [[Ivan Di Liberti]], _Towards Higher Topology_,  J. Pure Appl. Algebra 226 (2022)

*  [[Ivan Di Liberti]], _Formal Model Theory & Higher Topology_,  ArXiv 2020.

[[!redirects ionads]]

[[!redirects ionaid]]


Ionut Ciocan-Fontanine

* [web](http://www.math.umn.edu/~ciocan/)

* [[dg-scheme]]
* [[derived moduli space]]

[[!redirects Ciocan-Fontanine]]
[[!redirects I. Ciocan-Fontanine]]


Iosif Bena

## Selected writings

On [[black rings]] in [[D=5 supergravity]]:

* [[Iosif Bena]], [[Per Kraus]], _Microscopic description of black rings in AdS/CFT_ JHEP 12 (2004) 070 ([hep-th/0408186](http://arxiv.org/abs/hep-th/0408186))

On non-uniqueness of [[black rings]]:

* [[Iosif Bena]], [[Simon Ross]], [[Nicholas Warner]], _Coiffured Black Rings_, Class.Quant.Grav. 31 (2014) 165015 ([arXiv:1405.5217](https://arxiv.org/abs/1405.5217))

On [[D1-D5-brane bound states]]:

* [[Iosif Bena]], [[Per Kraus]], _Microstates of the D1-D5-KK system_ Phys. Rev. D72 (2005) 025007 ([hep-th/0503053](http://arxiv.org/abs/hep-th/0503053))

On [[AdS-CFT duality|AdS-CFT dual]] [[type II supergravity]]-solutions corresponding to [[D1-D5-P bound states]] ("[[superstrata]]"):

* [[Iosif Bena]], [[Stefano Giusto]], [[Rodolfo Russo]], [[Masaki Shigemori]], [[Nicholas P. Warner]], *Habemus Superstratum! A constructive proof of the existence of superstrata*, JHEP 1505 (2015) 110 [[arXiv:1503.01463](https://arxiv.org/abs/1503.01463)]

Discussion of [[D-brane polarization]] in context of the [[AdS/CFT correspondence]] and [[AdS/QCD correspondence]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

* [[Iosif Bena]], Aleksey Nudelman, _Exotic polarizations of D2 branes and oblique vacua of $(S)YM_{2+1}$_, Phys. Rev. D62 (2000) 126007 ([arXiv:hep-th/0006102](https://arxiv.org/abs/hep-th/0006102))

* [[Iosif Bena]], Stanislav Kuperstein, _Brane polarization is no cure for tachyons_, JHEP09 (2015) 112 ([arXiv:1504.00656](https://arxiv.org/abs/1504.00656))

Discussion of [[D4-branes]] [[polarized brane|polarizing]] into [[NS5-branes]]:

* [[Iosif Bena]], Calin Ciocarlie, _Exact $\mathcal{N}=2$ Supergravity Solutions With Polarized Branes_, Phys.Rev. D70 (2004) 086005 ([arXiv:hep-th/0212252](https://arxiv.org/abs/hep-th/0212252))

* [[Iosif Bena]], [[Radu Roiban]], _$\mathcal{N}=1*$ in 5 dimensions: Dijkgraaf-Vafa meets Polchinski-Strassler_, 	JHEP 0311 (2003) 001 ([arXiv:hep-th/0308013](https://arxiv.org/abs/hep-th/0308013))

On the [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]]) [[3-sphere]]-shape:

* [[Iosif Bena]], _The M-theory dual of a 3 dimensional theory with reduced supersymmetry_, Phys. Rev. D62:126006, 2000 ([arXiv:hep-th/0004142](https://arxiv.org/abs/hep-th/0004142))

* [[Iosif Bena]], [[Nicholas Warner]], _A harmonic family of dielectric flow solutions with maximal supersymmetry_, JHEP 0412:021, 2004 ([arXiv:hep-th/0406145](https://arxiv.org/abs/hep-th/0406145))

* [[Iosif Bena]], [[Mariana Graña]], Stanislav Kuperstein, Stefano Massai, _Tachyonic Anti-M2 Branes_, JHEP 1406:173, 2014 ([arXiv:1402.2294](https://arxiv.org/abs/1402.2294))

[[polarized brane|Polarization]] of [[M5-branes]] into [[MK6]]:

* [[Iosif Bena]], Diana Vaman, _The polarization of M5 branes and little string theories with reduced supersymmetry_, JHEP 0111 (2001) 032 ([arXiv:hep-th/0101064](https://arxiv.org/abs/hep-th/0101064))

On the relation of [[polarized branes]] to [[giant gravitons]]:

* [[Iosif Bena]], Douglas Smith,
_Towards the solution to the giant graviton puzzle_, 	Phys.Rev. D71 (2005) 025005 ([arXiv:hep-th/0401173](https://arxiv.org/abs/hep-th/0401173))

On (in-)stability of non-[[supersymmetry|supersymmetric]] [[AdS spacetime|AdS]] [[string theory vacua|vacua in string theory]]:

* [[Iosif Bena]], [[Krzysztof Pilch]], [[Nicholas Warner]], _Brane-Jet Instabilities_,  J. High Energ. Phys. 2020, 91 (2020) ([arXiv:2003.02851](https://arxiv.org/abs/2003.02851))

On [[M2-M5-brane bound states]]:

* [[Iosif Bena]], [[Anthony Houppe]], [[Dimitrios Toulikas]], [[Nicholas P. Warner]], *Maze Topiary in Supergravity* [[arXiv:2312.02286](https://arxiv.org/abs/2312.02286)]

category: people


IoT

* wikipedia [internet of things](https://en.wikipedia.org/wiki/Internet_of_things)
* K. Christidis, M. Devetsikiotis, _Blockchains and Smart Contracts for the Internet of Things_,  IEEE Access __4__, 2292-2303 (2016) [doi](https://doi.org/10.1109/ACCESS.2016.2566339) (open access)
* C Roulin et al. _On the activity privacy of blockchain for IoT_ [arxiv/1812.08970](https://arxiv.org/abs/1812.08970)
* Z. Bao et al. _IoTChain: a three-tier bockchain-based IoT security architecture_ [arxiv/1806.02008](https://arxiv.org/abs/1806.02008)
* [slock.it](https://slock.it)

category: computer science

[[!redirects Internet of Things]]
[[!redirects internet of things]]


IPC-property

#Idea#

The _IPC-property_ ("inductive limit product commutation property") is a technical condition on a [[category]] $A$ which ensures that [[presheaf|presheaves]] with values in $A$ have a good notion of [[sheafification]].

#Definition#

A [[category]] $A$ is said to satisfy the **IPC-property** if

* it admits small [[product|products]] and [[filtered category|filtered]] [[colimit|colimits]];

* for

  * every family $\{I_s\}_{s \in S}$ of [[small category|small]] [[filtered category|filtered]] categories

  * and for any family $\{\alpha_s : I_s \to A\}$ of [[functor|functors]]

  * indexed by a [[Grothendieck universe|small set]] $S$

  the canonical morphism

  $$
    colim \left(
      \prod_s I_s \stackrel{k \mapsto \prod_s \alpha_s(\pi_s(j))}{\to} A
    \right)
    \to
    \prod_s colim \alpha_s
  $$

  is an [[isomorphism]].

#References#

The IPC property is definition 3.1.10 in

* Kashiwara-Schapira, [[Categories and Sheaves]].

It is invoked for [[sheafification]] in section 17.4 there.


Irakli Patchkoria

* [webpage](http://www.math.uni-bonn.de/people/irpatchk/)

## Selected writings

On [[proper equivariant homotopy theory]]:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_ ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

## Related $n$Lab entries

* [[global equivariant stable homotopy theory]]

category: people


Irene Dorfman

*  Algebraic Aspects of Integrable Systems, In Memory of Irene Dorfman, Editors: A. S. Fokas, I. M. Gelfand Progress in Nonlinear Differential Equations and Their Applications (PNLDE) 26, Birkhäuser 1997

## Related entries

* [[Courant Lie 2-algebroid]]
* [[Dirac structure]]
* [[integrable system]]

category: people


IRIF

IRIF, which stands for  the _Research Institute on the Foundations of Computer Science_, is a research laboratory of CNRS and Université de Paris, also hosting two Inria project-teams.

* [webpage](https://www.irif.fr/)


Irina Kogan

* [webpage](http://www4.ncsu.edu/~iakogan/)

category: people


irrational number

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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

# Irrational numbers
* table of contents
{: toc}

## Idea

An irrational number is of course a [[number]] that is not [[rational number|rational]].  As such, the concept is perhaps uninteresting.  However, the term 'irrational number' is often used for an irrational *[[real number|real]]* number; in this case, it is interesting to consider such numbers for two reasons:

*  Historically, it was an important discovery that irrational real numbers exist.
*  The [[topological space]] of all irrational real numbers is interesting in [[constructive analysis]], [[computability]] theory, and [[descriptive set theory]].

Of course, there are also various theorems about general classes of numbers that distinguish rational from irrational numbers.

## Definition

### In the real numbers

An [[affine function]] on the real numbers consists of a function $f$ and real number [[coefficients]] $a$ and $b$ such that for all real numbers $x$, $f(x) = a x + b$. The [[constant function]] at zero $\lambda x.0$ is an affine function where $a = 0$ and $b = 0$. A real number is **irrational** if for all affine functions $f$ with [[integer]] coefficients, if $f(x) = 0$, then $f = \lambda x.0$. This is equivalent to saying that a real number is irrational if for all rational numbers $a$, $x \neq a$.

Alternatively, a [[real number]] $x$ is **irrational** if given any [[rational number]] $a$ (thought of as a real number), the [[absolute value]] ${|x - a|}$ is [[positive number|positive]].

These two definitions are equivalent in [[classical mathematics]]. However, these two definitions no longer coincide in [[constructive mathematics]]; the former definition of irrational number is called **weakly irrational** while the latter definition is called **strongly irrational** or **strictly irrational**. Strongly irrational numbers are most commonly used in constructive mathematics, since it uses the apartness relation or strict order relation of the real numbers, which, unlike equality, is what is detected of the real numbers in constructive mathematics.

The [[set]] of irrational real numbers (a [[subset]] of the set of real numbers) is variously denoted $\mathbb{I}$, $\mathbb{J}$, or $\mathbb{B}$ (in various fonts).  The $\mathbb{I}$ and $\mathbb{J}$ stand for 'irrational', while the $\mathbb{B}$ stands for 'Baire' (see the next paragraph).  Here we will use $\mathbb{J}$.

We may give $\mathbb{J}$ a [[topological structure|topology]] as a [[subspace]] of the [[real line]] $\mathbb{R}$.  With this topology, $\mathbb{J}$ is sometimes called __[[Baire space of irrational numbers|Baire space]]__; however, one uses a different [[uniform structure]].  (This should be distinguished from the sense of [[Baire space]] as a space to which the [[Baire category theorem]] applies; however, $\mathbb{J}$ is an example of such a space.)

### In Archimedean integral domains

Let $R$ be an [[Archimedean integral domain]] with the [[integers]] $\mathbb{Z} \subseteq R$ being a integral subdomain of $R$.

An element $r \in R$ is irrational if for all $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, $\vert a \vert \gt 0$ and $\vert a \cdot r - b \vert \gt 0$.

The set of __irrational numbers__ in $R$ is defined as
$$\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert \gt 0) \wedge (\vert a \cdot r - b \vert \gt 0) \}$$

### In integral domains with a p-adic norm

Let $R$ be an [[integral domain]] with a [[p-adic norm]] $\vert(-)\vert_p$ for a [[prime number]] $p$, with the [[integers]] $\mathbb{Z} \subseteq R$ being a integral subdomain of $R$.

An element $r \in R$ is irrational if for all $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$, $\vert a \vert_p \gt 0$ and $\vert a \cdot r - b \vert_p \gt 0$.

The set of __irrational numbers__ in $R$ is defined as
$$\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert_p \gt 0) \wedge (\vert a \cdot r - b \vert_p \gt 0) \}$$

## History

The followers of [[Pythagoras]] believed that 'All is number', meaning what we now call (positive) [[natural numbers]].  In [[geometry]], this meant that any two lengths (or other geometric magnitudes) $x$ and $y$ are [[commensurable quantities|commensurable]] in the sense that there exists a [[unit]] length $u$ such that $x = m u$ and $y = n u$ for some natural numbers $m$ and $n$.  Identifying the ratios of geometric magnitudes with (positive) [[real numbers]], this becomes the claim that every real number is rational.  The discovery that this is *false* is also attributed to the Pythagoreans (but the legends of punishment for this secret date from several hundred years later).  Greek mathematicians developed further the theory of irrational numbers, up to the general theory of magnitudes (which we may now regard as a theory of [[positive real numbers]]) attributed to [[Eudoxus]] in Book X of [[Euclid's Elements]].

Mathematicians coming from the cultures of the Islamic Golden Age (particularly [[Abu Kamil]]) were the first to treat irrational numbers algebraically as numbers (rather than geometrically as ratios of magnitudes); they applied the [[algebra]] of [[Al-Khwarizmi]] to [[square roots]], cube [[roots]], etc.  (Ultimately, [[Omar Khayyam]] developed a general method to find the real roots of any cubic [[polynomial]].)  However, they seem to have implicitly believed that all real numbers were expressible using such roots ([[radical number]]s), which we now know is false even for some algebraic numbers, such as the real root of $x^5 + 2 x + 1$.  In any case, they only used such numbers.

Later, European mathematicians of the early modern era (particularly [[Gerolamo Cardano|Cardano]], [[Tartaglia]], and [[Lodovico Ferrari|Ferrari]]) had begun work with [[imaginary number]]s, which are necessarily irrational.  Following this, [[Johann Lambert|Lambert]] and [[Adrien-Marie Legendre|Legendre]] succeeded in proving the irrationality of [[pi]], [[e]], and their powers, which ultimately led to the conjecture that they were transcendental (whereas radical numbers, and even the root of $x^5 + 2x + 1$, are by definition [[algebraic number|algebraic]]); this conjecture was later established by [[Charles Hermite|Hermite]] and [[Ferdinand von Lindemann|Lindemann]]. Around this time, [[Leonhard Euler|Euler]] and [[Joseph-Louis Lagrange|Lagrange]] popularized [[continued fraction]]s (see below) to study both rational and irrational numbers.

During the [[arithmetization of analysis]] in the 19th century, people sometimes wrote of the problem of 'defining irrational numbers'.  The actual issue here was defining real numbers in general; one could define rational numbers algebraically, leaving only the irrational numbers as the problem.  However, this may be a red herring; one could just as easily define [[algebraic number]]s algebraically and say that the problem is defining transcendental numbers; indeed, it was only with the discovery that such numbers as $\pi$ and $\mathrm{e}$ are irrational that work on this problem came to life.  On the other hand, it\'s not clear that anybody could completely work out the [[order]] properties of algebraic numbers without already coming upon [[Richard Dedekind|Dedekind]]\'s solution.  In any case, specific irrational algebraic numbers such as $\sqrt{2}$ posed no difficulty to the [[finitist mathematics|finitist]] methods used by such algebraists as [[Leopold Kronecker]].

To this day, there are various specific real numbers (such as $\pi + \mathrm{e}$, the [[Euler number|Euler–Mascheroni constant]] $\gamma$, etc.) whose rationality or irrationality is unknown.  In [[constructive mathematics]], this makes it unproved that these numbers are rational or irrational (although the [[double negation]] of this statement can be proved for any real number).  The question of whether ${\sqrt{2}}^{\sqrt{2}}$ is rational or irrational is part of a famous illustration of the nature of constructive vs nonconstructive proof.  (Namely, there is a cheap and easy nonconstructive proof that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational: let $b$ be $\sqrt{2}$ and let $a$ be either $\sqrt{2}$ or ${\sqrt{2}}^{\sqrt{2}}$, depending on whether the latter is rational or irrational.  A constructive proof that decides which of these is the case is much harder: ${\sqrt{2}}^{\sqrt{2}}$ turns out to be irrational, by a constructive version of the [[Gelfond-Schneider theorem|Gelfond–Schneider theorem]].[^fine])

[^fine]: Of course, one could also take $a = \sqrt{2}$ and $b = 2\frac{\log 3}{\log 2})$, which are both irrational by easy constructive proofs, if one is after definite irrational numbers $a$ and $b$ such that $a^b$ is rational (here the result is $a^b = 3$). The irrationality of $\frac{\log 3}{\log 2}$ follows, as easily as that of $\sqrt{2}$, from the [[fundamental theorem of arithmetic]]: there do not exist nonzero integers $p, q$ such that $2^p = 3^q$.

## Properties

The Baire space $\mathbb{J}$ is [[homeomorphic]] to the [[product space]] $\mathbb{N}^{\mathbb{N}}$ of $\aleph_0$ copies of the [[discrete space]] of [[natural numbers]].  The homeomorphism is given by [[continued fraction]]s (see below).

Every [[inhabited space|inhabited]] [[Polish space]] is a [[quotient space]] of $\mathbb{J}$, and $\mathbb{J}$ is itself a Polish space.

As a subset of the real line, $\mathbb{J}$ is a [[full set]] (meaning that its complement, the set of rational numbers, is [[null set|null]]).

[[Cantor space]] may be identified with a [[subspace]] of $\mathbb{J}$, consisting of those irrational numbers whose continued fraction expansion consists only of $1$ and $2$ (but this does not agree with the usual inclusions into $\mathbb{R}$).

The [[fan theorem]] states precisely that $\mathbb{J}$ (when thought of as a [[topological space]]) is [[sober space|sober]] or that $\mathbb{J}$ (when thought of as a [[locale]]) is [[topological locale|topological/spatial/has enough points]].  This is true in [[classical mathematics]] and in [[intuitionistic mathematics]] but fails in other forms of [[constructive mathematics]].

## Continued fractions

(Main article: [[continued fraction]].)

Let $[a_0;a_1,a_2,a_3,\ldots]$ be an [[infinite sequence]] of [[integers]], all positive except (possibly) $a_0$.  We interpret this as the number
$$ a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}} .$$
By truncating this expression after $a_i$, we produce a rational number; altogether, this is an infinite sequence of rational numbers.

+-- {: .un_theorem}
###### Theorems

This is a [[Cauchy sequence]] whose limit is irrational.  Furthermore, every irrational number has a unique representation in this way.  Yet more, the [[bijection]] thus shown between $\mathbb{J}$ and the infinitary [[cartesian product]] $\mathbb{Z} \times \mathbb{N}^+ \times \mathbb{N}^+ \times \mathbb{N}^+ \times \cdots$ is a [[homeomorphism]] when the two sets are given their usual [[topological structure|topologies]].
=--

The usual proofs of these theorems are entirely [[constructive mathematics|constructive]].  Accordingly, in the [[foundations of mathematics]], one may define Baire space either as the space of irrational numbers or as the infinite product $\mathbb{N}^{\mathbb{N}}$.  However, to treat Baire space as a [[uniform space]] or as a [[metric space]], one uses the structure from $\mathbb{N}^{\mathbb{N}}$.

## Examples

* [[pi]]

* [[e]]

* [[golden ratio]]

## References

*  Wikipedia (English):
   *  [Baire space (set theory)](http://en.wikipedia.org/wiki/Baire_space_%28set_theory%29)
   *  [Irrational numbers → History](http://en.wikipedia.org/wiki/Irrational_number#History)

[[!redirects irrational number]]
[[!redirects irrational numbers]]

[[!redirects irrational real number]]
[[!redirects irrational real numbers]]
[[!redirects irrational real]]
[[!redirects irrational reals]]

[[!redirects weakly irrational number]]
[[!redirects weakly irrational numbers]]

[[!redirects weakly irrational real number]]
[[!redirects weakly irrational real numbers]]
[[!redirects weakly irrational real]]
[[!redirects weakly irrational reals]]

[[!redirects strongly irrational number]]
[[!redirects strongly irrational numbers]]

[[!redirects strongly irrational real number]]
[[!redirects strongly irrational real numbers]]
[[!redirects strongly irrational real]]
[[!redirects strongly irrational reals]]

[[!redirects strictly irrational number]]
[[!redirects strictly irrational numbers]]

[[!redirects strictly irrational real number]]
[[!redirects strictly irrational real numbers]]
[[!redirects strictly irrational real]]
[[!redirects strictly irrational reals]]

[[!redirects set of irrational numbers]]
[[!redirects space of irrational numbers]]


irreducible character

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

#Contents#
* table of contents
{:toc}

## Definition

A [[character of a linear representation]] (of a [[group]]) is called *irreducible* if that [[linear representation]] is an [[irreducible representation]].

## Related concepts

* [[class function]]

* [[irreducible representation]]

## References

See also:

* Wikipedia, *[Character theory](https://en.wikipedia.org/wiki/Character_theory)*

[[!redirects irreducible characters]]