opposite category

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#### Duality
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## Idea

For a [[category]] $C$, its _opposite category_ $C^{op}$ is the category obtained by formally reversing the direction of all its [[morphisms]] (while retaining their original composition law).

Categories generalize (are a [[horizontal categorification]] of) [[monoids]], [[groups]] and [[algebras]], and forming the opposite category corresponds to forming the opposite of a group, of a monoid, of an algebra.

## Definition

### In category theory

For a [[category]] $C$, the **opposite category** $C^{op}$ has the same [[objects]] as $C$, but a [[morphism]] $f : x \to y$ in $C^{op}$ is the same as a morphism $f : y \to x$ in $C$, and a composite of morphisms $g f$ in $C^{op}$ is defined to be the composite $f g$ in $C$.

More precisely, $C_{\mathrm{obj}}$ and $C_{\mathrm{mor}}$ are, respectively, the collections of [[objects]] and of [[morphisms]] of $C$, and if the structure maps of $C$ are

* source and target: $s_C,t_C : C_{\mathrm{mor}} \to C_{\mathrm{obj}}$

* identity-assignment: $i_C : C_{\mathrm{obj}} \to C_{\mathrm{mor}}$

* composition: $\circ_C : C_{\mathrm{mor}} \times_{C_{\mathrm{obj}}} C_{\mathrm{mor}} \to C_{\mathrm{mor}}$

then $C^{op}$ is the category with

* the same (isomorphic) collections of objects and morphisms

  $(C^{op})_{\mathrm{obj}} := C_{\mathrm{obj}}$

  $(C^{op})_{\mathrm{mor}} := C_{\mathrm{mor}}$

* the same identity-assigning map

  $i_{C^{op}} := i_C$

* _switched_ source and target maps

  $s_{C^{op}} := t_C$

  $t_{C^{op}} := s_C$

* the _same_ composition operation,

  $\circ_{C^{op}} := \circ_{C}$.

  or more precisely, the composition operation of $C^{op}$ is

  $$
    \circ_{C^{op}}
    :
    C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{op}} C_{\mathrm{mor}}^{op}
    =
    C_{\mathrm{mor}} {}_{t} \times_{s} C_{\mathrm{mor}}
    \stackrel{\simeq}{\to}
    C_{\mathrm{mor}} {}_{s} \times_{t} C_{\mathrm{mor}}
    \stackrel{\circ}{\to}
    C_{\mathrm{mor}}
    \,,
  $$

  where the isomorphism in the middle is the unique one induced from the universality of the [[pullback]].

Notice that hence the composition law does _not_ change when passing to the opposite category. Only the interpretation of in which direction the arrows point does change. So forming the opposite category is a completely formal process. Nevertheless, due to the switch of source and target, the opposite category $C^{op}$ is usually far from being [[equivalence of categories|equivalent]] to $C$. See the examples below.

If $V$ is a monoidal category, then $V^{op}$ is equivalent to $(\Sigma V)^{co}$ where $\Sigma V$ is the [[delooping]] of $V$, i.e. $V$ viewed as a one-object [[bicategory]] and ${co}$ is the [[opposite 2-category|opposite on 2-cells]]

### Opposite functors

Given [[categories]] $C$ and $D$,
the __opposite functor__ of a [[functor]] $F:C\to D$
is the functor $F^{op}:C^{op}\to D^{op}$
such that $F^{op}_{obj}=F_{obj}$ and $F^{op}_{mor}=F_{mor}$.

In the literature, $F^{op}$ is often confused with $F$.
This is unfortunate, since (for example)
[[natural transformations]] $F^{op}\to G^{op}$ (of functors $C^{op}\to D^{op}$)
can be identified with natural transformations $G\to F$ (and not $F\to G$).

### Opposite natural transformations

Given [[categories]] $C$ and $D$,
[[functors]] $F,G:C\to D$,
the __opposite natural transformation__
of a [[natural transformation]] $t:F\to G$
is the natural transformation $t^{op}:G^{op}\to F^{op}$,
induced by the same map $C_{obj}\to D_{mor}$ as $t$.

Again, in the literature $t^{op}$ is often confused with $t$.

### The oppositization 2-functor

The above three constructions of the opposite category,
opposite functor, and opposite natural transformation
combine together into the __oppositization 2-functor__
$$Cat^{co}\to Cat,$$
where $Cat^{co}$ denotes the 2-cell dual of the [[2-category]] $Cat$,
with the direction of 2-morphisms reversed
and the direction of 1-morphisms preserved.

### In enriched category theory ##

The definition has a direct generalization to  [[enriched category theory]].

For $V$ a [[symmetric monoidal category]] and $C$ a $V$-[[enriched category]] the **opposite $V$-enriched category** $C^{op}$ is defined to be the $V$-enriched category with the same objects as $C$ and with

$$
  C^{op}(c,d) := C(d,c)
$$

and composition given by

$$
  C^{op}(b,c)\otimes C^{op}(a,b)
  :=
  C(c,b) \otimes C(b,a)
  \stackrel{\sigma}{\to}
  C(b,a) \otimes C(c,b)
  \stackrel{\circ_C}{\to}
  C_{c,a}
  =:
  C^{op}(a,c)
  \,.
$$

The unit maps $j_a : I \to C^{op}(a,a)$ are those of $C$ under the identification $C^{op}(a,a) = C(a,a)$.

Note that the [[braiding]] of $V$ is used in defining composition for $C^{op}$.  So, we cannot define the opposite of a $V$-enriched category if $V$ is merely a [[monoidal category]], though $V$-enriched categories still make perfect sense in this case.  If $V$ is a [[braided monoidal category]] there are (at least) two ways to define "$C^{op}$", resulting in two different "opposite categories": we can use either the braiding or the inverse braiding.  If $V$ is [[symmetric monoidal category|symmetric]] these two definitions coincide.

The opposite category can be regarded as a [[dual object]] of $C$ in the [[monoidal bicategory]] $V Prof$ of $V$-categories and $V$-[[profunctors]].  (Note that this does not characterize $C^{op}$ up to equivalence, but only up to [[Morita equivalence]], i.e. up to [[Cauchy completion]].)  When $V$ is symmetric, then $V Prof$ is also symmetric monoidal, so there is only one notion of dual object.  When $V$ is braided, then $V Prof$ is not symmetric and has two notions of dual: a left dual and a right dual.  These are exactly the two different opposite categories referred to above (the "left opposite" and "right opposite").

### In higher category theory

See

* [[opposite 2-category]]

* [[opposite (∞,1)-category]]

## Properties

The [[nerve]] $N(C^{op})$ of $C^{op}$ is the [[simplicial set]] that is degreewise the same as $N(C)$, but in each degree with the order of the face and the order of the degeneracy maps reversed. See [[opposite quasi-category]] for more details.

## Classes of examples

### Opposite group
 {#OppositeGroup}

For $G = (S, \cdot)$ a [[group]] (or [[monoid]] or [[associative algebra]], etc.) with product operation

$$
  \cdot_G : S \times S \to S
$$

the **opposite group** $G^{op}$ is the group whose underlying [[set]] (underlying object, underlying vector space, etc.) is the same as that of $G$

$$
  S^{op} := S
$$

but whose product operation is that of $G$ but combined with a switch of the order of the arguments:

$$
  \cdot_{G^{op}} : S \times S \stackrel{\sigma}{\to} S \times S \stackrel{\cdot_G}{\to} S
  \,.
$$

So for $g,h \in S$ two elements we have that their product in the opposite group is

$$
  g \cdot_{G^{op}} h := h \cdot_{G} g
  \,.
$$

Now, the group $G$ may be thought of as the pointed one-object [[delooping]] [[groupoid]] $\mathbf{B}G$ which is the groupoid with a single object, with $S$ as its set of morphisms, and with $\cdot_G$ its composition operation.

Under this identification of groups with one-object categories, passing to the opposite category corresponds precisely to passing to the opposite group

$$
  (\mathbf{B}G)^{op} = \mathbf{B}(G^{op})
  \,.
$$

### Opposite of the opposite

The opposite of an opposite category is the original category:

$$
  (C^{op})^{op} = C
  \,.
$$

This is also true for $V$-enriched categories when $V$ is symmetric monoidal, but not when $V$ is merely braided.  However, in the latter case we can say $(C^{op1})^{op2} = C = (C^{op2})^{op1}$, i.e. the two different notions of "opposite category" are inverse to each other (as is always the case for left and right dualization operations in a non-symmetric monoidal (bi)category).

### Co-algebraic structures

Every algebraic structure in a category, for instance the notion of [[monoid]] in a [[monoidal category]] $C$, has a co-version, where in the original definition the direction of all morphisms is reversed -- for instance the co-version of a monoid is a [[comonoid]]. Of an [[algebra]] its a [[coalgebra]], etc.

One may express this succinctly by saying that a co-structure in $C$ is an original structure in $C^{op}$. For instance a [[comonoid]] in $C$ is a [[monoid]] in $C^{op}$.

### Duality

Passing to the opposite category is a realization of abstract [[duality]].

This goes as far as _defining_ some entities as objects in an opposite category--in particular, all generalizations of [[geometry]] which characterize [[space and quantity|spaces]] in terms of [[algebra|algebras]]. The idea of [[noncommutative geometry]] is essentially to define a category of _spaces_ as the opposite category of a category of algebras. Similarly, a [[locale]] is opposite to a [[frame]].

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Are there examples where algebras are defined as dual to spaces?
=--

Another example is the definition of the category of $L_\infty$-[[Lie infinity-algebroid|algebroids]] as that opposite to quasi-free differential graded algebras, identifying every $L_\infty$-algebra with its [[duality|dual]] [[Chevalley-Eilenberg algebra]].

## Specific examples

### Opposite of $Set$ and $FinSet$

The [[power set]]-[[functor]]

$$
  \mathcal{P} \;\colon\; Set^{op} \to Bool
$$

constitutes an [[equivalence of categories]] from the opposite category of [[Set]] to that of [[complete atomic Boolean algebras]]. See at _[Set -- Properties -- Opposite category and Boolean algebras](Set#OppositeCategory)_

Restricted to [[finite sets]] this says that  the opposite of the category [[FinSet]] of [[finite sets]] is [[equivalence of categories|equivalent]] to the category of finite [[boolean algebras]]

$$
  FinSet^{op} \simeq FinBoolAlg
  \,.
$$

See at _[FinSet -- Properties -- Opposite category](FinSet#OppositeCategory)_. See at _[[Stone duality]]_ for more.

## Related concepts

* [[opposite 2-category]]

* [[opposite model category]]

* [[opposite (infinity,1)-category]]

* [[opposite adjunction]]

## References

* [[Saunders MacLane]], §II.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

For the definition in enriched category theory see page 12 of

* [[Max Kelly]], _Basic concepts of enriched category theory_ ([pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

[[!redirects opposite categories]]

[[!redirects opposite functor]]
[[!redirects opposite functors]]

[[!redirects dual category]]
[[!redirects dual categories]]

[[!redirects op]]


opposite internal relation

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## Definition

Let $C$ be a [[finitely complete category]]. This means that $C$ is symmetric, and for objects $X$ and $Y$ there exist isomorphisms $B_{X, Y}: X \times Y \cong Y \times X$ and $B_{Y, X}: Y \times X \cong X \times Y$ such that $B_{X, Y} \circ B_{Y, X} = id_{X \times Y}$ and $B_{Y, X} \circ B_{X, Y} = id_{Y \times X}$. Given an [[internal relation]] $R\stackrel{(s,t)}\hookrightarrow X \times Y$, the [[pullback]] of the internal relation $(s,t)$ along the [[braiding]] $B_{Y, X}$ is the **opposite internal relation** [[span]]
$$R \stackrel{\dagger_{R^\op,R}}\leftarrow R^\op\stackrel{(t,s)}\hookrightarrow Y \times X$$
with a morphism $\dagger_{R,R^\op}:R \to R^\op$ such that $\dagger_{R,R^\op} \circ \dagger_{R^\op,R} = id_{R^\op}$ and $\dagger_{R^\op,R} \circ \dagger_{R,R^\op} = id_{R}$.

## See also

* [[internal relation]]

* [[opposite relation]]

* [[internal antisymmetric relation]]

[[!redirects opposite internal relations]]


opposite magma

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# Opposite magmas
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## Idea

The _opposite_ of a magma -- hence of a set with a binary operation $(x,y) \mapsto x y$ --  has the same underlying set of elements, but binary operation changed by _reversing the order_ of the factors: $(x,y) \mapsto y x$.

This is often considered for the case that $A$ a [[monoid]], [[group]], [[ring]], or algebra ([[nonassociative algebra|nonassociative]] or [[associative algebra|associative]]), in which case one speaks of the _opposite group_, _opposite ring_, etc.

## Definitions

### In $Set$

Let $A$ be a [[magma]], that is a [[set]] ${|A|}$ equipped with a [[binary operation]]

$$
  \array{
    {\left\vert A \right\vert} \times
    {\left\vert A \right\vert}
    &\longrightarrow&
    {\left\vert A \right\vert}
    \\
    (x,y)
    &\mapsto&
    x \cdot y
    \mathrlap{
      \coloneqq
    {\color{blue}x} {\color{red}y}
    }
  }
$$

written as multiplication or juxtaposition.

Then the _opposite magna_ $A^{op}$ (also denoted $A^*$ or $A^\perp$) has the same underlying set

$$
  \left\vert A^{op} \right\vert
  \;\coloneqq\;
  \left\vert A \right\vert
$$

but [[binary operation]] that of $A$ but with the [[ordering]] in the [[pair]] of arguments reversed:

\[
  \label{elementDefinition}
  \array{
    {\left\vert A^{op} \right\vert}
      \times
    {\left\vert A^{op} \right\vert}
    &\longrightarrow&
    {\left\vert A^{op} \right\vert}
    \\
    (x,y)
    &\mapsto&
    x * y
    \mathrlap{
      \coloneqq
    {\color{red}y}{\color{blue}x}
    }
    \,.
  }
\]

If $A$ is a [[monoid]] or a [[group]] (or [[semigroup]], [[quasigroup]], [[quasigroup|loop]], etc), the same definition applies, and we see that $A^op$ is again a monoid or a group (etc).

If $A$ is a [[ring]] or a $K$-[[nonassociative algebra|algebra]], the same definition applies, and we see that $A^op$ is again a ring or a $K$-algebra (including such special cases of algebra as an [[associative algebra]], [[Lie algebra]], etc).  However, one can also interpret this situation as internal to [[Ab]] or $K\,$[[Mod]]; see below.

### In other categories

The notion of magma makes sense [[internalisation|in]] any [[monoidal category]] $C$.  The notion of opposite does not make sense in this general context, because we must switch the order of the variables $x$ and $y$ in (eq:elementDefinition).  It does make sense in a [[braided monoidal category]], although now there are two ways to write it, depending on whether we use the [[braiding]] or its [[inverse morphism|inverse]] to switch the variables.  In a [[symmetric monoidal category]], the definition not only makes sense but gives the same result either way.

In particular, a magma object in $K\,$[[Mod]] is a [[nonassociative algebra]] over $K$, a monoid object in $K Mod$ is an [[associative algebra]] over $K$, and a monoid object in [[Ab]] is a [[ring]].  So all of these have opposites.

## Commutative magmas

If $A$ is [[commutativity|commutative]], then $A^op \cong A$.  In fact, this [[isomorphism]] lives [[over category|over]] $Set$ (or over the underlying monoidal category $C$), so we may write $A^op = A$ to denote this.

## Categorifications

The concept of monoid may be [[horizontal categorification|oidified]] to that of [[category]]; the concept of opposite monoid is then oidified to that of [[opposite category]].

The concept of monoid may also be [[vertical categorification|categorified]] to that of [[monoidal category]]; the concept of opposite monoid is then categorified to that of [[opposite monoidal category]].

In particular, a monoidal category $A$ has *two* kinds of opposites: one as a mere category (an oidified monoid) and one as a monoidal object (a categorified monoid).  We denote the first as $A^op$ and the second as $A^co$.

If we categorify *and* oidify, then we get the concept of [[2-category]].  Again, a $2$-category $A$ has $2$ kinds of opposites, again denoted $A^op$ and $A^co$.  So $A^op$ reverses the [[1-morphisms]] while $A^co$ reverses the [[2-morphisms]].  See [[opposite 2-category]].

An $n$-[[n-category|category]] has $n$ kinds of opposites.  See (or write) [[opposite n-category]].  A [[monoidal n-category]] has $n + 1$ kinds of opposites.

## Related concepts

* under [[delooping]] a [[monoid]] to a pointed one-object [[category]], passing to the opposite monoid corresponds to passing to the [[opposite category]].

* [[antihomomorphism]]

[[!redirects opposite magma]]
[[!redirects opposite magmas]]

[[!redirects opposite monoid]]
[[!redirects opposite monoids]]

[[!redirects opposite group]]
[[!redirects opposite groups]]

[[!redirects opposite ring]]
[[!redirects opposite rings]]

[[!redirects opposite algebra]]
[[!redirects opposite algebras]]


opposite model structure

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## Definition

\begin{definition}\label{OppositeModelCategories}
**(opposite model categories)** \linebreak

If a [[category]] $C$ carries a [[model category]] structure, then the [[opposite category]] $C^{op}$ carries the **opposite model structure**:

* its [[weak equivalences]] are those morphisms whose dual was a weak equivalence in $C$,

* its [[fibrations]] are those morphisms that were cofibrations in $C$

* its [[cofibrations]] are those that were fibrations in $C$.

\end{definition}

## Properties

\begin{prop}\label{OppositeQuillenAdjunction}
**([[opposite Quillen adjunction]])** \linebreak
  Given a [[Quillen adjunction]]
  $$
    \mathcal{D}
      \underoverset
        {\underset{R}{\longrightarrow}}
        {\overset{L}{\longleftarrow}}
        {\;\;\;\;\; \bot_{\mathrlap{{}_{Qu}}} \;\;\;\;\;}
    \mathcal{C}
    \,,
  $$
  its [[opposite adjunction]] is a Quillen adjunction
  $$
    \mathcal{D}^{op}
      \underoverset
        {\underset{L^{op}}{\longrightarrow}}
        {\overset{R^{op}}{\longleftarrow}}
        {\;\;\;\;\; \bot_{\mathrlap{{}_{Qu}}} \;\;\;\;\;}
    \mathcal{C}^{op}
  $$
  between the opposite model categories (Def. \ref{OppositeModelCategories}).
\end{prop}

## Related concepts

* [[slice model structure]]

* [[opposite category]], [[opposite 2-category]]

* [[opposite (∞,1)-category]]

## References

Textbook accounts:

* {#Hirschhorn02} [[Philip Hirschhorn]], Proposition 7.1.10 of: _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

[[!redirects opposite model structures]]

[[!redirects opposite model category]]
[[!redirects opposite model categories]]

[[!redirects opposite Quillen adjunction]]
[[!redirects opposite Quillen adjunctions]]


opposite poset

## Definition

Given a [[poset]] (or [[proset]]) $P$, its __opposite__ (or __dual__, __inverse__, __converse__, __reverse__, etc), denoted $P^{op}$ (among other ways), is the poset (or proset) with the same [[underlying set]], with $x \leq y$ in $P^op$ iff $y \leq x$ (equivalently, $x \geq y$) in the original $P$.  We say that $\geq$ (the order relation in $P^op$ is the __opposite order__.  This is a special case of both an [[opposite relation]] ($\geq$ compared to $\leq$) and an [[opposite category]] ($P^op$ compared to $P$).

### In homotopy type theory

Given a [[preorder]] or (0,1)-category $P \coloneqq (F(P), \leq)$, an __opposite preorder__ or __opposite (0,1)-category__ is a preorder $P^\op \coloneqq (F(P), \geq)$ with $\geq$ defined as

$$a:F(P), b:F(P) \vdash a \geq b \coloneqq b \leq a $$

where $F$ is the forgetful functor that gets the underlying type for a preorder.

## See also

* [[poset]]

* [[opposite relation]]

* [[opposite magma]]

* [[opposite category]]

* [[antitone]]

[[!redirects opposite poset]]
[[!redirects opposite posets]]
[[!redirects opposite proset]]
[[!redirects opposite prosets]]
[[!redirects opposite order]]
[[!redirects opposite orders]]
[[!redirects opposite preorder]]
[[!redirects opposite preorders]]

[[!redirects dual poset]]
[[!redirects dual posets]]
[[!redirects dual proset]]
[[!redirects dual prosets]]
[[!redirects dual order]]
[[!redirects dual orders]]
[[!redirects dual preorder]]
[[!redirects dual preorders]]

[[!redirects inverse poset]]
[[!redirects inverse posets]]
[[!redirects inverse proset]]
[[!redirects inverse posets]]
[[!redirects inverse order]]
[[!redirects inverse orders]]
[[!redirects inverse preorder]]
[[!redirects inverse preorders]]

[[!redirects converse poset]]
[[!redirects converse posets]]
[[!redirects converse proset]]
[[!redirects converse posets]]
[[!redirects converse order]]
[[!redirects converse orders]]
[[!redirects converse preorder]]
[[!redirects converse preorders]]

[[!redirects reverse poset]]
[[!redirects reverse posets]]
[[!redirects reverse proset]]
[[!redirects reverse prosets]]
[[!redirects reverse order]]
[[!redirects reverse orders]]
[[!redirects reverse preorder]]
[[!redirects reverse preorders]]

[[!redirects opposite (0,1)-category]]
[[!redirects opposite (0,1)-categories]]


opposite quasi-category

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## Idea

The general notion of [[opposite (∞,1)-category]] leads to a notion of **opposite of a quasi-category** , when [[(∞,1)-categories]] are incarnated as [[quasi-categories]].

So the notion of **opposite** of a [[quasi-category]] generalizes the notion of [[opposite category]] from [[category theory]].

## Definition

Under the [[relation between quasi-categories and simplicial categories]] the opposite quasi-category is that corresponding to the obvious opposite [[SSet]]-[[enriched category]]. Concretely in terms of the [[simplicial set]] $S$ underlying the quasi-category, this amounts to reversing the order of the [[simplicial identities|face and degeneracy maps]]:

$$
  S_n^{op} := S_n
$$

$$
  (d_i : S_n^{op} \to S_{n-1}^{op}) :=
  (d_{n-i} : S_n \to S_{n-1})
$$

$$
  (s_i : S_n^{op} \to S_{n+1}^{op})
  :=
  (s_{n-i} : S_n \to S_{n+1})
  \,.
$$

## References

Secton 1.2.1 in

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects opposite quasi-categories]]

[[!redirects opposite (∞,1)-category]]
[[!redirects opposite-(∞,1)-category]]

[[!redirects opposite (∞,1)-categories]]
[[!redirects opposite-(∞,1)-categories]]

[[!redirects opposite (infinity,1)-category]]
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[[!redirects opposite (infinity,1)-categories]]
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opposite relation

# Opposite relations
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{: toc}

## Definition

Given a binary [[relation]] $R$ from $X$ to $Y$, its __opposite relation__ (or __dual__, __inverse__, __converse__, __reverse__, etc) is a relation $R^{op}$ from $Y$ to $X$ as follows:

*  $b$ is $R^{op}$-related to $a$ if and only if $a$ is $R$-related to $b$.

Note that $(R^{op})^{op} = R$.

The operation $op$ is part of the requirements for [[Rel]] to be an [[allegory]].

## Examples

If $f$ is a [[function]] thought as a [[functional relation|functional]] [[entire relation|entire]] relation, then $f^{op}$ is also a function if and only if $f$ is a [[bijection]]; in that case, $f^{op}$ is the [[inverse function|inverse]] of $f$.

More generally, we have the following:

| If $R$ is ... | | then $R^{op}$ is ... |
| ------------- |-| -------------------- |
| functional    | | injective            |
| entire        | | surjective           |
| injective     | | functional           |
| surjective    | | entire               |

If $R$ is a [[partial order]] (or even a [[preorder]]), then so is $R^{op}$; so each [[poset]] (or [[proset]]) has an [[opposite poset]] (or proset), which is a special case of an [[opposite category]].

## See also

* [[opposite internal relation]]

[[!redirects opposite relation]]
[[!redirects opposite relations]]
[[!redirects opposite of a relation]]
[[!redirects opposites of a relation]]
[[!redirects opposites of relations]]

[[!redirects dual relation]]
[[!redirects dual relations]]
[[!redirects dual of a relation]]
[[!redirects duals of a relation]]
[[!redirects duals of relations]]

[[!redirects inverse relation]]
[[!redirects inverse relations]]
[[!redirects inverse of a relation]]
[[!redirects inverses of a relation]]
[[!redirects inverses of relations]]

[[!redirects converse relation]]
[[!redirects converse relations]]
[[!redirects converse of a relation]]
[[!redirects converses of a relation]]
[[!redirects converses of relations]]

[[!redirects reverse relation]]
[[!redirects reverse relations]]
[[!redirects reverse of a relation]]
[[!redirects reverses of a relation]]
[[!redirects reverses of relations]]


optic (in computer science)

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{:toc}

## Idea

**Optics** are constructions used in [[computer science]] as bidirectional data accessors. They include [[lens (in computer science)|lenses]] and prisms, used to access and modify, respectively, components of product data types and components of coproduct data types.

Unlike [[lens (in computer science)|lenses]], in general optics do not require a cartesian structure to be present. That's replaced by a bi-[[actegory|actegorical]] structure and a quotient which 'ties' residual and incoming views.

## Definition

Let $(\mathbf M, I, \otimes)$ be a symmetric monoidal category. Let $\mathbf C$ and $\mathbf D$ be $\mathbf M$-[[actegories]], i.e. there are pseudoactions $\bullet$ and $\ast$ of $\mathbf M$ acts on $\mathbf C$ and $\mathbf D$, respectively.

\begin{definition}
The category of $(\mathbf C, \mathbf D)$-mixed optics $\mathrm{Optic}_{\bullet, \ast}(\mathbf C, \mathbf D)$ has objects given by pairs $(X : \mathbf C,Y : \mathbf D)$ and hom-sets given by
$$
  \mathrm{Optic}_{\bullet, \ast}(\mathbf C, \mathbf D)((S,T),(A,B)) = \int^{M : \mathbf M} \mathbf C(S, M \bullet A) \times \mathbf D(M \ast B, T)
$$
Optics are morphisms in this category. See [Riley 2018](#Riley18) for the composition rule.
\end{definition}

\begin{remark}
Concretely, an optic $(S,T) \to (A,B)$ is specified by a triple of $(l, M, r)$ where $M : \mathbf M$ is called *residual*, $l : S \to M \bullet A$ is called 'view' or 'get' or 'forward part' or 'left part', and $r : M \ast B \to T$ is called 'update' or 'put' or 'backward part' or 'right part'.
\end{remark}

\begin{remark}
  In the non-mixed case $\mathbf M = \mathbf C = \mathbf D$, the category of optics is defined exactly as the 'double' of $\mathbf C$ as introduced by Pastro and Street in [PS07](#PS07). Their theory has been extended to the mixed case in [CEGLMPR20](#CEGLMPR20). This provides the fundamental link between optics and the theory of [[Tambara modules]], which gives to mixed-optics the alternative name of 'profunctor optics'. See ['Profunctor representation'](#Profunctor+representation).
\end{remark}

The [[coend]] in the above definition has the effect of making two optics $(l, M, r)$ and $(l', M', r')$ (with same boundaries) equal if there exists a morphism $\alpha : M \to M'$ such that $(\alpha' \bullet A) \circ l = l'$ or a morphism $\beta:M' \to M$ such that $r = r' \circ (\beta \ast B)$.

One says optics are defined _up to sliding_ along the residual, a terminology suggested by the graphical representation of optics as open diagrams, see ([Román](#Roman)).

## Profunctor representation

Let $(\mathbf M, I, \otimes)$ be a monoidal V-category with two monoidal actions $(\bullet) \colon \mathbf{M} \otimes \mathbf{C} \to \mathbf{C}$ and $(\ast) \colon \mathbf{M} \otimes \mathbf{D} \to \mathbf{D}$.
A generalized **Tambara module** is a V-profunctor $P \colon \mathbf{C}^{op} \otimes \mathbf{D} \to \mathbf{V}$ endowed with a family of morphisms
  \[t_{M,A,B} \colon P(A,B) \to P(M \bullet A, M \ast B)\]
V-natural in $A \in \mathbf{C}$ and $B \in \mathbf{D}$ and V-dinatural in $M \in \mathbf{M}$, which additionally satisfies some axioms, see [PS07](#PS07).

Generalized Tambara modules are the copresheaves of the category of optics, see [CEGLMPR20](#CEGLMPR20).

## References

A brief account of the purpose of optics is in:

* [[Bartosz Milewski]], _Optics for the Working Mathematician_, ([blog post](https://bartoszmilewski.com/2021/09/08/optics-for-the-working-mathematician/))

A similar, but more detailed account, is in:

* Emily Pillmore, [[Mario Román]], _Profunctor Optics: The Categorical View_, ([blog post](https://golem.ph.utexas.edu/category/2020/01/profunctor_optics_the_categori.html#more))

Articles:

* {#Riley18} [[Mitchell Riley]], *Categories of optics*, [[arXiv:1809.00738](https://arxiv.org/abs/1809.00738), video:[YT](https://www.youtube.com/watch?v=Qy7Y4-mgwbw)]

* {#CEGLMPR20} [[Bryce Clarke]], Derek Elkins, Jeremy Gibbons, [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, [[Mario Román]], _Profunctor optics, a categorical update_, 2020, ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))

* {#Roman} [[Mario Román]], _Open Diagrams via Coend Calculus_, ([arXiv:2004.04526](https://arxiv.org/abs/2004.04526))

[[!redirects optics (in computer science)]]


optics

> For the concept of the same name in [[computer science]] see at _[[optic (in computer science)]]_.

***

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### Context
#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

**Optics** is the study of propagation of [[light]], i.e. of [[electromagnetic waves]], especially in dependence of the ambient medium, of light [[reflection]] etc.

In the limiting case of essentially [[plane waves]] (light rays) one speaks of [[geometrical optics]].

## Related concepts

* [[electromagnetic wave]]

* [[electromagnetism]], [[QED]]

* [[photon]]

## References

Textbooks:

* Ariel Lipson, Stephen G. Lipso, Henry Lipson, _Optical Physics_, Cambridge University Press ([publisher page](https://www.cambridge.org/ae/academic/subjects/physics/optics-optoelectronics-and-photonics/optical-physics-4th-edition?format=HB&isbn=9780521493451))

See also

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


optimization theory

#Contents#
* table of contents
{:toc}

## Idea

(...)

## References

Textbook account:

* Hubertus Th. Jongen, Klaus Meer, Eberhard Triesch,  *Optimization Theory*, Springer 2004 ([doi:10.1007/b130886](https://link.springer.com/book/10.1007/b130886))

See also

* Wikipedia, *[Mathematical optimization](https://en.wikipedia.org/wiki/Mathematical_optimization)*

* MathWorld, *[Optimization theory](https://mathworld.wolfram.com/OptimizationTheory.html)*

[[!redirects mathematical optimization]]


orbifold

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###Context###
#### Higher Geometry
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[[!include higher geometry - contents]]
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#### Higher Lie theory
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[[!include infinity-Lie theory - contents]]
=--
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#Contents#
* table of contents
 {:toc}

## Idea

An [[orbifold]] is much like a [[smooth manifold]] but possibly with [[singularities]] of the form of [[fixed points]] of [[finite group|finite]] [[group]]-[[actions]].

\begin{imagefromfile}
  "file_name": "OrbifoldTypes.jpg",
  "width": 240,
  "float": "right",
  "margin": {
    "top": -30,
    "right": 0,
    "bottom": 10,
    "left": 20,
    "unit": "px"
  },
  "caption": "(from [Hyde-Ramsden-Robins 14](#HydeRamsdenRobins14))"
\end{imagefromfile}

Where a [[smooth manifold]] is a [[space]] locally modeled on [[Cartesian spaces]]/[[Euclidean spaces]] $\mathbb{R}^n$, an _orbifold_ is, more generally, a [[space]] that is locally modeled on [[Lie groupoid|smooth]] [[action groupoids]] ([[homotopy quotients]]) $\mathbb{R}^n\sslash G$ of a [[finite group]] $G$ [[action|acting]] on a [[Cartesian space]].

This turns out to be broadly captured ([Moerdijk-Pronk 97](#MoerdijkPronk97), [Moerdijk 02](#Moerdijk02)) by saying that an orbifold is a [[proper groupoid|proper]] [[étale groupoid|étale]] [[Lie groupoid]]. ([[Morita equivalence|Morita equivalent]] Lie groupoids correspond to the same orbifolds.)

The word _orbifold_ was introduced in ([Thurston 1992](#Thurston)), while the original name was _$V$-manifold_ ([Satake](#Satake)), and was taken in a more restrictive sense, assuming that the [[actions]] of [[finite groups]] on the charts are always [[effective group action|effective]]. Nowadays these are called _effective orbifolds_ and those which are global quotients by a finite group are _[[global quotient orbifolds]]_.

There is also a notion of finite stabilizers in [[algebraic geometry]]. A singular variety is called an (algebraic) _orbifold_ if it has only so-called _orbifold singularities_.

## Definition

An orbifold is a stack presented by an [[orbifold groupoid]].

## Properties

### General

* One can consider a [[bicategory]] of proper étale Lie groupoids and the orbifolds will be the objects of certain bicategorical [[localization]] of this bicategory (a result of [Moerdijk-Pronk 97](#MoerdijkPronk97)).

* Equivalently, every orbifold is globally a quotient of a smooth manifold by an [[action]] of finite-dimensional [[Lie group]] with finite [[stabilizer subgroup|stabilizers]] in each point. (eg ([Adem-Leida-Ruan 2007](#ALR07)), Corollary 1.24)

### Global quotient orbifolds
 {#GlobalQuotientOrbifolds}

In ([ALR 07, theorem 1.23](#ALR07)) it is asserted that every effective orbifold $X$ (paracompact, Hausdorff) is isomorphic to a [[global quotient orbifold]], specifically to a global quotient of $O(n)$ (where $n$ is the dimension of $X$) acting on the [[frame bundle]] of $X$ (which is a manifold).

### (Co)homology

It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to [[orbifold Euler characteristics]], [[orbifold cohomology]] etc. One of the constructions which is useful in this respect is the [[inertia orbifold]] (the inertia stack of the original orbifold) which gives rise to "twisted sectors" in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.

## Examples

* Some basic building blocks of orbifolds:

  The quotient of a  [[ball]] by a [[discrete group|discrete]] [[subgroup]] of the [[special orthogonal group]] of rotations is an orbifold, and orbifolds may be obtained by cutting out balls from ordinary [[smooth manifolds]] and gluing in these orbifold quotients.

* The [[moduli stack of elliptic curves]] over the [[complex numbers]] is an orbifold, being the [[homotopy quotient]] of the [[upper half plane]] by the [[special linear group]] acting by [[Möbius transformations]].

* For $\mathcal{G}$ any orbifold, then the [[mapping space]] $\mathcal{G}^{\Pi(S^1)} = \mathcal{G}^{B\mathbb{Z}}$ is again an orbifold, called the [[inertia orbifold]].

* [[G2-orbifolds]]

* [[lens spaces]]

* [[ADE singularities]]

* [[metric cones]]

## Related concepts

* [[presentable orbifold]], [[good orbifold]], [[very good orbifold]]

* [[étale groupoid]], [[differentiable stack]]

* [[orbispace]], [[topological stack]]

* [[Riemannian orbifold]]

  * [[flat orbifold]]

* [[Lorentzian orbifold]]

* [[complex orbifold]]

* [[symplectic orbifold]]

* [[super-orbifold]]

* [[discrete torsion]]

* [[motivation for higher differential geometry]]

* [[orbifold Euler characteristic]]

* [[orbifold K-theory]], [[orbifold differential K-theory]]

Orbifolds are in [[differential geometry]] what [[Deligne-Mumford stacks]] are in [[algebraic geometry]]. See also at _[[geometric invariant theory]]_ and _[[GIT-stable point]]_.

Orbifolds may be regarded as a kind of _[[stratified spaces]]_.

See also

* [[étale groupoid]]

* [[stable map]]

* [[orbifold cohomology]]

* [[Gromov-Witten theory]]

Orbifolds in [[string theory]]:

* [[fractional D-brane]]

  [[permutation D-brane]]

* [[ADE singularity]]

* [[orientifold]]

## References

### General

The original articles:

* {#Satake} [[Ichiro Satake]], _On a generalisation of the notion of manifold_, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 ([doi:10.1073/pnas.42.6.359]( https://doi.org/10.1073/pnas.42.6.359))

* [[Ichiro Satake]], _The Gauss–Bonnet theorem for $V$-manifolds_, J. Math. Soc. Japan 9 (1957), 464–492 ([euclid:1261153826](https://projecteuclid.org/euclid.jmsj/1261153826))

* {#Thurston} [[William Thurston]], _Three-dimensional geometry and topology,_ preliminary draft, University of Minnesota, Minnesota, (1992)

  which in completed and revised form is available as his book:

  _The Geometry and Topology of Three-Manifolds;_ ([web](http://library.msri.org/books/gt3m/))

  in particular the orbifold discussion is in [chapter 13](http://library.msri.org/books/gt3m/PDF/13.pdf)

* [[André Haefliger]], _Groupoides d'holonomie et classifiants_, Astérisque no. 116 (1984), p. 70-97 ([numdam:AST_1984__116__70_0/](http://www.numdam.org/item/AST_1984__116__70_0/))

and specifically for orbifolds in [[complex geometry]]:

* [[Walter Lewis Baily]], _On the quotient of an analytic manifold by a group of analytic homeomorphisms_, PNAS 40 (9) 804-808 (1954) ([doi:10.1073/pnas.40.9.804](https://doi.org/10.1073/pnas.40.9.804))

* [[Walter Lewis Baily]], _The Decomposition Theorem for V-Manifolds_, American Journal of Mathematics Vol. 78, No. 4 (Oct., 1956), pp. 862-888 ([jstor:2372472](https://www.jstor.org/stable/2372472))

For careful comparative review of the definitions in these original articles see [IKZ 10](#IKZ10).

Survey of basic orbifold theory:

* Daryl Cooper, Craig Hodgson, Steve Kerckhoff, _Three-dimensional Orbifolds and Cone-Manifolds_,  MSJ Memoirs Volume 5, 2000 ([pdf](https://web.math.ucsb.edu/~cooper/37.pdf), [euclid:1389985812](https://projecteuclid.org/euclid.msjm/1389985812))

* [[Ieke Moerdijk]], [[Janez Mrčun]], Section 2.4 of: _[[Introduction to foliations and Lie groupoids]]_, Cambridge Studies in Advanced Mathematics  __91__, 2003. x+173 pp. ISBN: 0-521-83197-0 ([doi:10.1017/CBO9780511615450](https://doi.org/10.1017/CBO9780511615450))

* {#BoyerGalicki07} [[Charles Boyer]], [[Krzysztof Galicki]], Chapter 4 of: _Sasakian Geometry_, Oxford Mathematical Monographs, Oxford University Press, 2007 ([doi:10.1093/acprof:oso/9780198564959.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198564959.001.0001/acprof-9780198564959))

* {#Kaye07} Adam Kaye, _Two-Dimensional Orbifolds_, 2007 ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Kaye.pdf))

* Michael Davis, _Lectures on orbifolds and reflection groups_, 2008 ([pdf](https://math.osu.edu/sites/math.osu.edu/files/08-05-MRI-preprint.pdf))

* {#Porti09} Joan Porti, _An introduction to orbifolds_, 2009 ([pdf](http://mat.uab.es/~porti/orbifoldLeiden.pdf))

* {#Snowden11} Andrew Snowden, _Introduction to orbifolds_, 2011 ([pdf](https://ocw.mit.edu/courses/mathematics/18-904-seminar-in-topology-spring-2011/final-paper/MIT18_904S11_finlOrbifolds.pdf))

* {#AdemKlaus} [[Alejandro Adem]], Michele Klaus, _Lectures on orbifolds and group cohomology_ ([pdf](http://www.math.ubc.ca/~adem/hangzhou.pdf), [[AdemKlausOrbifolds.pdf:file]])

* Francisco C. Caramello Jr, _Introduction to orbifolds_ ([arXiv:1909.08699](https://arxiv.org/abs/1909.08699))

See also

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

  (which is mainly tailored toward [Thurston's approach](#Thurston)).

Textbook account:

* {#Ratcliffe06} [[John Ratcliffe]], _Geometric Orbifolds_, chapter 13 in _Foundations of Hyperbolic Manifolds_, Graduate Texts in Mathematics 149, Springer 2006 ([doi:10.1007/978-0-387-47322-2](https://doi.org/10.1007/978-0-387-47322-2), pdf)

* [[Michael Kapovich]], Chapter 6 of: _Hyperbolic Manifolds and Discrete Groups_,  Modern Birkhäuser Classics, Birkhäuser 2008 ([doi:10.1007/978-0-8176-4913-5](https://link.springer.com/book/10.1007/978-0-8176-4913-5))

Application to [[moduli spaces of curves]] and [[moduli spaces of Riemann surfaces]]:

* Lizhen Ji, [[Shing-Tung Yau]], _Transformation Groups and Moduli Spaces of Curves_, International Press of Boston 2011 ([ISBN:9781571462237](https://www.intlpress.com/site/pub/pages/books/items/00000353/index.php))

On [[Riemannian orbifolds]]:

* Christian Lange, _Orbifolds from a metric viewpoint_ ([arXiv:1801.03472](https://arxiv.org/abs/1801.03472))

* Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, _Teichmüller theory and collapse of flat manifolds_, Annali di Matematica (2018) 197: 1247 ([arXiv:1705.08431](https://arxiv.org/abs/1705.08431), [doi:10.1007/s10231-017-0723-7](https://doi.org/10.1007/s10231-017-0723-7))

* {#HydeRamsdenRobins14} S. T. Hyde, S. J. Ramsden and V. Robins, _Unification and classification of two-dimensional crystalline patterns using orbifolds_, Acta Cryst. (2014). A70, 319-337 ([doi:10.1107/S205327331400549X](https://doi.org/10.1107/S205327331400549X))

Survey of applications in [[mathematical physics]] and notably in [[string theory]]:

* {#AdemMoravaRuan02} [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310 American Mathematical Society, 2002

Orbifolds often appear as [[moduli spaces]] in differential geometric setting:

* [[Joel W. Robbin]], [[Dietmar Salamon|Dietmar A. Salamon]], _A construction of the Deligne--Mumford [[orbifold]]_, J. Eur. Math. Society 8, Nº 4 (2006) 611--699, arXiv:[math/0407090](https://arxiv.org/abs/math/0407090) [MR2009d:32012](http://www.ams.org/mathscinet-getitem?mr=2009d:32012), _Corrigendum_, J. Eur. Math. Soc. (JEMS)  __9__  (2007),  no. 4, 901--905, [doi](https://doi.org/10.4171/JEMS/101)

The generalization of orbifolds to _weighted [[branched manifolds]]_ is discussed in

* [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 ([project euclid](http://projecteuclid.org/euclid.jsg/1180135649)).

On [[orbifolds]], [[orbifold cohomology]] and specifically on [[Chen-Ruan cohomology]] and [[orbifold K-theory]]:

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

### As Lie groupoids
 {#ReferencesAsLieGroupoids}

Discussion of orbifolds as [[Lie groupoids]]/[[differentiable stacks]]:

* {#MoerdijkPronk97} [[Ieke Moerdijk]], [[Dorette Pronk]], _Orbifolds, sheaves and groupoids_, K-theory 12 3-21 (1997) ([pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf), [doi:10.4171/LEM/56-3-4](http://dx.doi.org/10.4171/LEM/56-3-4))

* {#Moerdijk02} [[Ieke Moerdijk]], _Orbifolds as Groupoids: an Introduction_, in: [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]] (eds.) _[[Orbifolds in Mathematics and Physics]]_, Contemporary Math 310, AMS (2002), 205–222 ([arXiv:math.DG/0203100](http://arxiv.org/abs/math.DG/0203100))

* {#Lerman08} [[Eugene Lerman]], *Orbifolds as stacks?*, Enseign. Math. **56** 3-4 (2010) 315-363 [[arXiv:0806.4160](http://arxiv.org/abs/0806.4160), [doi:10.4171/LEM/56-3-4](http://dx.doi.org/10.4171/LEM/56-3-4)]

Review:

* Alexander Amenta, _The Geometry of Orbifolds via Lie Groupoids_, ANU 2012 ([arXiv:1309.6367](https://arxiv.org/abs/1309.6367))

Analogous discussion for topological orbifolds as [[topological stacks]]:

* Vesta Coufal, [[Dorette Pronk]], Carmen Rovi, [[Laura Scull]], Courtney Thatcher, _Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach_, Contemporary Mathematics 641 (2015): 135-166 ([arXiv:1401.4772](https://arxiv.org/abs/1401.4772))

Discussion of the corresponding perspective in [[algebraic geometry]], via [[Deligne-Mumford stacks]]:

* {#Kresch09} [[Andrew Kresch]], _On the geometry of Deligne-Mumford stacks_ ([doi:10.5167/uzh-21342](https://doi.org/10.5167/uzh-21342), [pdf](https://www.zora.uzh.ch/id/eprint/21342/1/geodm.pdf)), in:  D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) _Algebraic Geometry: Seattle 2005_, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 ([pspum-80-1](https://bookstore.ams.org/pspum-80-1))

The [[mapping stacks]] of orbifolds are discussed in

* [[Weimin Chen]], *On a notion of maps between orbifolds, I. Function spaces*, Commun. Contemp. Math. 8 (2006), no. 5, 569–620 ([arXiv:math/0603671](https://arxiv.org/abs/math/0603671), [doi:10.1142/S0219199706002246](https://doi.org/10.1142/S0219199706002246)).

* {#RobertsVozzo18} [[David Roberts]], [[Raymond Vozzo]], _The Smooth Hom-Stack of an Orbifold_, In : Wood D., de Gier J., Praeger C., Tao T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham (2018) ([arXiv:1610.05904](https://arxiv.org/abs/1610.05904), [doi:10.1007/978-3-319-72299-3_3](https://doi.org/10.1007/978-3-319-72299-3_3))

Discussion of [[principal bundles]] and [[fiber bundles]] over orbifolds:

* [[Camille Laurent-Gengoux]], [[Jean-Louis Tu]], [[Ping Xu]], _Chern-Weil map for principal bundles over groupoids_, Math. Z. 255, 451–491 (2007) ([arXiv:math/0401420](https://arxiv.org/abs/math/0401420), [doi:10.1007/s00209-006-0004-4](https://doi.org/10.1007/s00209-006-0004-4))

* Christopher Seaton, _Characteristic Classes of Bad Orbifold Vector Bundles_, Journal of Geometry and Physics 57 (2007), no. 11, 2365--2371 ([arXiv:math/0606665](https://arxiv.org/abs/math/0606665))

An expected relation of orbifolds ([[orbispaces]]) to [[global equivariant homotopy theory]]:

* [[Stefan Schwede]], _Orbispaces, orthogonal spaces, and the universal compact Lie group_ ([arXiv:1711.06019](https://arxiv.org/abs/1711.06019)) (on the relation to [[orbispaces]]/[[topological stacks]])

### As diffeological spaces
  {#ReferencesAsDiffeologicalSpaces}

On [[orbifolds]] regarded as naive local [[quotient spaces]] (instead of [[homotopy quotients]]/[[Lie groupoids]]/[[differentiable stacks]]) but as such formed in [[diffeological spaces]]:

* {#IKZ10} [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, _Orbifolds as diffeologies_, Transactions of the American Mathematical Society 362 (2010), 2811-2831 ([arXiv:math/0501093](https://arxiv.org/abs/math/0501093))

* {#Watts15} [[Jordan Watts]], _The Differential Structure of an Orbifold_, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 ([arXiv:1503.01740](https://arxiv.org/abs/1503.01740))

and as [[stratified space|stratified]] [[diffeological spaces]]:

* [[Serap Gürer]], [[Patrick Iglesias-Zemmour]], *Orbifolds as stratified diffeologies*, Differential Geometry and its Applications **86** (2023) 101969 [[doi:10.1016/j.difgeo.2022.101969](https://doi.org/10.1016/j.difgeo.2022.101969)]

On this approach seen in the broader context of [[cohesive homotopy theory|cohesive]] [[higher differential geometry]]:

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

* [[David Jaz Myers]], *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* [[arXiv:2205.15887](https://arxiv.org/abs/2205.15887)]

### Orbifold cobordism

Orbifold [[cobordisms]] are discussed in

* K. S. Druschel, _Oriented Orbifold Cobordism_, Pacific J. Math., 164(2) (1994), 299-319 ([doi:10.2140/pjm.1994.164.299](http://dx.doi.org/10.2140/pjm.1994.164.299), [pdf](https://msp.org/pjm/1994/164-2/pjm-v164-n2-p04-p.pdf))

* K. S. Druschel, _The Cobordism of Oriented Three Dimensional Orbifolds_, Pacific J. Math., bf 193(1) (2000), 45-55.

* Andres Angel, _Orbifold cobordism_ ([pdf](http://www.math.uni-bonn.de/people/aangel79/Orbifold%20cobordism.pdf))

See also at _[[orbifold cobordism]]_.

[[tangential structure]] on [[orbifolds]] (in the context of [[factorization homology]]):

* [[Tim Weelinck]], _Equivariant factorization homology of global quotient orbifolds_ ([arXiv:1810.12021](https://arxiv.org/abs/1810.12021), [talk pdf](https://www.maths.ed.ac.uk/~tweelinck/efhqsp.pdf))

* John Pardon, _Orbifold bordism and duality for finite orbispectra_ ([arXiv:2006.12702](https://arxiv.org/abs/2006.12702))

### In string theory
 {#ReferencesInStringTheory}

\begin{imagefromfile}
  "file_name": "GreenStringOrbifold.jpg",
  "width": 700,
  "margin": {
    "top": -30,
    "right": 0,
    "bottom": 10,
    "left": 20,
    "unit": "px"
  },
  "caption": "(from [Green 86](string+theory#Green86))"
\end{imagefromfile}

In [[perturbative string theory]], orbifolds as [[target spaces]] for a [[string]] [[sigma-model]] were first considered in

* {#DixonHarveyVafaWitten85} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds_, Nuclear Physics B Volume 261, 1985, Pages 678-686 (doi:10.1016/0550-3213(85)90593-0)

* {#DixonHarveyVafaWitten86} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds (II)_, Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (doi:10.1016/0550-3213(86)90287-7)

and then further developed notably in

* {#DVVV89} [[Robbert Dijkgraaf]], [[Cumrun Vafa]], [[Erik Verlinde]], [[Herman Verlinde]], _The operator algebra of orbifold models_, Comm. Math. Phys. Volume 123, Number 3 (1989), 485-526 ([euclid:1104178892](https://projecteuclid.org/euclid.cmp/1104178892))

* [[Eric Zaslow]], _Topological orbifold models and quantum cohomology rings_, Comm. Math. Phys. 156 (1993), no. 2, 301–331.

See also:

* Stefano Giaccari, [[Roberto Volpato]], *A fresh view on string orbifolds* [[arXiv:2210.10034](https://arxiv.org/abs/2210.10034)]

Discussion of [[blow-up]] of orbifold [[singularities]] in string theory:

* [[Paul Aspinwall]], _Resolution of Orbifold Singularities in String Theory_ ([arXiv:hep-th/9403123](https://arxiv.org/abs/hep-th/9403123))

In terms of [[vertex operator algebras]]:

* [[Yi-Zhi Huang]], _Representation theory of vertex operator algebras and orbifold conformal field theory_ ([arXiv:2004.01172](https://arxiv.org/abs/2004.01172))

For [[orbifolds]] in [[string theory]] see also the references at

* _[[Riemannian orbifold]]_, _[[toroidal orbifold]]_

* _[[fractional D-brane]]_

* _[[Gepner model]]_

* _[[orientifold]]_

* _[[RR-field tadpole cancellation]]_

Review of orbifolds in the context of string [[KK-compactifications]] and [[intersecting D-brane models]]:

* {#BailibLove99} D. Bailin, A. Love, _Orbifold compactifications of string theory_, Phys. Rept. 315 (1999) 285-408 (doi:10.1016/S0370-1573(98)00126-4, [spire:504382](https://inspirehep.net/literature/504382))

* {#Wendland01} [[Katrin Wendland]], _Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry_, in _[[Orbifolds in Mathematics and Physics]]_ ([arXiv:hep-th/0112006](https://arxiv.org/abs/hep-th/0112006))

* {#Giedt02} Joel Giedt, _Heterotic Orbifolds_ ([arXiv:hep-ph/0204315](https://arxiv.org/abs/hep-ph/0204315))

* [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, _Resolved Toroidal Orbifolds and their Orientifolds_, Adv.Theor.Math.Phys.12:67-183, 2008 ([arXiv:hep-th/0609014](https://arxiv.org/abs/hep-th/0609014))

* {#Reffert07} Susanne Reffert, _The Geometer's Toolkit to String Compactifications_, lectures at _[String and M theory approaches to particle physics and cosmology](https://www.ggi.infn.it/showevent.pl?id=11)_, 2007 ([arXiv:0706.1310](https://arxiv.org/abs/0706.1310))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], Chapter 8 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012 ([doi:10.1017/CBO9781139018951](https://doi.org/10.1017/CBO9781139018951))

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Chapter 10.5 _Toroidal orbifolds_,  of _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013

and for orbifolds of [[G2-manifolds]] for [[M-theory on G2-manifolds]]:

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv. Theor. Math. Phys. **3** (1999) 227-248 [[arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205)]

* {#Reidegeld15} [[Frank Reidegeld]], _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ [[arXiv:1512.05114](http://arxiv.org/abs/1512.05114)]

* [[Frank Reidegeld]], _$G_2$-orbifolds with ADE-singularities_ [[pdf](https://core.ac.uk/download/pdf/159317626.pdf)]

and for [[heterotic string theory|heterotic]] [[string phenomenology]]:

* [[Saul Ramos-Sanchez]], [[Michael Ratz]], *Heterotic Orbifold Models*, in *[[Handbook of Quantum Gravity]]*, Springer (2024) [[arXiv:2401.03125](https://arxiv.org/abs/2401.03125), [doi:10.1007/978-981-19-3079-9](https://doi.org/10.1007/978-981-19-3079-9)]

For [[topological strings]] the [[path integral as a pull-push transform]] for target orbifolds -- in analogy to what [[Gromov-Witten theory]] is for [[Deligne-Mumford stacks]] -- has first been considered in

* {#ChenRuan01} [[Weimin Chen]], [[Yongbin Ruan]], _Orbifold Gromov-Witten Theory_, in _[[Orbifolds in Mathematics and Physics]]_ (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 ([arXiv:math/0103156](http://arxiv.org/abs/math/0103156))
 A review with further pointers is in

* {#Abramovich05} [[Dan Abramovich]], _Lectures on Gromov-Witten invariants of orbifolds_ ([arXiv:math/0512372](http://arxiv.org/abs/math/0512372))

On orbifolds by [[2-groups]] in view of [[sigma-models]] inspired from [[string theory]]:

* [[Alonso Perez-Lona]], [[Eric Sharpe]], *Three-dimensional orbifolds by 2-groups* [[arXiv:2303.16220](https://arxiv.org/abs/2303.16220)]

category: Lie theory

[[!redirects orbifolds]]

[[!redirects effective orbifold]]
[[!redirects effective orbifolds]]


orbifold cobordism

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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[cobordisms]] of [[orbifolds]].

## References

* K. S. Druschel, _Oriented Orbifold Cobordism_, Pacific J. Math., 164(2) (1994), 299-319.

* K. S. Druschel, _The Cobordism of Oriented Three Dimensional Orbifolds_, Pacific J. Math., bf 193(1) (2000), 45-55.

* Andres Angel, _Orbifold cobordism_ ([pdf](http://www.math.uni-bonn.de/people/aangel79/Orbifold%20cobordism.pdf))

[[!redirects orbifold cobordisms]]


orbifold cohomology

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#### Cohomology
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=--
#### Higher geometry
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[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Since the crucial extra [[structures]] carried by an [[orbifold]] are

1. geometric structure (e.g. [[topology|topological]], [[algebraic geometry|algebro-geometric]], [[differential geometry|differential-geometric]], [[supergeometry|super-geometric]], etc.)

1. [[singularities]]

the [[cohomology]] of orbifolds should be such as to provide [[invariants]] which are sensitive not just to the underlying plain [[homotopy type]] of an orbifold (its [[shape modality|shape]]) but also to this extra structure. This means that _orbifold cohomology_ should, respectively, unify

1. geometric cohomology (e.g. [[sheaf cohomology|sheaf]] [[hypercohomology]], [[differential cohomology]], etc.)

1. [[equivariant cohomology]] in its fine form of [[Bredon cohomology]]

in the sense that [[differential cohomology|geometric cohomology]] is recovered away from the orbifold [[singularities]] and [[equivariant cohomology]] is recovered right at the [[singularities]], while globally orbifold cohomology provides a unification of these two aspects.

Since any concept of [[cohomology]] (as discussed there) is effectively equivalent to the choice of ambient [[(∞,1)-topos]], the question of defining orbifold cohomology is closely related to the question of how exactly to define the [[(∞,1)-category]] of orbifolds (usually a [[(2,1)-category]]) in the first place. This question is notoriously more subtle than the simple intuitive idea of orbifolds might suggest, as witnessed by the convoluted history of the concept (see e.g. [Lerman 08, Introduction](orbifold#Lerman08)).

\linebreak

### Deficiency of orbifolds as geometric groupoids

A proposal popular among [[Lie theory|Lie theorists]] ([Moerdijk-Pronk 97](orbifold#MoerdijkPronk97)) is to regard an orbifold with local charts $U_i \in G_i Actions$ ([[actions]] of some [[group]] on some local model [[space]]) as the [[geometric stack]] obtained by gluing the corresponding [[homotopy quotients]]/[[quotient stacks]] $U_i \!\sslash \!G_i$. If $\mathbf{H}$ is the ambient [[cohesive (∞,1)-topos]] in which this takes place (for instance $\mathbf{H} = $ [[Smooth∞Groupoids]], [[SuperFormalSmooth∞Groupoids]], etc.) and if $G \in Grp \overset{Disc}{\hookrightarrow} Grp(\mathbf{H})$ is a [[discrete group]] in which all the [[isotropy groups]] of the orbifold are contained, this gives an object

$$
  \left(
  \array{
    \mathcal{X}
    \\
    \downarrow^{\mathrlap{faith}}
    \\
    \mathbf{B}G
  }
  \phantom{{}^{faith}}
  \right)
  \;\in\;
  \left(
    \mathbf{H}_{/\mathbf{B}G}
  \right)_{\leq 0}
  \hookrightarrow
  \mathbf{H}_{/\mathbf{B}G}
$$

in the [[slice (∞,1)-topos]] over the [[delooping]] $\mathbf{B}G = \ast \sslash G$ of $G$, which is still a [[0-truncated object]], reflecting that as a [[functor]] of [[groupoids]] the morphism $\mathcal{X} \to \mathbf{B}G$ is a [[faithful functor]].

Accordingly, if this is -- or were -- the correct formalization of the nature of [[orbifolds]] $\mathcal{X}$, then the corresponding orbifold cohomology has [[coefficients]] given by objects $\mathcal{A} \in \mathbf{H}_{/\mathbf{B}G}$ and cohomology sets being the [[connected components]] of the [[(∞,1)-categorical hom-spaces]]

$$
  H_{\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big)
  \;\coloneqq\;
  \pi_0 \mathbf{H}_{/\mathbf{B}G}\big( \mathcal{X}, \mathcal{A}\big)
  \,.
$$

This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects _some_ equivariance aspect. For example if $\mathcal{X} = \ast \sslash G$ is the one-point orbifold with singularity given by a finite group $G$, and if $V \in G Representations$ is a [[linear representation]], with $K(V,n)\sslash G \in \infty Groupoids_{/\mathbf{B}G} \overset{Disc}{\hookrightarrow} \mathbf{H}_{/\mathbf{B}G}$ its [[Eilenberg-MacLane space]], then

$$
  H_{\mathbf{B}G}\big(
    \mathbf{B}G, K(V,n)\sslash G
  \big)
  \;\simeq\;
  H^n_{grp}(G,V)
$$

is the [[group cohomology]] of $G$ with [[coefficients]] in $V$.

However, this definition does _not_ reflect [[Bredon cohomology|Bredon]]-[[equivariant cohomology]] around the orbifold singularities. Instead, it really given (geometric/stacky refinement) of _[[cohomology with local coefficients]]_.

$\,$

### Lift of orbifolds to singular-cohesive homotopy theory

Hence the proposal of [Moerdijk-Pronk 97](orbifold#MoerdijkPronk97), that an [[orbifold]] should be regarded as a certain [[geometric stack]], is missing something. It was briefly suggested in [Schwede 17, Introduction](#Schwede17), [Schwede 18, p. ix-x](#Schwede18) that the missing aspect is provided by [[global equivariant homotopy theory]], but details seem to have been left open.

Here we discuss how to define the required orbifold cohomology in detail and in general. We combine the [[differential cohesion]] for the geometric aspect with the cohesion of [[global equivariant homotopy theory]] that was observed and highlighted in [Rezk 14](#Rezk14).

The following may serve as intuition for the issue with the nature of orbifolds:

Envision the picture of an orbifold singularity and hold a mathemagical magnifying glass over the singular point. Under this magnification you can see resolved the singular point as a fuzzy fattened point, to be called $\mathbb{B}G$.

Removing the magnifying glass, what one sees with the bare eye depends on how one squints:

* The physicist says that what he sees is a singular point, but a point after all. This is the plain [[quotient]]  $\ast = \ast / G$.

* The Lie geometer says that what she sees is a point transforming under the $G$-[[action]] that fixes it, hence the [[homotopy quotient]] [[groupoid]] $\mathbf{B}G =\ast \sslash G$.

These are two [[adjoint modality|opposite extreme aspects]] of the orbifold singularity $\mathbb{B}G$, but the orbifold singularity itself is more than both of these aspects. The real nature of an orbifold singularity is in fact a point, not a big [[classifying space]] $\mathbf{B} G$ (recall that already $\mathbf{B}\mathbb{Z}_2 = \mathbb{R}P^\infty$), but it is a point that also remembers the group [[action]], for that characterizes how the singularity is being singular:

$$
   \array{
       &&
       { \text{orbifold singularity} \atop {\mathbb{B}G} }
      \\
      & {}^{\mathllap{\boxed{\lt}}}\swarrow
      &
       {{\phantom{A}} \atop {  \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \boxed{\subset} }}
     \\
     { \text{plain quotient} \atop {\ast = \ast/G} }
     &&
     &&
     { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } }
   }
$$

This "unity of opposites" may be captured by the [[modalities]] on the [[singular-cohesive (infinity,1)-topos|singular-cohesive $\infty$-topos]]  of [[singular-smooth infinity-groupoid|singular-smooth $\infty$-groupoids]]. Its intrinsic [[cohomology]] accomodates a good notion of orbifold cohomology ([SaSc 2020](#SatiSchreiber20)).

## Related concepts

* [[orbifold]]

* [[Gromov-Witten theory]]

* [[orbifold K-theory]], [[equivariant K-theory]]

* [[twisted ad-equivariant Tate K-theory]]

* [[equivariant elliptic cohomology]]

History

According to [Abramovich 05, p. 42](#Abramovich05):

> On December 7, 1995 [[Maxim Kontsevich]] delivered a history-making lecture at Orsay, titled _String Cohomology_. This is what is now know, after [Chen-Ruan 00](#ChenRuan00), as _orbifold cohomology_, Kontsevich's lecture notes described the orbifold and  [[quantum cohomology]] of a global quotient [[orbifold]]. Twisted sectors, the age grading, and a version of orbifold stable maps for global quotients are all there.

The same lecture also introduced _[[motivic integration]]_.

## References

[[!include traditional orbifold cohomology -- references]]

### Proper orbifold cohomology

Discussion of orbifold cohomology in the context of [[Bredon cohomology]]:

* {#PronkScull07} [[Dorette Pronk]], [[Laura Scull]], _Translation Groupoids and Orbifold Bredon Cohomology_, Canad. J. Math. 62(2010), 614-645 ([arXiv:0705.3249](https://arxiv.org/abs/0705.3249), [doi:10.4153/CJM-2010-024-1](https://doi.org/10.4153/CJM-2010-024-1))

The suggestion to regard orbifold cohomology in [[global equivariant homotopy theory]]:

* {#Schwede17} [[Stefan Schwede]], Introduction of: _Orbispaces, orthogonal spaces, and the universal compact Lie group_, Mathematische Zeitschrift 294 (2020), 71-107 ([arXiv:1711.06019](https://arxiv.org/abs/1711.06019))

* {#Schwede18} [[Stefan Schwede]], p. ix-x of: _[[Global homotopy theory]]_, New Mathematical Monographs, 34 Cambridge University Press, 2018 ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

Spelling out this suggestion of [Schwede 17, Intro](#Schwede17), [Schwede 1, p. ix-x8](#Schwede18):

* {#Juran20} [[Branko Juran]], _Orbifolds, Orbispaces and Global Homotopy Theory_ ([arXiv:2006.12374](https://arxiv.org/abs/2006.12374))

Based on results of or related to:

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

The observation of the [[cohesion of global- over G-equivariant homotopy theory]] is due to:

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_, 2014

A formulation of orbifold cohomology in the [[singular-cohesive (infinity,1)-topos|singular-cohesive $\infty$-topos]] of [[singular-smooth infinity-groupoids|singular-smooth $\infty$-groupoids]] is then discussed in:

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

On the resulting [[cohesive homotopy type theory]] with a pair of commuting cohesive structures:

* [[David Jaz Myers]], [[Mitchell Riley]], *Commuting Cohesions* [[arXiv:2301.13780](https://arxiv.org/abs/2301.13780)]

[[!redirects orbifold cohomologies]]

[[!redirects orbifold cohomology group]]
[[!redirects orbifold cohomology groups]]

[[!redirects orbispace cohomology]]
[[!redirects orbispace cohomologies]]

[[!redirects orbispace cohomology group]]
[[!redirects orbispace cohomology groups]]


orbifold differential K-theory

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

#Contents#
* table of contents
{:toc}

## Idea

Orbifold differential K-theory is a generalization of [[differential K-theory]] from [[smooth manifolds]] to [[orbifolds]], hence enhancement of [[orbifold K-theory]] to [[differential cohomology]]. On [[global quotient orbifolds]] this is typically supposed to reduce to [[equivariant differential K-theory]].

## Related concepts

* [[equivariant differential K-theory]]

* [[equivariant ordinary differential cohomology]]

## References

* [[Richard Szabo]], [[Alessandro Valentino]], _Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory_, Commun. Math. Phys. 294:647-702, 2010 ([arXiv:0710.2773](https://arxiv.org/abs/0710.2773))

* [[Ulrich Bunke]], [[Thomas Schick]], _Differential orbifold K-Theory_, J. Noncommut. Geom. 7 (2013), no. 4, 1027-1104 ([arXiv:0905.4181](https://arxiv.org/abs/0905.4181))

* {#BunkeSchick10} [[Ulrich Bunke]], [[Thomas Schick]], Section 9 of: _Differential K-theory. A survey_, Global differential geometry, Springer, Heidelberg (2012), 303-358 ([arXiv:1011.6663](https://arxiv.org/abs/1011.6663), [doi:10.1088/1126-6708/1998/12/019](http://dx.doi.org/10.1088/1126-6708/1998/12/019))

[[!redirects differential orbifold K-theory]]


orbifold Euler characteristic

## Idea

[[Euler characteristic]] of [[orbifolds]]

[[!redirects orbifold Euler characteristics]]

[[!redirects Euler characteristic of an orbifold]]
[[!redirects Euler characteristics of an orbifold]]


orbifold groupoid

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=--
=--
=--

#Contents#
* table of contents
{: toc}

##Idea

A  [[groupoid]] [[presented stack|presenting]] an [[orbifold]] (as a [[stack]]) is called an **orbifold groupoid** if it satisfies certain properties. Since in the literature there are different notions of orbifolds, there are different sets of properties the presenting groupoids are required to satisfy.

The most common definition requires the presenting groupoid to be [[étale groupoid|étale]] and [[proper groupoid|proper]]. Note that these properties can be defined not only in smooth- or topological context. See for the moment [[open map]], the section on infinitesimal cohesion in [[cohesive (infinity,1)-topos]], and [[proper geometric morphism]].

## Definitions

+-- {: .num_defn}
###### Definition
(sse (smooth stable étale) groupoid ([^Mc Duff])

An **sse groupoid** is a groupoid being

* [[étale groupoid|étale]]

* [[stable groupoid|stable]]

* [[Lie groupoid|Lie]]
=--

+-- {: .num_defn}

###### Definition
(ep (étale proper) groupoid )[^Mc Duff]

An **ep groupoid** is a groupoid being

* [[étale groupoid|étale]]

* [[stable groupoid|stable]]

* [[Lie groupoid|Lie]]

* [[proper groupoid|proper]]

Since this is the most common choice of axioms a groupoid presenting an orbifold is meant to satisfy it is the default class of orbifold groupoids.

Note that this definition is redundant since properness implies stability.

Note that what [^H. Hofer] calls ''ep-groupoid'' is ''étale and proper'' but not Lie in the ordinary sense: Since the underlying category is that of sc-spaces, étale becomes sc-étale and also ''proper'' is understood in some modified sense.

=--

There is further terminology applicable to orbifold groupoids:

+-- {: .num_defn}
###### Definition
([^Mc Duff])
Let $C$ be a groupoid, let $|C|=C_0/C_1$ denote its [[orbit space]].

$C$ is called

* **nonsingular** if every $C(c,c)$ is trivial

* **effective** / **faithful**  if for ever $c\in C_0$ and for every $f\in C(c,c)$ and for every neighborhood $f\in V\subset C_1$ of $f$ there is an $f^'\in V$ such that $s(f^')\not =t(f^')$

* **(path)connected** if $|C|$ is (path)connected.
=--

+-- {: .num_defn}
###### Definition
(wnb (weighted nonsingular branched) groupoid  )[^Mc Duff]

A **wnb (weighted nonsingular branched) groupoid** is a pair $(C,\Lambda)$ where $C$ is a oriented nonsingular sse Lie groupoid and $\Lambda:|C|_H\to (0,\infty)$ is a weighting function satisfying:

For each $p\in |C|_H$ there is an open neighborhood $p\in N:=N(p)\subset |C|_H$ of $p$ and disjoint open subsets $U_1 ,...,U_j\subset \pi^{-1}_H(N)\subset C_0$ -called **local branches** and positive weights $m_1 ,...,m_j$ such that

1. (covering) $\pi^{-1}_H(N)=|U_1|\cup ...\cup |U_j|\subset |C|$

1. (local regularity) all projections $\pi_H:U_i\to |U_i|_H$ is a homeomorphism onto a relatively closed subset of $N$.

1. (weighting) $\Lambda(q)=\Sigma_{i:q\in |U_i|_H}m_i$ for all $q\in N$

where $|C|_H$ denotes the maximal Hausdorff quotient of $|C|$ (If $C$ is proper we have $|C|=|C|_H$ and $\pi_H:C_0\to |C|_H$ the canonical projection. Points $p\in |C|_H$ having more than one inverse image are called **branch points**.

The tuple $(N,U_i,m_i)=(N^p,U^p_i,m^p_i)$ is called a **local branching structure at** $p$. $C$ is called **compact** if $|C|_H$ is.

Note that here the properness axiom is relaxed.
=--

#### Relation of the axioms

Given an ep groupoid, the properness axiom implies that the orbit space of the orbifold groupoid is Hausdorff.

Properness implies stability.

## Related concepts

* [[stack]]

* [[Lie groupoid]]

* [[orbifold]]

## References

The original paper is

* [[Ieke Moerdijk]], [[Dorette A. Pronk]], _Orbifolds, Sheaves and Groupoids_.  K-Theory 12:1 (1997), 3-21.  [doi](https://doi.org/10.1023/a:1007767628271).

An expository account is in

* [[Ieke Moerdijk]], _Orbifolds as groupoids: an introduction_.  Orbifolds in Mathematics and Physics, Contemporary Mathematics (2002), 205-222.  [doi:10.1090/conm/310/05405](http://dx.doi.org/10.1090/conm/310/05405), [arXiv:math/0203100](https://arxiv.org/abs/math/0203100).

Other sources:

[^Mc Duff]: [[Dusa McDuff]], _Groupoids, branched manifolds and multisections_.  Journal of Symplectic Geometry 4:3 (2006), 259-315.
 [doi:10.4310/jsg.2006.v4.n3.a2](http://dx.doi.org/10.4310/jsg.2006.v4.n3.a2).

[^H. Hofer]: [[Helmut Hofer]], _Polyfolds and Fredholm Theory_.  Oxford University Press (2017) ([doi:10.1093/oso/9780198784913.003.0004](http://dx.doi.org/10.1093/oso/9780198784913.003.0004))

category: Lie theory

[[!redirects orbifold groupoids]]
[[!redirects orbifold groupoid]]


orbifold K-theory

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Orbifold K-theory_ is [[K-theory]] (typically: [[topological K-theory]]) of [[orbifolds]], typically meant to reduce to [[equivariant K-theory]] on [[global quotient orbifolds]].

## Definition

In general it is subtle to decide whether a given [[orbifold cohomology]]-theory $E^\bullet(\mathcal{X})$ is equivalently the $G$-[[equivariant cohomology]] of a [[topological G-space]] $X$ for any realization of $\mathcal{X}$ as a [[global quotient orbifold]] of $X$ by $G$ (as highlighted in[Pronk-Scull 07](#PronkScull07)). But for [[topological K-theory|topological]] [[equivariant K-theory]] this is the case, by [Pronk-Scull 07, Prop. 4.1](#PronkScull07).

Therefore it makes sense to _define_ the K-theory of an [[orbifold]] $\mathcal{X}$ which is equivalent to a [[global quotient orbifold]]

$$
  \mathcal{X} \simeq \prec(X \!\sslash\! G)
$$

to be the $G$-[[equivariant K-theory]] of $X$:

$$
  K^\bullet(\mathcal{X})
  \;\coloneqq\;
  K_G^\bullet(X)
  \,.
$$

This is the approach taken in [AdemRuan 01, Def. 3.4](#AdemRuan01)

## Related concepts

* [[equivariant K-theory]]

* [[twisted equivariant K-theory]]

* [[orbifold differential K-theory]]

* [[twisted ad-equivariant Tate K-theory]]

  * [[equivariant elliptic cohomology]]

## References

### General

The definition originates, via [[bundle gerbes]] and [[bundle gerbe modules]] on [[Lie groupoids]], in:

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. **245** (2004) 449-489.  ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))

The definition of the K-theory of [[global quotient orbifolds]] as the [[twisted equivariant K-theory]] of the [[universal covering space]] appears in

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 $[$[arXiv:math/0107168](https://arxiv.org/abs/math/0107168), [doi:10.1007/s00220-003-0849-x](https://doi.org/10.1007/s00220-003-0849-x)$]$

Review in

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], Section 3 of: _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

The general [[proof]] that this is well-defined (independent of the realization of the orbifold as a [[global quotient orbifold|global quotient]]) is due to

* {#PronkScull07} [[Dorette Pronk]], [[Laura Scull]], Section 4 of: _Translation Groupoids and Orbifold Bredon Cohomology_, Canad. J. Math. 62(2010), 614-645 ([arXiv:0705.3249](https://arxiv.org/abs/0705.3249), [doi:10.4153/CJM-2010-024-1](https://doi.org/10.4153/CJM-2010-024-1))

A more geometric model of orbifold K-theory in terms of [[bundles]] of [[Fredholm operators]] over [[Lie groupoids]]/[[differentiable stacks]]:

* {#FreedHopkinsTeleman07} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Loop groups and twisted K-theory I_, Journal of Topology, **4** 4 [[arXiv:0711.1906](https://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019)]

Review in:

* {#Freed12} [[Daniel Freed]], Lecture 1 of: _Lectures on twisted K-theory and orientifolds_, lecures at _[K-Theory and Quantum Fields](http://www.esi.ac.at/activities/events/2012/k-theory-and-quantum-fields)_, ESI 2012 ([[FreedESI2012.pdf:file]])

* [[Joost Nuiten]], Section 3.2.2 of: _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_ MSc thesis, Utrecht, August 2013 ([pdf](http://ncatlab.org/schreiber/files/thesisNuiten.pdf))

* [[Valentin Zakharevich]], Sections 2.2, 2.3 of: _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf), [[ZakharevichKTheoryAndVerlindeRing.pdf:file]])

The claim that these two definitions are equivalent, in that this groupoid K-theory reduces to [[equivariant K-theory]] on [[global quotient orbifolds]], is [Freed-Hopkins-Teleman 07, Prop. 3.5](#FreedHopkinsTeleman07).

Another definition of K-theory of orbifolds ("full orbifold K-theory") is due to

* Tyler J. Jarvis, Ralph Kaufmann, Takashi Kimura, _Stringy K-theory and the Chern character_, Invent. math. 168, 23–81 (2007) ([arXiv:math/0502280](https://arxiv.org/abs/math/0502280), [doi:10.1007/s00222-006-0026-x](https://doi.org/10.1007/s00222-006-0026-x))

proven there to coincide with [Adem-Ruan 01](#AdemRuan01) on global quotients.

More on this in:

* Rebecca Goldin, Megumi Harada, Tara S. Holm, Takashi Kimura, _The full orbifold K-theory of abelian symplectic quotients_, Journal of K-theory, Volume 8, Issue 2 (2011) pp. 339-362 ([arXiv:0812.4964](https://arxiv.org/abs/0812.4964), [doi:10.1017/is010005021jkt118](https://doi.org/10.1017/is010005021jkt118))

### Via global equivariant homotopy theory

The suggestion ([Schwede 17, Intro](orbifold+cohomology#Schwede17), [Schwede 18, p. ix-x](orbifold+cohomology#Schwede18)) that [[orbifolds]] should be regarded as [[orbispaces]] in [[global equivariant homotopy theory]] and then their [[orbifold cohomology]] be given by [[equivariant cohomology]] with [[coefficients]] in [[global equivariant spectra]] is worked out for ([[Bredon cohomology]] and) [[orbifold K-theory]] in:

* Branko Juran, _Orbifolds, Orbispaces and Global Homotopy Theory_ ([arXiv:2006.12374](https://arxiv.org/abs/2006.12374))

Example 5.31 there shows that on [[global quotient orbifolds]] this is again equivalent to the previous definitions.

### Stringy product

* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64.  ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))

* {#BecerraUribe09} [[Edward Becerra]], [[Bernardo Uribe]], _Stringy product on twisted orbifold K-theory for abelian quotients_, Trans. Amer. Math. Soc. 361 (2009), 5781-5803  ([arXiv:0706.3229](https://arxiv.org/abs/0706.3229), [doi:10.1090/S0002-9947-09-04760-6](https://doi.org/10.1090/S0002-9947-09-04760-6))

* Jianxun Hu, [[Bai-Ling Wang]], _Delocalized Chern character for stringy orbifold K-theory_, Trans. Amer. Math. Soc. 365 (2013), 6309-6341  ([arXiv:1110.0953](https://arxiv.org/abs/1110.0953), [doi:10.1090/S0002-9947-2013-05834-5](https://doi.org/10.1090/S0002-9947-2013-05834-5))


Orbifolds in Mathematics and Physics

This page collects pointers and material related to the book collection

* [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]] (eds.)

  **Orbifolds in Mathematics and Physics**,

  Contemporary Mathematics 310

  American Mathematical Society, 2002

  ([ISBN:978-0-8218-2990-5](https://bookstore.ams.org/conm-310))

on [[orbifolds]], in [[mathematics]] and [[mathematical physics]], notably in [[string theory]].

#Contents#
* table of contents
{:toc}

## Orbifold Gromov-Witten Theory (p. 25)

On [[Gromov-Witten theory]] for orbifolds:

* {#ChenRuan01} Weimin Chen, [[Yongbin Ruan]], _Orbifold Gromov-Witten Theory_, in _[[Orbifolds in Mathematics and Physics]]_ (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 ([arXiv:math/0103156](http://arxiv.org/abs/math/0103156))

## On orbifold elliptic genus (p. 87-105)

On orbifold [[elliptic genera]]:

* Chongying Dong, [[Kefeng Liu]], Xiaonan Ma, _On orbifold elliptic genus_ ([pdf](https://webusers.imj-prg.fr/~xiaonan.ma/mypubli/dlm02.pdf)), p. 87-105 In: [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310 AMS (2002)([ams:conm-310](https://bookstore.ams.org/conm-310))

## Loop groupoids, gerbes, and twisted sectors on orbifold (p. 163-184)

On [[inertia orbifolds]] ([[loop spaces]] for [[orbifolds]]):

* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Loop groupoids, gerbes, and twisted sectors on orbifolds_, in: *[[Orbifolds in Mathematics and Physics]]*, Madison, WI, 2001, Contemp. Math. __310__, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184, ([math.AT/0110207](http://arxiv.org/abs/math/0110207), [ISBN:978-0-8218-2990-5](https://bookstore.ams.org/conm-310), [MR2004c:58043](http://www.ams.org/mathscinet-getitem?mr=1950946))

## Orbifolds as Groupoids: an Introduction (p. 205)

On orbifolds modeled as [[Lie groupoids]]:

* {#Moerdijk02} [[Ieke Moerdijk]], _Orbifolds as Groupoids: an Introduction_, [[Alejandro Adem]], [[Jack Morava]], Yongbin Ruan (eds.) _[[Orbifolds in Mathematics and Physics]]_, Contemporary Math 310 , AMS (2002), 205–222 ([arXiv:math.DG/0203100](http://arxiv.org/abs/math.DG/0203100))

## Discrete Torsion, Quotient Stacks, and String Orbifolds (p. 301)

On [[discrete torsion]]:

* {#Sharpe01} [[Eric Sharpe]],  _Discrete Torsion, Quotient Stacks, and String Orbifolds_, in _[[Orbifolds in Mathematics and Physics]]_ ([arXiv:math/0110156](http://arxiv.org/abs/math/0110156))

## Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry (p. 333)

On [[K3]], [[KK-compactification|compactifications]]:

* {#Wendland01} [[Katrin Wendland]], _Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry_, in _[[Orbifolds in Mathematics and Physics]]_ ([arXiv:hep-th/0112006](https://arxiv.org/abs/hep-th/0112006))

## ...

(...)

category: reference


orbispace

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## Idea

An _orbispace_ is a [[space]], specifically a [[topological stack]],  that is locally modeled on the [[homotopy quotient]]/[[action groupoid]] of a [[locally compact topological space]] by the  [[action]] of a [[compact topological group]]. Thus orbispaces are to [[topological spaces]] roughly  what [[orbifolds]] are to [[manifolds]].

As to what this means precisely, there is a good deal of variance in the literature:

Following [Haefliger 90](#Haefliger90), early references on orbispaces essentially mean just the [[topological space|topological]] version of _[[orbifold]]_ (i.e. without considering [[smooth structure]]), which, in the language of [[étale stacks]], means to consider [[topological groupoids]]/[[topological stacks]] instead of [[Lie groupoids]]/[[differentiable stacks]]  ([Haefliger 84](#Haefliger84), [Haefliger 91, 5](#Haefliger91), [Chen 01](#Chen01), [Henriques 01](#Henriques01), [Henriques 05](#Henriques05)). Hence this use of the term "orbispace" is such as to complete the following patter:

$$
  \array{
    \text{smooth manifold}
    &
    \text{topological manifold}
    \\
    \text{orbifold} & {\color{blue}\text{orbispace}}
    \\
    \text{Lie groupoid} & \text{topological groupoid}
    \\
    \text{differentiable stack} & \text{topological stack}
  }
$$

However, in [Henriques-Gepner 07](#HenriquesGepner07) it was suggested that orbispaces should be these topological groupoids but regarded in [[global homotopy theory]] as the [[(∞,1)-presheaves]] on a [[global orbit category]] which they represent:

$$
  \array{
    TopologicalGroupoids
    &\overset{\prec}{\longrightarrow}&
    Orbispaces
    \\
    \mathcal{X}
    &\mapsto&
    (G \mapsto Maps(\mathbf{B}G, \mathcal{X}))
  }
$$

This use of the term has become adopted among ([[equivariant homotopy theory|equivariant]]) [[homotopy theory|homotopy theorists]] (e.g. [Rezk 14](#Rezk14), [Koerschgen 16](#Koerschgen16), [Schwede 17](#Schwede17), [Lurie EllIII](#LurieEllIII), [Juran 20](#Juran20)). But even here there are at least two variants of the definition to be distinguished, depending on whether the morphisms in the [[global orbit category]] are taken to be general or only [[injective map|injective]] [[group homomorphisms]].

## Definition

Write $Orb$ for the [[global orbit category]]. Then its [[(∞,1)-presheaf (∞,1)-category]] $PSh_\infty(Orb)$ is the [[(∞,1)-category]] of _orbispaces_. ([Henriques-Gepner 07](#HenriquesGepner07), [Rezk 14, remark 1.5.1](#Rezk14))

## Properties

### Relation of global equivariant homotopy theory

By the main theorem of ([Henriques-Gepner 07, (4)](#HenriquesGepner07)) the [[(∞,1)-presheaves]] on the global [[orbit category]] are equivalently "cellular" [[topological stacks]]/[[topological groupoids]] ("[[orbispaces]]"), we might write this as

$$
  TopGrpd^{cell} \simeq PSh_\infty(Orb)
  \,.
$$

Hence cellular topological stacks are equivalently the objects of [[equivariant homotopy theory]].

See also ([Rezk 14, p. 4 and section 7](#Rezk14))

### Relation to $G$-equivariant homotopy theory

Fixing a [[compact topological group]] $G$ and writing $\mathbf{B}G \simeq \ast // G$ for its [[delooping]] stack (the moduli stack of $G$-[[principal bundles]]), then the [[slice (infinity,1)-category|slice]] homotopy theory of topological stacks over $\mathbf{B}G$ on the [[representable morphisms]] (those inducing closed monos on isotropy groups) is equivalently that of [[topological G-spaces]] (with their G-equivariant homotoy theoretical structure, see at _[[equivariant Whitehead theorem]]_):

$$
  TopGrpd_{/\mathbf{B}G}^{reprs.} \simeq G Spaces
$$

([Henriques-Gepner 07, p.7](#HenriquesGepner07))

## Related concepts

* [[Elmendorf's theorem]]

* [[orbispace cohomology]]

[[!include equivariant homotopy theory -- table]]

## References

Orbispaces as the topological version of [[orbifolds]] (in particular as [[topological groupoids]]/[[topological stacks]]):

* {#Haefliger84} [[André Haefliger]], _Groupoides d'holonomie et classifiants_, Astérisque no. 116  (1984), p. 70-97 ([numdam:AST_1984__116__70_0](http://www.numdam.org/item/?id=AST_1984__116__70_0))

* {#Haefliger90} [[André Haefliger]], _Orbi-Espaces_,  In: E. Ghys, P. de la Harpe (eds.), _Sur les Groupes Hyperboliques d’après Mikhael Gromov_, Progress in Mathematics, vol 83. Birkhäuser 1990 ([doi:10.1007/978-1-4684-9167-8_11](https://doi.org/10.1007/978-1-4684-9167-8_11))

  (apparently the term is first used here?)

* {#Haefliger91} [[André Haefliger]], _Complexes of Groups and Orbihedra_, in: E. Ghys, A. Haefliger, A Verjovsky (eds.), _Proceedings of the Group Theory from a Geometrical Viewpoint_, ICTP, Trieste, Italy , 26 March – 6 April 1990_, World Scientific 1991 ([doi:10.1142/1235](https://doi.org/10.1142/1235))

* {#Chen01} [[Weimin Chen]], _A homotopy theory of orbispaces_ ([arXiv:math/0102020](http://arxiv.org/abs/math/0102020))

* {#Henriques01} [[André Henriques]], _Orbispaces and orbifolds from the point of view of the Borel construction, a new definition_ ([arXiv:0112006](http://arxiv.org/abs/math/0112006))

* {#Henriques05} [[André Henriques]], _Vector bundles on orbispaces_ (2005) ([pdf](http://andreghenriques.com/PDF/orbiabstract.pdf), [[HenriquesVectorBundlesOnOrbispaces.pdf:file]])

The idea to regard these topological groupoids as in [[global homotopy theory]] via the [[(infinity,1)-presheaves]] on a [[global orbit category]] which they represent is due to

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

developed further in

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

* {#Koerschgen16} [[Alexander Körschgen]], _A Comparison of two Models of Orbispaces_, Homology, Homotopy and Applications, vol. 20(1), 2018, pp. 329--358 ([arXiv:1612.04267](https://arxiv.org/abs/1612.04267))

* {#Schwede17} [[Stefan Schwede]], _Orbispaces, orthogonal spaces, and the universal compact Lie group_, Mathematische Zeitschrift 294 (2020), 71-107 ([arXiv:1711.06019](https://arxiv.org/abs/1711.06019))

* {#Juran20} [[Branko Juran]], _Orbifolds, Orbispaces and Global Homotopy Theory_ [[arXiv:2006.12374](https://arxiv.org/abs/2006.12374)]

See also

* {#LurieEllIII} [[Jacob Lurie]], Section 3 of _Elliptic cohomology III: Tempered Cohomology_, 2019 ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

[[!redirects orbispaces]]


orbit

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## Definition

### Discrete case

Given an [[action]] $G\times X\to X$ of a ([[discrete group|discrete]]) [[group]] $G$ on a set $X$, any [[set]] of the form $G x = \{g x|g\in G\}$ for a fixed $x\in X$ is called an __orbit__ of the action, or the __$G$-orbit through the point $x$__.  The set $X$ is a [[disjoint union]] of its orbits.

### Category of orbits

The _category of orbits_ of a [[group]] $G$ is the full subcategory
of the category of sets with an action of $G$.

Since any orbit of $G$ is isomorphic to the orbit $G/H$
for some group $H$, the category of $G$-orbits admits the following
alternative description: its objects are subgroups $H$ of $G$
and morphisms $H_1\to H_2$ are elements $[g]\in G/H_2$
such that $H_1\subset g H_2g^{-1}$.

In particular, the group of automorphisms of a $G$-orbit $G/H$
is $N_G(H)/H$, where $N_G(H)$ is the [[normalizer]] of $H$ in $G$.

### Topological case

If $G$ is a [[topological group]], $X$ a [[topological space]] and the action [[continuous map|continuous]], then one can distinguish [[closed subset|closed]] orbits from those which are not. Even when one starts with $G,X$ [[Hausdorff space|Hausdorff]], the space of orbits is typically non-Hausdorff.
(This problem is one of the motivations of the [[noncommutative geometry]] of Connes' school.)

If the original space is [[paracompact space|paracompact]] Hausdorff, then every orbit $G x$ as a topological $G$-space is isomorphic to $G/H$, where $H$ is the [[stabilizer subgroup]] of $x$.

## Examples

* An orbit of a [[cyclic group|cyclic]] [[subgroup]] of a [[permutation group]] is called a *[[permutation cycle]]*.

## Related concepts

* [[orbit category]], [[orbit type]]

* [[coadjoint orbit]]

* [[coinvariant]]

* [[coset space]]

* [[orbit method]]

* [[slice theorem]]

## References

Textbook accounts:

* {#Bredon72} [[Glen Bredon]], Sections I.3, I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

[[!redirects orbits]]

[[!redirects orbit space]]
[[!redirects orbit spaces]]