groupoid object in an (infinity,1)-category

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###Context###
#### Group Theory
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[[!include group theory - contents]]
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#### Internal $(\infty,1)$-Categories
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[[!include internal infinity-categories contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[group object]] in an ordinary [[category]] $C$ with [[pullback]]s is an [[internalization|internal]] [[group]]. More generally, there is the notion of an [[internal groupoid]] in a category $C$.

By the logic of [[vertical categorification]], an **internal $\infty$-group** or **[[internal ∞-groupoid]]** may be defined as a group(oid) object internal to an [[(∞,1)-category]] $C$ with [[(∞,1)-pullbacks]]. As described there, in full generality this involves not only a weakening of the usual [[associativity]] and [[unit laws]] up to [[homotopy]], but requires specification of [[coherence]] laws of these homotopies up to higher homotopy, and so on.

A _group object_ in an [[(∞,1)-category]] generalizes and unifies two familiar concepts:

* it is the generalization of the notion of groupal [[Jim Stasheff|Stasheff]] $A_\infty$-[[A-infinity-space|space]] from [[Top]] to more general [[(∞,1)-sheaf (∞,1)-toposes]]: an object that comes equipped with an associative and invertible [[monoid]] structure, up to [[coherence|coherent]] [[homotopy]], and possibly only [[horizontal categorification|partially defined]] (see also [[looping and delooping]] for more on this) ;

* it generalizes the notion of _[[equivalence relation]]_ -- or rather the [[internalization|internal]] notion of _[[congruence]]_  -- from [[category theory]] to [[(∞,1)-category theory]].

Of particular relevance are such group objects that define [[groupoid objects in an (∞,1)-topos are effective|effective quotients]]

* these are [[delooping|deloopable]];

* these generalize the notion of [[regular epimorphism]];

* these serve to characterize [[regular category|regular (∞,1)-categories]] -- such as [[∞-stack]] [[(∞,1)-topoi]] -- as those where every such object is an [[quotient object|effective quotient]].

A _groupoid object_ is then accordingly the [[horizontal categorification|many-object]] version of a [[group object]].

But notice the following. Since this is defined [[internalization|internal]] to an [[(∞,1)-category]], externally these look like genuine [[∞-groupoid]] and [[∞-group]] objects. For instance a [[group object]] in a [[(2,1)-category]] such as [[Grpd]] is, externally, a [[2-group]].

Also notice that if the ambient $(\infty,1)$-category is in fact an [[(∞,1)-topos]], then every object in there may already be thought of as an "∞-groupoid with geometric structure" (see for instance the discussion at [[cohesive (∞,1)-topos]], but this is true more generally).
The relation between the _internal groupoid objects_ then and the objects themselves is (an [[horizontal categorification|oid-ification]]) of that of [[looping and delooping]]. Notably for $G$ any internal group object (externally an [[∞-group]]) the corresponging ordinary object is its [[delooping]] object $\mathbf{B}G$, and every [[pointed object|pointed]] [[n-connected object in an (infinity,1)-topos|connected]] object in the $(\infty,1)$-topos arises this way from an internal group object.

A groupoid object

$$
 \cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0
$$

being **[[groupoid objects in an (∞,1)-topos are effective|effective]]** means that it is the [[Cech nerve]]

$$
 \cdots
  C_0 \times_{C_0//C_1} C_0 \times_{C_0//C_1} C_0
  \stackrel{\to}{\stackrel{\to}{\to}}
  C_0 \times_{C_0//C_1} C_0
 \stackrel{\to}{\to} C_0
$$

of its [[quotient stack]] $C_0 \sslash C_1$ (the  [[(∞,1)-colimit]] over its diagram)

$$
 \cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0
  \to
  C_0//C_1 := colim_i C_i
  \,.
$$

Accordingly, groupoid objects in an $(\infty,1)$-category play a central role in the theory of [[principal ∞-bundles]].

Notice that one of the four characterizing properties of an [[(∞,1)-topos]] by the higher analog of the [[Giraud theorem]] is that all [[groupoid objects in an (infinity,1)-topos are effective]].

## Definition (complete Segal-space style)

### Groupoid object

The following definition follows in style the definition of a [[complete Segal space]] object.

+-- {: .num_defn #GroupoidObjectsViaGroupoidalSegalCondition}
###### Definition
**(groupoidal [[Segal conditions]])**
\linebreak

For $C$ an [[(∞,1)-category]], a **groupoid object** in $C$ is a [[simplicial object in an (∞,1)-category]]

$$
  A : \Delta^{op} \to C
$$

such that for all partitions $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we have that

$$
  \array{
    A([n]) &\to & A(S)
    \\
    \downarrow && \downarrow
    \\
    A(S') &\to& A(\{s\})
  }
$$

is a [[(∞,1)-pullback]] diagram in $C$. Here, by a partition $S \cup S'$ of $[n]$ that share precisely one vertex $s$, we mean two subsets $S$ and $S'$ of $\{0,1,\ldots,n\}$ whose union is $\{0,1,\ldots,n\}$ and whose intersection is the singleton $\{s\}$. The linear order on $[n]$ then restricts to the linear order on $S$ and $S'$.

The $(\infty,1)$-category of groupoid objects in $C$ is the full [[sub-(∞,1)-category]]

$$
  Grpd(C) \hookrightarrow Func(\Delta^{op}, C)
$$

of the [[(∞,1)-category of (∞,1)-functors]] on those objects that are groupoid objects.

=--

This is [[Higher Topos Theory|HTT, prop. 6.1.2.6, item 4'']] with [[Higher Topos Theory|HTT, def. 6.1.2.7]].

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

If one requires the above condition only for those partitions that are order-preserving, then this yields the definition of a (pre-)[[category object in an (∞,1)-category]].

=--

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

A groupoid object $A : \Delta^{op} \to C$ is the **[[Cech nerve]]** of a morphism $A_0 \to B$ if $A$ is the restriction of an augmented simplicial object $A^+ : \Delta^{op}_a \to C$ with $A^+_0 \to A^+_{-1}$ as the morphism $A_0 \to B$, such that the sub-[[diagram]]

  $$
    \array{
      A^+_1 &\to& A^+_0
      \\
      \downarrow && \downarrow
      \\
      A^+_0 &\to& A^+_{-1}
    }
  $$

  of $A^+$ is a [[(∞,1)-pullback]] diagram in $C$.

=--

This is [[Higher Topos Theory|HTT, below prop. 6.1.2.11]].

If $A$ is the [[Cech nerve]] of a morphism $A_0 \to A_{-1}$
then the groupoid object is [[delooping|deloopable]] in the groupoid sense.

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

A groupoid object $A : \Delta^{op} \to C$ is an **effective [[quotient object]]** if the [[(∞,1)-colimit]] diagram $A^+ : \Delta_a^{op} \to C$ exists, such that $A$ is the [[Cech nerve]] of $A^+_0 \to A^+_{-1}$, i.e. of $A_0 \to \lim_\to A_\bullet$.

=--

### Group object

A **group object** is a groupoid object $U : \Delta^{op} \to C$ for which $U_0 \simeq *$ is a [[terminal object in a quasi-category|terminal object]].

([[Higher Topos Theory|HTT, def. 7.2.2.1]]).

It follows ([[Higher Topos Theory|HTT, prop. 7.2.2.4]]) that a group object is of the form

$$
  U = \left(
      \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
  \,.
$$

### Relation to $(\infty,1)$-quotients {#RelationToQuotients}

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

A groupoid object in an $(\infty,1)$-category

$$
\left(  \cdots  A_2  \stackrel{\to}{\stackrel{\to}{\to}}
   A_1 \stackrel{\to}{\to} A_0
  \right)
$$

is the [[(∞,1)-category]] analog of an internal [[equivalence relation]] on $A_0$, which is just a pair of morphisms

$$
   R \stackrel{\to}{\to} A_0
   \,.
$$

The [[colimit]] ([[coequalizer]]) of the latter diagram is the [[quotient]] of $A_0$ by the relation $R$.

Analogously, the [[(∞,1)-colimit]]

$$
  \lim_\to (\Delta^{op} \stackrel{A}{\to} C)
$$

over the simplicial diagram $A : \Delta^{op} \to C$ is the corresponding **$(\infty,1)$-quotient**.

If we are given a [[model category]] [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] $C$, then this [[(∞,1)-colimit]] is presented by a [[homotopy colimit]] over the corresponding simplicial diagraM a **homotopy quotient** .

=--

## Properties

### Equivalent characterizations
 {#EquivalentCharacterizations}

We state in prop. \ref{EquivalentCharacterizationsViaSlicesAndPowering} below a list of equivalent conditions that characterize a [[simplicial object in an (∞,1)-category]] as a groupoid object. This uses the following basic notions, which we review here for convenience.

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

For $K \in $ [[sSet]] a [[simplicial set]], write $\Delta_{/K}$ for its [[category of simplices]]. For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ a simplicial object, write

$$
  X[K] \colon \Delta^{op}_{/K} \to \Delta^{op} \stackrel{X}{\to} \mathcal{C}
$$

for the precomposition of $X_\bullet$ with the canonical projection. Moreover, write

$$
  X(K) \coloneqq \underset{\leftarrow}{\lim} X[K]
$$

for the [[(∞,1)-limit]] over this composite [[(∞,1)-functor]] in $\mathcal{C}$ (if it exists). (Notice: square brackets for the composite functor, round brackets for its $(\infty,1)$-limit.)

=--

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

For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent

1. the induced morphism of cone $(\infty,1)$-categories $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an [[equivalence of (∞,1)-categories]];

1. the induced morphism of [[(∞,1)-limits]] $X(K) \to X(K')$ is an [[equivalence in an (∞,1)-category|equivalence]].

=--

(The first perspective is used in ([Lurie](#Lurie)), the second in ([Lurie2](#Lurie2)).)

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

In one direction: the limit is the [[terminal object in an (∞,1)-category|terminal object]] in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as ([Lurie, prop. 4.1.1.8](#Lurie))).  Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

=--

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

Let $\mathcal{C}$ be an $(\infty,1)$-category incarnated explicitly as a [[quasi-category]]. Then a [[simplicial object in an (∞,1)-category|simplicial object in]] $\mathcal{C}$ is a groupoid object if the following equivalent conditions hold.

1. If $K \to K'$ is a morphism in [[sSet]] which is a [[weak homotopy equivalence]] and a [[bijection]] on [[vertices]], then the induced morphism on [[slice-(∞,1)-categories]]

   $$
     \mathcal{C}_{/X[K]}
     \to
     \mathcal{C}_{/X[K']}
   $$

   is an [[equivalence of (∞,1)-categories]] (a [[weak equivalence]] in the [[model structure for quasi-categories]]).

1. For every $n \geq 2$ and every $0 \leq i \leq n$, the morphism $\mathcal{C}_{/X[\Delta^n]} \to \mathcal{C}_{/X[\Lambda^n_i]}$ is an weak equivalence in the [[model structure for quasi-categories]]

1. (...)

Using remark \ref{EquivalenceOfConeCategoriesAndLimits} this means equivalently that the simplicial object $X_\bullet$ is a groupoid precisely if the following

1. $X_\bullet$ satisfies the ordinary [[Segal conditions]] and the morphism $X(\Delta^2) \to X(\Lambda^2_0)$ is an [[equivalence in an (∞,1)-category|equivalence]].

1. (...)

=--

The first items appear as ([Lurie, prop. 6.1.2.6](#Lurie)). The second ones appear in the proofs of ([Lurie2, prop. 1.1.8, lemma 1.2.25](#Lurie2)).

### The $(\infty,1)$-category of groupoid objects

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

The $(\infty,1)$-category of groupoid objects in $C$ is a [[reflective sub-(∞,1)-category]]

$$
  Grpd(C)
  \stackrel{\overset{}{\leftarrow}}{\hookrightarrow}
  Func(\Delta^{op}, C)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 6.1.2.9]]. In nice cases the image of this reflective subcategory are the effective epimorphisms:

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

If $C = \mathbf{H}$ in an [[(∞,1)-semitopos]] there is a natural [[equivalence of (∞,1)-categories]]

$$
  Grpd(\mathbf{H})
  \simeq
  (\mathbf{H}^I)_{eff}
$$

between the $(\infty,1)$-category of groupoid objects in $\mathbf{H}$ and the full [[sub-(∞,1)-category]] of the [[arrow category]] of $\mathbf{H}$ (the [[(∞,1)-functor (∞,1)-category]] $Func(\Delta[1], \mathbf{H})$) on the [[effective epimorphism in an (∞,1)-category|effective epimorphisms]].

=--

This appears below [[Higher Topos Theory|HTT, cor. 6.2.3.5]].

### Cech nerves

Write $\Delta_a$ for the augmented [[simplex category]] (including the object $[-1]$).

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

An augmented simplicial object $A^+ : \Delta_a^{op} \to C$ is the right [[Kan extension]] of its restriction to $[-1]$ and $[0]$

$$
  \array{
    \{[-1] \leftarrow [0]\} &\stackrel{A^+|_{\leq 0}}{\to}& C
    \\
    \downarrow & \nearrow_{\mathrlap{A^+}}
    \\
    \Delta_a^{op}
  }
$$

precisley if $A^+|_{\geq 0}$ is a groupoid object in $C$ and the [[diagram]]

$$
  \array{
    A_1 &\to& A_0
    \\
    \downarrow && \downarrow
    \\
    A_0 &\to& A_{-1}
  }
$$

is a [[(∞,1)-pullback]] in $C$.

=--

$A$ is called the **[[Cech nerve]]** of $A_0 \to A_{-1}$ if the equivalent conditions of this proposition are satisfied.

### Effective quotients
  {#Effective}

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

In $C = $ [[∞Grpd]] every groupoid object is an effective quotient, def. \ref{EffectiveGroupoid}.

=--

This is [[Higher Topos Theory|HTT, below remark 6.1.2.15]] and [[Higher Topos Theory|HTT, cor. 6.1.3.20]].

More generally, this is true for every [[(∞,1)-topos]].

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

In $C$ is an [[(∞,1)-topos]], then every groupoid object in $C$ is an effective quotient, def. \ref{EffectiveGroupoid}.

=--

This is [[Higher Topos Theory|HTT, theorem 6.1.0.6 (4) iv)]].

### Delooping
  {#Delooping}

For $\mathcal{X}$ an [[(∞,1)-category]] with [[(∞,1)-pullback]]s and for
$x : * \to X$ a [[pointed object|pointed]] object in $\mathcal{X}$, its [[loop space object]] at $x$ is the [[(∞,1)-pullback]]

$$
  \Omega_x X := {*} \prod_{X} {*}
$$

hence the object universally filling the diagram

$$
  \array{
    \Omega_x X &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{x}}
    \\
    * &\stackrel{x}{\to}& X
  }
  \,.
$$

Since this is the beginning of the [[Cech nerve]] of $* \to X$, $\Omega_x X$ is naturally equipped with the structure of an $\infty$-group object in $\mathcal{X}$.

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

Let $\mathcal{X}$ be an [[(∞,1)-topos]]. Then the operation of forming [[loop space]] objects constitutes an [[equivalence of (∞,1)-categories]]

$$
  \Omega : PointedConnected(\mathcal{X}) \stackrel{\simeq}{\to}
   Grp(\mathcal{X})
$$

from the full [[sub-(∞,1)-category]] of the [[over-(∞,1)-category|under-(∞,1)-category]] $*/\mathcal{X}$ of [[pointed object]]s on those that are also [[0-connected]] (hence those that have an essentially unique point) with the $(\infty,1)$-category of group objects in $\mathcal{X}$.

=--

This is [[Higher Topos Theory|HTT, lemma 7.2.2.11 (1)]]

The inverse to $\Omega$ we write

$$
  \mathbf{B} : Grp(\mathcal{X}) \to PointedConnected(\mathcal{X})
  \,.
$$

For $G \in Grp(\mathcal{X})$ we call $\mathbf{B}G$ its [[delooping]].

## Examples

### Group objects in an $(\infty,1)$-topos

When the ambient [[(∞,1)-category]] is an [[(∞,1)-topos]] then -- by the $\infty$-Giraud axioms -- all groupoid objects are [[quotient object|effective]], meaning that for

$$
  \mathbf{B}G = \lim_{\to} U_\bullet
$$

the [[(∞,1)-colimit]] over the group object $U_\bullet$ we have that $U_\bullet$ is reproduced as the Cech nerve of $* \to \mathbf{B}G$

$$
  \left(
      \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *
  \right)
  \simeq
  \left(
      \cdots
   {*}\times_{\mathbf{B}G}{*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\stackrel{\to}{\to}} {*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\to} *
  \right)
  \,.
$$

The object $\mathbf{B}G$ is the [[delooping]] object of the group object $G$.

For more on this see also [[principal ∞-bundle]].

### Models for group objects in $\infty Grpd$ {#ModelsInInfGrpd}

There is a [[model category]] structure that presents the [[(∞,1)-category]] of group objects in [[∞Grpd]]: the [[∞-group]]s.

* The group objects $G$ themselves are modeled by a model structure on the category $sGrp$ of [[simplicial group]]s.

* Their delooping spaces $\mathbf{B}G$ are modeled by a model structure on the category $sSet_0$ of [[simplicial set]]s with a single vertex.

The operation of forming [[loop space object]]s constitutes a [[Quillen equivalence]] between these two model structures

$$
  \Omega : sSet_0 \stackrel{\simeq_{Quillen}}{\to} sGrp
  \,.
$$

The Quillen equivalence itself is in section 6 there.

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

There exists the [[transferred model structure]] on the category $sGrp$ of [[simplicial group]]s along the [[forgetful functor]]

$$
  U : sGrp \to sSet_{Quillen}
$$

to the standard [[model structure on simplicial sets]].

This means that a morphism in $sGrp$ is a

* weak equivalences

* or fibration

precisely if it is so in $sSet_{Quillen}$.

=--

This appears as ([GoerssJardine, ch V, theorem. 2.3](GoerssJardine)).

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

There is a [[model structure on reduced simplicial sets]] $sSet_0$ ([[simplicial sets]] with a single vertex) whose

* weak equivalences

* and cofibrations

are those in the standard [[model structure on simplicial sets]].

=--

This appears as ([GoerssJardine, ch V, prop. 6.2](GoerssJardine)).

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

The simplicial loop space functor $G$ and the delooping functor $\bar W(-)$ (discussed at [[simplicial group]]) constitute a [[Quillen equivalence]]

$$
  (G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_0
  \,.
$$

The $(G \dashv \bar W)$-[[unit of an adjunction|unit]] and counit are weak equivalences:

$$
  X \stackrel{\simeq}{\to} \bar W G X
$$

$$
  G \bar W G \stackrel{\simeq}{\to} G
  \,.
$$

=--

This appears as ([GoerssJardine, ch. V prop. 6.3](#GoerssJardine)).

## Related concepts

* [[∞-group]]

* [[groupoid object]], [[enriched groupoid]], [[simplicial groupoid]]

* [[groupoid objects in an (∞,1)-topos are effective]]

* [[monoid]]. [[monoid object]], [[monoid object in an (∞,1)-category]]

* [[group]], [[group object]]

* [[ring]], [[ring object]]

* [[looping and delooping]]

## References

Groupoid objects in $(\infty,1)$-categories are the topic of section 6.1.2 in

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

Model category presentations of groupoid objects in $\infty Grpd$ by groupoidal [[complete Segal spaces]] are discussed in

* [[Julia Bergner]],

  _Adding inverses to diagrams encoding algebraic structures_, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. ([arXiv:0610291](http://arxiv.org/abs/math/0610291))

  _Adding inverses to diagrams II: Invertible homotopy theories are spaces_, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. ([web](http://www.intlpress.com/hha/v10/n2/a9/), [arXiv:0710.2254](http://arxiv.org/abs/0710.2254))

A standard textbook reference on $\infty$-groups in the [[classical model structure on simplicial sets]] is

* {#GoerssJardine} [[Paul Goerss]], [[Rick Jardine]], chapter V of _[[Simplicial homotopy theory]]_ [chapter V](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-5.dvi).

Discussion from the point of view of [[category objects in an (∞,1)-category]] is in

* {#Lurie2} [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

[[!redirects groupoid object in an (∞,1)-category]]
[[!redirects groupoid objects in an (∞,1)-category]]
[[!redirects groupoid objects in an (infinity,1)-category]]
[[!redirects group object in an (infinity,1)-category]]
[[!redirects group object in an (∞,1)-category]]
[[!redirects group objects in an (infinity,1)-category]]
[[!redirects group objects in an (∞,1)-category]]

[[!redirects groupoid object in an infinity1-category]]


groupoid objects in an (∞,1)-topos are effective

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[higher category theory|higher]] generalization ([[categorification]]) of how all [[equivalence relations]] in a [[Grothendieck topos]] have [[effective quotients]], so in an [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-sheaf]] [[(infinity,1)-topos|$\infty$-topos]] all "$\infty$-equivalence relations", constituted by [[groupoid object in an (infinity,1)-category|$\infty$-groupoid objects]] have [[effective epimorphism in an (infinity,1)-category|effective $\infty$-quotients]]. Where the former statement is one of the [[Giraud axioms]] that characterize [[sheaf toposes]], the latter is one of the [[Giraud-Rezk-Lurie axioms]] that characterize [[(infinity,1)-category of (infinity,1)-sheaves|$\infty$-sheaf]] [[(infinity,1)-topos|$\infty$-toposes]].

## Statement

In an [[(infinity,1)-topos|$\infty$-topos]] $\mathbf{H}$, let $X_\bullet \,\colon\, \Delta^{op} \xrightarrow{\phantom{---}} \mathbf{H}$ be a [[simplicial object]] and write

\[
  \label{InfinityQuotientCoprojection}
  X_0
  \xrightarrow{\;\; q \;\;}
  \left\vert X_\bullet \right\vert
  \,\coloneqq\,
  \underset{\underset{[n] \in \Delta^{op}}{\longrightarrow}}{\lim} X_n
  \;\;\;
  \in
  \;
  \mathbf{H}
\]

for the *$\infty$-quotient coprojection*, i.e. induced universal morphism from $X_0$ to the [[(infinity,1)-colimit|$\infty$-colimit]]. Then:

\begin{proposition}
\label{GroupoidObjectsInAnInfinityToposAreEffective}
**(groupoid objects in an $\infty$-topos are effective)**
\linebreak
If $X_\bullet$ is a [[groupoid object in an (infinity,1)-category|groupoid object]] in that it satisfies the groupoidal [[Segal conditions]], then the $\infty$-quotient coprojection (eq:InfinityQuotientCoprojection) is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] in that $X_\bullet$ [[equivalence in an (infinity,1)-category|equivalent]] to its [[Cech nerve]]:

$$
  \underset{[n] \in \Delta^{op}}{\prod}
  \Big(
    X_n
    \;\simeq\;
    \underset{n \; \text{factors}}
    {
      \underbrace{
        X_0
          \underset{\left\vert X_\bullet\right\vert}{\times}
           \cdots
          \underset{\left\vert X_\bullet\right\vert}{\times}
        X_0
      }
    }
  \Big)
$$
\end{proposition}
([Lurie 2009, Def. 6.1.2.14, Prop. 6.1.0.6 (3.iv)](#Lurie09))

Morever, this correspondence extends to morphisms:
\begin{proposition}
\label{EquivalenceBetweenGroupoidsAndEffectiveEpimorphisms}
**(equivalence between groupoids and effective epimoirphisms)**
\linebreak
In any [[(infinity,1)-topos|$\infty$-topos]] $\mathbf{H}$
the correspondence of Prop. \ref{GroupoidObjectsInAnInfinityToposAreEffective}
extends to an  [[equivalence of (infinity,1)-categories|equivalence of $\infty$-categories]]
\[
  \label{EquivalenceOfEffectiveEpimorphisms}
  Grpd(\mathbf{H})
  \underoverset
    {
       \underset{
         (-)_0
          \to
         \underset{\longrightarrow}{\lim}(-)
       }
       {\longrightarrow}
    }
    {
       \overset{
         (-)^{\times_\bullet}
       }
       {\longleftarrow}
    }
    {\sim}
  \big(
    \mathbf{H}^{\Delta[1]}
  \big)_{eff}
\]
between
the [[groupoid objects in an (infinity,1)-category|groupoid objects in $\mathbf{H}$]]
and
the [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of the [[arrow category]] on the [[effective epimorphism in an (infinity,1)-category|effective epimorphisms]].
\end{proposition}
([Lurie 2009, below Cor. 6.2.3.5](#Lurie09))

Here the [[(infinity,1)-functors|$\infty$-functors]] are the [[restriction]] of  $L \colon Func(\Delta^{op}, \mathbf{H}) \to \Func([1], \mathbf{H})$ which sends a [[simplicial object]] to the universal morphism from $X_0$ to its [[(infinity,1)-colimit|$\infty$-colimit]] and $R \colon \Func([1], \mathbf{H}) \to Func(\Delta^{op}, \mathbf{H})$ sends a morphism to its [[Cech nerve]].

This follows since the correspondence in both directions is computed by taking a Kan extension to $Func(\Delta_+^{op}, \mathbf{H})$ followed by a restriction, and this identifies both sides with the same full subcategory of $Func(\Delta_+^{op}, \mathbf{H})$.

\begin{remark}
\label{InterpretationInTermsOfInfinityStacksWithAtlases}
**(interpretation in terms of $\infty$-stacks equipped with atlases)**
\linebreak
  In as far as every object $\mathcal{X} \,\in\, \mathbf{H}$ in an [[(infinity,1)-topos|$\infty$-topos]] may be thought of an an [[infinity-stack|$\infty$-stack]], an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] $X \twoheadrightarrow \mathcal{X}$ is an *[[atlas]]* for this [[infinity-stack|$\infty$-stack]] and its [[Cech nerve]] is the $\infty$-groupoid that *presents* the $\infty$-stack.

For example, if $\mathbf{H} \,=\,$ [[D-topological infinity-groupoids|$DTopGrpd_\infty$]] or $=$ [[smooth infinity-groupoid|$SmthGrpd_\infty$]] and $X \twoheadrightarrow \mathcal{X}$ is such that the [[Cech nerve]] takes values in [[0-truncated]] [[concrete objects]] (using that these are [[cohesive (infinity,1)-topos|cohesive $\infty$-toposes]]), then this recovers the traditional terminology in the field of [[topological groupoids]]/[[topological stacks]] and [[diffeological groupoids]]/[[differentiable stacks]].

From this perspective, Prop. \ref{EquivalenceBetweenGroupoidsAndEffectiveEpimorphisms} says that all [[(infinity,1)-toposes|$\infty$-toposes]] verify the expected relation between [[internal groupoids]] and [[geometric stacks]] equipped with [[atlases]].
For example, in this traditional terminology, a "[[Morita morphism]]" or "[[Hilsum-Skandalis morphism]]" between [[internal groupoids]] $X_\bullet$, $Y_\bullet$ is a morphism between their associated $\infty$-stacks $\left\vert X_\bullet \right\vert \to \left\vert Y_\bullet\right\vert$, and, conversely, Prop. \ref{EquivalenceBetweenGroupoidsAndEffectiveEpimorphisms} says that a morphism of $\infty$-stacks equipped with compatible [[atlases]] becomes equivalently a morphism of presenting groupoids.

See [p. 27](https://arxiv.org/pdf/2008.01101.pdf#page=27) in [[schreiber:Proper Orbifold Cohomology|Sati & Schreiber 2020]].
\end{remark}

## Literature

* {#Lurie09} [[Jacob Lurie]], *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

[[!redirects groupoid objects in an infinity1-topos are effective]]

[[!redirects groupoid objects in an (infinity,1)-topos are effective]]


groupoid of Lie-algebra valued forms

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

For $\mathfrak{g}$ a [[Lie algebra]], the _groupoid of $\mathfrak{g}$-valued forms_ is the [[groupoid]] whose [[object]]s are [[differential form|differential 1-form]]s with values on $\mathfrak{g}$, and whose [[morphism]]s are [[gauge transformation]]s between these.

This carries the structure of a generalized [[Lie groupoid]] $\mathbf{B}G_{conn}$ , which is a differential refinement of the [[delooping]] [[Lie groupoid]] $\mathbf{B}G$ of the [[Lie group]] $G$ corresponding to $\mathfrak{g}$:

its $U$-parameterized smooth families of objects are _[[Lie algebra]] valued [[differential form]]s_ on $U$. Its $U$-parameterized families of morphisms are gauge transformations of these forms by $G$-valued [[smooth function]]s on $U$.

A [[cocycle]] with coefficients in $\mathbf{B}G_{conn}$ is a [[connection on a bundle]].

For more discussion of this see ∞-Lie groupoid -- Lie groups.

## Definition

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

For $G$ a [[Lie group]] the **groupoid of $Lie(G)$-valued differential forms** is as a [[groupoid]] [[internal category|internal to]] [[smooth spaces]], the [[sheaf]] of groupoids
$$
  \bar \mathbf{B}G := G TrivBund_\nabla(-) := [P_1(-), \mathbf{B}G]
$$
that to a smooth test space $U \in Diff$ assigns the [[functor category]] $[P_1(U),\mathbf{B}G]$ of smooth functors (functors internal to [[smooth spaces]]) from the [[path groupoid]] $P_1(U)$ of $U$ to the one-object [[delooping]] groupoid $\mathbf{B}G$.
=--

## Properties

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

The groupoid $\bar \mathbf{B}G$ is canonically equivalent to the smooth groupoid where

* objects are smooth $\mathfrak{g}$-valued 1-forms $A \in \Omega^1(U, \mathfrak{g})$;
* morphisms $h : A \to A'$ are given by smooth $G$-valued functions $h \in C^\infty(U,G)$ such that
  $$
    A' = Ad_h(A) - h^* \bar \theta
  $$

Here $\bar \theta$ is the right invariant [[Maurer-Cartan form]] on $G$. A common way to write this is   $A' = Ad_h(A) + h d h^{-1}$.

=--

A proof is in [SchrWalI](https://arxiv.org/abs/0705.0452).

### Differential nonabelian cohomology

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

The [[cohomology]] with coefficients in $\bar \mathbf{B}G$ classifies $G$-[[principal bundles]] [[connection on a bundle]] with connection.

More is true: there is a natural canonical equivalence of groupoids
$$
  \mathbf{H}_{Diff}(X, \bar \mathbf{B}G)
  \simeq
  G Bund_\nabla(X)
  \,.
$$
=--

* There is the obvious projection
  $$
    \bar \mathbf{B}G \to \mathbf{B}G
    \,.
  $$

* Lifting a $G$-cocycle through this projection to a differential $G$-cocycle means equipping it with a [[connection on a bundle|connection]].

For $G = U(n)$ these differential cocycles model the [[Yang-Mills field]] in physics.

* For $G = U(1)$ the sheaf $\bar \mathbf{B}U(1)(-)$ coincides with the the [[Deligne cohomology|Deligne complex]] in degree 2, $\bar \mathbf{B}U(1)\simeq \mathbb{Z}(2)_D^\infty$, as described there.

  * This corresponds to the [[electromagnetic field]].

## Related concepts

* **groupoid of Lie-algebra valued forms**

* [[2-groupoid of Lie 2-algebra valued forms]]

* [[3-groupoid of Lie 3-algebra valued forms]]

* [[∞-groupoid of ∞-Lie-algebra valued forms]]

## References

Details are in

* [[Urs Schreiber|U.S.]], [[Konrad Waldorf]], _Parallel Transport and Functors_ (web)

The definition in terms of differential forms is def 4.6 there. The equivalence to $[P_1(-), \mathbf{B}G]$ is proposition 4.7.

See also [[∞-Chern-Weil theory introduction]]

[[!redirects Lie-algebra valued 1-form]]
[[!redirects Lie-algebra valued 1-forms]]
[[!redirects Lie-algebra valued differential 1-form]]
[[!redirects Lie-algebra valued differential 1-forms]]

[[!redirects groupoid of Lie algebra valued forms]]

[[!redirects groupoid of Lie-algebra valued 1-forms]]

[[!redirects Lie algebra valued 1-forms]]
[[!redirects Lie algebra valued 1-form]]

[[!redirects Lie-algebra valued differential forms]]

[[!redirects Lie algebra-valued form]]
[[!redirects Lie algebra-valued 1-form]]
[[!redirects groupoid of Lie algebra-valued forms]]

[[!redirects groupoid of Lie algebra-valued 1-forms]]
[[!redirects groupoid of Lie algebra valued 1-forms]]

[[!redirects Lie algebra-valued 1-forms]]

[[!redirects Lie algebra-valued differential form]]
[[!redirects Lie algebra-valued differential forms]]

[[!redirects Lie algebra valued form]]
[[!redirects Lie algebra valued differential form]]
[[!redirects Lie algebra valued forms]]
[[!redirects Lie algebra valued differential forms]]

[[!redirects Lie algebra valued differential 1-forms]]

[[!redirects Lie algebra-valued differential 1-form]]


groupoid principal bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Where a [[principal bundle]] with [[structure group]] $G$ (typically at least a [[topological group]]) is a [[fiber bundle]] with [[typical fiber]] $G$ and with [[action]]/[[transition functions]] given by multiplication in $G$, the notion of a *groupoid principal bundle* generalizes this to *structure [[groupoids]]* $\mathcal{G}$ (typically at least [[topological groupoids]]) and [[typical fibers]] being the [[source]]-[[fibers]] of $\mathcal{G}$ (the [[subspaces]] of [[morphisms]] all emanating from a given [[object]]), with [[action]]/[[transition functions]] given by [[composition]] of [[morphisms]] in $\mathcal{G}$.

Under the identification of [[groups]] with [[pointed object|pointed]] [[connected groupoid|connected]] [[groupoids]] (see at [[looping and delooping]]), groupoid principal bundles subsume ordinary [[principal bundles]].

Conversely, a general groupoid principal bundle has [[typical fibers]] that may ([[isomorphism|isomorphically]]) vary even over connected base spaces in a controlled way.

## Definition

For $G$ a [[group object]] in some [[(∞,1)-topos]] $\mathbf{H}$ (for instance $\mathbf{H} = $ [[Smooth∞Groupoids]] for smooth [[Lie groupoid]]-bundles), and $\mathbf{B}G$ the corresponding [[delooping]] object, $G$-principal bundles are the [[(∞,1)-pullback]]s of the form

$$
  \array{
    P &\to& *
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{B}G
  }
  \,.
$$

One equivalently (with non-negligible but conventional chance of confusion of terminology) calls such $P \to X$ a $\mathbf{B}G$-**groupoid principal bundle.

So more generally, for $\mathcal{G}$ any groupoid object with collection $\mathcal{G}_0$ of [[object]]s, the $(\infty,1)$-pullbacks

$$
  \array{
    P &\to& \mathcal{G}_0
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathcal{G}
  }
$$

are **groupoid principal bundles** .

## Examples

###Lie groupoid principal bundles

For $\mathbf{H}$ = $\infty LieGrpd$ and $G:=G_1 \Rightarrow G_0$ a [[Lie groupoid]], a $G$-principal bundle is locally of the form

$$
  U_i \times \mathcal{G}_{x_i}
$$

for $\mathcal{G}_{x_i}$ the target fiber over an object $x_i$, or equivalently of the form $U_i\times_{G_0} G_1$.

More precisely, we have the following sub-definition for **Lie groupoid principal bundles**: Given a [[Lie groupoid]] $G=G_1 \Rightarrow G_0$,  a $G$-principal bundle over a manifold $M$ (with right $G$ action) is a surjective submersion $P\xrightarrow{\pi} M$ together with a moment map $P\xrightarrow{\rho} G_0$ (sometimes also called an anchor), such that there is a (right) **$G$-action** on $P$, that is, there is map $\Phi: P\times_{G_0} G_1 \to P$, such that

* $\pi(\Phi(p, g))=\pi(p)$,

* $ \rho(\Phi(p,g))=s(g)$,

* $\Phi(\Phi(p, g_1), g_2)=\Phi(p, g_1 g_2)$.

Moreover, we require the $G$-action to be free and proper, that is,

* [free] $P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is injective,

* [proper]$P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is a proper map , i.e. the preimage of a compact is compact.

These two conditions are equivalent to the fact that the [[shear map]]

$$ P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times_M P$$

is an [[isomorphism]].

## Applications

* Regarded as groupoid [[bibundles]], groupoid principal bundles serve as models for [[Morita morphisms]] between [[Lie groupoids]].

## Related concepts

* [[principal bundle]] / [[torsor]] / **groupoid principal bundle**

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]]

* [[higher differential geometry]]

## References

* [[Ieke Moerdijk]], [[Janez Mrcun]] _Introduction to Foliations and Lie Groupoids_, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003) ([doi:10.1017/CBO9780511615450](https://doi.org/10.1017/CBO9780511615450))

[[!redirects groupoid principal bundles]]


groupoid quantale

## Idea

Let $G$ be a group or a discrete groupoid. Recall that an associative [[ring]] is a [[semigroup]] (if unital then monoid) in the category of abelian groups and a [[quantale]] a semigroup in the category of [[sup-lattice]]s.

While [[groupoid algebra|group(oid) ring]] $\mathbb{Z}G$ is the free abelian group with basis $G$ (resp. $Mor(G)$) with convolution product making it a ring, the groupoid quantale is the free sup-lattice with basis $G$ and the convolution product making it a quantale.

## Construction-definition

The free sup-lattice on a set $X$ is the power set $P(X)$ (set of subsets of $X$) with the infinite union and finite intersection as sup $\Vee$ (infinite join) and meet $\wedge$ respectively; or equivalently the set $2^X$ of boolean functions on $X$.
If $X = Mor(C)$ is a morphism set of a category (special cases: semigroup, group, groupoid)
then $P(Mor(C))$ has the multiplication

$$
 A B = \{ a b \,|\, a\in A, b\in B, a\,\,\, and\,\,\, b\,\,\, compose \}
$$
or, in terms of functions (the characteristic function of the subsets of $C$) of $f,g\in 2^{Mor(C)}$, the __convolution product__

$$
(f \ast g)(c) = \Vee \{ f(a)\wedge f(b)\,|\, c = a b \}
$$

This way, $2^{Mor(C)}$ becomes a quantale denoted simply $2^C$,
which may be called a category quantale. This
quantale is unital; namely the unit in $P(Mor(C))$ is the set of units $Ob(C) = C_0$ of the quantale which is canonically identified with a subset of $Mor(C)$. The unit for the convolution is, equivalently, the characteristic function of $C_0$.

If $C = G$ is a groupoid, then there is an additional involution $\star$ on $P(Mor(G))$ where

$$
X^\star = \{ g^{-1} \,|\, g\in X \}
$$

for $X \subset Mor(G)$. In terms of the convolution quantale $2^G$,

$$
f^*(g) = f(g^{-1}), \,\,\,\,f\in 2^{Mor(G)}, g\in G.
$$

This groupoid quantale is therefore an involutive unital quantale.

## Literature

* [[Pedro Resende]], _Étale groupoids and their quantales_, Advances in Mathematics 208 (2007) 147–209 [doi](http://dx.doi.org/10.1016/j.aim.2006.02.004)


groupoid representation

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

#Contentsrepre
* table of contents
{:toc}

## Idea

A _groupoid representation_ is a [[representation]] of a [[groupoid]].

## Definition

+-- {: .num_defn #GroupoidRepresentation}
###### Definition
**([[groupoid representation]])

Let $\mathcal{G}$ be a [[groupoid]]. Then:

A [[linear representation]] of $\mathcal{G}$ is a groupoid homomorphism ([[functor]])

$$
  \rho \;\colon\; \mathcal{G} \longrightarrow Core(Vect)
$$

to the groupoid [[core]] of the category [[Vect]] of [[vector spaces]] ([this example](groupoid#CoreGroupoid)). Hence this is

1. For each object $x$ of $\mathcal{G}$ a [[vector space]] $V_x$;

1. for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a
   [[linear map]] $\rho(f) \;\colon\; V_x \to V_y$

such that

1. (respect for composition) for all composable morphisms $x \overset{f}{\to}y \overset{g}{\to} z$ in the groupoid we have an [[equality]]

   $$
     \rho(g) \circ \rho(f) = \rho(g \circ f)
   $$

1. (respect for identities) for each object $x$ of the groupoid we have an equality

   $$
     \rho(id_x) = id_{V_x}
     \,.
   $$

Similarly a _[[permutation representation]]_ of $\mathcal{G}$ is a groupoid homomorphism ([[functor]])

$$
  \rho \;\colon\; \mathcal{G} \longrightarrow Core(Set)
$$

to the groupoid core of [[Set]]. Hence this is

1. For each object $x$ of $\mathcal{G}$ a [[set]] $S_x$;

1. for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a
   [[function]] $\rho(f) \;\colon\; S_x \to S_y$

such that composition and identities are respected, as above.

For $\rho_1$ and $\rho_2$ two such representations, then a homomorphism of representations

$$
  \phi \;\colon\; \rho_1 \longrightarrow \rho_2
$$

is a [[natural transformation]] between these functors, hence is

* for each object $x$ of the groupoid a (linear) function

  $$
    (V_1)_x \overset{\phi(x)}{\longrightarrow} (V_2)_x
  $$

* such that for all morphisms $x \overset{f}{\longrightarrow} y$
  we have

  $$
    \phi(y) \circ \rho_1(f) = \rho_2(x) \circ \phi(x)
    \phantom{AAAAAA}
    \array{
      (V_1)_x &\overset{\phi(x)}{\longrightarrow}& (V_2)_x
      \\
      {}^{\mathllap{\rho_1(f)}}\downarrow && \downarrow^{\mathrlap{\phi_2(f)}}
      \\
      (V_1)_y &\underset{\phi(y)}{\longrightarrow}& (V_2)_y
    }
  $$

A permutation representation of $\mathcal{G}$ is often called a "$\mathcal{G}$-set" (see at _[[G-set]]_) and the category of permutation representations is also often denoted

$$
  \mathcal{G}Set
  \phantom{AAAAA}
  \text{or}
  \phantom{AAAAA}
  Set^{\mathcal{G}}
$$

=--

## Properties

+-- {: .num_cor #GroupoidRepresentationsAreProductsOfGroupRepresentations}
###### Corollary
**([[groupoid representations]] are [[product category|products]] of [[group representations]])

Assuming the [[axiom of choice]] then the following holds:

Let $\mathcal{G}$ be a [[groupoid]]. Then its [[category of representations|category of]] groupoid representations is [[equivalence of categories|equivalent]] to the [[product category]] indexed by the set of [[connected components]] $\pi_0(\mathcal{G})$ ([this def.](groupoid#GroupoidConnectedComponents)) of [[group representations]]  of the [[automorphism group]] $G_i \coloneqq Aut_{\mathcal{G}}(x_i)$ ([this def.](groupoid#InGrupoidAutomorphismGroup)) for $x_i$ any object in the $i$th connected component:

$$
  Rep_{Grpd}(\mathcal{G})
   \;\simeq\;
  \underset{i \in \pi_0(\mathcal{G})}{\prod} Rep(G_i)
  \,.
$$

=--

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

Let $\mathcal{C}$ be the category that the representation is on. Then by definition

$$
  Rep_{Grpd}(\mathcal{G}) = Hom( \mathcal{G} , \mathcal{C}  )
  \,.
$$

Consider the injection functor of the [[skeleton]] (from [this lemma](groupoid#DeloopingGroupoidEquivalence))

$$
  inc
  \;\colon\;
  \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i
  \overset{}{\longrightarrow}
  \mathcal{G}
  \,.
$$

By [this lemma](groupoid#HmotopiesWithMorphismsHorizontaComposition) the pre-composition with this constitutes a functor

$$
  inc^\ast \;\colon\; Hom( \mathcal{G}, \mathcal{C} ) \longrightarrow Hom( \underset{i \in \pi_0(\mathcal{G})}{\sqcup} B G_i, \mathcal{C} )
$$

and by combining [this lemma](groupoid#DeloopingGroupoidEquivalence) with [this lemma](groupoid#HorizontalCompositionWithHomotopyIsNaturalTransformation)
this is an [[equivalence of categories]]. Finally, by [this example](groupoid#GroupoidRepresentationOfDeloopingGroupoid) the
category on the right is the product of group representation categories as claimed.

=--

## Examples

+-- {: .num_example #GroupoidRepresentationOfDeloopingGroupoid}
###### Example
**([[groupoid representation]] of [[delooping]] groupoid is [[group representation]])

If $B G$ is the [[delooping]] groupoid of a [[group]] $G$ ([this example](groupoid#GroupoidFromDelooping)),
then a [[groupoid representation]] of $B G$ according to def. \ref{GroupoidRepresentation} is
equivalently a [[group representation]] of the group $G$:

$$
  Rep_{Grpd}(B G) \simeq Rep(G)
  \,.
$$

=--

+-- {: .num_example }
###### Example
**([[fundamental theorem of covering spaces]])**

For $X$ a [[topological space]] then forming [[monodromy]] is a [[functor]] from the [[category of covering spaces]] over $X$ to that of [[permutation representations]] of the [[fundamental groupoid]] of $X$:

$$
  Fib \;\colon\; Cov(C)  \longrightarrow Set^{\Pi_1(X)}
  \,.
$$

If $X$ is [[locally path-connected topological space|locally path connected]] and [[semi-locally simply connected topological space|semi-locally simply connected]], then this is an [[equivalence of categories]]. See at _[[fundamental theorem of covering spaces]]_ for details.

=--

[[!redirects groupoid representations]]

[[!redirects representation of groupoids]]
[[!redirects representations of groupoids]]


groupoid-principal infinity-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of a $G$-[[principal ∞-bundle]] over an [[∞-group]] $G$ as $G$ is generalized to a [[groupoid object in an (∞,1)-category]].

## Definition

For $\mathbf{H}$ an [[(∞,1)-topos]] and $\mathcal{G}_\bullet \in Grp_\infty(\mathbf{H})$ a [[groupoid object in an (∞,1)-category]], a **$\mathcal{G}_\bullet$-principal $\infty$-bundle** over $X$ is

* a morphism $P \to X$

* equipped with an anchor $a \colon P \to \mathcal{G}_0$ and a [[groupoid ∞-action]] of $\mathcal{G}_\bullet$ on $(P,a)$ over $X$;

* such that $P \to X \simeq (P//\mathcal{G})$ is the corresponding quotient map.

## Related concepts

* [[∞-action]], [[principal ∞-bundle]]

* [[bibundle]]

[[!redirects groupoid-principal bundle]]
[[!redirects groupoid-principal bundles]]

[[!redirects groupoid-principal infinity-bundles]]

[[!redirects ∞-groupoid-principal ∞-bundle]]
[[!redirects ∞-groupoid-principal ∞-bundles]]

[[!redirects groupoid-principal ∞-bundle]]
[[!redirects groupoid-principal ∞-bundles]]

[[!redirects infinity-groupoid-principal infinity-bundle]]
[[!redirects infinity-groupoid-principal infinity-bundles]]


groupoidification

#Idea#

__Groupoidification__ is a program based on the observation that the operation of pull-pushing bundles of [[groupoid|groupoids]]

$$
  \array{
     \Psi
     \\
     \downarrow
     \\
     X
  }
$$

through [[span|spans]]

$$
  \array{
    && S
    \\
    & \swarrow && \searrow
    \\
    X &&&& Y
  }
$$

of [[groupoid|groupoids]] becomes a linear map acting on vector spaces after taking [[groupoid cardinality]] -- after "degroupoidification".

From another perspective, these over-groupoids are an [[example for geometric function objects]] as considered in the context of [[geometric function theory]].

#Remarks#

* Groupoidification is closely related to -- or maybe even synonymous to -- [[geometric function theory]].

De-groupoidification is similar to passing to [[motivic functions]].

#References#

John Baez keeps a web page with relevant links and background material

* John Baez _Groupoidification_ ([web](http://math.ucr.edu/home/baez/groupoidification/))

In particular there are the articles in preparation

* John Baez, Alexander Hoffnung, Christopher Walker, _Higher-dimensional algebra VII: Groupoidification_, [arxiv/0908.4305](http://arxiv.org/abs/0908.4305)

* John Baez, Alexander Hoffnung, _Higher-dimensional algebra VIII: The Hecke Bicategory_, ([pdf](http://math.ucr.edu/home/baez/hecke.pdf))

* _Towards topological groupoidification_ ([pdf](https://pdfs.semanticscholar.org/presentation/b157/90badcd7350ba0551f6eb362f5912780c5dd.pdf))

#Relation to representation theory#

Groupoidification seems to be a central underlying governing principle in [[representation theory]] in its incarnation in  [[geometric function theory]].

#Relation to quantization#

Groupoidification in particular seems to illuminate structures encountered in the context of [[quantum field theory]]. Discussions of groupoidification in the context of QFT are

* Jeffrey Morton, _Categorified algebra and quantum mechanics_, Theory and Application of Categories 16 (2006), 785-854 ([arXiv](http://xxx.lanl.gov/abs/math.QA/0601458), [tac](http://www.emis.de/journals/TAC/volumes/16/29/16-29abs.html))

* Jeffrey C. Morton, _2-Vector Spaces and Groupoids_ ([arXiv](http://arxiv.org/abs/0810.2361))

* [pdf](http://www.theoreticalatlas.net/talks/gpd-khovanov.pdf)

Some related remarks are in

* Schreiber: [[schreiber:Nonabelian cocycles and their sigma model QFTs]]

* Cafe: [new_paper_on_the_hecke_bicateg](http://golem.ph.utexas.edu/category/2010/04/new_paper_on_the_hecke_bicateg.html)


groupoids > history

< [[groupoids]]
[[!redirects groupoids -- history]]


Groupprops

*[Groupprops](https://groupprops.subwiki.org/wiki/Main_Page)* is a public [wiki](http://en.wikipedia.org/wiki/Wiki) on [[group theory]].

## Related projects

* [[GroupNames]]

category: reference


Grover's algorithm

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A search [[algorithm]] for [[quantum computers]] which is expected to produce quadratic speedup (from $O(N)$ to $O(\sqrt{N})$) compared to classical computers.

## Related concepts

* [[Shor's algorithm]]

* [[Deutsch-Jozsa algorithm]]

* [[quantum complexity theory]]

* [[quantum supremacy]]

## References

Due to:

* [[Lov K. Grover]], *A fast quantum mechanical algorithm for database search*, STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing (1996) 212-219 [[arXiv:quant-ph/9605043](https://arxiv.org/abs/quant-ph/9605043), [doi:10.1145/237814.237866](https://doi.org/10.1145/237814.237866)]

Survey:

* Renato Portugal, *Basic Quantum Algorithms* [[arXiv:2201.10574](https://arxiv.org/abs/2201.10574)]

Lecture notes:

* [[Scott Aaronson]], Chapters 22-24 of: *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

Realization in [[experiment]] with [[quantum error correction|error detecting codes]] realizing [[quantum advantage]]:

* Bibek Pokharel, Daniel Lidar, *Better-than-classical Grover search via quantum error detection and suppression* [[arXiv:2211.04543](https://arxiv.org/abs/2211.04543)]

Implementation in [[Quipper]]:

* Safat Siddiqui, Mohammed Jahirul Islam, Omar Shehab, §4.4 in: *Five Quantum Algorithms Using Quipper* [[arXiv:1406.4481](https://arxiv.org/abs/1406.4481)]

See also

* Wikipedia, _[Grover's algorithm](https://en.wikipedia.org/wiki/Grover%27s_algorithm)_

Discussion in terms of [[quantum information theory via dagger-compact categories]]:

* [[Jamie Vicary]], Section 3 of: _The Topology of Quantum Algorithms_, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 93-102 ([arXiv:1209.3917](https://arxiv.org/abs/1209.3917))

Review in:

* Ali Nabi Duman, _The Topology of Grover Algorithm_ ([arXiv:1405.4425](https://arxiv.org/abs/1405.4425))

* Kamil Khadiev, §§5-9 in: *Lecture Notes on Quantum Algorithms* [[arXiv:2212.14205](https://arxiv.org/abs/2212.14205)]


Grp

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

#Contents#
* table of contents
{:toc}

## Definition

__$Grp$__ is the [[category]] with [[group|groups]] as [[objects]] and [[group homomorphisms]] as [[morphisms]].

Similarly there there is the [[full subcategory]] $FinGrp \hookrightarrow Grp$ of [[finite groups]].

More generally, if $\mathcal{C}$ is any [[category]] with [[finite products]], there is a [[category]] $Grp(\mathcal{C})$ of [[group objects]] in $\mathcal{C}$. This category if [[equivalence of categories|equivalent]] to the category $Prod(T_{Grp}, \mathcal{C})$ of product-preserving functors from the [[Lawvere theory]] for groups to $\mathcal{C}$.  For instance for $\mathcal{C} = $ [[SmthMfd]] this yields the category of [[Lie groups]]; while for $\mathcal{C} = $ [[Set]] it reduces again to the default case.

##Remarks##

If one associates to a group $G$ its [[delooping]] one-object [[groupoid]] $B G$, it is sometimes of interest to regard the collection of groups instead as a [[2-category]], namely as the [[full subcategory|full]] sub-$2$-category of [[Grpd]] on one-object groupoids. In this case the $2$-morphisms between homomorphisms of groups come from "intertwiners": [[inner automorphism]]s of the target group -- hence this 2-category is _not_ equivalent to the 1-category of groups.

On the other hand, if we regard $Grp$ as a full sub-$2$-category of $Grpd_*$, the $2$-category of *[[pointed object|pointed]]* groupoids, then this is [[locally discrete 2-category|locally discrete]] and [[equivalence of categories|equivalent]] to the ordinary [[1-category]] $Grp$.  This is because the only *pointed* intertwiner between two homomorphisms is the identity.

Precisely analogous statements hold for the category [[Alg]] of algebras.

## Properties

(In this section, all statements about $Grp$ are valid more generally for $Grp(E)$ where $E$ is a [[topos]] with a [[natural numbers object]].)

The category $Grp$ is one of the prototypical examples of a [[semiabelian category]], and so enjoys some nice properties. For example, it is [[regular category|regular]] and even [[Barr-exact category|exact]], and [[protomodular category|protomodular]] so that one can expect a certain battery of [diagram chasing lemmas](/nlab/show/diagram+chasing+lemmas+-+contents) to hold in it.

The category of groups is also [[balanced category|balanced]]. This follows from a somewhat remarkable theorem:

+-- {: .num_theorem #eq}
###### Theorem
Every [[monomorphism]] in $Grp$ is an [[equalizer]].
=--

The proofs most commonly seen in the literature are elementary but nonconstructive; a typical example may be found [here](/nlab/show/regular+monomorphism#grpreg) at [[regular monomorphism]]. Here we give a [[constructive mathematics|constructive]] proof.

+-- {: .proof}
###### Proof
Let $i: H \to G$ be monic, and let $\pi: G \to G/H$ be the canonical [[surjective function]] $g \mapsto g H$. Let $A$ be the [[free object|free]] [[abelian group]] on $G/H$ with $j: G/H \to A$ the canonical injection, and let $A^G$ denote the set of functions $f: G \to A$, with the pointwise abelian group structure inherited from $A$. This carries a $G$-[[module]] structure defined by

$$(g \cdot f)(g') = f(g' g).$$

For any $f \in A^G$, the function $d_f: G \to A^G$ defined by $d_f(g) = g f - f$ defines a [[derivation]], i.e., a map satisfying the equation $d_f(g g') = g d_f(g') + d_f(g)$. Consider now the wreath product, i.e., the [[semidirect product]] $A^G \rtimes G$, where the multiplication is defined by $(f, g) \cdot (f', g') = (g f' + f, g g')$. By the derivation equation, we have a homomorphism $\phi: G \to A^G \rtimes G$ defined by $\phi(g) \coloneqq (d_{j \pi}(g), g)$, and there is a second homomorphism $\psi: G \to A^G \rtimes G$ defined by $\psi(g) \coloneqq (0, g)$. We claim that $i: H \to G$ is the equalizer of the pair $\phi, \psi$. For,

$$\array{
d_{j\pi}(g) = 0 & \text{iff} & (\forall_{g': G})\; g\cdot j\pi(g') = j\pi(g') \\
 & \text{iff} & (\forall_{g': G})\; j\pi(g' g) = j\pi(g') \\
 & \text{iff} & (\forall_{g': G})\; j(g' g H) = j(g' H) \\
 & \text{iff} & (\forall_{g': G})\; g' g H = g' H \\
 & \text{iff} & g H = H \\
 & \text{iff} & g \in H.
}$$

(All we needed was *some* injection $j: G/H \to A$ into an abelian group; embedding into the free abelian group is a pretty canonical choice.)
=--

+-- {: .num_remark}
###### Remark
This proof can be adapted to show that monomorphisms in the category of [[finite set|finite]] groups (group objects in $FinSet$) are also equalizers. All that needs to be modified is the choice of $A$, which we could take to be a free $\mathbb{F}_2$-[[vector space]] generated by $G/H$.
=--

+-- {: .num_remark}
###### Remark
The proof requires that the [[monad|unit]] $X \to U F X$ of the free abelian group monad (or the free $\mathbb{F}_2$ vector space monad) is monic, constructively. This is not entirely obvious, and is not a well-known fact, but details can be found in this [MO thread](https://mathoverflow.net/questions/302516/constructively-is-the-unit-of-the-free-abelian-group-monad-on-sets-injective/302519).
=--

+-- {: .num_cor}
###### Corollary
The category of groups is [[balanced category|balanced]]: every [[epimorphism|epic]] mono is an [[isomorphism]].
=--

+-- {: .proof}
###### Proof
This follows because an epic equalizer is an equalizer of two maps that (by epi-ness) must be the same, hence the equalizer is an isomorphism.
=--

+-- {: .num_cor}
###### Corollary
Every epimorphism in the category of groups is a coequalizer.
=--

+-- {: .proof}
###### Proof
Since every morphism $f: G \to H$ factors as a *regular* epi $p: G \to G/\ker(f)$ followed by a mono $i$, having $f$ epi implies $i$ is a epic mono. Epic monos $i$ being isomorphisms, $f$ is then forced to be regular epic as well.
=--

+-- {: .num_remark}
###### Remark
Despite the fact that every morphism in $Grp$ factors as an epi followed by a regular mono, it is not true that $Grp^{op}$ is regular. Indeed, (regular) monos are in $Grp$ not stable under pushouts. This follows essentially from the plenitude of [[simple objects]] in $Grp$: if $H$ is not simple but embeds in a simple group $G$, then there is a nontrivial quotient $H \to H/N$, and in the pushout diagram

$$\array{
H & \hookrightarrow & G \\
\downarrow & & \downarrow \\
H/N & \to & P
}$$

the object $P$ will be a proper quotient of $G$ and therefore $P \cong 1$, so that the pushout of the mono $H \to G$ which is $H/N \to 1$ fails to be mono.
=--

## Related entries

* [[Ab]], [[TopGrp]]

## References

An axiomatization of the category of groups is given in

* Pierre Leroux, *Une Caracterisation de la Categorie des Groupes*, Canadian Mathematical Bulletin, Volume 15, Issue 3, September 1972, pp. 375 - 380 ([DOI:10.4153/CMB-1972-069-6](https://doi.org/10.4153/CMB-1972-069-6))

[[!redirects Grps]]

[[!redirects Group]]
[[!redirects Groups]]

[[!redirects FinGrp]]

[[!redirects category of groups]]

category: category


Grpd

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

#Contents#
* table of contents
{:toc}

## Definition

The [[(2,1)-category]] $Grpd$ is the [[2-category]] whose

* [[object]]s are [[small category|small]] [[groupoid]]s;

* [[morphism]]s are [[functor]]s;

* [[2-morphism]]s are [[natural transformation]]s, which are necessarily [[natural isomorphism]]s.

This is the full sub-2-category of [[Cat]] on those [[categories]] that are groupoids.

## Properties

### Presentation

One may regard $Grpd$ also just as a 1-category by ignoring the natural isomorphisms between functors. This 1-category may be equipped with the [[natural model structure on groupoids]] to provide a 1-categorical [[presentable (infinity,1)-category|presentation]] of the full $(2,1)$-category.

## Related categories

* [[Pos]]

* [[Set]]

* **$Grpd$**, [[∞Grpd]]

* [[Cat]], [[(∞,1)Cat]]

* [[(∞,n)Cat]]

## References

On the [[locally cartesian closed category|local cartesian closure]] of the [[2-category]] of groupoids (and the failure of this property for the [[1-category]] version):

* [[Erik Palmgren]], *Groupoids and local cartesian closure* (2003) [[[Palmgren-GroupoidsAndLCC.pdf:file]]]

Discussion of $Grpd$ as (with some tweaks) [[categorical semantics]] for [[homotopy type theory]] (the "groupoid model", precursor to the simplicial set model):

* [[Martin Hofmann]], [[Thomas Streicher]], *The groupoid model refutes uniqueness of identity proofs*, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994) [[doi:10.1109/LICS.1994.316071](https://doi.org/10.1109/LICS.1994.316071)]

* {#HofmannStreicher98} [[Martin Hofmann]], [[Thomas Streicher]], _The groupoid interpretation of type theory_, in: [[Giovanni Sambin]] et al. (eds.), *Twenty-five years of constructive type theory*, Proceedings of a congress, Venice, Italy, October 19-21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. **36** (1998) 83-111   [[ISBN:9780198501275](https://global.oup.com/academic/product/twenty-five-years-of-constructive-type-theory-9780198501275), [ps](http://www.mathematik.tu-darmstadt.de/~streicher/venedig.ps.gz), [[HofmannStreicherGroupoidInterpretation.pdf:file]]]

* [[Martin Hofmann]], *The groupoid interpretation of type theory, a personal retrospective*, talk at *[[HoTT at DMV2015]]* (2015) [[[HofmannDMV.pdf|slides:file]]]

* Ethan Lewis, Max Bohnet, *The groupoid model of type theory*, seminar notes (2017) [[pdf](https://staff.fnwi.uva.nl/b.vandenberg3/Onderwijs/Homotopy_Type_Theory_2017/HoTT-bohnet-lewis-handout.pdf), [[LewisBohnet.GroupoidModelOfTT.pdf:file]]]

category: category

[[!redirects Gpd]]
[[!redirects category of groupoids]]

[[!redirects Groupoids]]


This page collects material related to

* [[Johann Fichte]],

  _Grundlage der gesammten Wissenschaftslehre_

  1794

  ([web archive](https://archive.org/details/grundlagedergesa00fich))

#Contents#
* table of contents
{:toc}

## Grundsätze der gesammten Wissenschaftslehre

### §1 Erster, schlechthin unbedingter Grundsatz
 {#ErsterGrundsatz}

Fichte starts with the [[first law of thought]]:

> 1) Den Satz _$A$ ist $A$_ (soviel als $A = A$ ...) giebt jeder zu.

> 6) Durch den Satz $A = A$ wird _geurtheilt_

In between is discussion that for "$A = A$" to make sense, first of all $A$ has to be at all. In [[type theory]] syntax for [[identity types]] this is the [[judgement]]

$$
  A \colon Type \vdash refl_A \colon A = A
$$

("Given $A$ is at all (is a type), then $A = A$.")

Later [[Georg Hegel]] in ([[Science of Logic]], [§875](https://ncatlab.org/nlab/show/Science+of+Logic#875)) called this the _[first original law of thought](https://ncatlab.org/nlab/show/Science+of+Logic#FirstOriginalLawOfThought)_.

Thomas Hughes, in _The Ideal Theory of Berkeley and the Real World_, Part II, Section XV, [Footnote, p. 38](https://archive.org/stream/idealtheoryberk01hughgoog#page/n48/mode/2up) calls this the first of the four _primary laws of thought_.

See also [Wikipedia -- Law of thought -- Four laws](https://en.wikipedia.org/wiki/Law_of_thought#Four_laws).

Next Fichte argues that the only $A$ whose existence is given is...

> Das Ich setzt sich selbst, und es ist, vermöge dieses bloßen Setzens durchs sich selbst; und umgekehrt: Das Ich ist, und es setzt sein Sein, vermöge seines bloßen Seins. - Es ist zugleich das Handelnde, und das Produkt der Handlung; das Tätige, und das, was durch die Tätigkeit hervorgebracht wird; Handlung, und Tat sind Eins und ebendasselbe; und daher ist das: Ich bin Ausdruck einer Tathandlung.

According to ([Hegel, Science of Logic, §1291b](Science+of+Logic#1291b)) this is closely related to the pure _Notion_ as discussed there.

### §2 Zweiter, seinem Gehalte nach bedingter Grundsatz

### §3 Dritter, seiner Form nach bedingter Grundsatz

## Related entries

* [[idealism]]

*  _[[Grundriss des Eigenthümlichen der Wissenschaftslehre]]_.

* [[Hegel]], _[[Science of Logic]]_

## References

* Wikipedia, _[Grundlage der gesamten Wissenschaftslehre](https://de.wikipedia.org/wiki/Grundlage_der_gesamten_Wissenschaftslehre)_

[[!redirects Grundlage der gesamten Wissenschaftslehre]]

category: reference


Grundriss des Eigenthümlichen der Wissenschaftslehre

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
This entry collects material related to

* [[Johann Fichte]]

  _Grundriss des Eigentümlichen der Wissenschaftslehre, in Rücksicht auf das theoretische Vermögen_

  1795

#Contents#
* table of contents
{:toc}

## § 1. Begriff der besonderen theoretischen Wissenschaftslehre

## § 2. Erster Lehrsatz: Das aufgezeigte Factum wird gesetzt: durch Empfindung, oder Deduction der Empfindung

## § 3. Zweiter Lehrsatz: Das empfindende wird gesetzt durch Anschauung oder: Deduction der Anschauung

## § 4. Die Anschauung wird bestimmt in der Zeit, und das angeschaute im Raume

### I

### II

### III

#### IV

> {#4IVUnendlichKleinsterTeilDesRaumes} Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume;

[[infinitesimal neighbourhood]], [[synthetic differential geometry]]

#### V

#### VI

#### VII

#### VIII

## Related entries

* _[[Grundlage der gesamten Wissenschaftslehre]]_

category: reference


Grundzüge der Mengenlehre

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

This page is to compile pointers related to:

* [[Felix Hausdorff]]:

  \linebreak

  **Grundzüge der Mengenlehre**

  \linebreak

  First edition: Leipzig: Veit (1914) [ISBN:978-0-8284-0061-9]

  Second edition (abridged and augmented): _Mengenlehre_, zweite umgearbeitete Auflage. Berlin: de Gruyter.  1927.

  Third edition: 1935.

  Reprinted by Chelsea Publishing Company (1944, 1949, 1965)

  [archive:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)

laying out early foundations of [[set theory]] and [[topology]]/[[topological spaces]].

See also:

* Wikipedia, *[Grundzüge der Mengenlehre](https://en.wikipedia.org/wiki/Grundz%C3%BCge_der_Mengenlehre)*

* [[Hausdorff space]] (introduced in the first edition)

category: reference

[[!redirects Grundzuege der Mengenlehre]]


Gruppenpest

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

#Contents#
* table of contents
{:toc}

## Idea

In the late 1920s, [[Eugene Wigner]] and [[Hermann Weyl]] highlighted the role that [[group theory]] and [[representation theory]] play in the analysis of [[quantum mechanics]], for instance in the analysis of [[atomic spectrum|atomic spectra]]. While many applications of [[groups]] and their [[representations]] to [[quantum physics]] had more or less explicitly been observed before, Wigner and Weyl stood out as making the mathematical formalism fully explicit. This attitude was not well received by some of their colleagues, who felt that this formal [[mathematics]] had no place in [[physics]]. In particular [[Erwin Schrödinger]] is said ([Wigner 1981](#Wigner81)) to have spoken of the _**Gruppenpest**_ (German for "plague of group theory") which ought to be abandoned.

In his [autobiography](#JS75) [[John Slater]], an MIT physicist, claimed:

>It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their "Gruppenpest": the pest of the group theory... The authors of the "Gruppenpest" wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practical consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest.

> I had what I can only describe as a feeling of outrage at the turn which the subject had taken...

> As soon as this [Slater's] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as "Slater has slain the 'Gruppenpest'". I believe that no other piece of work I have done was so universally popular.

Eventually this resistance vanished and turned into its opposite in theoretical fundamental physics: in the classification of [[fundamental particles]] by [[unitary representations of the Poincaré group]] introduced by [[Hermann Weyl]], in the description of [[gauge theory]] in terms of [[associated bundle]]s given by [[representations]] of [[gauge groups]].

Today almost the first thing that one wants to know about any [[theory (physics)|physical theory]] is its [[global symmetry]]-group and [[gauge group]] and the their relevant [[representations]].

## References

Historical texts:

* [[Hermann Weyl]], *Quantenmechanik und Gruppentheorie*, Zeitschrift für Physik **46** (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

* [[Hermann Weyl]], *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig, (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics* Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

* [[Eugene P. Wigner]]: *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), [pdf](https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf)]

* [[Eugene P. Wigner]]: *Group theory: And its application to the quantum mechanics of atomic spectra*, 5, Academic
Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]

Transcript of an interview with Wigner where he mentions Schrödinger's remark on the _Gruppenpest_:

* {#Wigner81} Interview with [[Eugene Wigner]], by Lillian Hoddeson, Gordon Baym and Frederick Seitz at the New Yorker Hotel January 24 (198) [[Niels Bohr Library and Archives](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4965)]

History and review:

* [[Brian G. Wybourne]], *The "The Gruppen Pest" yesterday, today, and tomorrow*, International Journal of Quantum Chemistry Symp. **7** (1973) 35-43 [[doi:10.1002/qua.560070706](https://doi.org/10.1002/qua.560070706)]

* Erhard Scholz, _Introducing groups into quantum theory (1926-1930)_, Historia Mathematica **33** 4 (2006) 440-490 [[arXiv:math/0409571](https://arxiv.org/abs/math/0409571), [doi:10.1016/j.hm.2005.11.007](https://doi.org/10.1016/j.hm.2005.11.007)]

* Erhard Scholz, _Weyl entering the 'new' quantum mechanics discourse_, in: C. Joas, C. Lehner, J. Renn (eds.), *HQ-1: Conference on the History of Quantum Physics*, MPI History of Science Berlin **350** II (2008) 253--271. (Berlin July 2--6, 2007) [[[Scholz-WeylEntering.pdf:file]]]

* Arianna Borrelli, Bretislav Friedrich, _Eugene Wigner and the bliss of the "Gruppenpest"_ (2011) [[[Borrelli-Friedrich_BlissOfGruppenpest.pdf:file]]]

* Christophe Eckes, _Weyl and the mathematisation of Quantum Mechanics individual and collective perspectives_ (2020) [[[Eckes-WeylMath.pdf:file]]]

See also:

* {#JS75} [[John Slater]], *Solid-State and Molecular Theory: A Scientific Biography*, Wiley (1975) [[ark:/13960/t07x0h23t](https://archive.org/details/solidstatemolecu00slat/)]

  > (in the context of [[solid state physics]] and [[molecule|molecular]] theory)


Grégory Ginot

* [website](http://www.math.jussieu.fr/~ginot/)

## Selected writings

On [[factorization algebras]]:

* [[Gregory Ginot]], _Notes on factorization algebras, factorization homology and applications_ ([arxiv/1307.5213](https://arxiv.org/abs/1307.5213))

On higher [[Hochschild homology]]:

* [[Gregory Ginot]], [[Thomas Tradler]], [[Mahmoud Zeinalian]], _Derived higher Hochschild homology, topological chiral homology and factorization algebras_ ([arxiv/1011.6483](http://arxiv.org/abs/1011.6483))

On [[differentiable stacks]]:

* [[Grégory Ginot]], _Introduction to Differentiable Stacks (and gerbes, moduli spaces ...)_, 2013 ([pdf](https://webusers.imj-prg.fr/~gregory.ginot/papers/DiffStacksIGG2013.pdf), [[GinotDifferentiableStacks.pdf:file]])

On [[level set]] [[persistent homology]] and [[sheaf theory]]:

* {#BerkoukGinotOudot19} [[Nicolas Berkouk]], [[Grégory Ginot]], [[Steve Oudot]], *Level-sets persistence and sheaf theory* &lbrack[arXiv:1907.09759](https://arxiv.org/abs/1907.09759), [talk](Center+for+Quantum+and+Topological+Systems#GinotAtHomotopicalPerspectivesOnTDAJune2022)]

On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:

* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]

category: people

[[!redirects Gregory Ginot]]


gs-monoidal category

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

\tableofcontents

## Idea

A *gs-monoidal category* is a [[symmetric monoidal category]] that [[supply|supplies]] [[cocommutative comonoids|cocommutative]] [[comonoid objects]].

## Related concepts

For now, see §6.6 of [Corradini & Gadducci 1999](#CG99).

## References

The notion originates under the name *S-monoidal categories*, motivated by [[term]] [[rewriting]], in:

* [[Fabio Gadducci]], _On the algebraic approach to concurrent term rewriting_, PhD thesis (1996) [[pdf](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1277313dd2c4fce640780b7472221156579e392f), [[Gadducci-AlgebraicRewriting.pdf:file]]]

after which it was renamed to *gs-monoidal categories*, in:

* {#CG99} [[Andrea Corradini]], [[Fabio Gadducci]], _An algebraic presentation of term graphs, via gs-monoidal categories_, Applied Categorical Structures **7** (1999) 299-331 [[doi:10.1023/A:1008647417502](https://doi.org/10.1023/A:1008647417502)]

Discussion in [[category theory|category theoretic]] [[probability theory]]:

* [[Tobias Fritz]], [[Wendong Liang]], *Free gs-monoidal categories and free Markov categories*, Appl. Categ. Structures **31** 21 (2023)  [[arXiv:2204.02284](https://arxiv.org/abs/2204.02284), [doi:10.1007/s10485-023-09717-0](https://doi.org/10.1007/s10485-023-09717-0)]

The notion was reintroduced under the name **CD-categories** (for **c**opy and **d**iscard) in:

* Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian inversion via string diagrams_, Mathematical Structures in Computer Science **29** 7 (2019) 938-971 [[arXiv:1709.00322](https://arxiv.org/abs/1709.00322), [doi:10.1017/S0960129518000488](https://doi.org/10.1017/S0960129518000488)]

[[!redirects gs-monoidal categories]]
[[!redirects CD-category]]
[[!redirects CD-categories]]


GSO projection

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## Idea

In the construction of the [[NSR string|NSR]] [[superstring]] as a [[2d SCFT]] from a [[supersymmetry|supersymmetric]] [[sigma-model]] one needs to remove one half of the [[sigma-model]]-excitations in order to achieve that the resulting [[2d SCFT]] satisfies [[modular invariance]], hence that it is defined not just on the [[complex plane]] but also on [[tori]]. This halfing of the naive sigma-model spectrum is called the _GSO projection_, after [GSO 77](#GSO77).

Depending on how the GSO projection is chosen, one gets [[type IIA string theory]], [[type IIB string theory]] or [[type 0 string theory]].

A good account is in [Majer, sections 1.3 and 2](#Majer).

## Properties

### Local spacetime supersymmetry - supergravity

The GSO projection to the [[type II superstring]] together with the [[Goddard-Thorn no-ghost theorem]] implies that the [[effective field theory]] encoded in the [[string scattering amplitudes]] ([[S-matrix]]) is a [[supergravity]] theory ([Green-Schwarz 81](#GreenSchwarz81)).

This way the mere assumption of strings with [[fermions]] in their spectrum (originally just called "[[spinning strings]]") _implies local_ [[supersymmetry]]_ in [[target space|target]] [[spacetime]]. For more on this see [[string theory FAQ]]: _[Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)_

### Beyond modular invariance

One should really check that the [[2d SCFT]] is well defined on _all_ [[worldsheet]] [[genus of a surface|genera]], not just on genus 1 as given by [[modular invariance]].

Because [[correlators]] at all the higher genera enter the [[string perturbation series]] at higher [[loop order]].

But this is subtle for the [[NSR superstring]], see at _[[super Riemann surface]]_ for more on this.

A full construction of [[2d CFTs]] at all [[genus of a surface|genera]] has been achieved only for bosonic [[rational 2d CFTs]], see at _[[FRS-theorem on rational 2d CFT]]_.

### The Green-Schwarz string

The [[Green-Schwarz superstring]] [[sigma-model]] gives an alternative description of the [[superstring]], where [[spacetime]] [[supersymmetry]] is manifest throughout the construction. No "GSO projection" is needed here, and no analog of the [[Goddard-Thorn no-ghost theorem]] needs to be proven. The downside is that generalization to higher [[genus of a surface|genera]] and to [[curved spacetimes]] is less immediate or unclear.

## Related entries

* [[Goddard-Thorn no-ghost theorem]]

* [[string theory FAQ]]: _[Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)_

## References

The concept is due to

* {#GSO77} F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supersymmetry, supergravity theories and the dual spinor model_, Nucl. Phys, B122, 253 (1977)

* {#GSO77} F. Gliozzi, [[Joël Scherk]], D. I. Olive, _Supersymmetry, Supergravity Theories and the Dual Spinor Model_, Nucl. Phys. B 122 (1977), 253 ([spire](http://inspirehep.net/record/111434))

Review includes

* {#Majer} Imre Majer, sections 1.3 and 2 of: _Superstrings_ ([pdf](http://edu.itp.phys.ethz.ch/fs13/cft/SuperS2_Majer.pdf), [[MajerSuperstrings.pdf:file]])

* _String primer_ around p. 43 in ([arXiv:hep-th/9810240](http://arxiv.org/abs/hep-th/9810240))

See also

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

That the GSO projection implies [[spacetime]] [[supersymmetry]] is due to

* {#GreenSchwarz81} [[Michael Green]], [[John Schwarz]],

  _Supersymmetrical dual string theory_ , Nucl. Phys. B181 (1981) 502;

  _Supersymmetrical string theories_ , Phys. Lett. 109B (1982) 444.

Superconformal invariance of the spinning string was discussed in

* {#Howe79} [[Paul Howe]], _Super Weyl transformations in two dimensions_ J. Phys. 12 (1979) 393

A review of the history of these developments is in

* [[John Schwarz]], _String theory origins of supersymmetry_, Nucl. Phys. Proc. Suppl. 101 (2001) 54-61 ([arXiv:hep-th/0011078](http://arxiv.org/abs/hep-th/0011078))

Interpretation of the [[GSO projection]] as a [[sum]] over [[spin structures]]:

* [[Nathan Seiberg]], [[Edward Witten]], _Spin Structures in String Theory_, Nucl. Phys. B276 (1986) 272 ([spire:227193](http://inspirehep.net/record/227193))

For discussion in the context of [[D-branes]] ([[boundary conformal field theory]]) see

* [[Jürgen Fuchs]], [[Christoph Schweigert]], J. Walcher, _Projections in string theory and boundary states for Gepner models_, Nucl.Phys. B588 (2000) 110-148 ([arXiv:hep-th/0003298](https://arxiv.org/abs/hep-th/0003298))

Relation of the [[GSO projection]] to the [[K-theory classification of topological phases of matter]]:

* Justin Kaidi, Julio Parra-Martinez, [[Yuji Tachikawa]], _Topological Superconductors on Superstring Worldsheets_ ([arXiv:1911.11780](https://arxiv.org/abs/1911.11780))

[[!redirects GSO projections]]


guarded recursion

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#Contents#
* table of contents
{:toc}

##Idea

**Guarded recursion** is a form of [[recursion]] that ensures that solutions of self-referential descriptions exist. This is achieved by "guarding" the recursive occurrence of the object under consideration using a unary [[modality]] $\blacktriangleright$ and pronounced "later".

The *later modality* was first considered by Nakano ([Nakano 2000](#Nakano00)) and used to  ensure *productivity* of [[coinduction|coinductively]] defined [[programs]].

For example, for an object $A$, the [[type]] of *guarded streams* is the unique solution to the equation
\[
\text{Str}_g A \cong A \times \blacktriangleright \text{Str}_g A
\]

The intended meaning is that the head of the stream is available now while the tail is only available after one step of computation.

(There are other forms of guarded recursion which act through syntactic restriction.)

## Related concepts

* [[synthetic guarded domain theory]]

## References

* Robert Atkey, Conor McBride, _Productive Coprogramming with Guarded Recursion_, ([Productive Coprogramming with Guarded Recursion](https://bentnib.org/productive.pdf))

* Aleš Bizjak, _On Semantics and Applications of Guarded Recursion_, ([PhD thesis](https://abizjak.github.io/documents/thesis/semantics-applications-gr.pdf))

* Ranald Clouston, Aleš Bizjak, Hans Bugge Grathwohl, Lars Birkedal, _The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types_, ([arXiv:1606.09455](https://arxiv.org/abs/1606.09455))

* {#Nakano00} Nakano Hiroshi, 2000, _A Modality for Recursion_. In Logic in Computer Science (LICS’00). ([pdf](https://pdfs.semanticscholar.org/a177/47f98e5b821f03ec8be858794f2f83a683b7.pdf))

* [[Lars Birkedal]], Ales Bizjak, Ranald Clouston, Hans Bugge Grathwohl, [[Bas Spitters]], Andrea Vezzosi, _Guarded cubical type theory_ [arxiv](https://arxiv.org/abs/1611.09263)

* [[Jon Sterling]], [[Robert Harper]], *Guarded Computational Type Theory*, LICS '18 (2018) 879–888 [[arxiv:1804.09098](https://arxiv.org/abs/1804.09098), [doi:10.1145/3209108.3209153](https://doi.org/10.1145/3209108.3209153)]

A general calculus for dependent modal type theories.

*  Ranald Clouston, Bassel Mannaa, Rasmus Ejlers Møgelberg, Andrew M. Pitts, [[Bas Spitters]], _Modal Dependent Type Theory and Dependent Right Adjoints_ [arxiv](https://arxiv.org/abs/1804.05236)

[[!redirects guarded recursions]]

[[!redirects guarded recursive]]


Gudrun Hiller

* [personal page](https://people.het.physik.tu-dortmund.de/~ghiller/)

* [Institute page](https://physik.tu-dortmund.de/en/research/research-focus-particle-physics/prof-dr-gudrun-hiller/)

## Selected writings

Early (first) proposal to look for [[flavour anomalies]] in [[B-meson]] decays:

* {#Hiller02} [[Gudrun Hiller]], * $b \to s \ell^+ \ell^-$ decays in and beyond the Standard Model*, AIP Conf. Proc. 549:614-618, 2002 ([arXiv:hep-ph/0008092](https://arxiv.org/abs/hep-ph/0008092))

* [[Gudrun Hiller]], *Physics reach of rare b-decays*, Nucl. Phys. Proc. Suppl. 115 (2003) 76-81 ([arXiv:hep-ph/0207121](https://arxiv.org/abs/hep-ph/0207121))

* [[Gudrun Hiller]], *Phenomenology of New Physics* ([arXiv:hep-ph/0308180](https://arxiv.org/abs/hep-ph/0308180))

On [[flavour anomalies]],

possibly explained via [[leptoquarks]]:

* {#HLN21} [[Gudrun Hiller]], Dennis Loose, Ivan Nišandžić, _Flavorful leptoquarks at the LHC and beyond: Spin 1_ ([arXiv:2103.12724](https://arxiv.org/abs/2103.12724))

possibly explained via [[Z'-bosons]] and in view of [[Higgs field]] [metastability](Higgs+field#MassAndVacuumInstability):

* Rigo Bause, [[Gudrun Hiller]], Tim Höhne, Daniel F. Litim, Tom Steudtner, *B-Anomalies from flavorful $U(1)'$ extensions, safely* ([arXiv:2109.06201](https://arxiv.org/abs/2109.06201))

On [[vacuum]] stability of the [[Higgs field]] in the [[standard model of particle physics]]:

* [[Gudrun Hiller]], Tim Höhne, Daniel F. Litim, Tom Steudtner, *Vacuum Stability in the Standard Model and Beyond* [[arXiv:2401.08811](https://arxiv.org/abs/2401.08811)]


Guenter Harder

__Günter Harder__ is an algebraic geometer, now a professor emeritus at [Max Planck Institut for Mathematics](http://www.mpim-bonn.mpg.de/) in Bonn. [[Harder-Narasimhan filtration]] is named after him.

category: people

[[!redirects Guenter Harder]]
[[!redirects Günter Harder]]
[[!redirects Gunter Harder]]


Guglielmo Lockhart

## Selected writings

On [[M-strings]]:

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))

On [[E-strings]]:

* {#HLV14} [[Babak Haghighat]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _Fusing E-string to heterotic string: $E + E \to H$_, Phys. Rev. D 90, 126012 (2014) ([arXiv:1406.0850](https://arxiv.org/abs/1406.0850))

and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

category: people


guide to noncommutative algebraic geometry literature

This entry provides commented references on the topic of [[derived noncommutative algebraic geometry]].

***

* [[Dmitri Orlov]], _Equivalences of derived categories and K3 surfaces_, 1996, [arXiv](http://arxiv.org/abs/alg-geom/9606006).

The original representability theorem: every [[fully faithful]] [[triangulated functor]] between the [[derived categories]] of [[perfect complexes]] on two [[smooth]] [[projective varieties]] is representable by some [[integral kernel]] (i.e. is a [[Fourier-Mukai functor]]).

* [[Alexei Bondal]], [[Dmitri Orlov]], _Reconstruction of a variety from the derived category and groups of auto-equivalences_, 1997, [arXiv](http://arxiv.org/abs/alg-geom/9712029).

If a [[smooth]] [[projective variety]] has the property that its [[canonical sheaf]] is [[ample]] or anti-ample, then one may reconstruct it up to isomorphism from its bounded [[derived category]] of [[coherent sheaves]].

* [[Alexei Bondal]], [[Dmitri Orlov]], _Derived categories of coherent sheaves on abelian varieties and equivalences between them_, [arXiv](http://arxiv.org/abs/alg-geom/9712017).

A criterion for the equivalence of the [[derived categories]] of [[coherent sheaves]] on two [[abelian varieties]].

* [[Alexei Bondal]], [[Michel van den Bergh]], _Generators and representability of functors in commutative and noncommutative geometry_, 2002, [arXiv](http://arxiv.org/abs/math/0204218).

The bounded [[derived category of coherent sheaves]] on a [[smooth]] [[projective variety]] admits a strong [[generator]].

* [[Bertrand Toen]], _Homotopy theory of dg-categories and derived Morita theory_, 2004, [arXiv](http://arxiv.org/abs/math/0408337)

The [[homotopy category]] of the [[model category]] of [[dg-categories]], where [[weak equivalences]] are quasi-equivalences, is studied.  It is shown to admit [[internal homs]] and the morphism sets are in bijection with right quasi-representable [[bimodules]].  Among other applications is a dg-version of Orlov's representability theorem: it is shown that morphisms between the [[dg-categories]] of [[perfect complexes]] on [[smooth]] [[proper schemes]] are in bijection with [[perfect complexes]] on the product.  An analogous statement for [[quasi-coherent complexes]] is also shown.

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and motives_, 2005, [arXiv](http://arxiv.org/abs/math/0512620).

The [[Chow motive]] of a [[smooth]] [[projective variety]] may be reconstructed up to [[Tate twists]] from the bounded [[derived category]] of [[coherent sheaves]].

* [[Alexander Kuznetsov]], _Base change for semiorthogonal decompositions_, 2007, [arXiv](http://arxiv.org/abs/0711.1734).

* [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral transforms and Drinfeld centers in derived algebraic geometry_, 2008, ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157)).

An [[(infinity,1)-category|(infinity,1)-categorical]] version of Orlov's representability theorem, which is further generalized to (smooth proper) [[derived stacks]].

* [[Alexander Kuznetsov]], _Hochschild homology and semiorthogonal decompositions_, 2009, [arXiv](http://arxiv.org/abs/0904.4330).

* [[Valery Lunts]], [[Dmitri Orlov]], _Uniqueness of enhancement for triangulated categories_, 2009, [arXiv](http://arxiv.org/abs/0908.4187).

Uniqueness results are established for [[dg-enhancements]] of [[triangulated categories]].  In particular the results are applied to the [[derived categories]] of [[perfect complexes]], [[coherent sheaves]], [[quasi-coherent sheaves]], on schemes.  This is then used to deduce from (Toen 2004) representability theorems on the level of [[triangulated categories]], in the non-smooth case.

* [[Alexander Kuznetsov]], [[Valery Lunts]], _Categorical resolution of irrational singularities_, 2012, [arXiv](http://arxiv.org/abs/1212.6170).

Describes how to glue two [[dg-categories]] in such a way that the [[homotopy category]] of the result has a [[semiorthogonal decomposition]] into two components, which are the respective [[homotopy categories]] of the original [[dg-categories]].

* [[Sergei Gorchinskiy]], [[Dmitri Orlov]], _Geometric phantom categories_, 2012, [arXiv](http://arxiv.org/pdf/1209.6183v2.pdf).

If an [[admissible subcategory]] of the bounded [[derived category]] of [[coherent sheaves]] on a [[smooth]] [[projective variety]] is a [[phantom]], has vanishing [[Hochschild homology]] and [[Grothendieck group]], then its [[noncommutative motive]] vanishes.  In particular its higher [[K-theory]] also vanishes.

* [[Beatriz Rodriguez Gonzalez]], _A derivability criterion based on the existence of adjunctions_, 2012, [arXiv:1202.3359](http://arxiv.org/abs/1202.3359).

It is proved that the [[internal hom]] of Ho(DGCat) constructed in (Toën 2004) is in fact the right [[derived functor]] of the internal hom of DGCat.

* [[David Ben-Zvi]], [[David Nadler]], Anatoly Preygel, _Integral transforms for coherent sheaves_ ([arXiv:1312.7164](http://arxiv.org/abs/1312.7164))

The [[(infinity,1)-category|(infinity,1)-categorical]] representability theorem of (Ben-Zvi-Francis-Nadler 2008) is extended to the non-smooth case.


Guido Mislin

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

* [inbstitute page](https://people.math.ethz.ch/~mislin/)

## Selected writings

On [[equivariant K-theory]] and the [[Baum-Connes conjecture]]:

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

category: people


Guifre Vidal

* [webpage](https://www.perimeterinstitute.ca/people/guifre-vidal)

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Guifr%C3%A9_Vidal)

## Selected writings

On [[tree tensor networks]] related to [[entanglement entropy]]:

* [[Luca Tagliacozzo]], G. Evenbly, [[Guifre Vidal]],
_Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law_, Phys. Rev. B 80, 235127 (2009) ([arXiv:0903.5017](https://arxiv.org/abs/0903.5017))

On [[tensor networks]] with [[global symmetry]]:

* Sukhwinder Singh, Robert N. C. Pfeifer, [[Guifre Vidal]], _Tensor network decompositions in the presence of a global symmetry_, Phys. Rev. A82:050301, 2010 ([arXiv:0907.2994](https://arxiv.org/abs/0907.2994))

* Sukhwinder Singh, Robert N. C. Pfeifer, [[Guifre Vidal]], _Tensor network states and algorithms in the presence of a global $U(1)$ symmetry_, Phys. Rev. B 83, 115125 (2011) ([arXix:1008.4774](https://arxiv.org/abs/1008.4774))

* {#SinghVidal12} Sukhwinder Singh, [[Guifre Vidal]], _Tensor network states and algorithms in the presence of a global $SU(2)$ symmetry_, Phys. Rev. B 86, 195114 (2012) ([arXiv:1208.3919](https://arxiv.org/abs/1208.3919))

category: people


Guilherme Frederico Lima

Guilherme Frederico Lima de Carvalho e Silva

PhD Candidate in Pure Mathematics - University of Cambridge

* [webpage](https://puc-rio.academia.edu/GuilhermeFredericoLima)

## stuff

* _From Essential Inclusions to Local Geometric Morphisms_, talk at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris  ([video](https://www.youtube.com/watch?v=YsoGN91Rh_s))

  on [[essential geometric morphisms]] and [[local geometric morphisms]]

category: people

[[!redirects Guilherme Frederico Lima de Carvalho e Silva]]

[[!redirects Guilherme Lima]]


Guillaume Bossard

## Selected writings

On [[E11]]-[[exceptional field theory]]:

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))

On maximal [[gauged supergravity|gauged]] [[D=2 supergravity]] via [[KK-reduction]] of [[D=11 supergravity]]:

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]

category: people


Guillaume Brunerie

* [home page](https://guillaumebrunerie.github.io/)

## Selected writings

On [[synthetic homotopy theory]] in [[homotopy type theory]]:

* [[Guillaume Brunerie]],  [[Dan Licata]], [[Peter LeFanu Lumsdaine]]: _Homotopy theory in type theory_, 2013 ([pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/))

On [[homotopy groups of spheres in HoTT]]:

* [[Guillaume Brunerie]]: *[[On the homotopy groups of spheres in homotopy type theory]]* $[$[arxiv:1606.05916](https://arxiv.org/abs/1606.05916)$]$

On [[ordinary cohomology in homotopy type theory]]:

* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$

  > (in [[cubical type theory|cubical]] [[Agda]])

## Related entries

* [[initiality conjecture]]

category: people