group presentation

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# Presentations of groups
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## Idea

In trying to study a [[group]] $G$, one way to proceed is to

* look for a set of generating elements;

* look for 'relations' between those elements.

The problem is partially how one is to interpret this second part. To do this we need to look at words in the generators and hence at the [[free group]] on the set of generators.

For example, if we have the symmetric group $S_3$, or, isomorphically, the dihedral group (of symmetries of a triangle) which we will call $D_3$ (following the geometric convention (see the Wikipedia article on dihedral groups), then we have 6 elements, and they are all able to be got as products of a transposition and a three cycle (or alternatively as a reflection and a rotation).  If we call the three cycle $a$ and the transposition $b$, we have

$$S_3=\{1,a,a^2,b,a b,a^2b\}.$$

What about the other words: $ba$ for instance.  If one calculates $b a$ in $S_3$ and then looks up what you get you have $ba = a^2b$, so that there is a relation between these two words. This however is not all. What about $b a b a a b a a  a b b a a $? That is a word in the $a$s and $b$s so should represent something in $S_3$. in fact if you think in the geometric interpretation of $S_3$ as being the dihedral group, $D_3$, you can pick up  a triangle and interpret that word as a list of instructions for moving the triangle. You then find out what this word is out of the 6 possible given forms.

For a presentation, you give a set of generators $X$, so there will be an epimorphism from $F(X)$ to $G$, and then you try to find a description for the kernel of that epimorphism, which we will denote by $N$.  The description of this normal subgroup $N$ is as the [[normal closure]] of a set $R$ of relations, i.e. words in the generators or, equally validly, elements in the free group on $X$.

## Definition

A presentation of a group, $G$, is a [[coequalizer|coequalizer diagram]] $F R \rightrightarrows F X \twoheadrightarrow G$, where $F X$ is the free group on a set of *generators*, and $F R$ one on a set of *relations* (or *relators*, depending on how the relations are specified).

This is not quite the usual `classical' form of the definition, so we will take it apart to show the relationship.

A  presentation is thus   given by a pair of sets, $X$ and $R$, written $\langle X: R\rangle$ such that setting $F=F(X)= \langle X\rangle =\langle X:\emptyset\rangle$ to be the [[free group]] on the set $X$ and $N$ the [[normal subgroup|normal closure]] of the set of relators $R$, there is a *specified* isomorphism from $F/N$ to $G$.

The specified isomorphism is often omitted, as often the set $X$ of generators is chosen as a subset of the set of elements of $G$.  In this case, the universal properties of free groups and quotients, there is a unique map $F\to G$ which restricts to the inclusion of $X$, and thereby at most one map $F/N \to G$ which does so; this map is then the one asserted to be an isomorphism.

In general it is necessary to proceed otherwise, however, and to give a specific function from a set $X$ to the set of elements of $G$.  This function then induces a group homomorphism from $F=\langle X\rangle$ to $G$, and if this is a surjection, then we can find some $N$ (generators for its kernel) to produce a presentation of $G$.  Without this extra data, certain of the manipulations of a group presntation look decidedly suspect, for instance a substitution which results in two copies of a generator being given. It is also much easier to work with morphisms of presentations if the full data is recorded.

## Examples of presentations

### The standard presentation

 A standard if somewhat trivial example of a presentation is given by the _standard presentation_ of a group, $G$.  We take $X= \{x_g\mid g\in G,g\neq 1\}$, to be a set in bijective correspondence with the underlying set of $G$. (You can take $X$ equal to that set if you like, but sometimes it is better to have a distinct set, for instance, it make for an easier notation for the description of certain morphisms.) The set of relations will be

$$R =\{x_g.x_h= x_{gh}\mid g,h \in G\},$$

so as a _set_ is just a copy of $G\times G$ as the relations are indexed by pairs of elements of $G$.

### [[cyclic group|Cyclic groups]]

Cyclic group, $C_n$, of order $n$ has presentation $\langle a : a^n\rangle$.  There are many different functions from the (singleton) set of generators to  $C_n$ that will give a suitable presentation in the fuller sense.

### The [[symmetric group]] of order 6

  The group of permutations of three letters, $S_3$, has a presentation

$$\langle a,b : a^3, b^2,(a b)^2 \rangle.$$

 Here $a$ corresponds to the 3-cycle, $(123)$, whilst $b$ corresponds to any transposition.

### [[knot group|Knot groups]]

 The [[trefoil knot|trefoil]] [[knot group]] has two useful presentations:

  * $\langle a,b : a^3= b^2\rangle$, which displays the fact that the trefoil is a (2,3)-[[torus knot]]; and

  * $\langle x,y : x y x = y x y\rangle$, which shows the link between this group and the Artin [[braid group]], $Br3$.

### [[Coxeter groups]]

 At the entry on [[Coxeter groups]] one finds the following:

**Definition** A _Coxeter matrix_ over an index [[set]] $I$ is a symmetric [[matrix]]

$$M \colon I \times I \to \{1, 2, 3, \ldots, \infty\}$$

such that $M(i, i) = 1$ for all $i \in I$, else $M(i, j) \gt 1$. Writing $m_{i,j} = M(i, j)$, the associated **Coxeter group** $W_M$ is the group presented as having generators $s_i$, $i \in I$, and relations

$$(s_i s_j)^{m_{i, j}} = 1$$

for all $i, j \in I$, whenever $m_{i, j} \neq \infty$. In other words, $m_{i, j}$ is the order of $s_i s_j$ (as is easily shown), and these orders determine the group.

We thus have an infinite family of group presentations.

## Discussion

*  *'Relations' and 'relators'*: In the discussion of $S_3$ above we had a **relation** $b a = a^2b$. so we are relating two words of $F(X)$. It is often the case that instead of relations we use **relators**, in other words a relation of form $r = 1$, where $r$ is a word in the generators. In the example $ b a = a^2b$ can be easily shown to imply and be implied by $ a b a b = 1$.

*  Given a group presentation as above, we have a
short exact sequence,

$$1\to N \to F \to G \to 1,$$

where $F = F(X)$, the free group on the set $X$, $R$ is a
subset of $F$ and $N = N(R)$ is the normal closure in $F$ of the set $R$.  The group
$F$ acts on $N$ by conjugation: ${}^u c = ucu^{-1}, for  c\in N, u \in F$ and the
elements of $N$ are words in the conjugates of the elements of $R$:

$$c = {}^{u_1}(r_1^{\varepsilon_1}){}^{u_2}(r_2^{\varepsilon_2})\ldots
{}^{u_n}(r_n^{\varepsilon_n})$$

where each $\varepsilon_i$ is $ + 1$ or $ - 1$.  One also says such elements are **consequences** of
$R$.  Heuristically an [[identity among the relations]] of $\mathcal{P}$ is such an element $c$
which equals 1.

##Transformations of a group presentation##

Given a group presentation, it is natural to perform transformations using substitutions, say adding in one new symbol for a string of generators, and adjusting the presentation accordingly.  The valid transformations that do not change the group being presented are formalised as the [[Tietze transformation]]s.

## Combinatorial group theory

The study of group presentations, their transformations etc. forms part of [[combinatorial group theory]].

## Presentations of monoids and other algebraic and categorical structures.

The theory of group presentations generalises to that of presentations of monoids and then to general [[rewriting]] systems.

## Related concepts

* [[finitely presentable group]]

* [[combinatorial group theory]]

* [[Andrews-Curtis conjecture]]

## References

Textbook accounts:

* [[Wilhelm Magnus]], A. Karras, D. Solitar, *Combinatorial group theory: Presentation of groups in terms of generators and relations*, Courier Corporation (1966, 2004) [[ISBN:0-486-43830-9](https://www.maa.org/press/maa-reviews/combinatorial-group-theory-presentations-of-groups-in-terms-of-generators-and-relations), [GoogleBooks](https://books.google.com/books?hl=en&lr=&id=1LW4s1RDRHQC&oi=fnd&pg=PP1&ots=WszNa-0hEz&sig=Hn_MTbyLcEreBBNZ8jJtQUVKlZk)]

* [[David Lawrence Johnson]], *Topics in Theory of Group Presentations*, London Mathematical Society Lecture notes series **42**, Cambridge Univ. Press (1980) [[doi:10.1017/CBO9780511629303](https://doi.org/10.1017/CBO9780511629303)]

* [[David Lawrence Johnson]], *Presentations of Groups*, London Mathematical Society Student Texts **15**, Cambridge Univ. Press (1990) [[doi:10.1017/CBO9781139168410](https://doi.org/10.1017/CBO9781139168410)]

* [[N. D. Gilbert]] and [[T. Porter]], Knots and Surfaces, Oxford U.P., 1994.

In the context of [[combinatorial group theory]]:

* [[Roger C. Lyndon]], [[Paul E. Schupp]], _Combinatorial group theory_, Springer, 1977; [toc](http://elib.tu-darmstadt.de/tocs/94684782.pdf).

Lecture notes:

* Derek Holt (notes by Florian Bouyer), *Presentation of Groups* (2013) [[pdf](https://warwick.ac.uk/fac/sci/maths/people/staff/fbouyer/presentation_of_group.pdf)]

On (finitely) presented groups as [[fundamental groups]] of ([[finite cell complex|finite]]) [[simplicial complexes]]/[[CW-complexes]]:

* [[Joseph J. Rotman]], around Thm. 7.34 in: *An Introduction to Algebraic Topology*, Graduate Texts in Mathematics **119** (1988) $[$[doi:10.1007/978-1-4612-4576-6](https://doi.org/10.1007/978-1-4612-4576-6)$]$

* Behrooz Mashayekhy, Hanieh Mirebrahimi, *Some Properties of Finitely Presented Groups with Topological Viewpoints*, International Journal of Mathematics, Game Theory and Algebra **18** 6 (2010) 511-515 [[arXiv:1012.1744](https://arxiv.org/abs/1012.1744)]

On [[rewriting]] in group presentations:

* [[Nick Gilbert]], Emma Antonia McDougall, _Groupoids and the algebra of rewriting in group presentations_, Beiträge zur Algebra und Geometrie **62** 3 (2021) 1-17 [[doi:10.1007/s13366-020-00531-6](https://doi.org/10.1007/s13366-020-00531-6), [arXiv:1901.04348](https://arxiv.org/abs/1901.04348)]

category: group theory

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group scheme

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#### Geometry
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# Contents
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## Definition

A __group scheme__ is a [[group object]] in the category of [[schemes]] (or in a category of *some* schemes as for instance that of schemes over a fixed base scheme); in particular a group scheme is a [[group functor]]. As explained at [[group object]] there are two equivalent ways of realizing this:

One way is to define it as a functor $G:C Ring\to Set$ equipped with a transformation $m:G\times G\to G$ satisfying the properties spelled out at [[group object]].

The other way is to define it as a functor $Sch\to Grp$ from the category of schemes to that of (discrete) groups whose composition with the forgetful functor $Grp\to Set$ is representable.

Grothendieck emphasized the study of schemes over a fixed base scheme. Following this idea in the [[functor of points]] formalism, a group scheme over a scheme $X$ is a functor

$$
G: (Sch /X)^{op} \to Grp
$$

A **morphism of group schemes** $f:G\to H$ is a morphism of schemes that is a group homomorphism on any choice of values of points. This is more easily stated by saying that a morphism of group schemes must be a [[natural transformation]] between the [[functor of points]]; i.e. $f$ is required to be a natural transformation of functors with values in the category $Grp$ of groups (instead of with values in Set); an equivalent way to state this is that $f$ needs to satisfy $f m=m(f\times f)$ if $m:G\times G\to G$ denotes the group law on $G$.

This construction generalizes to [[ind-object|ind-schemes]] (as for example [[formal scheme|formal schemes]]) to that of a [[formal group scheme]].

## Overview

Let $k$ be some base field. We start with the **constant group scheme** $E_k$ defined by some classical group $E$ which gives in every component just the group $E$. Next we visit the notion of **étale group scheme**. This is not itself constant but becomes so by scalar extension to the separable closure $k_sep$ of $k$. The importance of étale affine is that the category of them is equivalent to that of [[nLab:Galois module|Galois modules]] by $E\mapsto E \otimes_k k_sep=\cup_{K/k \,sep\,fin} E(K)$

So far these examples ''do nothing'' with the (multiplicative and additive) structure of the ring in which we evaluate our group scheme.
But the next instances do this: We define the **additive-** and the **[[multiplicative group]] scheme** by $\alpha_k: R\mapsto R^+$ and $\mu_k:R\mapsto R^\times$ sending a $k$-ring to to its underlying additive- and multiplicative group, respectively. These have the ''function rings'' $O_k(\alpha_k)=k[T]$ and $O_(\mu_k)=K[T,T^{-1}]$ and since $(O_k\dashv Spec_k):k.Ring\stackrel{Spec_k}{\to} k.Aff$ we note that our basic building blocks $\alpha_k$ and $\mu_k$ are in fact representable $k$-functors aka. **[[affine group schemes]]**. We observe that we have $k.Gr(\mu_k,\alpha_k)=0$ and call in generalization of this property any group scheme $G$ satisfying $k.Gr(G,\alpha_k)=0$ **multiplicative group scheme**. (We could have also the idea to call $G$ satisfying $k.Gr(\mu_k,G)=0$ ''additive'' but I didn't see this.)
By some computation of the hom spaces $k.Gr(G,\mu_k)$ involving co- and birings we see that these are again always values of a representable $k$-functor $D(G)(-) \coloneqq (-).Gr(G\otimes_k (-),\mu_{(-)})$; this functor we call the **[[Cartier dual]] of $G$**. If for example $G$ is a **[[finite group scheme]]** $D(G)$ also is, and moreover $D$ is a contravariant autoequivalence (''duality'') of $k.fin.comm.Grp$; in general it is also a **duality** in some specific sense.
By taking the Cartier dual $D(E_k)$ of a constant group scheme we obtain the notion of a **diagonalizable group scheme**. To justify this naming we compute some value $D(E_k)(R)=hom_{Grp}(E_k\otimes_k R, \mu_R)\simeq hom_{Grp}(E_k,R^\times)\simeq hom_Alg(k[E_k],R)$ where $k[E_k]$ denotes the [[group algebra]] of $E_k$ and the last isomorphism is due to the universal property of group rings; we observe that the last equality tells us that $G=Spec\,k[E_k]$ and recall that a $\zeta\in E_k\subset k[E_k]$ is called a [[nLab:character]] of $G$ (and one calls a group generated by these ''diagonalizable'').
Revisiting the condition $k.Gr(H,\alpha_k)=0$ by which we defined multiplicative group schemes and considering a group scheme $G$ satisfying this condition for all **sub group-schemes** $H$ of $G$ we arrive at the notion of **unipotent group scheme**. By the structure theorem of **decomposition of affine groups** we can proof that $G$ is unipotent iff the **completion of group schemes** (which gives us-by the usual technic of [[completion]]- a **formal (group) scheme** $\hat X$ if $X$ is a group scheme) of the Cartier dual of $G$, i.e. $\hat D(G)$ is a **connected formal group scheme** also called **local group scheme** since a local group scheme $Q=Spec_k A$ is defined to be the [[spectrum]] of a local ring; this requirement in turn is equivalent to $Q(K)=hom(A,K)=\{0\}$ hence the first name ''connected''. There is also a connection between connected and étale schemes: For any formal group scheme there is an essentially unique [[exact sequence]]

$$0\to G^\circ\to G\to \pi_0(G)\to 0\label{connectedetaledecomposition}$$

where $G^\circ$ is connected and $\pi_0(G)$ is étale.

{#SuchDecomposition} Such decomposition in exact sequences we obtain in further cases: $0\to G^{ex}\to G\to G_{ex}\to 0$ where

|$k$-group | $G^{ex}$ | $G_{ex}$ | | |
|-------|-------|-------|-------|--------|
| [[formal scheme|formal]] | [[connected object|connected]] | [[étale group scheme|étale]] | | p.34 |
| finite | [[ infinitesimal object | infinitesimal]] | étale | splits if $k$ is perfect | p.35 |
| [[nLab:ring spectrum|affine]] | multiplicative | [[unipotent group scheme|unipotent]] | splits if $k$ is perfect | p.39 |
| | connected  | $G_{red}$ is [[smooth scheme|smooth]] | $G/G_{red}$ is infinitesimal |p.43 |  |

where a **smooth (group) scheme** is defined to be the spectrum of a finite dimensional (over k) [[nLab:power series]] algebra, a (group) scheme is called **finite (group) scheme** if we restrict in all necessary definitions to $k$-ring which are finite dimensional $k$-[[nLab:vector space|vector spaces]], and a (group) scheme is called **infinitesimal (group) scheme** if it is finite and local. If  moreover $k$ is a perfect field any finite affine $k$-group $G$ is in a unique way the product of four subgroups $G=a\times b\times c\times d$ where $a\in Fem_k$ is a formal étale multiplicative $k$ group, $b\in Feu_k$ is a formal étale unipotent $k$ group, $c\in Fim_k$ is a formal infinitesimal multiplicative $k$ group, and $d\in Fem_k$ is a infinitesimal unipotent $k$ group.

If we now shift our focus to colimits- or more generally to codirected systems of [[finite group schemes]], in particular the notion of **[[p-divisible group]]** is an extensively studied case because the $p$-divisible group $G(p)$ of a group scheme encodes information on the **[[p-torsion]]** of the group scheme $G$. To appreciate the definition of $G(p)$ we first recall that for any group scheme $G$ we have the [[nLab:Frobenius morphism|relative Frobenius morphism]] $F_G:G\to G^{(p)}$ to distinguish it from the absolute Frobenius morphism $F^{abs}_G:G\to G$ which is induced by the Frobenius morphism of the underlying ring $k$. The passage to the relative Frobenius is necessary since in general it is not true that the absolute Frobenius respects the base scheme. Now we define $G[p^n]:=ker\; F^n_G$ where the kernel is taken of the Frobenius iterated $n$-times and the codirected system

$$G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$

is then called the $p$-divisible group of $G$. As **cardinality** (in group theory also called **rank**) of this objects we have $card(G[p^j])=p^{j\cdot h}$ for some $h\in \mathbb{N}$; this $h$ is called the **[[height of a group scheme|height]] of $G$**. Moreover we have (p1) the $G[p^i]$ are finite group schemes (we assumed this by definition), (p2) the sequences of the form $0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0$ are exact, (p3) $G=\cup_j ker\, p^j\cdot id_G$ and one can show that if we start with any codirected system $(G_i)_{i\in \mathbb{N}}$ satisfying (p1)(p2) we have that $colim_i G_i$ satisfies (p3) and $ker( F^n_G)\simeq G_n$ - in other words the properties (p1)(p2) give an equivalent **alternative definition of $p$-divisible groups** (and (p3) leads some authors to ''identify'' $G$ and $G(p)$). Basic examples of $p$-divisible groups are $(\mathbb{Q}_p/\mathbb{Z}_p)^h_k$ which is (up to isomorphism) the unique example of a constant $p$-divisible group of height $h$ and $A(p)$ where $A$ is a commutative [[nLab:variety]] with a group law (aka. **algebraic group**). $A(p)$ is called **[[nLab:the Barsotti-Tate group of an abelian variety]]**; if the dimension of $A$ is $g$ the height of $A(p)$ is $2g$.
Now, what about decomposition of $p$-divisible groups? We have even one more equivalent **''exactness'' characterization of $p$-divisible formal groups** by: $G$ is $p$-divisible iff in the connected-étale decomposition given by the exact sequence displayed in (1) we have ,($p1^\prime$), $\pi_0(G)(\overline k)\simeq (\mathbb{Q}_p/\mathbb{Z}_p)^r$ for some $r\in \mathbb{N}$ and ,($p2^\prime$), $G^\circ$ is **of finite type** (= the spectrum of a Noetherian ring), smooth, and the kernel of its [[Verschiebung morphism]] (this is the left adjoint the Frobenius morphism) is finite. Of course this characterization of $p$-divisiblity by exact sequences gives rise to propositions on dimensions and subgroups of $p$-divisible groups.

(...)

In cases where $k$ is a field of prime [[nLab:characteristic]] $p$, there is some special $k$-functor which is a group functor and even a ring functor (a $k$-functor equipped with a ring structure) - namely the functor $W:k.Ring. comm\to \lambda.Ring\hookrightarrow Set$ whose image is the category $\Lambda$ of **[[nLab:lambda ring|lambda-rings]]**; the objects $W(R)$ of $\Lambda$ are also called **[[nLab:Witt vectors]]** since they are infinite sequences of elements of $R$ (this justifies at least ''vectors''). $W$ possesses a left adjoint $(V\dasv W)$ forgetting the lambda-structure and the [[couniversal property]] associated to this adjunction states that for a $k$-ring $R$ we have that $W(R)$ is the couniversal object such that all so called **Witt polynomials** $w_n(x_0,\dots x_n):=x_0^{p^n}+p\cdot x_1^{p^{n-1}}+p^2 \cdot x_2^{p^{n-2}}+\dots+p^n\cdot x_n$ are ring homomorphisms. For this special $k$-group $W$ we revisit some construction we have done above for general $k$-groups: we firstly make the eponymous remark that the Verschiebung morphism $V_W(R):(a_1,a_2,\dots,a_n,\dots)\mapsto (0,a_1,a_2,\dots,a_n,\dots)$ is given by shifting (German: Verschiebung) one component to the right. By abstract nonsense we have also Frobenius. An important proposition concerning the ring of Witt vectors is that for a perfect field $k$, $W(k)$ is a **discrete valuation ring**. The next construction we visit with $W(R)$ is **Cartier duality of finite Witt groups** (here we forget that $W(R)$ is even a ring): For this note that the **ring of finite Witt vectors $W_fin(R)$** is an ideal in $W(R)$ and we have Frobenius and Verschiebung also in this truncated case; more precisely we have for each $n$ a Frobenius $F_{W_n}:W_n\to W_n$ where $W_n(R)$ denotes the **ring of Witt vectors of length $n$**. With this notation we find $ker(F^m_{W_n})\simeq D(ker(F^n_{W_n})$.

Since $W(k)$ is a ring we can ask of its modules in general; however there is in particular one $W(k)$-module of interest which is called the **Dieudonné module $M(G)$ of $G$**. It can be defined in two equivalent ways: 1. as a $W(k)$-module $M$ equipped with two endomorphisms of $F$ and $V$ satisfying the ''Witt-Frobenius identities''

(WF1): $FV=VF=p$

(WF2): $Fw=w^{(p)} F$

(WF3): $w V=V w^{(p)}$

or 2. as a left module over the **Dieudonné ring** which is the (noncommutative ring) generated by $W(k)$ and two variables $F$ and $V$ satisfying (WF1)(WF2)(WF3) in which case every element of $D_k$ can uniquely be written as a finite sum

$$\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i$$

(...)

## Examples

* For a field $k$ the terminal $k$-scheme $Sp_k k$ is a group scheme in a unique way.

* An [[affine scheme|affine]] group scheme. Affine group [[variety|varieties]] are called [[linear algebraic group|linear algebraic groups]].

* Complete group varieties are called [[abelian variety|abelian varieties]].

* Given any group $G$, one can form the [[constant group scheme]] $G_X$ over $X$.

* [[etale group scheme]] is the spectrum of a commutative [[Hopf algebra]]. In this case the multiplication- resp. inversion- reps. unit map are given by [[comultiplication]] reps. [[antipodism]] resp. [[counit]] in the Hopf algebra.

* The functor $\mu:=\mathbb{G}_m$ is a group scheme given by $\mathbb{G}_m(S)=\Gamma(S, \mathcal{O}_S)^\times$. A scheme is sent to the invertible elements of its global functions. This group scheme is called *the multiplicative group scheme*. In context of [[p-divisible group|p-divisible groups]] the kernels of the $k$-group scheme endomorphisms of $\mathbb{G}_m$ defined by $(-)^n:x\mapsto x^n$ for an integer $n$ are of particular interest. These kernels give the group schemes of the $n$-th [[root of unity]].

* [[diagonalizable group scheme]]. Note that the multiplicative group scheme is diagonalizable.

* [[multiplicative group scheme]] also called *group scheme of multiplicative type*. Every diagonalizable group scheme is in particular of multiplicative type.

* The *[[additive group]] scheme* assigns to a ring its additive group. Also here the kernels of the powering-by-n map are of interest. These kernels give the group schemes of the $n$-th [[nilpotent object|nilpotent element]].

* Group schemes can be constructed by [[restriction of scalars]].

* The functor $\alpha:=\mathbb{G}_a$ is a group scheme given by $\mathbb{G}_a(S)=\Gamma(S, \mathcal{O}_S)$ the additive group of the ring of global functions. This group scheme is called *the additive group scheme*.

* [[connected group scheme]] (is synonymous to [[local group scheme]])

* [[unipotent group scheme]] (these are [[Cartier duality|Cartier duals]] of local group schemes)

* the kernel of any group scheme morphism is a group scheme.

* Every [[algebraic group]] is in particular a group scheme.

## Properties

### Cartier duality
(main article: [[Cartier duality]])

Suppose now that $G$ is a finite flat commutative group scheme (over $X$). The __Cartier dual__ of $G$ is given by the functor $G^D(S)=Hom (G\otimes S, \mathbb{G}_m \otimes S)$. The Hom is taken in the category of group schemes over $S$.

For example, $\alpha_p^D\simeq \alpha_p$.

### Dieudonné module
(main article: [[Dieudonné module]])

There are certain correspondences ([Theorem Acu](#thmAcu)--[Theorem Fftc](#thmFftc)) between certain categories of group schemes and certain categories of Dieudonné modules.

+-- {: .num_defn}
###### Definition
A *Dieudonné module* is a module over the [[Dieudonné ring]] $D_k$ of a field $k$ of prime [[characteristic]] $p$.
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+-- {: .num_defn}
###### Definition
The Dieudonné ring of $k$ is the ring generated by two objects $F,V$ subject to the relations

$$FV=VF=p$$

$$Fw=w^\sigma F$$

$$w V=V w^\sigma$$

where

$$\sigma:\begin{cases}
W(k)\to W(k)
\\
(w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots)
\end{cases}$$

denotes the endomorphism of the [[Witt ring]] $W(k)$ of $k$ given by raising each component of the [[Witt vectors]] to the $p$-th power; this means that $\sigma$ is component-wise given by the [[Frobenius endomorphism]] of the file $k$.
=--

The Dieudonné ring is a $\mathbb{Z}$-[[graded ring]] where the degree $n$-part is the $1$-[[dimension|dimensional]] [[free object|free module]] generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$

+-- {: .num_theorem #thmAcu}
###### Theorem
([III.5](#Demazure), $Acu_k\simeq Tor_V D_kMod$)

(see also [[Dieudonné module]] for more details concerning this theorem)

Let $k$ be a [[perfect field]] of prime characteristic $p$. Since $k$ is perfect Frobenius is an automorphism.

On the left we have the category of [[affine scheme|affine]] commutative [[unipotent group scheme|unipotent group schemes]]. On the right we have the category of all [[Dieudonné ring|D_k]][[Dieudonné module|-modules]] of $V$-torsion. The (contravariant) equivalence is given by

$$M:\begin{cases}
Acu_k&\to& Tor_V D_kMod
\\
G&\mapsto&colim_n Acu_k(G,W_{nk})
\end{cases}$$

where we recall that how the colimit of the hom space can be multiplied by the generators of the [[Dieudonné ring]].
=--

+-- {: .num_theorem}
###### Theorem
([III.6](#Demazure), $Feu_k\simeq Tor_V D_kMod$)

=--

+-- {: .num_theorem}
###### Theorem
([III.6](#Demazure), $Fiu_k\simeq Tor_F D_kMod$)

=--

+-- {: .num_theorem}
###### Theorem
([III.8](#Demazure), $Torf_p\simeq (fin W(k) Mod,F,V)$)

=--

+-- {: .num_theorem #thmFftc}
###### Theorem
([III.9](#Demazure), $Fftc\simeq \hat D_k Mod_{fin.len.quot}$)

=--

## Related concepts

* [[affine group scheme]]

* [[abelian group scheme]]

* [[dual abelian group scheme]]

* [[spectral group scheme]]

## References

* M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, _Schemas en groupes_, i.e. SGA III-1, III-2, III-3

* Michel Demazure, [[Pierre Gabriel|P. Gabriel]], _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

* Michel [[Demazure, lectures on p-divisible groups]] [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups){#Demazure}

* W. Waterhouse, _Introduction to affine group schemes_, GTM 66, Springer 1979.

* D. Mumford, _Abelian varieties_, 1970, 1985.

* J. C. Jantzen, _Representations of algebraic groups_, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

[[!redirects group scheme]]
[[!redirects group schemes]]

[[!redirects Cartier dual]]
[[!redirects Cartier duals]]

[[!redirects group schemes]]

[[!redirects height of a group scheme]]


group stack

#Contents#
* table of contents
{:toc}

## Idea

A _group stack_ is a _[[group object]]_ in the [[2-category]] of [[stacks]], hence a [[2-group]] equipped with [[higher geometry]] structure.

## Examples

* [[smooth 2-group]]

[[!redirects group stacks]]


group T - complex > history

< [[group T - complex]]
[[!redirects group T - complex -- history]]


group T-complex

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

##Idea##
A **group $T$-complex** is ''a [[simplicial T-complex]] internal to the category of groups''. They were first studied in Nick Ashley's thesis (see references below for the published version)

More precisely:

##Definition##

A **group $T$-complex** is a pair,  $(G,T)$, in which $G$ is a [[simplicial group]] and $T$ is a graded subgroup  of $G$ consisting of _thin_ elements,  and which satisfies the conditions:

* Every degenerate element is thin.
* Every [[horn]] in $G$ has a unique thin filler.
* A thin filler of a thin box also has its last face thin.

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

Let $D = (D_n)_{n\geq 1}$ be the graded subgroup of $G$ generated by the
images of the degeneracy maps, $s_i :G_n \rightarrow G_{n+1}$, for all $i$ and
$n$,  then any box in $G$ has a  filler in $D$.
=--

+--{.proof}
######Proof
The algorithmic formulae
used when proving that any simplicial group is a Kan complex (cf., entry on  [[simplicial group]]) give a filler defined as a product of degenerate copies of the faces of the given box, so is in $D_n$.
=--
+--{.un_proposition}
######Proposition
If $(G,T)$ is a group $T$-complex then $T = D$.
=--

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

To see this, we note that
as every degenerate element is this, $D \subseteq T$. Conversely if $t \in T_n$, then it
fills the box made up of $ (  - , d_1t, \ldots, d_n t )$.  This, in turn, has a
filler, $d$, in  $D$, but, as this filler is also thin, it must be that $t = d$,
since thin fillers are uniquely determined.
=--
This is neat.  It says there is basically only one possible group $T$-complex structure on a given simplicial group.  The next result (again by Ashley) shows that not all simplicial groups carry such a structure.

+--{.un_proposition}
######Proposition
If $G$ is a simplicial group, then $(G,D)$ is a group $T$-complex if and
only if $NG\cap D$ is the trivial graded subgroup.
=--
+--{.proof}
######Proof
One way around, this is
nearly trivial.  If $(G,D)$ is a group $T$-complex and $x\in NG_n$, then $x$
fills a box $(-, 1, \ldots, 1)$, so if $x\in NG_n\cap D_n$, $x$ must itself
be the thin filler, however 1 is also a thin filler for this box, so $x = 1$ as
required.

Conversely if $NG\cap D = \{1\}$, then we must check the other two axioms as the first is trivial.  As any box has a standard filler in $D$, we only have to
check uniqueness, but if $x$ and $y $ are in $D_n$, and  both fill the same box
(with the $k^{th}$ face missing) then $z = xy^{-1}$ fills a box with 1s on all
faces (and the $k^{th}$ face missing).

If $k = 0$, then as $z \in NG_n\cap D_n$, we have $z = 1$ and $x$ and $y$ are
equal.  If $k \gt 0$, assume that if $\ell \lt k$ and $z \in D_n \cap
\bigcap_{i\neq \ell} Ker\, d_i$ then $z = 1$, (i.e, that we have uniqueness up to
at least the $(k-1)^{st}$ case).  Consider $w = zs_{k-1}d_k z^{-1}$.  This is
still in $D_n$ and $d_i w = 1$ unless $i = k-1$, hence by assumption $w = 1$.
Of course, this implies that $z = s_{k-1}d_kz$, but then $d_{k-1} z = d_k z$.  We
know that $d_{k-1} z = 1$, so $d_k z = 1$ and $z = 1$, i.e., $x = y$ and we have
uniqueness at the next stage.

To verify the third axiom, assume that $x \in D_{n + 1}$ and
each $d_i x \in D_n$ for $i \neq k$, then we can assume that $k = 0$, since
otherwise we can skew the situation around as before to get that to be true,
verify it in that case and 'skew' it back again later.

Suppose therefore that $d_i x \in D_n$ for all $0 \lt i \lt n$.  As $x$ must be the degenerate filler given by the standard method, we can calculate $x$ as follows:

let $w_n = s_{n-1}d_n x$, $w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i $ for $i =
1$, then $x = w_1$.  We can therefore check that $d_0x \in D_n$ as
required.
=--

##The group(oid) T-complex associated to a simplicial group(oid)##

This suggests that, given an arbitrary simplicial group(oid), $G$, we could form a quotient which would be a group(oid) $T$-complex, simply by dividing out by the subgroupoids, $NG_n\cap D_n$. We would need check that the face and degeneracy maps worked correctly, that the result did not somehow generate some new 'thin' elements, etc.  In fact the idea does not work because of a much more elementary problem, which is already apparent,  even in the case of simplicial groups,

* $NG_n\cap D_n$ need not be a normal subgroup of $G_n$!

A variant of the idea does however work. We concentrate on the case of simplicial groups.  The extension to simplicial groupoids is then straightforward.  We need the Conduché non-Abelian version of the [[decomposition theorem for simplicial groups]] that makes up an important part of the Dold-Kan correspondence.  With that we note the fairly obvious point that when we divide out by a graded subgroup in a simplicial group, then it has effects in all dimensions due to the face and degeneracy maps, so if we kill elements in $NG_n\cap D_n$ in all dimensions, we must also kill $d_0(NG_{n+1}\cap D_{n+1})$ and all the $s_k(NG_{n-1}\cap D_{n-1})$.  The end result of this is a group(oid) $T$-complex whose Moore complex has a [[crossed complex]] structure in a natural way.

It is easier to give the corresponding formula for the equivalent crossed complex to this group $T$-complex. Explicitly we  can form
$${C}(G)_{n+1} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},$$

in higher dimensions with at its 'bottom end', the crossed module,

 $$\frac{\mathcal{N}G_1}{d_0(\mathcal{N}G_2\cap D_2)} \to \mathcal{N}G_0$$

with $\partial$ induced from the boundary in the [[Moore complex]].  This gives a crossed complex.

##References##

* N. Ashley, _Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan_, Dissertations Math., 165, (1989), 11 – 58.
[[!redirects group T - complex]]


group theory

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

#Contents#
* table of contents
{:toc}

## Idea

The study of [[groups]].

## Related entries

* [[group]], [[Grp]]

* [[combinatorial group theory]], [[geometric group theory]]

* [[discrete group]], [[finite group]]

* [[permutation group]], [[symmetric group]]

* [[Cayley's theorem]]

* [[group action]], [[linear representation]]

* [[representation theory]]

* [[higher group character]]

* [[Erlangen program]] and [[Klein 2-geometry]]

* [[GAP]]

There are many generalizations and related structures; for example the vertical categorifications of groups like [[2-group]]s, horizontal categorifications as [[groupoids]], groups with structure, like [[topological group]]s, [[Lie group]]s, thus also [[Lie groupoids]], [[Lie infinity-groupoids]]; and noncommutative generalizations like [[quantum groups]]. Lie and algebraic group(oid)s have their infinitesimal precursors like formal groups, local Lie groups, tangent [[Lie algebras]], tangent [[Lie algebroids]] etc. In the smooth context the relation between Lie groupoids and Lie algebroids is the subject of [[Lie theory]].

## Literature

Lecture notes

* [[James Milne]], *Group theory* (2021) [[web](https://www.jmilne.org/math/CourseNotes/gt.html), [pdf](https://www.jmilne.org/math/CourseNotes/GT.pdf)]

Textbook accounts

in general

* [[Joseph J. Rotman]], *An Introduction to the Theory of Groups*, Springer (1995) [[doi:10.1007/978-1-4612-4176-8](https://doi.org/10.1007/978-1-4612-4176-8), [pdf](https://eclass.uoa.gr/modules/document/file.php/MATH676/Rotman%20An%20introduction%20to%20the%20theory%20of%20groups.pdf)]

in a more general context of [[algebra]]:

* [[Anthony Knapp]], *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]

and in relation to applications in ([[quantum physics|quantum]]) [[physics]] (cf. "[[Gruppenpest]]"):

* [[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)]

* {#Wigner31} [[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)]

* {#Sternberg94} [[Shlomo Sternberg]], *Group Theory and Physics*, Cambridge University Press 1994 ([ISBN:9780521558853](https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853))

See also:

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

Formalization in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]])

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 4 of: *[[Symmetry]]* (2021) [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]

and implementation in [[Agda]]:

* [[UniMath project]]: [agda-unimath/src/group-theory](https://github.com/UniMath/agda-unimath/tree/master/src/group-theory)

On aspects of group theory seen inside [[homotopy theory]]/[[infinity-group|$\infty$-group theory]]:

* [[Roman Mikhailov]], *Homotopy patterns in group theory*,  Proceedings of the [ICM 2022](https://icm2022.org) ([arXiv:2111.00737](https://arxiv.org/abs/2111.00737))

[[!redirects Group theory]]

category: group theory


group theory - contents

**[[group theory]]**

* [[group]], [[∞-group]]
* [[group object]], [[group object in an (∞,1)-category]]
* [[abelian group]], [[spectrum]]
* [[super abelian group]]
* [[group action]], [[∞-action]]
* [[representation]], [[∞-representation]]
* [[progroup]]
* [[homogeneous space]]

**Classical groups**

* [[general linear group]]

* [[unitary group]]

  * [[special unitary group]]. [[projective unitary group]]

* [[orthogonal group]]

  * [[special orthogonal group]]

* [[symplectic group]]

**Finite groups**

* [[finite group]]

* [[symmetric group]], [[cyclic group]],  [[braid group]]

* [[classification of finite simple groups]]

* [[sporadic finite simple groups]]

  * [[Monster group]], [[Mathieu group]]

**Group schemes**

* [[algebraic group]]

* [[abelian variety]]

**Topological groups**

* [[topological group]]

* [[compact topological group]], [[locally compact topological group]]

* [[maximal compact subgroup]]

* [[string group]]

**Lie groups**

* [[Lie group]]

* [[compact Lie group]]

* [[Kac-Moody group]]

**Super-Lie groups**

* [[super Lie group]]

* [[super Euclidean group]]

**Higher groups**

* [[2-group]]

  * [[crossed module]], [[strict 2-group]]

* [[n-group]]

* [[∞-group]]

  * [[simplicial group]]

  * [[crossed complex]]

  * [[k-tuply groupal n-groupoid]]

  * [[spectrum]]

* [[circle n-group]], [[string 2-group]], [[fivebrane Lie 6-group]]

**Cohomology and Extensions**

* [[group cohomology]]

* [[group extension]],

* [[∞-group extension]], [[Ext-group]]

**Related concepts**

* [[quantum group]]


Group theory > history

< [[Group theory]]
[[!redirects Group theory -- history]]


groupal model for universal principal infinity-bundles

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

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a model for an [[∞-group]], there is often a model for the [[universal principal ∞-bundle]] $\mathbf{E}G$ that itself carries a group structure such that the canonical inclusion $G \to \mathbf{E}G$ is a homomorphism of group objects. This extra groupal structure is important for various constructions.

## For ordinary groups {#OrdinaryGroups}

For $G$ an ordinary bare [[group]], the [[action groupoid]] $\mathbf{E}G = G//G$ of the right multiplcation action of $G$ on itself

$$
  G//G
  =
  \left(
  G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}}
  G
  \right)
$$

is [[contractible]]. We may think of this as the groupoid $INN(G) \subset AUT(G)$ of [[inner automorphism]]s inside the [[automorphism 2-group]] of $G$. But it is also simply isomorphic to the [[codiscrete groupoid]] on the set underlying $G$. In this latter form the 2-group structure on $\mathbf{E}G$ is manifest, which corresponds to the
[[crossed module]] $(G \stackrel{Id}{\to} G)$ .

It is also manifest then that this fits with the one-object [[delooping]] groupoid $\mathbf{B}G$ of $G$ into a sequence

$$
  G \to \mathbf{E}G \to \mathbf{B}G
  \,,
$$

where the first morphism is a homomorphism of [[strict 2-group]]s.

Regarded as a sequence of morphisms in the model [[Kan complex|KanCplx]] of the [[(∞,1)-topos]] [[∞-Grpd]] this _is_ already a model for the universal $G$-bundle.

If $G$ here is refined to a [[Lie group]] or [[topological group]] then this is a sequence of [[∞-Lie groupoid]]s or [[topological ∞-groupoid]]s, respectively, and also then, this _is_ already a model for the universal $G$-principal bundle, as discussed at ?LieGrpd -- the universal G-principal bundle.

By applying the [[geometric realization]] functor

$$
  |-| : \infty Grpd \stackrel{\simeq}{\to} Top
$$

we obtain a sequence of topological spaces

$$
  G \to \mathcal{E}G \to \mathcal{B}G
  \,,
$$

where $\mathcal{E}G$ carries the structure of a [[topological group]] and the morphism $G \to \mathcal{E}G$ is a topological group homomorphism. For bare groups $G$ and under mild assumptions also for general topological groups $G$, this groupal topological model for the universal $G$-bundle obtained from the realization of the groupoid $G//G$ was consider in [Segal68](#Segal).

## For strict 2-groups

For $G \in $ [[∞Grpd]] a [[strict 2-group]] a groupal model for $\mathbf{E}G$ was given in [RobertsSchr07](#RobertsSchreiber) generalizing the $INN(G) \subset AUT(G)$ construction mentioned above. This yields a weak 3-group structure on $\mathbf{E}G$ (A [[Gray-group]]).

In [Roberts07](#RobertsUniversal) it is observed that there is also an analog of $codisc(G)$ and that this yields a strict group structure on $\mathbf{E}G$. In fact, this strictly groupal model of $\mathbf{E}G$ turns out to be isomorphic to the standard model for the universal [[simplicial principal bundle]] traditionally denoted $W G$. And this statement generalizes...

## For $\infty$-Groups

Every [[∞-group]] may be modeled by a [[simplicial group]] $G$. There is a standard [[Kan complex]] model for the universal $G$-[[simplicial principal bundle]] $\mathbf{E}G \to \mathbf{B}G$ denoted $W G \to \bar W G$.

This standard model $W G$ does happen to have the structure of a [[simplicial group]] itself, and this structure is compatible with that of $G$ in that the canonical inclusion $G \to W G$ is a homomorphisms of simplicial groups.

This sounds like a statement that must be well known, but it was maybe not in the literature. The simplicial group structure is spelled out explicitly in [Roberts07](#RobertsUniversal)

## For $\infty$-groups in an arbitrary $(\infty,1)$-topos

Since the $W$ construction discussed above is functorial, this generalizes to prresheaves of simplicial groups and hence gives models for group objects and groupal universal [[principal ∞-bundle]]s in  [[(∞,1)-topos]]es modeled by the [[model structure on simplicial presheaves]].

(...)

## For $\infty$-Lie groups

In the [[(∞,1)-topos]] of [[smooth ∞-groupoids]] we can obtain [[∞-groups]] by [[Lie integration]] of [[L-∞-algebra|$L_\infty$-algebras]].

Corresponding to this is a construction of Lie-integrated groupal universal principal $\infty$-bundles:

for $\mathfrak{g}$ an $L_\infty$-algebra, there is an $L_\infty$-algebra $inn(\mathfrak{g})$, defined such that its [[Chevalley-Eilenberg algebra]] is the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$:

$$
  CE\big(inn(\mathfrak{g})\big)
    =
  W\big(\mathfrak{g}\big)
  \,.
$$

Under [[Lie integration]] this gives a groupal model for the universal principal $\infty$-bundle over the [[∞-Lie group]] that integrates $\mathfrak{g}$.

This is described at

The Lie-integrated universal principal ∞-bundle

## Related concepts

* [[principal ∞-bundle]]

* [[universal principal ∞-bundle]]

* [[simplicial classifying space]]

* [[universal connection]]

## References

The observation that for $G$ an ordinary [[group]], its [[action groupoid]] sequence $G \to G//G \to \mathbf{B}G$ -- which is the [[strict 2-group]] coming from the [[crossed module]] $(G \stackrel{Id}{\to} G)$ - maps under the [[nerve]] to the universal $G$-bundle appeared in

* G. B. Segal, _Classifying spaces and spectral sequences_ , Publ. Math. IHES
No. 34 (1968) pp. 105–112 {#Segal}

A weak 3-group structure on $G \to \mathbf{E}G$ for $G$ a [[strict 2-group]] is descibed in

* [[David Roberts]], [[Urs Schreiber]], _The inner automorphism 3-group of a strict 2-group_ , Journal of Homotopy and Related Structures, vol. 3(1) ([arXiv:0708.1741](http://arxiv.org/abs/0708.1741))
  {#RobertsSchreiber}

The [[simplicial group]] structure on $G \to \mathbf{E}G$ for $G$ a general [[simplicial group]] is stated explicitly in

* [[David Roberts]], _The universal simplicial bundle is a simplicial group_, [New York Journal of Mathematics](http://nyjm.albany.edu/), Volume 19 (2013) 51-60, [journal version](http://nyjm.albany.edu/j/2013/19-5.html), [arXiv:1204.4886](http://arxiv.org/abs/1204.4886).
  {#RobertsUniversal}

A general abstract construction of this simplicial group structure is discussed in

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parameterized spaces_, ([arXiv:1203.2460](http://arxiv.org/abs/1203.2460))

The use of [[L-∞-algebra]]s $inn(\mathfrak{g})$ as $L_\infty$-algebraic models for universal $\mathfrak{g}$-principal bundle (evident as it is) was considered as such in

* [[nLab:Hisham Sati]], [[nLab:Urs Schreiber|U.S.]], [[nLab:Jim Stasheff]],
  _$L_\infty$-algebra connections_ in Fauser (eds.) Recent Developments in QFT, Birkhäuser ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

[[!redirects groupal model for universal principal ∞-bundle]]
[[!redirects groupal model for universal principal ∞-bundles]]
[[!redirects groupal models for universal principal ∞-bundles]]

[[!redirects groupal models for universal principal infinity-bundles]]

[[!redirects groupal model for universal principal simplicial complex]]
[[!redirects groupal models for universal principal simplicial complex]]


grouplike A3-space

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

## Contents ##
* table of contents
{:toc}

## Idea ##

The invertible version of the [[A3-space]] up to homotopy, without any higher coherences for inverses.

## Definition ##
A __grouplike $A_3$-space__ or __grouplike $A_3$-algebra in homotopy types__ or __H-group__ consists of

* A type $A$,
* A basepoint $e:A$
* A binary operation $\mu : A \to A \to A$
* A unary operation $\iota: A \to A$
* A left unitor
$$\lambda_u:\prod_{(a:A)} \mu(e,a)=a$$
* A right unitor
$$\rho_u:\prod_{(a:A)} \mu(a,e)=a$$
* An asssociator
$$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$$
* A left invertor
$$l:\prod_{(a:A)} \mu(\iota(a), a)=e$$
* A right invertor
$$r:\prod_{(a:A)} \mu(a,\iota(a))=e$$

One could also speak of grouplike $A_3$-spaces where the existence of left and right inverses are mere property rather than structure, which is a grouplike $A_3$-space as defined above with additional structure specifying that the types $\prod_{(a:A)} \mu(\iota(a), a)=e$ and $\prod_{(a:A)} \mu(a,\iota(a))=e$ are [[contractible]]:

$$c_l: \sum_{l:\prod_{(a:A)} \mu(\iota(a), a)=e} \prod_{b:\prod_{(a:A)} \mu(\iota(a), a)=e} l = b$$
$$c_r: \sum_{r:\prod_{(a:A)} \mu(a,\iota(a))=e} \prod_{b:\prod_{(a:A)} \mu(a,\iota(a))=e} r = b$$

or equivalently,

$$c_l:\prod_{(a:A)} \sum_{(l:\mu(\iota(a), a)=e)} \prod_{(b:\mu(\iota(a), a)=e)} l = b$$
$$c_r:\prod_{(a:A)} \sum_{(r:\mu(a,\iota(a))=e)} \prod_{(r:\mu(a,\iota(a))=e)} r = b$$

## Examples ##

* The [[integers]] are an grouplike $A_3$-space.

* Every [[loop space]] is naturally an grouplike $A_3$-space with [[path]] concatenation as the operation. In fact every [[loop space]] is a $\infty$-group.

* A [[group]] is a 0-truncated grouplike $A_3$-space.

## See also ##

* [[higher algebra]]
* [[A3-space]]
* [[group]]


grouplike element

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

## Definition

### General definition in a coring

Given an $A$-[[coring]] $(C,\Delta,\epsilon)$ (a [[comonoid]] in the category of $A$-$A$-[[bimodule]]s for a $k$-[[algebra]] $A$)  (example: any $k$-algebra) a __semi-grouplike__ element in $A$ is any $g\in C$ such that

$$
  \Delta(g) = g\otimes g
  \,.
$$

A __grouplike__ (or group-like) element is a semi-grouplike one such that $\epsilon(g) = 1$.

**Proposition.** An $A$-coring $(C,\Delta,\epsilon)$ has a grouplike element iff $A$ is a right or left $C$-comodule.

*Proof.* Given a grouplike element $g\in C$, one defines a right coaction $\rho = \rho_g : A \to A\otimes_A C\cong C$ by the formula

$$
\rho(a) = 1_A \otimes_A ga = ga
$$

it is clear that this is a map of $A$-bimodules. Now $(\rho\otimes id_C)(1_A\otimes_A ga) = g\otimes_A 1_A\otimes ga = g\otimes ga$, while $(id\otimes \Delta_C)(1_A\otimes ga) = 1_A \otimes_A g\otimes ga = g\otimes ga$ hence the coassociativity and similarly for the counit.

Conversely, let $(A,\rho)$ be a right $C$-comodule. Then one checks that $\rho(1_A)\in A\otimes_A C \cong C$ is a grouplike.

For the left comodules the story is similar, e.g. $\rho(a) = ag$.
### Special case: grouplike elements in coalgebras

Every [[coalgebra]] is special case of a [[coring]].

The grouplike elements in a $k$-[[Hopf algebra]] form a [[group]]. (Can this fact be [[categorification|categorified]] ??)

## Relation to differential graded algebras

For corings with a (sometimes semi-)grouplike element one can define many useful notions which do not exist for general corings.

For example, given a semi-grouplike element $g$, the [[tensor algebra]] $\Omega C = \oplus_i \Omega^i C$ of the coring $C$, where $\Omega^i C = C\otimes_A \ldots \otimes_A C$ ($i$ times) over $A$ can be equipped with a [[differential]] $d$ of degree $+1$ in a canonical way making it a [[differential graded algebra]]:

in degree $0$, one defines

$$
  d a = g a - a g
$$

and in higher degree

$$
  d(c_1\otimes\ldots\otimes c_n) = g\otimes c_1\otimes\ldots\otimes c_n + (-1)^{n+1}c_1\otimes\ldots\otimes c_n\otimes g + \sum_{i=1}^n c_1\otimes\ldots\otimes c_{i-1}\otimes \Delta(c_i)\otimes c_{i+1}\otimes\ldots \otimes c_n\,.
$$

In fact, by a result in

* A. V. Roiter, _Matrix problems and representations of BOCS's_; in Lec. Notes. Math. 831, 288--324 (1980)

[[semi-free differential graded algebras]] are in bijective correspondence with corings with a group-like element. Moreover flat [[connection for coring|connections]] for a semi-free dga are in $1$-$1$ correspondence with the [[comodule]]s over the corresponding coring with a group-like element.

A special case of this construction is when $g = 1\otimes_R 1$ and $C$ is the [[Sweedler coring]] for a $k$-algebra extension $R\to S$. The dga obtained is the classical Amitsur complex $\Omega(S/R)$ for that extension; for this reason the complex $\Omega C = \Omega(C,g)$ above for any coring $C$ and semi-grouplike $g$ is sometimes said to be an _[[Amitsur complex]]_.

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.

* C. Menini, D. Ştefan, _Descent theory and Amitsur cohomology of triples_,
J. Algebra __266__ (2003), no. 1, 261--304.

* [[T. Brzeziński]], _Flat connections and comodules_, [math.QA/0608170](http://arxiv.org/abs/math/0608170)

* [[T. Brzeziński]], _Galois structures_, Warszawa 2007/8 course, part III, [pdf](http://toknotes.mimuw.edu.pl/sem7/files/Brzezinski_gs.pdf), [ps](http://toknotes.mimuw.edu.pl/sem7/files/Brzezinski_gs.ps)

[[!redirects group-like element]]


grouplike geometric morphism

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## Idea

...

## Definition

A geometric morphism $p:\mathcal{E} \to\mathcal{S}$ (, or the $\mathcal{S}$-topos $\mathcal{E}$ it corresponds to) is called _grouplike_ if for every $\mathcal{S}$-topos $q:\mathcal{F} \to\mathcal{S}$ the category of 1-cells from $q$ to $p$ in the 2-category $\mathbf{Top}/\mathcal{S}$ of $\mathcal{S}$-toposes is a [[groupoid]].

## Properties

...

## Related entries

* [[local geometric morphism]]

* [[totally connected geometric morphism]]

## Reference

* {#JT02}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.2_ , Oxford UP 2002. (C3.6.9; p.703, 710)

[[!redirects grouplike geometric morphisms]]


GroupNames

*[GroupNames](https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html)* is a public [wiki](http://en.wikipedia.org/wiki/Wiki) on [[group theory]].

## Related projects

* [[Groupprops]]

category: reference