crossed n-cube

#Idea#

* A [[crossed module]] is the 'algebraic core' of a $\cat^1$-[[cat-1-group|group]], in as much as within a $\cat^1$-group we can find a crossed module in a simple way from which the whole of the $\cat^1$-group in all its glory can be reconstructed.

* A [[crossed square]] is similarly the 'algebraic core' of a $\cat^2$[[cat-2-group|group]].

* A _crossed $n$-cube_ should be the 'algebraic core' of a $\cat^n$-[[cat-n-group|group]].

A crossed square in the Guin-Valery Loday specification has quite a few axioms, many more than those for a crossed module. Does that mean that crossed $n$-cubes will be defined with a very large number of axioms (perhaps dependent on $n$)? No (unless you think that 11 is large!)

Finally a particular example of a [[crossed square]] is given by a group with two [[normal subgroup]]s, and their intersection together with the commutator map.  It would be appropriate if a group together with $n$ normal subgroups, all the  intersections and all the commutator maps, formed an example of a crossed $n$-cube.  It does.  The axioms below govern the $h$-maps, by using analogues of the relationships between commutator maps in such a $n$-cube of inclusions of intersections.

The result (definition due to Graham Ellis and Richard Steiner, reference below) is a set of 11 axioms. That is all, and these objects model all homotopy $(n+1)$-[[homotopy n-type|types]].

#Definition#

We denote by $\langle n \rangle$ the set $\{1,2,\ldots, n\}$.

A _crossed $n$-cube_, $M$, is a family of  groups,  $\{M_A: A \subseteq  \langle n \rangle\}$,
together with homomorphisms,  $\mu_i :M_A \rightarrow  M_{A-\{i\}}$, for $i \in  \langle n \rangle
, A \subseteq  \langle n \rangle$, and functions, $h:M_{A} \times  M_{B} \rightarrow
M_{A\cup B}$, for $A, B \subseteq  \langle n \rangle$, such that if $^{a}b$ denotes $h(a,b)b$
for  $a \in M_{A}$ and $b \in M_{B}$ with  $A \subseteq  B$, then for  $a,
a^\prime \in M_{A},\, b, b^\prime \in M_{B},\, c \in  M_{C}$  and  $i, j \in
\langle n \rangle$, the following axioms hold:

1. $\mu_i a = a$ if $i \notin A$

1. $\mu_i\mu_j a = \mu_j\mu_i a$

1. $\mu_i h(a,b) = h(\mu_i a,\mu_i b)$

1. $h(a,b) = h(\mu_i a,b) = h(a,\mu_i b)$ if $i \in A \cap B$

1. $h(a,a^\prime ) = [a,a^\prime ]$

1. $h(a,b) = h(b,a)^{-1}$

1. $h(a,b) = 1$ if $a = 1$ or $b = 1$

1. $h(aa^\prime ,b) = {}^{a}h(a^\prime ,b)h(a,b)$

1. $h(a,bb^\prime )  =  h(a,b){ }^b h(a,b^\prime )$

1. ${ }^{a}h(h(a^{-1},b),c)^{c}h(h(c^{-1},a),b)^{b}h(h(b^{-1},c),a)  =  1$

1. ${ }^{a}h(b,c)  =  h(^{a}b,^{a} c)$  if $A \subseteq  B \cap  C$.

A morphism of crossed $n$-cubes
$$f :\{M_{A}\} \rightarrow \{M^{\prime}_{A}\}$$
is a family of homomorphisms, $\{f_{A}: M_{A} \rightarrow  M^{\prime}_A \,|\, A \subseteq  \langle n \rangle\}$, which commute with the maps, $\mu_i$, and the functions, $h$.

This gives us a category, $Crs^{n} $, equivalent to that of $\cat^n$-[[cat-n-group|groups]].

# Homotopical example#

 The  _fundamental crossed $n$-cube of groups functor_  $\Pi '$ is defined from  $n$-cubes of pointed spaces to crossed $n$-cubes of
groups:  $\Pi  'X_{*}$   is simply  the crossed $n$-cube of
groups equivalent to the cat$^n$-group  $\Pi X_{*}$. It is easier
to identify  $\Pi '$  in classical terms in the case $X_{*}$   is
the $n$-cube  of spaces arising from a pointed  $(n +
1)$-ad  $\mathcal{X}  = (X;X_1,\ldots ,X_n)$. That is, let  $X_{
\langle n \rangle } = X$  and for  $A$  properly contained in
$\langle n \rangle$ let   $X_A = \bigcap _{i \not\in A}  X_i$. Then
$M = \Pi '\mathcal{X}$   is given as follows (Ellis and Steiner, 1987):
$M_{\emptyset}  = \pi_1(X_\emptyset )$; if  $A = {i_1,\ldots ,i_r}$,
in the right order, then $M$ is the homotopy  $(r + 1)$-ad group
$\pi _{r+1}(X _A;X_A \cap X_{i_1} ,\ldots ,X_A   \cap  X_{i_r} )$;
the maps $\mu$ are given by the usual boundary maps;  the
$h$-functions are  given by generalised Whitehead products.

Note
that whereas these separate elements of structure had all been
considered previously, the aim of this theory is to consider the
whole structure, despite its apparent complications. The equivalence of categories is a convincing reason for supposing that the axioms for a crossed $n$-cube of groups are a complete axiomatisation of this homotopical structure, as was not previously known.

# Algebraic example#
Let us put a bit of flesh on the example given in the introduction.  Suppose that  $G$ is a group and $N_1, N_2, .., N_n$ are $n$-[[normal subgroup]]s of $G$ (there may be repeats).

Now define for $A \subseteq  \langle n \rangle$, $M_A = \bigcap\{N_i\mid i \in A\}$, and for $A\in M_A$, $b\in M_B$, $h(a,b) = [a,b]$, the [[commutator]] of $a$ and $b$ in $G$.  The result is a crossed $n$-cube (sometimes called the _inclusion crossed $n$-cube_ determined by the normal $n$-ad of subgroups).

#Simplicial example#
If instead of a space you start with a [[simplicial group]] $G$, as model for a connected homotopy type, then there is a crossed $n$-cube generalising the [[crossed square]] given in terms of the [[Moore complex]].

#References#

* [[G. J. Ellis]] and [[R. Steiner]], _Higher dimensional crossed modules and the homotopy groups of
(n + 1)-ads_, J. Pure Appl. Alg., 46, (1987), 117–136.

* [[Ronnie Brown|R. Brown]] _Computing homotopy types using crossed n-cubes of groups_, Adams  Memorial Symposium on
Algebraic Topology}, Vol 1, edited N. Ray and G  Walker,
Cambridge University Press, 1992, 187-210.  [Here is a link to a revised version with hyperref](http://groupoids.org.uk/pdffiles/ADAMSVT.pdf)

*  [[Ronnie Brown|R. Brown]] _Homotopical excision (work to be done!)_ Presentation at Higher Structures Lisbon, July 2017. [RB's preprints](http://groupoids.org.uk/brownpr.html)


crossed product algebra

#Contents#
* table of contents
{:toc}

## Smash product algebra

### Definition

Given a $k$-[[bialgebra]] $H$, and a left [[Hopf action]] $\triangleright$ of $H$ on a $k$-algebra $A$, one defines the **crossed product algebra** $A\sharp H$ (in [[Hopf algebra]] literature also called the **smash product algebra** or Hopf smash product; distinguish from the rather different [[smash product]] in topology) as the $k$-algebra whose underlying vector space is $A\otimes H$ and the product is given by

$$
(a\otimes h)(a'\otimes h') =
\sum a (h_{(1)}\triangleright a')\otimes h_{(2)}h'.
$$

The idea is that if the bialgebra $H$ is in fact a [[Hopf algebra]] embedded as $1\otimes H\subset A\sharp H$ -- whatever the product in the latter is (but assumed to satisfy $(a\otimes 1)(1\otimes h) = a\otimes h$) -- and if the action is inner within $A\sharp H$, i.e. $h\triangleright a = \sum h_{(1)} a S(h_{(2)})$, then we have

$$
  \sum (h_{(1)}\triangleright a) h_{(2)} = \sum h_{(1)} a S(h_{(2)}) h_{(3)} = \sum h_{(1)} a \epsilon(h_{(2)})= h a
  \,,
$$

and hence the formula for the product above is a tautology: $a h a' h' = a(h_{(1)}\triangleright a') h_{(2)} h'$.

Similarly, given a right Hopf action of $H$ on $A$, one defines the crossed product algebra $H\sharp A$ whose underlying space is $H\otimes A$. The left and right versions are isomorphic if $H$ has an invertible antipode; this extends the correspondence between the left and right actions obtained by composing with the antipode map.

### Properties

Every smash product algebra of the form $A\sharp H$ is naturally equipped with a [[monomorphism]] $A\mapsto A\sharp 1\hookrightarrow A\sharp H$ of algebras and with a right $H$-coaction $a\otimes h\mapsto a\otimes \Delta(h)\in (A\sharp H)\otimes H$ making $A\sharp H$ into a right $H$-comodule algebra. Map $\gamma: h\mapsto 1\otimes h$, $H\hookrightarrow A\sharp H$ is then a map of right $H$-comodule algebra (where the coaction on $H$ is $\Delta$), and $A\otimes 1\subset A\sharp H$ is the subalgebra of $H$-coinvariants.

If $H$ is a Hopf algebra, then the homomorphism $\gamma$ is a convolution invertible linear map with convolution inverse $\gamma^{-1}$ defined by $\gamma^{-1}(h)=\gamma(Sh)$ for $h\in H$, where $S$ is the antipode of $H$. Conversely,

**Proposition** _Let $H$ be a Hopf algebra, $E$ a right $H$-comodule algebra, and $\gamma:H\to E$ a map of right $H$-comodule algebra. Clearly $H$ acts on $E^{co H}$ by $h\triangleright a = \sum \gamma(h_{(1)}) a\gamma(Sh_{(2)})$ for $a\in E^{co H}$ and $h\in H$, where the product on the right-hand side is in $E$. Conclusion:
$E\cong E^{co H}\sharp H$ where the smash product is with respect to that action.

## Cocycled crossed product

There is also a more general cocycled crossed product. For a bialgebra $H$ and an algebra $U$, if we consider the category $C(U,H)$ of extensions $U\hookrightarrow E$ which are compatibly left $U$-modules and right $H$-comodules, and where $U=E^{\mathrm{co}H}$, then the crossed product algebras are the canonical representatives of [[cleft Hopf-Galois extension]]s  which are a more invariant concept.

Let $U$ be an algebra, $H$ a Hopf algebra, $\triangleright : H\otimes U\to U$ a [[measuring]], i.e. a $k$-linear map satisfying $h\triangleright(u v)=\sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright v)$ for all $h\in H$, $u,v\in U$, and which we assume unital, i.e. $h\triangleright 1 = \epsilon(h)1$ for all $h\in H$. We do _not_ assume that $\triangleright$ is an action.

Let further a (convolution) *invertible*  $k$-linear map $\sigma \in Hom_k(H\otimes H,U)$ be given.

We say that $\sigma$ is a **2-cocycle** (relative to the measuring $\triangleright$) if the following two cocycle identities hold

$$
h\triangleright (k\triangleright u) = \sum \sigma(h_{(1)},k_{(1)})
((h_{(2)}k_{(2)})\triangleright u) \sigma^{-1}(h_{(3)},k_{(3)})
$$

$$
\sum [h_{(1)}\triangleright\sigma(k_{(1)},m_{(1)})]\sigma(h_{(2)},k_{(2)}m_{(2)})=\sum \sigma(h_{(1)},k_{(1)})\sigma(h_{(2)}k_{(2)},m)
\,.
$$

These identities clearly generalize the classical [[factor system]]s in group theory (linearly extended to the case of group algebras, for the finite groups at least). Therefore it is an  example of a nonabelian cocycle in Hopf algebra theory. However, its role in the general theory is less well understood than the group case.

Define the **cocycled crossed product** on $U\otimes H$ by

$$
(u \sharp h)(v\sharp k) = \sum u (h_{(1)}\triangleright v)
\sigma(h_{(2)},k_{(1)})\sharp h_{(3)} k_{(2)}
$$

for all $h,k\in H$, $u,v\in U$. The cocycled crossed product is an associative algebra iff $\sigma$ is a cocycle.

If so, we call $U\sharp_\sigma H$ **cocycled crossed product algebra**. Map $1\otimes\Delta_H:U\sharp_\sigma H\to (U\sharp_\sigma H)\otimes H$ is a right $H$-coaction, making $U\sharp_\sigma H$ into a right $H$-comodule algebra, which is cleft extensions]] are always isomorphic (as $H$-extensions) to the cocycled crossed product algebras.

If $\sigma(h,k)=\epsilon(h)\epsilon(k)1_U$ then we say that $\sigma$ is a trivial cocycle and then the compatibility conditions above reduce to demanding that the measuring $\triangleright$ is an action. The cocycled crossed product then reduces to the usual smash product algebra.

**Theorem.** Suppose we are given two measurings $\triangleright,\triangleright':H\otimes U\to A$ with cocycles $\sigma, \tau$ respectively. Then there exists an isomorphism of $H$-extensions of $U$, $i: U\sharp_\sigma H\cong U\sharp_\tau H$ (i.e. an isomorphism of $k$-algebras, left $U$-modules and right $H$-comodules) iff there is an invertible element $f\in Hom_k(H,U)$ such that for all $u\in U$, $h,k\in H$

$$h\triangleright' u = \sum f^{-1}(h_{(1)})(h_{(2)}\triangleright u) f(h_{(3)}),$$

$$\tau(h,k) = \sum f^{-1}(h_{(1)})[h_{(2)}\triangleright f^{-1}(k_{(1)})]\sigma(h_{(3)},k_{(2)})f(h_{(4)}k_{(3)}).$$

The isomorphism $i$ is then given by
$$i(u\sharp_\sigma h) = \sum u f(h_{(1)})\sharp_\tau h_{(2)}$$

## Literature

Related $n$Lab entres include [[crossed product C*-algebra]], [[noncommutative torsor]], [[Hopf-Galois extension]]

* Y. Doi, M. Takeuchi, _Cleft comodule algebras for a bialgebra_, Comm. Alg. __14__ (1986) 801--818
* Y. Doi, _Equivalent crossed products for a Hopf
algebra_, Comm. Alg. __17__ (1989), 3053--3085, [MR91k:16027](http://www.ams.org/mathscinet-getitem?mr=1030610), [doi](ttp://dx.doi.org/10.1080/00927878908823895)

* S. Montgomery, _Hopf algebras and their actions on
rings_, CBMS Regional Conference Series in Mathematics __82__,
AMS 1993.
* S. Majid, _Foundations of quantum group theory_, Cambridge University Press 1995.

category: algebra, noncommutative geometry

[[!redirects crossed product algebras]]
[[!redirects Hopf smash product]]
[[!redirects smash product algebra]]
[[!redirects cleft extension]]
[[!redirects cleft Hopf-Galois extension]]
[[!redirects cleft comodule algebra]]


crossed product C*-algebra

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### Context
#### Operator algebra
+-- {: .hide}
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=--
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[crossed product algebra|crossed product]] [[C*-algebra]].

## Properties

### Equivalence to convolution algebra

If the $A$ is a [[C*-algebra]] [[algebra of functions|of functions]] on some [[topological space]] $X$ and the [[action]] of $G$ on $A$ is induced from an [[action]] on $X$, then in good cases the crossed product algebra is equivalent to the [[convolution algebra]] of the [[action groupoid]].

e.g. ([Land 13, prop. 4.2](#Land13), [Soltan, section 5.1](#Soltan), [Bos 07, example 12.2.8](#Bos07)) also ([Connes 06, around p. 13](#Connes06))

## Related concepts

* [[Baum-Connes conjecture]]

## References

A standard textbook account (with an eye towards [[KK-theory]]) is section 10 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_

Original articles include

* M. A. Rieffel, _Proper actions of groups on C∗-algebras_ Mappings of operator algebras (Philadelphia, PA, 1988), 141–182, Progr. Math. 84, Birkhäuser Boston, Boston, MA, 1990. ([pdf](http://math.berkeley.edu/~rieffel/papers/properactions.pdf))

* Jonathan Henry Brown, _Proper actions of groupoids on $C^*$-algebras_ ([arXiv:0907.5570](http://arxiv.org/abs/0907.5570))

Detailed lecture notes are in

* Piotr Soltan, _$C^\ast$-Algebras, group actions and crossed products_, lecture notes ([pdf](http://www.fuw.edu.pl/~psoltan/prace/C*-algebras,%20group%20actions%20and%20crossed%20products%20%28lecture%20notes%29.pdf))
 {#Soltan}

A survey of [[noncommutative geometry]] with a bunch of examples of crossed product $C^\ast$-algebras is

* [[Alain Connes]], _A walk in the non-commutative garden_ ([arXiv:math/0601054](http://arxiv.org/abs/math/0601054))
 {#Connes06}

Further discussion in the context of the [[Baum-Connes conjecture]] is in

* Markus Land, _The Analytical Assembly Map and Index Theory_, ([arXiv:1306.5657](http://arxiv.org/abs/1306.5657))
 {#Land13}

Applications to [[geometric quantization]] and specifically [[geometric quantization by push-forward]] are indicated in

* Roger Bos, _Groupoids in geometric quantization_, 2007 ([pdf](http://www.math.ru.nl/~landsman/ProefschriftRogier.pdf))
 {#Bos07}

(...)

[[!redirects crossed product C-star algebra]]
[[!redirects crossed product C*-algebras]]
[[!redirects crossed product C-star algebras]]


crossed profunctor

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

## Definition

Given [[crossed modules]] $\mathbb{X}=(\partial_X : X_1\to X_0)$, $\mathbb{Y}=(\partial_Y:Y_1\to Y_0)$ (actions surpressed from the notation), a __crossed profunctor__

$$
  \mathbb{X}\to\mathbb{Y}
$$

consists of a group $P$ and morphisms $x_1:X_1\to P$, $x_0:P\to X_0$, $y_1:Y_1\to P$, $y_0:P\to Y_0$

$$
  \array{
    X_1 &&&& Y_1
    \\
    & \searrow^{\mathrlap{x_1}} & & {}^{\mathllap{y_1}}\swarrow
    \\
    {}^{\mathllap{\partial_X}}\downarrow && P &&
    \downarrow^{\mathrlap{\partial_Y}}
    \\
    & \swarrow_{\mathrlap{x_0}} && {}_{\mathllap{y_0}}\searrow
    \\
    X_0 &&&& Y_0
  }
$$

such that

* the two triangles commute, $x_0\circ x_1=\partial_X$, $y_0\circ y_1=\partial_Y$,

* the diagonals compose to identities
  $y_0\circ x_1 = 1$, $x_0\circ y_1 = 1$,

* and

  $$
    x_1 ({}^{x_0(p)}x) = p x_1(x) p^{-1},
    \,\,\,\,\,y_1   ({}^{y_0(p)}y) = p y_1(y)p^{-1}
    \,,
  $$

  where $x\in X_1$, $p\in P$, $y\in Y_1$.

The complex $Y_1\to P\to X_0$ is called the NE-SW complex, and $X_1\to P\to Y_0$ is called the NW-SE complex.

## Related concepts

If to the above definition was added the property that the SE-NW sequence $Y_1\to P\to X_0$ is _exact_ in that it is a (nonabelian in general) [[group extension]], this is called a [[butterfly]]. See there for more details.

+-- {: .query}
Alex asks: Is there any relation between a [[profunctor]] and a Crossed Profunctor? Just as one could look at a [[crossed module]] as a [[module]] with "twisted multiplication" (according to the crossed module page), is there some analogous "twisting" of a profunctor to obtain a crossed profunctor?

Beppe says: no. Actually the name "crossed profunctor" was not a happy choice.
A crossed profunctor is just the normalized version  of a profunctor, as a
crossed module is the normalization of a groupoid. This extends to a strong biequivalence between the bicategory of crossed modules, crossed profunctors and their morphisms (in Grp) and that of crossed modules, profunctors and their morphisms.

This result holds also in intrinsic settings, say when we consider internal categories in a semi-abelian category.

=--

## References

This notion appeared in

* M. Jibladze, _Coefficients for cohomology of "large" categories_ , pp. 169--179, in H. Inassaridze (Ed.), K-theory and Homological Algebra (A Seminar held at the Razmadze Mathematical Institute in Tbilisi, Georgia, USSR 1987-88), Lec. Notes in Math. 1437, Springer 1990 (*[[jibladzeCoeffLargeCats.djvu:file]]*).


crossed square

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

## Idea

A [[crossed module]] is a bit like a [[normal subgroup]] ... without being a [[subgroup]]. In fact if a crossed module has a boundary map which is a monomorphism then it is isomorphic to the inclusion crossed module of a normal subgroup.

Crossed modules model all connected [[homotopy 2-type]]s (which by the [[looping and delooping]]-theorem means: all [[2-group]]s).  _Crossed squares_ model all connected [[homotopy 3-type]]s (hence all [[3-group]]s) and correspond in much the same way to pairs of normal subgroups.

Suppose $G$ is a group and $M$ and $N$ are normal subgroups of $G$; then of course, so is $M \cap N$.  Put these groups in a square, with the inclusion maps between them. Finally note that if $m \in M$ and $n \in N$, then $[m,n]$ is in the intersection $M \cap N$. This gives you a crossed square with $h$-map $h(m,n) = [m,n]$.  Removing the  condition that the inclusions are inclusions (!) gives the general form.

(The definition that follows is that given by Guin-Valery and Loday in their paper (see references).  Another definition can be given that is just the case $n = 2$ of that of [[crossed n-cube]], for which see that entry.

## Definition

A **crossed square**

$$\array{& L & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}$$

consists of four morphisms of groups $\lambda: L \to M$, $\lambda': L \to N$, $\mu: M \to P$, $\nu: N \to P$,  such that $\nu \lambda'= \mu \lambda$ together with [[action]]s of the group $P$ on $L, M, N$ on the left, conventionally, (and hence actions of $M$ on $L$ and $N$ via $\mu$ and of $N$ on $L$ and $M$ via $\mu$) and a function $h: M \times N \to L$.

This structure
shall satisfy the following axioms:

1. the maps $\lambda, \lambda '$ preserve the actions of $P$; further, with the
given actions, the maps $\lambda, \lambda',\mu, \nu$ and $\kappa = \mu\lambda = \mu
'\lambda '$ are crossed modules;
1. $\lambda h(m,n)=m\,^n (m^{-1}),\lambda 'h(m,n)= {^m}n\,n^{-1}$
1. $h(\lambda l, n)= l\,^n (l^{-1}), h(m,\lambda 'l)= {^m}l\, l^{-1}$
1. $ h(mm',n) = {^m}h(m',n) h(m,n), h(m,nn') = h(m,n) \, ^n h(m,n')$;
1. $h(^p m, {^p n})= {^p}h(m,n)$;

for all $l\in L, \,m, m'\in M,\, n,n'\in N$ and $p\in P$.

The similarity of these axioms to [[commutator]] identities is no accident (see below).

This should be thought of as a _crossed module of crossed modules_ (in either direction!). For instance horizontally:
$$\array{& L & & M & \\
         \lambda^\prime & \downarrow &\to &\downarrow & \mu\\
          &N & & P & \\
}$$
The image of this morphism is a normal sub-crossed module of $(M,P,\mu)$, so we can form a quotient
$$\overline{\mu} : M/\lambda L \to P/\nu N,$$
and this is a crossed module, and is the kernel crossed module of this (horizontal) morphism.

## Examples

### Homotopical example

The classical homotopical example $\Pi(X;A,B)$ is determined by a pointed [[triad]] $(X; A,B)$ where $A,B \subseteq X$, and $P = \pi_1(A \cap B)$, $M = \pi_2(A, A \cap B), N = \pi_2(B, A \cap B)$ and $L=\pi_3(X; A,B)$. The operations of $P$ are the standard ones and  $h$ is the generalised [[Whitehead product]]. (The conventions may be slightly different from the standard ones in homotopy theory.)  This can be generalised to a functor $\Pi$ from squares of pointed spaces to crossed squares.

Ellis uses this construction in

* G.J. Ellis, Crossed squares and combinatorial homotopy, Math. Z., 461 (1993) 93–110,

where the fact that that the crossed square associated to a triad is defined directly in terms of certain homotopy classes is important.

The fact that there is a [[van Kampen theorem|van Kampen type theorem]] for $\Pi$ implies that one calculates some nonabelian groups. It also implies that one is calculating some (pointed) homotopy 3-types.

### Algebraic example

The example hinted at above has $P$ a group, $M$ and $N$ normal subgroups, and $L = M \cap N$,

$$\array{& M\cap N & {\to}^\lambda & M & \\
         \lambda^\prime & \downarrow &&\downarrow & \mu\\
          &N & {\to}_{\nu}& P & \\
}$$

with all maps the evident inclusions, all actions by conjugation, and $h : M \times N \to M\cap N$ given by $h(m,n) = [m,n]$.

### Simplicial group example

If we replace each group in the algebraic example by a simplicial group, we would have a simplicial crossed square, now apply the connected component functor to that and you get a crossed square, and in fact any crossed square can be constructed up to isomorphism in this way.

If we start with a simplicial group, $G$, using the [[decalage]] functor we can construct a simplicial group and two normal subgroups and thus get to the previous situation. The result can be interpreted in terms of the [[Moore complex]] as follows:
$$\array{& \frac{NG_2}{d_0(NG_3)}& {\to} & Ker d_1 & \\
          & \downarrow &&\downarrow & \\
          &Ker d_2 & {\to}& G_1 & \\
}$$

The two morphisms with codomain $G_1$ are inclusions, the other two are induced by $d_0$. The $h$-map can be explicitly given. It can be found in

T. Porter, _n-types of simplicial groups and crossed n-cubes_, Topology, 32, (1993), 5 – 24,

which also contains the discussion of the generalisation to [[crossed n-cube|crossed n-cubes]]

## Properties

### Relation to $cat^2$-groups

A crossed module $\mu: M \to P$ determines a $cat^1$-structure on the [[semidirect product]] group $M \rtimes P$. Thus to say that the above crossed square is a _crossed module of crossed modules_ suggests that we should ask for $L \rtimes N \to M \rtimes P$ to be a crossed module, so that there is an action which allows the _big group_ $G = (L \rtimes N) \rtimes (M \rtimes P)$ to be a $cat^1$-group. Then $G$ becomes a _$cat^2$-[[cat-2-group|group]]_. The $h$-map of the crossed square derives from a commutator in $G$.

This equivalence between crossed squares and $cat^2$-groups confirms the completeness of the axioms for crossed squares. Notice also that to prove a diagram of crossed squares is a [[colimit]] diagram, it looks as if you have to make appallingly detailed verifications of axioms. It is much easier to prove the corresponding diagram of $cat^2$-groups is a colimit!

This theme of using two equivalent categories, one for conjecture and proof, the other for calculation and application to traditional invariants, runs through the story of [[higher homotopy van Kampen theorems]].

### Relation fo 2-crossed modules

In his work on homotopy $n$-types, [[Loday]] gave a construction of what he called a 'mapping cone' for a [[crossed square]]. Conduch\'{e} later noticed that this naturally had the structure of a [[2-crossed module]],

## Related concepts

* [[group]]

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

* [[3-group]], [[2-crossed module]] / **crossed square**, [[differential 2-crossed module]]

* [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]

##References

* D. Guin-Walery and [[J.-L. Loday]], 1981, _Obstructions à l'excision en K-théorie algèbrique_, in Evanston Conference on Algebraic K-theory, 1980, volume 854 of Lecture Notes in Maths., 179 – 216, Springer.

*  [[G.J. Ellis]], Crossed squares and combinatorial homotopy, Math. Z., 461 (1993) 93–110,

*  [[T. Porter]], _n-types of simplicial groups and crossed n-cubes_, Topology, 32, (1993), 5 – 24,

* [[D. Conduché]], Simplicial Crossed Modules and Mapping Cones, Georgian Math. J., 10, (2003), 623–636.

[[!redirects crossed squares]]


crossed squares > history

< [[crossed squares]]
[[!redirects crossed squares -- history]]


crossing number

Let $K$ be a knot.

+--{: .un_defn}
######Definition
The _crossing number_, $c(K)$, of $K$ is the minimum number of crossings in a diagram in the isotopy class of $K$.
=--

The crossing number is thus the number of crossings in the simplest picture of a knot.  A diagram of a knot $K$ with exactly $c(K)$ crossings is called a _minimal_ diagram.

#####Examples

* $c(unknot) = 0$;

* $c(trefoil) = 3$;

* $c(figure-8) = 4$.

* if a diagram has $1$ or $2$ crossings it represents the unknot, so there are no knots with $c(K) = 1$ or $2$.

The crossing number is related to the [[unknotting number]], but in quite a subtle way.

In the books by Burde and Zeischang (1985) and Kauffman (1987), the tables of knots are arranged according to crossing number.  (Choices have been made of one [[mirror image]] or the other.)
Given some arbitrary diagram, the crossing number of the knot that it respresents may be hard to determine.


crossing symmetry

## Related concepts

* [[Feynman amplitude]]

* [[S-matrix]]

## References

* Vadim Kaplunovsky, _Corssing symmetry_ ([pdf](http://bolvan.ph.utexas.edu/~vadim/classes/11f/crossing.pdf))

See also

* Wikipedia, _Crossing (physics)_

[[!redirects crossing symmetries]]


cryptography

#Contents#
* table of contents
{:toc}

## Idea

In [[computer science]], _cryptography_ is the art and science of securing communication and information through the use of [[mathematics|mathematical techniques]].

Its roots trace back to ancient civilizations where secret codes were employed to protect sensitive messages. The earliest known use of cryptography is attributed to the ancient Egyptians, who used [hieroglyphs to inscribe messages](https://www.redhat.com/en/blog/brief-history-cryptography) on monuments in a way that could only be understood by those with the knowledge of the secret key.

Today, cryptography relies on a wide variety of mathematical topics. Here is a non-exhaustive list of fields and topics that are particularly relevant to cryptography:

- **Number Theory** (modular arithmetic, prime numbers, and group theory in the context of modular arithmetic)

- **Abstract Algebra** (groups, rings, and fields such as Galois fields)

- **Algebraic Geometry** (elliptic curve cryptography and algebraic varieties over finite fields)

- **Combinatorics** (combinatorial designs and error-correcting codes)

- **Probability Theory** (randomized algorithms in cryptography and probability distributions relevant to cryptographic protocols

- **Graph Theory** (graph-based cryptographic algorithms
 and network flow algorithms in the context of network security)

- **Topology** (algebraic topology and homological algebra)

- **Logic** (in the context of formal verification and proof systems)

- **Complex Analysis** (analytic number theory)

- **Functional Analysis** (for signal processing and cryptographic protocols)

## Symmetric  and asymmetric cryptography

There are two fundamental types of cryptography: _symmetric_ and _asymmetric_. Symmetric cryptography involves the use of a single secret key $k$ for both encryption $E_k:M \to C$ and decryption $D_k:C \to M$. This key must be securely shared between the communicating parties.

\[
\begin{array}{ccccc}
\mathsf{Bob} &\longrightarrow& \mathsf{Channel} & \longrightarrow &\mathsf{Alice}\\
m & \longrightarrow & E_k(m) & \longrightarrow & D_{k}(E_k(m)) = m
\end{array}
\]

On the other hand, asymmetric cryptography uses a pair of keys – a public key $k$ for encryption $E_k:M \to C$ and a private key $k'$ for decryption $D_{k'}:C \to M$. This asymmetric key pair allows for secure communication without the need for a shared secret, offering enhanced security in various scenarios.

\[
\begin{array}{ccccc}
\mathsf{Bob} &\longrightarrow& \mathsf{Channel} & \longrightarrow &\mathsf{Alice}\\
m & \longrightarrow & E_k(m) & \longrightarrow & D_{k'}(E_k(m)) = m
\end{array}
\]

In practice, Alice would generate both keys $k'$ and $k$ and would send the public key $k$ to Bob.

## Security and attacks

Security in cryptography is a constant battle against potential attacks. Common attacks include brute force, where an attacker systematically tries all possible keys, and cryptanalysis, which involves exploiting vulnerabilities in the cryptographic algorithm.

Modern cryptographic systems aim to withstand these attacks, often relying on the complexity of mathematical problems, such as factoring large numbers or solving discrete logarithms, to ensure security.

### Complexity classes

[[complexity class|Complexity classes]] play a pivotal role in cryptography, providing a framework to analyze the efficiency and security of cryptographic algorithms.

The study of computational complexity helps assess the computational resources required to break a cryptographic scheme, distinguishing between feasible and infeasible attacks.

### Invertibility principle

In cryptography, the concept of invertibility is essential for ensuring secure decryption. When plaintext undergoes encryption through the encryption algorithm $E_k:M \to C$, the efficacy of the system depends on the ease of decryption for authorized users and the complexity it imposes on unauthorized attempts.

[[bijection|Bijections]] (or [[inverse|isomorphisms]]) $E:M \to C$ whose inverses $D = E^{-1}:C \to M$ are hard to compute provide practical examples of this principle. Such bijections $(E,D)$ establish a one-to-one relationship between plaintext $M$ and ciphertext $C$, enabling straightforward decryption given suitable information (constituting the potential private key $k$) while posing a significant challenge for unauthorized attempts.

Mathematically, we could imagine that Alice and Bob have access to a groupoid $\mathcal{G}$ and a category $\mathcal{C}$, respectively, such that $\mathcal{C}$ is embedded in $\mathcal{G}$ via an inclusion functor.
\[
\mathcal{C} \hookrightarrow \mathcal{G}
\]
The security of a cryptosystem would then rely on the difficulty of reconstructing this inclusion using an algorithmic completion procedure.

These mathematical principles form the foundation for secure cryptosystems, ensuring that only authorized users can successfully decrypt and access sensitive information.

\begin{remark}
Note that there is flexibility in considering weaker notions of isomorphisms, especially if Alice can employ these weaker isomorphisms effectively in the decryption of the message $m$ from the data $D_{k'}(E_k(m))$.
\[
D_{k'}(E_k(m)) \leftrightarrows m
\]
In category theory, examples of these weaker notions can be derived from higher category theory. This includes concepts such as [[adjoint equivalence|adjoint equivalences]], [[algebras for an endofunctor|algebras]], [[coalgebras|coalgebras]], and [[retractions|retractions]].
\end{remark}

## Post-quantum cryptography

With the advent of [[quantum computation|quantum computers]], there is growing concern about the potential threat they pose to existing cryptographic systems. Post-[[quantum cryptography]] is an evolving field that explores algorithms resistant to quantum attacks.

Researchers are developing cryptographic schemes that rely on mathematical problems difficult for both classical and quantum computers to solve, ensuring the continued security of communication in the post-quantum era.

### Useful links

* [NTRU](https://en.wikipedia.org/wiki/NTRU)

* [LWE](https://en.wikipedia.org/wiki/Learning_with_errors) and [RLWE](https://en.wikipedia.org/wiki/Ring_learning_with_errors)

* [[braid group cryptography -- references|Braid group cryptography]]

## Homomorphic cryptography

[[homomorphism|Homomorphic]] cryptography takes a unique approach by allowing the processing of operations $\square$ on encrypted data without decrypting it first.

\[
D(E(m_1) \square E(m_2)) = m_1 \square m_2
\]

This field aims to preserve privacy while still enabling meaningful computations on sensitive information.

Fully homomorphic cryptography, [a subset of homomorphic cryptography](https://en.wikipedia.org/wiki/Homomorphic_encryption), extends this concept to support arbitrary computations on encrypted data, opening new possibilities for secure and privacy-preserving data processing in various applications, including cloud computing and data analytics.

### Related concepts

* [[group|Group]]-[[homomorphism|homomorphisms]]

* [[ring|Ring]]-[[homomorphism|homomorphisms]]

* [[functor|Functors]]

* [[monoidal functor|Monoidal functors]]

## Related entries

* [[arithmetic cryptography]]

* [[digital identity]],

* [[information theory]],

* [[quantum cryptography]], [[quantum information theory]]

* [[quantum computing]],

* [[blockchain]]

* zoranskoda: [[zoranskoda:computer security]], [[zoranskoda:blockchain]], [[zoranskoda:blockchain security]], [[zoranskoda:digital identity]]

## References

### General

* Wikipedia, _[Outline of cryptography](https://en.wikipedia.org/wiki/Outline_of_cryptography)_, [public-key cryptography](https://en.wikipedia.org/wiki/Public-key_cryptography), [cryptographic protocol](https://en.wikipedia.org/wiki/Cryptographic_protocol), [authentication](https://en.wikipedia.org/wiki/Authentication), [zero-knowledge proof](https://en.wikipedia.org/wiki/Zero-knowledge_proof), [non-interactive zero-knowledge proof](https://en.wikipedia.org/wiki/Non-interactive_zero-knowledge_proof), [OAuth](https://en.wikipedia.org/wiki/OAuth), [homomorphic encryption](https://en.wikipedia.org/wiki/Homomorphic_encryption)

* post-quantum cryptography at [nist.gov](https://csrc.nist.gov/Projects/Post-Quantum-Cryptography)

[[probability theory|probabilistic]]:

* Shafi Goldwasser, Silvio Micali, _Probabilistic encryption_, Journal of Computer and System Sciences 28:2 (1984) 270-299

### Verified software
 {#VerifiedSoftwareReferences}

On [[verified software]] for cryptography:

* [[Mihhail Aizatulin]], Franccois Dupressoir, Andrew D. Gordon, Jan Jürjens, *Verifying Cryptographic Code in C: Some Experience and the Csec Challenge* ([pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/MSR-TR-2011-118.pdf))

* [[Mihhail Aizatulin]], Andy Gordon, Jan Jürjens, *Extracting and Verifying Cryptographic Models from C Protocol Code by Symbolic Execution*, CCS '11 Proceedings of the 18th ACM Conference on Computer and Communications Security 2011 ([pdf](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/11/extracting.pdf))

* Matthias Berg, *Formal Verification of Cryptographic Security Proofs*, 20013 ([pdf](https://publikationen.sulb.uni-saarland.de/bitstream/20.500.11880/26584/1/thesis_berg.pdf), [doi:10.22028/D291-26528](http://dx.doi.org/10.22028/D291-26528))

* [[Adam Petcher]], *A Foundational Proof Framework for Cryptography*, 2014 ([pdf](https://dash.harvard.edu/bitstream/handle/1/17463136/PETCHER-DISSERTATION-2015.pdf), [harvard:17463136](https://dash.harvard.edu/handle/1/17463136))

* [[Adam Petcher]], Greg Morrisett, *The Foundational Cryptography Framework*,  In: R. Focardi, A. Myers (eds.) *Principles of Security and Trust* POST 2015. Lecture Notes in Computer Science, vol 9036. Springer, Berlin, Heidelberg ([doi:10.1007/978-3-662-46666-7_4](https://doi.org/10.1007/978-3-662-46666-7_4))

* Andrew W. Appel, *Verification of a Cryptographic Primitive: SHA-256*, ACM Transactions on Programming Languages and SystemsApril 2015 Article No.: 7 ([doi:10.1145/2701415](https://doi.org/10.1145/2701415), [pdf](https://www.cs.princeton.edu/~appel/papers/verif-sha.pdf))

* Andres Erbsen, Jade Philipoom, [[Jason Gross]], Robert Sloan, [[Adam Chlipala]], *Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises*, [2019 IEEE Symposium on Security and Privacy (SP)](https://ieeexplore.ieee.org/xpl/conhome/8826229/proceeding) $[$[doi:10.1109/SP.2019.00005](https://doi.org/10.1109/SP.2019.00005)$]$

* [[Mihhail Aizatulin]], *Verifying Cryptographic Security Implementations in C Using Automated Model Extraction* ([arXiv:2001.00806](https://arxiv.org/abs/2001.00806))

On [[type theory]] for [[verified programming|verified]] cryptography:

* Cédric Fournet, Karthikeyan Bhargavan, Andrew D. Gordon, *Cryptographic Verification by Typing for a Sample Protocol Implementation*, In: Aldini A., Gorrieri R. (eds) Foundations of Security Analysis and Design VI. FOSAD 2011. Lecture Notes in Computer Science, vol 6858. Springer (2011) ([doi:10.1007/978-3-642-23082-0_3]( https://doi.org/10.1007/978-3-642-23082-0_3))

* Cédric Fournet, Markulf Kohlweiss, [[Pierre-Yves Strub]], *Modular code-based cryptographic verification*, CCS '11: Proceedings of the 18th ACM conference on Computer and communications securityOctober 2011 Pages 341–350 ([doi:10.1145/2046707.2046746](https://doi.org/10.1145/2046707.2046746))

On [[homotopy type theory]] for [[verified programming|verified]] cryptography:

* [[Paventhan Vivekanandan]], *A Homotopical Approach to Cryptography*, talk at [FCS 2018](https://www.andrew.cmu.edu/user/liminjia/events/fcs2018/papers/s33.pdf) ([pdf](https://www.andrew.cmu.edu/user/liminjia/events/fcs2018/papers/s33.pdf), [easychair:GLtQ#](https://easychair.org/smart-slide/slide/GLtQ#)), In: Charles Morisset and Limin Jia (eds.) FCS Informal Proceedings  **[55](http://t-news.cn/Floc2018/FLoC2018-pages/volume55.html)** (2018)

* [[Paventhan Vivekanandan]], *HoTT-Crypt: A Study in Homotopy Type Theory based on Cryptography*, Kalpa Publications in Computing Volume 9, 2018, Pages 75-90 ([doi:10.29007/tvpp](https://doi.org/10.29007/tvpp), [web slides](https://easychair.org/smart-slide/slide/WSST#), [[VivekanandanHoTTCryptography.pdf:file]])

category: applications, computer science

[[!redirects encryption]]

[[!redirects braid group cryptography]]


crystal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

> This entry is about the notion of "crystal" in [[solid state physics]]. For the notion in [[algebraic geometry]] see at *[[crystal (algebraic geometry)]]*.

***

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]] a *crystal* is a [[bound state]] of [[atomic nuclei]] and [[electrons]] (much as [[atoms]] and [[molecules]] are, too, but) in which the nuclei occupy a discrete "regular lattice" which is [[invariant]] -- as a [[discrete space|discrete]] [[subset]] of [[Euclidean space]] and as far as the crystal extends -- under a [[discrete group|discrete]] [[subgroup]] of the [[Euclidean group]] of [[translation group|translations]], [[reflections]] and [[rotation group|rotations]].

These [[subgroups]] are also known as *[[crystallographic groups]]* and are used to classify crytalline structures.

In addition to these *spatial symmetries*, the [[dynamics]] of the [[electrons]] in a crystalline material may also have [[internal symmetries]] and/or [[CPT theorem|CPT-symmetries]] (like [[time-reversal symmetry]]) which together with the crystallographic symmetry affect the physical properties of the crystalline material, for instance whether it behaves like a [[conductor]]/[[semi-conductor]]/[[insulator]] with respect to [[electric currents]] or whether it admits [[topological phases of matter]] which are "[[symmetry protected topological phase|protected]]" by these symmetries.

## Related concepts

* [[crystallographic group]]

* [[Bravais lattice]]

* [[Brillouin torus]]

* [[crystalline topological insulator]]

* [[photonic crystal]]

## References

Historical articles identifying the point-[[lattice (discrete subgroup)|lattice]] nature of crystals:

* {#Bravais1850}  [[Auguste Bravais]], *Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L'espace*, J. Ecole Polytech. 19 (1850) 1 $[$[ark:12148/bpt6k96124j](https://gallica.bnf.fr/ark:/12148/bpt6k96124j)$]$

* {#Barlow1883} [[William Barlow]], *Probable Nature of the Internal Symmetry of Crystals*, Nature **29** (1883) 186–188 $[$[doi:10.1038/029186a0](https://doi.org/10.1038/029186a0)$]$

Textbook accounts:

* {#Tilley2020} Richard Tilley, *Crystals and Crystal Structure*, Wiley (2020) [[ISBN:978-1-119-54838-6](https://www.wiley.com/en-us/Crystals+and+Crystal+Structures,+2nd+Edition-p-9781119548386)]

* [[Sheng San Li]], Chapter 1 of: *Semiconductor Physical Electronics*, Springer (2006)   $[$[doi:10.1007/0-387-37766-2](https://link.springer.com/book/10.1007/0-387-37766-2)$]$

  > (in [[semiconductor]]-theory)

With focus on the [[mathematics]] of [[crystallographic groups]]:

* [[Harold Hilton]], *Mathematical crystallography and the theory of groups of movements*, Oxford Clarendon Press (1903) $[$[web](https://archive.org/details/mathematicalcry03hiltgoog/page/n6/mode/2up)$]$

* {#Engel86} [[Peter Engel]], *Geometric Crystallography -- An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$[doi:10.1007/978-94-009-4760-3](https://doi.org/10.1007/978-94-009-4760-3)$]$

See also:

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

[[!redirects crystals]]

[[!redirects crystallography]]


crystal (algebraic geometry)

> This entry is about the notion of "crystal" in [[algebraic geometry]]. For the notion in [[solid state physics]] see at *[[crystal]]*.

***

# Contents
* table of contents
{:toc}

## Disambiguation

There are few mutually unrelated notions denoted by "crystal" in mathematics.

One can of course talk about mathematical models of physical crystals and their geometry.

 Another, is an intermediary notion leading to [[crystal basis|crystal bases]] of [[Kashiwara]] and of Lusztig, thus one associates a crystal to a [[quantized enveloping algebra]]. Finally, there are **crystals due Grothendieck** to which this entry is dedicated.

## Overview

[[Alexander Grothendieck|Grothendieck]]'s differential calculus is based on the [[infinitesimal object|infinitesimal]] thickenings of a diagonal of a space. If one takes a completion, then there is a filtration on infinitesimals there. On the other hand, this theory also provides a concept of a [[regular differential operator]] which is also filtered (by the degree). There is a fundamental duality between the infinitesimals and regular differential operators which is compatible with the two filtrations, in fact the duality is a perfect pairing on each filtered level. This pairing gives for example that a [[D-module]] corresponds to a flat [[connection on a bundle|connection]] on a usual [[quasicoherent sheaf]].

Infinitesimal version of flat connection in algebraic geometry is a [[Grothendieck connection]]. Flat connections can infinitesimally also be described as the [[descent]] data on [[de Rham space]]s called the (co)stratification. There is a [[site]] (the [[crystalline site]]) which formalizes these descent data. It can be explained in many ways, including intuitively in the sense of infinitesimal elements in a [[scheme]].

The **crystals of quasicoherent sheaves** are the [[quasicoherent sheaf|quasicoherent sheaves]] of modules over the crystalline site and are in correspondence with usual quasicoherent sheaves over the underlying scheme with flat connection.

But Grothendieck considered not only descent for quasicoherent sheaves but also for [[affine scheme]]s. This nonlinear version is harder and unlike descent for quasicoherent sheaves, it does not have a noncommutative generalization. Cf. [[p-connection]].

>([[Zoran Skoda|Zoran]]: we should find an exact reference from EGA or so for the descent for affine schemes).

Moreover this has also a crystalline version: **crystals of affine schemes**. This corresponds to a nonlinear version of [[D-modules]], called **[[D-schemes]]** (also called [[diffieties]] by Vinogradov). As D-modules correspond to solutions of systems of linear [[differential equations]], D-schemes correspond to systems of nonlinear ones. One has also analytic version (analytic D-spaces). One can do more general crystals, e.g. of affine schemes.

## Related entries

* [[crystalline cohomology]]

* [[Grothendieck connection]], [[D-module]], [[p-connection]]

* [[quasicrystal]]

## References

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

* [[Dennis Gaitsgory]], Nick Rozenblyum, _Crystals and D-modules_, [arxiv/1111.2087](http://arxiv.org/abs/1111.2087)
* A. [[Beilinson]], V. Drinfel'd, _Chiral algebras_ contains a section on D-schemes.

* A. [[Grothendieck]], _Crystals and the de Rham cohomology of schemes_, in "Dix exposes sur la cohomologie des schemas"

* [[Clark Barwick]], _$\mathcal{D}$-crystals_, notes from 2006 talk, [pdf](https://www.maths.ed.ac.uk/~cbarwick/papers/D-crys.pdf)

* mathoverflow: [The Infinitesimal topos in positive characteristic](http://mathoverflow.net/questions/15795/the-infinitesimal-topos-in-positive-characteristic)


crystal basis

Crystal bases are a construction in the [[representation theory]] of [[quantum groups]] (which in a specialization exist hence for usual [[Lie groups]] as well): roughly speaking they provide a uniform description not only of irreducible finite-dimensional modules but also a uniform choice of bases in all of them as well as in tensor products.

There are two mutually dual versions due [[Masaki Kashiwara]] and [[George Lusztig]].

# History and references

The terminology is due to the involved $q=0$ limit of quantum groups used in the construction (the classical case is $q=1$). In a thermodynamic parlance zero temperature would involve passing to crystalization.

* [[Masaki Kashiwara]], _Crystalizing the q-analogue of universal enveloping algebras_, Commun. Math. Phys. __133__ (1990) 249-260, [proj euclic](http://projecteuclid.org/euclid.cmp/1104201397), [pdf](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104201397)

* M. Kashiwara, _On crystal bases_, Representations of Groups, Proc. of the 1994 Annual Seminar of the Canadian Math. Soc. Ban 16 (1995) 155--197, Amer. Math. Soc., Providence, RI. ([pdf](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/on-crystal.pdf), [ps.gz](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/on-crystal.ps.gz))

* Masaki Kashiwara, _Global crystal bases of quantum groups_,  Duke Math. J.  69  (1993),  no. 2, 455--485, [link](http://projecteuclid.org/euclid.dmj/1077293577).
*  M. Kashiwara, _The crystal base and Littelmann's refined Demazure character formula_, Duke Math. J. 1993.
* Masaki Kashiwara, Yoshihisa Saito, _Geometric construction of crystal bases_, Duke Math. J. 1996, [pdf cached](http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=47F46C7CD4CE2825986E88F8A4C6BCE9?doi=10.1.1.11.9068&rep=rep1&type=pdf)
* S-J.Kang, M. Kashiwara, K.Misra, _Crystal bases of Verma modules for quantum affine Lie algebras_, Compositio Math. __92__ (1994) 299--325, [numdam](http://www.numdam.org/item?id=CM_1994__92_3_299_0)
* Hong and Kang, _Introduction to quantum groups and crystal bases_, Grad. Studies in Math. __42__, AMS 2002, 307 pp.
* John R. Stembridge, _A local characterization of simply-laced crystals_, Trans. Amer. Math. Soc. __355__ (2003), 4807--4823.
[[!redirects crystal basis]]


crystalline cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

**Crystalline cohomology** is the [[abelian sheaf cohomology]] with respect to the _[[crystalline site]]_ of a [[scheme]]. Hence, put more generally, it is the [[cohomology]] of [[de Rham spaces]]/[[coreduced objects]].

Crystalline cohomology serves to refine the notion of [[de Rham cohomology]] for [[schemes]].

Crystalline cohomology is in particular a [[Weil cohomology]] and is generalized by the notion of [[rigid cohomology]].

## Definition

Let $X$ be a [[scheme]] over a base $S$. The **[[crystalline site]]** $Cris(X/S)$ of $X$ is

* the [[category]] whose objects are all nilpotent $S$-immersions $U \hookrightarrow T$, where $U$ is an open set of $X$ and and the ideal on $T$ defining this immersion being endowed with a nilpotent [[divided power structure]] (...details...).;

* the [[Grothendieck topology]] on this category is the Zariski topology.

If $S$ is of characteristic 0, then $Cris(X/S)$ coincides with the **infinitesimal site** of $X$. (...details...).

## Properties

### Relation to de Rham space

Crystalline cohomology of $X$ is the [[cohomology]] of the [[de Rham space]]
of $X$. See there for more.

### Relation to de Rham cohomology

* [[comparison theorem (crystalline cohomology)]]

### Relation to differential homotopy type theory

In [[differential homotopy type theory]] the [[infinitesimal flat modality]]
sends [[coefficients]] to coefficients for crystalline cohomology.

## Related concepts

* [[p-adic cohomology]]

* [[de Rham cohomology]]

* [[Weil cohomology]]

  * [[étale cohomology]]

## References

Related entries: [[crystal]], [[infinitesimal site]], [[rigid cohomology]], [[Dieudonné module]], [[Monsky-Washnitzer cohomology]], [[Grothendieck connection]]

An original account of the definition of the crystalline topos is [section 7, page 299](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf#page=299) of

* [[Alexander Grothendieck]], _Crystals and de Rham cohomology of schemes_ , chapter IX in _Dix Exposes sur la cohomologie des schemas ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf))

A more recent account is

* [[Luc Illusie]], _Crystalline cohomology_, in _Motives_, Proc. Sympos. Pure Math., vol. 55, part 1, Amer. Math. Soc. Providence, RI, 1994, 43–70.

Discussion of this in the modern context of [[higher geometry]]/[[D-geometry]] is in

* {#LurieCrystals} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

A [[p-adic cohomology]] for [[varieties]] in [[characteristic]] $p$ it was it was discussed in

* [[Pierre Berthelot]], _Cohomologie cristalline des schémas de caractéristique $p \gt 0$, Lecture Notes in Mathematics, Vol. 407, Springer- Verlag, Berlin, 1974. ([doi:10.1007/BFb0068636](https://link.springer.com/book/10.1007/BFb0068636),  MR 0384804)

* [[Pierre Berthelot]], [[Arthur Ogus]], _$F$-Isocrystals and de Rham Cohomology_, I, Invent. math. 72, 1983, pp. 159-199 ([eudml:143016](https://eudml.org/doc/143016))

Review:

* [[Pierre Berthelot]], [[Arthur Ogus]], _Notes on crystalline cohomology_, Princeton Univ. Press 1978. vi+243, ([ISBN:0-691-08218-9](https://press.princeton.edu/books/hardcover/9780691648323/notes-on-crystalline-cohomology-mn-21))

Discussion of this in terms of [[Cech cohomology]] is in

* [[Bhargav Bhatt]], [[Aise Johan de Jong]], _Crystalline cohomology and de Rham cohomology_ ([pdf](http://www.math.columbia.edu/~dejong/papers/crystalline-comparison.pdf))

See also

* wikipedia [crystalline cohomology](http://en.wikipedia.org/wiki/Crystalline_cohomology)


crystalline differential operator

## Definition

For $X$ a [[smooth scheme]] over a [[field]] the [[sheaf]] $\mathcal{D} = \mathcal{D}_X$ of _crystalline differential operators_ is the the "enveloping algebroid of the [[tangent Lie algebroid]]" of $X$: to an affine $U \to X$ it assigns the algebra that is generated over $\mathcal{O}(U)$ from the $\mathcal{O}(U)$-[[module]] $Vect(U)$ of [[vector field]]s ([[derivation]]s of $\mathcal{O}(U)$), subject to the relations

$$
  v_1 \cdot v_2 - v_2 \cdot v_1 = [v_1, v_2]
$$

and

$$
  v_1 \cdot f - f \cdot v_1 = v_1(f)
$$

for all $v_1, v_2 \in Vect(U)$ and $f \in \mathcal{O}(U)$.

If the field $k$ has [[characteristic]] 0 this is the ordinary sheaf of differential operators

## Related concepts

* [[crystalline site]], [[crystalline cohomology]]

* [[D-module]]

## References

for instance section 3.1 of

* Roman Bezrukavnikov, _Noncommutative Counterparts of the Springer
Resolution_ ([pdf](http://www.cirm.univ-mrs.fr/videos/2006/exposes/17w2/Bezrukavnikov.pdf))


crystalline site

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

> under construction

#Contents#
* table of contents
{:toc}

## Definition

Let $C_{et}$ be the [[etale site]] of complex [[scheme]]s of finite type. For $X$ a [[scheme]], its **infinitesimal site** $Cris(X)$ is the [[big site]] $C_{et}/X_{dR}$ of the [[de Rham space]] $X_{dR} : C_{et} \to Set$:

the site whose objects are pairs $(Spec A, (Spec A)_{red} \to X)$ of an affine $Spec A$ and a morphism from its reduced part ($(Spec A)_{red} = Spec (A/I)$ for $I$ the [[nilradical]] of $A$) into $X$.

More generally, for positive [[characteristic]], the definition is more involved than that.

## Properties

The [[abelian sheaf cohomology]] over $Cris(X)$ is the [[crystalline cohomology]] of $X$.

## Logical characterization

Let $k$ be a ring. Let $k \to R$ be a finitely presented $k$-algebra. Then the big infinitesimal topos of the $Spec(k)$-scheme $Spec(R)$ [[classifying topos|classifies]] the theory of commutative squares of ring homomorphisms
$$
  \array{A & \rightarrow & B \\
         \uparrow &&\uparrow \\
         k & \rightarrow& R
}$$
where the rings $A$ and $B$ are local, the top arrow $A \to B$ is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent) This result is due to ([Hutzler 2018](#Hutzler18)). By the conditions on the top morphism, it is enough to require that $A$ or $B$ is local.

Routine arguments, to be made explicit in a further revision of this entry, allow to generalize this description to the non-affine case. Let $f \colon X \to S$ be a scheme over $S$. Assume that $X$ is locally of finite presentation over $S$. Then the big infinitesimal topos of the $S$-scheme $X$ classifies, as a $Sh(X)$-topos, the $Sh(X)$-theory of commutative squares of ring homomorphisms
$$
  \array{A & \rightarrow & B \\
         \uparrow &&\uparrow \\
         f^{-1}\mathcal{O}_S & \rightarrow& \mathcal{O}_X
}$$
where the rings $A$ and $B$ are local, the top arrow $A \to B$ is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent), and both vertical arrows are local (i.e. reflect invertibility).

## Related concepts

* [[crystalline cohomology]]

* [[crystalline differential operator]]

## References

An original account of the definition of the crystalline topos is [section 7, page 299](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf#page=299) of

* [[Alexander Grothendieck]], _Crystals and de Rham cohomology of schemes_ , chapter IX in _Dix Exposes sur la cohomologie des schema_ ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf))

A review of some aspects:

* {#LurieCrystal} [[Jacob Lurie]], *Notes on crystals and algebraic $\mathcal{D}$-modules* (2010) [[[Lurie-NotesOnCrystals.pdf:file]]]

and on page 7 of

* [[Carlos Simpson]], [[Constantin Teleman]], _deRham theorem for $\infty$-stacks_ ([pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf))

In the article

* Arthur Ogus, _Cohomology of the infinitesimal site_ Annales scientifiques de l'École Normale Supérieure, Sér. 4, 8 no. 3 (1975), p. 295-318 ([numdam](http://www.numdam.org/item?id=ASENS_1975_4_8_3_295_0))

it is shown that if $X$ is proper over an algebraically closed field $k$ of [[characteristic]] $p$, and embeds into a smooth scheme over $k$, then the infinitesimal cohomology of $X$ coincides with [[etale cohomology]] with coefficients in $k$  (or more generally $W_n(k)$ if we work with the infinitesimal site of $X$ over $W_n(k)$).

The result about the geometric theory classified by the big infinitesimal topos appears in

* {#Hutzler18}[[Matthias Hutzler]], _Internal language and classified theories of
toposes in algebraic geometry_, Master's thesis at the University of Augsburg, 2018, [GitLab](https://gitlab.com/MatthiasHu/master-thesis/blob/master/thesis.pdf), [pdf download](https://gitlab.com/MatthiasHu/master-thesis/raw/master/thesis.pdf?inline=false)

* [[Matthias Hutzler]], _Syntactic presentations for glued toposes and for crystalline toposes_, Phd. diss. Universität Augsburg 2021. ([arXiv:2206.11244](https://arxiv.org/abs/2206.11244))

[[!redirects infinitesimal site]]
[[!redirects crystaline site]]


crystallographic group

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Crystallographic groups (also: "space groups") are [[symmetry groups]] of [[crystals]].

The use of crystallographic groups for the study of [[crystals]] (e.g. [Hilton 1903](#Hilton03), [Engel 1986](crystal#Engel86)) is much in the spirit of [[Klein geometry]]/the [[Erlanger program]] (see for instance [Weyl 1938](Erlanger+program#Weyl38); [Grünbaum & Shephard 2010](Erlanger+program#GruenbaumShephard10)).

Concretely, the [[quotient space]]/[[quotient orbifold]] of the space of [[wave vectors]]/[[momenta]] in a [[crystal]] lattice by the (dual) [[crystallographic group]] is the [[Brillouin torus]](-[[orbifold]]), in terms of which most of [[condensed matter theory]] is formulated (see for instance the [[electron energy bands]], the [[valence bundle]], and the [[K-theory classification of topological phases]]).

## Definition

A **crystallographic group** or **space group** in [[dimension]] $n$  is a [[subgroup]] of the corresponding [[Euclidean group]], hence of the [[isometry group]] of [[Euclidean space]] $\mathbb{R}^n$, that contains a (maximal) [[lattice in a vector space|lattice]] in $\mathbb{R}^n$ as a [[subgroup]], and is contained within the [[automorphism group]] of that lattice. In other words, it is a subgroup of the automorphism group of the lattice that contains all the [[translations]] by elements of the lattice itself.

Equivalently, a crystallographic group on a [[Euclidean space]] $E$ is a [[discrete group|discrete]] [[subgroup]] $S \subset Iso(E)$ of the [[isometry group]] of $E$ (its [[Euclidean group]]) that contains a [[lattice (discrete subgroup)|lattice]] $N \subset E \subset Iso(E)$ of [[translation group|translations]] as a [[normal subgroup]] $N \subset S$, such that the corresponding [[quotient group]], called the _[[point group]]_ of the crystallographic group, is a subgroup $G \coloneqq S/N \;\subset\; O(E)$ of the [[orthogonal group]].

In short, a crystallographic groups is exhibited by an inclusion of [[short exact sequences]] of ([[nonabelian group|non-abelian]]) [[groups]], as follows:

\[
  \label{CrystallographicInclusionOfShortExactSequences}
  \array{
    &
    1 && 1
    \\
    & \downarrow && \downarrow
    \\
    {\text{normal subgroup}
    \atop
    \text{lattice of translations}}
    &
    N &\subset& E
    &
    {\text{translation} \atop \text{group}}
    \\
    &
    \big\downarrow && \big\downarrow
    \\
    {\text{crystallographic} \atop \text{group}}
    &
    S &\subset& Iso(E)
    &
    {\text{Euclidean}
    \atop
    \text{isometry group}}
    \\
    &
    \big\downarrow && \big\downarrow
    \\
    {\text{point} \atop \text{group}}
    &
    G &\subset& O(E)
    &
    {\text{orthogonal} \atop \text{group}}
    \\
    &
    \downarrow && \downarrow
    \\
    &
    1 && 1
  }
\]

{#BieberbachTheorem} This transparent description of crystallographic groups is essentially the content of *[[Bieberbach's first theorem]]* (following [Bieberbach 1910](#Bieberbach1910), see [Farkas 1981, Thm. 14](#Farkas81), [Charlap 1986, Thm. I 3.1](#Charlap86), [Engel 1986, Thm. 3.1](#Engel86) for modern accounts and find a concise statement in [Tolcachier 19, Thm. 2.3](#Tolcachier19), also implicitly in [Freed-Moore 13, (0.2)](#FreedMoore13)). Historically, the earlier definition of "crystallographic group" (eg. [Farkas 81, Sec. 3](#Farkas81)) just required that it be any [[discrete group|discrete]] [[subgroup]] of $Iso(E)$ such that the resulting [[quotient space|quotient]] [[topological group]] be [[compact topological group|compact]]. Bieberbach's first theorem (or the statement now known under this name) says that this already implies 1. that the translation subgroup is a full lattice, and 2. that the point group is finite.

If the crystallographic [[short exact sequence]] on the left of (eq:CrystallographicInclusionOfShortExactSequences) [[split exact sequence|splits]], hence if the space group $S \simeq G \ltimes N$ is the [[semidirect product]] of the [[point group]] with the translational lattice, then $S$ is called a **symmorphic space group**.

The [[conjugacy class]] of a crystallographic group (under [[conjugation]] by [[O(n)]]) represents the *[[Bravais lattice]]* of the corresponding crystal lattice.

## Properties

### Classification
 {#Classification}

In 2 [[dimensions]], there are precisely 17 crystallographic groups, which are distinct up to [[isomorphism]]; these are known as the _[[wallpaper groups]]_.

In 3 dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. The classification of space groups has been carried out up to 6 dimensions.

On the classification of symmorphic space groups see also [this MO comment](https://mathoverflow.net/q/77682/381).

From [Chuprunov-Kuntsevich 88](#ChuprunovKuntsevich88):

> Let us make a brief survey of the main achievements in the $n$-dimensional crystallography that have been amply covered in the literature on the subject [15-17]. When Fedorov and Schoenflies had completed the derivation of 230 space group types of crystals it was natural to consider a possibility of derivation of corresponding groups in higher dimensions. In 1911-12 Bieberbach and Frobenius developed a general theory of the group symmetry of the n-dimensional lattices and proved the existence of a finite number of nonisomorphous space groups in the n-dimensional Euclidean space with an arbitrary number of n. Basing on this general theory, in 1948 Zassenhaus suggested an algorithm to derive the n-dimensional space groups as extensions of the translation subgroups of these groups using point groups. About 1950 Hermann gave a complete description of the possible crystallographic symmetry operations in higher dimensions and discussed the lattices of maximal symmetry and their crystal classes. In 1951 Hurley found 222 geometric crystal classes in the four-dimensional Euclidean space making use of the 1889 work by Goursat who had enumerated the classes of finite groups of the real 4 × 4 matrixes. Later this number was corrected to 227. At present classification of crystallographic groups in the four-dimensional Euclidean space is completed in the main. A complete list of 4783 types of four-dimensional space groups was computed in 1973 and given in an excellent monograph ["Crystallographic groups of four- dimensional space" by Brown et al.](#BrownBulowNeubuserWondratschekZassenhaus78) [15]. These groups were derived on the base of the nine maximal arithmetic crystal classes, derived by Dade in 1965, which allowed one to determine all of the 710 four-dimensional arithmetic classes and to calculate the normalizers of finite groups of the unimodular 4 x 4 matrixes needed for the Zassenhaus algorithm. The monograph [15] is of interest not only by having a complete description of all classes of the four-dimensional crystallographic groups but also by taking a deeper approach to the system of classification of the n-dimensional crystallographic groups, as well as by giving characteristic properties of the four-dimensional crystallographic groups in comparison with that in lower dimensions. One of these properties is enantiomorphizm exhibited not only by the space group types but also by Bravais types of lattices, arithmetic classes and geometric classes. For the first time this phenomenon was found by Shtogrin [18].

> The n-dimensional mathematical crystallography is still in progess. Ryshkov [19] determined all maximal arithmetic crystal classes of five-dimensional Euclidean space. Some categories of five- and six-dimensional "small" groups isomorphic to the three-dimensional groups of symmetry, anti- symmetry, two-fold antisymmetry, p- and p'-symmetry were derived by Palistrant [20]. Some aspects of the mathematical theory applied to the n-dimensional crystallography were considered [15, 21].

### Compact flat orbifolds
 {#CompactFlatOrbifolds}

+-- {: .num_prop #InducedPointGroupActionOnTorus}
###### Proposition
**(induced point group action on torus)**

The assumption that the crystallographic translation group $N \subset S$ is a [[normal subgroup]]

$$
  1 \to N \longrightarrow S \longrightarrow G \to 1
$$

implies that the [[action]] of the [[point group]] $G = S/N$ descends to the [[torus]] [[quotient space]] $E/N$

$$
  \array{
    E
      &\overset{g}{\longrightarrow}&
    E
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    E/N
      &\underset{g}{\longrightarrow}&
    E/N
  }
$$

=--

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

By the definition of [[quotient space]], the condition for this to be the case is that for all $x \in E$ we have $g(n(x)) = n'(g(x))$, or equivalently $g(n(g^{-1}(y))) = n'(y)$, which is implied by $N$ being a [[normal subgroup]]: $g N g^{-1} = N$.

=--

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

The further [[homotopy quotient]] $(E/N)\sslash G$ of the [[torus]] $E/N$ by this induced [[action]] of the [[point group]] $G$ is a [[compact topological space|compact]] [[flat orbifold]], and most compact flat orbifolds arise this way.

=--

## Related concepts

* [[symmetry group]]

* [[Brillouin torus]]

* [[Bravais lattice]]

* [[topological crystalline insulator]]

## References

Review:

* {#Hilton03} [[Harold Hilton]], *Mathematical crystallography and the theory of groups of movements*, Oxford Clarendon Press (1903) $[$[web](https://archive.org/details/mathematicalcry03hiltgoog/page/n6/mode/2up)$]$

* {#Engel86} [[Peter Engel]], *Geometric Crystallography -- An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$[doi:10.1007/978-94-009-4760-3](https://doi.org/10.1007/978-94-009-4760-3)$]$

* [[Willard Miller]], Chapter 2 "The Crystallographic Groups" in : *Symmetry Groups and Their Applications*, Pure and Applied Mathematics **50** (1972) 16-60 $[$doi:10.1016/S0079-8169(08)60959-9$]$

* {#BrownBulowNeubuserWondratschekZassenhaus78} H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, _Crystallographic Groups of Four-Dimensional Space_, John Wiley, New York, 1978.

* {#Farkas81} Daniel R. Farkas, _Crystallographic groups and their mathematics_, Rocky Mountain J. Math. Volume 11, Number 4 (1981), 511-552 ([doi:10.1216/RMJ-1981-11-4-511](https://projecteuclid.org/euclid.rmjm/1250128489))

* {#ChuprunovKuntsevich88} E. V. Chuprunov, T. S. Kuntsevich, _$n$-Dimensional space groups and regular point systems_, Comput. Math. Applic. Vol. 16, No. 5-8, pp. 537-543, 1988 (doi:10.1016/0898-1221(88)90243-X)

* D. Weigel, T. Phan and R. Veysseyre, _Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space_, Acta Cryst. (1987). A43, 294-304 ([doi:10.1107/S0108767387099367](https://doi.org/10.1107/S0108767387099367))

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], _Twisted equivariant matter_, Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

* {#Tolcachier19} Alejandro Tolcachier, *Holonomy groups of compact flat solvmanifolds*, Geometriae Dedicata **209** (2020) 95–117  $[$[arXiv:1907.02021](https://arxiv.org/abs/1907.02021), [doi:10.1007/s10711-020-00524-8](https://doi.org/10.1007/s10711-020-00524-8)$]$

* [[GAP]] package, _The Crystallographic Groups Catalog_ ([web](http://www.math.rwth-aachen.de/~Greg.Gamble/gap4r3/pkg/crystcat/htm/CHAP001.htm))

Bieberbach's original articles:

* {#Bieberbach1910} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen des $n$ dimensionalen Euklidischen Raumes mit einem endlichen Fundamentalbereich*, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1910) 75-84 $[$[dml:58754](https://eudml.org/doc/58754)$]$

* {#Bieberbach1911} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung)*, Mathematische Annalen **70** (1911) 297–336  $[$[doi:10.1007/BF01564500](https://doi.org/10.1007/BF01564500)$]$

* {#Bieberbach1912} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen der Euklidischen Räume (ZweiteAbhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich*, Mathematische Annalen  **72** (1912) 400-412 $[$[doi:10.1007/BF01456724]( https://doi.org/10.1007/BF01456724)$]$

A monograph on the topic:

* {#Charlap86} [[Leonard S. Charlap]], *Bieberbach Groups and Flat Manifolds*, Springer (1986) $[$[doi:10.1007/978-1-4613-8687-2](https://doi.org/10.1007/978-1-4613-8687-2)$]$

See also

* [[eom]], _[Crystallographic group](https://www.encyclopediaofmath.org/index.php/Crystallographic_group)_

* [[Groupprops]], _[Space group](https://groupprops.subwiki.org/wiki/Space_group)_

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

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

[[!redirects crystallographic groups]]

[[!redirects crystallographic point group]]
[[!redirects crystallographic point groups]]

[[!redirects crystallographic symmetry]]
[[!redirects crystallographic symmetries]]

[[!redirects space group]]
[[!redirects space groups]]

[[!redirects point group]]
[[!redirects point groups]]

[[!redirects symmorphic space group]]
[[!redirects symmorphic space groups]]

[[!redirects symmorphic crystallographic group]]
[[!redirects symmorphic crystallographic groups]]

[[!redirects representation torus]]
[[!redirects representation tori]]

[[!redirects Bieberbach's first theorem]]

[[!redirects Bieberbach's theorem]]
[[!redirects Bieberbach's theorems]]


Ctirad Klimčik > history

see at *[[Ctirad Klimčík]]*


Ctirad Klimčík

Ctirad Klimčik is a Slovak mathematical physicist.

* [prabook entry](https://prabook.com/web/ctirad.klimcik/157572)

* [publications](http://arxiv.org/find/hep-th/1/au:+Klimcik_C/0/1/0/all/0/1)

#### Selected talks

* _Superintegrability, Killing symmetry and point particle T-duality_, at conference Integrability, Dualities and Deformations, Berlin 2022 [yt](https://www.youtube.com/watch?v=kvZXwgbMtCg)

> We report on a recent progress in the point particle T-duality story. We argue that the ideas  related to Killing symmetry, integrability and deformation play an important role not only in a string T-duality story but also in its point particle counterpart. Applying those ideas we find, quite remarkably, that the  T-duality seems to be a more widespread phenomenon in the context of the point particle dynamics  than in the string one. In this talk, we focus on superintegrable spherically symmetric electro-gravitational backgrounds in n dimensions where we can produce explicit new examples of point particle T-duality having an interesting physical consequences. In particular, the dynamics of a charged particle scattered by a repulsive electric potential in a flat space is T-dual to the dynamics of scattering in the space of constant negative curvature. Thus knowing just the exact Hamiltonian dynamics of the scattered particle cannot give us an information about the curvature of the space in which the particle moves.

category: people

## Selected writings

On examples of [[universal spacetimes]]:

* [[Daniele Amati]], [[Ctirad Klimčík]], _Nonperturbative computation of the Weyl anomaly for a class of nontrivial backgrounds_, Phys. Lett. B, 219:443–447, 1989 ([spire:269390](inspirehep.net/record/269390), 10.1016/0370-2693(89)91092-7)

Introducing the [[fuzzy 4-sphere]]:

* [[Harald Grosse]], [[Ctirad Klimčík]], P. Prešnajder, _On finite 4D quantum field theory in non-commutative geometry_, Commun. Math. Phys.180:429-438, 1996 ([arXiv:hep-th/9602115](https://arxiv.org/abs/hep-th/9602115))

On [[T-duality]] and the [[moment map]]:

* [[Ctirad Klimcik]], [[Pavol Ševera]], _[[T-duality]] and the moment map_, IHES/P/96/70, [hep-th/9610198](http://arxiv.org/abs/hep-th/9610198); _Poisson-Lie T-duality: open strings and D-branes_, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, [hep-th/9512124](http://arxiv.org/abs/hep-th/9512124)

Introducing [[nonabelian T-duality]]:

* {#KlimcikSevera95} [[Ctirad Klimčík]], [[Pavol Ševera]], _Dual non-Abelian duality and the Drinfeld double_, Physics Letters B 351:4 (1995) 455-462 (doi:10.1016/0370-2693(95)00451-P)

On [[Poisson-Lie T-duality]]:

* [[Ctirad Klimčik]], [[Pavol Ševera]], _Poisson-Lie T-duality: open strings and D-branes_, CERN-TH/95-339. Phys.Lett. B376 (1996) 82--89, [hep-th/9512124](https://arxiv.org/abs/hep-th/9512124)

## Related entries

* [[Poisson-Lie T-duality]]

* [[moment map]]

* [[universal spacetime]]

category: people

[[!redirects Ctirad Klimcik]]
[[!redirects Ctirad Klimčik]]
[[!redirects C. Klimčík]]
[[!redirects C. Klimcik]]


cube

# The $n$-cube
* tic
{: toc}

## Idea

The __$n$-dimensional cube__, or simply __$n$-cube__, is a generalisation of the ordinary cube (or $3$-cube) to arbitrary dimensions.  It comes in many guises.

## As a cubical set

The standard cubical $n$-cube is the [[cubical set]] represented (as a [[presheaf]]) by the object $[n]$ in the [[cube category]].

## As a topological space

The standard topological $n$-cube is the space $[0,1]^n$, where $[0,1]$ is the [[unit interval]]. In general, any closed $n$-cube is the [[topological product]] of [[closed intervals]]. The collection of topological cubes forms a topological [[cocubical set]].

There are also open $n$-cubes, which contain all points of the closed $n$-cube which are [[apartness relation|apart from]] the boundary. Open $n$-cubes are the [[topological product]] of [[open intervals]]. The standard topological open $n$-cube is the space $(0,1)^n$, where $(0,1)$ is the [[open interval|open]] [[unit interval]].

The open $n$-cubes are the [[balls]] in $n$-dimensional Cartesian space $\mathrm{R}^n$ with respect to the [[metric]] derived from the [[supremum norm]] $\Vert x \Vert_\infty$ on $\mathrm{R}^n$.

## related entries

* [[cubical set]]

* [[cubical type theory]]

## References

Discussion of the 3-cube as a [[Platonic solid]]:

* {#Klein1884} [[Felix Klein]], chapter I.5 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

[[!redirects cubes]]
[[!redirects cubical]]
[[!redirects n-cube]]
[[!redirects k-cube]]
[[!redirects n-cubes]]
[[!redirects k-cubes]]


cubic curve

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

#Contents#
* table of contents
{:toc}

## Definition

For $S$ a [[scheme]], a _cubic curve_ over $S$ is a scheme $p \colon X \to S$ over $S$ equipped with a [[section]] $e \colon S \to X$ and such that [[Zariski topology|Zariski locally]] on $S$, $X$ is given by a solution in [[projective space]] $\mathbb{P}_S^2$ of an [[equation]] of the form

$$
  y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6
$$

(the _Weierstrass equation_) such that $e \colon S \to X$ is the line at infinity.

Equivalently this says that $p$ is a [[proper morphism|proper]] [[flat morphism]] with a section contained in the [[smooth locus]] whose [[fibers]] are geometrically integral curves of [[arithmetic genus]] one.

A [[non-singular algebraic variety|non-singular]] solution to this equation is an [[elliptic curve]] (see there for more).
Write $\mathcal{M}_{cub}$ for the [[moduli stack]] of such [[cubic curves]]. Then the [[moduli stack of elliptic curves]] is the non-vanishing locus of the [[discriminant]]  $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$

(e.g. [Mathew, section 3](#Mathew))

[[!include moduli stack of curves -- table]]

## Properties

### Covers

There is an eight-fold cover of $\mathcal{M}_{cub}$ [[localization of a ring|localized]] at $2$ ([Mathew 13, section 4.2](#Mathew13)) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the $\mathbb{Z}_2$-action on [[KU]] whose [[homotopy fixed points]] are [[KO]]).

## References

Reviews for the case that 2 and 3 are invertible include

* [[Balázs Szendrői]], _Cubic curves: a short survey_  ([pdf](http://people.maths.ox.ac.uk/szendroi/cubic.pdf))

and specifically over the complex numbers:

* {#Hain08} [[Richard Hain]], section 5 of _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

Discussion of the general case in the context of the construction of [[tmf]] is in

* {#Mathew13} [[Akhil Mathew]], _The homology of $tmf$_ ([arXiv:1305.6100](http://arxiv.org/abs/1305.6100))

reviewed in

* {#Mathew} [[Akhil Mathew]], section 3 of: _The homotopy groups of $TMF$_ ([pdf](http://math.mit.edu/~sglasman/tmfhomotopy.pdf))

[[!redirects cubic curves]]

[[!redirects moduli stack of cubic curves]]

[[!redirects Weierstrass equation]]


cubic function

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## Definition

### In commutative rings

For a [[commutative ring]] $R$, a **cubic function** is a [[function]]  $f \colon R \to R$ with [[elements]] $a \in R$, $b \in R$, $c \in R$, $d \in R$ such that for all $x \in R$,

$$f(x) = a \cdot x^3 + b \cdot x^2 + c \cdot x + d$$

where $x^2$ and $x^3$ is the canonical [[square function]] and [[cube function]] of the [[multiplication|multiplicative]] [[monoid]].

## See also

* [[linear function]], [[quadratic function]], [[polynomial function]]

* [[depressed cubic function]]

* [[real cubic function]]

## References

See also:

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

[[!redirects cubic functions]]


cubical category

The term **cubical category** has at least two common meanings.  Thus, to avoid ambiguity, it is perhaps better to avoid it entirely and use an equivalent, unambiguous term for the particular meaning one has in mind.

* The **cubical category** $\Box$ is the domain category for the presheaf category of [[cubical set|cubical sets]].  To avoid ambiguity, $\Box$ may also be called the **[[cube category]]** (which see, for its definition) or the **cubical indexing category**.

* A **cubical category** is a [[cubical object]] in [[Cat]] (that is, a functor from $\Box^{op}$ to $Cat$), just like a cubical set is a cubical object in [[Set]], a cubical space is a cubical object in [[Top]], a cubical abelian group is a cubical object in [[Ab]], and so on.  To avoid ambiguity, **[[cubical object in Cat|cubical objects in Cat]]** may be called exactly that.

A third use of the adjective "cubical" with a type of category is the following.

* A **[[cubical ∞-category]]** is a [[cubical set]] equipped with operations giving a way to compose cubes with each other, in the same way that an ordinary [[∞-category]] is a [[globular set]] equipped with composition operations.


cubical geometric realisation

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\section{Idea}

The [[category of cubical sets]], just as with any reasonable category of [[geometric shapes for higher structures|shapes for higher structures]], admits a [[geometric realization|geometric realisation]] functor to the [[category of topological spaces]]. The universal property of the [[category of cubes]] allows for a particularly economical and canonical construction of this functor compared to some other such categories of shapes.

\section{Construction}

We make use of the notation at [[category of cubes]] and [[cubical set]], and denote the [[category of topological spaces]] by $\mathsf{Top}$. Let $I$ denote the topological [[unit interval]], let $\bullet$ denote the topological [[point]], let $0 : \bullet \rightarrow I$ and $1: \bullet \rightarrow I$ be the [[maps|continuous map]] which pick out the endpoints $0$ and $1$ of $I$ respectively, and let $t : I \rightarrow \bullet$ be the canonical map.

\begin{notn} We denote by $\left| - \right|_{\leq 1}$ the [[functor]] $\square_{\leq 1} \rightarrow \mathsf{Top}$ given by $I^{0} \mapsto \bullet$, $I^{1} \mapsto I$, $i_{0} \mapsto 0$, $i_{1} \mapsto 1$, and $p \mapsto t$. \end{notn}

\begin{notn} We denote by $\left| - \right|_{\square}: \square \rightarrow \mathsf{Top}$ the canonical functor determined by $\left| - \right|_{\leq 1}$, the [[cartesian monoidal category|cartesian monoidal structure]] on $\mathsf{Top}$, and the universal property of $\square$. \end{notn}

\begin{defn} The _geometric realisation_ functor $\mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Top}$ is the canonical functor determined by $\left| - \right|_{\square}$, the fact that $\mathsf{Top}$ is co-complete, and the universal property of $\mathsf{Set}^{\square^{op}}$. \end{defn}

## Related concepts

* [[topological realization]]

* [[cubulation]]

[[!redirects cubical geometric realization]]
[[!redirects geometric realisation of cubical sets]]
[[!redirects geometric realization of cubical sets]]

[[!redirects cubical topological realization]]
[[!redirects topological realisation of cubical sets]]
[[!redirects topological realization of cubical sets]]

[[!redirects topological cubical realization of a cubical set]]


cubical group

A _cubical group_ is a group object in the category of cubical sets.

Unlike in the category of simplicial sets, a cubical group need not be a fibrant cubical set. However a group object in the category of cubical sets with connections is a fibrant cubical set.

## Reference##

* Tonks, A.P., Cubical groups which are Kan, J. Pure Appl. Algebra, 81 (1992) 83--87.


cubical Kan complex

\tableofcontents

\section{Definition}

We shall make use of notions introduced at [[cubical set]], and of the notation of this page.

\begin{defn} A [[cubical Kan complex]] is a [[cubical set]] $X$ equipped with the following structure: for every integer $n \geq 1$, every integer $1 \leq i \leq n$, every integer $0 \leq \epsilon \leq 1$, and every morphism $f : \sqcap^{n,i,\epsilon} \rightarrow X$ of cubical sets, there is a morphism $g : \square^{n} \rightarrow X$ of cubical sets such that the following diagram in $\mathsf{Set}^{\square^{op}}$ commutes.

$$
   \array{
      \sqcap^{n,i,\epsilon}                         &                                            & \\
      \mathllap{i_{i,\epsilon}} \downarrow  & \overset{f}{\searrow}    & \\
      \square^{n}                        & \underset{g}{\rightarrow} & X
   }
$$
\end{defn}

\section{Homotopy groups}

Cubical Kan complexes admit a notion of homotopy group. The theory of these homotopy groups can be developed analogously to the theory of homotopy groups of a topological space. See [[homotopy groups of a cubical Kan complex]].

[[!redirects cubical kan complex]]


cubical object

#Contents#
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## Idea

A **cubical object** in a [[category]] $C$ is a [[functor]]

$$
  F : \Box^{op} \to C
$$

from the [[opposite category]] of the [[cube category]] $\Box$ into $C$.

If $C = Set$ then this is called a [[cubical set]].

Functors

$$
  F : \Box \to C
$$

from the cube category itself are **co-cubical object**s in $C$.

## Related concepts

* [[simplicial object]].

* ´[[cubical set]]

* [[model structure on cubical presheaves]]

[[!redirects cubical objects]]

[[!redirects co-cubical object]]
[[!redirects cocubical object]]


cubical path type

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## Idea

Cubical path types are a form of [[path types]] in [[dependent type theory]] used in [[cubical type theories]].

A major difference between cubical path types and Martin-Löf identity types is the behaviour of the J rule. In Martin-Löf identity types the J rule holds up to definitional equality, but for cubical path types, the J rule only holds up to a path.

Another difference is that [[transport]] generally behaves better with cubical path types.  Certain rules for the computation of transports in concrete type families hold up to definitional equality for cubical path types, but only up to an identification for Martin-Löf identity types.

## Rules

The rules for (dependent) cubical path types are as follows

* Formation

$$\frac{\Gamma, i:I \vdash A \; \mathrm{type} \quad \Gamma \vdash a:[0/i]A \quad \Gamma \vdash b:[1/i]A}{\Gamma \vdash \mathrm{path}_{i.A}(a,b) \; \mathrm{type}}$$

* Introduction

$$\frac{\Gamma, i:I \vdash a:A}{\Gamma \vdash \lambda i.a:\mathrm{path}_{i.A}([0/i]a,[1/i]a)}$$

* Elimination

$$\frac{\Gamma \vdash p:\mathrm{path}_{i.A}(a,b) \quad \Gamma \vdash r:I}{\Gamma \vdash p(r):[r/i]A [\partial(r) \to [r \equiv 0 \to a \vert r \equiv 1 \to b]]}$$

* Computation

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash (\lambda i.a)(r) \equiv [r/i]a:[r/i]A}$$

* Uniqueness (optional)

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash p \equiv \lambda i.p(i) : \mathrm{path}_{i.A}(a,b)}$$

In addition, there are coercion and composition operations which make the path type behave like an identity type:

* Coercion:

$$\frac{\Gamma, i:I \vdash A \; \mathrm{type} \quad \Gamma \vdash a:[r/i]A}{\Gamma \vdash \mathrm{coe}^{r \rightsquigarrow r'}_{i.A} a:[r'/i]A \; [r' \equiv r \to a]}$$

* Composition:

...

## Regularity

In general, cubical path types in cubical type theory do not satisfy the J rule definitionally. However one could force the J rule to hold definitionally by appending a coercion [[regularity]] rule:

$$\frac{\Gamma, i:I, j:I \vdash A \equiv [j/i]A \; \mathrm{type}}{\Gamma \vdash \mathrm{coe}^{r \rightsquigarrow r'}_{i.A} a \equiv a:[r'/i]A}$$

to the type theory.

## See also

* [[cubical type theory]]

* [[identity type]]

* [[heterogeneous identity type]]

## References

On why cubical path types and Martin-Löf identity types are different:

* [[Andrew Swan]], *Separating Path and Identity Types in Presheaf Models of Univalent Type Theory*, ([arXiv:1808.00920](https://arxiv.org/abs/1808.00920))

On cubical path types in [[XTT]]:

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In Herman Geuvers, editor, _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).

[[!redirects cubical path]]
[[!redirects cubical paths]]

[[!redirects cubical path type]]
[[!redirects cubical path types]]

[[!redirects cubical identity type]]
[[!redirects cubical identity types]]