chain homology and cohomology

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[homological algebra]], for $V_\bullet \in Ch_\bullet(\mathcal{A})$ a _[[chain complex]]_, its _chain [[homology group]]_ in degree $n$ is akin to the $n$-th [[homotopy groups]] of a topological space. It is defined to be the [[quotient]] of the $n$-[[cycles]] by the $n$-[[boundaries]] in $V_\bullet$.

[[duality|Dually]], for $V^\bullet \in Ch^\bullet(\mathcal{A})$ a [[cochain complex]], its _cochain [[cohomology group]]_ in degree $n$ is the quotient of the $n$-[[cocycles]] by the $n$-[[coboundaries]].

Basic examples are the [[singular homology]] and [[singular cohomology]] of a topological space, which are the (co)chain (co)homology of the [[singular complex]].

Chain homology and cochain cohomology constitute the basic invariants of (co)chain complexes. A [[quasi-isomorphism]] is a [[chain map]] between chain complexes that induces isomorphisms on all chain homology groups, akin to a [[weak homotopy equivalence]]. A [[category of chain complexes]] equipped with quasi-isomorphisms as [[weak equivalences]] is a presentation for the [[stable (infinity,1)-category]] of chain complexes.

## Definition

Let $\mathcal{A}$ be an [[abelian category]] such as that of $R$-[[modules]] over a [[commutative ring]] $R$. For $R = \mathbb{Z}$ the [[integers]] this is the category [[Ab]] of [[abelian groups]]. For $R = k$ a [[field]], this is the category [[Vect]] of [[vector spaces]] over $k$.

Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] in $\mathcal{A}$. Write $Ch^\bullet(\mathcal{A})$ for the [[category of cochain complexes]] in $\mathcal{A}$.

We label [[differentials]] in a chain complex as follows:

$$
  V_\bullet
  =
  [
   \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \to \cdots
  ]
$$

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

For $V_\bullet \in Ch_\bullet(\mathcal{A})$ a [[chain complex]] and $n \in \mathbb{Z}$, the **chain homology** $H_n(V)$ of $V$ in degree $n$ is the [[abelian group]]

$$
  H_n(V) \coloneqq \frac{Z_n(V)}{B_n(V)} = \frac{ker(\partial_{n-1})}{im(\partial_n)}
$$

given by the [[quotient]] ([[cokernel]]) of the group of $n$-[[cycles]] by that of $n$-[[boundaries]] in $V_\bullet$.

=--

## Properties

### Functoriality

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

For all $n \in \mathbb{N}$ forming chain homology extends to a [[functor]]
from the [[category of chain complexes]] in $\mathcal{A}$ to $\mathcal{A}$ itself

$$
  H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
  \,.
$$

=--

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

One checks that [[chain homotopy]] (see there) respects [[cycles]] and [[boundaries]].

=--

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

Chain homology commutes with [[direct product]] of chain complexes:

$$
  H_n(\prod_i C^{(i)})
  \simeq
  \prod_i H_n(C^{(i)})
  \,.
$$

Similarly for [[direct sum]].

=--

### Respect for direct sums and filtered colimits
 {#RespectForDirectSum}

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

The chain homology functor preserves [[direct sums]]:

for $A,B \in Ch_\bullet$ and $n \in \mathbb{Z}$, the canonical morphism

$$
  H_n(A \oplus B) \to H_n(A) \oplus H_n(B)
$$

is an [[isomorphism]].

=--

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

The chain homology functor preserves [[filtered colimits]]:

for $A \colon I \to Ch_\bullet$ a [[filtered category|filtered]] [[diagram]] and $n \in \mathbb{Z}$, the canonical morphism

$$
  H_n(\underset{\to_i}{\lim} A_i) \to \underset{\to_i}{\lim} H_n(A_i)
$$

is an [[isomorphism]].

=--

This is spelled out for instance as ([Hopkins-Mathew , prop. 23.1](#HopkinsMathew)).

## Examples

* For $X$ a [[topological space]] and $Sing X$ its [[singular simplicial complex]], write $N \mathbb{Z}[Sing X]$ for the [[normalized chain complex]] of the [[simplicial abelian group]] that is degreewise the [[free abelian group]] on $Sing X$. The resulting chain homology is the _[[singular homology]]_ of $X$

  $$
    H_\bullet( N \mathbb{Z}[Sing X]) \simeq H_\bullet(X, \mathbb{Z})
    \,.
  $$

* [[Koszul homology]]

* [[Ext]], [[Tor]]

## In the context of homotopy theory
 {#InTheContextOfHomotopyTheory}

We discuss here the notion of (co)homology of a [[chain complex]] from a more abstract point of view of [[homotopy theory]], using the [[nPOV]] on _[[cohomology]]_ as discussed there.

A [[chain complex]] in non-negative degree is, under the [[Dold-Kan correspondence]] a [[homological algebra]] model for a particularly nice [[topological space]] or [[∞-groupoid]]: namely one with an abelian group structure on it, a [[simplicial abelian group]].
Accordingly, an unbounded (arbitrary) [[chain complex]] is a model for a [[spectrum]] with abelian group structure.

One consequence of this embedding

$$
  N : Ch_+ \to \infty Grpd
$$

induced by the Dold-Kan [[nerve]] is that it allows to think of [[chain complexes]] as objects in the [[(∞,1)-topos]] [[∞Grpd]] or equivalently [[Top]]. Every [[(∞,1)-topos]] comes with a notion of [[homotopy]] and [[cohomology]] and so such abstract notions get induced on chain complexes.

Of course there is an independent, age-old definition of [[homology]] of chain complexes and, by dualization, of cohomology of cochain complexes.

This entry describes how these standard definition of chain homology and cohomology follow from the general [[(∞,1)-topos]] nonsense described at [[cohomology]] and [[homotopy]].

The main statement is that

* the naïve [[homology]] groups of a [[chain complex]] are really its [[homotopy groups]], in the abstract sense (i.e. with the chain complex regarded as a model for a space/$\infty$-groupoid);

* the naïve cohomology groups of a cochain complex are really the abstract [[cohomology groups]] of the dual [[chain complex]].

### Preliminaries

Before discussing chain homology and cohomology, we fix some terms and notation.

#### Eilenberg-MacLane objects

In a given [[(∞,1)-topos]] there is a notion of [[homotopy]] and [[cohomology]] for every (co-)coefficient object $A$ ($B$).

The particular case of [[chain complex]] [[homology]] is only the special case induced from coefficients given by the corresponding [[Eilenberg-MacLane objects]].

Assume for simplicity here and in the following that we are talking about non-negatively graded chain complexes of vector spaces over some field $k$. Then for every $n \in \mathbb{N}$ write

$$
  \begin{aligned}
    \mathbf{B}^n k &:=
    (
      \cdots
      \to
      \mathbf{B}^n k_n
      \to
      \cdots
       \to \stackrel{\partial}{\to} \mathbf{B}^n k_1
      \stackrel{\partial}{\to} \mathbf{B}^n k_0)
    \\
    &=
    (  \cdots \to k \to \cdots \to 0 \to 0 )
  \end{aligned}
$$

for the $n$th [[Eilenberg-MacLane object]].

Notice that this is often also denoted $k[n]$ or $k[-n]$ or $K(k,n)$.

#### Homotopy and cohomology

With the [[Dold-Kan correspondence]] understood, embedding
chain complexes into [[∞-groupoids]], for any chain complexes $X_\bullet$, $A_\bullet$ and $B_\bullet$ we obtain

* the $\infty$-groupoid

  $$
    \mathbf{H}_{\infty Grpd}(X_\bullet, A_\bullet)
  $$

  whose
  * objects are the $A$-valued [[cocycle]]s on $X$;
  * morphisms are the coboundaries between these [[cocycle]]s;
  * 2-morphisms are the coboundaries between coboundaries
  * etc.
  and where the elements of $\pi_0 \mathbf{H}(X_\bullet,A_\bullet)$ are the cohomology classes

* the $\infty$-groupoids

  $$
    \mathbf{H}_{\infty Grpd}(B_\bullet, X_\bullet)
  $$

  whose
  * objects are the $B$-co-valued cycles on $X$;
  * morphisms are the boundaries between these cycles;
  * 2-morphisms are the boundaries between boundaries
  * etc.
  and where the elements of $\pi_0 \mathbf{H}(B_\bullet,X_\bullet)$ are the homotopy classes

### Chain homology as homotopy

For $X_\bullet := V_\bullet$ any [[chain complex]]
and $H_n(V_\bullet)$ its ordinary chain [[homology]] in degree $n$, we have

$$
  H_n(V_\bullet) \simeq
  \pi_0 \mathbf{H}(\mathbf{B}^n k_\bullet, V_\bullet)
  \,.
$$

A cycle $c : \mathbf{B}^n k_\bullet \to V_\bullet$
is a chain map

$$
  \array{
    \cdots
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    k
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    \cdots
    \\
    && \downarrow
    && \downarrow^{c_n}
    && \downarrow
    \\
    \cdots
    &\stackrel{\partial}{\to}&
    V_{n+1}
    &\stackrel{\partial}{\to}&
    V_n
    &\stackrel{\partial}{\to}&
    V_{n-1}
    &\stackrel{\partial}{\to}&
    \cdots
  }
$$

Such chain maps are clearly in bijection with those elements $c_n \in V_n$ that are in the kernel of
$V_n \stackrel{\partial}{\to} V_{n-1}$
in that $\partial c_n = 0$.

A boundary $\lambda : c \to C'$ is a chain homotopy

$$
  \array{
    \cdots
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    k
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    \cdots
    \\
    && &
    {}^{\lambda_n}\swarrow
    \\
    \cdots
    &\stackrel{\partial}{\to}&
    V_{n+1}
    &\stackrel{\partial}{\to}&
    V_n
    &\stackrel{\partial}{\to}&
    V_{n-1}
    &\stackrel{\partial}{\to}&
    \cdots
  }
$$

such that $c' = c + \partial \lambda$.

(...)

### Cohomology of cochain complexes

The ordinary notion of cohomology of a [[chain complex|cochain complex]]
is the special case of cohomology in the
[[stable (∞,1)-category|stable (∞,1)-]] [[category of chain complexes]].

For $V^\bullet$ a cochain complex let

$$
  \begin{aligned}
      X &:= V_\bullet = (V^\bullet)^*
      \\
      &=
      (\cdots \to X_{n+1} \stackrel{\partial}{\to}
      X_n \stackrel{\partial}{\to} X_{n-1} \to \cdots)
      \\
      & :=
      (\cdots \to V_{n+1} \stackrel{\partial}{\to}
      V_n \stackrel{\partial}{\to} V_{n-1} \to \cdots)
  \end{aligned}
$$

be the corresponding dual [[chain complex]]. Let

$$
  \begin{aligned}
     A  &:=  \mathbf{B}^n I
     \\
     &=
     (\cdots \to A_{n+1}  \to A_n \to A_{n-1} \to \cdots )
     \\
      & =
     (\cdots \to 0  \to I \to 0 \to \cdots )
  \end{aligned}
$$

be the [[chain complex]] with the
tensor unit (the [[ground field]], say) in degree $n$ and
trivial elsewhere. Then
$$
  \begin{aligned}
    \mathbf{H}(X,A)
    &=
    Ch(V_\bullet, \mathbf{B}^n I)
  \end{aligned}
$$

has

* as objects chain morphisms
  $c : V_\bullet \to \mathbf{B}^n I$
  $$
    \array{
      \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}&
      V_{n}
      &\stackrel{\partial}{\to}& V_{n-1}
      &\to&
      \cdots
      \\
      &&  \downarrow^{c_{n+1}}
      && \downarrow^{c_{n}}
      && \downarrow^{c_{n-1}}
      \\
      \cdots &\to& 0 &\stackrel{\partial}{\to}&
      I
      &\stackrel{\partial}{\to}& 0
      &\to&
      \cdots
    }
    \,.
  $$
  These are in canonical bijection with
  the elements in the kernel of $d_{n}$ of the dual
  cochain complex $V^\bullet = [V_\bullet,I]$.

* as morphism chain homotopies $\lambda : c \to c'$

  $$
    \array{
      \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}&
      V_{n}
      &\stackrel{\partial}{\to}& V_{n-1}
      &\to&
      \cdots
      \\
      &&
      &&
      &{}^{\lambda}\swarrow&
      \\
      \cdots &\to& 0 &\stackrel{\partial}{\to}&
      I
      &\stackrel{\partial}{\to}& 0
      &\to&
      \cdots
    }
    \,.
  $$

Comparing with the general definition of cocycles and coboudnaries from above, one confirms that

* the **cocycles** are the chain maps
  $$
    V_\bullet \to I[n]_\bullet
  $$

* the **coboundaries** are the chain homotopies between these chain maps.

* the **coboundaries of coboundaries** are the second order chain homotopies between these chain homotopies.

* etc.

## References

[[!include cohomology -- early references]]

Review of basics

* [[Charles Weibel]], Section 1.1: _[[An Introduction to Homological Algebra]]_

* {#HopkinsMathew} [[Michael Hopkins]] (notes by [[Akhil Mathew]]), _algebraic topology -- Lectures_ ([pdf](http://people.fas.harvard.edu/~amathew/ATnotes.pdf))

[[!redirects chain homology]]
[[!redirects cochain homology]]
[[!redirects chain cohomology]]
[[!redirects cochain cohomology]]

[[!redirects chain homology group]]
[[!redirects chain homology groups]]


chain homotopy

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _chain homotopy_ is a [[homotopy]] in a [[category of chain complexes]] with respect to the standard [[interval object in chain complexes]].

Sometimes a chain homotopy is called a _homotopy operator_. This is the terminology common for instance in the [standard proof](Poincare+lemma#Statement) of the [[Poincaré lemma]].

## Definition

Let $\mathcal{A} = $ [[Ab]] be the [[abelian category]] of [[abelian groups]]. Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] in $\mathcal{A}$.

A chain homotopy is a [[homotopy]] in $Ch_\bullet(\mathcal{A})$. We first give the explicit definition, the more abstract characterization is below in prop. \ref{AsALeftHomotopy}.

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

A _chain homotopy_ $\psi : f \Rightarrow g$ between two [[chain maps]] $f,g : C_\bullet \to D_\bullet$ in $Ch_\bullet(\mathcal{A})$ is a sequence of morphisms

$$
  \{ (\psi_n : C_n \to D_{n+1}) \in \mathcal{A} | n \in \mathbb{N} \}
$$

in $\mathcal{A}$ such that

$$
  f_n - g_n = \partial^D \circ  \psi_n + \psi_{n-1} \circ \partial^C
  \,.
$$

=--

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

It may be useful to illustrate this with the following graphics, which however is _not_ a [[commuting diagram]]:

$$
  \array{
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    C_{n+1} &\stackrel{f_{n+1} - g_{n+1}}{\to}& D_{n+1}
    \\
    \downarrow^{\mathrlap{\partial^C_{n}}}
    &\nearrow_{\mathrlap{\psi_{n}}}&
    \downarrow^{\mathrlap{\partial^D_{n}}}
    \\
    C_n &\stackrel{f_n - g_n}{\to}& D_n
    \\
    \downarrow^{\mathrlap{\partial^C_{n-1}}}
    &\nearrow_{\mathrlap{\psi_{n-1}}}&
    \downarrow^{\mathrlap{\partial^D_{n-1}}}
    \\
    C_{n-1} &\stackrel{f_{n-1} - g_{n-1}}{\to}& D_{n-1}
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
  }
  \,.
$$

=--

Instead, a way to encode chain homotopies by genuine diagrammatics is below in prop. \ref{AsALeftHomotopy}.

## Properties

### In terms of general homotopy
 {#RelationToLeft}

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

Let

$$
  I_\bullet \coloneqq N_\bullet(C(\Delta[1]))
$$

be the [[normalized chain complex]] in $\mathcal{A}$ of the [[chain on a simplicial set|simplicial chains]] on the simplicial 1-[[simplex]]:

$$
  I_\bullet =
  [
    \cdots \to 0 \to 0 \to \mathbb{1} \stackrel{(id,-id)}{\to}
   \mathbb{1} \oplus \mathbb{1}
  ]
  \,.
$$

=--

This is the standard [[interval in chain complexes]]. Indeed it is manifestly the "abelianization" of the standard interval object in [[sSet]]/[[Top]].

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

A chain homotopy $\psi : f \Rightarrow g$ is equivalently a [[commuting diagram]]

$$
  \array{
    C_\bullet
    \\
    \downarrow & \searrow^{\mathrlap{f}}
    \\
    I_\bullet\otimes C_\bullet
    &\stackrel{(f,g,\psi)}{\to}&
    D_\bullet
    \\
    \uparrow & \nearrow_{\mathrlap{g}}
    \\
    C_\bullet
  }
$$

in $Ch_\bullet(\mathcal{A})$, hence a genuine [[left homotopy]] with respect to the [[interval object in chain complexes]].

=--

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

For notational simplicity we discuss this in $\mathcal{A} = $ [[Ab]].

Observe that $N_\bullet(\mathbb{Z}(\Delta[1]))$ is the chain complex

$$
  (  \cdots  \to 0   \to 0 \to \mathbb{Z} \stackrel{(id,-id)}{\to}  \mathbb{Z} \oplus \mathbb{Z} \to 0 \to 0 \to \cdots)
$$

where the term $\mathbb{Z} \oplus \mathbb{Z}$ is in degree 0: this is the [[free abelian group]] on the set $\{0,1\}$ of 0-simplices in $\Delta[1]$. The other copy of $\mathbb{Z}$ is the free abelian group on the single non-degenerate edge in $\Delta[1]$. All other cells of $\Delta[1]$ are degenerate and hence do not contribute to the [[normalized chain complex]]. The single nontrivial [[differential]] sends $1 \in \mathbb{Z}$ to $(1,-1) \in \mathbb{Z} \oplus \mathbb{Z}$, reflecting the fact that one of the vertices is the 0-[[boundary]] and the other is the 1-boundary of the single nontrivial edge.

It follows that the [[tensor product of chain complexes]] $C_\bullet \otimes I_\bullet$ is

$$
  \cdots
  \to
  C_2 \oplus C_{2} \oplus C_1
  \to C_1 \oplus C_{1} \oplus C_0 \to  C_0 \oplus C_0 \oplus C_{-1} \to \cdots
 \,.
$$

Therefore a chain map $(f,g,\psi) :  C_\bullet \otimes I_\bullet \to D_\bullet$ that restricted to the two copies of $C_\bullet$ is $f$ and $g$, respectively,  is characterized by a collection of commuting diagrams

$$
  \array{
    C_{n+1}\oplus C_{n+1} \oplus C_{n}
      &\stackrel{(f_{n+1},g_{n+1}, \psi_n)}{\to}& D_{n+1}
    \\
    {}^{\mathllap{}}\downarrow && \downarrow^{\mathrlap{\partial^D}}
    \\
     C_{n} \oplus C_{n} \oplus C_{n-1} &\stackrel{(f_n,g_n,\psi_{n-1})}{\to}
    &
    D_n
  }
  \,.
$$

On the elements $(1,0,0)$ and $(0,1,0)$ in the top left this reduces to the chain map condition for $f$ and $g$, respectively. On the element $(0,0,1)$ this is the equation for the chain homotopy

$$
  f_n - g_n - \psi_{n-1} d_C = d_D \psi_{n}
  \,.
$$

=--

### Homotopy equivalence

Let $C_\bullet, D_\bullet \in Ch_\bullet(\mathcal{A})$
be two chain complexes.

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

Define the [[relation]] _chain homotopic_ on
$Hom(C_\bullet, D_\bullet)$ by

$$
  (f \sim g)
    \Leftrightarrow
  \exists (\psi : f \Rightarrow g)
  \,.
$$

=--

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

Chain homotopy is an [[equivalence relation]] on $Hom(C_\bullet,D_\bullet)$.

=--

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

Write $Hom(C_\bullet,D_\bullet)_{\sim}$ for the [[quotient]] of the [[hom set]] $Hom(C_\bullet,D_\bullet)$ by chain homotopy.

=--

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

This quotient is compatible with [[composition]] of [[chain maps]].

=--

Accordingly the following [[category]] exists:

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

Write $\mathcal{K}(\mathcal{A})$ for the category whose
[[objects]] are those of $Ch_\bullet(\mathcal{A})$, and whose [[morphisms]]
are chain homotopy classes of chain maps:

$$
  Hom_{\mathcal{K}(\mathcal{A})}(C_\bullet, D_\bullet)
  \coloneqq
  Hom_{Ch_\bullet(\mathcal{A})}(C_\bullet, D_\bullet)_\sim
  \,.
$$

This is usually called the **[[homotopy category of chain complexes]]**
in $\mathcal{A}$.

=--

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

Beware, as discussed there, that another category that would deserve to
carry this name instead is called the _[[derived category]]_ of $\mathcal{A}$.
In the derived category one also quotients out chain homotopy, but one allows that first the [[domain]] of the two chain maps $f$ and $g$ is refined along a [[quasi-isomorphism]].

=--

## Related concepts

* [[null homotopy]]

* [[category of chain complexes]]

  * [[chain complex]]

  * [[chain map]], [[quasi-isomorphism]]

  * **chain homotopy**

* [[homotopy category of chain complexes]]

## References

Historical articles:

* Tibor Rado , *A Remark on Chain-Homotopy*, Proceedings of the American Mathematical Society Vol. 2, No. 3 (Jun., 1951), pp. 458-463 ([jstor:2031776](https://www.jstor.org/stable/2031776), [doi:10.2307/2031776](https://doi.org/10.2307/2031776))

Review:

* [[Charles Weibel]], Section 1.4 of: _[[An Introduction to Homological Algebra]]_

[[!redirects cochain homotopy]]
[[!redirects chain homotopies]]
[[!redirects cochain homotopies]]

[[!redirects homotopy operator]]
[[!redirects homotopy operators]]


chain homotopy equivalence

A [[homotopy equivalence]] of [[chain complexes]], hence a [[chain map]] that has an [[inverse]] up to [[chain homotopy]].

These are the [[weak equivalences]] in the [[Hurewicz model structure on chain complexes]].

[[!redirects chain homotopy equivalences]]


chain map

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

#Contents#
* table of contents
{:toc}

## Idea

A _chain map_ is a [[homomorphism]] of [[chain complexes]].
Chain complexes with chain maps between them form the [[category of chain complexes]].

## Definition

Let $V_\bullet, W_\bullet \in Ch_\bullet(\mathcal{A})$ be two  [[chain complexes]] in some ambient [[additive category]] $\mathcal{A}$ (often assumed to be an [[abelian category]]).

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

A **chain map** $f : V_\bullet \to W_\bullet$ is a collection of [[morphism]] $\{f_n : V_n \to W_n\}_{n \in \mathbb{Z}}$ in $\mathcal{A}$ such that all the [[diagrams]]

$$
  \array{
    V_{n+1} &\stackrel{d^V_n}{\to}& V_n
    \\
    \downarrow^{\mathrlap{f_{n+1}}}
    &&
    \downarrow^{\mathrlap{f_{n}}}
    \\
    W_{n+1} &\stackrel{d^W_n}{\to} & W_n
  }
$$

[[commuting diagram|commute]], hence such that all the [[equations]]

$$
  f_n \circ d^V_n = d^W_{n} \circ f_{n+1}
$$

hold.

=--

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

A chain map $f$ induces for each $n \in \mathbb{Z}$ a morphism $H_n(f)$ on [[homology groups]], see prop. \ref{OnHomology} below. If these are all [[isomorphisms]], then $f$ is called a _[[quasi-isomorphism]]_.

=--

## Properties

### On homology
 {#OnHomology}

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

For $f : C_\bullet \to D_\bullet$ a chain map, it respects [[boundaries]] and [[cycles]], so that for all $n \in \mathbb{Z}$ it restricts to a morphism

$$
  B_n(f) : B_n(C_\bullet) \to B_n(D_\bullet)
$$

and

$$
  Z_n(f) : Z_n(C_\bullet) \to Z_n(D_\bullet)
  \,.
$$

In particular it also respects [[chain homology]]

$$
  H_n(f) : H_n(C_\bullet) \to H_n(D_\bullet)
  \,.
$$

=--

+-- {: .num_cor}
###### Corollary

Conversely this means that taking [[chain homology]] is a [[functor]]

$$
  H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
$$

from the [[category of chain complexes]] in $\mathcal{A}$ to $\mathcal{A}$ itself.

=--

In fact this is a universal [[delta-functor]].

## Related concepts

* [[category of chain complexes]]

  * [[chain complex]]

  * **chain map**, [[quasi-isomorphism]]

  * [[chain homotopy]], [[null homotopy]]

## References

A basic discussion is for instance in section 1.1 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_.

A more comprehensive discussion is in section 11 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_,

[[!redirects chain maps]]

[[!redirects cochain map]]
[[!redirects cochain maps]]


chain rule

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Statement

The **chain rule** is the statement that [[differentiation]] $d : Diff \to Diff$  is a [[functor]] on [[Diff]]:

given two [[smooth function]]s between [[smooth manifold]]s $f : X \to Y$ and $g : Y \to Z$ we have

$$
  d(g \circ f) :
  T X
  \stackrel{d f}{\to}
  T Y
  \stackrel{d g}{\to}
  T Z
  \,.
$$

If one thinks of a [[tangent vector]] $v \in T_x X$ to be an equivalence class of a smooth path $\gamma_v : [-\epsilon,\epsilon] \to X$, for some $\epsilon\gt 0$, with $\gamma(0) = x$, then the chain rule is the [[associativity]] of the composite

$$
  [-\epsilon,\epsilon] \stackrel{\gamma_v}{\to} X \stackrel{f}{\to} Y \stackrel{g}{\to} Z
  \,.
$$

Bracketed as $(g \circ f)\circ \gamma_v$ this represents $d(g \circ f)(v)$. Bracketed as $g \circ (f \circ \gamma_v)$ is represents $d g (d f (v))$.

Alternatively, in a context of [[synthetic differential geometry]] where with $D$ being the [[infinitesimal space|infinitesimal interval]] we may _identify_ $v$ with $v : D \to X$, the chain rule is the associativity of

$$
  D \stackrel{v}{\to} X \stackrel{f}{\to} Y \stackrel{g}{\to} Z
  \,.
$$

## Examples

### Elementary calculus

Let $X = Y = Z = \mathbb{R}$ the [[real line]]. Then the [[tangent bundle]] $T X$ is canonically identified with $\mathbb{R} \times \mathbb{R}$.

Given two functions, $f, g : \mathbb{R} \to \mathbb{R}$ their derivatives are traditionally regarded again as functions $f', g' : \mathbb{R} \to \mathbb{R}$, though strictly speaking we are to think of them as the maps

$$
  d f, d g : \mathbb{R} \times \mathbb{R} = T \mathbb{R}  \to T \mathbb{R}  = \mathbb{R} \times \mathbb{R}
$$

given by

$$
  d f : (x,v) \mapsto (f(x), v f'(x))
$$

and

$$
  d g : (x,v) \mapsto (g(x), v g'(x))
  \,.
$$

The composite

$$
  d (g \circ f) : \mathbb{R} \times \mathbb{R} =
  T \mathbb{R} \to T \mathbb{R} = \mathbb{R} \times \mathbb{R}
$$

is therefore the map

$$
  d(g \circ f) : (x,v) \mapsto (f(x), v f'(x))
   \mapsto
     (g(f(x)), v f'(x) g'(f(x)))
  \,.
$$

Therefore we have

$$
  (g \circ f)'(x) = f'(x) g'(f(x))
  \,.
$$

This is the form in which the chain rule is usually introduced in elementary calculus.

#### Remarks

While the chain rule is of great theoretical importance, it is completely unnecessary for the working out of derivatives of elementary functions in ordinary calculus (including the multivariable case).  Every result giving the derivative of an elementary function corresponds to a rule (along the lines of the [[product rule]]) for the operation of that function to any expression.  For example, instead of learning that
$$ \frac{\mathrm{d}}{\mathrm{d}x} (\sin x) = \cos x $$
and then applying both this fact *and* the chain rule to find the derivative of an expression of the form $\sin u$, just learn
$$ \frac{\mathrm{d}}{\mathrm{d}x} (\sin u) = \cos u \,\frac{\mathrm{d}u}{\mathrm{d}x} $$
and apply this rule directly; the original fact is the special case in which $u \coloneqq x$.  Even better, learn the rule as
$$ \mathrm{d}(\sin u) = \cos u \,\mathrm{d}u ;$$
then it applies without further modification to multivariable calculus (as well as implicit differentiation, related rates, integration by substitution, and other stock features of one-variable calculus).

The chain rule could still be used in the *proof* of this 'sine rule'.  Even so, it is quite possible to prove the sine rule directly (much as one proves the product rule directly rather than using the two-variable chain rule and the partial derivatives of the function $x, y \mapsto x y$).  In any case, the chain rule is not directly needed when working out specific derivatives.  As a *rule* of differentiation, the chain rule is needed only when an unspecified [[differentiable function]] $f$ appears, and then may be given in the form
$$ \frac{\mathrm{d}}{\mathrm{d}x} \bigl(f(u)\bigr) = f'(u) \,\frac{\mathrm{d}u}{\mathrm{d}x} $$
or
$$ \mathrm{d}\bigl(f(u)\bigr) = f'(u) \,\mathrm{d}u $$
to match other rules.

Although the chain rule is often written as

\[
  \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u}
  \,
  \frac{\mathrm{d}u}{\mathrm{d}x} ,\label{trivial}
\]

this is an oversimplification.  Upon choosing an independent variable, it is possible (and easy) to give a rigorous definition of the [[derivative|differential]] $\mathrm{d}u$, and then (eq:trivial) is a triviality (assuming no division by zero).  However, with such a definition of differential, (eq:trivial) is *not* the chain rule!  The reason is that, when (eq:trivial) is used as a mnemonic for the chain rule, ${\mathrm{d}u}/{\mathrm{d}x}$ uses $x$ as the independent variable, but ${\mathrm{d}y}/{\mathrm{d}u}$ uses $u$.  Either may be chosen to define differentials[^invertibility], but one must (a priori) be consistent.  Now, it so happens that the choice of independent variable is entirely irrelevant; differentials have the same meaning no matter which independent variable is used.  But *this* fact requires proof; *it* is the chain rule (or at least a prerequisite for using (eq:trivial) as the chain rule), and it is *not* a triviality.  In this form, the chain rule is also known as **Cauchy's invariant rule**.

[^invertibility]:  Well, either may be chosen as long as their differentials are nowhere zero, which is exactly what must be true for (1) to make sense.  More precisely, given that $x$ works as an independent variable, the Chain Rule tells us that $u$ works just as well so long as $\mathrm{d}u$ (as defined using $x$) is nowhere zero.  This may be related to the easy but wrong proof of the Chain Rule that founders on a division by zero in exactly the same place (where $\mathrm{d}u$ would be), although I don't see a direct connection.

### Formal algebra {#FormalAlgebra}

The chain rule can also be discussed as a piece of formal algebra of [[power series]] (over a general [[commutative ring]] $A$). We present a conceptual proof based on considerations of [[synthetic differential geometry|SDG]].

The statement is that if $q, p \in A[ [x] ]$ are power series, with $0$ the $0^{th}$ (constant) [[coefficient]] of $p$, then $(q \circ p)'(x) = q'(p(x))p'(x)$ under standard definitions.

Let $D = A[y]/(y^2)$ be the [[representing object]] for [[derivations]]. Let $\delta: A[ [x] ] \to A[ [x] ] \otimes_A D \cong A[ [x] ][y]/(y^2)$  be the unique topological $A$-algebra map (under the $(x)$-[[adic topologies]]) that sends $x$ to $x + y$.  (If it helps, think $\delta(q) = q(x + y)$.)  For $p \in A[ [x] ]$, define $p'$ via the equation $\delta(p) = p(x) + p'(x)y$.

Let $\pi: A[ [x] ] \otimes_A D \to A[ [x] ] \otimes_A D$ be the unique [[topological algebra]] map taking $x$ to $p(x)$ and $y$ to $p'(x)y$.  Let $- \circ p: A[ [x] ] \to A[ [x] ]$ denote the unique topological algebra map that takes $x$ to $p$. Then the diagram

$$\array{
A[ [x] ] & \stackrel{\delta}{\to} &  A[ [x] ] \otimes_A D \\
\mathllap{- \circ p} \downarrow & & \downarrow \mathrlap{\pi} \\
A[ [x] ] & \underset{\delta}{\to} & A[ [x] ] \otimes_A D
}$$

commutes in the category of topological algebras, since the two legs agree when evaluated at the generator $x$. But then, evaluating each leg at a power series $q \in A [ [x]]$, we have

$$[\delta(- \circ p)](q) = \delta(q \circ p) = (q \circ p)(x) + (q \circ p)'(x)y$$

and

$$[\pi \delta](q) = \pi(\delta(q)) = \pi(q(x) + q'(x)y) = q(p(x)) + q'(p(x))(p'(x)y)$$

whence the coefficients of $y$ agree: $(q \circ p)'(x) = q'(p(x))p'(x)$.

## Related concepts

* [[natural bundle]]

* [[total derivative]]

* [[(∞,1)-category theory|(∞,1)-]][[categorification]]: [[Goodwillie chain rule]]

## References

A [[synthetic mathematics|synthetic]] formalization of the chain rule in [[differential cohesion|differential]] [[cohesive homotopy type theory]] is given in

* {#Wellen18} [[Felix Wellen]], Lemma 3.5 of _[[schreiber:thesis Wellen|Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

See also

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

On the [[Goodwillie chain rule]] in [[Goodwillie calculus]]:

* John R. Klein, [[John Rognes]], _A chain rule in the calculus of homotopy functors_, Geom. Topol. 6 (2002) 853-887 ([arXiv:math/0301079](http://arxiv.org/abs/math/0301079))

* [[Jacob Lurie]], section 6.3 of _[[Higher Algebra]]_

[[!redirects chain rule]]
[[!redirects chain law]]
[[!redirects Cauchy's invariant rule]]


chains and cochains - table

| $H_n = Z_n/B_n$  | **([[chain homology|chain]]-)[[homology]]** | **([[cochain cohomology|cochain]]-)[[cohomology]]** | $H^n = Z^n/B^n$ |
|--|--|--|--|
| $C_n$ | [[chain]] | [[cochain]] | $C^n$ |
| $Z_n \subset C_n$ | [[cycle]] | [[cocycle]] | $Z^n \subset C^n$ |
| $B_n \subset C_n$ | [[boundary]] | [[coboundary]] | $B^n \subset C^n$ |


Chaitanya Leena Subramaniam

* [website](https://www.chaitanyals.site/)

## Publications

* [[Cédric Ho Thanh]], [[Chaitanya Leena Subramaniam]] *Opetopic algebras I: Algebraic structures on opetopic sets* ([arXiv:1911.00907](https://arxiv.org/abs/1911.00907))

## Talks

* [[Chaitanya Leena Subramaniam]], *Dependently typed algebraic theories and their homotopy algebras*, [[Homotopy Type Theory Electronic Seminar Talks]], 17 March 2022 ([video](https://www.youtube.com/watch?v=NAT02hbwOUs), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Subramaniam-2022-03-17-HoTTEST.pdf))

category: people


Chaitin's incompleteness theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Chaitin incompleteness theorem_ is a kind of [[incompleteness theorem]] in the context of [[complexity theory]]. It says roughly that within any sufficiently strong [[formal logic|formal system]] $S$, there is an upper bound on provable complexity: a [[natural number]] $L$ such that for no given string of data is there is a [[formal proof]] in $S$ that its [[Kolmogorov complexity]] exceeds $L$.

Note that $L$ is not an upper bound on complexity; on the contrary, there are only finitely many strings with complexity $L$ or less, and all of the infinitely many others must have complexity larger than $L$.  The system $S$ may even be capable of proving this, but it still cannot prove any particular example.

## References

* [[Gregory Chaitin]], _Algorithmic Information Theory_, 2003. (first edition 1987) ([Internet Archive](https://archive.org/details/algorithmicinfor00chai))

* [[Gregory Chaitin]], _The Limits of Mathematics_, 1994 ([arXiv:chao-dyn/9407003](http://arxiv.org/abs/chao-dyn/9407003))

Expositions and surveys include

* Wikipedia, _[Chaitin's incompleteness theorem](http://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin.27s_incompleteness_theorem)_

* [[John Baez]], _[Surprises in logic](http://math.ucr.edu/home/baez/surprises.html)_, 2011

[[!redirects Chaitin incompletenss theorem]]
[[!redirects Chaitin incompleteness theorem]]
[[!redirects Chaitin's incompleteness theorem]]
[[!redirects Chaitin's incompleteness theorem]]
[[!redirects Chaitin\'s incompleteness theorem]]


Chan-Paton bundle

> under construction

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[string theory]], the [[background gauge field]] for the [[open string]] [[sigma-model]] over a [[D-brane]] in [[bosonic string theory]] or [[type II string theory]] is a unitary [[principal bundle]] [[connection on a bundle|with connection]], or rather, by the Kapustin-part of the [[Freed-Witten-Kapustin anomaly cancellation]] mechanism, a [[twisted bundle|twisted unitary bundle]], whose twist is the restriction of the ambient [[B-field]] to the [[D-brane]].

The first hint for the existence of such [[background gauge fields]] for the [[open string]] 2d-[[sigma-model]] comes from the fact that the open string's endpoint can naturally be taken to carry labels $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-[[gauge field]], the $(i,j)$-excitation being the given [[matrix]] element of a $U(n)$-valued connection 1-form $A$.

This original argument goes back work by Chan and Paton. Accordingly one speaks of _Chan-Paton factors_ and _Chan-Paton bundles_ .

## Definition

We discuss the Chan-Paton gauge field and its [[quantum anomaly cancellation]] in [[extended prequantum field theory]].

Throughout we write $\mathbf{H} = $ [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]].

### The $B$-field as a prequantum 2-bundle

For $X$ a [[type II supergravity]] [[spacetime]], the [[B-field]] is a map

$$
  \nabla_B \;\colon\; X \to \mathbf{B}^2 U(1)
  \,.
$$

If $X = G$ is a [[Lie group]], this is the [[prequantum 2-bundle]] of $G$-[[Chern-Simons theory]]. Viewed as such we are to find a canonical [[∞-action]] of the [[circle 2-group]] $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, form the corresponding [[associated ∞-bundle]] and regard the [[sections]] of that as the [[prequantum 2-states]] of the theory.

The Chan-Paton gauge field is such a prequantum 2-state.

### The Chan-Paton gauge field

We discuss the [[Chan-Paton gauge fields]] over [[D-branes]]
in [[bosonic string theory]] and over $Spin^c$-D-branes in [[type II string theory]].

We fix throughout a natural number $n \in \mathbb{N}$, the _[[rank]]_ of the Chan-Paton gauge field.

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

The [[extension of groups|extension]] of [[Lie groups]]

$$
  U(1) \to U(n) \to PU(n)
$$

exhibiting the [[unitary group]] as a [[circle group]]-extension of the [[projective unitary group]] sits in a long [[homotopy fiber sequence]] of [[smooth ∞-groupoids]] of the form

$$
  U(1) \to U(n) \to PU(n) \to \mathbf{B}U(1)
  \to \mathbf{B}U(n) \to \mathbf{B}PU(n) \stackrel{\mathbf{dd}_n}{\to}
  \mathbf{B}^2 U(1)
  \,,
$$

where for $G$ a [[Lie group]] $\mathbf{B}G$ is its [[delooping]] [[Lie groupoid]], hence the [[moduli stack]] of $G$-[[principal bundles]], and where similarly $\mathbf{B}^2 U(1)$ is the [[moduli ∞-stack|moduli 2-stack]] of [[circle 2-group]] [[principal 2-bundles]] ([[bundle gerbes]]).

=--

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

Here

$$
  \mathbf{dd}_n \;\colon\; \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)
$$

is a smooth refinement of the universal [[Dixmier-Douady class]]

$$
  dd_n \;\colon\; B PU(n) \to K(\mathbb{Z}, 3)
$$

in that under [[geometric realization of cohesive ∞-groupoids]] ${\vert- \vert} \colon$ [[Smooth∞Grpd]] $\to$ [[∞Grpd]] we have

$$
  {\vert \mathbf{dd}_n \vert}
  \simeq
  dd_n
  \,.
$$

=--

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

By the discussion at _[[∞-action]]_ the [[homotopy fiber sequence]] in prop. \ref{TheLongSequenceOfTheProjectiveUnitaryExtension}

$$
  \array{
    \mathbf{B} U(n)
    &\to&
    \mathbf{B} PU(n)
    \\
    && \downarrow
    \\
    && \mathbf{B}^2 U(1)
  }
$$

in $\mathbf{H}$
exhibits a smooth[[∞-action]] of the [[circle 2-group]] on the [[moduli stack]] $\mathbf{B}U(n)$ and it exhibits an equivalence

$$
  \mathbf{B} PU(n) \simeq (\mathbf{B}U(n))//(\mathbf{B} U(1))
$$

of the moduli stack of projective unitary bundles with the [[∞-quotient]] of this [[∞-action]].

=--

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

For $X \in \mathbf{H}$ a [[smooth manifold]] and $\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1)$ modulating a [[circle 2-group]]-[[principal 2-bundle]], maps

$$
  \mathbf{c} \to \mathbf{dd}_n
$$

in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^2 U(1)}$, hence [[diagrams]] of the form

$$
  \array{
    X &&\stackrel{}{\to}&& \mathbf{B} PU(n)
    \\
    & {}_{\mathllap{\mathbf{c}}}\searrow
    &\swArrow&
    \swarrow_{\mathrlap{\mathbf{dd}_n}}
    \\
    && \mathbf{B}^2 U(1)
  }
$$

in $\mathbf{H}$ are equivalently rank-$n$ unitary [[twisted bundles]] on $X$, with the twist being the class $[\mathbf{c}] \in H^3(X, \mathbb{Z})$.

=--

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

There is a further differential refinement

$$
  \array{
     (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}
     &\stackrel{\widehat \mathbf{dd}_n}{\to}&
     \mathbf{B}^2 U(1)_{conn}
     \\
     \downarrow && \downarrow
     \\
     (\mathbf{B}U(n))//(\mathbf{B}U(1))
     &\stackrel{\widehat \mathbf{dd}_n}{\to}&
     \mathbf{B}^2 U(1)
  }
  \,,
$$

where $\mathbf{B}^2 U(1)_{conn}$ is the universal moduli 2-stack of [[circle n-bundle with connection|circle 2-bundles with connection]] ([[bundle gerbes]] with connection).

=--

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

Write

$$
  \left(
  \left(\mathbf{B}U\left(n\right)//\mathbf{B}U\left(1\right)\right)_{conn}
   \stackrel{\mathbf{Fields}}{\to}
  \mathbf{B}^2 U\left(1\right)_{conn}
  \right)
  \;\;
  \in \mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}
$$

for the differential smooth universal Dixmier-Douady class of prop. \ref{DifferentialRefinementOfSMoothDDClass}, regarded as an object in the [[slice (∞,1)-topos]] over $\mathbf{B}^2 U(1)_{conn}$.

=--

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

Let

$$
  \iota_X
    \;\colon\;
  Q \hookrightarrow X
$$

be an inclusion of [[smooth manifolds]] or of [[orbifolds]], to be thought of as a [[D-brane]] [[worldvolume]] $Q$ inside an ambient [[spacetime]] $X$.

Then a **field configuration** of a _[[B-field]]_ on $X$ together with a compatible rank-$n$ **Chan-Paton gauge field** on the [[D-brane]] is a map

$$
  \phi
   \;\colon\;
  \iota_X \to \mathbf{Fields}
$$

in the [[arrow (∞,1)-topos]] $\mathbf{H}^{(\Delta^1)}$, hence a [[diagram]] in $\mathbf{H}$ of the form

$$
  \array{
    Q &\stackrel{\nabla_{gauge}}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))
    \\
    {}^{\iota_X}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{dd}_n}}
    \\
    X &\stackrel{\nabla_B}{\to}& \mathbf{B}^2 U(1)_{conn}
  }
$$

=--

This identifies a  [[twisted bundle]] with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk [[B-field]].

This relation is the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] cancellation for the [[bosonic string]] or else for the [[type II string]] on $Spin^c$ D-branes. ([FSS](#FSS))

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

If we regard the [[B-field]] as a [[background field]] for the [[Chan-Paton gauge field]], then remark \ref{PullbackAlongGeneralizedLocalDiffeomorphisms} determines along which maps of the B-field the Chan-Paton gauge field may be transformed.

$$
  \array{
    Y &\stackrel{}{\to}& X &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn}
    \\
    & \searrow & \downarrow & \swarrow
    \\
    &&\mathbf{B}^2 U(1)_{conn}
  }
  \,.
$$

On the local connection forms this acts as

$$
  A \mapsto A + \alpha
  \,.
$$

$$
  B \mapsto B + d \alpha
$$

This is the famous gauge transformation law known from the string theory literature.

=--

### The open string sigma-model

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

The [[D-brane]] inclusion $Q \stackrel{\iota_X}{\to} X$ is the [[target space]] for an [[open string]] with [[worldsheet]] $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a [[field (physics)|field]] configuration of the open string [[sigma-model]] is a map

$$
  \phi \;\colon\; \iota_\Sigma \to \iota_X
$$

in $\mathbf{H}^{\Delta^1}$, hence a [[diagram]] of the form

$$
  \array{
    \partial \Sigma &\stackrel{\phi_{bdr}}{\to}& Q
    \\
    \downarrow^{\mathrlap{\iota_\Sigma}}
     &\swArrow&
    \downarrow^{\mathrlap{\iota_X}}
    \\
    \Sigma &\stackrel{\phi_{bulk}}{\to}& X
  }
  \,.
$$

For $X$ and $Q$ ordinary [[manifolds]] just says that a field configuration is a map $\phi_{bulk} \;\colon\; \Sigma \to X$ subject to the constraint that it takes the [[boundary]] of $\Sigma$ to $Q$. This means that this is a [[trajectory]] of an [[open string]] in $X$ whose endpoints are constrained to sit on the [[D-brane]] $Q \hookrightarrow X$.

If however $X$ is more generally an [[orbifold]], then the [[homotopy]] filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called "orbifold twisted sectors" of open string configurations.

=--

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

The [[moduli stack]] $[\iota_\Sigma, \iota_X]$ of such field configurations is the [[homotopy pullback]]

$$
  \array{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    \\
    \downarrow &\swArrow& \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
  }
  \,.
$$

=--

### The anomaly-free open string coupling to the Chan-Paton gauge field

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

For $\Sigma$ a [[smooth manifold]] with [[boundary]] $\partial \Sigma$ of [[dimension]] $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a [[circle n-bundle with connection]] on some $X \in \mathbf{H}$, then the [[transgression]] of $\nabla$ to the [[mapping space]] $[\Sigma, X]$ yields a [[section]] of the [[complex line bundle]] [[associated bundle|associated]] to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram

$$
  \array{
    [\Sigma, X]
      &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{[\partial \Sigma, X]}}
    &&
    \downarrow^{\mathrlap{\overline{\rho}}_{conn}}
    \\
    [\partial \Sigma, X]
    &\stackrel{\exp(2 \pi i \int_{\partial \Sigma})}{\to}&
    \mathbf{B} U(1)_{conn}
  }
  \,.
$$

=--

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

This is the _[[higher parallel transport]]_ of the $n$-connection $\nabla$ over maps $\Sigma \to X$.

=--

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

The operation of forming the [[holonomy]] of a twisted unitary connection around a curve fits into a [[diagram]] in $\mathbf{H}$ of the form

$$
  \array{
    [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}]
    &\stackrel{hol_{S^1}}{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{[S^1, \widehat\mathbf{dd}_n]}}
     &\swArrow_{\simeq}&
    \downarrow^{\mathrlap{\overline{\rho}_{conn}}}
    \\
    [S^1, \mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{S^1})}{\to}&
    \mathbf{B}U(1)_{conn}
  }
  \,.
$$

=--

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

By the discussion at _[[∞-action]]_ the diagram in prop. \ref{TheTwistedHolonomyMapOnTwistedUnitaryBundles} says in particular that forming traced [[holonomy]] of twisted unitary bundles constitutes a [[section]] of the [[complex line bundle]] on the [[moduli stack]] of twisted unitary connection on the circle which is the [[associated bundle]] to the [[transgression]] $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ of the universal differential [[Dixmier-Douady class]].

=--

It follows that on the moduli space of the open string [[sigma-model]] of prop. \ref{TheTypeIIOpenStringSigmaModelModuliStackOfFields} above there are two $\mathbb{C}//U(1)$-valued [[action functionals]] coming from the bulk field and the boundary field

$$
  \array{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    &\stackrel{exp(2 \pi i \int_{\Sigma}[\Sigma, \nabla_B]  ) }{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
    \\
    \downarrow^{\mathrlap{hol_{S^1}([S^1, \nabla_{gauge}])}}
    \\
    \mathbb{C}//U(1)_{conn}
  }
  \,.
$$

Neither is a well-defined $\mathbb{C}$-valued function by itself. But by [[pasting]] the above diagrams, we see that both these constitute [[sections]] of the same [[complex line bundle]] on the moduli stack of fields:

$$
  \array{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    &\stackrel{[\Sigma, \nabla_B]}{\to}&
    [S^1, \mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow
    && && \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
    \\
    \downarrow^{\mathrlap{[S^1, \nabla_{gauge}]}} &&  & \searrow^{\mathrlap{[S^1, \nabla_B]}}
    & && \downarrow
    \\
    [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] & &\stackrel{[S^1, \widehat \mathbf{dd}_n]}{\to}& & [S^1, \mathbf{B}^2 U(1)_{conn}]
    \\
    \downarrow^{\mathrlap{hol_{S^1}}}
      && && &  \searrow^{\mathrlap{\exp(2 \pi i \int_{S^1}(-))}}
    \\
    \mathbb{C}//U(1)_{conn} &\to& &\to& &\to&  \mathbf{B}U(1)_{conn}
  }
  \,.
$$

Therefore the product action functional is a well-defined function

$$
  [\iota_\Sigma, \iota_X]
  \stackrel{
    \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] )
    \cdot
    hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1}
  }{\to}
  U(1)
  \,.
$$

This is the [[Freed-Witten-Kapustin anomaly|Kapustin anomaly]]-free [[action functional]] of the [[open string]].

## Related concepts

* [[B-field]]

* [[twisted K-theory]]

* [[Freed-Witten anomaly cancellation]]

* [[Dirac-Born-Infeld action]]

## References

In the traditional physicist's string theory introductions one finds Chan-Paton bundles discussed for instance

* [[Clifford Johnson]], section 2.4 of _D-Brane primer_ ([arXiv:hep-th/0007170](http://arxiv.org/abs/hep-th/0007170))

* David Tong, around p. 66 of _Lectures on string theory_ ([arxiv/0908.0333](http://front.math.ucdavis.edu/0908.0333))

These lectures tend to ignore most of the global subtleties though. For traditional discussion of the  _[[Freed-Witten-Kapustin anomaly]]_, see there. The above account in terms of [[higher geometry]] and [[extended prequantum field theory]] is due to  section 5.4 of

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_ .

Lecture notes along these lines are at

* _[[geometry of physics]]_

Discussion of the derivation of the  [[Yang-Mills theory]] on the D-brane from open [[string scattering amplitudes]]/[[string field theory]] includes

* [[David Gross]], [[Edward Witten]], _Superstring modifications of Einstein's equations_,  Nuclear Physics B Volume 277, 1986, Pages 1-10

and for the non-abelian case:

* Semyon Klevtsov, _Yang-Mills theory from String field theory on D-branes_ ([pdf](http://www.mi.uni-koeln.de/~klevtsov/cargese02.pdf))

[[!redirects Chan-Paton bundles]]

[[!redirects Chan-Paton vector bundle]]
[[!redirects Chan-Paton vector bundles]]

[[!redirects Chan-Paton gauge field]]
[[!redirects Chan-Paton gauge fields]]

[[!redirects Chan-Paton gauge bundle]]
[[!redirects Chan-Paton gauge bundles]]

[[!redirects Chan-Paton factor]]
[[!redirects Chan-Paton factors]]


Chandra Observatory

+-- {: .rightHandSide}
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### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Related entries

[[!include experiments -- contents]]

## References

* [home page](http://chandra.harvard.edu/)

* Wikipedia, _[Chandra X-ray observatory](https://en.wikipedia.org/wiki/Chandra_X-ray_Observatory)_

Observation of [[spinning black holes]] (near-[[extremal black holes|extremal]]):

* Xinyu Dai, Shaun Steele, Eduardo Guerras, Christopher W. Morgan, Bin Chen, _Constraining Quasar Relativistic Reflection Regions and Spins with Microlensing_ ([arXiv:1901.06007](https://arxiv.org/abs/1901.06007))

  exposition in:

  [[Chandra Observatory]], _[Chandra Views Spinning Black Holes Across Cosmic Sea](https://scitechdaily.com/chandra-views-spinning-black-holes-across-cosmic-sea/)_, July 5 2019


Chandrasekhar Ramanathan

* [institute page](https://faculty-directory.dartmouth.edu/chandrasekhar-ramanathan)

* [GoogleScholar page](https://scholar.google.com/citations?user=EBSaHLkAAAAJ&hl=en)

## Selected writings

On [[nuclear magnetic resonance]] on the [[nitrogen-vacancy center in diamond]] and related systems in [[quantum sensing]]:

* Santiago Bussandri, Daphna Shimon, [[Asif Equbal]], Yuhang Ren, Susumu Takahashi, [[Chandrasekhar Ramanathan]], Songi Han,  *P1 center electron spin clusters are prevalent in type Ib diamond* [[arXiv:2311.05396](https://arxiv.org/abs/2311.05396)]

category: people

[[!redirects Sekhar Ramanathan]]


Chandrashekar Devchand

* [webpage](http://users.math.uni-potsdam.de/~devchand/)

category: people


change of enriching category

+-- {: .rightHandSide}
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###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given two [[monoidal categories]] $(V_1,\otimes_1, I_1)$ and $(V_2, \otimes_2, I_2)$ used for enrichment in [[enriched category theory]] (for instance two [[Bénabou cosmoi]]), a [[lax monoidal functor]] between them

$$
  F
  \;\colon\;
  (V_1, \otimes_1, I_1)
    \longrightarrow
  (V_2, \otimes_2, I_2)
$$

canonically induces a [[2-functor]]

$$
  F_\ast \;\colon\; V_1 Cat \longrightarrow V_2 Cat
$$

between [[categories of V-enriched categories]] sending a $V_1$-[[enriched category]] $\mathcal{C}$ to the $V_2$-[[enriched category]] $F_\ast(\mathcal{C})$ such that

1. $F_\ast(\mathcal{C})$ has the same [[objects]] as $\mathcal{C}$;

1. the [[hom-objects]] of $F_\ast(\mathcal{C})$ are the [[images]] of the hom-objects of $\mathcal{C}$ under $F$:

   $$
     F_\ast(\mathcal{C})(x,y)
     \;\coloneqq\;
     F\big(\mathcal{C}(x,y)\big)
   $$

1. the [[composition]], [[unit]], [[associator]] and [[unitor]] [[morphisms]] in $F_\ast(\mathcal{C})$ are the [[images]] of those of $\mathcal{C}$ composed with the structure morphisms of the [[lax monoidal functor|lax monoidal]]-[[structure]] on $F$.

([Eilenberg-Kelly 65](#EilenbergKelly65))

This operation $F_\ast$ is sometimes called **change of enriching category** or **change of enriching base** or just **change of base** (e.g. [Crutwell 14, chapter 4](#Crutwell14), [Riehl 14, lemma 3.4.3](#Riehl14)). But notice that, despite some vague similarity, this is different from [[base change]] of [[slice categories]].

In the special case that $\mathcal{C}$ has a single object and is hence (if thought of as [[pointed object|pointed]] by that object) equivalently a [[monoid object]] in $V_1$, this statement reduces to the statement that lax monoidal functors preserve [[monoids]] ([this Prop.](monoidal+functor#MonoidsToMonoidsByLaxMonoidal))

## Generalizations and enhancements

* The operation of change of enriching category is functorial from [[MonCat]] to [[2Cat]].  In particular, any [[monoidal adjunction]] $V_1\rightleftarrows V_2$ gives rise to a [[2-adjunction]] $V_1 Cat\rightleftarrows V_2 Cat$ (and also for profunctors).

* $V$-enriched categories can be defined more generally when $V$ is a [[multicategory]], and any functor $F:V_1\to V_2$ between multicategories induces a change of enrichment 2-functor.  Note that a functor of multicategorise between the underlying multicategories of two monoidal categories corresponds to a *lax* monoidal functor, as in the original version above.  The multicategorical version also includes change of enrichment between [[closed categories]].

* A different generalization is when $V$ is a [[bicategory]], yielding [[bicategory-enriched categories]].  Any lax functor of bicategories induces a similar functor between its 2-categories of enriched categories.

* When $V_1$ and $V_2$ are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories $V_i$-[[Prof]] of $V_i$-categories and $V_i$-[[profunctors]] exist, change of enrichment also induces a [[lax functor]] $V_1 Prof \to V_2 Prof$.  To make this version functorial on $MonCat_{cocomplete}$, we need to consider $V_i Prof$ as [[double categories]], yielding a functor from [[MonCat]] to [[DblCat]].

* Finally, a generalization subsuming all of these is that for any [[virtual double category]] $V$ we can construct another virtual double category $V Prof$, and this construction is functorial.

## Examples

+-- {: .num_example #Underlying}
###### Example
For any a monoidal category $V$, the functor $V(I,-): V \to Set$ is lax monoidal, hence induces a 2-functor from $V Cat$ to $Cat$.  This assigns to any $V$-enriched category, $\mathcal{C}$, its [[underlying ordinary category]], usually denoted $\mathcal{C}_0$, defined by $\mathcal{C}_0(x,y) = V(I, hom(x,y))$.
=--

+-- {: .num_example #FreeCats}
###### Example
If $V$ is a [[cocomplete category|cocomplete]] monoidal category, the functor $V(I,-)$ above has a [[left adjoint]] that takes a set $X$ to the [[copower]] of $X$ copies of $I$.  The resulting 2-functor $Cat \to V Cat$ takes an ordinary category to the "free" $V$-category it generates.  Such $V$-categories are used, for instance, in subsuming "conical" [[limits]] under enriched [[weighted limits]].

By functoriality, the adjunction between these two functors gives rise to a 2-adjunction $Cat \rightleftarrows V Cat$.
=--

+-- {: .num_example #GenUnderlying}
###### Example
More generally, if an adjunction $F:V_1 \rightleftarrows V_2:U$ can be regarded as a [[free-forgetful adjunction]], then the corresponding adjunction between enriched categories can also be so regarded.  For instance, if $V_1 = Top$ and $V_2 = G Top$ for some [[topological group]] $G$, then the right adjoint $V_2 Cat \to V_1 Cat$ takes the "underlying topologically enriched category" of a category enriched in $G$-spaces (its morphisms are the fixed-point spaces of the original $G$-space-enriched category; we think of these as the "$G$-equivariant maps" while the original spaces consisted of not-necessarily-equivariant maps acted on by conjugation).  Similarly, any category enriched in [[spectra]] has an underlying category enriched in spaces (whose hom-spaces are the 0-spaces of the original hom-spectra), any [[dg-category]] has an underlying [[Ab-category]] (whose morphisms are the degree-0 ones in the original category), and so on.
=--

+-- {: .num_example #RealSing}
###### Example
The [[geometric realization]] and [[total singular complex]] adjunction $Real:SSet \rightleftarrows Top : Sing$ between [[simplicial sets]] and [[topological spaces]] induces another adjunction $SSet Cat \rightleftarrows Top Cat$ between [[simplicially enriched categories]] and [[topologically enriched categories]].
=--

+-- {: .num_example #HomotopyCat}
###### Example
If $V=$[[SSet]] or [[Top]], then the set of connected components $\pi_0:V\to Set$ is lax monoidal.  The resulting change of enrichment functor takes a $V$-category $C$ to its "naive homotopy category" $h C(x,y) = \pi_0(C(x,y))$ obtained by "identifying homotopic morphisms".

Similarly, the [[fundamental groupoid]] functor $\Pi_1:V\to Gpd$ is lax monoidal, so any $V$-category has an underlying [[(2,1)-category]].
=--

+-- {: .num_example #EnrichedHomotopyCat}
###### Example
An enhancement of the last example is that if $V$ is any [[monoidal model category]], then its [[homotopy category]] $Ho(V)$ comes with a lax monoidal functor $\gamma : V \to Ho(V)$.  Thus any $V$-category $C$ has an underlying $Ho(V)$-enriched "homotopy category" $h C$.  Usually the underlying ordinary category of $h C$ is the naive homotopy category from the previous example.
=--

+-- {: .num_example #Tau1}
###### Example
The nerve functor $N:Cat \to SSet$ preserves products and has a left adjoint $\tau_1$ that also preserves products.  Thus, we have a change of enrichment adjunction in which any [[2-category]] can be regarded as a [[simplicially enriched category]], and similarly any simplicially enriched category has a "homotopy 2-category".  The latter plays an important role in the theory of [[quasi-categories]].
=--

+-- {: .num_example #PolyMorphisms}
###### Example
**([[poly-morphisms]])**

For $F = P \;\colon\; Set \to Set$ the [[power set]]-[[functor]], the change of base functor $P_\ast \;\colon\;
 Cat \to Cat$ sends plain [[categories]] to plain categories. For $C$ any category, the morphisms of $C^{poly} \coloneqq P\ast(C)$ are called the _[[poly-morphisms]]_ of $C$ in [Mochizuki 12, section 0](poly-morphism#Mochizuki12).

=--

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[Max Kelly]], *Closed categories*,  Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer (1966) [[doi:10.1007/978-3-642-99902-4_22](https://doi.org/10.1007/978-3-642-99902-4_22)]

* {#Crutwell14} [[Geoff Cruttwell]], chapter 4 of: _Normed spaces and the Change of Base for Enriched Categories_, 2014 ([pdf](https://www.reluctantm.com/gcruttw/publications/thesis4.pdf))

* {#Riehl14} [[Emily Riehl]], lemma 3.4.3 inL _[[Categorical Homotopy Theory]]_, Cambridge University Press (2014)

[[!redirects changes of enriching category]]
[[!redirects changes of enriching categories]]
[[!redirects change of enriching categories]]

[[!redirects change of enrichment]]
[[!redirects changes of enrichment]]
[[!redirects changes of enrichments]]

[[!redirects change of enriching base]]
[[!redirects changes of enriching base]]
[[!redirects changes of enriching bases]]

[[!redirects change of base in enriched category theory]]
[[!redirects changes of base in enriched category theory]]
[[!redirects changes of bases in enriched category theory]]


change of integration variables

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

## Related concepts

* [[pullback of differential forms]]

* [[pullback of distributions]]

## References

* Eric Weisstein, _[Change of variables theorem](http://mathworld.wolfram.com/ChangeofVariablesTheorem.html)_

[[!redirects changes of integration variables]]

[[!redirects changing integration variables]]


change of rings theorem

#Contents#
* table of contents
{:toc}

## Statement

Let $A$ be a [[Hopf algebra]] over a [[field]] $k$, and let $B$ be a normal sub Hopf algebra of $A$. Then

$$
  Cotor_A(B,k) \simeq Cotor_{A//B}(k,k)
  \,.
$$

This is used in computations with the [[Adams spectral sequence]].

## References

Due to

* A. Liulevicius, _Notes on Homotopy of Thom Spectra_, Amer. J. of Math., 86
(1964), 1-16.

Textbook accounts include

* {#Kochman96} [[Stanley Kochman]], prop. 3.7.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Ravenel} [[Doug Ravenel]], theorems A1.1.18, A1.1.20 and A1.1.13 in _[[Complex cobordism and stable homotopy groups of spheres]]_ Academic Press 1986


Changzheng Li

* [personal page](https://math.sysu.edu.cn/gagp/czli)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=155889)

## Selected writings

On the [[Peterson isomorphism]] identifying [[quantum cohomology rings]] of [[flag varieties]] with a [[Pontryagin ring]]:

* [[Naichung Conan Leung]], [[Changzheng Li]], *Gromov-Witten invariants for $G/B$ and Pontryagin product for $\Omega K$*, Transactions of the American Mathematical Society **364** 5 (2012) 2567-2599  [[web](https://www.ams.org/journals/tran/2012-364-05/S0002-9947-2012-05438-9/), [arXiv:0810.4859](https://arxiv.org/abs/0810.4859), [jstor:41524936](https://www.jstor.org/stable/41524936)]

On [[quantum cohomology rings]] of [[flag varieties]] ([[quantum Schubert calculus]]):

* [[Naichung Conan Leung]], [[Changzheng Li]], *An update of quantum cohomology of homogeneous varieties* [[arXiv:1407.5905](https://arxiv.org/abs/1407.5905)]

On the variant for [[Pontryagin products]] not in [[ordinary homology]] but in [[topological K-theory|topological]] [[K-homology]]:

* [[Thomas Lam]], [[Changzheng Li]], [[Leonardo C. Mihalcea]], [[Mark Shimozono]], *A conjectural Peterson isomorphism in K-theory*, Journal of Algebra **513** (2018) 326-343 [[doi:10.1016/j.jalgebra.2018.07.029](https://doi.org/10.1016/j.jalgebra.2018.07.029), [arXiv:1705.03435](https://arxiv.org/abs/1705.03435)]

category: people


chaos

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

(...)

### For mathematical structures
 {#ForMathematicalStructure}

According to [Lawvere 1984](#Lawvere84), [[codiscrete objects]], i.e. those in the [[image]] of a [[right adjoint]] to a [[forgetful functor]], may be though of as "chaotic" with repect to whatever [[mathematical structure]] was forgotten by the forgetful functor.

This convention matches/subsumes more-or-less common  terminology such as *[[chaotic topology]]* or *[[chaotic groupoid]]*.

## Related concepts

* [[dynamical system]]

* [[chaotic topology]]

* [[chaotic groupoid]]

## References

### General

See also

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

### Concerning mathematical structure
 {#ConcerningMathematicalStructures}

The terminology "chaotic" for [[codiscrete objects]] goes back to

* [[SGA4|SGA4-1, 1.1.4]] (1972)

where it is used for [[Grothendieck topologies]], from which it was, apparently, adapted to [[chaotic topologies]] on [[topological spaces]].

General formalization of the concept in terms of [[right adjoints]] to [[forgetful functors]] is due to:

* {#Lawvere84} [[William Lawvere]], _Functorial Remarks on the General Concept of Chaos_, IMA Preprint 87, 1984 ([pdf](https://www.ima.umn.edu/sites/default/files/87s.pdf))

[[!redirects chaos theory]]


chaotic cosmic inflation

+-- {: .rightHandSide}
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###Context###
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[physics]] of [[cosmology]], what is called _chaotic cosmic inflation_ ([Linde 83](#Linde83)) or _chaotic [[eternal inflation]]_ is one variant [[model (physics)|model]] of the [[theory (physics)|physics]] of [[cosmic inflation]].

The name comes about since the model naturally predicts an ambient [[spacetime]] [[manifold]] in which local regions stochastically undergo [[cosmic inflation]]. The idea is that one such region would be what we observe from the inside as our [[observable universe]].  This large-scale picture of chaotic inflation has later come to be referred to as a _[[multiverse]]_ (but see the caveats there).

While the theory of [[cosmic inflation]] in general is in excellent agreement with astrophysical [[experiment]], chaotic inflation, which generically predicts large values of the tensor-to-scalar ratio $r$ is disfavored by recent measurements which show low upper bounds on $r$ ([PlanckCollaboration 13](#PlanckCollaboration13), [BICEP2-Keck-Planck 15](#BICEPKeckPlanck15)). (These low values of $r$ instead prefer the [[Starobinsky model of cosmic inflation]].)

## Description

Despite its name and its consequences, chaotic inflation is arguably the most uncontrived model of [[cosmic inflation]], in that (reviewed e.g. in [Linde 02, section II](#Linder02)) it takes the [[potential]] term $V$ of the hypothetical [[inflaton field]]  $\phi$ simply to be parabolic

$$
  V(\phi) \coloneqq \frac{m_\phi^2}{2} \phi^2
$$

hence to be the standard [[mass]] term of a [[scalar field]] ([[phi^2 interaction]]).

Whereas other inflationary models assume a more monotonely decreasing potential with and assume in an _ad hoc_ way that the inflaton field is to have a large amplitude "at the big bang" to then slowly "roll down" (at each point in spacetime, being a field!) its potential, the idea of ([Linde 83](#Linde83)) is that instead the inflaton by and large sits in its potential minimum in equilibrium, but that [[quantum fluctuations]] stochastically ("chaotically") drive it out of its minimum here and there. Wherever this happens [[cosmic inflation]] sets in and blows up the region of the ambient spacetime in which the inflaton happened to have fluctuated out of its equilibrium.

This idea of "chaotic" inflation driven by vacuum fluctuations of the inflator is in fact not tied to the simple square potential as above, and various variants have been considered, such as a [[Higgs field]]-like potential with a quartic term added.

## Implications

The idea of eternal cosmic inflation has been argued to provide a possible way to conceptualize the measured values of dimensionless  "physical constants", such as the [[fine structure constant]], the [[Yukawa couplings]] and notably the [[cosmological constant]]: for if in the fundamental [[theory (physics)|theory]] these parameters are not really constants but are dynamical [[field (physics)|fields]] that just happen to have constant value ("moduli fields") over large scales, then they might, so the argument, still vary from one "inflationary bubble" to the next. Thereby the idea of eternal inflation combined with that of a fundamental theory that has [[moduli fields]] (such as [[string theory]] with its [[landscape of string theory vacua]]) has been argued to put "physical constants" on the same footing as other more or less random phenomenological parameters, such as for instance the distance of our planets from the sun, etc.

For more on this see at _[[multiverse]]_.

## References

The original article is

* {#Linde83} [[Andrei Linde]], _Chaotic Inflation,_, Phys. Lett. 129B, 177 (1983).

Reviews include

* {#Linder02} [[Andrei Linde]], _Inflationary Theory versus Ekpyrotic/Cyclic Scenario_ ([arXiv:hep-th/0205259](http://de.arxiv.org/abs/hep-th/0205259))

* [[Alan Guth]], _Eternal inflation and its implications_ ([arXiv:hep-th/0702178](http://arxiv.org/abs/hep-th/0702178))

* [[Andrei Linde]], _Eternally Existing Self-Reproducing Chaotic Inflationary Universe_ Physics Letters B 175 (4): 395–400. ([pdf](http://www.stanford.edu/~alinde/Eternal86.pdf))

* Wikipedia, _[Eternal inflation](http://en.wikipedia.org/wiki/Eternal_inflation)_

The relevant experimental results (disfavoring the model) are due to

* {#PlanckCollaboration13} _Planck 2013 results. XXII. Constraints on inflation_ ([arXiv:1303.5082](http://arxiv.org/abs/1303.5082))

* {#BICEPKeckPlanck15} _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))

[[!redirects chaotic inflation]]


character

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

# Contents
* table of contents
{: toc}

## Definition

There are many notions of a character for an algebraic structure, often topologized.

### Character of a group
 {#CharacterOfAGroup}

For [[groups]], there are two different, but related, concepts which are both often just referred to as *characters* of a group:

1. *[multiplicative group characters](#MultiplicativeCharacterOfAGroup)*,

1. *[characters of linear representations of a group](#CharacterOfALinearRepresentation)*.

Here the first notion is the special case of the second notion for *1-[[dimension of a vector space|dimensional]]* [[linear representations]]. Since for [[abelian groups]] *all* [[irreducible representations]] over the [[complex numbers]] are 1-dimensional, in this case both notions agree, but in general they do not.

#### Multiplicative character of a group
 {#MultiplicativeCharacterOfAGroup}

A _multiplicative character_ on a ([[discrete group|discrete]]) [[group]] $G$ (a _[[group character]]_) is a [[group homomorphism]] from $G$ to the [[group of units]] $k^\times$ of the [[ground field]] $k$:

$$
  \chi
  \;\colon\;
  G
    \longrightarrow
  k^\times
$$

Since the [[codomain]] $k^\times$ is an [[abelian group]] we have that

1. such a [[group homomorphism]] is [[invariant]] under [[conjugation]], so that a group character descends to a [[function]] on the [[set]] of [[conjugacy classes]] of elements in $G$.

1. the collection of characters is itself an [[abelian group]] under pointwise multiplication, this is called the _[[character lattice]]_ $Hom(G,k^\times)$ of the group. Similarly the **cocharacter lattice** is $Hom(k^\times, G)$.

For more see at _[[group character]]_.

For [[topological groups]] one considers [[continuous map|continuous]] characters. Specifically, for a [[locally compact Hausdorff]] group $G$ (often further assumed to be an [[abelian group]]), a __character__ of $G$ is a continuous homomorphism to the [[circle]] group $\mathbb{R}/\mathbb{Z}$. If $G$ is [[profinite group|profinite]], then this is the same as an continuous homomorphism to the [[discrete space|discrete]] group $\mathbb{Q}/\mathbb{Z}$.  (See [MO](http://mathoverflow.net/questions/86089/two-definitions-of-character-of-topological-groups).)

#### Character of a linear representation of a group
 {#CharacterOfALinearRepresentation}

In [[representation theory]], one defines the _[[character of a linear representation]]_ $\rho\colon G\to End(V)$ to be the [[group character]] on $G$ given by $g \mapsto Tr \rho(g)$, whenever the [[trace]] in $V$ makes sense (e.g. when $V$ is [[finite-dimensional vector space|finite-dimensional]]).  Since such a function is [[invariant]] under [[conjugation action|conjugation]], one may equivalently consider it a function on the set of [[conjugacy classes]] of elements in $G$.

Sometimes we also extend a character linearly to the [[linear span|free vector space]] on the set of conjugacy classes.  This version of the character can be identified with the [[bicategorical trace]] of the identity map of the representation, considered as a $k[G]$-$k$-module.

There is a different notion of an _infinitesimal character_ in [[Harish–Chandra theory]] and also a notion of the _formal character_.

There are important formulas concerning characters in representation theory, like [[Weyl character formula]], Kirillov character formula, Demazure character formula and so on.  There is also a formula for the [[induced character]] of an [[induced representation]].

### Left and right characters on a ring over a ring

Let $A$ be a unital, not necessarily commutative, [[ring]] (or, more generally, $k$-algebra for $k$-commutative); then a monoid in a category of $A$-bimodules (which are respectively also compatibly $k$-modules), is called an $A$-ring. In other words an $A$-ring $B$ is an object in the [[coslice category]] $A\Ring$; it is thus a ring $B$ equipped with multiplication $\mu_B$ and a map $\eta: A\to B$ of rings.

A __left character__ of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (left $A$-linearity)
$\chi(\eta(a)b) = a\chi(b)$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi(b (\eta\circ\chi)(b'))$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

where we denoted multiplication in $A$ and in $B$ by concatenation. The conditions on $\chi$ can be restated as
the requirement that the map $B\otimes A\to A$ given by
$b\otimes a\mapsto \chi(b\eta(a))$ is a $B$-action extending
the left regular $A$-action (i.e. the multiplication on $A$
considered as a left action).

Dually, a __right character__ of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (right $A$-linearity)
$\chi(b\eta(a)) = \chi(b)a$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi((\eta\circ\chi)(b) b')$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

This is in turn equivalent to extending the right regular action of $A$ to the action of $B$ on $A$.

### Character of a topological space

The character $\chi(X,x)$ of a [[topological space]] $X$ at a point $x$ is the minimal cardinality of a local basis of neighborhoods of point $x$ (local [[topological basis|basis of topology]] on $X$) if it is infinite and aleph zero otherwise. The character of a topological space is the supremum of $\chi(X,x)$ when $x$ runs through $X$.

## Properties

* [[characters are cyclotomic integers]]

## Examples
 {#Examples}

* [[character table of 2D4=Dic2=Q8]]

* [[character table of 2T]]

* [[character table of 2O]]

* [[character table of 2I]]

* [[character table of GL(2,3)]]

## Related concepts

* [[irreducible character]]

* [[Weyl character formula]]

* [[Kac character formula]]

* [[Kac-Weyl character]]

* [[Cheeger-Simons differential character]]

* [[Chern root]], [[splitting principle]]

* [[character ring]]

* [[table of marks]]

## References

Character rings of [[compact Lie groups]] are discussed in

* [[Graeme Segal]], _The representation ring of a compact Lie group_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques __34__:1 (1968) 113-128 ([NUMDAM](http://archive.numdam.org/numdam-bin/fitem?id=PMIHES_1968__34__113_0))

* Masaru Takeuchi, _A remark on the character ring of a compact Lie group_, J. Math. Soc. Japan __23__:4 (1971), 555-705 ([Euclid](http://projecteuclid.org/euclid.jmsj/1259849785))

* Troels Roussauc Johansen, _Character Theory for Finite Groups and Compact Lie Groups_ [pdf](http://www.math.upb.de/~johansen/character-theory.pdf)

Discussion in the more general context of [[equivariant cohomology|equivariant]] [[complex oriented cohomology theory]] ([[transchromatic character]]) is in

* [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], _Generalized group characters and complex oriented cohomology theories_, J. Amer. Math. Soc. 13 (2000), 553-594 ([publisher](http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf))

for review see

* {#Raksit15} Arpon Raksit, _Characters in global equivariant homotopy theory_, 2015 [pdf](http://web.stanford.edu/~arpon/math/files/senior-thesis.pdf)

Examples of [[characters of linear representations]] of [[finite groups]] are discussed and listed at

* {#Montaldi08} James Montaldi, _[representations](http://www.maths.manchester.ac.uk/~jm/wiki/Representations/Representations)_, 2008

[[!redirects character]]
[[!redirects characters]]

[[!redirects character ring]]
[[!redirects character rings]]


character of a linear representation

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

#Contents#
* table of contents
{:toc}

## Idea

In [[representation theory]], one defines the _character of a [[linear representation]] $\rho\colon G\to End(V)$ to be the [[group character]] on $G$ given by $g \mapsto Tr \rho(g)$, whenever the [[trace]] in $V$ makes sense (e.g. when $V$ is [[finite-dimensional vector space|finite-dimensional]]).  Since such a function is invariant under conjugation, we may equivalently consider it a function on the set of [[conjugacy classes]] of elements in $G$.

Sometimes we also extend a character linearly to the free vector space on the set of conjugacy classes.  This version of the character can be identified with the [[bicategorical trace]] of the identity map of the representation, considered as a $k[G]$-$k$-module.

There is a different notion of an _infinitesimal character_ in [[Harish–Chandra theory]] and also a notion of the _formal character_.

There are important formulas concerning characters in representation theory, like [[Weyl character formula]], Kirillov character formula, Demazure character formula and so on.  There is also a formula for the [[induced character]] of an [[induced representation]].

## Properties

### The character homomorphism

+-- {: .num_prop #InCharZeroCharacterMorphismIsInjective}
###### Proposition
**(in [[characteristic zero]] the [[character homomorphism]] is [[injective function|injective]])**

For $G$ a [[finite group]] and $k$ a [[field]] of [[characteristic zero]], the [[character homomorphism]]

$$
  \array{
    Rep_k(G)_{/\sim}
    &\overset{\chi}{\longrightarrow}&
    k^{ConjCl(G)}
    \\
    V
      &\mapsto&
    \big(
      [g] \mapsto \chi_V(g)
    \mathrlap{
      \coloneqq \mathrm{tr}_V(g)
    \big)
    }
  }
$$

is [[injective function|injective]].

=--

(e.g. [tom Dieck 09, Theorem (2.1.3)](#tomDieck09))

### Schur inner product
 {#SchurInnerProduct}

+-- {: .num_prop #SchurInnerProductInTermsOfCharacters}
###### Proposition
**([Schur inner product](Schur's+lemma#InterpretationInCategoricalAlgebra) in terms of [[character of a linear representation|characters]])**

Let $G$ be a [[finite group]] of [[order of a group|order]] ${\vert G \vert} \in \mathbb{N}$, and let $V_1, V_2 \in Rep_k(G)$ be two [[linear representation]] over a [[ground field]] $k$ of [[characteristic zero]].

Then the [Schur inner product](Schur's+lemma#InterpretationInCategoricalAlgebra) of the two representation equals the [[convolution product]] of their [[character of a linear representation|characters]], normalized by the [[order of a group|order]] of $G$:

\[
  \label{OrthogonalityRelation}
  \left\langle
    V_1, V_2
  \right\rangle
  \;\coloneqq\;
  dim_k\Big(
    Hom\big(
      V_1, V_2
    \big)
  \Big)
  \;=\;
  \frac{1}{\left\vert G\right\vert}
  \underset{g \in G}{\sum}
  \chi_{V_1}(g^{-1})
    \cdot
  \chi_{V_2}(g)
  \,.
\]

=--

(e.g. [tom Dieck 09 (2.5)](#tomDieck09))

+-- {: .num_example #NormalizedSumOfCharacters}
###### Example
**(normalized [[sum]] of [[character of a linear representation|characters]] is [[fixed point space]]-[[dimension]])**

Let $G$ be a [[finite group]] of [[order of a group|order]] ${\vert G \vert} \in \mathbb{N}$, and let $V \in Rep_k(G)$ be a [[linear representation]] over a [[ground field]] $k$ of [[characteristic zero]].

Then the [[dimension]] of the [[fixed point space]] $V^G$ of $V$ under $G$ equals the normalized [[sum]] of the character values over all group elements:

\[
  \label{SumOverAllCharacterValues}
  dim\left( V^G \right)
  \;=\;
  \frac{1}{{\vert G\vert}}
  \underset{g \in G}{\sum}
  \chi_V(g)
  \,.
\]

In particular

1. if $V = \rho \neq \mathbf{1}$ is a non-[[trivial representation|trivial]] [[irreducible representation]] then

  $$
    \underset{g \in G}{\sum}
    \chi_{\rho}(g)
    \;=\;
    0
  $$

1. if $V = \mathbf{1}$ is the the [[trivial representation|trivial]] [[irreducible representation]] then

  $$
    \underset{g \in G}{\sum}
    \chi_{\mathbf{1}}(g)
    \;=\;
    \left\vert G\right\vert
  $$

=--

(e.g. [tom Dieck 09 (2.1)](#tomDieck09))

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

The first statement is a special case of Prop. \ref{SchurInnerProductInTermsOfCharacters} by observing that the [[fixed point space]] of a [[linear representation]] $V$ is the space of [[homomorphisms]] from the [[trivial representation|trivial]] [[irrep]]

$$
  dim\big( V^G\big)
  \;=\;
  dim\Big(
    \mathrm{Hom}\big( \mathbf{1}, V \big)
  \Big)
$$

and that the [[character of a linear representation|character]] of $\mathbf{1}$ is [[constant function|constant]] on $1 \in k$.

The second statement follows from this by [[Schur's lemma]], which says that $\langle \mathbf{1}, \rho\rangle = 0$ if the [[irrep]] $\rho \neq \mathbf{1}$.

The last statement may similarly be seen as the complementary special case of  (eq:SumOverAllCharacterValues) obtained from $\langle \mathbf{1},\mathbf{1}\rangle =1$, but of course it also follows directly from the fact that the [[character of a linear representation|character]] of the [[trivial representation|trivial]] [[irrep]] is [[constant function|constant]] on 1.

=--

### Schur orthogonality

See at *[[Schur orthogonality relations]]*.

### Special character values
 {#SpecialCharacterValues}

+-- {: .num_prop #CharactersAreCyclotomicIntegers}
###### Proposition
**([[characters are cyclotomic integers]])

Let $G$ be a [[finite group]], and let $V$ be a [[finite-dimensional vector space|finite-dimensional]] [[linear representation]] over a [[ground field]] $k$

Then the values of the [[character of a representation|character]] $\chi_V \colon G \to k$ of $V$ are [[cyclotomic integers]] over $k$, for some [[root of unity]].

=--

(see e.g. [Naik](#characters+are+cyclotomic+integers#Naik))

In particular it follows that:

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

If the [[ground field]] $k$ in Prop. \ref{CharactersAreCyclotomicIntegers} has [[characteristic zero]], then a [[character of a linear representation]] which takes values in the [[rational numbers]] $\mathbb{Q} \hookrightarrow k$ in fact already takes values in the [[integers]] $\mathbb{Z} \hookrightarrow\mathbb{Q} \hookrightarrow k$.

=--

(see e.g. [Yang](#characters+are+cyclotomic+integers#Yang))

## Related concepts

* [[equivariant K-theory]]

* [[fractional D-brane]]

## References

* {#tomDieck09} [[Tammo tom Dieck]], chapter 2 of _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

For more see the references at _[[representation theory]]_.

Character rings of [[compact Lie groups]] are discussed in

* [[Graeme Segal]], _The representation ring of a compact Lie group_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques
January 1968, Volume 34, Issue 1, pp 113-128 ([NUMDAM](http://archive.numdam.org/numdam-bin/fitem?id=PMIHES_1968__34__113_0))

* Masaru Tackeuchi, _A remark on the character ring of a compact Lie group_, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 ([Euclid](http://projecteuclid.org/euclid.jmsj/1259849785))

* Troels Roussauc Johansen, _Character Theory for Finite Groups and Compact Lie Groups_ [pdf](http://www.math.upb.de/~johansen/character-theory.pdf)

Discussion in the more general context of [[equivariant cohomology|equivariant]] [[complex oriented cohomology theory]] ([[transchromatic character]]) is in

* [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], _Generalized group characters and complex oriented cohomology theories_, J. Amer. Math. Soc. 13 (2000), 553-594 ([publisher](http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf))

for review see

* {#Raksit15} Arpon Raksit, _Characters in global equivariant homotopy theory_, 2015 [pdf](http://web.stanford.edu/~arpon/math/files/senior-thesis.pdf)

Examples of [[characters of linear representations]] of [[finite groups]] are discussed and listed at

* {#Montaldi08} [[James Montaldi]], _[representations](http://www.maths.manchester.ac.uk/~jm/wiki/Representations/Representations)_, 2008

[[!redirects characters of linear representations]]

[[!redirects character of a representation]]
[[!redirects characters of representations]]

[[!redirects character homomorphism]]
[[!redirects character homomorphisms]]

[[!redirects character morphism]]
[[!redirects character morphisms]]

[[!redirects character table]]
[[!redirects character tables]]


character table of 2D4=Dic2=Q8

**[[linear representation theory]] of [[binary dihedral group]] $2 D_4$**

$=$ [[dicyclic group]] $Dic_2$ $=$ [[quaternion group]] $Q_8$

$\,$

[[group order]]: ${\vert 2D_4\vert} = 8$

| [[conjugacy classes]]: |  1 |  2  |  4A | 4B | 4C |
|---------------------|----|-----|-----|----|----|
| their [[cardinality]]: | 1 |  1 | 2  | 2 | 2 |

$\,$

|  |  |
|--|--|
| [[splitting field]] | $\mathbb{Q}(\alpha, \beta)$ with $\alpha^2 + \beta^2 = -1$ |
| [[field]] generated by [[characters]] | $\mathbb{Q}$ |

**[[character table]] over [[splitting field]] $\mathbb{Q}(\alpha,\beta)$/[[complex numbers]] $\mathbb{C}$**

| [[irrep]] |  1 |  2  |  4A | 4B | 4C | [[Schur index]] |
|--------------------------------|----|-----|-----|----|----|----|
| $\rho_1$ | 1 |1 | 1 | 1  | 1 | 1 |
| $\rho_2$ | 1 |1 | -1 | 1  | -1 | 1 |
| $\rho_3$ | 1 |1 | 1 | -1  | -1 | 1 |
| $\rho_4$ | 1 |1 | -1 | -1  | 1 | 1 |
| $\rho_5$ | 2 | -2 | 0 | 0  | 0 | 2 |

**[[character table]] over [[rational numbers]] $\mathbb{Q}$/[[real numbers]] $\mathbb{R}$**

| [[irrep]] |  1 |  2  |  4A | 4B | 4C |
|--------------------------------|----|-----|-----|----|----|
| $\rho_1$ | 1 |1 | 1 | 1  | 1 |
| $\rho_2$ | 1 |1 | -1 | 1  | -1 |
| $\rho_3$ | 1 |1 | 1 | -1  | -1 |
| $\rho_4$ | 1 |1 | -1 | -1  | 1 |
| $\rho_5 \oplus \rho_5$ | 4 | -4 | 0 | 0  | 0 |

**References**

* [[GroupNames]], _[Q8](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Q8.html)_,

* [[Groupprops]], _[Linear representation theory of dicyclic groups](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dicyclic_groups)_

* [[James Montaldi]], _[Real representations -- Binary cubic -- Q8](http://www.maths.manchester.ac.uk/~jm/wiki/Representations/BinaryCubic)_

category: character tables


character table of 2I

**[[linear representation theory]] of [[binary icosahedral group]] $2 I$**

$\,$

[[group order]]: ${\vert 2I\vert} = 120$

| [[conjugacy classes]]: |  1 |  2  |  3 | 4 | 5A | 5B | 6 | 10A | 10B |
|------------------------|----|------|------|---|---|---|---|--|--|
| their [[cardinality]]: | 1 |  1 | 20  | 30 | 12 | 12 | 20 | 12 | 12 |

let $\phi \coloneqq \tfrac{1}{2}( 1 + \sqrt{5} )$ (the [[golden ratio]])

**[[character table]] over the [[complex numbers]] $\mathbb{C}$**

| [[irrep]] | 1 |  2  |  3 | 4 | 5A | 5B | 6 | 10A | 10B |
|-----------|---|-----|----|---|----|----|---|-----|-----|
| $\rho_1$  | 1 |  1  |  1 | 1 |  1 | 1  | 1 |  1  |  1  |
| $\rho_2$  | 2 | -2  | -1 | 0 | $\phi - 1$ | $-\phi$ | 1 | $\phi$ | $1 - \phi$ |
| $\rho_3$  | 2 | -2  | -1 | 0 | $-\phi$ | $\phi - 1$ | 1 | $1-\phi$ | $\phi$ |
| $\rho_4$  | 3 | 3  | 0 | -1 | $1 - \phi$ | $\phi$ | 0 | $\phi$ | $1-\phi$ |
| $\rho_5$  | 3 | 3  | 0 | -1 | $\phi$ | $1-\phi$ | 0 | $1-\phi$ | $\phi$ |
| $\rho_6$  | 4 | 4 |  1 | 0 | -1  | -1 | 1 |  -1 | -1 |
| $\rho_7$  | 4 | -4 |  1 | 0 | -1  | -1 | -1 | 1 | 1 |
| $\rho_8$  | 5 | 5 |  -1 | 1 | 0  | 0 | -1 | 0  | 0  |
| $\rho_9$  | 6 | -6 |   0 | 0 | 1  | 1 |  0 | -1 | -1 |

**References**

* [[GroupNames]], _SL(2,5)_

* Bockland, _Character tables and McKay quivers_ ([pdf](https://staff.fnwi.uva.nl/r.r.j.bocklandt/notes/kleinian.pdf))

category: character tables


character table of 2O

**[[linear representation theory]] of [[binary octahedral group]] $2 O$**

$\,$

[[group order]]: ${\vert 2O\vert} = 48$

| [[conjugacy classes]]: |  1 |  -1  |  $i$ | a | c | e | f | g |
|------------------------|----|------|------|---|---|---|---|--|
| their [[cardinality]]: | 1 |  1 | 6  | 8 | 8 | 6 | 6 | 12 |

**[[character table]] over the [[complex numbers]] $\mathbb{C}$**

| [[irrep]]  |  1 |  -1  |  $i$ | a | c | e | f | g |
|-----------|----|------|------|---|---|---|---|--|
| $\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\rho_2$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 |
| $\rho_3$ | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 |
| $\rho_4$ | 3 | 3 | -1 | 0 | 0 | 1 | 1 | -1 |
| $\rho_5$ | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 1 |
| $\rho_6$ | 2 | -2 | 0 | 1 | -1 | $\sqrt{2}$ | $-\sqrt{2}$ | 0 |
| $\rho_7$ | 2 | -2 | 0 | 1 | -1 | $-\sqrt{2}$ | $\sqrt{2}$ | 0 |
| $\rho_8$ | 4 | -4 | 0 | -1 | 1 | 0 | 0 | 0 |

**[[character table]] over the [[real numbers]] $\mathbb{R}$**

| [[irrep]]  |  1 |  -1  |  $i$ | a | c | e | f | g |
|-----------|----|------|------|---|---|---|---|--|
| $\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\rho_2$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 |
| $\rho_3$ | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 |
| $\rho_4$ | 3 | 3 | -1 | 0 | 0 | 1 | 1 | -1 |
| $\rho_5$ | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 1 |
| $\rho_6 \oplus \rho_6$ | 4 | -4 | 0 | 2 | -2 | $2 \sqrt{2}$ | $-2 \sqrt{2}$ | 0 |
| $\rho_7 \oplus \rho_7$ | 4 | -4 | 0 | 2 | -2 | $-2 \sqrt{2}$ | $2 \sqrt{2}$ | 0 |
| $\rho_8 \oplus \rho_8$ | 8 | -8 | 0 | -2 | 2 | 0 | 0 | 0 |

**References**

* [[GroupNames]], _CSU(2,3)_

* [[James Montaldi]], _[Real representations -- Binary cubic -- 2O](http://www.maths.manchester.ac.uk/~jm/wiki/Representations/BinaryCubic#BinOct)_

* [[Groupprops]], _[Linear representation theory of binary octahedral group](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_binary_octahedral_group)_

* Bockland, _Character tables and McKay quivers_ ([pdf](https://staff.fnwi.uva.nl/r.r.j.bocklandt/notes/kleinian.pdf))

category: character tables


character table of 2T

**[[linear representation theory]] of [[binary tetrahedral group]] $2 T$**

$\,$

[[group order]]: $\vert 2T\vert = 24$

| [[conjugacy classes]]: |  1 |  -1  |  $i$ | a | b | c | d |
|------------------------|----|------|------|---|---|---|---|
| their [[cardinality]]: | 1 |  1 | 6  | 4 | 4 | 4 | 4 |

$\,$

let $\zeta_3$ be a [[root of unity|third root of unity]], $(\zeta_3)^3 = 1$

e.g. $\zeta_3 = \tfrac{1}{2}(-1 + \sqrt{3} i)$,
notice that $\zeta_3 + \zeta_3^2 = 1$

**[[character table]] over the [[complex numbers]] $\mathbb{C}$**

| [[irrep]] |  1 |  -1  |  $i$ | a | b | c | d |
|-----------|----|------|------|---|---|---|---|
| $\rho_1$  |  1 |   1  |   1  | 1 | 1 | 1 | 1 |
| $\rho_2$  |  1 |   1  |   1  | $\zeta_3$ | $\zeta_3^2$ | $\zeta_3^2$ | $\zeta_3$ |
| $\rho_2^\ast$  |  1 |   1  |   1  | $\zeta^2_3$ | $\zeta_3$ | $\zeta_3$ | $\zeta_3^2$ |
| $\rho_3$  |  3 |   3  |   -1  | 0 | 0 | 0 | 0 |
| $\rho_4$  |  2 |  -2  |  0   | $\zeta_3$ | $\zeta_3^2$ | $-\zeta_3^2$ | $-\zeta_3$  |
| $\rho_4^\ast$  |  2 |  -2  |  0   | $\zeta_3^2$ | $\zeta_3$ | $-\zeta_3$ | $-\zeta_3^2$  |
| $\rho_5 $  |  2 |  -2  |  0   | 1 | 1 | -1 | -1  |

**[[character table]] over the [[real numbers]] $\mathbb{R}$**

| [[irrep]] |  1 |  -1  |  $i$ | a | b | c | d |
|-----------|----|------|------|---|---|---|---|
| $\rho_1$  |  1 |   1  |   1  | 1 | 1 | 1 | 1 |
| $\rho_2 \oplus \rho_2^\ast$  |  2 |   2  |   2  | -1 | -1 | -1 | -1 |
| $\rho_3$  |  3 |   3  |   -1  | 0 | 0 | 0 | 0 |
| $\rho_4 \oplus \rho_4^\ast$  |  4 |  -4  |  0   | -1 | -1 | 1 | 1  |
| $\rho_5 \oplus \rho_5$  |  4 |  -4  |  0   | 2 | 2 | -2 | -2  |

**References**

* [[GroupNames]], _SL(2,3)_

* [[James Montaldi]], _[Real representations -- Binary cubic -- 2T](http://www.maths.manchester.ac.uk/~jm/wiki/Representations/BinaryCubic#BinTet)_

* Bockland, _Character tables and McKay quivers_ ([pdf](https://staff.fnwi.uva.nl/r.r.j.bocklandt/notes/kleinian.pdf))

category: character tables


character table of GL(2,3)

**[[linear representation theory]] of the [[finite group|finite]] [[general linear group]] [[GL(2,3)]]**

$\,$

[[group order]]: ${\vert GL(2,3)\vert} = 48$

| [[conjugacy classes]]: |  1 |  2A  |  2B | 3 | 4 | 6 | 8A | 8B |
|------------------------|----|------|------|---|---|---|---|--|
| their [[cardinality]]: | 1 |  1 | 12  | 8 | 6 | 8 | 6 | 6 |

**[[character table]] over the [[complex numbers]] $\mathbb{C}$**

| [[irrep]]  | 1 |  2A  |  2B | 3 | 4 | 6 | 8A | 8B |
|-----------|----|------|------|---|---|---|---|--|
| $\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\rho_2$ | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 |
| $\rho_3$ | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 |
| $\rho_4$ | 2 | -2 | 0 | -1 | 0 | 1 | $-\sqrt{-2}$ | $\sqrt{-2}$ |
| $\rho_5$ | 2 | -2 | 0 | -1 | 0 | 1 | $\sqrt{-2}$ | $-\sqrt{-2}$ |
| $\rho_6$ | 3 | 3  | -1 | 0 | -1 | 0 | 1 |  1 |
| $\rho_7$ | 3 | 3 | 1 | 0 | -1 | 0 | -1 | -1 |
| $\rho_8$ | 4 | -4 | 0 | 1 | 0 | -1 | 0 | 0 |

**[[character table]] over the [[rational numbers]] $\mathbb{Q}$ and [[real numbers]] $\mathbb{R}$**

| [[irrep]]  | 1 |  2A  |  2B | 3 | 4 | 6 | 8A | 8B |
|-----------|----|------|------|---|---|---|---|--|
| $\rho_1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\rho_2$ | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 |
| $\rho_3$ | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 0 |
| $\rho_4 \oplus \rho_5$ | 4 | -4 | 0 | -2 | 0 | 2 | 0 | 0 |
| $\rho_6$ | 3 | 3  | -1 | 0 | -1 | 0 | 1 |  1 |
| $\rho_7$ | 3 | 3 | 1 | 0 | -1 | 0 | -1 | -1 |
| $\rho_8$ | 4 | -4 | 0 | 1 | 0 | -1 | 0 | 0 |

**References**

* [[GroupNames]], _GL(2,3)_

category: character tables


characteristic

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

# The characteristic of a field (etc)
* table of contents
{: toc}

## Idea

It is well known that you cannot divide by [[zero]], lest you be doomed to [[trivial ring|triviality]].  Conversely, in a [[field]], you *can* divide by anything *except* zero.  But this rule can be misleading, since it\'s possible that (even) an ordinary number can be zero when you don\'t expect it!  The _characteristic_ of a field states when (if ever) this happens.

It is straightforward to generalise from fields to other [[rings]], and even [[rigs]]. See also [[characteristic zero]].

## Definition

### For rings

Let $K$ be a [[rig]] (possibly a [[ring]], possibly a [[commutative ring]], possibly even a [[field]]).  Then there exists a unique [[homomorphism]] $\phi_K\colon \mathbb{N} \to K$ to $K$ from the [[initial object|initial]] rig, which is the rig $\mathbb{N}$ of [[natural numbers]].  The [[kernel]] of $\phi_K$ is an [[ideal]] of $\mathbb{N}$, which (by a well-known property of $\mathbb{N}$) is a [[principal ideal]] with a unique generator.  This generator is the __characteristic__ of $K$, denoted $\char K$.

If $K$ is a ring, then we may use $\phi_K\colon \mathbb{Z} \to K$ instead, where $\mathbb{Z}$ is the ring of [[integers]].  However, in this case, the kernel will usually have two generators, in which case we pick the positive one to get the same result as above.

### For $E_\infty$-rings

The concept of the characteristic has been generalized to [[E-∞ rings]] ([Szymik 12](#Szymik12), [Szymik 13](#Szymik13), [Baker 14](#Baker14)).

## Properties

If $n$ is a [[natural number]], then we suppress mention of $\phi_K$ to think of $n$ as an element of $K$.  If $K$ is a ring, then we do the same for a negative [[integer]] $n$.  We then have that $n = 0$ in $K$ if and only if $n$ is a multiple of $\char K$.

The characteristic of a field must be either [[zero]] or a [[prime number]]. Basically, this is because the kernel of $\phi_K$, for $K$ a field, must be a [[prime ideal]]. Similarly, the characteristic of an [[integral domain]] must be either zero or a prime number.

Every rig with positive characteristic is in fact a ring, since we have $\char K - 1 = -1$.  In other words, any rig other than a ring must have characteristic zero (although many rings also have that characteristic).

If there is any [[homomorphism]] at all between two fields, then they have the same characterstic. In other words, any [[field extension|extension]] of a field keeps the same characteristic.

## Examples

If $n$ is a positive natural number, then the characteristic of $\mathbb{N}/n = \mathbb{Z}/n$ is $n$.  This ring is always a [[commutative ring]], and it is a [[field]] if and only if $n$ is prime, in which case it is the [[prime field]] $\mathbb{F}_n$.  More generally, every [[finite field]] has positive prime characteristic.

For $n = 0$, $\mathbb{N}/0 = \mathbb{N}$, $\mathbb{Z}/0 = \mathbb{Z}$, and the prime field $\mathbb{F}_0 = \mathbb{Q}$ (the field of [[rational numbers]]) are no longer all the same, but they still all have characteristic $0$.  Every [[ordered field]] has characteristic $0$.  The [[real numbers]] and [[complex numbers]] each form fields of characteristic $0$.

## Related concepts

* [[freshman's dream]]

* [[characteristic zero]], [[positive characteristic]]

## References

Discussion for [[E-∞ rings]]:

* {#Szymik12} [[Markus Szymik]], _Commutative S-algebras of prime characteristics and applications to unoriented bordism_ ([arXiv:1211.3239](http://arxiv.org/abs/1211.3239))

* {#Szymik13} [[Markus Szymik]],  _String bordism and chromatic characteristics_ ([arXiv:1312.4658](http://arxiv.org/abs/1312.4658))

* {#Baker14} [[Andrew Baker]], _Characteristics for $E_\infty$ ring spectra_ ([arXiv:1405.3695](http://arxiv.org/abs/1405.3695))

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