+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The fundamental [[theorem]] of [[rational homotopy theory]] modeled by [[dgc-algebras]].

## Preliminaries

+-- {: .num_defn #NilpotententFiniteRationalHomotopyTypes}
###### Definition
**(nilpotent and finite rational homotopy types)**

Write

\[
  \label{NilpotentQFiniteHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the [[full subcategory]] of the [[classical homotopy category]]  ([[homotopy category]] of the [[classical model structure on simplicial sets]]) on those [[homotopy types]] $X$ which are

* [[connected topological space|connected]]: $\pi_0(X) = \ast$;

* [[nilpotent space|nilpotent]]: the [[fundamental group]] $\pi_1(X)$ is a [[nilpotent group]] and the higher [[homotopy groups]] are [[nilpotent modules]] over it;

* rationally [[of finite type]]: the [[rational cohomology]]-[[cohomology group|groups]] are [[finite number|finite]]-[[dimension|dimensional]], $dim_{\mathbb{Q}}\big( H^n(X;,\mathbb{Q}) \big) \lt \infty$, for all $n \in \mathbb{N}$.

and

\[
  \label{NilpotentFiniteRationalHomotopyTypes}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the further [[full subcategory]] on those [[homotopy types]] that are already [[rational spaces|rational]].

Similarly, write

\[
  \label{NilpotentFiniteTypedgcAlgebras}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)_{fin}^{\geq 1}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)
\]

for the [[full subcategory]] of the [[homotopy category]] of the [[projective model structure on connective dgc-algebras]] on those [[dgc-algebras]] $A$ over the [[rational numbers]] which are

* connected: $H^0(A) \simeq \mathbb{Q}$

* [[finite type]]: the [[cochain cohomology]] [[cohomology group|groups]] are [[finite number|finite]] [[dimension|dimensional]], $dim_{\mathbb{Q}}\big( H^n(A) \big) \lt \infty$, for all $n \in \mathbb{N}$.

=--

([Bousfield-Gugenheim 76, 9.2](#BousfieldGugenheim76))

## Statement

+-- {: .num_pro #FundamentalTheoremOfdgAlgebraicRationalHomotopyTheory}
###### Proposition
**([[fundamental theorem of dg-algebraic rational homotopy theory]])**

The [[derived adjunction]]

$$
  Ho
  \left(
    \big(
      DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
    \big)^{op}_{proj}
  \right)
  \underoverset
    {
      \underset
        {\;\;\; \mathbb{R} exp_{PL} \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot}
  Ho
  \big(
    SimplicialSets_{Qu}
  \big)
$$

of the [[Quillen adjunction between simplicial sets and connective dgc-algebras]] (whose [[left adjoint]] is the [[PL de Rham complex]]-[[functor]]) has the following properties:

* on connected, nilpotent rationally finite homotopy types $X$ (eq:NilpotentQFiniteHomotopyTypes) the [[derived adjunction unit]] is [[rationalization]]

  $$
    \array{
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
    &
      \overset{
      }{\longrightarrow}
    &
    Ho
    \big(
       SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
    \\
    X
    &\mapsto&
    \mathbb{R}\exp_{PL} \circ \Omega^\bullet_{PLdR}(X)
    }
  $$

  $$
    X
    \underoverset
      {\eta_X^{der}}
      {rationalization}
      {\longrightarrow}
    \mathbb{R}\exp_{PL} \circ \Omega^\bullet_{PLdR}(X)
  $$

* on the [[full subcategories]] of nilpotent and finite rational homotopy types from Def. \ref{NilpotententFiniteRationalHomotopyTypes} it restricts to an [[equivalence of categories]]:

  $$
    Ho
    \left(
      \big(
        DiffGradedCommAlgebras^{\geq 0}_{\mathbb{Q}}
      \big)^{op}_{proj}
    \right)^{\geq 1}_{fin}
    \underoverset
      {
        \underset
          {\;\;\; \mathbb{R} exp_{PL} \;\;\;}
          {\longrightarrow}
      }
      {
        \overset
          {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
          {\longleftarrow}
      }
      {\simeq}
    Ho
    \big(
      SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  $$

=--

([Bousfield-Gugenheim 76, Theorems 9.4 & 11.2](#BousfieldGugenheim76))

## Change of scalars
 {#ChangeOfScalars}

Often it is desireable to work with dgc-algebras not over the [[rational numbers]] but over the [[real numbers]], because these relate to [[de Rham complex|de Rham theory]] (e.g.: [[the PL de Rham complex of a smooth manifold is equivalent to the de Rham complex]]). While a [[PL de Rham complex]]-[[Quillen adjunction]] $\Omega^\bullet_{\mathrm{P}\!k\!\mathrm{LdR}} \dashv \exp_{\mathrm{P}\!k\!\mathrm{L}}$ ("piecewise $k$-linear") exists over all [[ground fields]] $k$ of [[characteristic zero]], with induced [[derived adjunction]]

$$
  Ho
  \Big(
    \big(
      DGCAlgebras^{\geq 0}_{k}
    \big)^{op}_{proj}
  \Big)
  \underoverset
    {\underset{\mathbb{R} exp_{\mathrm{P}\!k\!\mathrm{L}}}{\longrightarrow}}
    {\overset{\mathbb{L} \Omega^\bullet_{\mathrm{P}\!k\!\mathrm{LdR}}}{\longleftarrow}}
    {\bot}
  Ho
  \big(
    SimplicialSets_{Qu}
  \big)
  \,,
$$

this does not model $k$-[[localization]] of [[homotopy type|spaces]] unless $k = \mathbb{Q}$. However, it does still relate to [[rationalization]] under [[extension of scalars]], given by the [[derived adjunction]] (via [this Prop.](model+structure+on+dg-algebras#ExtensionOfScalarsQuillenAdjunction))
$$
  Ho
  \Big(
    \big(
      DGCAlgebras^{\geq 0}_{k}
    \big)^{op}_{proj}
  \Big)
  \underoverset
    {
      \underset{
        \mathbb{R}\big( (-)\otimes_{\mathbb{Q}}\mathbb{R} \big)
      }{
        \longrightarrow
      }
    }
    {
      \overset{
        \mathbb{L} res_{\mathbb{Q}}
      }{
      \longleftarrow
      }
    }
    {\bot}
  Ho
  \Big(
    \big(
      DGCAlgebras^{\geq 0}_{\mathbb{Q}}
    \big)^{op}_{proj}
  \Big)
  \,,
$$
in that the following holds:

\begin{proposition}\label{DerivedPLdeRhamFunctorsRelatedByChangeOfScalars}
  For $k$ be a [[field]] of [[characteristic zero]], the following [[diagram]] of [[derived functors]] [[commuting diagram|commutes]] up to [[natural isomorphism]]:

\begin{tikzcd}
    \mathrm{Ho}
    \Big(
      \big(
        \mathrm{dgcAlg}^{\geq 0 }_{\color{blue}\mathbb{Q}}
      \big)^{\mathrm{op}}_{\mathrm{proj}}
    \Big)
    \ar[
      dd,
      "{
        \mathbb{R}
        \big(
          (-) \otimes_{\mathbb{Q}} \mathbb{R}
        \big)
      }"{left}
    ]
    &&
    \;\;
    \mathrm{Ho}
    \big(
      \mathrm{sSets}_{\mathrm{Qu}}
    \big)^{\mathrm{fin}_{\mathbb{Q}}}
    \ar[
      ll,
      "
        \mathbb{L}
        \Omega^\bullet_{\mathrm{P}{\color{blue}\mathbb{Q}}\mathrm{LdR}}
      "{above}
    ]
    \ar[
      dd,-,
      shift right=1pt
    ]
    \ar[
      dd,-,
      shift left=1pt
    ]
    \\
    \\
    \mathrm{Ho}
    \Big(
    \big(
      \mathrm{dgcAlg}^{\geq 0}_{\color{blue}\mathbb{R}}
    \big)^{\mathrm{op}}_{\mathrm{proj}}
    \Big)
    &&
    \;\;
    \mathrm{Ho}
    \big(
      \mathrm{sSets}_{\mathrm{Qu}}
    \big)^{\mathrm{fin}_{\mathbb{Q}}}
    \ar[
      ll,
      "
        \mathbb{L}
        \Omega^\bullet_{\mathrm{P}{\color{blue}\mathbb{R}}\mathrm{LdR}}
      "{below}
    ]
\end{tikzcd}

\end{proposition}

This is effectivley the statement of [Bousfield&Gugenheim 1976, Lem. 11.7](#BousfieldGugenheim76).

This state of affairs may be recast as follows ([FSS 2020](#FSS20)):

For any [[field]] $k$ of [[characteristic zero]], abbreviate

$$
  L_k
  \;\coloneqq\;
  \mathbb{R} exp_{\mathrm{P}k\mathrm{L}}
    \circ
  \mathbb{L} \Omega^\bullet_{\mathrm{P}kLdR}
  \,,
$$

keeping in mind that this is a localization of spaces only if $k = \mathbb{Q}$.

Then for $X, A \in Ho(sSet)^{fin_{\mathbb{Q}}}_{\geq 1, nil}$ a [[pair]] of [[connected homotopy type|connected]] [[nilpotent homotopy type|nilpotent]] [ℚ-finite](finite+type#InRationalHomotopyTheory) [[homotopy types]], define the *$k$-[[Chern-Dold character]]* on the [[non-abelian cohomology|non-abelian]] $A$-cohomology of $X$ to be the [[cohomology operation]] induced by the [[derived adjunction unit]] of the [[PL de Rham complex|PL de Rham adjunction]] ([this Prop.](PL+de+Rham+complex#QuillenAdjunctionBetweenSimplicialSetsAndConnectivedgcAlgebras)):

\[
  \label{ChernDoldCharacter}
  ch^k_A(X)
  \;\colon\;
  H(X;\, A)
  \;\xrightarrow{
    \;\;
    H(X;\, \mathbb{D}\eta^{\mathrm{P}k\mathrm{L}_A})
    \;\;
  }\;
  H(X;\, L_k A)
  \,.
\]

Moreover, define *extension of scalars on [[non-abelian cohomology|non-abelian]] [[rational cohomology]]* to be the comoposite

\[
  \label{ExtensionOfScalarsOnChomolology}
  \array{
    H(X;\, L_{\mathbb{Q}}A)
    &\xrightarrow{ (-) \otimes_{{}_{\mathbb{Q}}} k   }&
    H(X;\, L_{k}A)
    \\
    {}^{\mathllap{
      \widetilde{(-)}
    }}
    \big\downarrow {}^{\mathrlap{\simeq}}
    &&
    {}^{\mathllap{\simeq}}
    \big\uparrow
    {}^{\mathrlap{
      \widetilde{(-)}
    }}
    \\
    H
    \big(
      \mathbb{D}\Omega^\bullet_{\mathrm{P}\mathbb{Q}\mathrm{LdR}}(X);
      \,
      \mathbb{D}\Omega^\bullet_{\mathrm{P}\mathbb{Q}\mathrm{LdR}}(A)
    \big)
    &
    \xrightarrow{
      \mathbb{D}
      \big(
        (-) \otimes_{{}_{\mathbb{Q}}} k
      \big)
    }
    &
    H
    \big(
      \mathbb{D}\Omega^\bullet_{\mathrm{P}k\mathrm{LdR}}(X);
      \,
      \mathbb{D}\Omega^\bullet_{\mathrm{P}k\mathrm{LdR}}(A)
    \big)
  }
\]

(where $H(-;\,-) \coloneqq Ho(-,\,-)$ denotes [[hom-sets]] in the respective [[homotopy category of a model category|homotopy category]])

of:

1. the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the [[derived adjunction|derived]] [[PL de Rham complex|PL de Rham adjunction]];

1. the corresponding hom-component of the [[derived functor]] of [[extension of scalars]] ([this Prop.](model+structure+on+dg-algebras#ExtensionOfScalarsQuillenAdjunction)).

(This is essentially the construction of "tensoring a homotopy type with $\mathbb{R}$" that is mentioned in [DGMS 1975, Footnote 5](#DGMS75).)

Then:

\begin{prop}\label{kChernDoldCharacterFactorsThroughRationalChernDoldCharacter}
  The $k$-Chern-Dold character (eq:ChernDoldCharacter)
  factors through the rational Chern-Dold character via the extension-of-scalars-transformation (eq:ExtensionOfScalarsOnChomolology).

$$
  ch^k_A(X)
  \;=\;
  \big(
    (-)\otimes_{{}_{\mathbb{Q}}} k
  \big)
  \,\circ\,
  ch^{\mathbb{Q}}_A(X)
  \,.
$$
\end{prop}
\begin{proof}
Consider the following [[diagram]] of [[hom-sets]] (shown for $k = \mathbb{R}$, just for definiteness):

\begin{imagefromfile}
    "file_name": "RationalAndRealCharacterMap20210708.jpg",
    "width": 740,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(diagram from [FSS 20](#FSS20))"
\end{imagefromfile}

Here:

* the [[commuting diagram|commutativity]] of the bottom part is Prop. \ref{DerivedPLdeRhamFunctorsRelatedByChangeOfScalars};

* the triangle on the left as well as as the outer diagram [[commuting diagram|commute]] by basic properties of [[adjoint functors]] ([this Lemma](geometry+of+physics+--+categories+and+toposes#ReExpressingMiddleFunctorInAdjointTriple));

* the square on the right commutes by definition (eq:ExtensionOfScalarsOnChomolology);

Together this implies that the top rectangle commutes, which is the claim to be shown.
\end{proof}

## Related concepts

* [[fundamental theorem of equivariant dg-algebraic rational homotopy theory]]

* [[real homotopy theory]]

## References

The full-blown equivalence first appears in

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], *[[On PL deRham theory and rational homotopy type]]*, Memoirs of the AMS **179** (1976) [[ams:memo-8-179](https://bookstore.ams.org/memo-8-179)]

Concise review (without [[model category]]-theory, but discussion as an [[sSet]]-[[enriched adjunction]] nonetheless) and generalization both to [[Borel-equivariant rational homotopy theory]] (of [[covering spaces]] of non-[[nilpotent spaces]]), as well as to [[real homotopy theory]]:

* {#BrownSzczarba95} [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*,  Elsevier, 1995 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009))

Further related discussion over the [[real numbers]]:

* {#DGMS75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], *Real homotopy theory of Kähler manifolds*,
Inventiones mathematicae volume 29, pages 245–274 (1975) ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

Re-derivation in a context of [[derived algebraic geometry]]:

* [[Jacob Lurie]], Prop. 1.3.3, Cor. 1.3.4 in DAG XIII: _[[Rational and p-Adic Homotopy Theory]]_, 2011 ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIII.pdf), [[Lurie_RationalHomotopyTheory.pdf:file]])

Review and interpretation in terms of [[non-abelian cohomology|non-abelian]] [[Chern-Dold character]]-theory:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!redirects fundamental theorem of dgc-algebraic rational homotopy theory]]


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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_theorem #FundamentalTheoremOfFinitelyGeneratedAbelianGroups}
###### Theorem
**(fundamental theorem of finitely generated abelian groups)**

Every [[finitely generated object|finitely generated]] [[abelian group]] $A$ is [[isomorphism|isomorphic]] to a [[direct sum]] of [[p-primary groups|p-primary]] [[cyclic groups]] $\mathbb{Z}/p^k \mathbb{Z}$ (for $p$ a [[prime number]] and $k$ a [[natural number]] ) and copies of the [[infinite cyclic group]] $\mathbb{Z}$:

$$
  A
  \;\simeq\;
  \mathbb{Z}^n
   \oplus
  \underset{i}{\bigoplus}
   \mathbb{Z}/p_i^{k_i} \mathbb{Z}
  \,.
$$

The summands of the form  $\mathbb{Z}/p^k \mathbb{Z}$ are also called the _[[p-primary group|p-primary]] components_ of $A$. Notice that the $p_i$ need not all be distinct.

**fundamental theorem of finite abelian groups**:

In particular every [[finite group|finite]] [[abelian group]] is of this form  for $n = 0$, hence is a [[direct sum]] of [[cyclic groups]].

**fundamental theorem of cyclic groups**:

In particular every [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ is a [[direct sum]] of cyclic groups of the form

$$
  \mathbb{Z}/n\mathbb{Z}
    \simeq
  \underset{i}{\bigoplus} \mathbb{Z}/ p_i^{k_i} \mathbb{Z}
$$

where all the $p_i$ are distinct and $k_i$ is the maximal power of the [[prime factor]] $p_i$ in the prime decomposition of $n$.

Specifically, for each natural number $d$ dividing $n$ it contains $\mathbb{Z}/d\mathbb{Z}$ as the [[subgroup]] generated by $n/d \in \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$. In fact the [[lattice of subgroups]] of $\mathbb{Z}/n\mathbb{Z}$ is the [[formal dual]] of the lattice of natural numbers $\leq n$ ordered by inclusion.

=--

(e.g. [Roman 12, theorem 13.4](#Roman12), [Navarro 03](#Navarro03)) for cyclic groups e.g. ([Aluffi 09, pages 83-84](#Aluffi09))

This is a special case of the _[[ structure theorem for finitely generated modules over a principal ideal domain]]_.

## Examples

The following examples may be useful for illustrative or instructional purposes.

+-- {: .num_example #TwoFiniteGroupsOfOrderp2}
###### Example

For $p$ a [[prime number]], there are, up to [[isomorphism]], two [[abelian groups]] of [[order of a group|order]] $p^2$, namely

$$
  (\mathbb{Z}/p\mathbb{Z}) \oplus (\mathbb{Z}/p\mathbb{Z})
$$

and

$$
  \mathbb{Z}/p^2 \mathbb{Z}
  \,.
$$

For $p = 2$ the first is the [[Klein 4-group]].

=--

+-- {: .num_example}
###### Example

For $p_1$ and $p_2$ two distinct [[prime numbers]], $p_1 \neq p_2$, then there is, up to isomorphism, precisely one [[abelian group]] of order $p_1 p_2$, namely

$$
  \mathbb{Z}/p_1 \mathbb{Z}
  \oplus
  \mathbb{Z}/p_2 \mathbb{Z}
  \,.
$$

This is equivalently the [[cyclic group]]

$$
  \mathbb{Z}/p_1 p_2 \mathbb{Z}
    \simeq
  \mathbb{Z}/p_1 \mathbb{Z}
    \oplus
  \mathbb{Z}/p_2 \mathbb{Z}
  \,.
$$

The isomorphism is given by sending $1$ to $(p_2,p_1)$.

=--

+-- {: .num_example }
###### Example

Moving up, for two distinct prime numbers $p_1$ and $p_2$, there are exactly two abelian groups of order $p_1^2 p_2$, namely $(\mathbb{Z}/p_1 \mathbb{Z})\oplus (\mathbb{Z}/p_1 \mathbb{Z}) \oplus (\mathbb{Z}/p_2 \mathbb{Z})$ and $(\mathbb{Z}/p_1^2 \mathbb{Z})\oplus (\mathbb{Z}/p_2 \mathbb{Z})$.  The latter is the [[cyclic group]] of order $p_1^2 p_2$.  For instance, $\mathbb{Z}/12\mathbb{Z} \cong (\mathbb{Z}/4 \mathbb{Z})\oplus (\mathbb{Z}/3 \mathbb{Z})$.

=--

+-- {: .num_example}
###### Example

Similarly, there are four abelian groups of order $p_1^2 p_2^2$, three abelian groups of order $p_1^3 p_2$, and so on.

More generally, theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups} may be used to compute exactly how many abelian groups there are of any finite [[order of a group|order]] $n$ (up to [[isomorphism]]): write down its [[prime factorization]], and then for each prime power $p^k$ appearing therein, consider how many ways it can be written as a product of positive powers of $p$.  That is, each [[partition]] of $k$ yields an abelian group of order $p^k$.  Since the choices can be made independently for each $p$, the numbers of such partitions for each $p$ are then multiplied.

Of all these abelian groups of order $n$, of course, one of them is the [[cyclic group]] of order $n$.  The fundamental theorem of cyclic groups says it is the one that involves the one-element partitions $k= [k]$, i.e. the cyclic groups of order $p^k$ for each $p$.

=--

## Graphical representation
 {#GraphicalRepresentation}

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

Theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups} says that for any [[prime number]] $p$, the [[p-primary group|p-primary part]] of any finitely generated abelian group is determined uniquely up to [[isomorphism]] by

* a total number $k \in \mathbb{N}$ of powers of $p$;

* a [[partition]] $k = k_1 + k_2 + \cdots + k_q$.

The corresponding [[p-primary group]] is

$$
  \underoverset{i = 1}{q}{\bigoplus} \mathbb{Z}/p^{k_i} \mathbb{Z}
  \,.
$$

In the context of [[Adams spectral sequences]] it is conventional to depict this information graphically by

* $k$ dots;

* of which sequences of length $k_i$ are connected by vertical lines, for $i \in \{1, \cdots, q\}$.

For example the graphical representation of the $p$-primary group

$$
  \mathbb{Z}/p\mathbb{Z}
    \oplus
  \mathbb{Z}/p\mathbb{Z}
    \oplus
  \mathbb{Z}/p^3 \mathbb{Z}
    \oplus
  \mathbb{Z}/p^4\mathbb{Z}
$$

is

$$
  \array{
      &&& \bullet
      \\
      && & \vert
      \\
      && \bullet & \bullet
      \\
      && \vert & \vert
      \\
      && \bullet & \bullet
      \\
      && \vert & \vert
      \\
     \bullet & \bullet & \bullet & \bullet
  }
  \,.
$$

This notation comes from the convention of drawing stable pages of [[multiplicative spectral sequence|multiplicative]] [[Adams spectral sequences]] and reading them as encoding the [extension problem](spectral+sequence#ExtensionProblem) for computing the homotopy groups that the spectral sequence converges to:

* a dot at the top of a vertical sequence of dots denotes the group $\mathbb{Z}/p\mathbb{Z}$;

* inductively, a dot vetically below a sequence of dots denotes a [[group extension]] of $\mathbb{Z}/p\mathbb{Z}$ by the group represented by the sequence of dots above;

* a vertical line between two dots means that the the generator of the group corresponding to the upper dot is, regarded after inclusion into the group extension, the product by $p$ of the generator of the group corresponding to the lower dot, regarded as represented by the generator of the group extension.

So for instance

$$
  \array{
    \bullet
    \\
    \vert
    \\
    \bullet
  }
$$

stands for an [[abelian group]] $A$ which forms a [[group extension]] of the form

$$
  \array{
    \mathbb{Z}/p\mathbb{Z}
    \\
    \downarrow
    \\
    A
    \\
    \downarrow
    \\
    \mathbb{Z}/p\mathbb{Z}
  }
$$

such that multiplication by $p$ takes the generator of the bottom copy of $\mathbb{Z}/p\mathbb{Z}$, regarded as represented by the generator of $A$, to the generator of the image of the top copy of $\mathbb{Z}/p\mathbb{Z}$.

This means that of the two possible choices of extensions (by example \ref{TwoFiniteGroupsOfOrderp2}) $A$ corresponds to the non-trivial extension $A = \mathbb{Z}/p^2\mathbb{Z}$. Because then in

$$
  \array{
    \mathbb{Z}/p\mathbb{Z} &
    \\
    \downarrow
    \\
    \mathbb{Z}/p^2 \mathbb{Z} &
    \\
    \downarrow
    \\
    \mathbb{Z}/p\mathbb{Z} &
  }
$$

the image of the generator 1 of the first group in the middle group is $p = p \cdot 1$.

Conversely, the notation

$$
  \array{
    \bullet
    \\
    \\
    \bullet
  }
$$

stands for an [[abelian group]] $A$ which forms a [[group extension]] of the form

$$
  \array{
    \mathbb{Z}/p\mathbb{Z}
    \\
    \downarrow
    \\
    A
    \\
    \downarrow
    \\
    \mathbb{Z}/p\mathbb{Z}
  }
$$

such that multiplication by $p$ of the generator of the top group in the middle group does _not_ yield the generator of the bottom group.

This means that of the two possible choices (by example \ref{TwoFiniteGroupsOfOrderp2}) $A$ corresponds to the _trivial_ extension $A = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$. Because then in

$$
  \array{
    \mathbb{Z}/p\mathbb{Z}
    \\
    \downarrow
    \\
    \mathbb{Z}/p\mathbb{Z}
      \oplus
    \mathbb{Z}/p\mathbb{Z}
    \\
    \downarrow
    \\
    \mathbb{Z}/p\mathbb{Z}
  }
$$

the generator 1 of the top group maps to the element $(1,0)$ in the middle group, and multiplication of that by $p$ is $(0,0)$ instead of $(0,1)$, where the latter is the generator of the bottom group.

Similarly

$$
  \array{
    \bullet
    \\
    \vert
    \\
    \bullet
    \\
    \vert
    \\
    \bullet
  }
$$

is to be read as the result of appending to the previous case a dot _below_, so that this now indicates a group extension of the form

$$
  \array{
    \mathbb{Z}/p^2 \mathbb{Z}
    \\
    \downarrow
    \\
    A
    \\
    \downarrow
    \\
    \mathbb{Z}/p \mathbb{Z}
  }
$$

such that $p$-times the generator of the bottom group, regarded as represented by the generator of the middle group, is the image of the generator of the top group. This is again the case for the unique non-trivial extension, and hence in this case the diagram stands for $A = \mathbb{Z}/p^3 \mathbb{Z}$.

And so on.

For example the stable page of the $\mathbb{F}_2$-[[classical Adams spectral sequence]] for computation of the [[p-primary group|2-primary part]] of the [[stable homotopy groups of spheres]] $\pi_{t-s}(\mathbb{S})$  has in ("internal") degree $t-s \leq 13$ the following non-trivial entries:

<img src="http://ncatlab.org/nlab/files/ClassicalAdamsSpectralSequence.jpg" width="600" >

(graphics taken from ([[Symmetric spectra|Schwede 12]])))

Ignoring here the diagonal lines (which denote multiplication by the element $h_1$ that encodes the additional [[ring]] structure on $\pi_\bullet(\mathbb{S})$ which here we are not concerned with) and applying the above prescription, we read off for instance that $\pi_3(\mathbb{S}) \simeq \mathbb{Z}/8\mathbb{Z}$ (because all three dots are connected) while $\pi_8(\mathbb{S}) \simeq \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z} $ (because here the two dots are not connected). In total

| $k =$ | 0 | 1 | 2 | 3 | 4  | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $\pi_k(\mathbb{S})_{(2)} = $ | $\mathbb{Z}_{(2)}$  | $\mathbb{Z}/2$  |  $\mathbb{Z}/2$ |  $\mathbb{Z}/8$ |  $0$ |  $0$ | $\mathbb{Z}/2$ |   $\mathbb{Z}/16$ | $(\mathbb{Z}/2)^2$ |  $(\mathbb{Z}/2)^3$ | $\mathbb{Z}/2$ | $\mathbb{Z}/8$ | $0$ | $0$ |

Here the only entry that needs further explanation is the one for $k = 0$. The symbol $\mathbb{Z}_{(2)}$ refers to the [[p-adic integers|2-adic integers]], i.e. for the [[limit]]

$$
  \mathbb{Z}_{(2)}
  =
  \underset{\longleftarrow}{\lim}_{n \geq 1} \mathbb{Z}/2^n \mathbb{Z}
  \,.
$$

This is not [[p-primary group|2-primary]], but it does arise when applying [[p-completion|2-completion]] of abelian groups to finitely generated abelian groups as in theorem \ref{FundamentalTheoremOfFinitelyGeneratedAbelianGroups}. For more on this see at _[Adams spectral sequence -- Convergence](Adams+spectral+sequence#Convergence)_. Here we just note why this 2-completion is associated with the infinite sequence of dots

$$
  \array{
    \vdots
    \\
    \vert
    \\
    \bullet
    \\
    \vert
    \\
    \bullet
    \\
    \vert
    \\
    \bullet
    \\
    \vert
    \\
    \bullet
  }
$$

as in the above figure. Namely by the above prescrption, this infinite sequence should encode an abelian group $A$ such that it is an [[group extension|extension]] of $\mathbb{Z}/p\mathbb{Z}$ by itself of the form

$$
  0 \to A \overset{p \cdot(-)}{\longrightarrow} A \longrightarrow \mathbb{Z}/p\mathbb{Z}
$$

(Because it is supposed to encode an extension of $\mathbb{Z}/p\mathbb{Z}$ by the group corresponding to the result of chopping off the lowest dot, which however in this case does not change the figure.)

Indeed, by [this lemma](p-adic+integer#pAdicIntegersAspExtensionofFpByThemselves) we have a short exact sequence

$$
  0
    \to
  \mathbb{Z}_{(p)}
    \overset{p \cdot (-)}{\longrightarrow}
  \mathbb{Z}_{(p)}
    \longrightarrow
  \mathbb{Z}/p\mathbb{Z}
    \to
  0
  \,.
$$

=--

## References

Textbook accounts:

* {#Roman12} Steven Roman, _Fundamentals of group theory_, Birkhäuser (2012)

* [[Joseph A. Gallian]], Section 11 of: *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020)  [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331)[pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)]

A new proof of the [[fundamental theorem of finite abelian groups]] was given in

* {#Navarro03} Gabriel Navarro, _On the fundamental theorem of finite abelian groups_, Amer. Math. Monthly, February 2003

reviewed in

* {#Sullivan} John Sullivan, _Classification of finite abelian groups_ ([pdf](http://www.isama.org/jms/m317/handouts/finabel.pdf))

See also

* Wikipedia, _[Finitely generated abelian group -- Primary decomposition](https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Primary_decomposition)_

* {#Aluffi09} Paolo Aluffi, _Algebra Chapter 0_, 2009 ([pdf](https://zr9558.files.wordpress.com/2013/11/algebra-chapter-0-aluffi.pdf))

[[!redirects fundamental theorem of finite abelian groups]]

[[!redirects fundamental theorem of cyclic groups]]


fundamental theorem of identity types

> This entry currently contains mistakes, see the discussion page [here](https://nforum.ncatlab.org/discussion/15838/fundamental-theorem-of-identity-types/?Focus=114709#Comment_114709).

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

\tableofcontents

## Definition

There are two different fundamental theorems of [[identity types]], depending on whether identity types are defined with plain identity induction or with based identity induction.

### With identity induction
 {#WithIdentityInduction}

The **fundamental theorem of identity types** states that given a [[type]] $A$, a [[type family]] $x:A, y:A \vdash B(x, y)$ and a [[dependent function|family of functions]]

$$
  x \colon A,
  y \colon A
  \;\vdash\;
  f(x, y)
  \,\colon\,
  (x =_A y)
    \to
  B(x, y)
$$

the following conditions are equivalent:

1. For each $x:A$ the [[dependent sum type]] of the type family $y:A \vdash B(x, y)$ is a [[contractible type]].
$$x:A \vdash \mathrm{ftid}(x):\mathrm{isContr}\left(\sum_{y:A} B(x, y)\right)$$

1. There is a family of [[equivalence in type theory|equivalences]]
$$x:A, y:A \vdash \mathrm{ftid}(x, y):(x =_A y) \simeq B(x, y)$$

1. $B(x, y)$ is an [[identity system]].

1. For each $x:A$ and $y:A$, the function $f(x, y)$ is an [[equivalence of types]]
$$x:A, y:A, \vdash \mathrm{ftid}(x, y):\mathrm{isEquiv}(f(x, y))$$

1. $f(x, y)$ is a [[retraction]]
$$x:A, y:A \vdash \mathrm{ftid}(x, y):B(x, y) \to (x =_A y)$$
$$x:A, y:A, r:B(x, y) \vdash G(x, y):f(x, y, \mathrm{ftid}(x, y, r)) =_{B(x, y)} r$$

1. $R(x, y)$ with the function $f(x, y)$ satisfies the [[universal property]] of the [[unary sum]] of $x =_A y$.

### With based identity induction

The **fundamental theorem of identity types** states that given a [[type]] $A$, an [[term|element]] $a:A$, a [[type family]] $x:A \vdash B(x)$ and a [[dependent function|family of functions]]

$$x:A \vdash f(x):(a =_A x) \to B(x)$$

the following conditions are equivalent:

1. The dependent sum type of the type family $x:A \vdash B(x)$ is a [[contractible type]].
$$\mathrm{ftid}:\mathrm{isContr}\left(\sum_{x:A} B(x)\right)$$

1. There is a family of equivalences
$$x:A \vdash \mathrm{ftid}(x):(a =_A x) \simeq B(x)$$

1. $B(x)$ equipped with $f(\mathrm{refl}_A(a)):B(a)$ is an [[identity system]].

1. For each $x:A$, the function $f(x)$ is an [[equivalence of types]]
$$x:A \vdash \mathrm{ftid}(x):\mathrm{isEquiv}(f(x))$$

1. For each $x:A$, $f(x)$ is a [[retraction]]
$$x:A \vdash \mathrm{ftid}(x):B(x) \to (a =_A x)$$
$$x:A, b:B(x) \vdash G(x):f(x, \mathrm{ftid}(x, r)) =_{B(x)} r$$

1. For each $x:A$, $B(x)$ with the function $f(x)$ satisfies the [[universal property]] of the [[unary sum]] of $a =_A x$.

## Proofs that the conditions are the same

### Proof that conditions 1 and 2 are the same

Suppose the equivalence type used in the second definition is a [[weak equivalence type]]. It is possible to show that definitions 1 and 2 are the same.

Definition 1 implies definition 2, because both $\sum_{x:A} (a =_A x)$ and $\sum_{x:A} B(x)$ are [[contractible types]], there is a [[weak equivalences of types]]
$$f(a):\left(\sum_{x:A} (a =_A x)\right) \simeq \left(\sum_{x:A} B(x)\right)$$

Thus, there is a family of functions
$$g(a):\left(\sum_{x:A} (a =_A x)\right) \to B(x)$$
indexed by $a:A$, defined by
$$g(a) \coloneqq \lambda z:\sum_{x:A} (a =_A x).\pi_2(f(a)(z))$$
and by currying this is the same as the function
$$g'(a):\prod_{x:A} (a =_A x) \to B(x)$$
defined by
$$g'(a) \coloneqq \lambda x:A.\lambda p:a =_A x.\pi_2(f(a)(x, p))$$

Then by theorem 11.1.3 of [Rijke22](#Rijke22), since $f(a)$ is an equivalence of types, then each $g'(a)(x)$ is an equivalence of types for all $x:A$. Thus we define $\mathrm{ftid}(x)$ to be $g'(a)(x)$.

Definition 2 implies definition 1, as we begin with a family of weak equivalences
$$x:A \vdash \mathrm{ftid}(x):(a =_A x) \simeq B(x)$$

or equivalently

$$\mathrm{ftid}:\prod_{x:A} (a =_A x) \simeq B(x)$$

We can get a weak equivalences
$$f':\left(\sum_{x:A} (a =_A x)\right) \simeq \left(\sum_{x:A} B(x)\right)$$
by defining
$$f' \coloneqq \mathrm{tot}^{-1}(\mathrm{ftid})$$

The type $\sum_{x:A} (a =_A x)$ is always contractible; this means the type $\sum_{x:A} B(x)$ is contractible as well, since the two types are equivalent to each other. Thus, one could construct a witnesses
$$\mathrm{ftid}':\mathrm{isContr}\left(\sum_{x:A} B(x)\right)$$

### Proof that conditions 2 and 4 are the same

Suppose the equivalence type used in the second definition is a [[weak equivalence type]]. Then definitions 2 and 3 are the same because given any type $A$ and $B$, there is a equivalence

$$\delta_\simeq(A, B):(A \simeq B) \simeq \prod_{f:A \to B} \mathrm{isEquiv}(f)$$

Thus, in one direction, we define
$$\mathrm{ftid}'(x) \coloneqq \delta_\simeq(a =_A x, B(x))^{-1}(f(x))$$

and in the other direction, we inductively define $\mathrm{ftid}'(x)$ by induction on reflexivity

$$\mathrm{ftid}'(a, \mathrm{refl}_A(a)) \coloneqq \delta_\simeq(a =_A a, B(a))(\mathrm{ftid}(a))(\mathrm{refl}_A(a))$$

### Proof that condition 4 implies condition 5

Suppose that for every $x:A$ we have a witness that the function $f(x)$ is an [[equivalence of types]]
$$x:A, \vdash \mathrm{ftid}(x, y):\mathrm{isEquiv}(f(x, x))$$

For every type $A$ and $B$, there is a family of functions
$$f:A \to B \vdash \mathrm{isEquivtoQInv}(f):\mathrm{isEquiv}(f) \to \mathrm{QInv}(f)$$

which takes a witness that $f$ is an equivalence to a [[quasi-inverse function]] of $f$. The [[retraction]] of $f(x)$ is represented by
$$\mathrm{isEquivtoQInv}(f(x))$$

### Proof that condition 5 implies condition 1

This proof is adapted from [[Dan Licata]] in [Licata 16](#Licata16):

Suppose that for every $x:A$ we have a function
$$\mathrm{ftid}(x):B(x) \to (a =_A x)$$
and a family of homotopies
$$G(x):f(x)(\mathrm{ftid}(x)(r) =_{B(x)} r$$
This exhibits $B(x)$ as a [[retract]] of $a =_A x$, hence $\sum_{x:A} B(x)$ as a [[retract]] of the [[contractible type]] $\sum_{x:A} a =_A x$, so there is an element
$$\mathrm{ftid}:\mathrm{isContr}\left(\sum_{x:A} B(x)\right)$$

## See also

* [[identity type]]

* [[identity system]]

* [[univalent reflexive graph]]

* [[univalent universe]]

## References

The fundamental theorem of identity types appears in section 11.2 of

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

Definition 5 arises as a generalization of this proof by [[Dan Licata]] for [[univalent universes]]:

* [[Dan Licata]], *weak univalence with "beta" implies full univalence* ([web](https://groups.google.com/forum/#!msg/homotopytypetheory/j2KBIvDw53s/YTDK4D0NFQAJ))


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

#Contents#
* table of contents
{:toc}

## Idea

The _fundamental theorem of Riemannian geometry_ is the [[theorem]] which states that on a [[pseudo-Riemannian manifold|pseudo]]-[[Riemannian manifold]] there is a unique metric-compatible [[affine connection]] with vanishing [[torsion]] -- the _[[Levi-Civita connection]]_.

In terms of [[integrability of G-structures]] this says that the [[torsion of a G-structure|torsion]] of an [[orthogonal structure]] vanishes, see at _[integrability of G-structures -- Examples -- Orthogonal structure](integrability+of+G-structures#ExamplesOrthogonalStructure)_.

## Related concepts

* [[integrability of G-structures]]


fundamental theorem of topos theory

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

#Contents#
* table of contents
{:toc}

## Idea

The "fundamental theorem" of [[topos theory]], in the terminology of [McLarty 1992](#McLarty92), asserts that for $\mathcal{T}$ any [[topos]] and $X \,\in\, \mathcal{T}$ any [[object]], also the [[slice category]] $\mathcal{T}_{/X}$ is a topos: the *[[slice topos]]*.

If $\mathcal{T} \,\simeq\, Sh(\mathcal{S})$ is a [[category of sheaves]], hence a [[Grothendieck topos]], then so its its slice: $\mathcal{T}_{/X} \,\simeq\, Sh\big( \mathcal{S}_{/X} \big)$ ([SGA4.1, p. 295](#SGA41)).

The analogous statement holds for [[slice (infinity,1)-category|slice $\infty$-categories]] of [[(infinity,1)-toposes|$\infty$-toposes]]: *[[slice (infinity,1)-toposes|slice $\infty$-toposes]]* ([Lurie 2009, Prop. 6.3.5.1](#Lurie09)).

The archetypical special case is that [[slice categories]] $PSh(\mathcal{S})_{/y(s)}$ of [[categories of presheaves]] over a [[representable functor|representable]] are [[equivalence of categories|equivalently]] categories of presheaves on the slice site $\mathcal{S}_{/s}$. This is exhibited by the [[functor]] which sends a [[bundle]] $E \to y(X)$ [[internalization|internal]] to [[presheaves]] to its system $U_X \mapsto \Gamma_U(E)$ of sets of [[local sections]]:

$$
  PSh(\mathcal{S})_{/y(X)}
  \underoverset
    {\sim}
    { \Gamma_{(-)}(-) }
    {\longrightarrow}
  PSh
  \big(
     \mathcal{S}_{/X}
  \big)
$$

The [[model structure on sSet-enriched presheaves|sSet-enriched]] [[derived functor]] of this construction yields the analogous statement for [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]], see at *[[slice of presheaves is presheaves on slice]]* for details.

## Related concepts

* [[slice of presheaves is presheaves on slice]]

* [[base change]]

* [[Giraud's theorem]]

## References

Discussion for [[Grothendieck toposes]]:

* {#SGA41}[[Michael Artin]], [[Alexander Grothendieck]], [[Jean-Louis Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas_ ([[SGA4]]) Tome 1: *Théorie des Topos* Springer **LNM** **269** (1972) ([doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf))

  (In particular, exposé III.5 and exposé IV.5 on the "induced topos" - _topos induit_ = slice topos)

Discussion in the generality of [[elementary toposes]]:

* {#McLarty92} [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

See also:

* Wikipedia, *[Fundamental theorem of topos theory](https://en.wikipedia.org/wiki/Fundamental_theorem_of_topos_theory)*

Discussion for [[slice (infinity,1)-toposes|slices of]] Grothendieck [[(infinity,1)-toposes|$\infty$-toposes]]:

* {#Lurie09} [[Jacob Lurie]], Section 6.3.5 of: *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

The terminology "fundamental theorem of $\infty$-topos theory" for this is used in

* [[Marco Vergura]], Rem. 2.2.10 in: *Localization Theory in an Infinity Topos*, 2019 [[pdf](https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=8583&context=etd), [ir.lib.uwo.ca:etd/6257](https://ir.lib.uwo.ca/etd/6257)]

  > (in a context of [[localization of an (infinity,1)-category|localizations]] motivated by [[modal homotopy type theory]])

[[!redirects fundamental theorem of (infinity,1)-topos theory]]
[[!redirects fundamental theorem of (∞,1)-topos theory]]


fundamental vector field

# Fundamental vector fields and differential forms
* table of contents
{: toc}

## Definitions

Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}\cong T_e G$, $M$ a $C^1$-[[differentiable manifold]] and
$$
\nu\colon G \times M\to M
$$
a left $C^1$-[[differentiable map|differentiable]] [[action]]. For every $m \in M$, denoted by $\nu_m\colon G\to M$ the $C^1$-differentiable map $\nu_m\colon g\mapsto \nu(g,m)$. If $A \in \mathfrak{g}$, the $C^1$-differentiable [[vector field]] on $M$ given by
$$
(\chi_A)_m = (T_{1_G}\nu_m)(A),\,\,\,\,m\in M,
$$
sometimes also denoted $A^\sharp$, is called the __fundamental vector field__ corresponding to $A$. If $s\colon I\to G$ is a curve around $s(0) = 1_G$ representing $A$, then $(\chi_A)_m$ is represented by $t \mapsto s(t) m$.

Analogously, one can define the fundamental vector field for the right actions.

There is a dual notion as well. Given a right $C^1$-differentiable $G$-manifold $E$, a $C^1$-differentiable $1$-[[differential 1-form|form]] $\omega$ with values in $\mathfrak{g}$,
$$
\omega \in \Gamma (T^* E\otimes\mathfrak{g})
$$
is called the __fundamental differential form__ corresponding to $A$, if for all $p\in E$, $\omega_p(\chi_A) = A$.

## Applications

The fundamental vector fields are important in study of differentiable actions and particularly useful in the basic study of [[Ehresmann connections]]. Indeed, the connection form is a fundamental form ("1st Ehresmann condition"), and there is a characterization of those fundamental forms which are connection forms ("2nd Ehresmann condition").

[[!redirects fundamental vector field]]
[[!redirects fundamental vector fields]]

[[!redirects fundamental form]]
[[!redirects fundamental forms]]
[[!redirects fundamental differential form]]
[[!redirects fundamental differential forms]]
[[!redirects fundamental differentiable form]]


funding of the nLab

\tableofcontents

\section{Overview}

The physical existence of the [[HomePage|nLab]] is funded by donations (see [below](#Donate)) which are received via a mechanism handled by the [Topos Institute](https://topos.institute/), with the kind administrative help of [[Brendan Fong]].

The nLab's [[nlabmeta:technical board]] decides about how to use these funds for paying server rent and sysadmin services (see [further below](#Expenditures)).

\section{Donate!}
  {#Donate}

If you like to support the $n$Lab financially, you are welcome to send a donation (a one-off fixed amount or a running monthly sum) through the following mechanism, kindly provided by the _[Topos Institute](https://topos.institute/)_ (thanks to [[Brendan Fong]]!):

Clicking on [the link](https://topos.institute/contact) you will be asked for your credit card details and the intended sum, as in any online transaction. It's quick and straightforward.

Please be sure to pick from the menu "My donation is for..." the field "**nLab Online Wiki Restricted Fund**":

Thanks for your support!

## Available funds
 {#AvailableFunds}

As of 05 May 2022 the funds available to the nLab due to donations are

* 7280.59 USD .

## Expenditures
 {#Expenditures}

Since then we have spent:

* 3 $\times$ 400 USD

  on payment of a temporary sysadmin.

* 504 EUR

  paying a software company to help with the server migration.

The full expected cost of (at least) $\sim$ 4000 EUR was saved, since the enlarged [[nlabmeta:technical board|technical team]] kindly took care of the necessary server migration themselves.


funding opportunities in mathematics

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

#Contents#
* table of contents
{:toc}

## Overview

This page contains a list of funding opportunities for researchers in pure mathematics at all levels: graduate students, post-docs, and university professors.

The intention of this list is to make it easier for advisors, administrators, and practitioners to identify funding opportunities for activities in pure mathematics.

## By Sponsoring Institution

### National Science Foundation

* [Standard grant](https://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5431&org=DMS&sel_org=DMS&from=fund)

* [NSF postdoc](https://www.nsf.gov/pubs/2021/nsf21573/nsf21573.pdf)

* [REU](https://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5517)

* [FRG](https://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5671&org=NSF)

* [RTG](https://beta.nsf.gov/funding/opportunities/research-training-groups-mathematical-sciences-rtg)

### National Security Agency

* conferences, workshops, REU

* [submission guidlines](https://www.nsa.gov/Research/Math-Sciences-Program/Proposal-Submission-Guidelines/)

### Simons Foundation

* [list of opportunities](https://www.simonsfoundation.org/funding-opportunities/?type=mps)

* [Collaboration grants](https://www.simonsfoundation.org/grant/collaboration-grants-for-mathematicians/)

* [Fellows](https://www.simonsfoundation.org/grant/simons-fellows-in-mathematics/)

### American Mathematical Society

* [travel grants, at different levels](https://www.ams.org/programs/travel-grants)

### Air Force Office of Scientific Research

* [Young Investigator](https://community.apan.org/wg/afosr/w/researchareas/12792/young-investigator-program-yip/)

### Sloan Fellowship

### Templeton Foundation

* [Reverse Mathematics, Constuctivism, and Combo](https://www.templeton.org/grant/reverse-mathematics-constructivism-and-combinatorics)

* [Infinite](https://www.templeton.org/grant/foundational-investigations-into-the-infinitefinite-in-mathematics)

### Department of Energy

### In Canada

* NSERC

* CMS

## Research Institute Opportunities

### American Institute of Mathematics

* [main page](https://aimath.org/)

* Workshops and SQuaREs

### Institute for Advanced Study

* Summer programs

### Mathematical Research Institute of Oberwolfach

* [research in pairs](https://www.mfo.de/scientific-program/long-term/research-in-pairs)

### Institut Henri Poincaré

* [research in Paris](http://www.ihp.fr/en/research-in-paris/submit)

### Clay Mathematics Institute

* [programs](https://www.claymath.org/programs)

## University Sponsored

### Emory

* URC

### University of Vermont

* Efromyson (for Graduate Students)


funny tensor product

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

# Funny tensor products

* table of contents
{: toc}

## Definition

For [[categories]] $C,D$, let $C \Rightarrow D$ be the category whose [[objects]] are [[functors]] from $C$ to $D$ and whose [[morphisms]] are [[unnatural transformations]].  This makes $Cat$ into a [[closed category]].  We can then define a [[tensor product]] by a [[universal property]] and make $Cat$ into a [[symmetric closed monoidal category]] $(Cat,\Box)$; this tensor product $\Box$ is called the **funny tensor product**.

As shown by [Foltz, Lair & Kelly 1980](#FoltzLairKelly80), this constitutes one of precisely two [[symmetric monoidal closed category|symmetric monoidal closed]] structures on the [[1-category]] [[Cat|$Cat$]]; the other, of course, is the [[cartesian closed category]] structure. Both the funny product and the cartesian product are [[semicartesian monoidal category|semicartesian monoidal]].

More explicitly, the category $C \Box D$ can be defined as the pushout

$$\array{
C_0 \times D_0 & \to & C_0 \times D \\
\downarrow &&\downarrow  \\
C \times D_0 & \to & C \Box D
}$$

where $C_0,D_0$ are the [[discrete categories]] of objects of $C,D$ and the maps are the inclusions.

The funny tensor product can also be generalized to [[higher categories]].

## Separate functoriality
 {#SeparateFunctoriality}

A functor $F : C \Box D \to E$ can be described as being a functor of 2-variables that is "separately" functorial in the $C$ and $D$ arguments, in analogy with separate continuity. (See at [[multifunctor]] -- [separately functorial maps](multifunctor#SeparatelyFunctorialMaps)).

That is, it has an action on objects $F : C_0 \times D_0 \to E_0$ and for each object $c \in C$, a functorial action $F(id_c,-) : D \to E$ and for each object $d \in D$, a functorial action $F(-,id_d) : C \to E$, both of which agree on objects with $F$.

Contrast this to a "jointly" functorial functor of 2-arguments, also known as a [[bifunctor]], which is equivalent to a functor from the cartesian product $F : C \times D \to E$ where we have to define for any $f : c \to c'$ and $g : d \to d'$ a morphism $F(f,g) : F(c,d) \to F(c',d')$. With a separately functorial $F$, there are two candidates for this morphism: $F(f,id) \circ F(id,g)$ and $F(id,g) \circ F(f,id)$ that are not in general equal.

For a simple example of a separately functorial action that is not a bifunctor, consider the identity functor on $2 \Box 2$ where $2$ is the [[interval category|walking arrow category]]. If we label one copy of $2$ as $\top \to \bot$ and the other as $l \to r$ then $2 \Box 2$ is a non-commuting square:

$$\array{
(\top,l) & \to & (\top,r) \\
\downarrow &&\downarrow  \\
(\bot,l) & \to & (\bot,r)
}$$

then viewing the identity as a functor of 2-arguments, we get an obvious separately functorial action, but since the square does not commute, it is not a bifunctor.

## Enrichment

Categories enriched in the funny tensor product monoidal structure are precisely [[sesquicategories]].

## Related pages

* [[Gray tensor product]]
* Separately functorial functors arise naturally when studying effectful languages where the two sequencings of $F(f,id)$ and $F(id,g)$ correspond to the choices of order of evaluation of the two functions. See [[premonoidal category]] for more: a strict premonoidal category is precisely a monoid object in the monoidal category $(Cat, \Box)$, while a general premonoidal category is a pseudomonoid in the monoidal 2-category $(Cat, \Box)$.

##References

* [[Mark Weber]], _Free products of higher operad algebras_, Theory and Applications of Categories **28** 2 (2013) 24-65. ([arXiv](https://arxiv.org/abs/0909.4722), [TAC](http://www.tac.mta.ca/tac/volumes/28/2/28-02abs.html))

This paper shows that there are just two [[symmetric closed monoidal category|symmetric closed monoidal]] structures on [[Cat]], the cartesian product and the funny tensor product:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]


Furio Honsell

* [personal page](https://users.dimi.uniud.it/~furio.honsell/)

* [Wikipedia page](https://it.wikipedia.org/wiki/Furio_Honsell) (Italian)

## Selected writings

On [[logical frameworks]]:

* [[Robert Harper]], [[Furio Honsell]], [[Gordon Plotkin]], _A framework for defining logics_, J. ACM __40__:1 (1993) 143--184 [[doi:10.1145/138027.138060](https://doi.org/10.1145/138027.138060)]

category: people


Furstenberg-Zimmer structure theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# Contents #
* table of contents
{: toc}

## Idea

A result in [[ergodic theory]] concerning the structure of [[measure]]-preserving [[dynamical systems]].

## References

* [[Terence Tao]], [Lecture: The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem](https://terrytao.wordpress.com/2008/03/05/254a-lecture-15-the-furstenberg-zimmer-structure-theorem-and-the-furstenberg-recurrence-theorem/)

* [[Asgar Jamneshan]], _An uncountable Furstenberg-Zimmer structure theory_ ([arXiv:2103.17167](https://arxiv.org/abs/2103.17167)).


fusion 2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

\tableofcontents

## Idea

A [[categorification]] of the notion of *[[fusion categories]]* to [[2-categories]].

## Properties
### Tannaka-Krein reconstruction theorem

The [[Tannaka duality]] establishes a correspondence between (multi-)[[fusion category|fusion categories]] and [[weak Hopf algebra|weak Hopf algebras]]. It is expected that an analogous statement holds for the higher categorification of this setting. The theorem corresponding to the specialization of this setting to fusion 2-categories and [[semisimple category|semisimple]] [[Hopf monoidal category|Hopf categories]] (as opposed to *multi-* fusion 2-categories and *weak* Hopf categories) appears in ([Green (2023)](#Green23)).

+-- {: .num_theorem}
###### Theorem
**(Tannaka-Krein reconstruction theorem for fusion 2-categories)**

There is a symmetric monoidal equivalence, contravariant at the level of 1-morphisms,
between the full subcategory of $3Vec/2Vec$ consisting of locally faithful functors and the 2-category of
2-Hopf Algebras. The natural transformations associated to this equivalence reconstruct a semisimple
Hopf category from its fusion 2-category of representations and fiber functor, and a fusion 2-category
with fiber functor $F$ from the Hopf category $End(F)$.

=--

## References

* [[Christopher L. Douglas]], [[David J. Reutter]], *Fusion 2-categories and a state-sum invariant for 4-manifolds* [[arXiv:1812.11933](https://arxiv.org/abs/1812.11933)]

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]]. *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories*. (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

* {#Green23} David Green, *Tannaka-Krein reconstruction for fusion 2-categories* (2023) [[arXiv:2309.05591](https://arxiv.org/abs/2309.05591)]

In the context of "[[generalized global symmetries]]":

* Wenjie Xi, Tian Lan, Longye Wang, Chenjie Wang, Wei-Qiang Chen, *On a class of fusion 2-category symmetry: condensation completion of braided fusion category* [[arXiv:2312.15947](https://arxiv.org/abs/2312.15947)]

[[!redirects fusion 2-categories]]


fusion bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The [[image]] of the [[transgression]] operation from [[bundle gerbes]] ([[bundle gerbe with connection|with connection]]) to [[complex line bundles]] ([[connection on a vector bundle|with connection]]) on the [[free loop space]] of their base space may be characterized ([Waldorf 2009](#Waldorf09), [2010](#Waldorf10a), [2011](#Waldorf10b)) as consisting of bundles which are suitably compatible with the "fusion" of [[pairs]] of [[loops]] along [[thin homotopy|thin]] [[trinions]].

\begin{imagefromfile}
    "file_name": "Waldorf-ThinTrinion.jpg",
    "width": "450",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Waldorf 2010](#Waldorf10)"
\end{imagefromfile}

## Definition

(...)

## Properties

### General properties

(...)

### Relation to string field theory?
 {#RelationToStringFieldTheory}

Curiously, the fusion operation that is formalized by the notion of fusion bundles is mathematically reminiscent of (and physically of essentially the same intuitive nature as)  the "star product" on closed string fields considered in [[string field theory]] [[Witten (1986),Fig. 20](string+field+theory#Witten86)]:

\begin{imagefromfile}
    "file_name": "Witten-ClosedStringFieldProduct.jpg",
    "width": "700",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

## Related concepts

* [[stringor bundle]]

## References

* {#Waldorf09} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps*, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 [[arXiv:0911.3212](https://arxiv.org/abs/0911.3212), [cahierstgdc:LIII](http://cahierstgdc.com/index.php/volumes/volume-liii-2012)]

* {#Waldorf10} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection*, Asian Journal of Mathematics **20** 1 (2016) 59-116 [[arXiv:1004.0031](https://arxiv.org/abs/1004.0031), [doi:10.4310/AJM.2016.v20.n1.a4](https://dx.doi.org/10.4310/AJM.2016.v20.n1.a4)]

* {#Waldorf11} [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles*, Advances in Mathematics **231** (2012) 3445-3472 [[arXiv:1109.0480](https://arxiv.org/abs/1109.0480), [doi:10.1016/j.aim.2012.08.016](https://doi.org/10.1016/j.aim.2012.08.016)]

[[!redirects fusion bundles]]


fusion category

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

#Contents#
* table of contents
{:toc}

## Definition ##

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

A **fusion category** is a [[rigid monoidal category|rigid]] [[semisimple category|semisimple]] [[linear category|linear]] ([[Vect]]-[[enriched category|enriched]]) [[monoidal category]] ("[[tensor category]]"), with only [[finite number|finitely]] many [[decategorification|isomorphism classes]] of [[simple objects]], such that the [[endomorphisms]] of the unit object form just the [[ground field]] $k$.

=--

Often one also assumes a [[braided monoidal category|braiding]] and speaks of a *braided fusion category*.

## Examples
 {#Examples}

The name "fusion category" comes from the central examples of structures whose canonical [[tensor product]] is called a "fusion product", notably [[representations]] of [[loop groups]] and of [[Hopf algebras]] and of [[vertex operator algebras]].

Simple examples:

\begin{example}
\label{GradedVectorSpaces}
**(graded vector spaces)**
\linebreak
For

* $G$ a [[finite group]],

* $\mathbb{K}$ a [[field]],

the category of [[finite-dimensional vector space|finite-dimensional]] $G$-[[graded vector spaces|graded]] $\mathbb{K}$-[[vector spaces]]

$$
  Vect_{G}^{fdim}
$$

is a fusion category, with [[tensor product]] given by the [[tensor product of vector spaces]] and the [[binary operation]] of the group:

$$
  V_g \otimes W_h
  \;\;
  \coloneqq
  \;\;
  (V \otimes W)_{g \cdot h}
  \,.
$$

More generally, for

* $\omega \,\in\, H^3_{Grp}(G, \mathbb{K}^\times)$ a 3-cocycle in the [[group cohomology]] of $G$ with [[coefficients]] in the [[group of units]] $\mathbb{K}^\times$,

the above construction but with [[associator]] multiplied by the 3-cocycle applied to the $G$-degrees of the 3 factors is again a fusion category

$$
  Vect_{G, \omega}^{fdim}
  \,.
$$
\end{example}
([Etingof, Nikshych & Ostrik 2005, item 1. on p. 584](#EtingofNikshychOstrik05))

\begin{example}
For $\mathbb{K}$ a [[field]] and
$G$ a [[finite group]] (or [finite super-group](supergroup#finite_supergroups)), whose [[order of a group|order]] is [[coprime integer|relatively prime]] to the [[characteristic of a field|characteristic]] of $\mathbb{K}$, then the [[category of representations]] $Rep(G, \mathbb{K})$ is a fusion category.
\end{example}
([Etingof, Nikshych & Ostrik 2005, item 2. on p. 584](#EtingofNikshychOstrik05))

## Properties

### Relation to weak Hopf algebras

Under [[Tannaka duality]], every fusion category $C$ arises as the [[representation category]] of a [[weak Hopf algebra]]. ([Ostrik](#Ostrik))

### Relation to pivotal and spherical categories

Fusion categories were first systematically studied by [Etingof, Nikshych & Ostrik 2005](#EtingofNikshychOstrik05). This paper listed many examples and proved many properties of fusion categories. One of the important conjectures made in that paper was the following:

+-- {: .num_theorem}
###### Conjecture (Etingof, Nikshych, and Ostrik)

Every fusion category admits a [[pivotal structure]].

=--

Providing a certain condition is satisfied, a pivotal structure on a fusion category can be shown to correspond to a 'twisted' monoidal natural endotransformation of the identity functor on the category, where the twisting is given by the [[pivotal symbols]].

### Relation to extended 3d TQFT
 {#RelationtoTQFT}

Given the data of a [[fusion category]] one can build a 3d [[extended TQFT]] by various means. This is explained by the fact, see below, that fusion categories are (probably precisely) the [[fully dualizable object]]s in the [[3-category]] $MonCat$ of [[monoidal categories]]. By the [[homotopy hypothesis]] this explains how they induce [[3d TQFT]]s.

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

Write $MonCat_{bim}$ for the [[(infinity,n)-category|(infinity,3)-category]] which has as

* objects [[monoidal categories]],

* morphism [[bimodule]]s of these,

* and so on.

=--

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

With its natural [[tensor product]], $MonCat$ is a [[symmetric monoidal (infinity,n)-category|symmetric monoidal (infinity,3)-category]].

=--

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

A monoidal category which is fusion is [[fully dualizable object|fully dualizable]] in the [[(infinity,n)-category|(infinity,3)-category]] $MonCat_{bim}$, def. \ref{MonCat}.

=--

This is due to ([Douglas & Schommer-Pries & Snyder 13](#DSPS13)).

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

Via the [[cobordism theorem]] prop. \ref{FusionCategoriesAreFullyDualizable} implies that fusion categories encode [[extended TQFTs]] on 3-dimensional [[framed manifold|framed]] [[cobordisms]], while their $O(3)$-[[homotopy fixed points]] encode extended 3d TQFTs on general (not framed) cobordisms.

These 3d TQFTs hence arise from similar algebraic data as the [[Turaev-Viro model]] and the [[Reshetikhin-Turaev construction]], however there are various slight differences. See ([Douglas & Schommer-Pries & Snyder 13, p. 5](#DSPS13)).

=--

## Suggestions ##

Here are three things such that it'd be awesome if they were sorted out on this page:

1. Kuperberg's theorem saying that [[abelian category|abelian]] semisimple implies [[linear category|linear]] over some field. [Finite, connected, semisimple, rigid tensor categories are linear](http://arxiv.org/abs/math/0209256)

2. Some correct version of the claim that abelian semisimple is the same as [[idempotent complete category|idempotent complete]] and nondegenerate. [Math Overflow question](http://mathoverflow.net/questions/245/are-abelian-nondegenerate-tensor-categories-semisimple)

3. Good notation distinguishing [[simple object|simple]] versus [[absolutely simple object|absolutely simple]] (is $End(V) = k$ or just $V$ has no nontrivial proper subobjects).

Together 1 and 2 let you go between the two different obvious notions of semisimple which seem a bit muddled here when I clicked through the links.

## Related concepts

* **fusion category**

  * [[unitary fusion category]]

  * [[fusion 2-category]]

  * [[fusion ring]], [[Frobenius-Perron dimension]]

* [[pivotal category]]

* [[spherical category]]

* [[modular tensor category]]

* [[Deligne's theorem on tensor categories]]

* [[graded fusion category]]

* [[Tambara-Yamagami category]]

## References ##

### General

Original articles:

* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series **162** 2 (2005) 581-642 [[arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926)]

* [[Pavel Etingof]] and [[Damien Calaque]], [Lectures on tensor categories](http://arxiv.org/abs/math/0401246).

* [[Michael Müger]], [Tensor categories: A selective guided tour](http://arxiv.org/abs/0804.3587).

* [[Pavel Etingof]], D. Nikshych and V. Ostrik, _Fusion categories and homotopy theory_ ,  Quantum Topology, 1(2010), 209-273. (Earlier version available as [ArXiv:0909.3140](http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.3140v2.pdf)

* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

Further review:

* [[Bruce Bartlett]], chapter 6 of
*[On unitary 2-representations of finite groups and topological quantum field theory](http://arxiv.org/abs/0901.3975)*.

On the [[Tannaka duality]] to [[weak Hopf algebras]]:

* Takahiro Hayashi, _A canonical Tannaka duality for finite seimisimple tensor categories_ ([arXiv:math/9904073](http://arxiv.org/abs/math/9904073))

* {#Ostrik} [[Victor Ostrik]], _Module categories, weak Hopf algebras and modular invariants_ ([arXiv:math/0111139](http://arxiv.org/abs/math/0111139))

The relation to [[3d TQFT]] clarified via the [[cobordism hypothesis]]:

* {#DSPS} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _The Structure of Fusion Categories via 3D TQFTs_ ([talk pdf](https://sites.google.com/site/chrisschommerpriesmath/Home/recent-and-upcoming-talks/UPennTalk.pdf?attredirects=0))

* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))

and for the case of [[modular tensor categories]]:

* [[Bruce Bartlett]], [[Christopher Douglas]], [[Chris Schommer-Pries]], [[Jamie Vicary]], _Modular categories as representations of the 3-dimensional bordism 2-category_ ([arXiv:1509.06811](http://arxiv.org/abs/1509.06811))

Discussion in terms of [[skein relations]]:

* Anup Poudel, [[Sachin J. Valera]], *Skein-Theoretic Methods for Unitary Fusion Categories* [[arXiv:2008.07129](https://arxiv.org/abs/2008.07129)]

See also:

* Math Overflow, _[Why are fusion categories interesting?](http://mathoverflow.net/questions/6180/why-are-fusion-categories-interesting)_ .

On a notion of [[fusion 2-categories]]:

* [[Christopher L. Douglas]], [[David J. Reutter]], *Fusion 2-categories and a state-sum invariant for 4-manifolds* [[arXiv:1812.11933](https://arxiv.org/abs/1812.11933)]

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]]. *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories*. (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

[[!include anyonic topological order via braided fusion categories -- references]]

[[!redirects fusion categories]]

[[!redirects braided fusion category]]
[[!redirects braided fusion categories]]

[[!redirects fusion]]


fusion ring

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

#Contents#
* table of contents
{:toc}

## Idea

The [[Grothendieck ring]] of every fusion category can be naturally viewed as an (algebraic) [[Z+ ring|fusion ring]]. The structure of this resulting ring is purely determined by the fusion rules of the staring category. Hence, it is known as the __fusion ring__ of the fusion category.

This ring encodes a large amount of the structure of the category, but not all of it. Namely, there are non-isomorphic fusion categories with isomorphic fusion rings. For example, the representation categories of the two non-isomorphic non-abelian groups of order eight have isomorphic fusion rings.

The strongest result in this direction is [[Ocneanu Rigidity]], which asserts that there are only finitely many isomorphism classes of categories which induce a given fusion ring. A standard approach towards classifications of fusion categories thus begins by first understanding fusion rings, and then understanding what categorifications can be imposed.

Such a choice of categorification amounts to a choice of associativity on the monoidal structure of your category. These associativity rules can be also described explicitely in terms of algebraic structure on your distinguished $\mathbb{Z}_+$ basis by using [[Wigner 6-j symbol | 6j symbols]].

## Definition

Let $\mathcal{C}$ be a [[fusion category]], i.e. a [[tensor category]] which is [[finite abelian category|finite]] and [[semisimple category | semisimple]] (i.e. it has a [[finite number]] of [[isomorphism classes]] $[X_i]$ of [[simple objects]], all finite [[direct sums]] of these exist, and every object is isomorphic to such).

Let $Gr(\mathcal{C})$ be the Grothendieck ring of $\mathcal{C}$. That is, the ring whose objects are isomorphism classes of objects in $\mathcal{C}$, whose addition is defined by $[X]+[Y]=[X\oplus Y]$, and whose multiplication is defined by $[X]\cdot [Y]=[X\otimes Y]$.

The set of simple objects of $\mathcal{C}$ give a [[Z+ ring |$\mathbb{Z}_+$-basis]] for $Gr(\mathcal{C})$. This follows immediately from the fact that every object is isomorphic to the direct sum of some integer number of copies of every simple objects.

The fact that the tensor unit is simple implies that $Gr(\mathcal{C})$ is moreover a unital $\mathbb{Z}_+$ring.

The rigidity structure $*$ on $\mathcal{C}$ induces an anti-involution on $Gr(\mathcal{C})$, which gives it the structure of a fusion ring.

## Related concepts

* [[Frobenius-Perron dimension]]

* [[Clebsch-Gordan coefficient]]

* [[Verlinde algebra]]

## References

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]] Tensor categories. Vol. 205. American Mathematical Soc., 2016.

* [[Sonia Natale]]. "On the classification of fusion categories." Proc. Int. Congr. Mathematicians, Rio de Janeiro (2018): 173-200.


future

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

# Past and future
* table of contents
{: toc}

## Idea

The _past_ of any physical event (object, system, etc) consists of everything that might (by the principles of [[causality]]) have potentially influenced that event; while its _future_ consists of everything that it might potentially influence.

## Definitions

Let $(X,g,o)$ be a [[spacetime]], that is a [[Lorentzian manifold]] $(X,g)$ equipped with [[time orientation]].  This time-orientation consists precisely of specification of which [[timelike]] and [[lightlike]] [[curves]] are _future-directed_ (and which are complementarily _past-directed_).

Let $x$ be a [[point]] in this spacetime.  Then:

* the **future** of $x$ is the subset $J^+(x)$ of all points of $X$ connected to $x$ by a future-directed [[timelike]] or [[lightlike]] [[curve]] *starting* at $x$;

* the **past** of $x$ is the subset $J^-(x)$ of all points of $X$ connected to $x$ by a future-directed [[timelike]] or [[lightlike]] [[curve]] *ending* at $x$.

Let $A$ be a more general [[subset]] of this spacetimes.  Then:

* the **future** of $A$ is the subset $J^+(A)$ defined as the [[union]] of $J^+(x)$ for all $x \in A$;

* the **past** of $A$ is the subset $J^-(A)$ defined as the [[union]] of $J^-(x)$ for all $x \in A$.

## Properties

A [[Cauchy surface]] $\Sigma$ in $(X,g)$ is a minimal subset of $X$ with the property that $X$ is the [[union]] of the future and past of $\Sigma$.  (Does this suffice to *define* Cauchy surfaces in the case of a Lorentzian manifold that admits a time-orientation?)

The operations $J^+$ and $J^-$ are (separately) [[Moore closures]] on the [[power set]] of $X$.  Stated explicitly (for $J^+$):

*  the future of $A$ contains $A$;
*  the future of the future of $A$ is simply the future of $A$;
*  if $A$ contains $B$, then the future of $A$ contains the future of $B$.

## Variations

Sometimes one wants to remove $x$ itself from $J^+(x)$ and $J^-(x)$ (or more precisely, to include $x$ only in the case of a [[closed timelike curve]] through $x$).  However, the operations $J^+$ and $J^-$ are not quite as mathematically well-behaved in this case.  (Note that $J^+(A)$ may still intersect $A$, or even contain all of $A$, even in the absence of closed timelike curves.)

## Related concepts

* [[future causal cone]], [[past causal cone]]

* [[advanced propagator]], [[retarded propagator]]

[[!redirects past]]
[[!redirects pasts]]
[[!redirects future]]
[[!redirects futures]]

[[!redirects causal future]]

[[!redirects causal past]]


future cone

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

# The future cone
* table of contents
{: toc}

## Idea

The future cone of a point $x$ is the set consisting of all points in the [[future]] of $x$.  (We usually interpret this condition weakly, so that $x$ itself belongs to its own future cone.)

Perhaps the terms *forward cone* resp. *backward cone* might be used as synonyms for the concept.

The dual concept is the _past cone_.

## In relativity theory

Given a [[Lorentzian manifold]] $(X,x)$ equipped with a [[time orientation]] then

* the _future cone_ at $x \in X$ is the subspace $V_x^+ \subset T_x X$ of the [[tangent space]] of $X$ consisting of all those [[tangent vectors]] which are future pointing;

* the __future__ of $x$ consists of all points $y$ such that there exists a future-directed curve (which may be timelike, lightlike, or a mixture) from $x$ to $y$.

If we may put a global time coordinate $t$ on the manifold (which is a stronger condition), then the future cone of $x$ consists of all points $y$ such that $t(y) \geq t(x)$ and $x$ and $y$ are not space-like separated.

For more see at _[[causal cone]]_.

## In directed homotopy theory

In the context of [[directed homotopy theory]] the **future cone** of a point, $x$, in a [[directed topological space]] is, more-or-less, the directed subspace of those points $y$ which are greater than $x$. In case the space might not be that 'nice' or, for instance, we have a local partial order rather than a global one, it is preferable to say that it is the directed space of points, $y$, for which there is a distinguished path from $x$ to $y$.

## Related concepts

* [[light cone]]

* [[conal manifold]]

[[!redirects future cone]]
[[!redirects future cones]]

[[!redirects past cone]]
[[!redirects past cones]]


fuzzy dark matter

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A model for [[dark matter]] made up of massive but extremely light [[particles]], whose [[de Broglie wavelength]] is at the scale of galaxies.

The idea is that on scales above that of galaxies, the predictions of fuzzy dark matter agree with the standard cold dark matter models that work exceptionally well on [[cosmology|cosmological scales]], while on scales of the size of galaxies the [[quantum physics|quantum]] properties of these light particles become relevant and change their effect just so as to fix the problems (see also [[MOND]]) that standard cold dark matter models have on these scales.

A natural candidate for such ultra-light particles are [[axions]].

This kind of model was brought up independently by several groups of authors (see [Lee 17](#Lee17) for historical survey) with early precursors going back as far as ([Baldeschi-Gelmini-Ruffini 83](#BaldeschiGelminiRuffini83)), and accordingly goes by a number of different names, including the following:

* _[[Bose-Einstein condensate]] (BEC) dark matter_ ([Sin 92](#Sin92), [Sikivie-Yang 09](#SikivieYang09))

* _[[superfluid]] dark matter_ ([Berezhiani-Khoury 15](#BerezhianiKhoury15), [Khoury 16](#Khoury16))

* _fuzzy dark matter_ ([Wu-Barkana-Gruzinov 00](#WuBarkanaGruzinov00))

* _ultra-light [[axion]]_ ([Erken-Sikivie-Tam-Yang 12](#ErkenSikivieTamYang12))

* _wave dark matter_

* _repulsive dark matter_

and more.

The suggestion that fuzzy dark matter induces the observed almost-flat galactic rotation curves ("[[MOND]]") seems to go back to ([Sin 92](#Sin92)). Further pointers are in ([Lee 17, p. 3](#Lee17)):

> There are many works explaining the rotation curves of dwarf [17, 23, 69], and large galaxies [29, 43, 70–78] in this model.

More recently, detection of the [[21cm hydrogen line]] from cosmic dawn indicates that star formation set in earlier than compatible with fuzzy dark matter models ([Nebrin 17](#Nebrin17), [Nebrin-Ghara-Mellema 18](#NebrinGharaMellema18)). This would rule out substantial contributions of fuzzy dark matter.

<br/>

## References
 {#References}

### General

Early precursors of the idea include

* {#BaldeschiGelminiRuffini83} M. R. Baldeschi, G. B. Gelmini, and R. Ruffini, Physics Letters B 122, 221 (1983).

The role of the [[Bose-Einstein condensate]] of [[axions]] on galactic scales was considered in

* {#Sin92} Sang-Jin Sin, _Late time Cosmological Phase Transition and Galactic Halo as Bose-liquid_, Phys.Rev.D50:3650-3654,1994 ([arXiv:hep-ph/9205208](https://arxiv.org/abs/hep-ph/9205208))

*  {#SikivieYang09} [[Pierre Sikivie]], Q. Yang, _Bose-Einstein Condensation of Dark Matter Axions_, Phys.Rev.Lett.103:111301, 2009 ([arXiv:0901.1106](https://arxiv.org/abs/0901.1106))

* {#ErkenSikivieTamYang12} O. Erken, [[Pierre Sikivie]], H. Tam, Q. Yang, _Cosmic axion thermalization_, Phys. Rev. D 85, 063520 2012 ([arXiv:1111.1157](https://arxiv.org/abs/1111.1157))

The proposal in the guise of "fuzzy dark matter" is originally due to

* {#WuBarkanaGruzinov00} Wayne Hu, Rennan Barkana, Andrei Gruzinov, _Cold and Fuzzy Dark Matter_, Phys.Rev.Lett. 85 (2000) 1158-1161 ([arXiv:astro-ph/0003365](https://arxiv.org/abs/astro-ph/0003365))

A detailed discussion is in

* {#HOTW16} [[Lam Hui]], [[Jeremiah Ostriker]], [[Scott Tremaine]], [[Edward Witten]], _On the hypothesis that cosmological dark matter is composed of ultra-light bosons_,  Phys. Rev. D 95, 043541 (2017) ([arXiv:1610.08297](https://arxiv.org/abs/1610.08297))

Review includes

* David J. E. Marsh, _Axion Cosmology_ ([arXiv:1510.07633](https://arxiv.org/abs/1510.07633))

* {#Lee17} Jae-Weon Lee, _Brief History of Ultra-light Scalar Dark Matter Models_ ([arXiv:1704.05057](https://arxiv.org/abs/1704.05057))

Computer simulation of [[structure formation]] with fuzzy dark matter:

* Philip Mocz et al., _First Star-Forming Structures in Fuzzy Cosmic Filaments_, Phys. Rev. Lett. 123, 141301 – 2019 ([doi:10.1103/PhysRevLett.123.141301](https://doi.org/10.1103/PhysRevLett.123.141301))

Some thoughts on the quantum [[measurement problem]] for fuzzy DM particles with huge macroscopic [[Compton wavelengths]] is in

* Adam D. Helfer, _Quantum measurement and fuzzy dark matter_ ([arXiv:1809.04946](https://arxiv.org/abs/1809.04946))

### Superfluid dark matter

Discussion of [[superfluid]] dark matter could be found in:

* {#HossenfelderMistele19} [[Sabine Hossenfelder]], [[Tobias Mistele]], *Strong lensing with superfluid dark matter*. Journal of Cosmology and Astroparticle Physics, Volume 2019, February 2019. ([doi:10.1088/1475-7516/2019/02/001](https://doi.org/10.1088/1475-7516/2019/02/001), [arXiv:1809.00840](https://arxiv.org/abs/1809.00840))

* {#Mistele22} [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Galactic mass-to-light ratios with superfluid dark matter*, Astronomy and Astrophysics, Volume 664, Article 40, August 2022 ([doi:10.1051/0004-6361/202243216](https://doi.org/10.1051/0004-6361/202243216), [arXiv:2201.07282](https://arxiv.org/abs/2201.07282))

### MOND Phenomenology

Discussion of how [[superfluid]] aspects of [[axion|axionic]] fuzzy dark matter reproduce [[MOND]] [[phenomenology]] is in

* {#BerezhianiKhoury15} [[Lasha Berezhiani]], [[Justin Khoury]], _Theory of Dark Matter Superfluidity_, Phys. Rev. D 92, 103510 (2015) ([arXiv:1507.01019](https://arxiv.org/abs/1507.01019))

* {#Khoury16} [[Justin Khoury]], _Another Path for the Emergence of Modified Galactic Dynamics from Dark Matter Superfluidity_, Phys. Rev. D 93, 103533 (2016) ([arXiv:1602.05961](https://arxiv.org/abs/1602.05961))

* JiJi Fan, _Ultralight Repulsive Dark Matter and BEC_ ([arXiv:1603.06580](https://arxiv.org/abs/1603.06580))

* Andrea Addazi, Antonino Marciano, _UV completion of a theory of Superfluid Dark Matter_ ([arXiv:1801.04083](https://arxiv.org/abs/1801.04083))

* [[Justin Khoury]], *Dark Matter Superfluidity* ([arXiv:2109.10928](https://arxiv.org/abs/2109.10928))

### Comparison and tension with experiment
 {#ReferencesTensionWithHydrogenLineSpectroscopy}

Comparison to [[experiment]] (observation):

* Bohua Li, Tanja Rindler-Daller, and Paul R. Shapiro, _Cosmological constraints on Bose-Einstein-condensed scalar field dark matter_, Phys. Rev. D 89, 083536, 2014 ([arXiv:1310.6061](https://arxiv.org/abs/1310.6061))

* Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst, _Cosmic structure as the quantum interference of a coherent dark wave_, Nature Physics 10, 496–499 (2014) ([doi:10.1038/nphys2996](http://www.nature.com/nphys/journal/v10/n7/abs/nphys2996.html))

* Nilanjan Banik, Adam J. Christopherson, [[Pierre Sikivie]], Elisa Maria Todarello, _New astrophysical bounds on ultralight axionlike particles_, Phys. Rev. D 95, 043542 (2017) ([arXiv:1701.04573](https://arxiv.org/abs/1701.04573))

Strong constraints on fuzzy dark matter from observation of the cosmic [[21cm hydrogen line]]  are claimed and discussed in

* Adam Lidz, [[Lam Hui]], _The Implications of a Pre-reionization 21 cm Absorption Signal for Fuzzy Dark Matter_, Phys. Rev. D 98, 023011 (2018) ([arXiv:1805.01253](https://arxiv.org/abs/1805.01253))

* {#Nebrin17} Olof Nebrin, _Cosmic Dawn in a Fuzzy Universe: Constraining the nature of Dark Matterwith 21 cm Cosmology_, Stockholm 2017 ([diva2:1195402](http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A1195402&dswid=-8037), [urn:nbn:se:su:diva-154861](http://su.diva-portal.org/smash/record.jsf?pid=diva2%3A1195402&dswid=6755))

* {#NebrinGharaMellema18} Olof Nebrin, Raghunath Ghara, Garrelt Mellema, _Fuzzy Dark Matter at Cosmic Dawn: New 21-cm Constraints_ ([arXiv:1812.09760](https://arxiv.org/abs/1812.09760), [reddit](https://www.reddit.com/r/Physics/comments/a9zeeb/my_first_paper_fuzzy_dark_matter_at_cosmic_dawn/))

Claim that the galaxy [[core-cusp problem]] is not resolved after all is discussed in

* Heling Deng, Mark P. Hertzberg, Mohammad Hossein Namjoo, Ali Masoumi, _Can Light Dark Matter Solve the Core-Cusp Problem?_ ([arXiv:1804.05921](https://arxiv.org/abs/1804.05921))

Claim that the fitting of the [[galaxy rotation curves]] is not resolved after all:

* Mariangela Lisanti, Matthew Moschella, Nadav Joseph Outmezguine, Oren Slone, _The Inconsistency of Superfluid Dark Matter with Milky Way Dynamics_ ([arXiv:1911.12365](https://arxiv.org/abs/1911.12365))

Claim that superfluid dark matter is in tension with weak gravitational lensing data:

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Superfluid dark matter in tension with weak gravitational lensing data* ([arXiv:2303.08560](https://arxiv.org/abs/2303.08560))

[[!redirects Bose-Einstein condensate dark matter]]
[[!redirects BEC dark matter]]

[[!redirects axion dark matter]]
[[!redirects superfluid dark matter]]


fuzzy funnel

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The microscopic geometry of transversal [[Dp-D(p+2)-brane intersections]] and [[Dp-D(p+4)-brane intersections]] look like warped [[non-commutative geometry|non-commutative]] [[metric cones]] on [[fuzzy spheres]] (namely on the spheres around the lower dimensional [[D-branes]] inside the higher dimensional D-branes). These have hence been called _fuzzy funnels_.

> graphics grabbed from [Fazzi 17, Fig. 3.14](#Fazzi17), taken in turn from
[Gaiotto-Tomassiello 14, Figure 5](#GaiottoTomassiello14)

> graphics grabbed from [Fazzi 17](#Fazzi17)

[[!include Dp-D(p+2)-brane intersections in fuzzy funnels -- section]]

## Properties

### Single trace observables as $\mathfrak{su}(2)$-weight systems on chord diagrams
  {#SingleTraceObservablesAsWeightSystemsOnChordDiagrams}

We discuss how the [[single trace observables]] on the [[fuzzy 2-sphere]]-sections  of [[Dp-D(p+2) brane intersection]] [[fuzzy funnels]] are given by [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] (following [Ramgoolam-Spence-Thomas 04](#RamgoolamSpenceThomas04), [McNamara-Papageorgakis 05](#McNamaraPapageorgakis05), see [McNamara 06, Section 4](#McNamara06) for review).

For more see at _[[weight systems on chord diagrams in physics]]_.

\linebreak

While in the commutative [[large N limit]], all powers of the [[radius]] function on the [[fuzzy 2-sphere]] are equal

$$
  \underset{N\to \infty}{\lim}
  \int_{S^2_N} R^{2 k}
  \;=\;
  4 \pi
  \,;
$$

for [[finite number|finite]] $N$ there is an ordering ambiguity: In fact, the number of functions on the [[fuzzy 2-sphere]] at [[finite number|finite]] $N$ that all go to the same function $R^{2k}$ in the [[large N limit]] grows rapidly with $k$.

At $k = 1$ there is the single radius observable (eq:RadiusOfFuzzy2Sphere)

$$
  \int_{S^2_N} R^2
  \;=\;
  \int_{S^2_N}
  \underset{i}{\sum} X_i \cdot X_i
  \;=\;
  4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }
$$

At $k = 2$ there are, under the integral (eq:FuzzyS2Integration), two radius observables:

1. $ \int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(Here we are using that under the integral/trace, a [[cyclic permutation]] of the factors in the integrand does not change the result).

Similarly for higher $k$, where the number of possible orderings increases rapidly. The [[combinatorics]] that appears here is familiar in [[knot theory]]:

Every ordering of operators, up to cyclic permutation, in the [[single trace observable]] $Tr(R^2)^n$ is encoded in a [[chord diagram]] and the value of the corresponding [[single trace observable]] is the value of the [[su(2)]]-[[Lie algebra weight system]] on this chord diagram.

## Related concepts

* [[Nahm's equation]]

* [[fuzzy sphere]]

* [[noncommutative torus]]

\linebreak

[[!include brane bound states -- table]]

## References

### General

* Rajsekhar Bhattacharyya, Robert de Mello Koch, _Fluctuating Fuzzy Funnels_, JHEP 0510 (2005) 036 ([arXiv:hep-th/0508131](https://arxiv.org/abs/hep-th/0508131))

### For D1-D3-brane intersections

On [[D1-D3 brane intersections]] as fuzzy funnels on [[fuzzy 2-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

* [[Robert Myers]], Section 4 of: _Nonabelian D-branes and Noncommutative Geometry_, J. Math. Phys. 42: 2781-2797, 2001 ([arXiv:hep-th/0106178](https://arxiv.org/abs/hep-th/0106178))

* [[Neil Constable]], [[Neil Lambert]], _Calibrations, Monopoles and Fuzzy Funnels_, Phys. Rev. D66 (2002) 065016 ([arXiv:hep-th/0206243](https://arxiv.org/abs/hep-th/0206243))

### For D3-D5 brane intersections

On [[D3-D5 brane intersections]] as fuzzy funnels on [[fuzzy 2-spheres]]:

* [[Davide Gaiotto]], [[Edward Witten]], Section 3.4.3 of: _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

* {#Fazzi17} [[Marco Fazzi]], Section 3.2.3 of: _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

### For D6-D8 brane intersections

On [[D6-D8 brane intersections]] as fuzzy funnels on [[fuzzy 2-spheres]]:

* {#GaiottoTomassiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP 12 (2014) 003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* [Fazzi 17, p. 41-46](#Fazzi17)

### For D1-D5-brane intersections

On [[D1-D5 brane intersections]] as fuzzy funnels on [[fuzzy 4-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _Non-abelian Brane Intersections_, JHEP 0106:023, 2001 ([arXiv:hep-th/0102080](https://arxiv.org/abs/hep-th/0102080))

### For D1-D7-brane intersections

On [[D1-D7 brane intersections]] as fuzzy funnels on [[fuzzy 6-spheres]]:

* [[Paul Cook]], Robert de Mello Koch, Jeff Murugan, _Non-Abelian BIonic Brane Intersections_,  Phys. Rev. D68:126007, 2003 ([arXiv:hep-th/0306250](https://arxiv.org/abs/hep-th/0306250))

### Single trace observables as weight systems on chord duagrams

Relation of [[single trace observables]] on [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] computing [[radii]] averages of [[fuzzy spheres]]:

* {#RamgoolamSpenceThomas04} [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* {#McNamaraPapageorgakis05} [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* {#McNamara06} [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

[[!redirects fuzzy funnels]]


fuzzy logic

# Fuzzy logic

* table of contents
{: toc}

## Idea

Fuzzy logic is an algebraic form of [[logic]] in which the [[truth values]] are [[real numbers]] in the [[closed space|closed]] [[interval]] between 0 and 1. It has been proposed as a way to model vagueness or uncertainty, partial truths, and imprecise information.

One proposed application is to resolve paradoxes of Sorites type. For example, the proposition "if $n$ is a very large number, then so is $n-1$" does not stand scrutiny if there is a sharp true-false distinction between "very large" and "not very large". The counter is that "very large" is inherently a fuzzy concept, steadily shading into gray as one descends from $n$ to $n-1$.

To be properly considered "a logic", fuzzy logic should be developed into a full-fledged deductive system (with rules of inference and so on), but historically this has been a source of difficulty. See the [SEP](#SEP) for a description of what has been achieved on this front.

It may also be questioned whether it makes sense to have degrees of certainty be totally ordered, as $[0, 1]$ is. Goguen for example allows consideration of frames or locales $L$ more general than $[0, 1]$, and presumably the same consideration would extend to quantale structures more general than those supported on $[0, 1]$. For some analysis of Goguen's framework, see [Barr 1986](#Barr86).

## Examples

### T-norm fuzzy logics

One class of fuzzy logics is the class of **[[t-norm]] fuzzy logics**. In this class, one starts by equipping the interval $[0, 1]$ with a [[t-norm]] structure

$$\otimes\colon [0, 1] \times [0, 1] \to [0, 1],$$

thought of as playing a role analogous to [[logical conjunction]] (but not requiring the condition of idempotence), such that $t \otimes -\colon [0, 1] \to [0, 1]$ is left-[[continuous function|continuous]] for any $t \in [0, 1]$, i.e., such that $\underset{s \to r^-}{\lim}\; t \otimes s = t \otimes r$.

Left continuity is equivalent to the condition that $t \otimes -$ preserves [[suprema]]. By the [[adjoint functor theorem]] for posets, this implies that $t \otimes -$ has a [[right adjoint]] $(t \Rightarrow -) \,\colon\  [0, 1] \to [0, 1]$, or in other words that

$$a \otimes b \leq c\;\;\; \Leftrightarrow\;\;\; a \leq b \Rightarrow c.$$
By considering the case $a = (b \Rightarrow c)$, one then easily derives the corresponding law of [[modus ponens]], $(b \Rightarrow c) \otimes b \;\leq\; c$.

The resulting binary operation $\Rightarrow$ is often referred to in the literature as a *residuum* associated with $\otimes$. In alternative language, this class of "fuzzy logics" is identified with the class of commutative [[quantale]] structures on $[0, 1]$.

Many examples of fuzzy logics are t-norm fuzzy logics, including [[basic fuzzy logic]], [[monoidal t-norm logic]], [[Gödel logic]], [[Łukasiewicz logic]], and [[product fuzzy logic]].

## Related entries

- [[Gödel logic]]

## References

* {#SEP} Stanford Encyclopedia of Philosophy, [Fuzzy Logic](https://plato.stanford.edu/entries/logic-fuzzy/)

* Wikipedia, [Fuzzy logic](https://en.wikipedia.org/wiki/Fuzzy_logic)

In relation to [[topos theory]]:

* {#Barr86} [[Michael Barr]], *Fuzzy set theory and topos theory*, Canad. Math. Bull. **29** 4 (1986) 501-508 [[doi:10.4153/CMB-1986-079-9](https://doi.org/10.4153/CMB-1986-079-9), [pdf](https://www.math.mcgill.ca/barr/papers/fuzzy.pdf), [[Barr-FuzzySetToposTheory.pdf:file]]]

In relation to [[linear logic]]:

* [[Michael Barr]], *Fuzzy models of linear logic*, Math. Structures Comp. Sci. **6** 3 (1996) 301–312 [[doi:10.1017/S096012950000102X](https://doi.org/10.1017/S096012950000102X), [pdf](https://www.math.mcgill.ca/barr/papers/fzymod.pdf), [[Barr-FuzzyLinearLogic.pdf:file]]]

[[!redirects fuzzy logic]]