tautological line bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The canonical [[line bundle]] over a [[projective space]] is sometimes called its "tautological line bundle". For more see at _[[classifying space]]_.

## Definition

In the following, let $k$ be a [[star-algebra|star]]-[[field]], possibly a [[skew-field]]

$$
  k \,\in\, StarSkewFields
  \,.
$$

In the context of [[algebraic topology]] or [[differential topology]], one is usually interested in $k$ being one of the three [[associative algebra|associative]] [[real normed division algebras]] ([[real numbers]], [[complex numbers]] or [[quaternions]]):

$$
  k \,\in\, \big\{ \mathbb{R}, \mathbb{C}, \mathbb{H}\big\}
$$

equipped with their canonical [[quaternionic conjugation|conjugation]] operations.

The relevance of the [[quaternions]] here is the reason to insist on the generality of [[skew-fields]] and hence of some pedantry about order of products and distinction of left/right [[actions]] in the following. (For [[octonions]] the notion of tautological lines should make some sense over the first [[octonionic projective space]] but not beyond.)

Let $n \in \mathbb{N}$ be a [[natural number]].

### Tautological line bundle and blow-up
  {#BareDefinition}

The _tautological $k$-line bundle_ over the [[projective space]] $k P^n$ is the following vertical [[bundle]] map:

\[
  \label{TautologicalBundleProjection}
  \array{
    \mathcal{L}_{k P^n}
    & \coloneqq &
    \frac{
      (k^{n+1} \setminus \{0\}) \times k^\ast
    }{
      k^\times
    }
    &
    \overset{
      [v,z]
      \mapsto
      \big(
        [v], v \cdot z
      \big)
    }{\hookrightarrow}
    &
    \frac{
      k^{n+1} \setminus \{0\}
    }{
      k^\times
    }
    \times
    k^{n+1}
    \\
    \big\downarrow
    &&
    \big\downarrow {}^{\mathrlap{
       \frac{id \times pt}{ k^\times }
    }}
    \\
    k P^n
    &=&
    \frac{
      (k^{n+1} \setminus \{0\}) \times \ast
    }{
      k^\times
    }
  }
\]

Here:

* $k^\times \,\coloneqq\, k \setminus \{0\}$ is the [[group of units]] of $k$;

* $k^{n+1} \coloneqq \underset{n+1\;summands}{\underbrace{k \oplus \cdots \oplus k}}$ is the canonical $n+1$-[[dimension|dimensional]] $k$-[[vector space]],

  whose elements we will also denote as [[lists]]

  \[
    \label{CoordinatesAsListsOfElements}
    v \,=\, (v_1, v_{rest}) \,=\, (v_1, v_2, \cdots, v_{n+1})
    \,,
  \]

  regarded with the _right_ $k^\times$-[[group action]]:

  \[
    \label{RightDirectMultiplicationAction}
    \array{
       k^{n+1} \times k^\times
       &
         \overset{ \;\;\;\; }{\longrightarrow}
       &
       k^{n+1}
       \\
       \big(
         (v_1, \, \cdots, \, v_{n+1}),
         \,
         z
       \big)
       &\mapsto&
       (v_1 \!\cdot\! z,\, \cdots, \, v_{n+1} \!\cdot\! z)
       \,,
    }
  \]

* $k^\ast$ is $k$ equipped with the right $k^\times$-action by _inverse multiplication from the left_:

  \[
    \label{DualActionOnk}
    \array{
      k^\times \times k^\ast
      &\longrightarrow&
      k^\ast
      \\
      (z,g) &\mapsto& z^{-1} \cdot g
    }
  \]

  (equivalently, (eq:RightDirectMultiplicationAction) and  (eq:DualActionOnk) are left actions of the [[opposite group]] $(k^\times)^{op}$)

* $\frac{(-) \times (-)}{k^\times}$ denotes the [[quotient space]] of a [[product space|produc]] of right $k^\times$-[[G-space|spaces]] by their [[diagonal action]];

* $[-]$ denotes its elements as [[equivalence classes]] of elements of the original space;

* so that, for $z \,\in\, k^\times$,

  $$
    [v,\,z]
      \;=\;
    [v,\, z \!\cdot\! 1]
      \;=\;
    [v \!\cdot\! z ,\, 1]
      \;\;\in\;\;
    \frac{
      (V \setminus \{0\}) \times k^\ast
    }{
      k^\times
    }
  $$

* and hence so that the quotient construction in (eq:TautologicalBundleProjection) is equivalently the $k$-[[fiber bundle|fiber]] _[[associated bundle]]_ to the $k^\times$-[[principal bundle]] (here "$T$" is for [[torsor]]):

\[
  \label{TautologicalPrincipalBundleOverProjectiveSpace}
  \array{
    T_{k P^n}
    & \coloneqq &
    k^{n+1} \setminus \{0\}
    \\
    \big\downarrow
    &&
    \big\downarrow {}^{\mathrlap{
      quotient\;map
    }}
    \\
    k P^n
    &=&
    (k^{n+1} \setminus \{0\}) / k^\times
    \,.
  }
\]

For $k$ a [[star algebra]], the quotient space (eq:TautologicalBundleProjection) becomes a $\mathbb{K}$-[[vector bundle]] (specifically: [[line bundle]]) via the residual left action of $\mathbb{K}$ on itself by _[[quaternionic conjugation|conjugate]] right multiplication_.

The [[line bundle]] (eq:TautologicalBundleProjection) is "tautological" in the sense that its [[fiber]] over a point labeled $[v]$ -- which may be regarded as the _name_ of the line [[linear span|spanned]] by the vector $v \in k^{n+1}$ -- consists of all the points $v \!\cdot\! z$ _on that line_ -- as made explicit by the horizontal map in (eq:TautologicalBundleProjection).

Often the tautological line bundle is referred to via the notation "$\mathcal{O}_k(-1)$", which in [[algebraic geometry]] is standard notation for its [[abelian sheaf]] of [[sections]] (see e.g. [Wirthmüller 12, p. 14 (16 of 67)](#Wirthmuller12)):

* "$\mathcal{L}_{k P^n}$" $\;\;\;\leftrightarrow\;\;\;$ "$\mathcal{O}_k(-1)$"

The [[corestriction]] of the horizontal map in (eq:TautologicalBundleProjection) to $k^{n+1}$

$$
    \mathcal{L}_{k P^n}
    \coloneqq
    \frac{
      (k^{n+1} \setminus \{0\}) \times k^\ast
    }{
      k^\times
    }
    \overset{
      [v,z]
      \mapsto
       v \cdot z
    }{\longrightarrow}
    k^{n+1}
$$

exhibits the total space of the tautological bundle as the "[[blow-up]]" of the origin of $k^{n+1}$.

{#Illustration} The following illustration shows the tautological [[real line bundle]] over 1-dimensional [[real projective space]], but the general picture is "the same", up to higher dimensionality of all spaces involved:

\begin{imagefromfile}
  "file_name": "TautologicalLineBundleOverRP1.jpg",
  "width": 700
\end{imagefromfile}

### Dual tautological line bundle and its Thom space
 {#DualTautologicalLineBundle}

The [[dual vector bundle|dual line bundle]] of the tautological line bundle (eq:TautologicalBundleProjection) is

\[
  \label{DualTautologicalBundleProjection}
  \array{
    \mathcal{L}^\ast_{k P^n}
    & \coloneqq &
    \frac{
      (k^{n+1} \setminus \{0\}) \times k
    }{
      k^\times
    }
    &
    \overset{
      [v,z]
      \mapsto
      [ (v , z) ]
    }{\hookrightarrow}
    &
    \frac{
      k^{n+2} \setminus \{0\}
    }{
      k^\times
    }
    &
    =
    \,
    k P^{n+1}
    \\
    \big\downarrow
    &&
    \big\downarrow {}^{\mathrlap{
       \frac{id \times pt}{ k^\times }
    }}
    \\
    k P^n
    &=&
    \frac{
      (k^{n+1} \setminus \{0\}) \times \ast
    }{
      k^\times
    }
  }
\]

with [[typical fiber]] $k$ instead of $k^\ast$, meaning that the [[action]] of $k^\times$ on the fibers is now the direct right multiplication action (eq:RightDirectMultiplicationAction), instead of the dual action (eq:DualActionOnk).

Often the dual tautological line bundle is referred to via the notation "$\mathcal{O}_k(1)$", which in [[algebraic geometry]] is standard notation for its [[abelian sheaf]] of [[sections]]:

* "$\mathcal{L}^\ast_{k P^n}$" $\;\;\;\leftrightarrow\;\;\;$ "$\mathcal{O}_k(1)$".

The horizontal map in (eq:DualTautologicalBundleProjection) [[embedding of topological spaces|embeds]] the [[complement]] of the single point $[(v\! = \! 0,\,1)] \in k P^{n+1}$. That point however is the [[limit of a sequence|limit]] as $z \to \infty$, hence is the image of the base point as (eq:DualTautologicalBundleProjection) [[extension|extends]] to a map on the [[Thom space]] of the dual tautological line bundle:

\[
  \label{ThomSpaceOfDualTautologicalBundleIsNextProjectiveSpace}
  Th
  \left(
    \mathcal{L}^\ast_{k P^n}
  \right)
    \underoverset
    {\simeq}
    {
      [v,z]
      \mapsto
      \left\{
      \array{
        [(0,1)] &\vert& z = \infty
        \\
        [ (v , z) ] &\vert& else
      }
      \right.
    }{\longrightarrow}
   k P^{n+1}
  \,.
\]

(E.g.: [Tamaki-Kono 06, Part III, Lemma 3.8](#TamakiKono06), bewaring that these authors secretly identify lines with dual lines -- as seen from the inner product used in the second but last line on p. 46. Also see [Conner-Floyd 66, Part I, Prop. 4.3](#ConnerFloyd66) for an alternative perspective in terms of [[coset spaces]].)

Notice how the inclusion of the point at $[v,\, z\!=\!\infty] \leftrightarrow [(v\!=\!0,\,1)]$ in the Thom space interplays with the condition that $v = 0$ is excluded in the base space.

Under the canonical inclusion of projective spaces $k P^n \hookrightarrow k P^{n+1}$ their dual tautological line bundles (eq:DualTautologicalBundleProjection) evidently [[pullback bundle|pullback]] to each other, and their total spaces compatibly include into each other:

$$
  \array{
    [v,z] &\mapsto& [(0,v),z]
    \\
    \mathcal{L}^\ast_{k P^n}
    &
    \overset{\;\;\;\;\;\;}{\hookrightarrow}
    &
    \mathcal{L}^\ast_{k P^{n+1}}
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    k P^n
    &
    \overset{\;\;\;\;\;\;}{\hookrightarrow}
    &
    k P^{n+1}
    &
    \\
    [v] &\mapsto& [(0,v)]
    \,.
  }
$$

Here the coordinate expressions make manifest that
the induced inclusions of the [[Thom spaces]] of the tautological line bundles recover,
under the identification (eq:ThomSpaceOfDualTautologicalBundleIsNextProjectiveSpace),
the canonical inclusion of the projective spaces:

\[
  \label{InclusionsOfThomSpacesOfTautologicalLineBundles}
  \array{
    [(v,z)]
    &\mapsto&
    [v,z]
    &\mapsto&
    [(0,v),z]
    &\mapsto&
    [(0,v,z)]
    \\
    k P^{n+1}
    &=&
    Th
    \big(
      \mathcal{L}^\ast_{k P^n}
    \big)
    &
    \overset{\;\;\;\;\;\;}{\hookrightarrow}
    &
    Th
    \big(
      \mathcal{L}^\ast_{k P^{n+1}}
    \big)
    &=&
    k P^{n+2}
    \\
    &&
    {}^{\mathllap{zero \atop section}}
    \big\uparrow
    &&
    \big\uparrow
    {}^{\mathrlap{zero \atop section}}
    \\
    k P^n
    &=&
    k P^n
    &
    \overset{\;\;\;\;\;\;}{\hookrightarrow}
    &
    k P^{n+1}
    &=&
    k P^{n+1}
    \\
    [v]
    &\mapsto&
    [v]
    &\mapsto&
    [(0,v)]
    &\mapsto&
    [(0,v)]
    \,.
  }
\]

Notice how, in this coordinatization, the projective spaces are horizontally included by adjoining a 0-coordinate to the _left_ of the list (eq:CoordinatesAsListsOfElements) and vertically by adjoining a 0-coordinate to the _right_.

It follows that under forming a suitable [[colimit]] over this diagram as $n \to \infty$, in a suitable category (typically in [[homotopy types]] of [[topological spaces]] if $k$ is a [[topological field]], see also [below](#AsAtopologicalLieBundle)), the infinite projective space wants to be equivalent to the Thom space of its dual tautological line bundle:

$$
  k P^\infty
    \;\simeq\;
  k P^{\infty + 1}
    \;\simeq\;
  Th
  \big(
    \mathcal{L}^\ast_{k P^\infty}
  \big)
  \,.
$$

See at _[[zero-section into Thom space of universal line bundle is weak equivalence]]_.

\linebreak

### As a topological line bundle
  {#AsAtopologicalLieBundle}

We make fully explicit how the tautological line bundle (eq:TautologicalBundleProjection) is a [[locally trivial bundle|locally trivial]] [[topological vector bundle]]. Hence regard now $k$ as a [[topological field]], either

* $k = \mathbb{R}$ the [[real numbers]],

* or $k = \mathbb{C}$ the [[complex numbers]]

equipped with their [[Euclidean space|Euclidean]] [[metric topology]].

+-- {: .num_defn #ToplogicalProjectiveSpace}
###### Definition
**(topological [[projective space]])**

Let $n \in \mathbb{N}$. Consider the [[Euclidean space]] $k^{n+1}$ equipped with its [[metric topology]], let $k^{n+1} \setminus \{0\} \subset k^{n+1}$ be the [[topological subspace]] which is the [[complement]] of the origin, and consider on its underlying set the [[equivalence relation]] which identifies two points if they differ by [[multiplication]] with some $c \in k$ (necessarily non-zero):

$$
  (\vec x_1 \sim \vec x_2)
   \;\Leftrightarrow\;
  \left(
    \underset{c \in k}{\exists}
    ( \vec x_2 = c \vec x_1 )
  \right)
  \,.
$$

The [[equivalence class]] $[\vec x]$ is traditionally denoted

$$
  [x_1 : x_2 : \cdots : x_{n+1}]
  \,.
$$

Then the _[[projective space]]_ $k P^n$ is the corresponding [[quotient topological space]]

$$
  k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim
  \,.
$$

=--

+-- {: .num_defn #TopologicalProjectiveSpaceStandardOpenCover}
###### Definition
**(standard open cover of topological projective space)**

For $n \in \mathbb{N}$ the _standard open cover_ of the projective space $k P^n$ (def. \ref{ToplogicalProjectiveSpace}) is

$$
  \left\{
    U_i \subset k P^n
  \right\}_{i \in \{1, \cdots, n+1\}}
$$

with

$$
  U_i
    \coloneqq
  \left\{
    [x_1 : \cdots : x_{n+1}] \in k P^n
    \;\vert\;
    x_i \neq 0
  \right\}
  \,.
$$

To see that this is an open cover:

1. This is a cover because with the orgin removed in $k^n \setminus \{0\}$ at every point $[x_1: \cdots : x_{n+1}]$
   at least one of the $x_i$ has to be non-vanishing.

1. These subsets are open in the [[quotient topology]] $k P^n = (k^n \setminus \{0\})/\sim$, since their [[pre-image]] under the quotient co-projection $k^{n+1} \setminus \{0\} \to k P^n$ coincides with the pre-image $(pr_i\circ\iota)^{-1}( k \setminus \{0\} )$ under the [[projection]] onto the $i$th coordinate in the [[product topological space]] $k^{n+1} = \underset{i \in \{1,\cdots, n\}}{\prod} k$ (where we write $k^n \setminus \{0\} \overset{\iota}{\hookrightarrow} k^n \overset{pr_i}{\to} k$).

=--

+-- {: .num_defn #TautologicalTopologicalLineBundle}
###### Definition
**(tautological topological line bundle)

For $k$ a [[topological field]] and $n \in \mathbb{N}$, the _tautological line bundle_ over the [[projective space]] $k P^n$ is topological $k$-[[line bundle]] whose total space is the following [[subspace]] of the [[product space]] of the [[projective space]] $k P^n$ with $k^n$:

$$
  T
   \coloneqq
  \left\{
    ( [x_1: \cdots : x_{n+1}], \vec v)
    \in
    k P^n \times k^{n+1}
    \;\vert\;
    \vec v \in \langle \vec x\rangle_k
  \right\}
  \,,
$$

where $\langle \vec x\rangle_k \subset k^{n+1}$ is the $k$-[[linear span]] of $\vec x$.

(The space $T$ is the space of pairs consisting of the "name" of a $k$-line in $k^{n+1}$ together with an element of that $k$-line)

This is a bundle over [[projective space]] by the projection function

$$
  \array{
    T &\overset{\pi}{\longrightarrow}& k P^n
    \\
    ([x_1: \cdots : x_{n+1}], \vec v) &\mapsto& [x_1: \cdots : x_{n+1}]
  }
  \,.
$$

=--

+-- {: .num_prop #WellDefinedTautologicalTopologicalLineBundle}
###### Proposition
**(tautological topological line bundle is well defined)**

The tautological line bundle in def. \ref{TautologicalTopologicalLineBundle} is well defined in that it indeed admits a [[local trivialization]].

=--

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

We claim that there is a local trivialization over the canonical cover of def. \ref{TopologicalProjectiveSpaceStandardOpenCover}. This is given for $i \in \{1, \cdots, n\}$ by

$$
  \array{
    U_i \times k &\overset{}{\longrightarrow}& T\vert_{U_i}
    \\
    ( [x_1 : \cdots x_{i-1}: 1 : x_{i+1} : \cdots : x_{n+1}] , c )
      &\mapsto&
    ( [x_1 : \cdots x_{i-1} : 1 : x_{i+1} : \cdots : x_{n+1} ], (c x_1, c x_2, \cdots , c x_{n+1}) )
  }
  \,.
$$

This is clearly a [[bijection]] of underlying sets.

To see that this function and its inverse function are continuous, hence that this is a [[homeomorphism]] notice that this map is the [[extension]] to the [[quotient topological space]] of the analogous map

$$
  \array{
    ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , c)
     &\mapsto&
    ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , (c x_1, \cdots c x_{i-1}, c, c x_{i+1}, \cdots, c x_{n+1}) )
  }
  \,.
$$

This is a [[polynomial]] function on [[Euclidean space]] and since [[polynomials are continuous]], this is continuous. Similarly the [[inverse function]] lifts to a [[rational function]] on a subspace of Euclidean space, and since [[rational functions are continuous]] on their domain of definition, also this lift is continuous.

Therefore by the [[universal property]] of the [[quotient topology]], also the original functions are continuous.

=--

## Examples

### Möbius strip

The tautological line bundle over the 1-dimensional [[real projective space]] $\mathbb{R}P^1$ is the [[Möbius strip]].

### Over Riemann sphere

The [[basic complex line bundle on the 2-sphere]] is the tautological [[complex line bundle]] over the [[complex projective space]] $\mathbb{C}P^1 \simeq S^2$ (the [[Riemann sphere]]).

This plays a key role in [[topological K-theory]] and more generally in [[complex oriented cohomology theory]].

## Related concepts

* [[canonical line bundle]], [[universal complex line bundle]]

* [[hyperplane line bundle]]

* [[tautological equivariant line bundle]]

* [[resolution of singularities]]

* [[Bott element]]

## References

Discussion with an eye towards [[complex-oriented cohomology theory]]:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], p. 21 (28 of 120) in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Chapter 3, Section 2 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([[GeneralizedCohomology.pdf:file]], [ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Lecture notes with an eye towards [[topological K-theory]]:

* {#Wirthmuller12} [[Klaus Wirthmüller]], section 2 of _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

See also:

* Wikipedia, _[Tautological bundle](https://en.wikipedia.org/wiki/Tautological_bundle)_

[[!redirects tautological line bundles]]


tautology

# Tautologies
* table of contents
{: toc}

## Idea

In common language, a *tautology* refers to a [[proposition]]  whose [[truth]] is felt to be "trivial" or "completely obvious" or "not requiring argument or highlighting", for instance since it follows just by unwinding definitions ("true by definition").

In [[classical logic|boolean]] [[propositional logic]], a _tautology_ is any [[proposition]] whose validity is unconditional --- independent of the validity of its propositional variables. Several generalizations are possible. On the one hand, the family of boolean tautologies is also the family of boolean [[theorems]]: a proposition is a boolean tautology iff it has a boolean proof. On the other hand, construing a boolean proposition as a [[universal quantification|universalized]] [[equation]] in the language of [[boolean algebras]], the boolean propositions are also the (universal, first-order) propositions of the form $P(x_1,...,x_n) = \top$ that are valid in every boolean algebra.

Of course, tautologies exist in other logics besides boolean logic, although boolean logic is perhaps the simplest nontrivial case.

## Definition

Given a [[logic]], a [[context]] $\Gamma$ within that logic, and a class of [[models]] of $\Gamma$, a __tautology__ is a [[proposition]] $\phi$ in $\Gamma$ that is true in all models; that is,
$$ \mathcal{M} \vDash \phi $$
for every model $\mathcal{M}$.

## Discussion

Compare the notion of __[[theorem]]__, sometimes called a _syntactic tautology_, which asks that $\phi$ be *provable* in $\Gamma$:

$$
 \Gamma
   \;\vdash\;
 \phi
  \,.
$$

In the best behaved cases, each context has a [[free model]] such that the theorems are precisely the tautologies for the free model.  (For example, in the case of boolean propositional logic, the free model for the context with $n$ variables is the [[free object|free]] [[boolean algebra]] on $n$ generators.)  Even without this, there may be a class of models that can be proved to be [[soundness theorem|sound]] (every theorem is a tautology) and [[completeness theorem|complete]] (every tautology is a theorem).

## Related concepts

[[!include mathematical statements --- contents]]

## References

See also:

* Wikipedia, *Tautology (language)*

* Wikipedia, *Tautology (logic)*

* Wikipedia, *Tautology (rule of inference)*

[[!redirects tautology]]
[[!redirects tautologies]]


Taylor Dupuy

[[!redirects Taylor Duypuy]]
[[!redirects Taylor Duypuy]]

* [webpage](http://www.uvm.edu/~tdupuy/)

## Selected writings

On [[Mochizuki's corollary 3.12]]:

* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_ [[arxiv:2004.13228](https://arxiv.org/abs/2004.13228)]

and on [[Szpiro's conjecture]]:

* [[Taylor Dupuy]], [[Anton Hilado]], _Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12_ [[arxiv:2004.13108](https://arxiv.org/abs/2004.13108)]

## Related entries

* [[anabelian geometry]]

* [[inter-universal Teichmüller theory]]

* [[initial Θ-data]]

* [[Mochizuki's corollary 3.12]]

category: people


Taylor Hughes

* [webpage](https://physics.illinois.edu/people/directory/profile/hughest)

## Selected writings

Predicting a [[quantum spin Hall effect]] in [[topological insulators]] based on mercury telluride (HgTe quantum wells):

* [[Andrei Bernevig]], [[Shou-Cheng Zhang]], _Quantum Spin Hall Effect_, Phys. Rev. Lett. 96, 106802 (2006) ([arXiv:cond-mat/0504147](https://arxiv.org/abs/cond-mat/0504147), [doi:10.1103/PhysRevLett.96.106802](https://doi.org/10.1103/PhysRevLett.96.106802))

* [[Andrei Bernevig]], [[Taylor Hughes]], [[Shou-Cheng Zhang]], _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science 15 December 2006: __314__, n. 5806, pp. 1757-1761 ([doi:10.1126/science.1133734](http://dx.doi.org/10.1126/science.1133734), [arXiv:cond-mat/0611399](https://arxiv.org/abs/cond-mat/0611399))

category: people


Taylor series

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

# Contents
* table of contents
{: toc}

## Idea

The *Taylor series* of a [[smooth function]] $f$ (at a given point $c$) is a [[formal power series]] (in $x - c$) whose [[partial sum|partial sums]] are the [[Taylor polynomials]] of $f$ (at $c$).  As the Taylor polynomials are approximations of $f$ by [[polynomials]] (up to a given degree), so the Taylor series is an approximation of $f$ by an [[analytic function]] (or at least an [[asymptotic expansion]] that attempts to be this).

See also [[Taylor's theorem]] for error estimates in the convergence of Taylor series.

## Definition

Let $f \in C^\infty(\mathbb{R})$ a [[smooth function]] with $n$th [[derivative]] $f^{(n)} \in C^\infty(\mathbb{R})$ and let $c$ be a [[real number]].

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

The **Taylor series** of $f$ at $c$ is the [[formal power series]]

$$
  \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(c) (x-c)^n
  \,.
$$

=--

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

For $f \in C^\infty(\mathbb{R})$ a [[smooth function]] with $n$th [[derivative]] $f^{(n)} \in C^\infty(\mathbb{R})$, its **Maclaurin series** is its Taylor series at [[zero]]:

$$
  \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n
  \,.
$$
=--

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

Similar definitions apply to functions on any [[Cartesian space]] or [[smooth manifold]].
=--

## Properties

Recall that a ([[partial function|partial]]) function $f$ is [[analytic function|analytic]] at $c$ (in the [[interior]] of $\dom f$) if there exists a [[power series]] $P$ at $c$ and a [[neighbourhood]] $U$ of $c$ such that, for all $x \in U \cap \dom f$, $P(x)$ converges to $f(x)$.

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

If $f$ is analytic at $c$, then the only power series witnessing this is the Taylor series of $f$ at $c$ (so in particular, the Taylor series exists; analytic functions are [[smooth function|smooth]]).
=--

In contrast, a smooth function need not be analytic; the classic counterexample is a [[bump function]].  In fact, the Taylor series of $f$ at $c$ might not converge to $f$ anywhere except at $c$, either because the Taylor series has vanishing [[radius of convergence]] or because it converges to something else (an analytic function with the same [[jet]] as $f$ but a different [[germ]]).  However, we can say this:

+-- {: .num_prop}
###### Proposition
**(Taylor series is [[asymptotic series]])**

The Taylor series of a [[smooth function]] $f$ is always an [[asymptotic expansion]] of $f$.

=--

This follows from the [[Hadamard lemma]], see [this example](asymptotic series#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries) for details.

+-- {: .num_theorem}
###### Theorem
**(Borel's theorem)**

The map

$$
  C^\infty(\mathbb{R}^{k+l})
  \to
  C^\infty(\mathbb{R}^k) [ [ X_1, \cdots X_l] ]
$$

obtained by forming Taylor series in $l$ variables is [[surjection|surjective]].
=--

In particular, every [[power series]] in one real variable is the Taylor series of some [[smooth function]] on the [[real line]] (even if it has vanishing radius of convergence and so is not the Taylor series of any analytic function).

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

The proof is reproduced for instance in [[Models for Smooth Infinitesimal Analysis|MSIA, I, 1.3]]
=--

See more at [[Borel's theorem]].

## Examples

* of the [[natural logarithm]]: see *[[Mercator series]]*

## Related concepts

* [[series]], [[asymptotic series]]

* [[analytic geometry]]

[[!include infinitesimal and local - table]]

* analogue in ([[stable homotopy theory|stable]]) [[homotopy theory]]/[[Goodwillie calculus]]: _[[Taylor tower]]_

[[!redirects Taylor series]]
[[!redirects Taylor series expansion]]
[[!redirects Mac Laurin series]]
[[!redirects MacLaurin series]]
[[!redirects Maclaurin series]]

[[!redirects Taylor expansion]]
[[!redirects Taylor expansions]]


Taylor's theorem

# Taylor\'s Theorem
* table of contents
{: toc}

## Idea

The [[Taylor polynomial|Taylor polynomials]] of a [[differentiable function]] approximate it by [[polynomial]] functions; the various versions of Taylor's Theorem describe how good this approximation is.  The limiting case of this is the [[Taylor series]].

## Preliminary definitions

Let $f$ be a [[partial function]] on a [[cartesian space]] $\mathbb{R}^d$ (or $\mathbb{C}^d$), let $c$ be a [[concrete point|point]] in the cartesian space, and let $k$ be a [[natural number]] (actually we can allow $k \geq -1$ also, using [[negative thinking]]).

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

If  $f$ is [[differentiable function|differentiable]] $k$ times at $c$, then the __Taylor polynomial__ of $f$ at $c$ with order $k$ is the unique [[polynomial]] in $d$ variables of degree at most $k$ whose derivatives at $c$ match those of $f$ up to order $k$.
=--

It is straightforward (by differentiating a polynomial [[ansatz]]) to find an explicit formula (in the variable $x$):
$$ T^k_{f,c}(x) = \sum_{n = 0}^k \frac{f^{(n)}(c) (x-c)^n}{n!} .$$
We have written this as if $f$ is a function of one variable; but interpret $n$ as a [[multi-index]] whose length $\ell$ satisfies $0 \leq \ell \leq k$, with $f^{(n)}$ the mixed [[partial derivative]] given by that multi-index, $n!$ the [[factorial]] of $\ell$, and $(x-c)^n$ a product of $\ell$ factors of the form $x_i - c_i$ (where $i$ is an index appearing in the multi-index $n$).  Then this works in any cartesian space.  (Note that we include all possible orderings of a multi-index as separate terms; this leads to repeated terms that, when combined, will cancel some of the factors in the factorial.)

In the case of a function of several variables, we can also manage the maximum degrees in the various variables separately, although nobody seems to bother with this.

## Statements

There are several different versions of Taylor\'s Theorem, all stating an extent to which a Taylor polynomial of $f$ at $c$, when evaluated at $x$, approximates $f(x)$.

Here is the simplest statement, which requires only continuity of $f$ (which we really only need for $k = 0$, since it\'s automatic for $k \geq 1$ and not actually necessary for $k = -1$):
+-- {: .num_theorem #limit}
###### Theorem

If $f$ is [[continuous function|continuous]] at $c$, then
$$ \lim_{x \to c} \frac{{f(x) - T^k_{f,c}(x)}}{{\|x - c\|}^k} = 0 .$$
=--
(The norm ${\|\cdot\|}$ can be left out in one variable, or placed in the numerator to handle all components of a vector-valued function at once.)

If $f^{(k)}$ has some continuity, then we get a version of Taylor\'s Theorem with an integral:
+-- {: .num_theorem #integral}
###### Theorem

If $f^{(k)}$ is [[absolutely continuous function|absolutely continuous]] on $[\min(x,c),\max(x,c)]$, then
$$ f(x) = T^k_{f,c}(x) + \int_{t=a}^x \frac{f^{(k+1)}(t) (x-t)^{k+1} \,\mathrm{d}t}{(k + 1)!} .$$
=--
(Note that $f^{(k+1)}$ is [[almost function|defined almost everywhere]] and [[integrable function|Lebesgue integrable]], because $f^{(k)}$ is absolutely continuous.)

This result is not directly very useful is one is using Taylor polynomials to approximate $f$ where one doesn\'t know its behaviour, but we have a corollary which can often be used:
+-- {: .num_theorem #estimate}
###### Theorem

If $f^{(k)}$ is [[absolutely continuous function|absolutely continuous]] on $[\min(x,c),\max(x,c)]$, then
$$ {|f(x) - T^k_{f,c}(x)}| \leq \frac{M {|x-c|}^{k+1}}{(k + 1)!} ,$$
where $M$ is any [[essentially bounded function|essential upper bound]] of $f^{(k+1)}$ on $[\min(x,c),\max(x,c)]$.
=--
In many cases, finding a good upper bound of $f^{(k+1)}$ can be reduced to solving $f^{(k+2)}(t) = 0$.

There are also versions that generalize the [[mean value theorem]]:
+-- {: .num_theorem #MVT}
###### Theorem

If $f^{(k)}$ is differentiable on $[\min(x,c),\max(x,c)]$, then, for some $t \in [\min(x,c),\max(x,c)]$,
$$ f(x) = T^k_{f,c}(x) + \frac{f^{(k+1)}(t) (x-c)^{k+1}}{(k + 1)!} .$$
=--

All of these can be generalized in a fairly straightforward way to functions of several variables.

## Related concepts

* [[Hadamard lemma]]

[[!redirects Taylor's theorem]]
[[!redirects Taylor's theorems]]
[[!redirects Taylor's theorem]]
[[!redirects Taylor's theorems]]
[[!redirects Taylor\'s theorem]]
[[!redirects Taylor\'s theorems]]

[[!redirects Taylor polynomial]]
[[!redirects Taylor polynomials]]


TCFT

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

#Contents#
* table of contents
{:toc}

## Idea

The term _topological conformal field theory_ (TCFT) is used for a  linearization or [[stabilization]] of something that is like a [[conformal field theory]] (CFT) up to [[homotopy]]. It is a notion somewhere half-way between a (2-dimensional) [[TQFT]] and a [[CFT]].

> (Actually, the remnant of conformal structure here should be just
> an artefact of  the way to parameterize the moduli space of surfaces.
> As the classification result by Lurie discussed below shows,
> TCFTs are really $(\infty,2)$-TFTs.)

This formalizes the physics notion of "the [[topological string]]", a _topologically twisted_ superconformal field theory, such as, notably, the [[A-model]] and the [[B-model]]. TCFTs are therefore a tool for formalizing [[homological mirror symmetry]].

Recall that an ordinary [[conformal field theory]] (CFT) is, in [[FQFT]]-language, a [[symmetric monoidal functor]] on a [[category]] $Bord_2^{conf}$ whose objects are disjoint unions of intervals and circles, and whose [[morphism]]s are [[Riemann surface]]s with these 1d manifolds as incoming and outgoing punctures.

Since Riemann surfaces form a well-understood [[moduli space]], one can turn this also into a [[Top]]-[[enriched category]], i.e. an [[(∞,1)-category]],
$Bord_{2}^{conf,top}$ whose [[hom-space]]s are these [[moduli space]]s of [[Riemann surface]]s with given 1d manifolds as incoming and outgoing punctures.

A "truly topological conformal field theory" would be an [[(∞,1)-functor]] of the form

$$
  Bord_2^{conf,top} \to \infty Grpd
$$

or similar. But what is actually called a "topological conformal field theory" is the linearization or [[stabilization]] of this:

in a TCFT, this [[(∞,1)-category]] of conformal [[cobordism]]s is replaced by a [[stable (∞,1)-category]] whose [[hom-object]]s (when modeled by a [[dg-category]]) are just the [[homology]] [[chain complex]]s of the original [[hom-space]]s.

Write $Bord_2^{conf,dg}$ for the resulting [[symmetric monoidal category|symmetric monoidal]] [[dg-category]] of Riemann cobordisms. Then a TCFT is a an homotopy-symmetric monoidal [[chain complex]]-[[enriched functor]]

$$
  F : Bord_2^{conf,dg} \to Ch_\bullet
$$

to the symmetric monoidal [[dg-category]] of chain complexes.

This means in particular that when two [[Riemann surface]]s $\Sigma_1$ and $\Sigma_2$ are homologous as chains in the [[moduli space]] of Riemann surfaces, then the TCFT will send them to two equivalent morphisms $f_{\Sigma_1}$ and $f_{\Sigma_2}$ of chain complexes between the in- and the output states. The equivalence between $f_{\Sigma_1}$ and $f_{\Sigma_2}$, however, is not unique neither up to equivalence. Rather, it funtorially depends on the 1-chain realizing the homology equivalence between $\Sigma_1$ and $\Sigma_2$ as 0-chains in the moduli space. In particular, two non-homologous 1-chains between $\Sigma_1$ and $\Sigma_2$ will in general lead
to non-equivalent equivalences between $f_{\Sigma_1}$ and $f_{\Sigma_2}$.

## Definition
 {#Definition}

According to [[On the Classification of Topological Field Theories|ClassTFT]]
the original definition of the domain for TCFTs can be formulated as follows (without reference to any conformal or Riemann structure).

**Definition**
The $(\infty,2)$-category $Bord^{nc}_2$ of non-compact 2-dimensional cobordism is defined as follows:

* The objects of $Bord^{nc}_2$ are oriented 0-manifolds.

* Given a pair of objects $X, Y \in Bord^{nc}_2$ , a 1-morphism from $X$ to $Y$ is an oriented bordism $B : X \to Y$.

* Given a pair of 1-morphsims $B,B' : X \to Y$ in $Bord^{nc}_2$, a 2-morphism from $B$ to $B'$ in $Bord^{nc}_2$ is
an oriented bordism $\Sigma: B \to B'$ (which is trivial along $X$ and $Y$) with the following property: every
connected component of $\Sigma$ has nonempty intersection with $B'$.

* Higher morphisms in $Bord^{nc}_2$ are given by (orientation preserving) diffeomorphisms, isotopies between
diffeomorphisms, and so forth.

Then, the [[cobordism hypothesis]]-theorem for $Bord^{nc}_2$ becomes

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

Let $C$ be a [[symmetric monoidal (∞,2)-category]]. Then
symmetric monoidal [[(∞,2)-functor]]s

$$
  Z : Bord^{nc}_2 \to C
$$

are equivalent to [[Calabi-Yau objects]] $A$ in $C$:
the functor $Z$ sends the point to $A$.

=--

This is [[On the Classification of Topological Field Theories|ClassTFT, theorem 4.2.11]]. One can "unfold" $Bord^{nc}_2$ and the theorem above, obtaining a statement in terms of [[symmetric monoidal (∞,1)-category|symmetric monoidal (∞,1)-categories]]. Actually it was the unfolded version to be proven first, ([Costello 04](#Costello04)).

in the particular case $C=Ch_\bullet$. We state it below in the general version given by [[Jacob Lurie]] in [[On the Classification of Topological Field Theories|ClassTFT]].

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

Let $\mathcal{OC}$ be the [[(infinity,1)-category]] of open-closed strings, described as follows:

1. objects are oriented 1-manifolds with boundary;

1. morphisms are oriented bordisms between 1-manifolds such that each
   connected component has non-vanishing intersection with the codomain
   1-manifold;

1. the higher morphisms are given by orientation preserving
   [[diffeomorphism]]s, isotopies between these, and so forth.

Write $\mathcal{O}$ for the full [[sub-(∞,1)-category]] on
disjoint unions of intervals (open strings sector).

=--

This is [[On the Classification of Topological Field Theories|ClassTFT, above theorem 4.2.13]].

## Classification
 {#Classification}

### With coefficients in (algebras in) chain complexes

The original statement of the classification result for TCFTs
concerned symmetric homotopy-monoidal functors $Bord_2^{conf,dg} \to Ch_\bullet$:

+-- {: .num_defn}
###### Definition
([[Kevin Costello|Costello]], following [[Maxim Kontsevich|Kontsevich]])

1. The category of open TCFTs with set $\Lambda$ of D-branes is equivalent to that of [[Calabi-Yau categories]] with set $\Lambda$ of objects.

1. The homology of the chain complex of closed states of the universal extension of an open TCFT to an open-closed TCFT is the [[Hochschild homology]] of the corresponding [[Calabi-Yau category]].

=--

In ([Costello 04](#Costello04)) this is proven using information about cell decompositions of the moduli space of punctured Riemann surfaces, thus effectively presenting $Bord_2^{conf,dg}$ by generators-and-relations, The then theorem amounts to noticing that representations of these generators and relations define the operations in an $A_\infty$-category with pairing operation.

### General version

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

Let $C$ be a [[symmetric monoidal (∞,1)-category]]. Then
symmetric monoidal [[(∞,1)-functor]]s

$$
  Z : \mathcal{O} \to C
$$

are equivalent to [[Calabi-Yau algebra]] [[Calabi-Yau object|objects]] $A$ in $C$:
the functor $Z$ sends the interval $[0,1]$ to $A$.

=--

This is the result of spring [Cos04](http://arxiv.org/abs/math/0412149)
reformulated and generalized according to
[[On the Classification of Topological Field Theories|ClassTFT, theorem 4.2.14]].

This is a special case of the general [[cobordism hypothesis]]-theorem.

The idea of the proof is that a topological open string theory, i.e., a symmetric monoidal [[(∞,1)-functor]] $Z : \mathcal{O} \to C$ has a [[Kan extension]] to an open-closed topological string theory, i.e., to a symmetric monoidal [[(∞,1)-functor]] $Z : \mathcal{OC} \to C$, which is the unfolded version of a symmetric monoidal [[(∞,2)-functor]] from $Bord^{nc}_2$ to a symmetric monoidal $(\infty,2)$-category $C'$.

## Worldsheet and effective background theories
 {#ActionFunctionals}

One imagines generally that one obtains TCFTs, in their formal definition given above, from worldsheet [[action functional]]s as familiar from the physics literature (such as on the [[A-model]] and the [[B-model]]) by performing the [[path integral]] and finding from it a collection of [[differential form]]s on [[moduli space]] of bosonic field configurations.

It seems there is at this point no literature giving a direct construction along these lines, but there is the following:

In [Cos06](http://arxiv.org/abs/math/0605647) is constructed from the geometric input datum of a generalized [[Calabi-Yau space]] $(X,Q)$ and it is shown that

1. there is a collection of differential forms $K_{g,h}(\cdots)$on the [[moduli space]] $\mathcal{M}_{g}^{h,n}$ of [[Riemann surface]]s such that these define a 2d TCFT;

   (In the discussion leading up to Lemma 4.5.1 there. The proof that this yields a TCFT is theorem 4.5.4.)

1. the partition function of the string perturbation series for the
   above TCFT is

   $$
     \sum_{{g,n \geq 0, h \gt 0} \atop {2g-2+h+\frac{n}{2}}}
     \lambda^{2g-2+h}N^h
     \frac{1}{n!}
     \int_{\mathcal{M}_g^{h,n}}
     K_{g,h}(a^\otimes n)
   $$

   which is shown to be the partition function of a background
   [[Chern-Simons theory]] coming from the [[action functional]]

   $$
     a \mapsto S(a) = \int_X \frac{1}{2} a Q a + \frac{1}{3}a^3
     \,.
   $$

So this constructs a 2d TCFT and shows that its [[effective field theory|effective]] background quantum field theory is a [[Chern-Simons theory]]. While the action functionl on the worldsheet itself, whose [[path integral]] should give the [[differential forms]] on [[moduli space]] considered above, is not explicitly considered here, this does formalizes at least some aspects of an observation that was earlier made in ([Witten 92](#Witten92)) where it was observed that Chern-Simons theory is the effective background string theory of 2d TFTs obtained from action functionals of the A-model and the B-model.

Similarly the effective background QFT of the [[B-model]] topological string can be identified. This is known as _[[Kodeira-Spencer gravity]]_ or as _[[BCOV theory]]_.

(See also at _[[world sheets for world sheets]]_ for a similar mechanism and see at _[[super 1-brane in 3d]]_ for related "physical" strings.)

So via the detour over the effective background field theory, this sort of shows that the physicist's A-model and B-model are indeed captured by the abstract [[FQFT]] definition of TCFT as given above.

## Related concepts

* [[A-model]], [[B-model]], [[Landau-Ginzburg model]], [[homological mirror symmetry]]

* [[HCFT]]

## References

The concept is essentially a formalization of what used to be called [[cohomological field theory]] in

* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

The definition was given independently by

* {#Getzler92} [[Ezra Getzler]], _Batalin-Vilkovisky algebras and two-dimensional topological field theories_ , Comm. Math. Phys. 159(2), 265–285 (1994) ([arXiv:hep-th/9212043](http://arxiv.org/abs/hep-th/9212043))

and

* [[Graeme Segal]], _Topological field theory_ , (1999), Notes of lectures at Stanford university. ([web](http://www.cgtp.duke.edu/ITP99/segal/)). See in particular [lecture 5](http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf) ("topological field theory with cochain values").

The classification of TCFTs by [[Calabi-Yau categories]] was discussed in

* {#Costello04} [[Kevin Costello]], _Topological conformal field theories and Calabi-Yau categories_, Advances in Mathematics, Volume 210, Issue 1, (2007), ([arXiv:math/0412149](http://arxiv.org/abs/math/0412149))

* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([arXiv:math/0509264](http://arxiv.org/abs/math/0509264))

following conjectures by [[Maxim Kontsevich]], e.g.

* [[Maxim Kontsevich]], _Homological algebra of mirror symmetry_ , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.

This classification is a precursor of the full [[cobordism hypothesis]]-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

Here are notes from a seminar on these definitions and results:

* [[Peter Teichner]] and [[Kevin Costello]] _TCFT seminar_ ([pdf notes](http://math.berkeley.edu/~cpries/Hot-Topics-07.pdf))

Discussion of the construction of TCFTs from differential forms on moduli space and the way this induces by "[[second quantization]]" effective background Chern-Simons theories is in

* {#Costello06} [[Kevin Costello]], _Topological conformal field theories and gauge theories_ ([arXiv:math/0605647](http://arxiv.org/abs/math/0605647))

formalizing at least aspects of the observations in

* {#Witten92} [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_ ([arXiv:hep-th/9207094](http://arxiv.org/abs/hep-th/9207094))

* P.A. Grassi, [[Giuseppe Policastro]], _Super-Chern-Simons Theory as Superstring Theory_ ([arXiv:hep-th/0412272](http://arxiv.org/abs/hep-th/0412272))

On how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]]:

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

* [[Kevin Costello]], [[Si Li]], _Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model_ ([arXiv:1201.4501](http://arxiv.org/abs/1201.4501))

* [[Si Li]], _BCOV theory on the elliptic curve and higher genus mirror symmetry_ ([arXiv:1112.4063](http://arxiv.org/abs/1112.4063))

* [[Si Li]], _Variation of Hodge structures, Frobenius manifolds and Gauge theory_ ([arXiv:1303.2782](http://arxiv.org/abs/1303.2782))

[[!redirects TCFTs]]

[[!redirects topological conformal field theory]]
[[!redirects topological conformal field theories]]


team logic

Team logic and team semantics

In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation.

## Literature

Some early ideas are from

* W. Hodges, _Compositional semantics for a language of imperfect information_, Log. J. IGPL 5, 539–563 (1997)

The framework is explicitly developed by J. Väänänen as an extension to [[dependence logic]] by classical negation.

* Jouko Väänänen, Interactive logic. Selected papers from the 7th Augustus de Morgan workshop, 281-302, Amsterdam University Press 2007
* Fan Yang, Jouko Väänänen, _Propositional team logics_,
Annals of Pure and Applied Logic __168__:7 (2017) 1406-1441 [doi](https://doi.org/10.1016/j.apal.2017.01.007)
* J. Kontinen, J. Väänänen, _On definability in dependence logic_, Journal of Logic, Language and Information, 18(3) (2009) 317–332.
* Juha Kontinen, Ville Nurmi, _Team logic and second-order logic_, Fundamenta Informaticae 106, no. 2-4, 259-272, 2011, [doi:10.3233/FI-2011-386](https://doi.org/10.3233/FI-2011-386)


technicolor

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of the [[standard model of particle physics]], _technicolor_ is a proposal for realizing [[electroweak symmetry breaking]] not via a [[Higgs boson]] which is a [[fundamental particle]], but by realizing it as a bound state of auxiliary [[fermions]] and introduing new [[gauge fields]].

## References

* Kenneth Lane, _Two Lectures on Technicolor_ ([arXiv:hep-ph/0202255](http://arxiv.org/abs/hep-ph/0202255))

* Christopher T. Hill, Elizabeth H. Simmons, _Strong Dynamics and Electroweak Symmetry Breaking_ ([arXiv:hep-ph/0203079](http://arxiv.org/abs/hep-ph/0203079))

* Robert Shrock, _Some Recent Results on Models of Dynamical Electroweak Symmetry Breaking_ ([arXiv:hep-ph/0703050](http://arxiv.org/abs/hep-ph/0703050))

* Adam Martin, _Technicolor Signals at the LHC_ ([arXiv:0812.1841](http://arxiv.org/abs/0812.1841))

* Wikipedia _[technicolor (physics)](http://en.wikipedia.org/wiki/Technicolor_%28physics%29)_


Ted Courant

Theodore James Courant is an American mathematician.

He got his PhD in 1987 from UC Berkeley, advised by [[Alan Weinstein]].

## Related entries

* [[Courant Lie 2-algebroid]]

* [[Dirac structure]]

category: people


Teena Gerhardt

* [webpage](http://www.math.msu.edu/~teena/Home.html)

## related $n$Lab entries

* [[algebraic K-theory]]

* [[topological cyclic homology]]

category: people


Teichmüller theory

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

#Contents#
* table of contents
{:toc}

## Idea

The _Teichmüller space_ $\mathcal{T}_{\Sigma}$ of a  ([[closed manifold|closed]]) 2-[[dimension|dimensional]] [[manifold]] $\Sigma$  is the [[moduli space of complex structures]] on $\Sigma$, where two complex structures are identified if they are taken into each other by a [[homeomorphism]] $\phi \colon \Sigma \to \Sigma$ which is [[isotopy|isotopic]] to the [[identity]].

Since the [[diffeomorphism]] class of  a [[closed manifold]] of dimension 2 is given by its [[genus]] $g\in \mathbb{Z}$, typically one speaks of the Teichmüller space $\mathcal{T}_g$ for each $g$. More generally one considers $\Sigma$s equipped with $n$ punctures/[[boundary]] circle and then writes $\mathcal{T}_{g,n}$.

Teichmüller space is a [[covering space]] for the [[moduli space of curves]] over the complex numbers (see [below](#RelationToModuliStackOfCurves)). When also the singular [[nodal curve]] is included one speaks of the _[[augmented Teichmüller space]]_ which in turn is a cover of the [[Deligne-Mumford compactification]] of the moduli space of complex curves.

## Properties

### Complex structure on Teichmüller space
 {#ComplexStructureOnTeichmüllerSpace}

Teichmüller space itself carries a [[complex structure]].

This was envisioned in ([Teichmüller 44](#Teichmuller44)) and proven in ([Ahlfors 60](#Ahlford60), [Bers 60](#Bers60)).

Review includes ([Schumacher, section 2](#Schumacher))

### Relation to moduli stack of complex curves / Riemann surfaces
 {#RelationToModuliStackOfCurves}

Quotienting the space of complex structures on $\Sigma$ by _all_ [[homeomorphisms]] produces what is called the [[moduli space of curves]] (over the complex numbers).
Closely related to the [[moduli space of conformal structures]] hence of  [[Riemann surfaces]].

Hence the Riemann moduli space is the [[orbifold quotient]] of Teichmüller space by the [[mapping class group]] of $\Sigma$. See ([Hubbard-Koch 13](#HubbardKoch13)).

## Related concepts

* [[Kodaira-Spencer theory]]

* [[moduli space of curves]]

* [[Grothendieck-Teichmüller group]]

* [[quantum Teichmüller theory]]

* [[p-adic Teichmüller theory]]

* [[inter-universal Teichmüller theory]]

* [[Outer space]]

* for version in [[supergeometry]] see at _[[super Riemann surface]]_

## References

The original articles are

* {#Teichmuller39} [[Oswald Teichmüller]], _Extremale quasikonforme Abbildungen und quadratische Differentiale_, Preußische Akademie der Wissenschaften, nat. Kl. 22 1-197 (1939)

* {#Teichmuller44} [[Oswald Teichmüller]], _Veränderliche Riemannsche Flächen_, Deutsche Math. 7 344-359 (1944). English translation: _Variable Riemann surfaces_, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp 787–803, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:[10.4171/117-1/19](https://doi.org/10.4171/117-1/19).

and see also

* Annette A'Campo-Neuen, Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos. _A commentary on Teichmüller's paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces)_, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp.804-814, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:[10.4171/117-1/20](https://doi.org/10.4171/117-1/20), [hal-00730238](https://hal.archives-ouvertes.fr/hal-00730238), arXiv:[1209.1882](https://arxiv.org/abs/1209.1882).

The complex structure on Teichmüller spaces was fully established in

* {#Ahlfors60} L. Ahlfors, _The complex analytic structure of the space of closed Riemann surfaces_, in _Analytic Functions_, Princeton University Press (1960)

* {#Bers60} [[Lipman Bers]], _Spaces of Riemann surfaces_, Proc. Int. Cong. of Mathematics 1958, Cambridge 1960

Reviews include

* {#Schumacher} Georg Schumacher, _The theory of Teichmüller spaces -- A view towards moduli spaces of Kähler manifolds_ ([pdf](http://www.mathematik.uni-marburg.de/~schumac/ges.pdf))

* Wikipedia, _[Teichmüller space](http://en.wikipedia.org/wiki/Teichm%C3%BCller_space)_

Further developments include

* [[Alexander Grothendieck]],  _Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes_ Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8 (Paris: Secrétariat Mathématique). (1960/1961) ([Numdam](http://www.numdam.org/item?id=SHC_1960-1961__13_1_A4_0))

* {#HubbardKoch13} [[John Hubbard]], [[Sarah Koch]], _An analytic construction of the Deligne-Mumford compactification of the moduli space of curves_ ([arXiv:1301.0062](http://arxiv.org/abs/1301.0062))

Relation to [[anti-de Sitter spacetime]]:

* Francesco Bonsante, Andrea Seppi, _Anti-de Sitter geometry and Teichmüller theory_ ([arXiv:2004.14414](https://arxiv.org/abs/2004.14414))

[[!redirects Teichmüller space]]
[[!redirects Teichmüller spaces]]

[[!redirects Teichmüller orbifold]]
[[!redirects Teichmüller orbifolds]]

[[!redirects Teichmüller stack]]
[[!redirects Teichmüller stacks]]

[[!redirects Teichmüller groupoid]]
[[!redirects Teichmüller groupoids]]


Teiko Heinosaari

* [institute page](https://www.utu.fi/en/people/teiko-heinosaari)

* [InSpire page](https://inspirehep.net/authors/1981744)

* [GoogleScholar page](https://scholar.google.com/citations?user=H4JUHQ4AAAAJ&hl=fi)

## Selected writings

On [[quantum physics]] with emphasis on [[quantum probability theory]]:

* [[Teiko Heinosaari]], [[Mário Ziman]], *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

category: people


Teimuraz Pirashvili

__Teimuraz Pirashvili__ is a Georgian mathematician (Razmadze Institute of the Georgian Academy of Sciences and University of Leicester) who has contributed to homotopical and [[homological algebra]], [[algebraic topology]], [[homotopy theory|homotopy]] and [[category theory]].

* [web](http://www2.le.ac.uk/departments/mathematics/extranet/staff-material/staff-profiles/tp59)

category: people

[[!redirects T. Pirashvili]]


Telegram Open Network

[Telegram](https://telegram.org) (see official [FAQ](https://telegram.org/faq) and [wikipedia](https://en.wikipedia.org/wiki/Telegram_(messaging_service)) is developing Telegram Open Network (TON) (see list.[wiki](https://list.wiki/Telegram_Open_Network_(TON) and wikipedia.[ru](https://ru.wikipedia.org/wiki/Telegram_Open_Network)), centered around a new generation (and [[high performance DLs|high performance]]) [[blockchain]] project. TON is envisioned to support for decentralised applications and [[smart contract]]s, distributed file-storage service (TON Storage), micropayments, a proxy service (TON proxy) and TON DNS. TON had large private ICO for large investors in early 2018 and is supposed to be fully operational in Q4 2019. According to press ([kod.ru](https://kod.ru/ton-test-version-apr-2019), [zfr.su](https://zfr.su/nachalsya-zakrytyy-beta-test-telegram-open-network), [cryptor.net](https://cryptor.net/news/pavel-durov-zapustil-testnet-telegram-open-network-3490), [beincrypto](https://beincrypto.com/telegram-launches-open-network-testnet-in-private-beta/), [kryptonews](https://www.kryptonews.org/2019/04/12/telegram-launches-open-network-testnet-in-private-beta), [other](https://ton-telegram.net/news/testnet-launched)), TON testnet has started, with access only to limited parties, on April 11, 2019. Due
SEC suit in US the original project has been discontinued in mid 2020. Some partial replica of original vision, building on available early 2020 TON software are being developed as FreeTON project. The SEC court proceedings are studied case in the paper

* Muisala et al., _Telegram: deconstructing one of the biggest blockchain cases of 2020_, [doi](https://doi.org/10.1108/JOIC-10-2020-0035)

Official announcement of halt of the project is

* Pavel Durov, [What was TON and why it is over](https://telegra.ph/What-Was-TON-And-Why-It-Is-Over-05-12)

TON blockchain's native cryptocurrency for original net was called Gram. The underlying blockchain protocol supports a number of innovative ideas helping scaling, including notably infinite sharding paradigm and instant hypercube routing. It employs a new [[virtual machine]] (TON VM), supporting algebraic types. TON has upgradeable  formal  blockchain  specifications and support for off-chain payment networks.

* TON whitepaper, 23 page [pdf](https://cdn.crowdfundinsider.com/wp-content/uploads/2018/03/TON-White-Paper-Telegram-ICO.pdf), mirrors [pdf](https://drive.google.com/file/d/1ucUeKg_NiR8RxNAonb8Q55jZha03WC0O/view) [pdf](https://icorating.com/upload/whitepaper/gNQ7e9z3lCGi519Wz8mmC0Kg8aA0goeZKAQ802vo.pdf)
* [[nlab:Nikolai Durov]]'s 132-page technical overview _Telegram Open Network_, Dec 2017 [pdf](https://toncoin.tech/TON-official-whitepaper.pdf), mirror [pdf](https://www.kriptovaluta.hr/wp-content/uploads/2018/03/TON-Technology.pdf); _Telegram Open Network Blockchain_, Sep 2018, 121 pp. [pdf](https://www.docdroid.net/qY4sQEv/telegram-open-network-blockchain-september-5-2018.pdf); _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf); _Catchain consensus: an outline_, [pdf](https://test.ton.org/catchain.pdf)

The whole thing will run on protocols used already by Telegram and use similar specs:

* specs: [TL language](https://core.telegram.org/mtproto/TL) ("Type Language", TON will use an upgrade, TL-B), [MTProto mobile protocol](https://core.telegram.org/mtproto)
* JAVA MTProto implementation [github](https://github.com/ex3ndr/telegram-mt)
* github [OpenTl.ClientApi](https://opentl.github.io/OpenTl.ClientApi), [mtproto2json](https://github.com/nikat/mtproto2json)

It will use Telegram Passport.

* Telegram Passport [manual](https://core.telegram.org/passport), [example](https://core.telegram.org/passport/example), [telepass.me](https://telepass.me), [guide](https://technandu.com)

> Telegram is introducing a new feature called Passport, a single sign-on option for services that need actual real-world IDs or documents. Passport will let users store their files and data in Telegram's cloud and pass it directly on to services that need that information (via Neowin).

See also [[digital identity]].

According to a press release [pdf](https://www.wirecard.com/uploads/tx_nenews/PM_2019_04_17_EN_TON_01.pdf) (accompanying event of partnership with Wirecard),

> [TON Labs](https://tonlabs.io) is a core infrastructure and ecosystem developer for TON (Telegram Open Network). Its secure and accessible TON.Dev and TON.Space environments will deliver a TON full node implementation, a versatile toolchain and a cutting-edge SDK, as well as novel ways users interact with fintech services. It will enable the developer community to create, deploy, distribute and manage TON chains for consumer and enterprise applications.

According to other [press](https://www.vedomosti.ru/technology/articles/2019/04/03/798189-razrabotchikam-uprostyat-sozdanie-prilozhenii), TON Labs developes their tools using technical documentation which Telegram distributed to their investors in Autumn 2019, but may also have other insider information making pre-release development effective.

[[!redirects TON]]


teleparallel gravity

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[first order formulation of gravity]] the field of [[gravity]] is modeled by a [[connection]] with values in the [[Poincaré Lie algebra]]. Such a connection locally decomposes into a $\mathbb{R}^{d,1}$-[[Lie algebra valued 1-form]] called the _[[vielbein]]_ and a [[special orthogonal Lie algebra|so(d,1)]]-[[Lie algebra valued 1-form|valued 1-form]] called the _[[spin connection]]_ piece.

There is a constraint on this data. Usually the $so(d,1)$-connection is required to be the [[Levi-Civita connection]] corresponding to the [[pseudo-Riemannian metric]] encoded by the vielbein, hence the unique metric compatible connection with vanishing [[torsion]] but in general non-vanishing curvature.

But to some extent one may instead impose a dual constraint and demand the connection to be the _[[Weitzenböck connection]]_, which has vanishing [[curvature]], but in general non-vanishing [[torsion]]. This formulation of gravity is called _teleparallel gravity_, because by flatness of the connection, its [[parallel transport]] is trivial and hence [[tangent vectors]] at different points of [[spacetime]] may consistently be compared without specifying a [[path groupoid|path]] between them, hence without first [[parallel transport|parallel transporting]] one to the other. This "parallelism at a distance" is what gives teleparallel gravity its name.

Notice that under some conditions a manifold being flat (i.e. admitting a [[flat connection]] on its tangent bundle) implies that it is a [[parallelizable manifold]], hence that the tangent bundle is actually trivializable. Sufficient such condition is notably that the flat manifold is also [[simply connected topological space|simply connected]], but this may be relaxed ([Thorpe 65](#Thorpe65)). Some authors may explicitly assume for teleparallel gravity that the tangent bundle is trivialized by the [[vielbein field]] (making it a [[framed manifold]]), though authors in the physics literature tend to be vague on this point.

At least without coupling to matter, the teleparallel formulation is locally classically equivalent to the ordinary formulation in [[first order formulation of gravity]] in that inserting in the [[Euler-Lagrange equations]] for the [[vielbein]]+[[Weitzenböck connection]] the expression of the latter in terms of the [[Levi-Civita connection]] one reaobtains the ordinary [[Einstein equations]].

Despite this equivalence, the map of the formal field content of teleparallel gravity to its physical interpretation has a rather different feel to it than in the ordinary formulation. Where ordinarily the field of gravity does not quite affect the motion of bodies like an ordinary [[force]] in Newtonian physics, but instead by [[curvature]] causing non-trivial [[geodesics]], in the teleparallel description the curvature does vanish just as in newtonian physics, and instead the [[torsion]] of the connection acts like a genuine Newtonian force.

## References

Review and survey:

* J. Pereira, _Torsion and the description of the gravitational interaction_ ([pdf slides](http://www.phys.sinica.edu.tw/~heptheory/files/pereira.pdf))

* R. Aldrovandi, J. G. Pereira, K. H. Vu, _Gravitation without the equivalence principle_ ([arXiv:gr-qc/0304106](http://arxiv.org/abs/gr-qc/0304106))

* Manuel Hohmann, *Teleparallel gravity*, in *Signatures and experimental searches for modified and quantum gravity* [[arXiv:2207.06438](https://arxiv.org/abs/2207.06438)]

Further discussion:

* R. Aldrovandi, J. G. Pereira, and K. H. Vu, _Selected topics in teleparallel gravity_ Brazilian Journal of Physics, vol. 34, no. 4A, (2004) ([pdf](http://www.sbfisica.org.br/bjp/files/v34_1374.pdf))

The [[Schwarzschild spacetime]] in the context of teleparallel gravity is discussed in

* Gheorghe Zet, _Schwarzschild solution on spacetime with torsion_ ([arXiv:gr-qc/0308078](http://arxiv.org/abs/gr-qc/0308078))

The relation between flat tangent connections and parallelizablity is discussed in

* {#Thorpe65} [[John Thorpe]], _Parallelizablility and flat manifolds_, 1965 ([[ThorpeParallelizable.pdf:file]])

Via [[multisymplectic geometry]] ([[De Donder-Weyl-Hamiltonian]]-formulation):

* [[Igor V. Kanatchikov]]. *The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization* (2023) [[arXiv:2308.10052](https://arxiv.org/abs/2308.10052)]


telescope conjecture

# The telescope conjecture
* table of contents
{: toc}

## Idea

In the [[stable homotopy category]] of $p$-local [[spectra]] for some [[prime number]] $p$, there are two collections of similarly behaved spectra: the [[Morava K-theories]] and spectra which are the telescopes of [[finite spectra]] under their $v_n$-maps. The former are usually denoted $K(n)$ and the latter are often denoted $T(n)$.  The _telescope conjecture_ is that [[localization]] at $K(n)$ and localization at $T(n)$ are [[Bousfield equivalence|(Bousfield) equivalent]].

The conjecture is known to be true for $n=1$. A disproof of this conjecture for $n\geq 2$ using algebraic K-theory has been released by Burklund-Hahn-Levy-Schlank.


telescopic localization

#Contents#
* table of contents
{:toc}

## Definition

For some [[prime number]] $p$, let $X$ be a [[finite spectrum|finite]] [[p-local spectrum]]. By the [[periodicity theorem]] this carries a $v_k$-self map

$$
  f \;\colon\; \Sigma^k X \longrightarrow X
  \,.
$$

The _telescopic localization_ of $X$ is the [[homotopy colimit]] $X[f^{-1}]$ of the [[direct sequence]]

$$
  X \stackrel{\Omega^k f}{\longrightarrow} \Sigma^{-k} X
  \stackrel{\Omega^{2k} f}{\longrightarrow} \Sigma^{-2k} X
  \stackrel{}{\longrightarrow}  \cdots
  \,.
$$

For $X$ in addition of type $\gt n$, this is a [[Bousfield localization of spectra]]. ([Lurie 10, prop. 1](#Lurie10))

## References

Lecture notes include

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010 , Lecture 28 _Telescopic localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture28.pdf))
 {#Lurie10}

[[!redirects telescopic localizations]]


temperature

## Related entries

* [[thermodynamics]]

* [[absolute zero]]

* [[inverse temperature]]

* [[thermal quantum field theory]], [[Euclidean field theory]]

  * [[KMS state]]

  * [[Wick rotation]]

* [[entropy]], [[energy]]

* [[intensive and extensive]]

## References

* Wikipedpia, _[Temperature](http://en.wikipedia.org/wiki/Temperature)_

[[!redirects temperatures]]


tempered distribution

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

#Contents#
* table of contents
{:toc}

## Idea

A _tempered (or Schwartz) distribution_ is a [[distribution]] $u\in\mathcal{D}'(\mathbb{R}^n)$ that does not "grow too fast" -- at most polynomial (or moderate/tempered) growth -- at infinity (in all directions); in particular it is only defined on $\mathbb{R}^n$, not on any open subset. Formally, a tempered distribution is a [[continuous linear functional]] on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ of [[smooth functions with rapidly decreasing derivatives]]. The space of tempered distributions (with its natural topology) is denoted $\mathcal{S}'(\mathbb{R}^n)$. The main property is that its [[Fourier transform of distributions|Fourier transform]] $\mathcal{F}u$ is well-defined, and is itself a tempered distribution; and that it naturally extends the standard Fourier transform $\mathcal{F}: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$. This makes tempered distributions the natural setting for solving (linear) partial differential equations.

## Definition

+-- {: .num_defn #TemperedDistribution}
###### Definition
**(tempered distributions)**

For $n \in \mathbb{N}$, a _[[tempered distribution]]_ on the [[Euclidean space]] $\mathbb{R}^n$ is a [[continuous linear functional]] on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ ([this def.](Schwartz+space#SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives)) of [[smooth functions with rapidly decreasing derivatives]].

The [[topological vector space]] of tempered distributions is denoted $\mathcal{S}'(\mathbb{R}^n)$.

=--

(e.g. [Hörmander 90, def. 7.1.7](#Hoermander90))

## Examples

+-- {: .num_example}
###### Example
**([[compactly supported distributions]] are [[tempered distributions]])**

Every [[compactly supported distribution]] is a [[tempered distribution]] (def. \ref{TemperedDistribution}), yielding an inclusion

$$
  \mathcal{E}'(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{S}'(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_example #SquareIntegrableFunctionsInduceTemperedDistributions}
###### Example
**([[square integrable functions]] induced [[tempered distributions]])**

Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th [[Lebesgue space]], e.g. for $p = 2$ this means that $f$ is a [[square integrable function]]. Then the operation of [[integration]] against the [[measure]] $f dvol$

$$
  g \mapsto \int_{x \in \mathbb{R}^n} g(x) f(x) dvol(x)
$$

is a [[tempered distribution]] (def. \ref{TemperedDistribution}).

=--

(e.g. [Hörmander 90, below lemma 7.1.8](#Hoermander90))

## Related concepts

* [[Fourier transform of distributions]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 7.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#Melrose03} [[Richard Melrose]], chapter 1 of _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

[[!redirects tempered distributions]]

category: functional analysis


Template page

This page is meant to provide a general example and template for new [[nLab:HomePage|nLab]]-pages. You can look at **[its source code](http://ncatlab.org/nlab/source/Template+page)** to see how the various parts are done. A minimal template is given first which can be copy-and-pasted into newly created pages.  See [[HowTo]] for more details.

category: meta
[[!redirects Template page]]
[[!redirects Template pages]]
[[!redirects template page]]
[[!redirects template pages]]
[[!redirects page template]]
[[!redirects page templates]]
[[!redirects Subobject Classifier a Generalization of the Classical Bit ]]

## Minimal Template Code

~~~~

# Contents (or put a title here)
* this block creates the table of contents, leave as is
{: toc}

## Idea

(...)

## Definition

(...)

## Properties

(...)

## Examples

(...)

## References

(...)

[[!redirects <put page name here>]]
[[!redirects <put page name here>s]]

~~~~

A more detailed example follows. Check out the source code [here](https://ncatlab.org/nlab/source/Template+page) to see how it's coded:

****

# Contents
* this block creates the table of contents
{: toc}

## Idea

It is an old observation that xyz. One notices that from the [[nPOV]] this is just an abc. This leads to the definition of a uvw. It is useful for doing klm and provides the basis for the more general theory of äöü.

## Abstract

A uvw is effectively a uv together with a w. Its main property is encoded in Somebody's Theorem which says that it consists of precisely three letters. The archetypical example of a uvw is $\mu \nu \omega$; details will be explained in the [special examples paragraph](/nlab/show/template+page#Specific_examples).

## Definition

As Jacques Distler said,

> See [more about definition/theorem/proof-environments](http://golem.ph.utexas.edu/wiki/instiki/show/Theorems).

\begin{definition}\label{uvwDef}
**(uvw)** \linebreak

A **uvw** is a UVW in which all letters are lower case.

\end{definition}

This may be summed up in the slogan:

+-- {: .standout}

A uvw is just what it looks like.

=--

## Properties

\begin{lemma}

Every uvw (Def. \ref{uvwDef}) contains at least one letter.

\end{lemma}

\begin{proof}
By inspection.
\end{proof}

\begin{proposition}
Every uvw contains strictly more than one letter.

\end{proposition}

\begin{proof}
Use the above lemma and continue counting:

\[
  1 + 1 = 2  \label{firstequation}
  \,.
\]
\end{proof}

\begin{theorem}
Every uvw (Def. \ref{uvwDef}) contains exactly three letters.

\end{theorem}

\begin{proof}

Along the lines of the above proposition, we use equation \eqref{firstequation} and then conclude with

$$
  2 + 1 = 3
  \,.
$$

Notice that this is indeed independent of in which order we sum up the letters, in that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    \mathbb{N}\times \mathbb{N} \times \mathbb{N}
    &
      \overset{Id \times + }{\longrightarrow}
    &
    \mathbb{N} \times \mathbb{N}
    \\
    {}^{\mathllap{+ \times Id}}
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{+}}
    \\
    \mathbb{N} \times \mathbb{N}
    &
      \underset{+}{\longrightarrow}
    &
    \mathbb{N}
  }
  \,.
$$

\end{proof}

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

No uvw contains more than three letters.
=--

## Examples

### Special cases

*  First case

*  Second case

*  Third case

### Specific examples
{#Specific_examples}

For ease of reference, we will number the examples.

\begin{example}\label{FirstExample}
The first example is obvious.
\end{example}

\begin{example}\label{SecondExample}
The second example is a slight variation of Exp. \ref{FirstExample}.
\end{example}

\begin{example}\label{ThirdExample}
The third example is completely different from both Exp. \ref{SecondExample}.
\end{example}

## References
 {#References}

The original definition appeared in section 3 of

* FirstName LastName, _Title_, Journal, **vol** num (year) pages-pages  ([arXiv:xxxxxx](http://put.url/here), [doi:yyyyyyy](http://another.url), [pdf](http://yet.another.url/)).

[[!redirects uvw]]
[[!redirects uvws]]


temporal algebra

# Temporal algebras
* table of contents
{: toc}

## Idea

A temporal algebra is a [[modal algebra]] with two operations that reflect the future and past modal operators of a [[temporal logic]].

## Definitions

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

A **temporal algebra** is a [[Boolean algebra]], $(\mathbb{B}, m_0,m_1)$, with operators, of type $2$,
with the condition that the operators are _conjugate_:

$m_0 x \cdot y = 0$ if, and only if, $m_1 y \cdot x = 0$.

Equivalently (and equationally) this can be written as

$x \leq l_0 m_1 x \cdot l_1 m_0 x$

where $l_i$ is the [[de Morgan duality|dual]] of $m_1$.
=--

[[!redirects temporal algebra]]
[[!redirects temporal algebras]]


temporal gauge

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

#Contents#
* table of contents
{:toc}

## Idea

A special kind of [[gauge fixing]] condition on a [[connection on a bundle]].

## Definition

Let $X = \mathbb{R}^n \times \mathbb{R}$ a product [[manifold]] and let $t : X \to \mathbb{R}$ be the projection on the last factor.

For $A$ a [[connection on a bundle]] on $X$, a _temporal gauge_ is a gauge in which

$$
  A_{\partial_t} = 0
  \,.
$$

## Related concepts

* [[gauge group]], [[gauge transformation]]

* [[gauge fixing]]

  * **temporal gauge**

  * [[Coulomb gauge]]

* [[uncertainty of fluxes]]


temporal logic

# Contents
* table of contents
{:toc}

## Idea

**Temporal logics**, as their name suggests, are [[formal logics]] that are meant to refer to the passage of [[time]].  They form a very large and important class of [[modal logics]], but here we will, to start with, only look at some simple cases. An important early example of a temporal logic is given by [[Arthur Prior]]'s **tense** logic.

## Basic temporal language, (BTL)

First the basic temporal language is built with two binary model operators, which instead of being denoted 'box' and 'diamond', are written $F$ and $P$.  The intended interpretation of these are:

* $F\phi$ is _$\phi$ will be true at some **future** time_;

* $P\phi$ is _$\phi$ was true at some **past** time.

The reason for the notation $F$ and $P$ should be clear.

* The dual of $F$ is $G$, so $G\phi = \neg F\neg \phi$. This means that we read $G\phi$ as 'at no future time is $\phi$ not true', i.e., $\phi$ is always **going** to be true. ($G$ for 'going'.)

* The dual of $P$ is written $H$, whence $H\phi = \neg P\neg \phi$ and $H\phi$ interprets as '$\phi$ **has** always been true'. ($H$ is for 'has' - which is a bit weak, but that is life!)

+-- {: .un_example}
###### Examples

*  If for all $\phi$ we have $P\phi \to G P\phi$, then _whatever has happened will always have happened_, which seems reasonable so this might be a suitable 'general truth' we might want a temporal logic to contain.

*  If we had in our logic that $F\phi \to F F\phi$, then this would means that if $\phi$ is true at some future time, then at some future time it will be true that $\phi$ will be true at some future time, so _between any two instants there must be a third_, and the general truth of this would say something about the 'internal structure' of time as modelled by such a logic.
=--

This basic temporal language does not yet really constitute a sensible logic that could model important features of 'time'. but we can still look at models so as to see what properties of the models can be identified as corresponding to seemingly feasible formulae in the language.

## Frames and models for BTL

A _frame_ for BTL will be a set, $T$, with two binary relations $R_F$ and $R_P$.  We have $R_F x y$ reads that 'at $x$, $y$ is in the future'  - but, in the ordinary common sense meaning of words, this should mean $R_P y x$ and conversely.  In other words $R_P$ should be the _converse_ or [[opposite relation]] of $R_F$.

(In general, if $R$ is a binary relation on a set $X$, then its [[opposite relation]] is defined by

$$(y,x) \in R^{op} \Leftrightarrow (x,y)\in R.)$$

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

A frame of the form $(T,R,R^{op})$ is called a _bidirectional frame_.
=--

If we have **any** model $\mathfrak{M} = (T,R,V)$ with a valuation, $V$, then we can interpret BTL in terms of it by specifying  the interpretation of $P$ in terms that of $R$.  explicitly we define

*  $\mathfrak{M}, t\models F\phi$ if and only if $\exists s (R t s \wedge \mathfrak{M}, s\models \phi);$

*  $\mathfrak{M}, t\models P\phi$ if and only if $\exists s (R s t \wedge \mathfrak{M}, s\models \phi)$,

but, of course, this is still a long way from having a logic that looks as if it captures 'time-like' behaviour. We could have models with 'circular time', and 'branching time'.  Both of these correspond to various situations in computational applications so should be kept in mind, both to be able to identify their occurrence and to build them in or to avoid them in the logics.

One condition it would be natural to impose is transitivity of $R_F$, since if $F\phi$ is true at some future time, then clearly, $\phi$ itself must also be true at some future time, i.e., later on still!  This leads one to ask that

$$(4) \quad\quad    F F\phi \to F\phi$$

should be in our logic,  (and similarly for $P$, but that will hold in the bidirectional models since if $R$ is transitive then so will be $R^{op}$).

### Periods as primitive entities

Instead of points, periods can be thought of as primitive entities. Therefore one may consider structures of the form $(P, \prec, \sqsubset)$, where $p\prec q$ means that the whole period $p$ precedes $q$, and $\sqsubset$ is the [[inclusion]] relation (see [Venema 2017](#Venema17)).

### Other structures

Other time structures are described in [(Hajnicz 1996, Chapter 2)](#Hajnicz96). In particular, a *metric cyclic point time structure* is defined as:

* a set $T$ of time moments,
* a set of distances between time points,
* a global order over $T$,
* a local order over $T$,
* a metric over $T$,
* the length of the semicircle.

It can be axiomatized by five axioms (totality, local antisymmetry, local linearity, local transitivity, and consistency) in the [[predicate logic|first-order predicate calculus]].

## Temporal type theory in terms of adjoints

It is possible to specify a temporal type theory in the context of [[adjoint logic]]. Consider a category $\mathcal{C}$, and an [[internal category]] given by $b, e: Time_1 \rightrightarrows Time_0$. Here we understand elements of $Time_1$ as time intervals, and $b$ and $e$ as marking their beginning and end points. We may choose to impose additional structure on $Time$, e.g., that it be an internal [[poset]], or a [[linear order]].

Now each arrow, $b$ and $e$, generates an adjoint triple, e.g., $\sum_b \dashv  b^{\ast} \dashv \prod_b$, formed of [[dependent sum]], [[base change]], [[dependent product]], going between the slices $\mathcal{C}/Time_0$ and $\mathcal{C}/Time_1$.

Then we find two adjunctions $\sum_b e^{\ast} \dashv \prod_e b^{\ast}$ and $\sum_e b^{\ast} \dashv \prod_b e^{\ast}$, e.g.,

$$
Hom(\sum_b e^{\ast} C(t), D(t)) = Hom(e^{\ast} C(t), b^{\ast} D(t)) = Hom(C(t), \prod_e b^{\ast} D(t)).
$$

Now consider for the moment that $C$ and $D$ are propositions. Then $\sum_b e^*C$ means "there is some interval beginning now and such that $C$ is true at its end", i.e. $F C$; while $\prod_e b^*D$ means "for all intervals ending now, $D$ is true at their beginning", i.e. $H D$. Hence our adjunction is $F \dashv H$. Similarly, interchanging $b$ and $e$, we find $P \dashv G$. Note that we don't have to assume the classical principle $G\phi = \neg F\neg \phi$ and $H\phi = \neg P\neg \phi$.

In the setting of dependent type theory, we do not need to restrict to propositions, but can treat the temporal operators on general time-dependent types. So if $People(t)$ is the type of people alive at $t$, $F People(t)$ is the type of people alive at a point in the future of $t$, and $G People(t)$ is a function from future times to people alive at that time, e.g., an element of this is 'The oldest person alive(t)'.

Since $Time$ is a category, we have in addition to the two projections $p,q:Time_1\times_{Time_0} Time_1 \to Time_1$, composition $c:Time_1\times_{Time_0} Time_1 \to Time_1$. This allows us to express more subtle temporal expressions, such as '$\phi$ has been true since a time when $\psi$ was true', denoted $\phi S \psi$.

$$\phi S \psi := \Sigma_e (b^* \psi \times \Pi_c (e p)^* \phi) $$

(note $e p = b q$).  That is, there is a subinterval ending now such that $\psi$ was true at its beginning and $\phi$ was true at all points inside it. Similarly, '$\phi$ will be true until a time when $\psi$ is true' is

$$\phi U \psi := \Sigma_b (e^* \psi \times \Pi_c (e p)^* \phi).$$

Temporal logicians have debated the relevant advantages of instant-based and interval-based approaches. Some have also considered hybrid approaches ([Balb11](#Balb11)). The analysis of this section suggests that working with intervals and instants together in the form of an internal category allows for a natural treatment via [[adjoint logic]].

An important construction in interval-based temporal logics is the **chop operator**, $C$ ([Venema 1991](#Venema91)). This applies to two interval properties, $\alpha$, $\beta$, so that $\alpha C \beta$ denotes the property of an interval that it may be divided into two sub-intervals such that $\alpha$ holds of the first and $\beta$ of the second. In our current framework we have $\alpha C \beta : \equiv \Sigma_c(p^{\ast}\alpha \times q^{\ast} \beta)$.

One of the consequences of taking $Time$ as an internal category is that the future includes the present, so that $\phi$ could be true now and at no other instant but we would have that $F \phi$ is true when it is supposed to say "$\phi$ will be true at some Future time". Similarly, we would have that $\phi U \psi$ holds now if $\psi$ and $\phi$ both hold now (in general, as defined above it requires $\phi$ to still hold at the instant when $\psi$ becomes true).

If we wish to change these consequences, we could let $Time_1$ be the $\lt$-intervals instead of the $\le$-ones. In other words, we could take $Time$ to be a [[semicategory]]. While this accords with standard practice, the original alternative has been proposed:

>The most common practice in temporal logic is to regard time as an irreflexive ordering, so that "henceforth", meaning "at all future times", does not refer to the present moment. On the other hand, the Greek philosopher Diodorus proposed that the necessary be identified with that which is now and will always be the case. This suggests a temporal interpretation of $\Box$ that is naturally formalised by using reflexive orderings. The same interpretation is adopted in the logic of concurrent programs to be discussed. ([Gold92, p. 44](#Gold92))

On the other hand, some temporal logicians look to represent both forms of 'henceforth'.

For an related approach to temporal type theory, see ([Schultz-Spivak 17](#SchSpi17)).

## Computation tree logic

In [[computer science]], logics have been devised for dealing with the behaviours of entities undergoing transitions between its states. Where in a linear temporal logic, operators are provided for describing events along a single computation path, in a branching-time logic the temporal operators quantify over the paths that are possible from a given state. The **computation tree logic** $CTL^{\ast}$ (pronounced "CTL star") combines both the branching-time operators of $CTL$ and the linear-time operators of $LTL$.

Starting from a [[transition system|finite-state system]], where a finite collection of states represented by the nodes of a graph are connected by labelled edges representing transitions, the path tree of the system can be generated. This presents as a [[tree]] all possible paths that the system may undergo from an initial state.

The Computation tree logic $CTL^{\ast}$ is then deployed to reason about the behaviour of the system. This allows for quantification, both universal and existential, over paths. It is also possible to express that a certain state will occur at some point in the future or always in the future.

Now beyond the $F$ and $G$ operators above, we can express new operators such as 'necessarily in the future' to convey that along every path emanating from the current state that some property will hold at some point, and 'possibly henceforth' to convey that there is at least one path along which some property holds at every point.

There is a [[Curry-Howard correspondence]] between linear-time temporal logic (LTL) and functional reactive programming (FRP) ([Jeffrey 12](#Jeffrey12), [Jeltsch 12](#Jeltsch12)).

## References

Generally this entry is based on

* P. Blackburn, M. de Rijke and Y. Vedema, _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

but see also:

*  A. Galton, [Temporal Logic, (Stanford Encyclopedia of Philosophy)] (http://plato.stanford.edu/entries/logic-temporal/)

* [[Patrick Blackburn]], _Tense, Temporal Reference and Tense Logic_, Journal of Semantics, 1994,11,
    pages 83--101, [on-line version](http://hylo.loria.fr/content/papers/files/tense.pdf)

* [[Arthur Prior]], 1957, _Time and Modality_, Oxford: Clarendon Press.

* [[Arthur Prior]], 1967, _Past, Present and Future_, Oxford: Clarendon Press.

* {#Balb11} Balbiani, P., Goranko, V. and Sciavicco, G., 2011, 'Two-sorted Point-Interval Temporal logics', in _Proc. of the 7th International Workshop on Methods for Modalities (M4M7)_ (Electronic Notes in Theoretical Computer Science: Volume 278), pp. 31–45.

* {#Gold92} [[Robert Goldblatt]],  _Logics of time and computation_, 1992, ([pdf](http://sul-derivatives.stanford.edu/derivative?CSNID=00003782&mediaType=application/pdf))

For the relationship between linear-time temporal logic and functional reactive programming, see

* {#Jeffrey12} Alan Jeffrey _LTL types FRP: Linear-time temporal logic propositions as types, proofs as functional reactive programs_, in: Proceedings of the Sixth Workshop on Programming Languages Meets Program Verification (PLPV ’12)(2012), pp. 49–60. ([pdf](http://ect.bell-labs.com/who/ajeffrey/papers/plpv12.pdf))

* {#Jeltsch12} Wolfgang Jeltsch, _Towards a Common Categorical Semantics for Linear-Time Temporal Logic and Functional Reactive Programming_, Electronic Notes in Theoretical Computer Science 286, pp. 229-242, ([doi](https://doi.org/10.1016/j.entcs.2012.08.015))

* {#Venema17} Yde Venema, _Temporal logic_. 2017. ([pdf](https://staff.science.uva.nl/y.venema/papers/TempLog.pdf))

For an interval-based approach see

* {#Venema91} Yde Venema, _A modal logic for chopping intervals_. Journal of Logic and Computation, 1(4), pp. 453–476, 1991.

For other time structures see

* {#Hajnicz96} Elżbieta Hajnicz, _Time Structures_. Springer 1996.

For versions of temporal type theory see

* {#SchSpi17} Patrick Schultz, David Spivak, _Temporal Type Theory: A topos-theoretic approach to systems and behavior_, ([arXiv:1710.10258](https://arxiv.org/abs/1710.10258))

* {#Corfield20} [[David Corfield]], _Modal homotopy type theory_, Oxford University Press 2020, Sec. 4.3 ([ISBN: 9780198853404](https://global.oup.com/academic/product/modal-homotopy-type-theory-9780198853404))

[[!redirects temporal logic]]
[[!redirects temporal logics]]


ten-fold way

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## Idea

Various structures in [[algebra]] and [[algebraic topology]] have a classification into 10 classes, more or less explicitly related to the 2 + 8 [[topological K-theory]]-[[groups]] of the [[point]] up to [[Bott periodicity]]: 2 for [[KU]] and 8 for [[KO]], which are unified in [[KR]].

The specific term "10-fold way" is a variation of the term "threefold way" used in [Dyson (1962)](#Dyson62), which is referred to as inspiration by [Heinzner, Huckleberry & Zirnbauer (2004)](#HHZ04) and [Zirnbauer (2010)](#Z10) in reference to work going back to [Altland & Zirnbauer (1997)](#AZ97). While the term "10-fold way" is now often attributed to these authors, they may not actually have used it as a term ([Zirnbauer (2010)](#Z10) finally speaks of the "10-way", at least).

The term became rather popular in the wake of the seminal suggestion by [Kitaev (2009)](#Kitaev09) that "free" [[topological phases of matter]], specifically free [[crystalline topological insulator|crystalline]] [[topological insulator]]-phases,  are classified by some form of [[topological K-theory]] (though [Kitaev (2009)](#Kitaev09) does not use the term "10-fold way", either): see at *[[K-theory classification of topological phases of matter]]* for more on this.

Kitaev's suggestion was made more precise by [Freed & Moore (2013)](#FreedMoore13) (who, however and ironically, doubted, on [p. 57](https://arxiv.org/pdf/1208.5055.pdf#page=57), its application to topological phases -- but see [pp. 2](https://arxiv.org/pdf/2206.13563.pdf#page=2) of [SS23](K-theory+classification+of+topological+phases+of+matter#SatiSchreiber23) for resolution) and it is these authors who very much amplify (and may have actually coined) the term "10-fold way". Moreover, they point out ([pp. 75](https://arxiv.org/pdf/1208.5055.pdf#page=75)) that a 10-fold classification is already contained in [Dyson (1962)](#Dyson62)(!), which the authors re-interpret as the classification of [[super division algebras]].

This 10-fold way of [[super-division algebras]] is further amplified in [[Moore (2013), p. 129](#Moore13)] and [Geiko & Moore (2021)](#GeikoMoore2021)

## Related concepts

* [[Bott periodicity]]

* [[K-theory classification of topological phases of matter]]

## References

* {#Dyson62} [[Freeman Dyson]], *The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics*, J. Math. Phys. **3** 1199 (1962) [[doi:10.1063/1.1703863](https://doi.org/10.1063/1.1703863)]

* {#AZ97} Alexander Altland and Martin R. Zirnbauer, *Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures*, Phys. Rev. B **55** (1997) 1142 [[doi:10.1103/PhysRevB.55.1142](https://doi.org/10.1103/PhysRevB.55.1142)]

* {#HHZ04} P. Heinzner, A. Huckleberry, M.R. Zirnbauer, *Symmetry classes of disordered fermions* [[arXiv:math-ph/0411040](https://arxiv.org/abs/math-ph/0411040)]

* {#Z10} Martin R. Zirnbauer, *Symmetry Classes* [[arXiv:1001.0722](https://arxiv.org/abs/1001.0722)]

* {#Kitaev09} [[Alexei Kitaev]], _Periodic table for topological insulators and superconductors_, talk at: L.D.Landau Memorial Conference "Advances in Theoretical Physics", June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) ([arXiv:0901.2686](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/10.1063/1.3149495))

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], *Twisted equivariant matter*, Ann. Henri Poincaré **14** (2013)  1927-2023 [[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/10.1007/s00023-013-0236-x)]

* {#Moore13} [[Gregory Moore]], _Quantum symmetries and compatible Hamiltonians_ (2013) [[pdf](http://www.physics.rutgers.edu/~gmoore/695Fall2013/CHAPTER1-QUANTUMSYMMETRY-OCT5.pdf)]

* {#GeikoMoore21} Roman Geiko, [[Gregory W. Moore]], _Dyson’s classification and real division superalgebras_, Journal of High Energy Physics **2021** 4 (2021) 299 [[doi: 10.1007/jhep04(2021)299](http://dx.doi.org/10.1007/jhep04%282021%29299)]

Review:

* [[John Baez]], _The Tenfold Way_ (2020) [[webpage](https://math.ucr.edu/home/baez/tenfold.html)]

[[!redirects tenfold way]]
[[!redirects ten fold way]]

[[!redirects 10-fold way]]
[[!redirects 10 fold way]]


tensor

> This is about tensor quantities in the sense of [[multilinear algebra]], [[differential geometry]] and [[physics]], as in [[tensor calculus]]. For the different notion of a tensor in [[enriched category theory]] see under _[[copower]]_.

***

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## Definition

Generally, a _tensor_ is an [[element]] of a [[tensor product]].

Traditionally this is considered in [[differential geometry]] for the following case:

for $X$ a [[manifold]], $T X$ the [[tangent bundle]], $T^* X$ the [[cotangent bundle]], $\Gamma(T X)$, $\Gamma(T^* X)$ their [[spaces of sections]] and $C(X)$ the [[associative algebra]] of [[functions]] on $X$, a **[[rank]]-$(p,q)$ tensor** or **tensor field** on $X$ is an element of the [[tensor product of modules]] over $C(X)$

$$
  t \in \Gamma(T X)^{\otimes_{C(X)}^p} \otimes_{C(X)} \Gamma(T^* X)^{\otimes^q_{C(X)}}
  \,.
$$

A rank $(p,0)$-tensor is also called a **covariant tensor** and a rank $(0,q)$-tensor a **contravariant tensor**.

## Examples

### General

(...)

### In differential geometry

* A [[vector field]] is a rank $(1,0)$-tensor field.

* A [[Riemannian metric]] is a symmetric rank $(0,2)$-tensor.

* A [[differential form]] of  degree $n$ is a skew-symmetric rank $(0,n)$-tensor.

* A [[Poisson tensor]] is a skew-symmetric tensor of rank $(2,0)$.

## Related concepts

* [[tensor calculus]], [[tensor network]], [[string diagram]]

* [[decomposable tensor]]

* [[field (physics)]]

* [[hypermatrix]]

## References

Discussion with an eye towards application in ([[particle physics|particle]]) [[physics]]:

* [[Theodore Frankel]], section 2.4 in: _[[The Geometry of Physics - An Introduction]]_

* [[Howard Georgi]], §10  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

[[!redirects tensors]]

[[!redirects tensor field]]
[[!redirects tensor fields]]

[[!redirects covariant tensor]]
[[!redirects contravariant tensor]]
[[!redirects covariant tensors]]
[[!redirects contravariant tensors]]


tensor (∞,1)-category

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## Idea

The [[(∞,1)-category]] refinement of the notion of _[[tensor category]]_:

a [[stable (∞,1)-category|stable]] [[symmetric monoidal (∞,1)-category]].

## Related concepts

* [[tensor triangulated category]]

* [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]

* [[higher linear algebra]]

[[!redirects tensor (∞,1)-categories]]

[[!redirects tensor (infinity,1)-category]]
[[!redirects tensor (infinity,1)-categories]]


tensor calculus

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## Idea

The yoga of handling [[tensors]], in particular contracting them with each other; and in the case of [[tensor products]] of sections of a [[tangent bundle]] and [[cotangent bundle]] also the operations of [[differentiation]] of tensors, hence the generalization of "[[calculus]]" ([[analysis]]) from [[vectors]] to [[tensors]].

See at _[[string diagram]]_ and _[[tensor network]]_ for more.

## Related entries

* [[Einstein summation convention]]

* [[tensor product]]

* [[tensor category]]

* [[string diagram]], [[tensor network]]

## Wikipedia

* Wikipedia, _[Tensor calculus](https://en.wikipedia.org/wiki/Tensor_calculus)_