t-structure

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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
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## Definition
 {#Definition}

Originally, t-structures were defined

* [on triangulated categories](#OnTriangulatedCategories)

These typically arise as [[homotopy category of an (infinity,1)-category|homotopy categories]] of t-structures

* [on stable $\infty$-categories](#InStableInfinityCategories).

### On triangulated categories
 {#OnTriangulatedCategories}

\begin{definition}\label{tStructure}
**(t-structure on a triangulated category)**
\linebreak
Let $C$ be a [[triangulated category]]. A _t-structure_ on $C$ is a [[pair]] $\mathfrak{t}=(C_{\ge 0}, C_{\le 0})$ of [[strictly full subcategories]]

$$
   C_{\geq 0}, C_{\leq 0} \hookrightarrow C
$$

such that

1. for all $X \in C_{\geq 0}$ and $Y \in C_{\leq 0}$ the [[hom object]] is the [[zero object]]: $Hom_{C}(X, Y[-1]) = 0$;

1. the subcategories are closed under [[suspension]]/desuspension: $C_{\geq 0}[1] \subset C_{\geq 0}$ and $C_{\leq 0}[-1] \subset C_{\leq 0}$.

1. For all [[objects]] $X \in C$ there is a [[fiber sequence]] (i.e. an [[exact triangle]]) $Y \to X \to Z$ with $Y \in C_{\geq 0}$ and $Z \in C_{\leq 0}[-1]$.

\end{definition}

\begin{definition}
\label{HeartOfATStructure}
Given a t-structure (Def. \ref{tStructure}), its _heart_ is the intersection

$$
  C_{\geq 0} \cap C_{\leq 0}
  \hookrightarrow
  C
  \,.
$$

\end{definition}

### On stable $\infty$-categories
 {#InStableInfinityCategories}

\begin{definition}\label{tStructureOnStableInfinityCategory}
**(t-structure in a stable $\infty$-category)**
\linebreak
  A *t-structure* on a [[stable (∞,1)-category]] $\mathcal{C}$ is a t-structure in the above sense (Def. \ref{tStructure}) on its [[underlying]] [[homotopy category of an (∞,1)-category|homotopy category]] (which is [[triangulated category|triangulated]], see [there](stable+infinity-category#TheTriangulatedHomotopyCategory)).
\end{definition}
([Lurie, Higher Algebra, Def. 1.2.1.4](#LurieHA))

Therefore, a t-structure on a stable $\infty$-category $\mathcal{C}$ is a system of [[full sub-(∞,1)-categories]] $\mathcal{C}_{\geq n}$, $\mathcal{C}_{\leq n}$, $n \in \mathbb{Z}$.

\begin{proposition}
  In this situation (Def. \ref{tStructureOnStableInfinityCategory})

1. the $\mathcal{C}_{\leq n} $ are [[reflective sub-(∞,1)-categories]]

   $$
     \mathcal{C}_{\leq n}
     \underoverset
       {\underset{}{\hookrightarrow}}
       {\overset{\tau_{\leq n}}{\longleftarrow}}
       {\;\; \bot \;\;}
     \mathcal{C}
   $$

1. the $\mathcal{C}_{\geq n}$ are [[coreflective sub-(∞,1)-categories]];

   $$
     \mathcal{C}
     \underoverset
       {\underset{\tau_{\geq n}}{\longrightarrow}}
       {\hookleftarrow}
       {\;\; \bot \;\;}
     \mathcal{C}_{\leq n}
   $$

\end{proposition}
([Lurie, Higher Algebra, Def. 1.2.1.5, Cor. 1.2.1.6, Ntn. 1.2.1.7](#LurieHA))

## Properties

### General

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

The heart of a stable $(\infty,1)$-category is an [[abelian category]].

=--

([BBD 82](#BeilinsonBernsteinDeligne82), [[Higher Algebra|Higher Algebra, remark 1.2.1.12]], [FL16, Ex. 4.1](#FL16) and [FLM19, §3.1](#FLM19))

### Relation to spectral sequences

If the heart (Def. \ref{HeartOfATStructure}) of a t-structure on a [[stable (∞,1)-category]] with [[sequential limits]] is an [[abelian category]], then the [[spectral sequence of a filtered stable homotopy type]] [converges](spectral+sequence#ConvergenceOfSpectralSequences) (see there).

### Relation to normal torsion theories
 {#RelationToNormalTorsionTheories}

In the [[(∞,1)-category theory]], $t$-structures arise as [[torsion theory|torsion/torsionfree]] classes associated with suitable [[orthogonal factorization system|factorization systems]] on [[stable ∞-categories]] $C$.

* In [[stable ∞-category]]-theory, the relevant sub-[[(∞,1)-categories]] are closed under de/suspension simply because they are (co-)[[reflective sub-(∞,1)-categories|reflective]], arising from co/[[reflective factorization systems]] on $C$.

* A _bireflective_ factorization system on a $\infty$-category $C$ consists of a [[orthogonal factorization system|factorization system]] $\mathbb{F}=(E,M)$ where both classes satisfy the [[two-out-of-three]] property.

* A bireflective factorization system $(E,M)$ on a stable $\infty$-category $C$ is called _normal_ if the diagram $S x\to x\to R x$ obtained from the reflection $R\colon C\to M/0$ and the coreflection $S\colon C\to *\!/E$ (where the category $M/\!* =\{A\mid (0\to A)\in M\}$ is obtained as $\Psi(E,M)$ under the adjunction $\Phi \dashv \Psi$ described at *[[reflective factorization system]]* and in [CHK85](#CHK85); see also [FL16, §1.1](#FL16)) is _exact_, meaning that the square in
$$
\begin{array}{cccccc}
0 &\to& S X &\to& X\\
&& \downarrow&&\downarrow\\
&& 0 &\to& R X\\
&& && \downarrow\\
&& && 0
\end{array}
$$
is a fiber sequence for any object $X$; see [FL16, Def 3.5 and Prop. 3.10](#FL16) for equivalent conditions for normality.

\begin{remark}
[CHK85](#CHK85) established a hierarchy between the three notions of simple, semi-exact and normal factorization system: in the setting of stable $\infty$-category the three notions turn out to be equivalent: see [FL16, Thm 3.11](#FL16).
\end{remark}

\begin{proposition}
\label{CorrespondenceWithNormalTorsionTheories}
There is a bijective correspondence between the class $TS( C )$ of $t$-structures and the class of normal torsion theories on a stable $\infty$-category $C$, induced by the following correspondence:

* On the one side, given a normal, bireflective factorization system $(E,M)$ on $C$ we define the two classes $(C_{\ge0}(\mathbb{F}), C_{\lt 0}(\mathbb{F}))$ of a $t$-structure $\mathfrak{t}(\mathbb{F})$ to be the torsion and torsionfree classes $(*\!/E, M/\!*)$  associated to the factorization $(E,M)$.

* On the other side, given a $t$-structure on $C$ we set
$$E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};$$
$$
  M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}.
$$

\end{proposition}

This is [FL16, Theorem 3.13](#FL16)

\begin{proposition}
There is a natural monotone action of the group $\mathbb{Z}$ of integers on the class $TS( C )$ (now confused with the class $FS_\nu( C )$ of normal torsion theories on $C$) given by the suspension functor: $\mathbb{F}=(E,M)$ goes to $\mathbb{F}[1] = (E[1], M[1])$.
\end{proposition}

This correspondence leads to study _families_ of $t$-structures $\{\mathbb{F}_i\}_{i\in I}$; more precisely, we are led to study _$\mathbb{Z}$-equivariant_ [[k-ary factorization system|multiple factorization systems]] $J\to TS( C )$.

\begin{proposition}
\label{StableTStructure}
Let $\mathfrak{t} \in TS(C)$ and $\mathbb{F}=(E,M)$ correspond each other under the above bijection (Prop. \ref{CorrespondenceWithNormalTorsionTheories}); then the following conditions are equivalent:

1. $\mathfrak{t}[1]=\mathfrak{t}$, i.e. $C_{\geq 1}= C_{\geq 0}$;

2. $C_{\geq 0}=*\!/E$ is a stable $\infty$-category;

3. the class $E$ is closed under pullback.

\end{proposition}

In each of these cases, we say that $\mathfrak{t}$ or $(E,M)$ is _stable_.

This is [FLM19, Theorem 6.3](#FLM19)

This results allows us to recognize _$t$-structures with stable classes_ precisely as those which are fixed in the natural $\mathbb{Z}$-action on $TS( C )$.

Two "extremal" choices of $\mathbb{Z}$-chains of $t$-structures draw a connection between two apparently separated constructions in the theory of derived categories: _Harder-Narashiman filtrations_ and _semiorthogonal decompositions_ on triangulated categories: we adopt the shorthand $\mathfrak{t}_{1,\dots, n}$ to denote the tuple $\mathfrak{t}_1\preceq \mathfrak{t}_2\preceq\cdots\preceq \mathfrak{t}_n$, each of the $\mathfrak{t}_i$ being a $t$-structure $((C_i)_{\ge 0}, (C_i)_{\lt 0})$ on $C$, and we denote similarly $\mathfrak{t}_\omega$. Then

* In the _stable case_ the tuple $t_{1,\dots, n}$ is endowed with a (monotone) $\mathbb{Z}$-action, and the map $\{0\lt 1\cdots\lt n\}\to TS( C )$ is equivariant with respect to this action; the absence of nontrivial $\mathbb{Z}$-actions on $\{0\lt 1\cdots\lt n\}$ forces each $t_i$ to be stable.
* In the _orbit case_ we consider an _infinite_ family $t_\omega$ of $t$-structures on $C$, obtained as the orbit of a fixed $(E_0, M_0)\in TS( C )$ with respect to the natural $\mathbb{Z}$-action.

### Towers

The HN-filtration induced by a $t$-structure and the factorization induced by a [[semiorthogonal decomposition]] on $C$ both are the byproduct of the _tower_ associated to a tuple $\mathfrak{t}_{1,\dots, n}$:

## Examples
 {#Examples}

The archetypical and historically motivating example (cf. [Gelfand & Manin (1996), IV.4 §1](#GelfandManin96)) is the following:

\begin{example}
\label{TStructureOnDerivedCategoryOfAbelianCategory}
**(canonical t-structure on the [[derived category]] of an [[abelian category]])**
\linebreak
  For $\mathcal{A}$ an [[abelian category]], its unbounded [[derived category]] $\mathcal{D}_\bullet(\mathcal{A})$

1. carries a t-structure (Def. \ref{tStructure}) for which $\mathcal{D}(\mathcal{A})_{\geq n}$ (rep. $\mathcal{D}(\mathcal{A})_{\leq n}$) is the [[full subcategory]] of [[objects]] presented by [[chain complexes]] $V_\bullet$ whose [[chain homology]]-[[homology groups|groups]] are [[trivial group|trivial]] in degrees $\lt n$ (resp. $\gt n$);

1. whose heart (Def. \ref{HeartOfATStructure}) is [[equivalence of categories|equivalent]] to $\mathcal{A}$ (embedded as the [[chain complexes]] which are concentrated in degree 0).

\end{example}

(eg. [Gelfand & Manin (1996), IV.4 §3](#GelfandManin96))

\begin{example}
\label{CanonicalTStructureOnSpectra}
**(canonical t-structure on spectra)**
\linebreak
The [[stable (infinity,1)-category of spectra]], $Spectra$, carries a canonical t-structure for which

* $Spectra_{\geq 0}$ is the sub-category of [[connective spectra]], with $\tau_{\geq 0} \colon Spectra \to Spectra_{\geq 0}$ the [[connective cover]]-construction.

* ...

\end{example}

(e.g. [Lurie, Higher Algebr, pp. 150](#LurieHA))

## Related concepts

* [[Bridgeland stability]]

* [[n-connective object]]

* [[connective cover]]

* [[theorem of the heart]]

## References

For [[triangulated categories]]:

the notion is due to

* {#BeilinsonBernsteinDeligne82} [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]

  > (otherwise introducing [[perverse sheaves]])

Further development:

* {#GelfandManin96} [[Sergei Gelfand]], [[Yuri Manin]], Section IV.4  of: *[[Methods of homological algebra]]*,  transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)]

* [[Donu Arapura]], _Triangulated categories and $t$-structures_ [[pdf](http://www.math.purdue.edu/~dvb/preprints/perv2.pdf)]

* [[Dan Abramovich]], [[Alexander Polishchuk]], *Sheaves of t-structures and valuative criteria for stable complexes*, J. reine angew. Math. **590** (2006) 89-130 [[arXiv:math/0309435](https://arxiv.org/abs/math/0309435), [doi:10.1515/CRELLE.2006.005](https://doi.org/10.1515/CRELLE.2006.005)]

* A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, _t-stabilities and t-structures on triangulated categories_, Izv. Ross. Akad. Nauk Ser. Mat. __68__ (2004), no. 4, 117-150

* [[Alexander Polishchuk]], *Constant families of t-structures on derived categories of coherent sheaves*, Moscow Math. J. __7__ (2007) 109-134 [[arXiv:math/0606013](https://arxiv.org/abs/math/0606013)]

* John Collins, [[Alexander Polishchuk]], _Gluing stability conditions_ [[arxiv/0902.0323](http://arxiv.org/abs/0902.0323)]

For [[stable (∞,1)-categories]]:

* {#LurieHA} [[Jacob Lurie]], Section 1.2.1 in: *[[Higher Algebra]]*

On [[reflective factorization systems]]:

* {#CHK85} C. Cassidy, M. Hébert, [[Max Kelly]], _Reflective subcategories, localizations, and factorization systems_,  J. Austral. Math Soc. (Series A) **38** (1985) 287-329 [[doi:10.1017/S1446788700023624](https://doi.org/10.1017/S1446788700023624)]

* {#RosickyTholen08} [[Jiri Rosicky]], [[Walter Tholen]], _Factorization, Fibration and Torsion_, Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 295-314  [[arXiv:0801.0063](http://arxiv.org/abs/0801.0063), [publisher](http://www.emis.de/journals/JHRS/volumes/2007/n2a14/)]

and on normal torsion theories in stable $\infty$-categories:

* {#FL16} [[Domenico Fiorenza]], [[Fosco Loregian]], *$t$-Structures are normal torsion theories*, Appl Categor Struct **24** (2016) 181–208 [[arxiv:1408.7003](http://arxiv.org/abs/1408.7003), [doi:10.1007/s10485-015-9393-z](https://doi.org/10.1007/s10485-015-9393-z)]

* {#FLM19} [[Domenico Fiorenza]], [[Fosco Loregian]], [[Giovanni Marchetti]], *Hearts and towers in stable $\infty$-categories*,  J. Homotopy Relat. Struct. **14**  (2019) 993–1042 [[arXiv:1501.04658](https://arxiv.org/abs/1501.04658), [doi:10.1007/s40062-019-00237-0](https://doi.org/10.1007/s40062-019-00237-0)]

* [[Fosco Loregian]], [[Simone Virili]] *Triangulated factorization systems and t-structures*, Journal of Algebra **550** (2020) 219-241 [[doi:10.1016/j.jalgebra.2019.12.021](https://doi.org/10.1016/j.jalgebra.2019.12.021)]

[[!redirects t-structures]]

[[!redirects t-structure on a stable (∞,1)-category]]
[[!redirects t-structures on a stable (∞,1)-category]]

[[!redirects t-structure on a stable (infinity,1)-category]]
[[!redirects t-structures on a stable (infinity,1)-category]]

[[!redirects heart]]
[[!redirects hearts]]

[[!redirects heart of a t-structure]]
[[!redirects hearts of t-structures]]

[[!redirects heart of a stable (∞,1)-category]]
[[!redirects hearts of stable (∞,1)-categories]]

[[!redirects heart of a stable (infinity,1)-category]]
[[!redirects hearts of stable (infinity,1)-categories]]


T. Daniel Brennan

* [personal page](http://www.physics.rutgers.edu/~tdanielbrennan/)

* [InSpire page](https://inspirehep.net/authors/1359061)

## Selected writings

On an analogue of the [[DBI action]] via [[TT deformation|$T {\bar T}$-deformation]] of 2d [[Yang-Mills theory]]:

* {#BrennanFerkoSethi20} [[T. Daniel Brennan]], [[Christian Ferko]], [[Savdeep Sethi]], *A Non-Abelian Analogue of DBI from $T \bar T$*, SciPost Phys. **8** 052 (2020) [[arXiv:1912.12389](https://arxiv.org/abs/1912.12389), [doi:10.21468/SciPostPhys.8.4.052](https://doi.org/10.21468/SciPostPhys.8.4.052)]

category: people

[[!redirects T. D. Brennan]]


T. R. Govindarajan

__T. R. Govindarajan__ is a theoretical physicist at Chennai (Madras), India.

Related pages: [[entanglement]]

category: people


category: type theory

This entry is about the article [T. Streicher - a model of type theory in simplicial sets - a brief introduction to Voevodsky' s homotopy type theory](http://www.mathematik.tu-darmstadt.de/~streicher/sstt.pdf).

> The aim of this note is to describe in an accessible way how simplicial
>sets organize into a model of Martin-Löf type theory. Moreover, we explain
>Voevodsky's Univalence Axiom which holds in this model and implements
>the idea that isomorphic types are identical

The topos $sSet=:H$ contains a [[natural-numbers object]]. $H$ is equipped with the [[model structure]] $(C,W,F)$ (...).

Families of types are defined to be [[Kan fibrations]]. The class $F$ of Kan fibrations has the following properties:

* It is closed under composition, pullback along arbitrary morphisms and contains all isomorphisms.

* It is closed under $\Pi$; i.e. if $a:A\to I$ and $b:B\to A$ are in $F$ then $\Pi_a(b)$ is in $F$.

## Identity on $X$

Let $X\in H$ be an object. Let

$$X\xrightarrow{r_X}Id(X)\xrightarrow{p_X}X\times X$$

be a factorization of the diagonal $\delta_X : X \xrightarrow{}X\times X$ into an acyclic cofibration $r_X\in C\cap W$ followed by a fibration $p_X\in F$. We interpret the fibration $p_X$ as

$$x,y:X\vdash I_X(x,y)$$

Analogous, for a Kan fibration $f:A\to I$ we factor the diagonal $\delta_f:A\to A\times_I A$ of $f$ in this way.

Now the problem is that such a factorization is not stable under pullback. To solve this problem we introduce the notion of  _type theoretic universe_ / _Martin-Löf universe.

## Universes in $sSet$ and lifting of Grothendieck universes into presheaf toposes

A _universe in $sSet$_ is defined to be a Kan fibration $p_U:\tilde U\to U$.

Such a universe in $sSet$ may be obtained by lifting a Grothendieck universe $\mathcal{U}\in Set$ to a type theoretic universe $p_U:\tilde U\to U$ in a presheaf topos $\hat C=Set^{C^{op}}$ on some site $C$ by exponentiating $\mathcal{U}$ with the component-wise dependent sum and then forming the generalized universal bundle. To be more precise this is done by defining

$$\label{1}
U(I)=\mathcal{U}^{(C/I)^{op}}
$$
$$\label{2}
U(\alpha)=\mathcal{U}^{\Sigma_\alpha^{op}}

$$

where for a morphism $\alpha$ the _dependent sum_ ${\Sigma_\alpha^{op}}=\alpha\circ (-)$ is given by postcomposition with $\alpha$. The idea behind this is that $\mathcal{U}^{(C/I)^{op}}$ is quivalent to the full subcategory of $\hat C/Y(I)$ on those maps whose fibers are $\mathcal{U}$-small. The presheaf $\tilde U$ is defined as

$$\label{3}\tilde U(I)=\{\langle A,a\rangle | A\in U(I)\,\text{ and }\,a\in A(id_I)\}$$

and

$$\label{4}\tilde U(\alpha)(\langle A,a\rangle)=\langle U(\alpha)(A),A(\alpha\to id_I)(a)\rangle$$

for $\alpha:J\to I$ in $C$. The map $p_U:\tilde U\to U$ sends $\langle A,a\rangle$ to $A$. $p_U$ is generic for maps with $\mathcal{U}$-small fibers. These maps are up to isomorphism precisely those which can be obtained as pullback of $p_U$ along some morphism in $\hat C$.

For $C=\Delta$ we adapt this idea in such a way that $p_U$ is generic for Kan fibrations with $\mathcal{U}$-small fibers. We redefine in this case

$$\label{formula5}U([n])=\{A\in \mathcal{U}^{\Delta/[n])^{op}} | P_A\,\text{ is a Kan fibration }\,\}$$

where $P_A:Elts(A)\to \Delta[n]$ is obtained from $A$ by the Grothendieck construction: For $A:\Delta[n]\to U$ it is the pullback $(P_A:Elts(A)\to \Delta[n]):=A^* p_U$ of $p_U$ along $A$. For morphisms $\alpha$ in $\Delta$ we define $U(\alpha)$ as above since Kan fibrations are stable under pullbacks. $\tilde U$ and $p_U$ are defined as above in (3) and (4) but with the modified $U$, formula (5).

+-- {: .num_theorem #1}
###### Theorem
The simplicial set $U$, formula (5) is a Kan complex
=--

+-- {: .proof}
###### Proof
V. Voevodsky some draft papers and Coq files - www.math.ias.edu/∼vladimir/Site3/Univalent Foundations.html
=--

+-- {: .num_theorem #2}
###### Theorem
For the simplicial set $U$, formula (5), the morphism $p_U:\tilde U\to U$ is universal for morphisms with $\mathcal{U}$-small fibers in that every morphism with $\mathcal{U}$-small fibers arises as a pullback of $p_U$ along some morphism in $H$.
=--

+-- {: .proof}
###### Proof
$p_U$ is a Kan fibration since if

$$\array{
\Lambda_k[n]&\xrightarrow{a}&\tilde U
\\
\downarrow^{i_k^n}&&\downarrow^{p_U}
\\\Delta[n]&\xrightarrow{A}&U
}$$
 is commutative, the pullback of $p_U$ along $A$ is by definition the Kan fibration $P_A:Elts(A)\to \Delta[n]$. This gives a lift $\Delta[n]\to Elts(A)$ which composed with the projection $Elts(A)\to \tilde U$ provides the searched diagonal filler $\Delta[n]\to \tilde U$.

$p_U$ is universal since a Kan fibration $a:A\to I$ with $\mathcal{U}$-small fibers is the pullback of $p_U$ along the morphism $A:I\to U$ sending $x\in I([n])$ to an $\mathcal{U}$-valued presheaf on $\Delta/[n]$ which is isomorphic to $x^* a$ by the Grothendieck construction.

$$\array{
A_x&\to&A&\to&\tilde U
\\
\downarrow^{x^* a}&&\downarrow^a&&\downarrow^{p_U}
\\
* &\xrightarrow{x}&I&\xrightarrow{A}&U
}$$
=--

+-- {: .num_cor}
###### Corollary
The type theoretic universe $p_U:\tilde U\to U$, Theorem \ref{2}, is closed under dependent sums, products and contains the natural-numbers object $N:=\Delta(\mathbb{N})$.
=--

## Identity types in the lifted Grothendieck universe $U$

We consider the factorization $\tilde U\xrightarrow{r_{\tilde U}}Id_{\tilde U}\xrightarrow{p_{\tilde U}}\tilde U\times_U \tilde U$ of the diagonal $\delta_{p_{\tilde U}}$ of $p_{\tilde U}$ into an acyclic cofibration $r_{\tilde U}\in C\cap W$ followed by a fibration $p_{\tilde U}\in F$.

### Interpretation of the eliminator for identity types

For interpreting the eliminator $J$ for $Id$-types we pull back the whole situation along the projection

$$\Gamma:=(C:Id_{\tilde U}\xrightarrow{U}U^* U, d:\Pi_U(r^*_{\tilde U} C))$$

(...)

## Voevodsky' s univalence axiom

Now we will see that Voevodsky's univalence axiom holds in the model described above.

Fix the following notation

$iscontr(X :Set) := (\Sigma_x :X )(\Pi_y:Y ) Id_X (x, y)$

$hfiber(X, Y :Set)(f :X \to Y )(y:Y ) := (\Sigma_x :X ) Id_Y (f (x), y)$

$isweq(X, Y :Set)(f :X \to Y ) := (\Pi_y:Y ) iscontr(hfiber(X, Y , f , y))$

$Weq(X, Y :Set) := (\Sigma_f :X \to Y ) isweq(X, Y , f )$

The eliminator $J$ for identity types induces a canonical map

$$eqweq(X,Y:Set)isweq(eqweq(X,Y))$$

The univalence axiom claims then that all maps $eqweq(X,Y)$ are themselves weak equivalences, i.e.

+-- {: .num_defn}
###### Definition
$$UnivAx:(\Pi X, Y:Set)isweq(eqweq(X,Y))$$
=--

+-- {: .num_remark}
###### Remark
The univalence axiom implies the _function extensionality principle_: for $f,g:X\to Y$ we have

$$(\Pi x:X)Id_Y(fx,gx))\to Id_{X\to Y}(f,g)$$
=--

Even without the univalence axiom we have the following result corresponding to the fact that in $sSet$ a morphism to a Kan complex is a weak equivalence iff it is a homotopy equivalence.

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

$isweq(X,Y)(f)$ is equivalent to

$$isiso(X,Y)(f):=(\Sigma g :Y\to X)((\Pi x:X)Id_X(g(fx),x))\times ((\Pi y:Y)Id_Y(f(gy),y))$$
=--


table - models for (infinity,1)-operads

The entries of the following table display [[model categories]] and  [[Quillen equivalences]] between these that [[presentable (∞,1)-category|present]] the [[(∞,1)-category]] of [[(∞,1)-categories]] (second table), of [[(∞,1)-operads]] (third table) and of $\mathcal{O}$-[[monoidal (∞,1)-categories]] (fourth table).

|  | general pattern |  | |  |
|--|--|--|-|--|
| strict [[enriched (∞,1)-category|enrichment]]|  | [[(∞,1)-category]]/[[(∞,1)-operad]] |  |  |
| $\downarrow$ | | $\downarrow$ | | |
| [[enriched (∞,1)-category]] | $\hookrightarrow$ | [[internal category in an (∞,1)-category|internal (∞,1)-category]] | | |
|  |  |  | | |
|  | **[[(∞,1)Cat]]** |  | |  |
| [[model structure on sSet-categories|SimplicialCategories]] | $-$[[homotopy coherent nerve]]$\to$ | [[model structure on simplicial sets|SimplicialSets]]/[[quasi-categories]] | | [[model structure for Cartesian fibrations|RelativeSimplicialSets]] |
| $\downarrow$[[nerve|simplicial nerve]] | | $\downarrow$ | | |
| [[model structure for Segal categories|SegalCategories]] | $\hookrightarrow$ | [[model structure for complete Segal spaces|CompleteSegalSpaces]] | | |
|  |  |  | | |
|  | **[[(∞,1)Operad]]** |  | | |
| [[model structure on simplicial operads|SimplicialOperads]] | $-$[[homotopy coherent dendroidal nerve]]$\to$ | [[model structure on dendroidal sets|DendroidalSets]] | | [[model structure for dendroidal Cartesian fibrations|RelativeDendroidalSets]] |
| $\downarrow$[[dendroidal set|dendroidal nerve]] | | $\downarrow$ | | |
| [[model structure for Segal operads|SegalOperads]] | $\hookrightarrow$ | [[model structure for dendroidal complete Segal spaces|DendroidalCompleteSegalSpaces]] | | |
|  |  |  | | |
|  | **$\mathcal{O}$[[Mon(∞,1)Cat]]** |  | | |
|  |  | [[model structure for dendroidal Cartesian fibrations|DendroidalCartesianFibrations]] | | |

[[!redirects table - models for (infinity,1)-categories]]

[[!redirects table - models for (∞,1)-categories]]
[[!redirects table - models for (∞,1)-operads]]


table of branes

**Table of [[brane|branes]] appearing in [[supergravity]]/[[string theory]]** (for classification see at _[[brane scan]]_).

|   [[brane]]    |    in [[supergravity]]          |[[charge]]d under [[gauge field]]|   has [[worldvolume]] theory
|----------------|---------------------------------|---|-----------------------------------------
|**[[black brane]]**|  [[supergravity]] | [[higher gauge field]] | [[SCFT]]
|**[[D-brane]]** |[[type II supergravity|type II]]  |[[RR-field]]| [[super Yang-Mills theory]]
|**$(D = 2n)$**  |[[type IIA supergravity|type IIA]]| $\,$ |         $\,$
| [[D(-2)-brane]]   |        $\,$                     | $\,$ |   |
| [[D0-brane]]   |        $\,$                     | $\,$ |   [[BFSS matrix model]]  |
| [[D2-brane]]   |        $\,$                     | $\,$ |      $\,$
| [[D4-brane]]   |        $\,$                     | $\,$ | [[D=5 super Yang-Mills theory]] with [[Khovanov homology]] [[observables]]
| [[D6-brane]]   |        $\,$                     | $\,$ |  [[D=7 super Yang-Mills theory]] |
| [[D8-brane]]   |        $\,$                     | $\,$ |   |
|**$(D = 2n+1)$**|[[type IIB supergravity|type IIB]]| $\,$ |          $\,$
| [[D(-1)-brane]]   |        $\,$                     | $\,$ | $\,$
| [[D1-brane]]   |        $\,$                     | $\,$ | 2d [[CFT]] with [[BH entropy]]
| [[D3-brane]]   |        $\,$                     | $\,$ | [[N=4 D=4 super Yang-Mills theory]]
| [[D5-brane]]   |        $\,$                     | $\,$ |          $\,$
| [[D7-brane]]   |        $\,$                     | $\,$ |          $\,$
| [[D9-brane]]   |        $\,$                     | $\,$ |          $\,$
| [[(p,q)-string]]   |        $\,$                     | $\,$ |          $\,$
| ([[D25-brane]])  | ([[bosonic string theory]]) | |  |
|**[[NS-brane]]**|[[supergravity|type I, II, heterotic]]|[[circle n-bundle with connection|circle n-connection]] |          $\,$
|  [[string]]    |           $\,$                  |[[B2-field]]|  [[2d SCFT]]
| [[NS5-brane]]  |           $\,$                  |[[B6-field]] | [[little string theory]]
|   **D-brane for [[topological string]]**    |           |   |   $\,$
|   [[A-brane]]    |           |   |   $\,$
|   [[B-brane]]    |           |   |   $\,$
|**[[M-brane]]** |[[11-dimensional supergravity|11D SuGra]]/[[M-theory]]|[[circle n-bundle with connection|circle n-connection]]|           $\,$
| [[M2-brane]]   |           $\,$                  |[[C3-field]]|         [[ABJM theory]], [[BLG model]]
| [[M5-brane]]   |           $\,$                  |[[C6-field]]| [[6d (2,0)-superconformal QFT]]
| [[M9-brane]]/[[O-plane|O9-plane]]   |                                 |            | [[heterotic string theory]] |
| [[M-wave]] |  |   | |
| [[topological M2-brane]] | [[topological M-theory]]                  |[[C3-field]] on [[G2-manifold]] | |
| [[topological M5-brane]] | $\,$                  | [[C6-field]] on [[G2-manifold]] | |
| **[[S-brane]]** |  |   | |
| [[SM2-brane]],<br/>[[membrane instanton]] |  |   | |
| [[M5-brane instanton]] |  |   | |
| [[D3-brane instanton]] |  |   | |
| **[[solitons]]** on [[M5-brane]] | [[6d (2,0)-superconformal QFT]] |  |  |
| [[self-dual string]]    |      |  [[self-dual higher gauge field|self-dual]] [[B-field]] |  |
| [[3-brane in 6d]] |   |    |   |


table of category theories

[[higher category theory]]

* [[(0,1)-category theory]]

* [[category theory|1-category theory]], [[(∞,1)-category theory]]

  [[enriched category theory]]

* [[2-category theory]], [[(∞,2)-category theory]]

* [[(∞,n)-category theory]]


table of marks

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

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #BurnsideCharacter}
###### Definition
**([[Burnside marks]] and [[Burnside character]])**

Let $G$ be a [[finite group]] and let $X \in G Set^{fin}$ any [[finite set|finite]] [[G-set]]. For
$[H]$ the [[conjugacy class]] of a [[subgroup]] $H \subset G$, the _$[H]$-mark on $X$_ is the [[cardinality]] of the [[set]] of $H$-[[fixed points]], hence the [[natural number]]

$$
  \phi_{[H]}(X)
  \;\coloneqq\;
  \left\vert X^{H} \right\vert
$$

of [[elements]] $x \in X$ such that for each $h \in H \subset G$ we have $h(x) = x$.

This construction extends to a [[ring homomorphism]]

\[
  \label{BurnsideCharacter}
  \array{
    \phi
    &\colon&
    A(G) &\overset{\phi}{\longrightarrow}& \mathbb{Z}^{Conj(G)}
    \\
    && [X] &\mapsto& \big( [H] \mapsto \left\vert X^H\right\vert\big)
  }
\]

from the [[Burnside ring]] to the ring of [[tuples]] of [[integers]] of length the number $Conj(G)$ of [[conjugacy classes]] of [[subgroups]] of $G$.

This morphism is also called the _Burnside character_ or _mark homomorphism_.

=--

+-- {: .num_remark #MarksInTermsOfHoms}
###### Remark
**(marks in terms of homs)**

Equivalent the set of $[H]$-marks of a [[G-set]] $X$ (Def. \ref{BurnsideCharacter}) is the [[hom-set]] in [[GSet]] from $G/H$ to $X$.

=--

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

This follows from a basic standard argument. For completeness, we make it explicit:

By [[transitive action|transitivity]] of the [[action]] on $G/H$ a $G$-equivariant function $f \colon G/H \to X$ is fully specified by its [[image]] $f([e]) \in X$ of the [[equivalence class]] $[e] \in G/H$ of the [[neutral element]]. Since this $[e] \in G/H$ is [[fixed point|fixed]] precisely by the elements in $H \subset G$ it may, again by $G$-equivariance, be mapped to any $H$-[[fixed point]] $f([e]) \in X^H \subset X$.

=--

Of particular interest are the marks of the [[transitive action|transitive]] [[G-sets]], i.e. those [[isomorphism|isomorphic]] to sets $G/H$ of [[coset]], for $H\subset G$ a [[subgroup]].  These arrange into a _table of marks_:

+-- {: .num_defn #TableOfMarks}
###### Definition
**([[table of marks]])**

The _table of Burnside marks_ (or _table of marks_, for short) of a [[finite group]] $G$ is the [[matrix]] indexed by [[conjugacy classes]] $[H]$ of [[subgroups]] $H \subset G$ whose $([H_i], [H_j])$-entry is the $[H_j]$-marks of $G/H_i$ (Def. \ref{BurnsideCharacter}), hence the number of [[fixed points]] of the [[action]] of $H_j$ on the [[coset space]] $G/H_i$:

$$
  M_{i j}
  \;\coloneqq\;
  \left\vert
    \left(G/H_i\right)^{H_j}
  \right\vert
  \,.
$$

=--

(e.g. [Pfeiffer 97](#Pfeiffer97), chapter _[The Burnside Ring and the Table of Marks](http://schmidt.ucg.ie/~goetz/pub/marks/node1.html#SECTION00010000000000000000)_)

+-- {: .num_remark #TableOfMarksInTermsOfHoms}
###### Remark
**(table of marks in terms of homs)**

The expression of marks in terms of homs (Remark \ref{MarksInTermsOfHoms}) means here that the table of marks (Def. \ref{TableOfMarks}) is equivalently given by

$$
  M_{i j}
  \;\coloneqq\;
  \left\vert
    \left(G/H_i\right)^{H_j}
  \right\vert
  \;=\;
  \left\vert Hom_{G Set}(G/H_j, G/H_i)\right\vert
  \,.
$$

=--

<br/>

## Properties
 {#Properties}

### Relation to characters
 {#RelationToCharacters}

The following Prop. \ref{BurnsideCharacterIsInjective} that the [[Burnside character]] plays the same role for finite [[G-sets]] as [[characters of representations]] play for finite-dimensional [[linear representations]], in that it faithfully reflects $G$-sets. In fact the marks of a $G$-set over [[cyclic group|cyclic]] [[subgroups]] coincides with the [[character of a linear representation|character]] of its [[permutation representation]] over any [[ground field]] (Prop. \ref{MarkHomomorphismIsCharactersOfPermutationRepresentation}) below.

+-- {: .num_prop #BurnsideCharacterIsInjective}
###### Proposition
**([[Burnside character]] is [[injective function|injective]])**

The Burnside character (eq:BurnsideCharacter) is [[injective function|injective]]. Hence any two [[finite set|finite]] [[G-sets]] are [[isomorphism|isomorphic]] precisely if they have the same Burnside marks (Def. \ref{BurnsideCharacter}).

=--

(e.g. [tomDieck 79, Prop. 1.2.2](#tomDieck79), [tomDieck 09, Prop. 5.1.1](#tomDieck09))

+-- {: .num_prop #MarkHomomorphismIsCharactersOfPermutationRepresentation}
###### Proposition
**([[mark homomorphism]] on [[cyclic groups]] agrees with [[character of a linear representation|characters]] of corresponding [[permutation representations]])**

For $S \in G Set_{fin}$ a [[finite set|finite]] [[G-set]], for $k$ any [[field]] and $k[S] \in Rep_k(G)$ the corresponding [[permutation representation]], the [[character of a representation|character]] $\chi_{k[S]}$ of the [[permutation representation]] at any $g \in G$ equals the [[Burnside marks]] (Def. \ref{BurnsideCharacter}) of $S$ under the [[cyclic group]] $\langle g\rangle \subset G$ [[generators and relations|generated]] by $g$:

$$
  \chi_{k[S]}\big( g \big)
  \;=\;
  \left\vert X^{\langle g \rangle} \right\vert
  \;\in\;
  \mathbb{Z} \longrightarrow k
  \,.
$$

Hence the [[mark homomorphism]] (Def. \ref{BurnsideCharacter}) of $G$-sets restricted to [[cyclic group|cyclic]] [[subgroups]] coincides with the [[character of a representation|characters]] of their [[permutation representations]].

This statement immediately generalizes from plain representations to [[virtual representations]], hence to the [[Burnside ring]].

=--

(e.g. [tom Dieck 09, (2.15)](#tomDieck09))

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

By definition of _[[character of a linear representation]]_, we have that

$$
  \chi_{k[S]}(g)
  =
  tr_{k[S]}(g)
$$

is the [[trace]] of the [[linear map|linear]] [[endomorphism]] $k[S] \overset{g}{\to} k[S]$ of the given [[permutation representation]].

Now the canonical $k$-[[linear basis]] for $k[S]$ is of course the [[set]] $S$ itself, and so

$$
  \begin{aligned}
    \chi_{k[S]}(g)
    & =
    \underset{
      s \in S
    }{\sum}
    \left\{
      \array{
         1 &\vert& g(s) = s
         \\
         0 &\vert& \text{otherwise}
      }
    \right.
    \\
    & =
    \left\vert S^g \right\vert
    \\
    &
    =
    \left\vert S^{\langle g \rangle} \right\vert
  \end{aligned}
$$

Here in the first step we spelled out the definition of [[trace]] in the canonical basis, and in the second step we observed that the [[fixed point set]] of a [[cyclic group]] equals that of any one of its generating elements.

=--

### Relation to Burnside product
 {#RelationToBurnsideProduct}

We discuss, in Prop. \ref{BurnsideProductInTermsOfTableOfMarks} below, how the [[table of marks]] encodes the product in the [[Burnside ring]] of the given [[finite group]] $G$. For this purpose we first consider two Lemma: Lemma \ref{LinearOrderOnConjugacyClassesOfSubgroups} and Lemma \ref{TableOfMarksIsInvertibleUpperTriangular}.

+-- {: .num_prop #LinearOrderOnConjugacyClassesOfSubgroups}
###### Lemma
**([[linear order]] on set of [[conjugacy classes]] of [[subgroups]])**

There exists a [[linear order]] $\leq_{lin}$ on the [[set]] of [[conjugacy classes]] $[H]$ of [[subgroups]] of $G$ such that a subgroup inclusion $H \subset H'$ implies that $[H] \leq_{lin} [H']$.

=--

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

This follows by the general existence of [[linear extensions of partial orders]] applied to the [[subgroup lattice]] of $G$.

=--

+-- {: .num_lemma #TableOfMarksIsInvertibleUpperTriangular}
###### Lemma
**([[table of marks]] is [[lower triangular matrix|lower triangular]] [[invertible matrix]])**

With respect to any [[linear order]] on the [[conjugacy classes]] of subgroups as in Lemma \ref{LinearOrderOnConjugacyClassesOfSubgroups},  the [[table of marks]] (Def. \ref{TableOfMarks}) becomes a [[lower triangular matrix]] over the [[integers]] with non-[[zero]] entries on the diagonal. In  particular, it is an [[invertible matrix]].

=--

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

That the [[subgroup]] $H_j \subset G$ has any [[fixed points]] in $G/H_i$ means that there is a $g \in G$ such that $h g H_i = g H_i$ for all $h \in H_j$, and thus that $g^{-1}H_{j}g$ is a subgroup of $H_{i}$, or in other words, that $H_{j}$ is conjugate to a subgroup of $H_{i}$. Hence

$$
  \big(
    M_{i j} \gt 0
  \big)
  \;\Rightarrow\;
  \big(
    [H_j] \leq_{lin} [H_i]
  \big)
  \,,
$$

and thus the matrix $M$ is [[lower triangular matrix|lower triangular]].

Since at least $H = 1_{G/H}$ is fixed by $H$, we moreover have that the diagonal entries are non-zero.

=--

In the following, given a [[G-set]] $G/H_i$ we write $[G/H_i] \in A(G)$ for its [[isomorphism class]], regarded as an element in the [[Burnside ring]].

+-- {: .num_defn #BurnsideMultiplicities}
###### Definition
**(Burnside multiplicities)**

Given a choice of [[linear order]] on the [[conjugacy classes]] of [[subgroups]] of $G$ (for instance as in Lemma \ref{LinearOrderOnConjugacyClassesOfSubgroups}), we say that the corresponding _structure constants_ of the [[Burnside ring]] (or _Burnside multiplicities_) are the [[natural numbers]]

\[
  n_{i j}^\ell \;\in\; \mathbb{N}
\]

uniquely defined by the [[equation]]

\[
  \label{BurnsideStructureConstants}
  [G/H_i] \times [G/H_j]
  \;=\;
  \underset{ \ell }{\sum}
  n_{i j}^\ell [G/H_\ell]
  \,.
\]

=--

+-- {: .num_prop #BurnsideProductInTermsOfTableOfMarks}
###### Proposition
**([[Burnside ring]] product in terms of [[table of marks]])**

The Burnside ring structure constants  $\left( n_{i j}^\ell\right)$ (Def. \ref{BurnsideMultiplicities}) are equal to the following algebraic expression in the [[table of marks]] $\left( M_{i j}\right)$ and its [[inverse matrix]] $\left( \left(M^{-1}\right)_{r s} \right)$ (which exists by Lemma \ref{TableOfMarksIsInvertibleUpperTriangular}):

$$
  n_{i j}^\ell
  \;=\;
  \underset{m}{\sum}
  M_{i m} \cdot M_{j m}
  \cdot
  (M^{-1})_{m \ell}
$$

=--

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

Let $t$ be the dimension of $M$, i.e. $M$ is a $t \times t$ matrix. For any $1 \leq m \leq t$, we compute as follows:

$$
  \begin{aligned}
    \underset{1 \leq \ell \leq t}{\sum}
    n_{i j}^\ell \cdot M_{\ell m}
    & =
    \underset{1 \leq \ell \leq t}{\sum}
    n_{i j}^\ell
    \cdot
    \left\vert
      Hom_{GSet}
      \big(
        G/H_m
        ,
        G/H_\ell
      \big)
    \right\vert
    \\
    & =
    \left\vert
      Hom_{GSet}
      \big(
        G/H_m
        ,
        \underset{\ell}{\sum}
        n_{i j}^\ell
        \cdot
        G/H_\ell
      \big)
    \right\vert
    \\
    & =
    \left\vert
      Hom_{GSet}\big(
       G/H_m,
       \;
       G/H_i \times G/H_j
      \big)
    \right\vert
    \\
    & =
    \left\vert
      Hom_{GSet}\big(
       G/H_m,
       \;
       G/H_i
      \big)
    \right\vert
    \cdot
    \left\vert
      Hom_{GSet}\big(
       G/H_m,
       \;
       G/H_j
      \big)
    \right\vert
    \\
    & =
    M_{i m} \cdot M_{j m}
  \end{aligned}
$$

Here the third step uses the defining equation (eq:BurnsideStructureConstants) of the structure constants $n_{i j}^\ell$, while all other steps use that the mark homomorphism is a [[ring homomorphism]], which we made manifest by expressing the marks via [[hom-sets]] (Remark \ref{TableOfMarksInTermsOfHoms}).

Thus we have that $\left( n_{i j}^{1}, n_{i j}^{2}, \ldots, n_{i j}^{t} \right)$ is a solution to the following system of equations.

$$
\begin{aligned}
    M_{i 1} \cdot M_{j 1} &= M_{1 1}x_{1} + \cdots + M_{t 1}x_{t} \\
    M_{i 2} \cdot M_{j 2} &= M_{1 2}x_{1} + \cdots + M_{t 2} x_{t} \\
                          &\vdots \\
   M_{i t} \cdot M_{j t} &= M_{1 t}x_{1} + \cdots + M_{t t} x_{t}
\end{aligned}
$$

But, since $M$ is invertible, the unique solution to this system of equations is given by the product of $M^{-1}$ and the transposition of $\left( M_{i 1} \cdot M_{j 1}, \ldots, M_{i t} \cdot M_{j t} \right)$. The claim follows immediately.

=--

\begin{corollary} The [[table of marks]] of a finite group determines its [[Burnside ring]]. That is to say, if the tables of marks of a pair of groups $G_{1}$ and $G_{2}$ are [[equality|equal]], then the Burnside ring of $G_{1}$ is [[isomorphism|isomorphic]] to the Burnside ring of $G_{2}$.
\end{corollary}

\begin{proof} The Burnside ring of a finite group is a free abelian group on the set $G / H_1, \ldots, G / H_t$, where $H_1, \ldots, H_t$ are representatives of the conjugacy classes of that group, equipped with a certain multiplication. Thus it suffices to check that the structure constants of the Burnside rings coincide, which is established by the previous proposition. \end{proof}

## Applications

* [[Burnside ring is equivariant stable cohomotopy of the point]]

## Related concepts

* [[character table]]

## References

The concept was introduced in

* {#Burnside1897} [[William Burnside]], chapter XII of _Theory of Groups of Finite Order_, 1897 ([pdf](http://www.gutenberg.org/files/40395/40395-pdf.pdf))

* {#Burnside01} [[William Burnside]], _On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups_, Proceedings of the American Mathematical Society 1901 ([doi:10.1112/plms/s1-34.1.159](https://doi.org/10.1112/plms/s1-34.1.159))

Modern textbook accounts and lecture notes:

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_ Lecture Notes in Mathematics 766 Springer 1979

* {#tomDieck09} [[Tammo tom Dieck]], sections 5.1 and 5.4 of _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

* {#LuxPahlings10} Klaus Lux, Herbert Pahlings, section 3.5 of _Representations of groups -- A computational approach_, Cambridge University Press 2010 ([author page](http://www.math.rwth-aachen.de/~RepresentationsOfGroups/), [publisher page](http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521768078))

Computer implementation is discussed in

* {#GAP} [GAP](https://www.gap-system.org/) [Reference Manual](https://www.gap-system.org/Manuals/doc/ref/chap0.html), _[70 Tables of Marks](https://www.gap-system.org/Manuals/doc/ref/chap70.html)_

See also

* {#Pfeiffer97} [[Götz Pfeiffer]], _The Subgroups of $M_{24}$, or How to Compute the Table of Marks of a Finite Group_, Experiment. Math. 6 (1997), no. 3, 247–270 ([doi:10.1080/10586458.1997.10504613](https://doi.org/10.1080/10586458.1997.10504613), [web](http://schmidt.ucg.ie/~goetz/pub/marks/marks.html))

* Liam Naughton, [[Götz Pfeiffer]], _Computing the table of marks of a cyclic extension_, Math. Comp. 81 (2012), no. 280, 2419–2438.

* Brendan Masterson, [[Götz Pfeiffer]], _On the Table of Marks of a Direct Product of Finite Groups_, Journal of Algebra Volume 499, 1 April 2018, Pages 610-644 ([arXiv:1704.03433](https://arxiv.org/abs/1704.03433))

[[!redirects tables of marks]]

[[!redirects Burnside mark]]
[[!redirects Burnside marks]]

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

[[!redirects mark homomorphism]]
[[!redirects mark homomorphisms]]


table of orthogonal groups and related

| [[group]] | symbol | [[universal cover]] | symbol | higher cover | symbol |
|-----------|--------|---------------------|--------|--------------|--------|
| [[orthogonal group]] | $\mathrm{O}(n)$ | [[Pin group]] | $Pin(n)$ | [[Tring group]] | $Tring(n)$ |
| [[special orthogonal group]] | $SO(n)$ | [[Spin group]] | $Spin(n)$ | [[String group]] | $String(n)$ |
|[[Lorentz group]] | $\mathrm{O}(n,1)$ | $\,$ | $Spin(n,1)$ | $\,$ | $\,$ |
|[[anti de Sitter group]] | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |
|[[conformal group]] | $\mathrm{O}(n+1,t+1)$ | $\,$  |             |      |      |
|[[Narain group]] | $O(n,n)$ | | | | |
|[[Poincaré group]] | $ISO(n,1)$ | [[Poincaré spin group]] | $\widehat {ISO}(n,1)$ | $\,$ | $\,$ |
|[[super Poincaré group]] | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |
|[[superconformal group]] |             |      |      |      |      |


taboo

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

# Taboos
* table of contents
{: toc}

## Idea

A **taboo**, for a particular flavor of mathematics or formal system, is a simple statement that is known to be not provable therein, and that can therefore be used to establish the unprovability of other statements without the need to descend into metamathematical considerations (such as syntactic analysis or construction of countermodels) or, in some cases, even to decide on a particular formal system to be working in.

For example, the [[law of excluded middle]] (LEM) is a taboo for [[constructive mathematics]].  Therefore, if some statement $P$ implies LEM, then one can be sure that $P$ is also not provable in constructive mathematics; this is known as a **Brouwerian counterexample**.

## Examples

### Constructive taboos

These taboos are unprovable in [[constructive mathematics]].

* [[law of excluded middle]]
* [[principle of omniscience]]
* [[axiom of choice]]
* [[supports split]]
* [[fan theorem]]
* [[choice operator]]
* [[presentation axiom]]
* [[dependent choice]]

### Weakly predicative-constructive taboos

These taboos are unprovable in (constructive) [[weakly predicative mathematics]].

* [[power sets]]
* [[axiom of separation|unbounded separation]]
* [[coinductive types]]
* [[propositional resizing]]
* [[type of all propositions]]

### Strongly predicative taboos

These taboos are unprovable in [[strongly predicative mathematics]].

* [[subset collection]]
* [[function sets]]
* [[axiom of replacement]]
* [[axiom of collection]]
* [[function types]]
* [[dependent function types]]

### Homotopical taboos

These taboos are unprovable in [[homotopy type theory]], even if we assume constructive taboos such as LEM and AC.

* any [[axiom of set truncation]]
  * [[uniqueness of identity proofs]]
  * [[axiom K]]
  * [[choice operator]]
  * [[axiom of choice]] for all types
  * [[axiom of finiteness]]

A taboo found in [[cubical type theory]]:

* [[boundary separation]]

### Constructive-homotopical taboos

These taboos are unprovable in constructive homotopy type theory, but they may be provable if we assume a constructive taboo *or* a homotopical taboo.

* [[sets cover]]
* [[Whitehead's principle]]

### Discrete cohesion taboos

These taboos are unprovable in discrete [[cohesive homotopy type theory]] (i.e. the setting for plain homotopy type theory):

* [[axiom of real cohesion]]
* [[axiom of A1-cohesion]]

## Related entries

* [[constructive set theory]]

## References
 {#References}

* {#Aczel08Slides} [[Peter Aczel]], slide 8 of: *Constructive Set Theory*, 2008 ([pdf](http://www.cs.man.ac.uk/~petera/mathlogaps-slides.pdf))

> A mathematical taboo is a statement that we may not want
to assume false, but we definitely do not want to be able to prove.

* {#AczelRathjen01} [[Peter Aczel]], [[Michael Rathjen]], Section 2.3 of: _Notes on Constructive Set theory_, 2001 ([pdf](https://events.math.unipd.it/3wftop/pdf/AczelRathjen.pdf), [[AczelRathjenCST.pdf:file]])

> Certain basic principles of classical mathematics are taboo for the constructive mathematician. Bishop called them *principles of omniscience*.

* {#Mandelkern89} [[Mark Mandelkern]], _Brouwerian Counterexamples_, Mathematics Magazine, vol. 62 no. 1, 1989 ([doi:10.2307/2689939](https://www.tandfonline.com/doi/abs/10.1080/0025570X.1989.11977404))

See also:

* [[Michael J. Beeson]], Footnote 3, p. 6 (27 of 484) in: *Foundations of Constructive Mathematics*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **3** 6, Springer 1985 ([doi:10.1007/978-3-642-68952-9](https://link.springer.com/book/10.1007/978-3-642-68952-9), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-642-68952-9.pdf))

> Peter Aczel has introduced the word *taboo* in this context. The  decidability of equality on the  reals is taboo in the sense that any proposition which has been shown to imply it will be regarded as essentially  non-constructive. When we refute a proposition, we show that it implies a
contradiction, an absurdity. If instead we show that it implies the decidability of equality on the reals, we have shown its essential non-constructivity, but in a weaker way than by actually refuting it. In other  words, $0 = 1$ is the most taboo of all taboos. There are several other  common taboos besides the decidability of equality, which we shall soon  encounter.

* [[Hannes Diener]], Section 0.3 and 0.4 of: *Constructive Reverse Mathematics*, 2018 ([arXiv:1804.05495](https://arxiv.org/abs/1804.05495), [dspace:ubsi/1306](https://dspace.ub.uni-siegen.de/handle/ubsi/1306))

>  the full axiom of choice is a definite constructive taboo

[[!redirects taboos]]
[[!redirects constructive taboo]]
[[!redirects constructive taboos]]
[[!redirects homotopical taboo]]
[[!redirects homotopical taboos]]
[[!redirects constructive-homotopical taboo]]
[[!redirects constructive-homotopical taboos]]
[[!redirects Brouwerian counterexample]]
[[!redirects Brouwerian counterexamples]]


tabular dagger 2-poset

[[!redirects tabular dagger 2-posets]]

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

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

## Idea ##

A dagger 2-poset with tabulations, such as a [[tabular allegory]].

## Definition ##

A **tabular dagger 2-poset** is a [[dagger 2-poset]] $C$ such that for every object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A,B)$, there is an object $\vert R \vert:Ob(C)$ and [[map in a dagger 2-poset|maps]] $f:Hom(\vert R \vert, A)$, $g:Hom(\vert R \vert, B)$, such that $R = f^\dagger \circ g$ and for every object $E:Ob(C)$ and maps $h:Hom(E,\vert R \vert)$ and $k:Hom(E,\vert R \vert)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply $h = k$.

## Properties ##

The [[category of maps]] of a tabular dagger 2-poset has all [[pullbacks]].

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a tabular dagger 2-poset.

## See also ##

* [[dagger 2-poset]]


tabulator

+-- {: .rightHandSide}
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###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

\tableofcontents

## Idea

A *tabulator* is a kind of [[double limit]], i.e. a [[limit]] in a [[double category]], generalizing the notion of the *[[graph of a profunctor]]* (also called the _cocollage_). Dually, the notion of a *cotabulator* generalizes that of the *[[cograph of a profunctor]]* (also called the _collage_).

## Definition

The _tabulator_ of a [[vertical morphism]] $u \colon A \nrightarrow B$ in a [[double category]] $\mathbb{D}$ consists of an object $Tu$ and a [[2-morphism]]
\begin{tikzcd}
Tu
\arrow[r, "p"]
\arrow[d, "\bullet"{anchor=center}, "id"']
\arrow[rd, phantom, "\tau_{u}"]
& A
\arrow[d, "\bullet"{anchor=center}, "u"]
\\
Tu
\arrow[r, "q"']
& B
\end{tikzcd}
with the universal properties:

1. For every [[2-morphism]]
\begin{tikzcd}
X
\arrow[r, "f"]
\arrow[d, "\bullet"{anchor=center}, "id"']
\arrow[rd, phantom, "\xi"]
& A
\arrow[d, "\bullet"{anchor=center}, "u"]
\\
X
\arrow[r, "g"']
& B
\end{tikzcd}
there is a unique horizontal morphism $x \colon X \rightarrow Tu$ such that
$\tau_{u} x = \xi$.

2. The tetrahedron condition holds (see [Paré, page 478](#Pare2011)).

If only the first universal property holds, we say that the vertical morphism has a _$1$-tabulator_.

## Properties

* A double category $\mathbb{D} = (D_{0}, D_{1})$ has all $1$-tabulators (resp. $1$-cotabulators) if and only if the [[identity morphisms|identity]]-assigning map $id \colon D_{0} \rightarrow D_{1}$ has a [[right adjoint]] (resp. [[left adjoint]]).

* A double category has all small [[double limits]] if and only if it has small double products, double equalisers, and tabulators.

* A tabulator in a [[2-category]], viewed as a [[double category]], is a [[power]] by the [[interval category]].

## References

The definition of a (co)tabulator was first introduced in Section 5.3 of:

* [[Marco Grandis]], [[Robert Pare]], _Limits in double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **40**, 1999. ([url](http://www.numdam.org/item/CTGDC_1999__40_3_162_0/))

The $2$-dimensional universal property of a tabulator is stated in Section 3 of:

* {#Pare2011} [[Robert Pare]], _Yoneda theory for double categories_, Theory and Applications of Categories_, **25**, 2011. ([url](http://www.tac.mta.ca/tac/volumes/25/17/25-17abs.html))

Other references on (co)tabulators include:

* [[Susan Niefield]], _Span, cospan, and other double categories_ Theory and Applications of Categories_, **26**, 2012. ([url](http://www.tac.mta.ca/tac/volumes/26/26/26-26abs.html))

* [[Marco Grandis]], _Higher Dimensional Categories. From Double to Multiple Categories_, 2019. ([doi:10.1142/11406](https://doi.org/10.1142/11406))

[[!redirects tabulators]]
[[!redirects cotabulator]]
[[!redirects cotabulators]]


TAC reprints series

The journal [Theory and Applications of Categories](http://www.tac.mta.ca/tac/) has an extremely useful reprint series. The full list of reprints is [here](http://www.tac.mta.ca/tac/reprints/index.html).


tachyon

+-- {: .rightHandSide}
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### Context
#### Vacua
+-- {: .hide}
[[!include vacua -- contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbation theory]] (in [[QFT]]) a _tachyon_ is an excitation/[[particle]] with _negative [[mass]] squared_, hence technically with [[imaginary number|imaginary]] mass.

Since the masses of particles in [[perturbation theory]] give the second derivative of the background potential energy at the point about which the [[perturbation series]] is developed, a tachyon in the spectrum indicates that the [[perturbation theory]] is set up about an _unstable_ [[vacuum]].

The [[dynamics]] of a tachyon will hence make it form a [[condensate]] at an actual local [[potential energy]] [[minimum]]. Once there it is no longer a tachyon, but a genuine [[particle]] excitation ("tachyon decay").

## Examples

### In bosonic string theory

The excitation of the [[open string]] in [[bosonic string theory]] famously contains a tachyon mode. By [[Sen's conjecture]] this indicates the instability of the space-filling [[D25-brane]] on which the ends of the open bosonic string propagate. Using [[string field theory]] to get a [[non-perturbative field theory|non-perturbative]] description of the situation one can follow the decay of the [[D25-brane]] to the "true" (stable) open [[bosonic string theory]] [[vacuum]]. (Where however analysis shows that this no longer contains open string excitations, so it is maybe better called a closed string vacuum.) See at _[[Sen's conjecture]]_ for more.

The fate of the [[closed string|closed]] bosonic string tachyon is more subtle. But see the [references below](#ReferencesInStringTheory).

### In superstring theory

Similarly, in [[superstring theory]] [[open string]] states between [[D-brane]]/[[anti D-brane]] pairs may be tachyonic, signallying the decay of these brane configurations: see *[[Sen's conjecture]]* [in superstring theory](Sen's+conjecture#InSuperstringTheory). This leads to the conjectured classification of [[D-brane charge]] in [[K-theory]] (see [there](D-brane#ReferencesKTheoryDescription)).

(...)

## Related concepts

[[!include states and observables -- content]]

* [[field (physics)]]

* [[inflaton field]]

## References

### General

The concept originates with

* Olexa‐Myron Bilaniuk, V. K. Deshpande, [[E. C. G. Sudarshan]], _"Meta" Relativity_, American Journal of Physics 30, 718 (1962) ([doi:10.1119/1.1941773](https://doi.org/10.1119/1.1941773))

* Olexa‐Myron Bilaniuk, [[E. C. George Sudarshan]], _Particles beyond the light barrier_, Physics Today 22, 5, 43 (1969) ([doi:10.1063/1.3035574](https://doi.org/10.1063/1.3035574))

See also

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

### In string theory
 {#ReferencesInStringTheory}

In addition to the open string tachyon [[condensation]] discussed in the references at _[[Sen's conjecture]]_, there is also work on the closed string tachyon [[condensation]] in the following articles.

* A. Adams, [[Joseph Polchinski]] and [[Eva Silverstein]], _Don't panic! Closed string tachyons in ALE space-times,_ JHEP 0110 (2001) 029 ([arXiv:hep-th/0108075](http://arxiv.org/abs/hep-th/0108075)).

* R. Rabadan and [[Joan Simon]], _M theory lift of brane anti-brane systems and localized closed string tachyons_, JHEP 0205 (2002) 045 ([arXiv:hep-th/0203243](http://arxiv.org/abs/hep-th/0203243)).

* [[Ángel Uranga]], _Localized instabilities at conifolds_, ([arXiv:hep-th/0204079](http://arxiv.org/abs/hep-th/0204079)).

* Shin Nakamura, _Closed-string Tachyon Condensation and the On-shell Effective Action of Open-string Tachyons_,  	Prog.Theor.Phys. 106 (2001) 989-1016, ([arXiv:hep-th/0105054](http://arxiv.org/abs/hep-th/0105054))

The following articles specifically identify closed string tachyon condensation as the disappearing of [[dimensions]] of [[spacetime]].

* Simeon Hellerman, X. Liu, _Dynamical dimension change in supercritical string theory_, JHEP0709:096,2007
([hep-th/0409071](http://arxiv.org/abs/hep-th/0612051))

* Simeon Hellerman, Ian Swanson, _Dimension-changing exact solutions of string theory_, JHEP0709:096,2007 ([arXiv:hep-th/0612051](http://arxiv.org/abs/hep-th/0612051))

* Mikel Berasaluce-González, Miguel Montero, Ander Retolaza, [[Ángel Uranga]], _Discrete gauge symmetries from (closed string) tachyon condensation_ ([arXiv:1305.6788](http://arxiv.org/abs/1305.6788))

[[!redirects tachyons]]

[[!redirects tachyon field]]
[[!redirects tachyon fields]]

[[!redirects tachyon condensation]]
[[!redirects tachyon condensations]]


Tadakatsu Sakai

* [webpage](http://www2.yukawa.kyoto-u.ac.jp/~tadakatsu.sakai/)

## Selected writings

On the [[Witten-Sakai-Sugimoto model]] ([[geometric engineering of QFT|geometric engineering]] of [[QCD]] via [[D4-D8-brane bound state]] [[intersecting D-brane models]]):

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Prog.Theor.Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _More on a holographic dual of QCD_, Prog.Theor.Phys.114:1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

* Hiroyuki Hata, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Shinichiro Yamato, _Baryons from instantons in holographic QCD_, Prog.Theor.Phys.117:1157, 2007 ([arXiv:hep-th/0701280](https://arxiv.org/abs/hep-th/0701280))

* [[Koji Hashimoto]], [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Holographic Baryons: Static Properties and Form Factors from Gauge/String Duality_, Prog.Theor.Phys.120:1093-1137, 2008 ([arXiv:0806.3122](https://arxiv.org/abs/0806.3122))

On the [[nuclear matrix model]]:

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

## Related $n$Lab entries

* [[AdS-QCD correspondence]]

* [[skyrmion]]

* [[hadron]],

  [[meson]], [[baryon]]

category: people


Tadao Oda

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tadao_Oda)

## Selected writings

On [[Gauss-Manin connections]]:

* [[Nicholas M. Katz]], [[Tadao Oda]], *On the differentiation of De Rham cohomology classes with respect to parameters*, J. Math. Kyoto Univ. **8** 2 (1968) 199-213 $[$[doi:10.1215/kjm/1250524135](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-8/issue-2/On-the-differentiation-of-De-Rham-cohomology-classes-with-respect/10.1215/kjm/1250524135.full)$]$

On [[algebraic analysis]] over [[complex manifolds]]:

* [[Tadao Oda]], *Introduction to algebraic analysis on complex manifolds*, Algebraic varieties and analytic varieties (Tokyo, 1981), 29–48, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam (1983) $[$[doi:10.2969/aspm/00110029](https://doi.org/10.2969/aspm/00110029), [MR85a:14010](http://www.ams.org/mathscinet-getitem?mr=715644)$]$

category: people


Tadao Tannaka

**Tadao Tannaka** was a Japanese mathematician.

* wikipedia: [de](http://de.wikipedia.org/wiki/Tadao_Tannaka), [en](http://en.wikipedia.org/wiki/Tadao_Tannaka), [ja](http://ja.wikipedia.org/wiki/%E6%B7%A1%E4%B8%AD%E5%BF%A0%E9%83%8E)

His early works were related mainly to Galois theory and number theory.

His famous classical paper on [[Tannaka duality]] proving the original [[Tannaka-Krein theorem]] is

* _Über den Dualitätssatz der nichtkommutativen topologischen Gruppen_, Tôhoku Math. J. 45 (1939) 1-12.  [PDF](https://www.jstage.jst.go.jp/article/tmj1911/45/0/45_0_1/_pdf/-char/ja).

Note that this article is in the first series of Tôhoku.

Tannaka was an editor of Tôhoku Math. J. for a longer period.

He also wrote a booklet in Japanese which treats a wider picture of dualities

* Sôtai genri.  [The principle of duality.] Iwanami Shoten, Tokyo, 1951. v+128 pp.

His biography has been published in Tôhoku Math. J. (see project [euclid](http://projecteuclid.org/euclid.tmj/1178228363))

## Related entries

* [[Tannaka duality]]

[[!redirects Tannaka]]
[[!redirects T. Tannaka]]


Tadashi Takayanagi

* [webpage](http://www2.yukawa.kyoto-u.ac.jp/~tadashi.takayanagi/eindex.html)

## Selected writings

On the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]]:

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Holographic Derivation of Entanglement Entropy from AdS/CFT_, Phys. Rev. Lett. 96:181602, 2006 ([arXiv:hep-th/0603001](https://arxiv.org/abs/hep-th/0603001))

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Aspects of Holographic Entanglement Entropy_, JHEP 0608:045, 2006 ([arXiv:hep-th/0605073](https://arxiv.org/abs/hep-th/0605073))

review:

* {#NishiokaRyuTakayanagi09} Tatsuma Nishioka, [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Holographic Entanglement Entropy: An Overview_, J.Phys.A42:504008,2009 ([arXiv:0905.0932](https://arxiv.org/abs/0905.0932))

Textbook on [[holographic entanglement entropy]]:

* [[Mukund Rangamani]], [[Tadashi Takayanagi]], *Holographic Entanglement Entropy*, Lecture Notes in Physics **931**, Springer (2017) [[doi:10.1007/978-3-319-52573-0](https://doi.org/10.1007/978-3-319-52573-0), [arXiv:1609.01287](https://arxiv.org/abs/1609.01287)]

On [[Chern-Simons theories]] arising from the [[higher WZ term]] on [[super p-brane]] [[worldvolumes]] (notably on [[D2-branes]]):

* Mitsutoshi Fujita, Wei Li, [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy_, JHEP 0906:066, 2009 ([arXiv:0901.0924](https://arxiv.org/abs/0901.0924))

On the [[K-theory classification of topological phases of matter]] translating under [[AdS/CFT duality in solid state physics]]  to the [[K-theory classification of D-brane charge]] in [[string theory]], allowing a dual description of the topological phases even at [[non-perturbative effects|strong coupling]] via [[AdS/CFT duality]]:

* [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from D-branes_, Phys. Lett. B693:175-179, 2010 ([arXiv:1001.0763](https://arxiv.org/abs/1001.0763))

* [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from String Theory_, Phys. Rev. D82:086014, 2010 ([arXiv:1007.4234](https://arxiv.org/abs/1007.4234))

* [[Andreas Karch]], Joseph Maciejko, [[Tadashi Takayanagi]], _Holographic fractional topological insulators in 2+1 and 1+1 dimensions_,  Phys. Rev. D 82, 126003 (2010) ([arXiv:1009.2991](https://arxiv.org/abs/1009.2991))

On [[AdS3/CFT2]] for [[boundary field theory]]:

* [[Tadashi Takayanagi]], Takahiro Uetoko, _Chern-Simons Gravity Dual of BCFT_ ([arXiv:2011.02513](https://arxiv.org/abs/2011.02513))

On the [[swampland cobordism conjecture]]:

* [[Hirosi Ooguri]], [[Tadashi Takayanagi]], _Cobordism Conjecture in AdS_ ([arXiv:2006.13953](https://arxiv.org/abs/2006.13953))

## Related $n$Lab entries

* [[holographic entanglement entropy]]

category: people


Tadasi Nakayama

* wikipedia entry

* [MathGenealogy entry](https://www.mathgenealogy.org/id.php?id=147654)

## Selected writings

On [[separable algebras]] and [[Frobenius algebras]]:

* {#EilenbergNakayama55} [[Samuel Eilenberg]], [[Tadasi Nakayama]], *On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings*, _Nagoya Math. J._ **9** (1955) 1-16  [[euclid](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-9/issue-none/On-the-dimension-of-modules-and-algebras-II-Frobenius-algebras/nmj/1118799677.pdf)]

category: people


Tadayuki Haraguchi

## Selected writings

On the [[adjunction between topological spaces and diffeological spaces]]:

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

On the [[model structure on Delta-generated topological spaces]] ([[D-topological spaces]]):

* {#Haraguchi13} [[Tadayuki Haraguchi]], _On model structure for coreflective subcategories of a model category_, Math. J. Okayama Univ. 57 (2015), 79–84 ([arXiv:1304.3622](http://arxiv.org/abs/1304.3622), [doi:10.18926/mjou/53040](http://doi.org/10.18926/mjou/53040), MR3289294, Zbl 1311.55027)

On a [[model structure on diffeological spaces]]:

* [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces_ ([arXiv:1311.5668](https://arxiv.org/abs/1311.5668))

* {#Haraguchi18} [[Tadayuki Haraguchi]], _Homotopy structures of smooth CW complexes_ ([arXiv:1811.06175](https://arxiv.org/abs/1811.06175))

* [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _On homotopy types of diffeological cell complexes_ ([arXiv:1912.05359](https://arxiv.org/abs/1912.05359))

* {#HaraguchiShimakawa20} [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces, I_ ([arXiv:2011.12842](https://arxiv.org/abs/2011.12842))

category: people


Tadayuki Watanabe

* [personal page](https://www.math.kyoto-u.ac.jp/~tadayuki.watanabe/)

* [institute page](https://www.math.kyoto-u.ac.jp/en/people/profile/tadayuki.watanabe)

## Selected writings

On counter-examples (using [[graph complexes]]) to the analogue of the [[Smale conjecture]] for the [[4-sphere]]:

* [[Tadayuki Watanabe]], *Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$* ([arXiv:1812.02448](https://arxiv.org/abs/1812.02448))

* [[Tadayuki Watanabe]], *Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (homological interpretation)* ([arXiv:2109.01609](https://arxiv.org/abs/2109.01609))

category: people


Tadeusz Balaban

* [webpage](http://www.math.rutgers.edu/component/comprofiler/userprofile/tbalaban)

## related $n$Lab entries

* [[lattice gauge theory]]

category: people


Tadeusz Józefiak

* [Mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=146524)

category: people

[[!redirects Tadeusz Jozefiak]]


tadpole

+-- {: .rightHandSide}
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###Context###
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+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In _[[perturbative quantum field theory]]_ a _tadpole_ is a contribution to the [[S-matrix]]/[[scattering amplitude]] corresponding to a [[Feynman diagram]] which contains an [[edge]] that connects some [[vertex]] with itself.

<img src="https://ncatlab.org/nlab/files/tadpole.png" width="70">

Typically tadpoles are required to be absent. For instance the construction of [[time-ordered products]] in [[causal perturbation theory]] via the "star product" induced by the [[Feynman propagator]] (see [there](S-matrix#ExistenceAndRenormalization)) implies that tadpoles are absent.

## Examples

* [[RR-field tadpole cancellation]] in [[type II string theory]]

* [[C-field tadpole cancellation]] in [[M-theory]]

## Related concepts

* [[tadpole cancellation]]

## References

Discussion in context of [[causal perturbation theory]]/[[perturbative AQFT]] is in

* {#Keller10} [[Kai Keller]], around p. 30 of of _Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization_, PhD thesis ([arXxiv:1006.2148](https://arxiv.org/abs/1006.2148))

[[!redirects tadpoles]]

[[!redirects tadpole diagram]]
[[!redirects tadpole diagrams]]

[[!redirects tadpole-diagram]]
[[!redirects tadpole-diagrams]]


tadpole cancellation

+-- {: .rightHandSide}
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###Context###
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]], _tadpole cancellation_ refers to the condition that, or the mechanism by which, [[tadpoles]] disappear from the [[Feynman perturbation series]].

## Examples

* [[RR-field tadpole cancellation]] in [[type II string theory]]

* [[C-field tadpole cancellation]] in [[M-theory]]

* [[24 branes transverse to K3]] in [[F-theory]] and [[HET-theory]]

## Related concepts

* [[anomaly cancellation]]

[[!redirects tadpole cancellations]]

[[!redirects tadpole anomaly cancellation]]
[[!redirects tadpole anomaly cancellations]]


Taejin Lee

* [GoogleScholar page](https://scholar.google.co.kr/citations?user=kG1cXtsAAAAJ&hl=ko)

## Related $n$Lab entries

* [[string field theory]]

* [[gauge enhancement]]

* [[BFSS matrix model]], [[IKKT matrix model]]

category: people


Tai Tsun Wu

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tai_Tsun_Wu)

* [Institute page](https://www.physics.harvard.edu/people/facpages/wu)

* [InSpire page](https://inspirehep.net/authors/983112)

## Selected writings:

On the [[Aharonov-Bohm effect]], generalized to non-abelian [[Yang-Mills theory]] and its description by [[connection on a bundle|connections]] on [[fiber bundles in physics]]:

* {#WuYang75} [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

On the [[Dirac monopole]]:

* [[Tai Tsun Wu]], [[Chen Ning Yang]], _Dirac monopole without strings: monopole harmonics_, Nuclear Physics B107:3 (1976) 365--380

* Tai Tsun Wu, Chen Ning Yang, _Dirac's monopole without strings: Classical Lagrangian theory_, Phys. Rev. D 14, 437 (1976)

category: people


Tai-Danae Bradley

* [webpage](https://www.math3ma.com/about)

## Selected writings

On [[general topology|general]] [[topology]] with emphasis on tools from [[category theory]]:

* {#BradleyBrysonTerilla20} [[Tai-Danae Bradley]], [[Tyler Bryson]], [[John Terilla]], _Topology -- A categorical approach_, MIT Press 2020 ([ISBN:9780262539357](https://mitpress.mit.edu/books/topology), [web version](https://topology.pubpub.org/))

category: people


Taichi Uemura

* [Homepage](http://www.kurims.kyoto-u.ac.jp/~uemura/)

## Selected writings

On [[fibration categories]]:

* [[Taichi Uemura]], *Fibred fibration categories*, 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (2017) $[$[arXiv:1602.08206](http://arxiv.org/abs/1602.08206), [doi:10.1109/LICS.2017.8005084](https://doi.org/10.1109/LICS.2017.8005084)$]$

On [[categorical semantics]]  [[relation between type theory and category theory|for type theories]] and presenting a solution to the [[initiality conjecture]] for type theories formulated in terms of second-order generalized algebraic theories:

* [[Taichi Uemura]], _A General Framework for the Semantics of Type Theory_ [[arXiv:1904.04097](https://arxiv.org/abs/1904.04097), talk slides: [pdf](https://hott.github.io/HoTT-2019/conf-slides/Uemura.pdf)]

* [[Taichi Uemura]], _Abstract and concrete type theories_, PhD thesis (2021) [[hdl:11245.1/41ff0b60-64d4-4003-8182-c244a9afab3b](https://hdl.handle.net/11245.1/41ff0b60-64d4-4003-8182-c244a9afab3b), [pdf](https://pure.uva.nl/ws/files/62028111/Thesis.pdf)]

category: people


Taichiro Kugo

* [webpage](http://www2.yukawa.kyoto-u.ac.jp/~taichiro.kugo/index-e.html)

## Selected writings

On [[hidden local symmetry]] in [[chiral perturbation theory]]/[[quantum hadrodynamics]]:

* [[Masako Bando]], [[Taichiro Kugo]], Koichi Yamawaki, _Nonlinear realization and hidden local symmetries_, Physics Reports Volume 164, Issues 4–5, July 1988, Pages 217-314 (doi:10.1016/0370-1573(88)90019-1)

* [[Masako Bando]], [[Taichiro Kugo]], S. Uehara, Koichi Yamawaki, and T. Yanagida, _Is the $\rho$ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?_, Phys. Rev. Lett. 54, 1215 (1985) ([doi:10.1103/PhysRevLett.54.1215](https://doi.org/10.1103/PhysRevLett.54.1215))

## Related $n$Lab entries

* [[supersymmetry and division algebras]]

category: people


Taimanov theorem

## Statement

Let $X$ be a $T_1$ space, $i\colon D \hookrightarrow X$ a dense subspace, $Y$ a compact Hausdorff space and $f\colon D \to Y$ be any continuous map. If for all disjoint closed $A, B \subseteq Y$ we have $\overline{i(f^{-1}(A))} \cap \overline{i(f^{-1}(B))} = \emptyset$, then there is a continuous extension of $f$ to $X$.

There is a variant where $X$ is arbitrary and $Y$ is $T_3$.

## Related entries

* [[Tychonoff theorem]]
* [[Tietze extension theorem]]
* [[Whitney extension theorem]]

## References

* A.D. Taĭmanov, _О распространении непрерывных отображений
топологических пространств_ (*On extension of continuous mappings of topological spaces.*) Mat. Sbornik N.S. 31(73), (1952). 459–463. [mathnet.ru page](http://mi.mathnet.ru/eng/msb/v73/i2/p459)

The result is stated, for instance, as Theorem 3.2.1 in

* Engelking, _General Topology_

* [Math.SE discussion](https://math.stackexchange.com/questions/312845/taimanovs-extension-theorem-collecting-applications)


Takaki Matsumoto

* [inspire page](https://inspirehep.net/authors/1950574)

## Selected writings

On [[M5-branes]] in the [[BMN matrix model]] in a limit where aspects of [[little string theory]] on [[NS5-branes]] becomes visible:

* [[Yuhma Asano]], [[Goro Ishiki]], [[Takaki Matsumoto]], [[Shinji Shimasaki]], [[Hiromasa Watanabe]], *On the existence of the NS5-brane limit of the plane wave matrix model*, PTEP 2023 (2023) 4, 043B01 $[$[arXiv:2211.13716](https://arxiv.org/abs/2211.13716), [doi:10.1093/ptep/ptad042](https://doi.org/10.1093/ptep/ptad042)$]$

category: people


Takao Matumoto

## Selected writings

On [[G-CW-complexes]] and the [[equivariant Whitehead theorem]] for [[compact Lie groups]] $G$:

* {#Matumoto71} [[Takao Matumoto]], _On $G$-CW complexes and a theorem of JHC Whitehead_, J. Fac. Sci. Univ. Tokyo Sect. IA **18** (1971) 363-374 [[irdb:00926/0001786419](https://irdb.nii.ac.jp/en/00926/0001786419), [[matumoto.pdf:file]]]

*  [[Takao Matumoto]], _Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 ([pdf](https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_action_common_download&item_id=39826&item_no=1&attribute_id=19&file_no=1), [[MatumotoEquivariantKTheory.pdf:file]])

On detection of [[weak homotopy equivalences]] and [[equivariant weak homotopy equivalences]] in terms of free homotopy sets:

* [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))

category: people


Takashi Kimura

* [webpage](http://math.bu.edu/people/kimura/)

## Selected writings

On the [[BRST complex]] (with focus on its [[cochain cohomology]]):

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], *The cohomology of BRST complexes* (1988) [[pdf](https://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/homol.pdf), [[FigueroaKimuar-BRSTCohomology.pdf:file]]]

and application to the [[NSR string|NSR]] [[superstring]]:

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], _The BRST cohomology of the NSR string: vanishing and ``no-ghost'' theorems_, Comm. Math. Phys. **124** 1 (1989) 105-132. [[euclid:cmp/1104179078](http://projecteuclid.org/euclid.cmp/1104179078)]

On [[Lie algebra weight systems]] arising from [[super Lie algebras]]:

* {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014))

category: people


Takashi Oka

* [webpage](https://www.pks.mpg.de/nonequilibrium-quantum-matter/members/takashi-oka)

## Selected writings

Interpretation in [[holographic QCD]] of the [[Schwinger effect]] as exhibited by the [[DBI-action]] on [[flavor branes]]:

* {#HashimotoOka13} [[Koji Hashimoto]], [[Takashi Oka]], _Vacuum Instability in Electric Fields via AdS/CFT: Euler-Heisenberg Lagrangian and Planckian Thermalization_, JHEP 10 (2013) 116 ([arXiv:1307.7423](https://arxiv.org/abs/1307.7423))

* {#HashimotoOkaSonoda14a} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD_, J. High Energ. Phys. 2014, 85 (2014) ([arXiv:1403.6336](https://arxiv.org/abs/1403.6336))

* {#HashimotoOkaSonoda14b} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Electromagnetic instability in holographic QCD_,  J. High Energ. Phys. 2015, 1 (2015) ([arXiv:1412.4254](https://arxiv.org/abs/1412.4254))

* [[Koji Hashimoto]], Shunichiro Kinoshita, Keiju Murata, [[Takashi Oka]], _Electric Field Quench in AdS/CFT_, J. High Energ. Phys. 2014 ([arXiv:1407.0798](https://arxiv.org/abs/1407.0798))

category: people


Takeshi Morita

## Selected writings

On the [[nuclear matrix model]] for [[baryons]]/[[nucleons]] the [[Witten-Sakai-Sugimoto model]]:

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

category: people


Takeshi Torii

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=66668)

category: people


Takeuchi product

Given an associative algebra $A$, with enveloping algebra $A^e = A\otimes A^{op}$, the Takeuchi product $\times_A$ is certain product
in the category of $A$-[[ring]]s.
It generalizes a construction of [[M. E. Sweedler]] where $A$ is commutative; Sweedler's article may be itself viewed in a sense a "generalization of the relative [[Brauer group]] and the associated theory".

Takeuchi product is used in the theory of associative [[bialgebroid]]s over noncommutative base.

* [[Mitsuhiro Takeuchi]], _Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan __29__, 459–492, 1977, [MR0506407](http://www.ams.org/mathscinet-getitem?mr=0506407), [euclid](http://projecteuclid.org/euclid.jmsj/1240432948)
* M. E. Sweedler, _Groups of simple algebras_, Publ. IHES __44__, 79--189, [MR51:587](http://www.ams.org/mathscinet-getitem?mr=51:587), [numdam](http://www.numdam.org/item?id=PMIHES_1974__44__79_0)
* [[T. Brzeziński]], G. Militaru, _Bialgebroids, $\times_{R}$-bialgebras and duality_, J. Algebra 251: 279-294, 2002, [math.QA/0012164](http://arxiv.org/abs/math/0012164)
* P. Schauenburg, _Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules_, Appl. Categ. Structures __6__ (1998), 193–222, [ps](http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/bnrsthb.ps) [doi](http://dx.doi.org/10.1023/A:1008608028634)

category: algebra


Takuro Mochizuki

* [web](http://www.kurims.kyoto-u.ac.jp/en/list/takuro.html)

* *Donaldson type invariants for algebraic surfaces: transition of moduli stacks*, Lec. Notes in Math. 1972, Springer 2009

* _Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules I, II_, AMS 2007

category: people


Talia Ringer

* [personal page](https://dependenttyp.es/)

## Selected writings

Using [[type equivalence]] in proof reuse/repair for [[certified programming]]

* [[Talia Ringer]], *Proof Repair*, Univ. Washington (2021) [[proquest:2568297410](https://www.proquest.com/docview/2568297410), [video](https://www.youtube.com/watch?v=_BkTrp44uBU)]

category: people