synthetic tangent bundle

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#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
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## Idea

In [[synthetic differential geometry]], the _[[tangent bundle]]_ of an [[object]] $X$ is the [[internal hom]] $X^{\mathbb{D}^1}$ out of the 1d first order [[infinitesimal disk]] $\mathbb{D}^1$, equipped with the projection to $X$ induced from the unique point $\ast \to \mathbb{D}^1$:

$$
  X^{(\ast \to \mathbb{D}^1)} \;\colon\; X^{(\mathbb{D}^1)} \longrightarrow X
  \,.
$$

(Here we are using that the [[internal hom|internal]] [[hom-functor]] $(-)^{(-)}$ is a [[contravariant functor]] in its "exponent" variable.)

This makes manifest and precise the intuitive idea that a [[tangent vector]] on $X$ is an "[[infinitesimal object|infinitesimal]] [[curve]]" in $X$, see also the [Examples](#Examples) below.

In this formulation the operation of [[differentiation]] is simply the [[internal hom]]-[[functor]]:

Given a function:

$$
  X \overset{f}{\longrightarrow} Y
  \,.
$$

its [[differential]] is its image under the [[internal hom]] $(-)^{(\mathbb{D}^1)}$:

$$
  X^{(\mathbb{D}^1)}
   \overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow}
  Y^{(\mathbb{D}^1)}
  \,.
$$

## Examples
 {#Examples}

In a standard model for [[synthetic differential geometry]]/[[differential cohesion]] such as the [[Cahiers topos]] $\mathbf{H}$, for $X \in SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ an ordinary [[smooth manifold]], its [[synthetic tangent bundle]] coincides with the traditional [[tangent bundle]] $T X \overset{p}{\to} X$:

$$
  \array{
    X^{(\mathbb{D}^1)} & \simeq & T X
    \\
    {}^{\mathllap{ X^{ (\ast \to \mathbb{D}^1) } }}\big\downarrow
      &&
    \big\downarrow{}^{ \mathrlap{p_X} }
    \\
    X^{\ast} & \simeq & X
  }
$$

Moreover, if $Y \in SmthMfd \hookrightarrow \mathbf{H}$ is another [[smooth manifold]], then a [[morphism]]

$$
  X \overset{\phantom{AA} f \phantom{AA} }{\longrightarrow} Y
$$

is equivalently a [[smooth function]] in the traditional sense (i.e. the external [[Yoneda embedding]]-[[functor]] $SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ is [[fully faithful functor|fully faithful]] ) and its [[image]] under the [[internal hom]] is its traditional [[differentiation]] $d f$:

$$
  \array{
    X^{(\mathbb{D}^1)}
      &\overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow}&
    Y^{(\mathbb{D}^1)}
    \\
    {}^\simeq\big\downarrow && \big\downarrow{}^\simeq
    \\
    T X &\overset{ \phantom{AA} d f \phantom{AA} }{\longrightarrow}&
    T Y
  }
$$

This way the evident [[functor|functoriality]] of the [[internal hom]] $(-)^{(\mathbb{D}^1)}$ is identified with the [[chain rule]] of traditional [[differentiation]].

For more on this see

* at _[Cahiers topos -- Synthetic tangent spaces](Cahiers+topos#RelationToSyntheticTangentSpaces)_.

* at _[[geometry of physics -- supergeometry]]_ the section _[Super mapping spaces](geometry+of+physics+--+supergeometry#SuperMappingSpaces)_

## Properties

### For Microlinear spaces

For $X$ a [[microlinear space]] the synthetic tangent bundle shares many of the expected properties of a [[tangent bundle]].

## Related concepts

* [[odd tangent bundle]] (analog in [[supergeometry]])

* [[kinematic tangent bundle]], [[operational tangent bundle]]

* [[jet bundle]] via [[jet comonad]]

## References

For lecture notes see

* [[geometry of physics -- supergeometry]] the section _[Super mapping spaces](geometry+of+physics+--+supergeometry#SuperMappingSpaces)_

[[!redirects synthetic tangent bundles]]
[[!redirects synthetic tangent space]]
[[!redirects synthetic tangent spaces]]


synthetic topology

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
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## Idea

**Synthetic topology**, like [[synthetic domain theory]], [[synthetic differential geometry]], and [[synthetic computability]], are part of [[synthetic mathematics]]. It uses the internal logic of a topos to develop a part of mathematics. In this case [[topology]]. This is closely related to topology via logic and [[abstract Stone duality]].

The formal system of [[type theory]] has semantics in many [[categories]], and in particular in many categories of "spaces". Thus types may be regarded not just as [[sets]] but as [[general topology|topological objects]].  Interestingly, a good deal of this "topology" can be detected intrinsically in type theory, often corresponding to the possible failure of principles of [[classical mathematics]].

## Dictionary

[[Martín Escardó]] has given the following translations between the two fields:

| general topology | type theory |
|--|--|
| [[topological space|space]] | [[type]] |
| [[continuous function]] | [[function]] |
| [[clopen set]] | [[decidable object|decidable]] set |
| [[open set]] | [[semi-decidable equality|semi-decidable]] set |
| [[closed set]] | set with semi-decidable [[complement]] |
| [[discrete space]] | type with [[decidable equality]] |
| [[Hausdorff space]] | type with semi-decidable [[inequality relation|inequality]] |
| [[convergent sequence]] | map out of $\mathbb{N}_\infty$ (see [below](#Semantics)) |
| [[compact set]] | exhaustively searchable set, in a finite number of steps |

It should be stressed that the concepts on the right are *not* the only ways to represent the topological concepts on the left in type theory.  For instance, in [[cohesive homotopy type theory]] there is a notion of "discrete space" that has nothing to do with decidable equality (in particular, in homotopy type theory a type with decidable equality is necessarily an [[h-set]], whereas discrete spaces don't need to be h-sets).

## Semantics {#Semantics}

There are many different topological semantics for type theory, but one which seems especially closely related to the above dictionary is the [[topological topos]].  For instance, in that case the internally defined set $\mathbb{N}_\infty$ (the set of infinite decreasing binary sequences) really does get interpreted semantically as "the generic convergent sequence".

## Implementation

Many of the results that have originated from this view have been implemented in an [[Agda]] [library](https://www.cs.bham.ac.uk/~mhe/agda/).

## Related pages

* [[dominance]]
* [[open subset]]
* [[overt space]]

## References

* [[Martín Escardó]], _Synthetic topology of data types and classical spaces_, 2004 ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf))

* [[Martín Escardó]], {#EscardoPopl2012} _The topology of seemingly impossible functional programs_, 2012 ([pdf](http://www.cs.bham.ac.uk/~mhe/.talks/popl2012/escardo-popl2012.pdf), almost same presentation with minor changes [pdf](https://cs.ioc.ee/ewscs/2012/escardo/slides.pdf))

* [[Andrej Bauer]], [[Davorin Lešnik]], _Metric Spaces in Synthetic Topology_, 2010 ([pdf](http://math.andrej.com/wp-content/uploads/2010/01/csms_in_synthtop.pdf))

* [[Davorin Lešnik]], _Synthetic Topology and Constructive Metric Spaces_, PhD, 2010 ([arXiv:2104.10399](https://arxiv.org/abs/2104.10399))

* [[Davorin Lešnik]], _Unified Approach to Real Numbers in Various Mathematical Settings_, 2014 ([arXiv:1402.6645](https://arxiv.org/abs/1402.6645))

* [[Steven Vickers]], _Locales and toposes as spaces, 2007 ([pdf](https://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf))

[[!redirects relation between type theory and topology]]


syntomic cohomology

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#### Cohomology
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#Contents#
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## Idea

_Syntomic cohomology_ is the [[abelian sheaf cohomology]] of the _syntomic site_ of a [[scheme]].
It is a $p$-[[p-adic|adic]] analogue of [[Deligne-Beilinson cohomology]].

Syntomic cohomology is closely related to the [[crystalline cohomology]] of that scheme and may be regarded as a [[p-adic numbers|$p$-adic]] [[absolute Hodge cohomology]].

## Construction via Prismatic Cohomology

The syntomic cohomology may also be obtained from [[prismatic cohomology]] ([Bhatt22](#Bhatt22)). Let $R$ be a p-adically complete ring and let $\Delta_{R}$ be its absolute prismatic cohomology. It has an action of the [[Frobenius morphism]] $\phi$. We also have a "Breuil Kisin twist" $\Delta\lbrace 1\rbrace$ and a [[filtration]] $\Fil_{N}^{\bullet}\Delta$ called the _Nygaard filtration_ (see see [Bhatt21](#Bhatt21), section 2). The syntomic cohomology $\mathbb{Z}_{p}(i)$ is then defined to be the [[fiber]]

$$\mathbb{Z}_{p}(i)(R)=\fib(\Fil_{N}^{i}\Delta_{R}\lbrace i\rbrace\xrightarrow{\phi_{i}-1}\Delta_{R}\lbrace i\rbrace)$$

This construction globalizes and may be applied to p-adic [[formal schemes]] instead of just p-adically complete rings (see [Bhatt21](#Bhatt21), Remark 2.14).

## Related

* [[Deligne-Beilinson cohomology]]

* [[p-adic Hodge theory]]

* [[prismatic cohomology]]

## References

The syntomic site was introduced in

* Jean-Marc Fontaine and [[William Messing]], _$p$-Adic periods and $p$-adic etale cohomology_ ([pdf](http://www.ams.org/publications/online-books/conm67-conm67-ch4.pdf))

A construction of syntomic cohomology via [[prismatic cohomology]] is briefly discussed in

* {#Bhatt21} [[Bhargav Bhatt]], _Algebraic Geometry in Mixed Characteristic_ ([arXiv:2112.12010](https://arxiv.org/abs/2112.12010))

* {#Bhatt22} [[Bhargav Bhatt]], _p-adic Hodge Theory and Applications: Connections to Algebraic Topology_ (Day 3 of Simons Lectures) ([YouTube](https://www.youtube.com/watch?v=OUDJZnuy8Jw))

Further developments are in

* Amnon Besser, _Syntomic regulators and $p$-adic integration I: rigid syntomic regulators_ ([pdf](http://www.math.uiuc.edu/K-theory/0298/reg.pdf))

The following shows that, just as [[Deligne-Beilinson cohomology]] may be interpreted as [[absolute Hodge cohomology]], syntomic cohomology may be interpreted as $p$-adic absolute Hodge cohomology.

* [[Frédéric Déglise]], [[Wiesława Nizioł]], _On $p$-adic absolute Hodge cohomology and syntomic coefficients, I_, [arXiv:1508.02567](http://arxiv.org/abs/1508.02567).


syntomic site

## Related concepts

[[fpqc-site]] $\to$ [[fppf-site]] $\to$ **syntomic site** $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]


syntopogenous space

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#### Topology
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# Syntopogenous spaces
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## Idea

A *syntopogenous space* is a common generalization of [[topological spaces]], [[proximity spaces]], and [[uniform spaces]].  The category of syntopogenous spaces includes $Top$, $Prox$, and $Unif$ as full subcategories whose intersection is fairly trivial.

## Definitions

### Topogenous relations

We work with binary [[relation]]s on the [[power set]] $P(X)$ of a [[set]] $X$.  As for [[proximity spaces]], there are three styles of such relation that are classically inter-definable, but in [[constructive mathematics]] the choice matters.  They are notated by $A \;\delta\; B$, $A\bowtie B$, and $A\ll B$; the relationship between them classically is that $\bowtie$ is the [[negation]] of $\delta$, while $A\ll B$ means $A \bowtie (X\setminus B)$.

A relation $\delta$ on $P(X)$ is called **topogenous** (or a **topogenous nearness**) if it satisfies:

1. *Nontriviality* or *reflexivity*: if $A \cap B$ is [[inhabited set|inhabited]], then $A \;\delta\; B$.

2. *Binary additivity*: $A \;\delta\; (B \cup C)$ if and only if $A \;\delta\; B$ or $A \;\delta\; C$; and $(A \cup B) \;\delta\; C$ if and only if $A \;\delta\; C$ or $B \;\delta\; C$.

3. *Nullary additivity*: it is never true that $A \;\delta\; \emptyset$ or $\emptyset \;\delta\; A$ for any $A$.

Note that the "if" directions of binary additivity are equivalent to *isotony*: if $A \supseteq B \;\delta\; C \subseteq D$ implies $A \;\delta\; D$.  We might specify these separately and call only the reverse direction of binary additivity 'binary additivity'.

A relation satisfying merely (2) and (3) is called a __topogeny__ (between $P(X)$ and itself) at [[toddtrimble:topogeny]]; it is slightly easier to work with the lattice of topogenies than the lattice of topogenous relations, but syntopogenous spaces are built only out of those topogenies that satisfy (1).

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###### Remark
As is shown at [[toddtrimble:topogeny]], a topogenous nearness can also be regarded as equivalent to a [[reflexive relation]] from $\beta X$ to $\beta X$ (see [[ultrafilter monad]]) in the [[pretopos]] of [[compact Hausdorff spaces]]. In more concrete terms: each topogeny $\delta$ is a union of "basic topogenies" which are those of the form $\mathcal{U} \times \mathcal{V} \subset P X \times P X$ where $\mathcal{U}, \mathcal{V}$ are ultrafilters, and the set of all pairs $(\mathcal{U}, \mathcal{V})$ whose product is contained in $\delta$ forms a closed subspace of $\beta X \times \beta X$.
=--

The axioms of a topogenous nearness can be reinterpreted in terms of $\bowtie$ to yield a **topogenous apartness**, satisfying

1. *Nontriviality* or *reflexivity*: if $A\bowtie B$, then $A \cap B = \emptyset$.

2. *Binary additivity*: $A \bowtie (B \cup C)$ if and only if $A \bowtie B$ and $A \bowtie C$; and $(A \cup B) \bowtie C$ if and only if $A \bowtie C$ and $B \bowtie C$.

3. *Nullary additivity*: it is always true that $A \bowtie \emptyset$ and $\emptyset \bowtie A$ for any $A$.

Similarly, in terms of $\ll$ we obtain a **topogenous order** or **topogenous neighborhood relation**:

1. *Nontriviality* or *reflexivity*: if $A\ll B$, then $A\subseteq B$.

2. *Binary additivity*: $A \ll (B \cap C)$ if and only if $A \ll B$ and $A \ll C$; and $(A \cup B) \ll C$ if and only if $A \ll C$ and $B \ll C$.

3. *Nullary additivity*: it is always true that $A \ll X$ and $\emptyset \ll A$ for any $A$.

The set of topogenous nearness relations on $X$, ordered by containment, is a [[complete lattice]], as is the set of topogenies.  (If we are working with topogenous apartnesses $\bowtie$ or orders $\ll$ instead of nearnesses $\delta$, then the order relation in these lattices is reversed).

* The [[least element]] is the *discrete topogenous relation*, defined by $A \;\delta\; B$ if and only if $A \cap B$ is [[inhabited subset|inhabited]]. (In the set of topogenies, the least element is the *trivial topogeny*, in which $A \;\delta\; B$ never holds.)
* The [[greatest element]] is the *codiscrete topogenous relation* (also the *cotrivial topogeny*), defined by $A \;\delta\; B$ if and only if both $A$ and $B$ are inhabited.
* The [[union]] of any inhabited set of topogenous relations is topogenous, hence is a [[join]].  To treat inhabited joins and the nullary join (the least element) uniformly, we can take the union of the topogenous relations and the discrete topogenous relation.  (With topogenies, we can just take the union, period.)
* The [[intersection]] of a [[directed set]] of topogenous relations (or topogenies) is again a topogenous relation (or topogeny) and so is a [[meet]].  (This is not provable constructively for topogenous nearnesses $\delta$, but the opposite facts for $\bowtie$ and $\ll$ are: directed unions of topogenous apartnesses and topogenous orders are again such.)
* In general, the intersection of two topogenous relations is not topogenous (or even a topogeny); however, they still have a meet: $A \;\delta\; B$ (where $\delta$ is the meet of $\delta_1$ and $\delta_2$) iff, whenever $A = \bigcup_{i=1}^n A_i$ and $B = \bigcup_{j=1}^m B_j$, there exist $i$ and $j$ such that $A_i \;\delta_1\; B_j$ and $A_i \;\delta_2\; B_j$.

From binary meets (just above), directed meets (above that) and nullary meets (the greatest element), we get all meets (even constructively).  But the [[meet]] of an arbitrary set $\mathcal{D}$ of topogenous relations (or topogenies) can still be easily described explicitly: we have $A \;(\bigwedge\mathcal{D})\;B$ if and only if whenever $A = \bigcup_{i=1}^n A_i$ and $B = \bigcup_{j=1}^m B_j$, there exist $i$ and $j$ such that $A_i \;\delta\; B_j$ for all $\delta\in\mathcal{D}$.

The [[opposite relation]] of a topogenous relation is again topogenous.  A topogenous relation (or topogeny) is called **symmetric** if it is equal to its opposite, i.e. if $A \;\delta\; B$ if and only if $B \;\delta\; A$.  This is easy to reexpress in terms of $\bowtie$.  For $\ll$ the corresponding fact would be $A \ll B$ if, or if and only if, $(X\setminus B)\ll (X\setminus A)$; but constructively the first is too weak and the second too strong.

A topogenous relation is called **perfect** if $A \;\delta\; B$ implies there exists an $x \in A$ with $\{x\} \;\delta\; B$.  It is called **biperfect** if both it and its opposite are perfect.  Of course, a symmetric perfect topogenous relation is automatically biperfect.  Similarly, a topogenous apartness is perfect if $\{x\}\bowtie B$ for all $x\in A$ implies $A\bowtie B$, and a topogenous order is perfect if $\{x\}\ll B$ for all $x\in A$ implies $A\ll B$.  Biperfection for a topogenous order is better expressed without reference to opposites as "if $A\ll B_i$ for all $i$, then $A\ll \bigcap_i B_i$".

### Syntopogenous spaces

A **syntopogeny** (or **syntopogenous structure**) on a set $X$ is a [[filter]] $\mathcal{O}$ of topogenous nearness relations such that

* For any $\delta\in \mathcal{O}$, there exists a $\delta'\in\mathcal{O}$ such that if $A,B\subseteq X$ have the property that for any $D\subseteq X$, either $A\;\delta'\; (X\setminus D)$ or $D\;\delta'\; B$, then $A\;\delta\; B$.

If we are working with apartnesses $\bowtie$ or orders $\ll$, then we instead ask for an [[ideal]] satisfying the analogous properties:

* For any $\bowtie \in \mathcal{O}$, there exists a $\bowtie'\in \mathcal{O}$ such that if $A\bowtie B$, then there exists $D\subseteq X$ such that $A\bowtie' (X\setminus D)$ and $D\bowtie' B$.

* For any $\ll\in\mathcal{O}$, there exists an $\ll'\in \mathcal{O}$ such that if $A\ll B$, then there exists $D\subseteq X$ such that $A\ll' D \ll' B$.

If we have been working with the lattice of topogenies rather than topogenous relations, then we must explicitly state here that every topogeny in $\mathcal{O}$ satisfies (1).  This requirement actually joins with the requirement above to form an [[unbiased definition]] that is nicely expressed in terms of topogenous orders $\ll$ (and tacitly using isotony to show the equivalence):

* For each [[natural number]] $n$ and each $\ll$ in the syntopogeny $\mathcal{O}$, there is an $\ll'$ in $\mathcal{O}$ such that, whenever $A \ll B$, there is a [[list]] $C_0, \ldots, C_n$ such that $A \subseteq C_0 \ll' C_1 \cdots \ll' C_n \subseteq B$.

A **basis** for a syntopogeny on $X$ is a [[filterbase]] in the complete lattice of topogenous relations (or of topogenies), such that the filter it generates is a syntopogeny.  When $X$ is equipped with a syntopogeny, it is called a **syntopogenous space**.  In constructive mathematics, where it matters whether we choose $\delta$, $\bowtie$, or $\ll$, we may speak for emphasis of *syntopogenous nearness spaces*, *syntopogenous apartness spaces*, or *syntopogenous neighborhood spaces*.

A syntopogeny is called **symmetric**, **perfect**, or **biperfect** if it admits a basis consisting of symmetric, perfect, or biperfect topogenous relations, respectively.  It is called **simple** if it admits a basis that is a [[singleton]] (which must then be a [[quasiproximity]], and a [[proximity]] in the symmetric case).

### Syntopogenous functions

If $\delta$ is a topogenous relation on $Y$ and $f:X\to Y$ is a function, then we have a topogenous relation $f^*\delta$ on $X$ defined by $A\;(f^*\delta)\;B$ iff $f(A) \;\delta\; f(B)$.  That is, $f^*\Delta = (\exists_f \times \exists_f)^{-1}(\delta)$, where $\exists_f : P(X) \to P(Y)$ is the [[quantifier|left adjoint]] of $f^{-1}:P(Y) \to P(X)$.

Now if $(X_1,\mathcal{O}_1)$ and $(X_2,\mathcal{O}_2)$ are syntopogenous spaces, a function $f:X_1\to X_2$ is called **syntopogenous**, or **syntopologically continuous**, if for any $\delta\in\mathcal{O}_2$, we have $f^*\delta\in\mathcal{O}_1$.  This defines the [[category]] $STpg$.

If $(Y,\mathcal{O})$ is a syntopogenous space, then the collection $\{ f^*\delta | \delta\in\mathcal{O}\}$ is a basis for a syntopogeny on $X$, which is the [[initial structure]] induced on $X$ by $f$.  The operation of taking initial structures, as a map from syntopogenies on $Y$ to syntopogenies on $X$, preserves opposites, simplicity, meets, symmetry, and perfectness.

More generally, if $(Y_i,\mathcal{O}_i)$ is a family of syntopogenous spaces and $f_i:X\to Y_i$ are functions, then the meet of the initial structures induced by all the $f_i$ is the initial structure induced by them jointly.  Thus, $STpg\to Set$ is a [[topological concrete category]].

## Relation to other topological structures

### Topological spaces

If $X$ is a [[topological space]], we define $A\;\delta\; B$ to hold if $A\cap \overline{B}$ is inhabited, where $\overline{B}$ denotes the [[closure]] of $B$.  This is a basis for a simple perfect syntopogeny.

Conversely, given a simple perfect syntopogeny, with singleton basis $\{\delta\}$, we define $\overline{B} = \{ x | \{x\}\;\delta\; B \}$; then this is a Kuratowski closure operator and hence defines a topology.

These constructions define an [[equivalence of categories]] between [[Top]] and the full subcategory of $STpg$ on the simple, perfect, syntopogenous spaces.

In constructive mathematics, the three kinds of (simple, perfect) syntopogenous spaces can be identified with three no-longer-equivalent notions of "topological space": the nearness ones correspond, as described above, with [[closure spaces]], the apartness ones with [[point-set apartness spaces]], and the neighborhood onse with ordinary [[topological spaces]].

### Proximity spaces

A simple symmetric syntopogeny is easily seen to be precisely a [[proximity]] (of the appropriate sort: nearness, apartness, or neighborhood).  In this way we have an equivalence of categories between $Prox$ and the full subcategory of $STpg$ on the simple, symmetric, syntopogenous spaces.

More generally, an arbitrary simple syntopogeny can be identified with a [[quasiproximity]], and we have an equivalence of categories between $QPrxo$ and the subcategory of simple syntopogenous spaces.

### Uniform spaces

If $\delta$ is a biperfect topogenous relation, then we have $A\;\delta\;B$ if and only if there exist $x\in A$ and $y\in B$ with $\{x\}\;\delta\;\{y\}$.  Therefore, $\delta$ is completely determined by a binary relation $U\subseteq X\times X$ on $X$, which contains the diagonal $\Delta_X$.  Conversely, any binary relation on $X$ containing the diagonal defines a biperfect topogenous relation.  (For a biperfect topogeny, remove the requirement that $U$ contain the diagonal.)

It follows that biperfect syntopogenies are equivalent to [[quasi-uniformities]], which are like [[uniformities]] but lack the symmetry axiom.  We have an equivalence of categories between $QUnif$ and the full subcategory of $STpg$ on the biperfect syntopogenous spaces, which easily restricts to an equivalence between $Unif$ and the category of symmetric, (bi)perfect topogenous spaces.

Constructively, this equivalence still holds for both nearness *and* neighborhood syntopogenous spaces.  In general, topogenous nearnesses and topogenous neighborhood relations are not equivalent, but a *biperfect* topogenous neighborhood relation is determined by giving for each $x$ the smallest set $B$ such that $\{x\}\ll B$, and as $x$ varies these sets again determine exactly a binary relation on $X$ (containing the diagonal).  Syntopogenous apartness spaces, on the other hand, correspond to [[uniform apartness spaces]].

### Preorders and symmetric prosets

A syntopogeny which is both simple and biperfect is determined uniquely by a single relation on $X$ which must be both [[reflexive relation|reflexive]] and [[transitive relation|transitive]], i.e. a [[preorder]].  Thus, the intersection $Top \cap QUnif$ inside $STpg$ is equivalent to $Preord$.

Of course, it follows that a simple, symmetric, (bi)perfect syntopogeny is determined uniquely by a relation on $X$ that is reflexive, transitive, and also symmetric -- i.e. an [[equivalence relation]].  Thus, the intersections $Top \cap Unif$, $Top \cap Prox$, and $Prox\cap (Q)Unif$ inside $STpg$ are all equivalent to the category $SymPreOrd$ of [[symmetric prosets]] (sets equipped with an equivalence relation).

## Some coreflections

In the preorder of topogenous relations on any set $X$, the following sub-preorders are coreflective:

* The symmetric elements.  The symmetric coreflection of $\delta$ is the meet $\delta^s \coloneqq \delta \wedge \delta^{op}$.
* The perfect elements.  The perfect coreflection of $\delta$ is defined by $A\;\delta^p\;B$ iff there exists $x\in A$ with $\{x\}\;\delta\;B$.
* The biperfect elements.  The byperfect coreflection of $\delta$ is defined by $A\;\delta^b\;B$ iff there exist $x\in A$ and $y\in B$ with $\{x\}\;\delta\;\{y\}$.

It follows that in the preorder of syntopogenous structures on $X$, the symmetric, perfect, and biperfect elements are also reflective; the coreflections are obtained by applying the above one to each topogenous relation in turn.  Moreover, the simple syntopogenous structures on $X$ are also coreflective; the coreflection just takes the intersection of all relations belonging to the filter (this is a directed intersection, hence automatically again a topogenous relation --- although constructively this only works in the dual cases of $\bowtie$ and $\ll$).

Finally, for any function $f:X\to Y$, the preimage function $f^*$, mapping syntopogenous structures on $Y$ to those on $X$, preserves all of these coreflections.  Therefore, the full subcategories of

* simple (i.e. quasi-proximity),
* symmetric,
* perfect, and
* biperfect (i.e. quasi-uniformity)

syntopogenous spaces are all coreflective in $STpg$, with coreflections written $(-)^t$, $(-)^s$, $(-)^p$, and $(-)^b$ respectively.

In general, of course, coreflections into distinct subcategories do not commute or even preserve each other's subcategories.  However, by construction, we see that the coreflections $(-)^s$, $(-)^p$, and $(-)^b$ all preserve simplicity.  Therefore, the full subcategories of

* simple symmetric (i.e. proximity),
* simple perfect (i.e. topological), and
* simple biperfect (i.e. preorders)

syntopogenous spaces are all coreflective in $STpg$, with coreflections $(-)^{t s}$, $(-)^{t p}$, and $(-)^{t b}$ respectively.  Finally, it is evident by construction that $(-)^b$ preserves symmetry, so the full subcategories of

* symmetric biperfect (i.e. uniformity), and
* simple symmetric biperfect (i.e. symmetric prosets)

syntopogenous spaces are also both coreflective in $STpg$, with coreflections $(-)^{s b}$ and $(-)^{t s b}$ respectively.

It is straightforward to verify the following.

1. When applied to a (quasi)-proximity space or a (quasi)-uniform space, the coreflection $(-)^{t p}$ into topological spaces computes the underlying topology of these structures, as usually defined.

2. When applied to a uniform space, the coreflection $(-)^{t s}$ computes its underlying proximity, as usually defined.  The same is true in the non-symmetric case for quasi-uniformities and quasi-proximities.

3. When applied to any syntopogenous space, the coreflection $(-)^{t b}$ computes the [[specialization order]] of its underlying topology (i.e. its image under $(-)^{t p}$).  In particular, this is the case for topological spaces, proximity spaces, and uniform spaces.

## Related concepts

* [[funcoid]]

[[!include generalized uniform structures - table]]

## References

* Ákos Császár, *Foundations of General Topology*, 1963

Császár uses topogenous orders $\ll$.  Császár also defines a syntopogenous structure to be what we have called a *basis* for one.  As usual, this is convenient for concreteness (especially in the simple case), but has the disadvantage that distinct structures can nevertheless be isomorphic via an identity function, i.e. the forgetful functor to $Set$ is not [[amnestic functor|amnestic]].  On this page, we have followed the traditional practice for other topological structures in choosing to make this functor amnestic.

[[!redirects syntopogenous space]]
[[!redirects syntopogenous spaces]]
[[!redirects syntopogenous structure]]
[[!redirects syntopogenous structures]]
[[!redirects syntopogeny]]
[[!redirects syntopogenies]]

[[!redirects symmetric syntopogenous space]]
[[!redirects symmetric syntopogenous spaces]]
[[!redirects symmetric syntopogenous structure]]
[[!redirects symmetric syntopogenous structures]]
[[!redirects symmetric syntopogeny]]
[[!redirects symmetric syntopogenies]]

[[!redirects topogenous relation]]
[[!redirects topogenous relations]]
[[!redirects topogenous nearness relation]]
[[!redirects topogenous nearness relations]]
[[!redirects topogenous nearness]]
[[!redirects topogenous nearnesses]]
[[!redirects topogenous apartness relation]]
[[!redirects topogenous apartness relations]]
[[!redirects topogenous apartness]]
[[!redirects topogenous apartnesses]]
[[!redirects topogenous order relation]]
[[!redirects topogenous order relations]]
[[!redirects topogenous order]]
[[!redirects topogenous orders]]

[[!redirects syntopogenous nearness space]]
[[!redirects syntopogenous nearness spaces]]
[[!redirects syntopogenous apartness space]]
[[!redirects syntopogenous apartness spaces]]
[[!redirects syntopogenous neighborhood space]]
[[!redirects syntopogenous neighborhood spaces]]


System F

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_System F_, also known as the _[[polymorphic]] [[lambda-calculus]]_ or _[[second-order logic|second-order]] [[lambda-calculus]]_, is an extension of [[simply typed lambda-calculus]] to include a [[type]] of [[impredicative]] [[universal quantification]].

## Semantics

Models of System F can be described as a form of [[hyperdoctrine]]:

1. A [[category]] K of *kinds* with [[cartesian products]].
2. A [[functor]] $P \colon K^o \to CCC$, i.e., a [[contravariant functor]] from $K$ to the category of [[cartesian closed categories]] and [[functors]] preserving the CC structure.
3. A [[representable functor|representing object]] $U \in K$, i.e., $K(-,U) \cong P_0$ where $P_0 : K^o \to Set$ is the [[presheaf]] of objects of the corresponding CCC.
4. Such that for any $\Delta \in K$, the functor $P(\pi) \colon P(\Delta) \to P(\Delta \times U)$ induced from the [[projection]] $\pi \colon \Delta \times U \to \Delta$ has a [[right adjoint]] $\forall \colon P(\Delta \times U) \to P(\Delta)$ satisfying a [[Beck-Chevalley condition]].

Typically, the objects of the category of *kinds* are restricted to be generated by [[Cartesian product|products]] of $U$.

Note that this models the $\beta\eta$ equivalence of System F only, not the more complex notion of [[relational parametricity]]. A *relationally parametric model* of System F is

1. A model of System F [[internal]] to the category of [[reflexive graphs]]
2. Such that the functor $(s,t) : P_e(\Delta_e) \to P_v(s(\Delta_e)) \times P_v(t(\Delta_e))$ is a [[fibration]].

Note that by taking the "vertex" component of such a model, we get an underlying ordinary "equational" model of System F, and so we can see a relatonally parametric model as an extension of an equational model.
The internalization to reflexive graphs models the relational structures: the "edge" components are the relations, the source/target maps ensure that each relation lies over an ordinary structure, and the reflexive edge models the notion of an identity/[[equality]] relation. The fibration requirement then provides a rich supply of relations: given a relation R and morphisms $f_s : B_s \to s(R)$ and $f_t : B_t \to t(R)$, the fibration structure defines a "weakest pre-relation" $R[f_s,f_t]$ with source and target $B_s,B_t$. Notably, if $R$ is an identity relation and one of $f_s,f_t$ is the identity, this is a kind of [[graph of a function|graph relation]] of a morphism. Compare the definition of a [[virtual equipment]], where the reflexive edges are analogous to the units and the fibration structure is analogous to the restrictions.

## References

The calculus now called *System F* was developed independently in the following works:

* {#Girard72} [[Jean-Yves Girard]], _Interprétation functionelle et élimination des coupures dans l'arithmétique d'ordre supérieure_, Ph.D. thesis, Université Paris VII (1972) [[pdf](http://www.cs.cmu.edu/afs/cs.cmu.edu/user/kw/www/scans/girard72thesis.pdf), [[Girard-InterpretationFunctionelle.pdf:file]]]

* [[John C. Reynolds]], _Towards a theory of type structure_, Programming Symposium, 1974 ([doi](https://doi.org/10.1007/3-540-06859-7_148))

The hyperdoctrine-like notion of model:

* [[R.A.G. Seely]], _Categorical Semantics for Higher Order Polymorphic Lambda Calculus_, Journal of Symbolic Logic, Vol. 52, Issue 4, December 1987 ([doi](https://doi.org/10.2307/2273831))

The reflexive graph/fibrational view of parametricity:

* Brian Dunphy and [[Uday Reddy]], _Parametric Limits_, Logic in Computer Science 1994 ([doi](https://doi.org/10.1109/LICS.2004.1319618))


system L

#Contents#
* table of contents
{:toc}

## Idea

The term "system L" may refer to (at least) two different concepts in [[formal logic]]:

* there is a [[syntax]] for [[sequent calculus]] introduced in ([Curien-Herbelin 00](#CurienHerbelin00)) under the name _$\lambda \bar \mu \mu$-calculus_, whose name then changed in time to "system L" in reference to the "L" in "LK", "LJ", from [[Gerhard Gentzen]]'s original terminology of sequent calculi;

* there is a [[natural deduction]] system of this name, see _[System L](http://en.wikipedia.org/wiki/System_L)_.

## References

### The sequent calculus

* {#CurienHerbelin00} Pierre-Louis Curien, [[Hugo Herbelin]], _The duality of computation_,  Proceeding
ICFP '00 Proceedings of the fifth ACM SIGPLAN international conference on Functional programming Pages 233 - 243  ([web](https://dl.acm.org/citation.cfm?id=351262))

* {#Spiwack14} [[Arnaud Spiwack]], _A dissection of L_ (2014) [[pdf](http://assert-false.science/arnaud/papers/A%20dissection%20of%20L.pdf)]

### The natural deduction system

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

[[!redirects System L]]


system of imprimitivity

#Contents#
* table of contents
{:toc}

## Definition.

Given a [[locally compact topological group]] $G$, a system of imprimitivity on $G$ consists of a

* unitary representation $\rho: G\to U(H)$ on a Hilbert space $H$

* a locally compact Hausdorff space $X$ with continuous left $G$-action

* a regular [[projection-valued measure]] $P:B(X)\to End H$ where $B(X)$ is the Borel $\sigma$-algebra of $X$

such that

$$
\rho(g)P(E)\rho(g)^{-1} = P(gE)
$$

for all $g\in G$ and $E\in B(X)$.

#### An approach via $\ast$-representations

In the above definition, one can replace the projection-valued measure $P$ by a $\ast$-representation $M : C_0(X)\to H$ of the $C^\ast$-algebra $C_0(X)$ by defining $M(f) = \int f dP$, then
$$
\rho(g)M(f)\rho(g^{-1}) = M(L_g f), \,\,\,\,L_g(f)(x) := f(g^{-1}x).
$$
On the other hand, any $M$ satisfying this property defines a regular projection-valued measure as above.

#### Extensions

Remark: A possible extension is to replace $X$ by a measurable space with a measurable left action of $G$.

## Applications

This concept is important in Mackey machinery and in the applications to the study of coherent states and Berezin quantization.

## References

* [[Gerald B. Folland]], Sec. 6.4 in: *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

[[!redirects systems of imprimitivity]]


Syu Kato

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

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

## Selected writings

On [[quantum cohomology]] [[topological K-theory|K-theory]] of [[flag varieties]] identified as a [[Pontrjagin product]]-structure on a corresponding Grassmannian:

* [[Syu Kato]], *Loop structure on equivariant K-theory of semi-infinite flag manifolds* $[$[arXiv:1805.01718](https://arxiv.org/abs/1805.01718)$]$

* {#Kato19} [[Syu Kato]], *On quantum $K$-groups of partial flag manifolds* $[$[arXiv:1906.09343](https://arxiv.org/abs/1906.09343)$]$

* [[Syu Kato]], *Darboux coordinates on the BFM spaces* $[$[arXiv:2008.01310](https://arxiv.org/abs/2008.01310)$]$

* {#Kato21} [[Syu Kato]], *Quantum $K$-groups on flag manifolds*, talk at *IMPANGA 20* (2021) $[$[pdf](https://www.impan.pl/konferencje/bcc/2020/20-impanga/impanga2021.pdf)$]$

category: people


syzygy

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#Contents#
* table of contents
{:toc}

## Idea

For $R$ a [[ring]] and $N$ an $R$-[[module]] which is finitely [[generators and relations|generated]] over $R$ on $n$ generators, the _syzygies_ are the _[[generators and relations|relations]]_ between these generators.

Higher order syzygies are relations between these relations, and so forth.

Similar definitions apply in non-additive contexts. In the theory of [[group presentation]]s, 2-dimensional [[homotopical syzygies]] are specific cellular representatives for [[identities among relations]] of a presentation of a group.

## Definition

Let $R$ be a [[ring]], $N$ an $R$-[[module]] generated on $n$ generators. Write

$$
  R^n \to N
$$

for the canonical projection from the [[free module]] over $R$ on $n$ generators to $N$, which takes these generators to their image in $N$.

The **module of syzygies** is the [[kernel]] of this morphism. This being a [[submodule]] of a [[free module]] it is itself free under suitable conditions on $R$, and hence the resulting [[exact sequence]] looks like

$$
  R^{n_1} \to R^{n} \to N
$$

relations/syzygies $\to$ generators $\to$ elements

Continuing in this way yields, under suitable assumptions on $R$, a [[projective resolution]] (actually a [[free resolution]]) of $N$ by syzygies and higher order syzygies.

$$
  \cdots \to R^{n_3} \to R^{n_2} \to R^{n_1} \to R^{n} \to N
  \,.
$$

## Properties

### Hilbert's syzygy theorem

For $k$ a [[field]] and $R = k[x_1, \cdots, x_n]$ the [[polynomial ring]] over $k$ in $n$ [[variables]], every finitely-generated $R$-module has a [[free resolution]] of length at most $n$.

## Related concepts

* [[projective resolution]]

* [[Koszul complex]]

Non-linear variants of the idea of syzygy are

* [[homotopical syzygy]]

* [[computad]]

and

* [[homological syzygy]].

The idea of a homotopical  $n$-syzygy is discussed at

* [[higher dimensional szyzgy]]

## References

An exposition is in

* Roger Wiegand, _What is... a syzygy?_ ([pdf](http://www.ams.org/notices/200604/what-is.pdf))

Lecture note discussion in a general context of [[projective resolutions]] in [[homological algebra]] includes

* E. L. Lady, _A course in homological algebra -- Syzygies, Projective dimension, regular sequences and depth_ (1997) ([pdf](http://www.math.hawaii.edu/~lee/homolog/proj-dim.pdf))

and section 4.5 of

* [[Pierre Schapira]], _Categories and homological algebra_, lecture notes (2011) ([pdf](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf))
 {#Schapira}

Discussion in the context of the [[Koszul complex]] is in

* Yasuhiro Shimoda, _On the syzygy part of Koszul homology on certain ideals_, J. Math. Kyoto Univ. (1984) ([EUCLID](http://projecteuclid.org/euclid.kjm/1250521386))

A useful source that discusses the use of syzygies in algebraic geometry,

* [[D.Eisenbud]], _The Geometry of Syzygies_, Graduate Texts in Mathematics, vol. 229, Springer, 2005.

[[!redirects syzygies]]

[[!redirects Hilbert's syzygy theorem]]


Sze-Tsen Hu

* [Wikipedia page](https://en.wikipedia.org/wiki/Sze-Tsen_Hu)

* Online Archive of California, *[Sze-Tsen Hu, Mathematics: Los Angeles ](http://texts.cdlib.org/view?docId=hb1r29n709&doc.view=frames&chunk.id=div00025&toc.depth=1&toc.id=)*

## Selected writings

On [[homotopy theory]]:

* {#SzeTsen59} [[Sze-Tsen Hu]], *Homotopy Theory*, Academic Press 1959 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hu2.pdf))

category: people


Szpiro's conjecture

A statement about [[elliptic curves]].

## Relation to other statements

In its general form, equivalent to the [[abc conjecture]].

If [[Mochizuki's corollary 3.12]] is true, then Szpiro's conjecture is also true.

## References

* Wikipedia, _[Szpiro's conjecture](http://en.wikipedia.org/wiki/Szpiro%27s_conjecture)_

The relation to the [[abc conjecture]] is discussed in

* Matt Baker (notes taken by William Stein), _Elliptic curves, the ABC conjecture, and points of small canonical height_ ([pdf](http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01.pdf))

The relation to [[Mochizuki's corollary 3.12]] is discussed in

* [[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)]

[[!redirects Szpiro conjecture]]


Séminaire Bourbaki

Séminaire Nicolas Bourbaki, or simply Séminaire Bourbaki,
is a seminar organized by the [[Bourbaki]] group in Paris from 1948.

Website: <https://www.bourbaki.fr/seminaires/>

## Related entries

* [[Nicolas Bourbaki]]

* [[Elements of Mathematics]]

[[!redirects Séminaire Nicolas Bourbaki]]
[[!redirects Bourbaki Seminar]]

category: reference


Sören Illman

* [webpage](https://www.ias.edu/scholars/s%C3%B6ren-illman)

* [Wikipedia entry](https://fi.wikipedia.org/wiki/S%C3%B6ren_Illman)

* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=1346)

## Selected writings

Introducing [[G-CW complexes]], [[Bredon-Illman cohomology]] and foundations of [[equivariant differential topology]]:

* {#Illman72a} [[Sören Illman]], *Equivariant algebraic topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* {#Illman72b} [[Sören Illman]], *Equivariant singular homology and cohomology for actions of compact lie groups* ([doi:10.1007/BFb0070055](https://doi.org/10.1007/BFb0070055)) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), *Proceedings of the Second Conference on Compact Transformation Groups* Lecture Notes in Mathematics, vol 298. Springer 1972 ([doi:10.1007/BFb0070029](https://link.springer.com/book/10.1007/BFb0070029))

* {#Illman73} [[Sören Illman]], *Equivariant algebraic topology*, Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 ([doi:10.5802/aif.458](https://doi.org/10.5802/aif.458))

* [[Sören Illman]], _Equivariant singular homology and cohomology_, Bull. Amer. Math. Soc. Volume 79, Number 1 (1973), 188-192 ([euclid:bams/1183534324](https://projecteuclid.org/euclid.bams/1183534324))

On [[equivariant differential topology]], proving the [[equivariant triangulation theorem]]:

* {#Illman78} [[Sören Illman]], _Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group_, Math. Ann. (1978) 233: 199 ([doi:10.1007/BF01405351](https://doi.org/10.1007/BF01405351))

* {#Illman83} [[Sören Illman]], _The Equivariant Triangulation Theorem for Actions of Compact Lie Groups_, Mathematische Annalen (1983) Volume: 262, page 487-502 ([doi:10.1007/BF01456063](https://doi.org/10.1007/BF01456063), [dml:163720](https://eudml.org/doc/163720))

* {#Illman00} [[Sören Illman]], _Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion_,  J. Reine Angew. Math. **524** (2000) 129–183.  [[doi:10.1515/crll.2000.054](https://doi.org/10.1515/crll.2000.054)]

## Related $n$Lab entries

* [[G-CW complex]]

* [[Bredon-Illman cohomology]]

* [[equivariant cellular approximation theorem]]

* [[equivariant Whitehead theorem]]

* [[equivariant triangulation theorem]]

* [[G-representation spheres are G-CW-complexes]]

category: people

[[!redirects Soren Illman]]


Søren Galatius

* [website](https://sorengalatius.com/)

## related entries

* [[Galatius-Madsen-Tillmann-Weiss theorem]]

* [[cobordism hypothesis]]

category: people

[[!redirects Soren Galatius]]


T-complex

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#Contents#
* table of contents
{:toc}

## Idea

A _$T$-complex_ is a [[geometric definition of higher categories|higher-dimensional combinatorial structure]] with a class of designated [[thin element]]s.  The concept can be made sense of for various [[geometric shapes for higher structures|shapes]].

In general, the requirements are:

1. Every degenerate element is thin.
1. Every hollow shape has a unique thin filler.
1. If a thin element has every face but one also thin, then the last face is thin as well.

The theory describes a version of [[∞-groupoid|higher groupoids]], rather than [[higher category theory|higher categories]] more generally. See however [[algebraic quasi-categories]] for more.

## Types of $T$-complexes

### Simplicial $T$-complexes

See

* [[simplicial T-complex]];

### Cubical $T$-complexes

See

* [[cubical T-complex]].

## Polyhedral sets and polyhedral $T$-complexes##

There is also a notion of polyhedral $T$-complex, defined in ([Jones, 1983](#Jones)). There are no degeneracies in this theory, but it does allow for the shapes quite, but not completely,  general forms of regular cell decompositions of cells. This gives a solution to the problem of defining general compositions. One has  to define:

1. What are the pieces that might be composable?
1. When are they composable?
1. What is their composite?
1. What are the axioms on the composition?

On the face of it, the last problem seems the hardest. It turns out that the last two $T$-complex axioms are sufficient!  Thus the geometry determines the algebra.

## References##

See the references at [[simplicial T-complex]].

Polyhedral $T$-complexes are discussed in

* David W. Jones, _A general theory of polyhedral sets and the corresponding
  $T$-complexes_. Dissertationes Math. (Rozprawy Mat.) 266 (1988) 110. [Scanned Thesis](http://pages.bangor.ac.uk/~mas010/pdffiles/jones1983polyTComplexes.pdf) or from [RBPhDsSupervised](http://www.bangor.ac.uk/~mas010/doctorates.html)


T-D-space

A [[topological space]] $X$ satisfies the __$T_D$ separation axiom__ if, given any [[element]] $a$ of $X$, the [[singleton subset]] $\{a\}$ is an [[open subspace]] in the [[induced topology]] on its own [[closed subspace|closure]].  In other words, for every point $a$, there is a [[neighbourhood]] $U$ of $a$ such that the [[intersection]] of $U$ and $Cl\{a\}$ is $\{a\}$.

[[!redirects T-D-space]]
[[!redirects T-D-spaces]]
[[!redirects TD-space]]
[[!redirects TD-spaces]]
[[!redirects T-D space]]
[[!redirects T-D spaces]]
[[!redirects TD space]]
[[!redirects TD spaces]]


T-duality

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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

A $d$-dimensional [[sigma-model]] is a [[quantum field theory]] that is induced from certain [[differential geometry|differential geometric]] and [[differential cohomology|differential cohomological]] data, to be thought of as encoding the background geometry on which quantum objects of dimension $d$ propagate.

The operation of **T-duality** is a map that interchanges pairs of such geometric data for 2-dimensional [[conformal field theory]] [[sigma-model]]s, such that the induced QFTs are equivalent.

More specifically the space of [[differential geometry|differential geometric]] data consisting of

* a smooth [[manifold]] $X$

* equipped with the structure of a $k$-torus [[bundle]] $X \simeq Y \times T^k$; -- the total space of this bundle modelling [[spacetime]];

* and equipped with a [[Riemannian metric]] $g$ -- modelling the field of [[gravity]];

* and a $U(1)$-[[gerbe]] with connection $G$; -- modelling the [[Kalb-Ramond field]];

* possibly a cocycle in [[differential K-theory]] modelling the [[RR-field]];

admits a certain operation that, roughly, inverts the Riemannian circumference of the torus fibers and mixes the metric with the gerbe data, such that the induced 2-dimensional [[sigma-model]] QFTs for these backgrounds are equivalent. This is the operation called **T-duality**.

This was noticed originally in the study of [[conformal field theory|conformal field theories]] in the context of [[string theory]]: the conformal field theory [[sigma-model]]s with target space $X$ turn out to be equivalent as [[quantum field theory|quantum field theories]] for T-dual backgrounds $(X,g,G)$ and $(X',g',G')$ (at least to the approximate degree to which these are realized as full CFTs in the first place).

Further generalisations let $X$ be a nontrivial torus bundle, but the T-dual is then generically a bundle of [[noncommutative torus|non-commutative tori]]. (cite Mathai, Rosenberg and Hannabus)

## Path integral heuristics deriving T-duality {#PathIntegral}

We indicate how one can see T-duality from formal manipulations
of the [[path integral]] for the [[string theory|string]] [[sigma-model]]. We look at the simplest situation, where the torus bundle in question is a trivial circle bundle over a [[Cartesian space]] carrying the metric induced from the standard flat metric on $\mathbb{R}^n$ and where there are no other nontrivial background fields. In fact, for the purpose of the following computation we can entirely ignore the base of this bundle and consider target space to be nothing but a circle. Since the [[sigma-model]] for this is on the worldsheet just the theory of a single free field with values in $S^1$, this is often also called the "free boson on the circle".

This means that the only geometric datum determining the background geometry is the circumference $2 \pi R$ of the fiber of the circle bundle. The statement of T-duality in this situation is that the 2-dimensional [[sigma-model]] on this background yields the same 2-dimensional [[CFT]] as that for this kind of background with circumference of the circle being $2 \pi 1/R$.

### A first rough look

A quick way to get an indication for this is to consider the center-of-mass energy of the string in such a circle-bundle background. In the simplified setup we mentioned before, a string on a circle of radius $R$ has
quantized momentum $p = \frac{\ell \in \mathbb{Z}}{R}$. In a state in which the string winds around the circle $m$ times and has $\ell$ quanta of kinetic momentum for propagation around the circle,  its energy is

$$
  E_0 = \sqrt{p^2 + M^2} = \sqrt{(\ell/R)^2 + (R m)^2 }
  \,,
$$

This energy is clearly invariant under exchanging

$$
  (R, (\ell, m)) \mapsto (\frac{1}{R}, (m,\ell))
  \,.
$$

This is of course far from being a proof that the corresponding two [[QFT]]s are equivalent, but it does already capture a good deal of the essence of what T-duality does and why it works.

In slightly more detail, but still at a very rough level, if we denote by

$$
  X : \Sigma \to S^1_R
$$

the $\sigma$-model field on the worldsheet $\Sigma = \mathbb{R}^2$ with values in target space $S^1_R$ then T-duality with respect to this circle may be thought of as exchanging worldsheet momentum $\partial_t X$ with worldsheet winding $\partial_\sigma X$.

This then also means that for the open string it exchanges von Neumann boundary conditions $\partial_\sigma X|_{\sigma = 0} = 0$ with Dirichlet boundary conditions $\partial_t X|_{\sigma = 0} = 0$. The first boundary condition is that describing an open string whose endpoints are free to propagate in worldsheet time, whereas the second boundary condition describes a situation where the endpoint of the string is fixed at some point in target space. In terms of the language of geometric target space data, a sigma-model with such a constraint is said to describe a [[D-brane]] in target space: the locus where the endpoints of the string are fixed. This is a first indication that the T-duality operation on geometric background also involves the [[RR-field]].

### The path integral

We follow [[Kentaro Hori]]'s [[path integral]] discussion of T-duality. Here the strategy is to consider a path integral over a certain space of auxiliary fields and show or argue that by "algebraically integrating out" some of these in two different ways, the path integral is equivalent to that over two different [[action functional]]s, which describe two T-dual geometric backgrounds.

Let the boundary components of the worldsheet $\Sigma$ be labeled by
$\partial \Sigma_{(1)}$.

We consider the following fields on the worldsheet:

* $\tilde X : \Sigma \to \mathbb{R}/(2\pi/R)\mathbb{Z} = S^1_{1/R}$ -- a circle-valued function; this is the standard $\sigma$-model field describing propagation of the string on the circle;

* $X_{i} : \partial \Sigma_{(i)} \to S^1_R$ -- the boundary values of this field;

* $b \in \Omega^1(\Sigma, \mathbb{R})$ -- a 1-form; this is the auxiliary field that will not contribute to the dynamics but serves to make the T-duality manifest.

Consider then the following [[action functional]] on this collection of fields given by the assignment

$$
  S'_E(\tilde X,b) = \frac{1}{2 \pi}
  \int_\Sigma
  \left(
     \frac{1}{2} b \wedge \star b  - b \wedge d \tilde X
  \right)
  -
  \frac{i}{2 \pi} \sum_{i = 1}^s
  \int_{\partial \Sigma_{(i)}}
  (\tilde X -a_i) d X_i
  \,,
$$

where the $(a_i)$ are a collection of real numbers.

We now want to formally perform the [[path integral]] over the fields in two different orders, which should give the same quantum field theories but in terms of different effective action functionals.

If we do first the path integral over the field $b$ then by the general formal rule of "algebraically integrating out a non-dynamical field" which says that we can evaluate this path integral that formally looks like a Gaussian integral by the usual formulas for Gaussian integrals, we obtain the action functional

$$
  \tilde S_E =
  \frac{1}{4 \pi}
  \int_\Sigma d \tilde X \wedge \star d \tilde X
  +
  \frac{i}{2\pi}
  \int_{\partial \Sigma_{(i)}}
  (\tilde X - a_i)
  d X_i
$$

then doing the integral over the boundary values $X_i$ yields

$$
  \tilde X|_{\partial \Sigma_i} = a_i
$$

This is the action functional for a $\sigma$-model on $S^1_{1/R}$ with a D-brane at $\tilde X = a_i$.

Now we evaluate the original path integral in a different way, this way first integrating over components of $\tilde X$. To do so, we imagine that we may re-encode the field $\tilde X$ in terms of its de Rham differential

$$
  d \tilde X = d f + \frac{2\pi}{R}\sum_A \eta_A \omega_A
$$

where $\eta_A$ are integers and

$$
  \{\omega_A\} \subset
   H^1(\Sigma, \mathbb{Z})
   \simeq
   \mathbb{Z}^{\oplus 2g + s - 1}
 \,.
$$

Then formally performing the path integral over $f$ yields $d b = 0$ and $b|_{\partial \Sigma} = d X_i$. It follows that $b = d X$ for some other field $X : \Sigma \to S^1_R$.

So we get the action

$$
  S_E = \frac{1}{4\pi} \int_\Sigma d X \wedge \star d X
  +
  \frac{i}{2 \pi}
   \sum_{i = 1}^s
   \int_{\partial \Sigma_i}
   a_i d X
$$

in terms of the field $X$.  This is the $\sigma$-model for string propagation on  $S^1_R$. with D-brane wrapped on  $S^1_R$ that carries on its worldvolume a [[gauge field]] given by a constant connection 1-form $a_i$.

## Topological T-duality

It turns out to be possible and useful to discuss just the _topological_ aspects of T-duality, meaning all the aspects that depend on the $X$ as a [[topological space]], on the topological class of the [[gerbe]] and of its 3-form curvature, but not on the [[Riemannian metric]] and not on the precise connection on the gerbe (there may be several inequivalent ones for a given curvature)!

This sub-phenomenon is discussed in more detail at [[topological T-duality]].

## Geometric T-duality

### In terms of generalized differential cohomology
 {#InDiffCohomology}

[[gauge field|Gauge field]]s are [[cocycle]]s in [[differential cohomology]]. The [[Kalb-Ramond field]] is given by degree-3 [[ordinary differential cohomology]], the differential refinement on degree-3 [[integral cohomology]]. The [[RR-field]] is given by [[differential K-theory]].

Induced by the morphisms $\mathbf{c}(n)$ in the [[fiber sequence]]s

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

is induced a notion of [[twisted cohomology]] which makes the Kalb-Ramond field act as a twist for [[twisted K-theory]].

In these terms, the setup of T-duality is a [[correspondence]] of [[Kalb-Ramond field]]s over [[spacetime]] [[torus]]-bundles $P \to X$ and $\hat P \to X$ that induces an integral transform

$$
  K_{diff}^{\bullet + \tau}(P)
  \to
  K_{diff}^{\bullet + \tau -1}(\hat P)
$$

of twisted differential K-theory classes.

This is an [[isomorphism]] -- the action of the T-duality isomorphism on the Kalb-Ramond field and the RR-field.

See ([KahleValentino](#KahleValentino)).

### In generalized complex geometry {#TdualityIngcg}

Another approach to the study of T-duality takes a somewhat complementary point of view and ignores the [[Eilenberg-MacLane spectrum|integral cohomology]] class in $H^3(X,\mathbb{Z})$ of the [[gerbe]] but does consider the [[Riemannian metric]].

In this context T-duality is described as an isomorphism of [[standard Courant algebroid]]s. This picture emerged in the study of [[generalized complex geometry]].

## Examples of T-dual pairs

### In Mirror symmetry

One special case of T-duality is [[mirror symmetry]].

* [[Andrew Strominger]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Mirror Symmetry is T-Duality_, Nucl. Phys. B 479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) [hep-th/9606040](http://arxiv.org/abs/hep-th/9606040)

### In Langlands duality

In some cases the passage to the [[Langlands dual group]] in the [[geometric Langlands correspondence]] can be understood as T-duality. ([Daenzer-vanErp](#DaenzerErp))

## Related concepts

[[duality in physics]], [[duality in string theory]]

* **T-duality**

  * [[topological T-duality]]

    * [[spherical T-duality]]

  * [[Poisson-Lie T-duality]]

  * [[differential T-duality]]

  * [[T-duality 2-group]]

* [[T-fold]], [[double field theory]]

* [[duality in string theory]]

  * [[U-duality]], [[S-duality]]

  * [[AdS/CFT correspondence]]

The geometry of the fiber product of two torus fiber bundles with a [[circle 2-bundle]] on it is sometimes referred to as _[[Bn-geometry]]_.

## References

The observation of T-duality is attributed to

* [[Thomas Buscher]], _A symmetry of the string background field equations,
Phys. Lett. B 194 (1987) 59 (doi:10.1016/0370-2693(87)90769-6)

* [[Thomas Buscher]], _Path integral derivation of quantum duality in nonlinear sigma models_, Phys. Lett. B 201 (1988) 466 (doi:10.1016/0370-2693(88)90602-8)

Precursors include (according to [Schwarz 96, first paragraph](#Schwarz96)):

* Keiji Kikkawa, Masami Yamasaki, _Casimir effects in superstring theories_, Physics Letters B Volume 149, Issues 4–5, 20 December 1984, Pages 357-360 (doi:10.1016/0370-2693(84)90423-4)

Discussion for the [[superstring]] is in

* M. Abou-Zeid, [[Chris Hull]], [[Ulf Lindström]], [[Martin Roček]], _T-Duality in $(2,1)$ Superspace_ ([arXiv:1901.00662](https://arxiv.org/abs/1901.00662))

Review:

* Amit Giveona, Massimo Porrati, Eliezer Rabinovicia, _Target space duality in string theory_, Physics Reports Volume 244, Issues 2–3, August 1994, Pages 77-202 (doi:10.1016/0370-1573(94)90070-1)

* E. Alvarez, [[Luis Alvarez-Gaume]], Y. Lozano, _An Introduction to T-Duality in String Theory_, Nucl.Phys.Proc.Suppl.41:1-20,1995 ([arXiv:hep-th/9410237](https://arxiv.org/abs/hep-th/9410237))

* [[Jonathan Rosenberg]], *Topology, $C^*$-algebras, and string duality*, Regional Conference Series in Mathematics **111**, Amer. Math. Soc. (2009)  [[doi:10.1090/cbms/111](https://doi.org/10.1090/cbms/111), [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:05632075&type=pdf&format=complete)]

  > (focus on [[topological T-duality]])

* {#West12} [[Peter West]], section 17.1 of _[[Introduction to Strings and Branes]]_, Cambridge University Press 2012

* [[Mathai Varghese]], _T-duality: A basic introduction_, 10th Geometry and Physics Conference _Quantum Geometry_, Anogia 2012 ([[Mathai12.pdf:file]])

* Jnanadeva Maharana, _The Worldsheet Perspective of T-duality Symmetry in String Theory_ ([arXiv:1302.1719](http://arxiv.org/abs/1302.1719))

* Mark Bugden, _A Tour of T-duality: Geometric and Topological Aspects of T-dualities_, ([arXiv:1904.03583](https://arxiv.org/abs/1904.03583))

Discussion at strong coupling ("[[F-theory]]") includes

* {#Schwarz96} [[John Schwarz]], _M Theory Extensions of T Duality_ ([arXiv:hep-th/9601077](https://arxiv.org/abs/hep-th/9601077))

* {#Russo97} [[Jorge Russo]], _T-duality in M-theory and supermembranes_, Phys.Lett. B400 (1997) 37-42 ([arXiv:hep-th/9701188](https://arxiv.org/abs/hep-th/9701188))

Dscussion in [[higher differential geometry]]:

* {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortsch. d. Phys. 2020 ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010))

  (relating [[Kaluza-Klein compactification]] on [[principal ∞-bundles]] to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]], ...)

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

Discussion of geometric T-duality in terms of some form of [[differential cohomology]]:

as an operation on [[twisted K-theory|twisted]] [[differential K-theory]]:

* {#KahleValentino} [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_

using [[adjusted Weil algebra|adjusted]] [[principal 2-connections]]:

* [[Hyungrok Kim]], [[Christian Saemann]], *Non-Geometric T-Duality as Higher Groupoid Bundles with Connections* *lbrack;[arXiv:2204.01783](https://arxiv.org/abs/2204.01783)]

* [[Hyungrok Kim]], [[Christian Saemann]], *T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections* [[arXiv:2303.16162](https://arxiv.org/abs/2303.16162)]

* [[Konrad Waldorf]], *Geometric T-duality: Buscher rules in general topology* [[arXiv:2207.11799](https://arxiv.org/abs/2207.11799)]

More physically oriented discussion of this is in

* [[Katrin Becker]], Aaron Bergman, _Geometric Aspects of D-branes and T-duality_ ([arXiv:0908.2249](http://arxiv.org/abs/0908.2249))

Geometric T-duality is identified as an [[isomorphism]] of [[standard Courant algebroid]]s ([[generalized complex geometry]]) in section 4 of

* [[Gil Cavalcanti]], [[Marco Gualtieri]], _Generalized complex geometry and T-duality_ ([arXiv:1106.1747](http://arxiv.org/abs/1106.1747))

Discussion of the [[sigma-model]] description of T-duality in this context includes

* [[Ulf Lindstöm]], [[Martin Rocek]], Itai Ryb, [[Rikard von Unge]], [[Maxim Zabzine]], _T-duality and Generalized Kähler Geometry_, JHEP 0802:056,2008 ([arXiv:0707.1696](http://arxiv.org/abs/0707.1696))

* Jonas Persson, _T-duality and Generalized Complex Geometry_ ([arXiv:hep-th/0612034](http://arxiv.org/abs/hep-th/0612034))

Further references are

* Willie Carl Merrell, _Application of superspace techniques to effective actions, complex geometry and T-duality in String theory_ ([pdf](http://www.lib.umd.edu/drum/bitstream/1903/6865/1/umi-umd-4355.pdf))

* Peggy Kao, _T-duality and Poisson-Lie T-duality in generalized geometry_ ([pdf](http://tpsrv.anu.edu.au/Members/bouwknegt/Kao.pdf))

Discussion of the infinitesimal T-duality geometry, replacing [[gerbes]] on [[torus]]-[[fiber bundles]] with the corresponding [[dg-manifolds]] is in

* [[Ernesto Lupercio]], Camilo Rengifo, [[Bernardo Uribe]], _T-duality and exceptional generalized geometry through symmetries of dg-manifolds_ ([arXiv:1208.6048](http://arxiv.org/abs/1208.6048))

For references on [[topological T-duality]] see there.

A relation to [[Langlands dual groups]]:

* {#DaenzerErp} [[Calder Daenzer]], [[Erik Van Erp]], *T-Duality for Langlands Dual Groups*, Advances in Theoretical and Mathematical Physics **18** 6 (2014) 1267–1285 [[arXiv:1211.0763](http://arxiv.org/abs/1211.0763), [doi:10.4310/ATMP.2014.v18.n6.a2](https://dx.doi.org/10.4310/ATMP.2014.v18.n6.a2)]

For [[RR-fields]]:

* {#Hori99} [[Kentaro Hori]], _D-branes, T-duality and Index theory_, Adv. Theor. Math. Phys. 3 (1999) 281 ([arXiv:hep-th/9902102](https://arxiv.org/abs/hep-th/9902102))

Discussion of T-duality that takes into account the [[super p-brane]] charges (i.e. the fermionic components of the [[RR-fields]]) on [[super spacetime]], hence also of [[Green-Schwarz action functionals]], includes the following:

* [[Mirjam Cvetic]], H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], _T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map_, Nucl.Phys.B573:149-176,2000 ([arXiv:hep-th/9907202](https://arxiv.org/abs/hep-th/9907202))

* Bogdan Kulik, Radu Roiban, _T-duality of the Green-Schwarz superstring_, JHEP 0209 (2002) 007 ([arXiv:hep-th/0012010](https://arxiv.org/abs/hep-th/0012010))

* {#BandosJulia03} [[Igor Bandos]], [[Bernard Julia]], _Superfield T-duality rules_, JHEP 0308 (2003) 032 ([arXiv:hep-th/0303075](https://arxiv.org/abs/hep-th/0303075))

  reviewed in _Superfield T-duality rules in ten dimensions with one isometry_ ([arXiv:hep-th/0312299](https://arxiv.org/abs/hep-th/0312299))

Lift of T-duality from [[string theory]] to a [[SL(2,Z)]]-[[U-duality]] acting on the [[M2-brane]]-[[Green-Schwarz action functional|sigma-model]]:

* [[Maria P. Garcia del Moral]], I. Martin, [[Alvaro Restuccia]], *Nonperturbative $SL(2,\mathbb{Z})$ $(p,q)$-strings manifestly realized on the quantum M2* [[arXiv:0802.0573](https://arxiv.org/abs/0802.0573)]

* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], *Aspects of the T-duality construction for the Supermembrane theory*, J. Phys.: Conf. Ser. **720** (2016) 012025 [[arXiv:1504.06907](https://arxiv.org/abs/1504.06907), [doi:10.1088/1742-6596/720/1/012025](https://doi.org/10.1088/1742-6596/720/1/012025)]

[[!redirects Buscher rule]]
[[!redirects Buscher rules]]


T-duality 2-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _T-duality 2-group_ is a [[smooth 2-group]] (or rather a class of such) which controls [[T-duality]] and [[T-folds]]. It is the [[string 2-group]] for the [[cup product]] [[universal characteristic class]] on [[fiber products]] of [[torus]]-[[fiber bundles]] with their dual [[torus]]-[[principal bundles]].

## Definition

For $T$ a [[torus]] and $\tilde T$ its dual torus, there is the [[cup product]] [[universal characteristic class]]

$$
  \langle -\cup -\rangle \;\colon\;B T \times B \tilde T \longrightarrow K(\mathbb{Z}, 4)
  \,.
$$

This has a smooth refinement to morphism of [[smooth groupoids]]/[[moduli stacks]]

$$
  \langle -\cup -\rangle \;\colon\;\mathbf{B} T \times \mathbf{B} \tilde T \longrightarrow \mathbf{B}^3 U(1)
  \,.
$$

In fact it has furthermore a differential refinement to a [[universal Chern-Simons circle 3-bundle with connection]]

$$
  \langle -\cup -\rangle_{conn} \;\colon\;(\mathbf{B} T \times \mathbf{B} \tilde T)_{conn} \longrightarrow \mathbf{B}^3 U(1)_{conn}
$$

of which the above is obtained by forgetting the connections ([FSS 12, section 3.2.1](#FSS12))

As such this is the [[local Lagrangian]] of abelian [[Chern-Simons theory]] with two abelian gauge field species (the [[diagonal]] is 3d abelian CS theory itself).

This universal class is suitably equivariant under the action of the integral T-duality group $O(n,n,\mathbb{Z})$, so that one may consider ([Nikolaus 14](#Nikolaus14))

$$
  \langle -\cup -\rangle \;\colon\;\mathbf{B} T \times \mathbf{B} \tilde T // O(n,n,\mathbb{Z}) \longrightarrow \mathbf{B}^3 U(1)
  \,.
$$

As for the [[string 2-group]], this defines an [[infinity-group extension]] (the looping of the [[homotopy fiber]] of this map) and this one may call the _T-duality 2-group_ as it controls T-duality pairs by the discussion at _[[T-Duality and Differential K-Theory]]_. Indeed, according to ([Nikolaus 14](#Nikolaus14)) the [[principal 2-bundles]] for this 2-group are the correct formalization of the concept of _[[T-folds]]_.

## Related concepts

* [[T-duality]], [[topological T-duality]]

* [[string 2-group]]

## References

* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_, Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 ([arXiv:1207.5449](http://arxiv.org/abs/1207.5449))

* {#Nikolaus14} [[Thomas Nikolaus]], _[[T-Duality in K-theory and Elliptic Cohomology]]_, talk at _String Geometry Network Meeting_, Feb 2014, ESI Vienna ([website](http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html))

* {#NikolausWaldorf18} [[Thomas Nikolaus]], [[Konrad Waldorf]], *Higher geometry for non-geometric T-duals*, Commun. Math. Phys. **374** (2020) 317-366 [[arXiv:1804.00677](https://arxiv.org/abs/1804.00677), [doi:10.1007/s00220-019-03496-3](https://doi.org/10.1007/s00220-019-03496-3)]

See also:

* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))

[[!redirects T-duality Lie 2-group]]
[[!redirects T-duality Lie 2-algebra]]


T-Duality and Differential K-Theory

> under construction

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

This entry is about the article

* {#KahleValentino} [[Alexander Kahle]], [[Alessandro Valentino]], _T-duality and Differential K-Theory_, Communications in Contemporary Mathematics, Volume 16, Issue 02, April 2014 ([arXiv:0912.2516](http://arxiv.org/abs/0912.2516))

on the identification of the correct description in [[differential cohomology]] ([[ordinary differential cohomology]] and [[differential K-theory]]) of [[T-duality]] or [[topological T-duality]]: [[differential T-duality]].

A review is in section 7.4 of

* {#BunkeSchick} [[Ulrich Bunke]], [[Thomas Schick]], _Differential K-theory. A survey_ ([arXiv:1011.6663](http://arxiv.org/abs/1011.6663)).

We try to indicate some of the content. There are two main ingredients:

* a [formalization of the setup](#FormalizationOfTheSetup)

which proposes a refinement of the ingredients of [[topological T-duality]] to [[differential cohomology]]; and then the

* [statement of differential T-duality](#StatementOfDifferentialTduality),

which is the assertion that this setup naturally induces the T-duality operation as an [[isomorphism]] on [[twisted differential K-theory]].

## Formalization of the setup
 {#FormalizationOfTheSetup}

Let $\Lambda \subset \mathbb{R}^n$ be a [[lattice]] (a [[discrete group|discrete]] subgroup of the [[abelian group]] of [[real number]]s to the $n$th [[cartesian product|cartesian power]]).

Write

$$
  \hat \Lambda := Hom_{Grp}(\Lambda, \mathbb{Z})
$$

for the dual lattice and

$$
  (-,-) : \Lambda \times \hat \Lambda \to \mathbb{Z}
$$

for the canonical pairing (the [[evaluation map]]). Notice that $\hat\Lambda$ is canonically identified with the lattice of $(\mathbb{R}^n)^*$ consisting of those linear functionals $\varphi:\mathbb{R}^n\to\mathbb{R}$ such that $\varphi(\Lambda)\subseteq \mathbb{Z}$. With this identification, the canonical pairing $\Lambda \times \hat \Lambda \to \mathbb{Z}$ can be seen as the restriction to $\Lambda \times \hat \Lambda$ of the canonical pairing $\mathbb{R}^n\otimes(\mathbb{R}^n)^*\to \mathbb{R}$.

The [[quotient]] $\mathbb{R}^n / \Lambda$ is a [[torus]].
A $\mathbb{R}^n/\Lambda$-[[principal bundle]] is a [[torus]]-bundle. Write $\mathcal{B}\mathbb{R}^n/\Lambda \in $ [[Top]] for the [[classifying space]] and $\mathbf{B} \mathbb{R}^n / \Lambda$ for its [[moduli space]]: the [[smooth infinity-groupoid|smooth groupoid]] [[delooping]] the [[Lie group]] $\mathbb{R}/\Lambda$.

Torus bundles on a [[smooth manifold]] $X$ are classified by $H^2(X, \Lambda)$. Following the discussion at [[smooth ∞-groupoid]] we write here $\mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda)$ for the [[groupoid]] of smooth torus bundles and _smooth_ bundle morphisms between. Write $\mathbf{H}_{conn}(X,\mathbf{B} \mathbb{R}^n /\Lambda)$ for the corresponding differential refinement to bundles with [[connection on a bundle|connection]].

For $A$ an abelian group, let $A[n]$ be the chain complex consisting of $A$ concentrated in degree $n$. Then the tensor product of [[chain complex]]es

$$
  \Lambda[2]\otimes\hat{\Lambda}[2]\cong (\Lambda\otimes\hat{\Lambda})[4]
$$
together with the map of complexes
$$
(\Lambda \otimes \hat \Lambda)[4]\stackrel{(-,-)}{\to} \mathbb{Z}[4]
$$
induces the [[cup product]]
$$
  H^k(X, \Lambda) \times
  H^l(X, \hat \Lambda)
  \to
  H^{k+l}(X, \Lambda \otimes \hat \Lambda)
   \stackrel{(-,-)}{\to}
  H^{k+l}(X, \mathbb{Z})
  \,.
$$
This can be refined to a pairing in [[differential cohomology]]
$$
  \bar{H}^k(X, \Lambda) \times
  \bar{H}^l(X, \hat \Lambda)
  \to
  \bar{H}^{k+l}(X, \Lambda \otimes \hat \Lambda)
   \stackrel{(-,-)}{\to}
  \bar{H}^{k+l}(X, \mathbb{Z})
  \,.
$$
by considering the [[Deligne cohomology|Deligne complex]] of sheaves
$$
\Lambda[2]^\infty_D:=(\Lambda\hookrightarrow C^\infty(-,\mathbb{R}^n)\xrightarrow{d_\Lambda} \Omega^1(-,\mathbb{R}^n)),
$$
where the differential $d_\Lambda$ is defined as follows: if $e_1,\dots e_n$ is a $\mathbb{Z}$-basis of $\Lambda$ and $e^1,\dots,e^n$ are the corresponding projections $e^i:\mathbb{R}^n\to \mathbb{R}$, then
$$
d_\Lambda=(d\circ e^i)\otimes e_i
$$
(this is independent of the chosen basis). The definition of $d_\Lambda$ is
clearly chosen so to have an isomorphism of complexes $(\mathbf{Z}[2]^\infty_D)^{\otimes n}\cong \Lambda[2]^\infty_D$ induced by the choice of a $\mathbb{Z}$-basis of $\Lambda$.

Write $(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}$ for the smooth groupoid associated by the [[Dold-Kan correspondence]] to the Deligne complex $\Lambda[2]^\infty_D$. Then we have the morphism of smooth groupoids
to a morphism
$$
(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}
$$
induced by the composition of morphisms of complexes
$$
\Lambda[2]^\infty_D\otimes \hat{\Lambda}[2]^\infty_D
\stackrel{\cup}{\to}
(\Lambda\otimes\hat\Lambda)[4]^\infty_D \stackrel{}{\to}\mathbb{Z}[4]^\infty_D,
$$
where $\cup$ is the [[Beilinson-Deligne cup-product]].

Notice that this is the one which defines [[higher dimensional Chern-Simons theory|abelian Chern-Simons theories]]. The higher [[holonomy]] of the  [[circle n-bundle with connection |circle 3-bundle with connection]] appearing here is the [[action functional]] of torus-[[Chern-Simons theory]].

Differential T-duality pairs form the homotopy fiber of the morphism
$(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}$
It relates differential cohomological structures on $\mathbb{R}^n / \Lambda$-principal bundles with that on certain dual $(\mathbb{R}^n)^* /\hat \Lambda$-principal bundles.

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

A **differential T-duality pair** is

* a [[smooth manifold]] $X$;

* a $\mathbb{R}^n/\Lambda$-[[principal bundle]] $P \to X$ with connection $\theta$ and a $\mathbb{R}^n / \hat \Lambda$-principal bundle $\hat P \to X$ with connection $\hat \theta$;

  such that the underlying topological class of the [[cup product]]
  $(P, \theta) \cup (\hat P, \hat \theta )$ is trivial;

* a choice of trivialization

  $$
    \sigma
     :
    (0,C) \stackrel{\simeq}{\to} (P, \theta) \cup (\hat P , \hat \theta)
    \,.
  $$

=--

This is ([KahleValentino, def. 2.1](#KahleValentino)). It is the evident differential generalization of the description in [[topological T-duality]] that appears for instance around (7.11) of ([BunkeSchick](#BunkeSchick)).

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

We may refine this naturally to a **[[2-groupoid]] of twisted T-duality pairs** $TDualityPairs(X)_{conn}$, the [[homotopy pullback]]

$$
  \array{
    TDualityPairs(X)_{conn} &\stackrel{tw}{\to}& H^4_{diff}(X)
    \\
    \downarrow &\swArrow_{\sigma}& \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda)
    &\stackrel{}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn}
  }
  \,.
$$

This is itself an example of [[twisted cohomology]] (as discussed there). (We use here the notation at _[[schreiber:differential cohomology in a cohesive topos]]_ .)

The differential T-duality pairs of def. \ref{DifferentialTDualityPair} are those elements $(P,\hat P, \sigma) \in TDualityPairs(X)_{conn}$ for which the [[twisted cohomology|twist]] $tw(P,\hat P, \sigma) \in H^4_{diff}(X)$ in [[ordinary differential cohomology]] has an underlying trivial [[circle n-bundle|circle 3-bundle]]. We could restrict the homotopy pullback to these, but it seems natural to include the full collection of twists. (Notice that these "twist" here are not the twists in "twisted K-theory", rather we are observing that already the notion of T-duality pairs itself is an example of cocycles in twisted cohomology in the general sense.)

Notice that the above analogous to the notion of _[[differential string structure]]s in $StringBund(X)_{tw,conn}$ over $X$: as discussed in detail there, this is the [[homotopy pullback]]

$$
  \array{
    StringBund(X)_{tw,conn} &\to& H^4_{diff}(X)
    \\
    \downarrow &\swArrow_{\sigma}& \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}Spin \times \mathbf{B} SU)
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \frac{1}{46}\hat \mathbf{c}}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn}
  }
  \,.
$$

In particular the [[looping and delooping|looping]] $TString$ of the [[homotopy fiber]]

$$
  \array{
    \mathbf{B}TString &\stackrel{}{\to}& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda
      &\stackrel{\langle - \cup -\rangle}{\to}&
    \mathbf{B}^3 U(1))_{conn}
  }
$$

has the right to be called the [[T-duality 2-group]] or similar. The [[principal 2-bundles]] for this are [[T-folds]] (see there).

=--

+-- {: .num_defn }
###### Definition (roughly)

Write $\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1))$ for a [[3-groupoid]] whose objects are cocycles in [[ordinary differential cohomology]] in degree 4, but whose morphisms need not preserve connections and are instead such that the [[automorphism]] [[2-groupoid]] of the 0-object is that of circle 2-bundles with connection $\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1))$.

=--

A 1-groupoid truncation of this idea is the object denoted $\mathcal{H}^p(X)$ in [KahleValentino, A.2](#KahleValentino).

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

In terms of the notion of [[differential function complex]] we should simply set

$$
  \mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1))
  :=
  filt_1 ( H \mathbb{Z}_4)^X
  \,.
$$

(Notice the $filt_1$ instead of $filt_0$. ) By [this proposition](http://ncatlab.org/nlab/show/differential%20function%20complex#homotopy_groups_24) this has the right properties.
=--

+-- {: .num_lemma }
###### Lemma

The choice $\sigma$ of the trivialization of the cup product of the two torus bundles induces canonically elments in degree 3 ordinary differential cohomology (two [[circle n-bundle with connection|circle 2-bundles with connection]]) on $P$ and on $\hat P$, respectively, whose pullbacks to the [[fiber product]] $P \times_X \hat P$ are equivalent there.

=--

This is ([KahleValentino, 2.2, 2.3](#KahleValentino)), where an explicit construction of the classes and their equivalence is given.

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

This is a special case of the general statement about extensions of higher bundles discussed here:

Let $A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ be the [[homotopy fiber]] of the pairing class $\mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)$. This leads to the long [[fiber sequence]] (as discussed there)

$$
  \cdots \to \mathbf{B}^2 U(1) \to A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)
$$

The characteristic map $X \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ of a pair of torus bundles $P , \hat P \to X$ factors through $A$ precisely if these form a T-duality pair. Such a factorization induces a $\mathbf{B} U(1)$-[[principal 2-bundle]] on the [[fiber product]] $P \times_X \hat P$. This follows from the following [[pasting diagram]] of [[homotopy pullback]]s

$$
  \array{
    P \times_X \hat P &\stackrel{\tilde \tau}{\to}& \mathbf{B}^2 U(1) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\to& A & \to & \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda
  }
  \,.
$$

The $\tilde \tau$ here is the class on the fiber product in question.

Notice that in the top left we indeed have $P \times_X \hat P$: the bottom left homotopy pullback of the product coefficients is equivalently given by the following pasting composite of homotopy pullbacks

$$
  \array{
     && && P \times_X \hat P
     \\
     && & \swarrow && \searrow
     \\
     && P &&&& \hat P
     \\
     & \swarrow && \searrow && \swarrow && \searrow
     \\
     * && && X && && *
     \\
     & \searrow && \swarrow && \searrow && \swarrow
     \\
     && \mathbf{B}\mathbb{R}^n / \Lambda
     && && \mathbf{B}\mathbb{R}^n / \hat \Lambda
  }
  \,.
$$

Notice also that this is again directly analogous to the situation for [[string structure]]s: as discussed there, a string structure on $X$ induces a $\mathbf{B}U(1)$-2-bundle on the total space of a $Spin$-principal bundle over $X$.

=--

## Statement of differential T-duality
 {#StatementOfDifferentialTDuality}

(...)

category: reference


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

This page records some information on the results announced in the talk

* {#Nikolaus14} [[Thomas Nikolaus]], _T-Duality in K-theory and Elliptic Cohomology_, talk at _String Geometry Network Meeting_, Feb 2014, ESI Vienna ([website](http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html))

The talk announced three results which are foundational for the topic of [[topological T-duality]].

#Contents#
* table of contents
{:toc}

## 1. T-duality equivalences in universal elliptic cohomology ($tmf$)

A famous result of the 1990s is that [[D-brane charges]], hence the information associated with the [[boundary conditions]] of [[open string|open]] [[superstrings]], are [[cocycles]] in [[twisted K-theory|twisted]] [[differential K-theory|K-theory]]. Since two [[target spaces]] related under [[T-duality]] are supposed to induce equivalent [[superstring]] [[sigma-models]], they should in particular have the same [[K-theory]]. This is indeed the case, a fact proven in full generality in [[topological T-duality]] (see there for references).

But [[T-duality]] is supposed to induce an equivalence not just of these boundary data, but for the full [[string theory]]. By the seminal results on the [[Witten genus]] it is known that the [[partition function]] of the [[superstring]] is an [[elliptic genus]], and by the seminal results on the [[string orientation of tmf]] it is known that this is just a shadow of an [[orientation in generalized cohomology|orientation]] in [[tmf]]-[[generalized (Eilenberg-Steenrod) cohomology]] theory (the "universal [[elliptic cohomology]]"). Ever since it is natural to expect that full string theories are actually classified not just in [[K-theory]] but in [[elliptic cohomology]] hence in [[tmf]].

> A properly developed theory of [[elliptic cohomology]] is likely to shed some light on what [[string theory]] really means. ([Witten 87, very last sentence](string+theory#EllipticGenera)).

Now ([Nikolaus 14](#Nikolaus14)) claims, under certain conditions, that indeed [[topological T-duality]] induces equivalences in [[tmf]] cohomology classes. The conditions are actually fairly mild, one sufficient condition is that the rank of the [[torus]] bundle is $\leq 6$. (This is already the most interesting case in applications in [[physics]].)

## 2. Precise definition of T-folds

See at _[[T-duality 2-group]]_

## 3. K-Cohomology of T-folds

See at _[[T-duality 2-group]]_

category: reference


T-fold

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

#Contents#
* table of contents
{:toc}

## Idea

A _T-fold_ ([Hull 04](#Hull04)) is supposed to be a kind of [[space]] that locally has [[charts]] which are [[Riemannian manifolds]] equipped with a [[B-field]] (i.e. a [[circle 2-bundle with connection]] or [[bundle gerbe with connection]]) but where the charts are glued together  not just by [[diffeomorphisms]] (as an ordinary [[smooth manifold]] is) but also by [[T-duality]] transformations along some [[torus]] [[fibers]].

The idea is that a T-fold is a [[target space]] for a [[string]] [[sigma-model]] that is only locally a [[Riemannian manifold]] but globally a more general kind of [[geometry]], due due [[duality in string theory]]. In the literature sometimes the term _[[non-geometric backgrounds]]_ is used for such "generalized geometric" backgrounds.

It is expected that T-folds should have a description in terms of spaces that locally are [[fiber products]] of one [[torus]] [[fiber bundle]] with its T-dual, as the correspondence spaces considered in [[topological T-duality]]. In the [[rational homotopy theory|rational]]/[[Lie theory|infinitesimal]] approximation this is derived from analysis of [[geometry of physics -- fundamental super p-branes|super p-brane WZ-terms ]] in [FSS 16](#FSS16). A proposal for a (non-supersymmetric) global description of T-folds as total spaces of [[principal 2-bundles]] for the [[T-duality 2-group]] is in [Nikolaus-Waldorf 18](#NikolausWaldorf18).

One may then consider [[local field theory]] on these double torus fibrations, and this should be closely related to what is called _[[double field theory]]_ ([Hull 06](#Hull06)).

## Related concepts

* [[type II geometry]]

* [[non-geometric string theory vacua]]

* [[doubled geometry]], [[type II geometry]]

* [[double field theory]]

## References

### General

The original idea:

* {#Hull04} [[Chris Hull]], _A Geometry for Non-Geometric String Backgrounds_, JHEP0510:065,2005 ([arXiv:hep-th/0406102](http://arxiv.org/abs/hep-th/0406102))

The relation to [[double field theory]] goes back to:

* {#Hull06} [[Chris Hull]], _Doubled geometry and T-folds_ JHEP0707:080,2007  ([arXiv:hep-th/0605149](http://arxiv.org/abs/hep-th/0605149))

Further developments:

* [[Chris Hull]], _Global Aspects of T-Duality, Gauged Sigma Models and T-Folds_ ([arXiv:hep-th/0604178](http://arxiv.org/abs/hep-th/0604178))

* [[Aaron Bergman]], Daniel Robbins, _Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes_ ([arXiv:0710.5158](http://arxiv.org/abs/0710.5158))

* [[Dieter Lüst]], Stefano Massai, Valentí Vall Camell, _The monodromy of T-folds and T-fects_ ([arXiv:1508.01193](http://arxiv.org/abs/1508.01193))

* Yoan Gautier, [[Chris Hull]], Dan Israël, _Heterotic/type II Duality and Non-Geometric Compactifications_ ([arXiv:1906.02165](https://arxiv.org/abs/1906.02165))

Discussion for [[nonabelian T-duality]]:

* Mark Bugden, _Non-abelian T-folds_ ([arXiv:1901.03782](https://arxiv.org/abs/1901.03782))

* Ladislav Hlavatý, Ivo Petr, _T-folds as Poisson-Lie plurals_ ([arXiv:2004.08387](https://arxiv.org/abs/2004.08387))

Comments in relation to [[topological T-duality]]:

* [[Peter Bouwknegt]], [[Ashwin S. Pande]], *Topological T-duality and T-folds*, Advances in Theoretical and Mathematical Physics **13** 5 (2009) [[arXiv:0810.4374](https://arxiv.org/abs/0810.4374), [doi:10.4310/ATMP.2009.v13.n5.a6](https://dx.doi.org/10.4310/ATMP.2009.v13.n5.a6)]

### Via higher geometry
 {#ReferencesViaHigherGeometry}

A global definition of [[T-folds]] (with [[T-duality]] understood as [[topological T-duality]]) in [[higher differential geometry]], concretely as [[principal 2-bundles]] for the [[T-duality 2-group]] (as described in _[[T-Duality and Differential K-Theory]]_) is proposed in

* {#Nikolaus14} [[Thomas Nikolaus]], _T-Duality in K-theory and elliptic cohomology_, talk at _String Geometry Network Meeting_, Feb 2014, ESI Vienna ([website](http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html))

* {#NikolausWaldorf18} [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_, Commun. Math. Phys. **374**  (2020) 317-366 [[arXiv:1804.00677](https://arxiv.org/abs/1804.00677), [doi:10.1007/s00220-019-03496-3](https://doi.org/10.1007/s00220-019-03496-3)]

The local [[superspace]] [[supergeometry]] of T-folds compatible with this proposal is derived from [[Green-Schwarz action functionals|fundamental super $p$-brane structure]] in:

* {#FSS16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, [ATMP Volume 22 (2018) Number 5](http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/) ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536), [doi:10.4310/ATMP.2018.v22.n5.a3](https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3))

The [[automorphism 2-group]] of the [[T-duality 2-group]] (and hence potentially the full structure of T-folds) is determined in:

* [[Konrad Waldorf]], *Categorical symmetries of T-duality* [[arXiv:2202.09187](https://arxiv.org/abs/2202.09187), [spire:2035224](https://inspirehep.net/literature/2035224)]

Introduction and review:

* [[Konrad Waldorf]], *Stacks, Gerbes and T-Duality*, talk notes, Adelaide (2019) [[pdf](http://math.konradwaldorf.de/www/link.php?lg=en&link=257&savename=Waldorf-Stacks-Gerbes-and-T-Duality.pdf)]

Proposals for further generalization ([[connection on a 2-bundle|2-connections]] and further [[non-geometric backgrounds]]):

* [[Hyungrok Kim]], [[Christian Saemann]], *Non-Geometric T-Duality as Higher Groupoid Bundles with Connections* [[arXiv:2204.01783](https://arxiv.org/abs/2204.01783)]

[[!redirects T-folds]]

[[!redirects super T-fold]]
[[!redirects super T-folds]]


T-Grothendieck construction

## Idea

The T-Grothendieck construction generalizes the [[displayed category]] construction to models of an [[algebraic theory]] T. The ordinary displayed category construction is the instance when T is the [[theory of categories]]. This construction forms an equivalence between models of T in the category of [[indexed sets]], and models of T in the [[arrow category]] of [[Set]]. The T-Grothendieck construction answers the following question: which objects classify morphisms of the models of an algebraic theory T?

## Definition

The T-Grothendieck construction relies on internalizing models of $T$ in the category of indexed sets and the arrow category of $Set$.
\begin{defn}
An indexed set is a function $f: X \to Set$. The set $X$ is called the indexing set, and the set $f(x)$ for $x \in X$ is called the fiber over $x$.
\end{defn}
\begin{defn}
Let $IndSet$ be the category whose objects are functions $a: X \to Set$ and a morphism from $a$ to $b: Y \to Set$ is a function $f: X \to Y$ and an $X$-indexed family of functions $\alpha_x: a(x) \to b(f(x))$. Let $Set^{\to}$ be the arrow category of Set.
\end{defn}

The T-Grothendieck construction is the forward direction of the following equivalence:

\begin{theorem}
There is an equivalence of categories
\[ \int_T : [T,IndSet] \cong [T,Set^{\to}]\]
\end{theorem}

\begin{proof}
There is an equivalence of categories
\[ IndSet \cong Set^{\to}\]
An indexed set $f : X \to Set$ is sent to the function $p: \int f \to X$ where
$\int f = \{ (x,a) | a \in f(x) \}$ and $p$ is the projection onto the first coordinate. The inverse mapping of this equivalence sends a function $f: Y \to X$ to its preimage mapping $f^{-1}: X \to Set$. The functor category is a [[2-functor]]
\[ [T,-] : Cat \to Cat \]
sending a category $C$ to the functor category $[T,C]$, sending a functor $F: C \to D$ to the functor from $[T,F]:[T,C]\to [T,D]$ given by composition with $F$, and sending a natural transformation $\alpha : F \Rightarrow G$ to the natural transformation $[T,\alpha]: [T,F] \Rightarrow [T,G]$ given by [[whiskering]] the functors in the image of $[T,F]$ by $\alpha$. Because every 2-functor preserves equivalences, the equivalence $IndSet \cong Set^{\to}$ is sent to the desired one.
\end{proof}

## Examples

* When T is the [[theory of categories]] we recover the [[displayed category]] construction. This relies on an equivalence
\[ [Th(Cat),IndSet] \cong Cat/Span_{lax} \]
where the latter category is the category of displayed categories i.e.\ lax functors into the [[bicategory of spans]] and functional natural transformations between them.
* When T is the theory of graphs, we get an equivalence
\[ Grph/USpan \cong Grph^{\to}\]
between the category of graph morphisms with codomain $USpan$, the graph whose vertices are sets and whose edges are spans and the arrow category of $Grph$. This example is like the previous example but with less structure.
* When T is the theory of monoids we get an equivalence
\[ [Th(Mon),IndSet] \cong Mon^{\to} \]
which unpacks morphisms of monoids into "indexed monoids" i.e.\ indexing monoids $M$ equipped with fiberwise monoids whose operations are coherent with the operations of $M$.


t-norm

# T-norms

* table of contents
{: toc}

## Definition

A **t-norm** is a [[semicartesian monoidal category|semicartesian]] [[symmetric monoidal category|commutative]] [[monoidal category|monoidal]] structure on the [[unit interval]] $[0,1]$ as a [[poset]].

That means it is a [[commutative monoid]] structure on the set $[0,1]$ that is order-preserving in each argument and with $1$ as the unit element.

## Related concepts

T-norms are used in [[fuzzy logic]].

## Examples

* $T(x,y) = \min(x,y)$.  This is the [[cartesian monoidal category|cartesian]] monoidal structure, also known as the *minimum t-norm* or the *Godel t-norm*.
* $T(x,y) = x y$, the *product t-norm*.
* $T(x,y) = \max(x+y-1,0)$, the *Lukasiewicz t-norm* (see [[Lukasiewicz logic]]).  This monoidal structure is moreover [[star-autonomous category|star-autonomous]] with the obvious involution $x^\ast = 1-x$.

[[!redirects t-norms]]