Tychonoff theorem for locales

# Contents
* table of contents
{:toc}

(This page is under construction... Will get back to this -- Todd.)

## Statement and introduction

The Tychonoff theorem for locales states that a small product of compact locales is compact. Remarkably, and in stark contrast to the Tychonoff theorem for compact spaces, this result requires only constructive reasoning for its proof; in particular, the axiom of choice (or some other choice principle) is not needed.

Our account of this result is adapted from [Johnstone and Vickers](#JV91), who provide a convenient conceptual niche for the Tychonoff theorem in terms of the closed category of preframes. Our plan here is to extract just enough from their account to give a relatively short proof of the Tychonoff theorem, without building up much of their theory of preframes -- which is of interest in its own right of course and a source of conceptual explanation.

The key result is the finitary case, that the product of *two* compact locales is compact, or equivalently that the coproduct of two compact frames is compact. For topological spaces the standard argument is well-known, and appears regularly in undergraduate courses, but crucially uses spatiality (existence of enough points) as a kind of "crutch". The reader might like to try his or her own hand at proving this result for general locales where the crutch is no longer available, to appreciate that a somewhat different approach may be needed even for the finitary case.

## Frames and the category of sup-lattices

The category of locales is by definition opposite to the category of frames; following Joyal and Tierney, we consider frames as idempotent commutative monoids in the symmetric monoidal closed (smc) category $Sup$ consisting of sup-lattices.

The category of frames is the full subcategory of the category of commutative [[quantales]] $(Q, \cdot, 1)$, consisting of the *idempotent* commutative quantales, meaning that $x \leq 1$ for all $x \in Q$ (the quantale unit $1$ is the top element $\top$), and $x \cdot x = x$ for all $x \in Q$.

+-- {: .num_prop}
###### Proposition
**(Joyal-Tierney)**
The coproduct of a collection of frames coincides with their coproduct in the category of commutative quantales.
=--

As commutative quantales are just commutative monoids in $(Sup, \otimes)$, the coproduct of two commutative quantales $P, Q$ is given by the tensor product $P \otimes Q$ in $Sup$. This is just the dual form of the general result recounted [here](/nlab/show/cocommutative+coalgebra#relation_to_cartesian_monoidal_categories). Thus the binary coproduct of two frames $X, Y$ is their sup-lattice tensor product $X \otimes Y$. The nullary coproduct is the monoidal unit in $Sup$. Furthermore,

+-- {: .num_theorem}
###### Theorem
**(Joyal-Tierney)**
The forgetful functor $Frame \to Sup$ preserves and reflects filtered colimits.
Thus the coproduct of a family of frames $\{F_i\}_{i \in S}$ may be computed as

$$\sum_{i \in S} F_i = colim_{finite\; F \subseteq S} \otimes_{i \in F} F_i$$

where the colimit is over the filtered diagram of finite subsets and inclusions between them.
=--

Much of the proof of the Tychonoff theorem will be concentrated in explicit manipulations with sup-lattice tensor products; we turn to this next.

## Symmetric monoidal closed structure for sup-lattices

The closed structure of $Sup$ has internal homs $[X, Y]$, the poset of sup-preserving maps between sup-lattices $X, Y$, which are sup-lattices. As sup-preserving maps between sup-lattices are left adjoints (by an adjoint functor theorem), we may also denote the internal hom as $Ladj(X, Y)$. Note that sup-lattices are inf-lattices (a poset $X$ is a sup-lattice iff $X^{op}$ is a sup-lattice), and it will be convenient also to consider $Radj(X, Y)$, the sup/inf-lattice of right adjoints $X \to Y$.

We specially point out the sup-lattice $\mathbf{2} = \{0 \leq 1\}$ (if we are generalizing from $Set$ to a topos, we use instead the subobject clasifier $\Omega$ which is an internal sup-lattice). This is the unit for the smc structure on $Sup$: we have $[\mathbf{2}, X] \cong X$. Its opposite $\mathbf{2}^{op}$ or $\Omega^{op}$ plays the role of dualizing object: homming into it, we calculate

$$Ladj(X, \mathbf{2}^{op}) \cong Radj(\mathbf{2}^{op}, X)^{op} \cong Ladj(\mathbf{2}, X^{op}) \cong X^{op}.$$

Here the first isomorphism takes a left adjoint $f: X \to \mathbf{2}^{op}$ to its right adjoint $g: \mathbf{2}^{op} \to X$ in the opposite direction, and it also reverses 2-cells, i.e., takes an inequality $f \leq f'$ of left adjoints to
a mated inequality $g' \leq g$ of right adjoints. The second isomorphism is a case of taking a functor $g: \mathbf{2}^{op} \to X$ to its opposite $g^{op}: \mathbf{2} \to X^{op}$; this reverses direction of 2-cells. The double dual embedding

$$X \to Ladj(Ladj(X, \mathbf{2}^{op}), \mathbf{2}^{op})$$

is thus identified (up to canonical isomorphism) with

$$X \stackrel{id}{\to} X^{op\; op} = X$$

making $\mathbf{2}^{op}$ a dualizing object: the smc category $Sup$ is a $\ast$-autonomous category, with $(-)^\ast = (-)^{op}: Sup^{op} \to Sup$.

Being a $\ast$-autonomous category, we have an isomorphism $X \otimes Y \cong [X, Y^{op}]^{op}$. This results from the isomorphisms

$$(X \otimes Y)^{op} \cong Ladj(X \otimes Y, \mathbf{2}^{op}) \cong Ladj(X, Ladj(Y, \mathbf{2}^{op})) \cong Ladj(X, Y^{op}).$$

This observation saves us from having to construct $X \otimes Y$ explicitly as a coequalizer of a free sup-lattice $P(X \times Y)$ (of $\mathbf{2}$-valued presheaves on the poset $X \times Y$). We may thus write

$$X \otimes Y \cong Radj(X^{op}, Y);$$

this isomorphism takes $w \in X \otimes Y$ to the right adjoint $\phi_w: X^{op} \to Y$ defined by $\phi_w(u) = \bigvee_{v: u \otimes v \leq w} v$. Equally well, there is a canonical isomorphism $X \otimes Y \cong Radj(Y^{op}, X)$ taking $w \in X \otimes Y$ to a right adjoint $\psi_w: Y^{op} \to X$, and the pair $(\phi_w, \psi_w)$ forms a contravariant Galois connection with $\psi_w^{op} \dashv \phi_w$.

+-- {: .num_example}
###### Example
Recall the classical proof that a product of two compact topological spaces $X, Y$ is compact. A standard lemma, sometimes called the *tube lemma*, is that for any open cover $A$ of $X \times Y$ and any $x \in X$, there is an open neighborhood $U$ of $x$ such that $U \times Y$ is finitely covered by $A$. Proof: Some finite subset $F \subseteq A$ covers $\{x\} \times Y$; put $W = \bigcup F$. For each $y \in Y$ there are neighborhoods $U$ of $x$ and $V$ of $y$ such that $U \otimes V \leq W$. The collection $\{V: x \in \psi_W(V)\}$ thus covers $Y$, hence there is a finite subcover $V_1, \ldots, V_n$. Putting $U = \bigwedge_i \psi_W(V_i)$, we have for all $i$ that $V_i \leq \phi_W(\psi_W(V_i)) \leq \phi_W(U)$ (the last inequality is by contravariance), hence $Y = \bigvee_i V_i \leq \phi_W(U)$, i.e., $U \times Y \leq W$.
=--

Any $w \in X \otimes Y$ can be represented as a supremum of elements of the form $u \otimes v$, but one virtue of the identification $X \otimes Y \cong [X, Y^{op}]^{op}: w \mapsto \phi_w$ is that it allows one to give a canonical such representation:

+-- {: .num_defn}
###### Definition
For sup-lattices $X, Y$, the *canonical representation* of $w \in X \otimes Y$ (as a sup of tensors $u \otimes v$ with $u \in X$, $v \in Y$) is $w = \bigvee_{u \in X} u \otimes \phi_w(u)$.
=--

## Preframe representations

+-- {: .num_defn}
###### Definition
A *preframe* is a poset with directed joins and finite meets that distribute over directed joins. A *preframe map* is a poset map that preserves finite meets and directed joins.
=--

A frame is *a fortiori* a preframe. As far as the Tychonoff theorem is concerned, the decisive advantage of preframes is that the notion of compactness of frames can be described categorically in terms of preframe maps.

Recall that a frame $X$ is *compact* if whenever its top element $\top$ is the join of a directed subset $J \subseteq X$, then $\top \in J$. Equivalently,

+-- {: .num_prop #char}
###### Proposition
Let $X$ be a frame, and let $\chi_\top: X \to \mathbf{2}$ be the characteristic map of the poset inclusion $\top: 1 \hookrightarrow X$. Then $X$ is *compact* iff $\chi_\top$ is a preframe map. (Of course $\phi_\top$ preserves finite meets for any frame, so the essential condition is preservation of directed joins.)
=--

An essential insight of Johnstone and Vickers is that since directed joins commute with finite meets, the theory of preframes is essentially a commutative theory, and thus the category of preframes forms a symmetric monoidal closed category; moreover, frames are exactly those preframes $X$ such that the binary join operation $\vee: X \times X \to X$ is a preframe map in each separate variable. Thus, frames can be described either as commutative monoids $(X, \wedge, \top)$ in the smc category of sup-lattices, or as commutative monoids $(X, \vee, \bot)$ in the smc category of preframes. In either case, coproducts of frames can be described as tensor products in either of these smc categories, with preframe tensors offering a description that is, roughly speaking, "De Morgan dual" to the sup-lattice tensor description.

This motivates a construction that is "De Morgan dual" to tensor products $u \otimes v$ for elements $u \in X, v \in Y$.

+-- {: .num_defn}
###### Definition
For elements $u \in X$ and $v \in Y$ in frames $X, Y$, define their *par* $u \parr v$ in the sup-lattice $X \otimes Y$ by the formula $u \parr v = (u \otimes \top_Y) \vee (\top_X \otimes v)$.
=--

The following proposition is easily checked.

+-- {: .num_prop #prop}
###### Proposition
**(Johnstone-Vickers)**
If $X, Y$ are frames, then

1. For each $u \in X$, the map $u \parr -: Y \to X \otimes Y$ is a preframe map, and similarly for each $v \in Y$ the map $- \parr v: X \to X \otimes Y$ is a preframe map.

1. The map $\parr: X \times Y \to X \otimes Y$ preserves arbitrary sups (jointly in the two arguments).

1. $u \otimes v = (u \parr \bot_Y) \wedge (\bot_X \parr v)$.
=--

+-- {: .num_prop}
###### Proposition
Given frames $X, Y$, any finite join of tensors $\bigvee_{i \in F} u_i \otimes v_i$ can be expressed as a finite meet of pars.
=--

+-- {: .proof}
###### Proof
For each subset $S \subseteq F$, put $u_S \coloneqq \bigvee_{i \in S} u_i$ and $v_S = \bigvee_{i \in S} v_i$, and let $\neg S$ denote the complement $F \setminus S$.

Exploiting Proposition \ref{prop} and finite distributivity in the frame $X \otimes Y$, we have

$$\array{
\bigvee_{i \in F} u_i \otimes v_i & = & \bigvee_{i \in F} (u_i \parr \bot_Y) \wedge (\bot_X \parr v_i) \\
 & = & \bigwedge_{f: F \to \{u, v\}} \left[(\bigvee_{i \in f^{-1}(u)} u_i \parr \bot_Y) \vee (\bigvee_{i \in f^{-1}(v)} \bot_X \parr v_i)\right] \\
 & = & \bigwedge_{S \subseteq F} u_S \parr v_{\neg S}
}$$

which proves the proposition.
=--

This leads to a canonical preframe representation of elements $w \in X \otimes Y$ in a frame coproduct, where $w$ is expressed as a directed join of finite meets of par expressions. We have

$$\array{
w & = & \bigvee_{u \in X} u \otimes \phi_w(u) \\
 & = & \bigvee_{finite\; F \subseteq X} \bigvee_{u \in F} u \otimes \phi_w(u) \\
 & = & \bigvee_{fin.\; F \subseteq X} \bigwedge_{S \subseteq F} \left[(\bigvee_{u \in S} u) \parr (\bigvee_{u \notin S} \phi_w(u))\right]
}$$

## Finitary Tychonoff theorem

Now suppose $X, Y$ are compact frames, which by Proposition \ref{char} means precisely that the characteristic maps $\chi_{\top_X}: X \to \mathbf{2}$ and $\chi_{\top_Y}: Y \to \mathbf{2}$ preserve directed joins and finite meets.

+-- {: .num_theorem}
###### Theorem
**(Finitary Tychonoff theorem)**
The frame $X \otimes Y$ is compact.
=--

Before giving the proof, we might as well say where it comes from. We have remarked earlier that the category of preframes is symmetric monoidal closed; let $\parr$ denote its monoidal product. Then, as $\mathbf{2}$ is a frame, the map $\vee: \mathbf{2} \times \mathbf{2} \to \mathbf{2}$ is a preframe map in each separate variable, and thus induces a canonical preframe map (which we again denote by $\vee$): $\mathbf{2} \parr \mathbf{2} \to \mathbf{2}$. Since $\chi_{\top_X}$ and $\chi_{\top_Y}$ are preframe maps by assumption, we may form a composite of preframe maps

$$X \parr Y \stackrel{\chi_{\top_X} \parr \chi_{\top_Y}}{\to} \mathbf{2} \parr \mathbf{2} \stackrel{\vee}{\to} \mathbf{2}.$$

Call this preframe map $h$. Now, as remarked earlier, the preframe tensor $X \parr Y$ gives the coproduct of $X$ and $Y$. Then one proves that $h(w) = 1$ implies $w = \top_{X \parr Y}$; that is to say, $h = \chi_{\top_{X \parr Y}}$, meaning that the coproduct of $X$ and $Y$ is compact.

Or so this is how Johnstone and Vickers proceed; our account circumvents the building up of the closed category infrastructure of preframes which lies in the conceptual background.

+-- {: .proof}
###### Proof
Exploiting the canonical preframe representation of elements $w \in X \otimes Y$ from the previous section, define $h: X \otimes Y \to \mathbf{2}$ by the formula

$$h(w) = \bigvee_{fin.\; F \subseteq X} \bigwedge_{S \subseteq F} \left[\chi_{\top_X}(\bigvee_{u \in S} u) \vee \chi_{\top_Y}(\bigvee_{u \notin S} \phi_w(u))\right]$$

Now $h$ preserves directed joins and finite meets, due to the fact that directed joins and finite meets commute with directed joins and finite meets, and that $\vee$ preserves directed joins and finite meets in separate variables, and that $w \mapsto \phi_w$ is a poset isomorphism (so preserves directed joins and finite meets).

Now suppose $h(w) = 1$. Then for some finite subset $F \subseteq X$ we have, for every subset $S \subseteq F$, that either $\bigvee_{u \in S} u = \top_X$ or $\bigvee_{u \notin S} \phi_w(u) = \top_Y$. But since $\top_X \parr v = \top_{X \otimes Y} = u \parr \top_Y$ for any $u \in X, v \in Y$, we conclude

$$\array{
w & = & \bigvee_{fin.\; F \subseteq X} \bigwedge_{S \subseteq F} (\bigvee_{u \in S} u) \parr (\bigvee_{u \notin S} \phi_w(u)) \\
 & = & \bigvee_{fin.\; F \subseteq X} \bigwedge_{S \subseteq F} \top_{X \otimes Y} \\
 & = & \top_{X \otimes Y}
}$$

Thus we have shown $h = \chi_{\top_{X \otimes Y}}$ is a preframe map.
=--

+-- {: .num_theorem}
###### Theorem
**(Tychonoff theorem)**
An arbitrary coproduct of compact frames $\sum_{i \in I} X_i$ is compact.
=--

+-- {: .proof}
###### Proof
We may assume all the frames $X_i$ are non-trivial, i.e. non-terminal. Then each coproduct inclusion $\otimes_{i \in F} X_i \hookrightarrow \otimes_{j \in F'} X_j$, where $F, F'$ are finite subsets of $I$ and $F \subseteq F'$, is an embedding.
(...)
=--

## References

Peter Johnstone and Steven Vickers, *Preframe presentation present*, in A. Carboni, M.C. Pedicchio, G. Rosolini (eds.) *Category Theory* (Proceedings, Como 1990), Springer Lectures Notes in Mathematics 1488 (1991), 193-212. ([online pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.438.8880&rep=rep1&type=pdf))
 {#JV91}

Citation to this [link](http://dml.cz/bitstream/handle/10338.dmlcz/106680/CommentatMathUnivCarol_029-1988-4_3.pdf) to be added...


Tyler Bryson

* [page at MIT Press](https://mitpress.mit.edu/contributors/tyler-bryson)

## Selected writings

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

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

category: people


Tyler Lawson

* [webpage](http://www.math.umn.edu/~tlawson/)

## Selected writings

On [[topological modular forms]] ([[tmf]]):

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_, Int. Math. Res. Not. (2013) ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

with [[level structure on an elliptic curve|level structure]]:

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_, Inventiones mathematicae volume 203, pages 359–416 (2016) ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394), [doi:10.1007/s00222-015-0589-5](https://doi.org/10.1007/s00222-015-0589-5))

On [[Bousfield localization of model categories]] ([[Bousfield localization of spectra]]):

* [[Tyler Lawson]], *An introduction to Bousfield localization* [[arXiv:2002.03888](https://arxiv.org/abs/2002.03888)] in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*  MSRI Book Series, Cambridge University Press (2022)

category: people


typal equality

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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#### Foundations
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[[!include foundations - contents]]
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\tableofcontents

## Idea

In [[dependent type theory]], the term *typal equality* is used to describe the use of the [[identity type]] to represent the notion of equality of terms, and is primarily used to distinguish from the notion of [[judgmental equality of terms]].

In dependent type theories with a separate [[proposition]] [[judgment]] $\phi \mathrm{prop}$, the term *typal equality* is also used to distinguish identity types from [[propositional equality]], which is the equality judged to be a proposition; i.e. $a =_A b \mathrm{prop}$. On the other hand, in dependent type theory without a separate [[proposition]] [[judgment]], propositions are usually defined to be certain types, and the term "propositional equality" is used as a synonym of typal equality in referring to identity types.  (In fact, "propositional equality" is the older and arguably still more common term in this context.)

## See also

* [[judgmental equality of terms]]
* [[propositional equality]]

[[!redirects typal equality]]
[[!redirects typally equal]]


type

> This entry is about the notion in [[type theory]]. For the unrelated notion of the same name in [[model theory]] see at _[[type (in model theory)]]_.

***

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In modern [[logic]], we understand that every [[variable]] should have a _type_, or _domain of discourse_ or be of some _sort_.  For instance we say that if a variable $n$ is constrained to be an [[integer]] then "$n$ is of integer type" or "of type $\mathbb{Z}$". The usual formal expression from [[set theory]] for this -- $n \in \mathbb{Z}$ -- is then often written $n \colon \mathbb{Z}$

We speak of _typed logic_ if this typing of variables is enforced by the [[metalanguage]]. In formulations of a [[theory]] the types are often called _sorts_. More generally, _[[type theory]]_ formalizes reasoning with such typed variables. See there for more

(Untyped logic may be seen as simply a special case, in which there is only a single unique type.  Thus, untyped logic has *one* type, not *no* type.)

## Definition

Reasoning with types is formalized in [[natural deduction]] (which in turn is formalized in a [[logical framework]] such as [[Elf]]).

Behaviour of types is specified by a 4-step set of rules

1. [[type formation]]

1. [[term introduction]]

1. [[term elimination]]

1. [[computation rules]]

## Properties

Deep relations between [[type theory]], [[category theory]] and [[computer science]] relate types to other notions, such as [[objects]] in a [[category]]. See at _[[computational trinitarianism]]_ for more on this.

## Related concepts

* [[category (philosophy)]]

* [[fibrant type]]

[[!include notions of type]]

[[!redirects types]]

[[!redirects domain of discourse]]
[[!redirects domains of discourse]]
[[!redirects universe of discourse]]
[[!redirects universes of discourse]]

[[!redirects sort]]
[[!redirects sorts]]


type (functional analysis)

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#### Functional analysis
### Contents {: .clickToReveal}
### Contents {: .clickToHide tabindex="0"}
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[[!include functional analysis - contents]]
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# Type and Cotype in Functional Analysis
* tic
{: toc}

## Idea

The **type** and **cotype** of a [[Banach space]] measure how far it is from being a [[Hilbert space]].  The definition is based on the observation, due to [[John von Neumann]], that a Banach space is a Hilbert space if and only if it satisfies the [[parallelogram identity]].  Recall that this states that in a Hilbert space,
$$
{\|x + y\|^2} + {\|x - y\|^2} = 2{\|x\|^2} + 2{\|y\|^2}.
$$
This can be thought of as a way of improving the triangle inequality, which relates $\|x \pm y\|$ to $\|x\|$ and $\|y\|$, by finding an equality relating $\|x \pm y\|$ to $\|x\|$ and $\|y\|$.

To measure the type and cotype of a Banach space, one takes the parallelogram identity and finds out how bad it gets.  Slightly more precisely, one tries to see what happens if one looks merely for an inequality, perhaps with a constant.  Since the equality can break in one of two ways, this leads to two notions.

## Definition ##

To define the type and cotype of a Banach space, we start with a finite family of vectors, say $\{x_1,\dots,x_n\}$.  Then we look for constants $T_2$ and $C_2$ such that the following inequalities are true:

$$
\begin{aligned}
\Average_\pm {\left\|\sum \pm x_i\right\|^2} &\le T_2^2 \sum {\|x_i\|^2}, \\
\Average_\pm {\left\|\sum \pm x_i\right\|^2} &\ge C_2^{-2} \sum {\|x_i\|^2}
\end{aligned}
$$

Here, the left-hand side is the average value over all choices of $\pm$ (so in the original parallelogram identity we have divided each side by $2$).

+-- {: .num_defn #type}
###### Definition ######
The smallest constant $T_2$ making the first inequality true for all finite sequences of vectors is the **type $2$ constant** of the space.

The smallest constant $C_2$ making the second inequality true for all finite sequences of vectors is the **cotype $2$ constant** of the space.

Either of these is allowed to be infinite.  A space is said to be **type $2$** if its type $2$ constant is finite.  Similarly, it is said to be **cotype $2$** if its cotype $2$ constant is finite.
=--

## Properties ##

If we consider all Banach spaces that are either of type $2$ or cotype $2$ then we find that these split into the two obvious classes with Hilbert spaces sitting plum in the middle.  Not only are Hilbert spaces the intersection of these classes, but also a continuous linear operator from a space of type $2$ into a space of cotype $2$ factors through a Hilbert space.
(This follows from a generalization/extension of Grothendieck's inequality.)

More precisely:

1. If a Banach space is of type $2$ and of cotype $2$ (ie both constants are finite) then it is a Hilbert space.  This is due to Kwapien.
2. Any bounded linear operator from a Banach space of type $2$ to a Banach space of cotype $2$ factors through a Hilbert space.

As type and cotype are isomorphism invariants, they can be used to distinguish between some Banach spaces.  See [[isomorphism classes of Banach spaces]] for more.

## Generalisations ##

In the inequalities for type $2$ and cotype $2$ it is possible to replace the $2$ by a real number $p$ and define the notions of *type $p$* and *cotype $p$*.  Taken as a whole, these provide more information and thus give a finer classification of Banach spaces.  In particular:

1. For $1 \le p \le 2$, the [[Lebesgue space]], $L_p$, has type $p$ and cotype $2$.
1. For $2 \le p \lt \infty$, the [[Lebesgue space]], $L_p$, has type $2$ and cotype $p$.

We also have the following properties:

1. For $r \lt p$, type $p$ implies type $r$.
1. For $r \gt p$, cotype $p$ implies cotype $r$.
1. Both type and cotype pass to subspaces.
1. Type passes to quotients, cotype does not in general but if the space has some type strictly larger than one then cotype does pass to quotients.
1. Type dualises to cotype, in that if $X$ has type $p$ then $X^*$ has cotype $p'$ (where $p^{-1} + {p'}^{-1} = 1$), but cotype does not dualise to type unless the space has some type strictly larger than one.

[[!redirects cotype (functional analysis)]]


type (in model theory)

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### Context
#### Model theory
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[[!include model theory - contents]]
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#Contents#
* table of contents
{:toc}

**Warning** on terminology: This is a different notion than that of _[[type]]_ in [[type theory]], which is what model theorists call _sorts_.

## Idea
The [[syntactic category]] of a Boolean coherent theory has a Boolean algebra structure on each of its subobject posets. Types are ultrafilters of the Boolean algebras of subobjects of the maximal objects in these syntactic categories (those corresponding to formulas of the form "$x=x$").

If one tries to add a global point to such a syntactic category (i.e. name a constant) "without changing anything", this point must have a sort (in the sense of type from type theory), and hence must pick out a principal ultrafilter of definable sets which live above it in the subobject lattice. Types tell you what kinds of global points you can add. When you have "all possible" such global points plus a homogeneity condition, you're in a saturated [[monster model]].

## Definition
Let $M$ be a model (in $\mathbf{Set}$) of a first-order theory. Let $ \overline{m} = (m_1, \dots, m_n)$ be a tuple of elements from that model. The _type_ of $\overline{m}$ is the [[ultrafilter]] of definable sets of the same _sort_ ([[type]] in the sense of type theory, sort in the sense of Makkai's FOLDS) as $\overline{m}$ containing $\overline{m}$.

Syntactically, if $x$ is a tuple of variables, an _$x$-type_ is a finitely consistent (hence consistent, by the [[compactness theorem]]) set of formulas whose tuple of free variables has the same sort (arity) as $s$. These correspond to filters of definable sets. A type corresponds to an _ultrafilter_ of definable sets if and only if it is a maximally consistent set of formulas; we call such a type _complete_.

By [[Stone duality]], the set of all complete $x$-types of a theory $T$ inherits a Stone topology, and is called the _Stone space_ $S_x(T)$ of the theory $T$.

There are some extension of this notion of type beyond first order, for example the notion of [[Galois type]]s in [[infinitary logic]], which is particularly important in the study of [[abstract elementary class]]es.

An important method toward classification of theories is study of the number of nonisomorphic models in each cardinality; its study is related to the study of types,
and the phenomenon of [[forking]] in model theory; that aspect is studied in [[stability theory]].

## Saturation and the monster model
If $m$ is a tuple of sort $s$, and $p$ is an $s$-type, we say that $m$ _realizes_ the type $p$ if the type of $m$ is $p$.

By the completeness theorem, we can always realize types in larger elementary extensions if they aren't already realized in your model. For example, this is how one usually obtains a nonstandard model of arithmetic, because one can form the type (with parameters coming from the $\mathbb{N}$, which is small) which says "any realization of this must not equal anything in $\mathbb{N}$."

More generally, let $A$ be a subset of the model $M$. If we also allow parameters from $A$ in our formulas, we can form _types over $A$_. We say that a model is $\kappa$-saturated if all types over $A$ for $|A| = \kappa$ are realized.

Saturated models always exist (maybe modulo some large cardinal assumptions.) For example, an $\aleph_0$-saturated elementary extension of the natural numbers $(\mathbb{N}, \leq)$ equipped with their usual ordering just looks like $\mathbb{N}$ with $\omega$-many copies of $(\mathbb{Z}, \leq)$ stacked on top, each of them comprising points at infinity.

In modern model theory it is customary to work in a Grothendieck universe-sized or class-sized _monster model_, so that all types over small parameter sets are realized. Inside a monster $\mathbb{M}$ things often get more explicitly geometric. For example, two tuples $a$ and $b$ in $\mathbb{M}$ will have the same type if and only if they are $\operatorname{Aut}(\mathbb{M})$-conjugate, i.e. lie in the same orbit of the automorphism group of the monster, and in the presence of two constants the syntactic category is Barr-exact (i.e. eliminates [[imaginary element]]s) if and only if all definable sets can be _coded_: for all $X \in \mathsf{Syn}(T),$ there exists a tuple $x$ such that any automorphism which fixes $X$ setwise fixes $x$ pointwise.

Stated in this form, it's easy to see that the [[theory of algebraically closed fields]] codes at least all finite parameter sets. This paves the way for [[model-theoretic Galois theory]].

## Examples

* In the theory of an equivalence relation with two infinite classes, the type (over the empty set) of a point in any model is isolated in the Stone space by the formula which says which equivalence class that point belongs to.

* Types generalize points by replacing them with their principal ultrafilters of definable sets. The simplest first order theories are generalizations of algebraic geometry where points correspond to special (e.g. maximal, prime) ideals in a ring; types generalize points in some of such cases. Other "spectral" intuition also applies.

* Finally, one may expand the model $M$ a larger model $M'$. The elements of automorphism group of the larger model will move the "points" around and a type will become an orbit for the larger model $M'$. This point of view is used especially for the generalization, so called [[Galois type]]s.

## Omitting types theorem

We start with some definitions:

* A theory $T$ _locally realizes_ a type $\Sigma$ if there is a formula $\varphi$ in the context of $\Sigma$ such that $\varphi$ is consistent with $T$ and for all $\sigma \in \Sigma$, $T \vDash \varphi \to \sigma$.

* $T$ _locally omits_ $\Sigma$ if and only if $T$ does not locally realize $\Sigma$.

Note that for a complete theory $T$, if $T$ has any model that omits $\Sigma$, then $T$ locally omits $\Sigma$. The omitting types theorem is a converse for any consistent theory in a countable language:

+-- {: .num_example #OmittingTypes}
###### Theorem
**(Omitting Types Theorem)**
Let $T$ by a consistent theory in a countable language, and let $\Sigma$ be a type. If $T$ locally omits $\Sigma$, then $T$ has a countable model that omits $\Sigma$.
=--

## Relation to measures

## Related concepts

* [[topos of types]]
* [[forking]]
* [[Galois type]]
* [[stability theory]]

## References

* Wikipedia, _[type (model theory)](http://en.wikipedia.org/wiki/Type_%28model_theory%29)_, [forking extension](http://en.wikipedia.org/wiki/Forking_extension)
* Siu-Ah Ng, _Forking_, (Springer Online) Encyclopedia of Mathematics. [web](http://www.encyclopediaofmath.org/index.php?title=Forking&oldid=19231) (note inside the wrong link (cross reference) to a type in the sense of type theory)
* Lou van den Dries, _An Introduction To Model-Theoretic Stability_, [online notes](http://www.karlin.mff.cuni.cz/~krajicek/vddries-stable.pdf)

[[!redirects type in model theory]]
[[!redirects omitting types theorem]]


type 0 string theory

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] _type 0_ refers to a kind of [[perturbative string theory]] where the [[strings]] themselves are [[superstrings]] (hence have [[supersymmetry]] on their [[worldsheet]]) but their [[target space|target]] [[spacetime]] [[effective field theory]] is not supersymmetric, in fact does not even contain any [[fermion fields]].

Technically, this comes from one particular choice of [[GSO projection]] of the [[superstring]] which breaks all target space supersymmetry, in contrast to the other choices of [[GSO projection]] which give the [[type IIA string theory]], [[type IIB string theory]].

## References

Application to a variant of [[AdS-CFT duality]], or rather [[AdS-QCD correspondence]]:

* {#GLMR00} Roberto Grena, Simone Lelli, Michele Maggiore, Anna Rissone, _Confinement, asymptotic freedom and renormalons in type 0 string duals_, JHEP 0007 (2000) 005 ([arXiv:hep-th/0005213](https://arxiv.org/abs/hep-th/0005213))

* Mohammad Akhond, Adi Armoni, Stefano Speziali, _Phases of $U(N_c)$ $QCD_3$ from Type 0 Strings and Seiberg Duality_ ([arxiv:1908.04324](https://arxiv.org/abs/1908.04324))

See also

* Wikipedia, _[Type 0 string theory](https://en.wikipedia.org/wiki/Type_0_string_theory)_


type checking

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

#Contents#
* table of contents
{:toc}

## Idea

In [[type theory]], *type checking* refers to the problem whether the [[judgment]] $\Gamma\vdash a \colon A$ is derivable in a given type theory.

## Properties

### Decidability

For instance, [[intensional type theory]] has [[decidability|decidable]] type checking, but [[extensional type theory]], which adds the rule of [[equality reflection]], does not. For an exposition of this result, see [Hofmann 1995](#Hofmann95).

In more expressive type theories, we can also ask whether [[judgmental equality]] (also called _computational_ or _definitional_ equality) is [[decidability|decidable]]. Since deciding the judgmental equality is often a prerequisite for deciding type checking, systems with decidable type checking often have decidable judgmental equality.

## Related entries

* [[software verification]]

## References

* {#Hofmann95} [[Martin Hofmann]], _Extensional concepts in intensional type theory_, Ph.D. dissertation, University of Edinburgh (1995). ([link](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/))

See also:

* Wikipedia, *[Type system](https://en.wikipedia.org/wiki/Type_system)*


type formation

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

#Contents#
* table of contents
{:toc}

## Idea

In [[type theory]] a _[[type formation rule]]_ is a [[natural deduction]] step roughly of the form

$$
  \frac{
    \vdash\; A \colon Type \;\;\; \vdash \;B \colon Type \;\;\; \cdots
  }{
    \vdash \; F(A,B,\cdots) \colon Type
  }
$$

which says that given [[types]] $A, B, \cdots $, there is a new type $F(A,B, \cdots)$.

For instance for [[product types]] it says that

$$
  \frac{\vdash \; A \colon Type \;\;\; \vdash \; B \colon Type}{\vdash \; A \times B \colon Type}
  \,.
$$

In [[natural deduction]] the type formation rule for a kind of type is the first in a quadruple of rules that come with every kind of type:

1. **type formation rule**

1. [[term introduction rule]]

1. [[term elimination rule]]

1. [[computation rule]]

## Properties

### Relation to category theory

Under the [[relation between type theory and category theory]], the [[categorical semantics]] of type formation essentially corresponds to certain [[universal constructions]] in [[category theory]].

## References

Introductory textbook account:

* [[Simon Thompson]], §4.2  4.3 in: *[[Type Theory and Functional Programming]]*, Addison-Wesley (1991) [ISBN:0-201-41667-0, [webpage](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP), [pdf](http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf)]

[[!redirects type formation]]
[[!redirects type formations]]

[[!redirects type former]]
[[!redirects type formers]]
[[!redirects type formation rule]]
[[!redirects type formation rules]]

[[!redirects type forming]]
[[!redirects type forming operation]]
[[!redirects type forming operations]]

[[!redirects type-forming]]
[[!redirects type-forming operation]]
[[!redirects type-forming operations]]

[[!redirects formation rule]]
[[!redirects formation rules]]


type I string theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _type I string theory_ is [[type IIB string theory]] on [[orientifold]] [[spacetimes]], hence on [[O9-planes]].

Its [[T-duality|T-dual]], called _type I' string theory_, is  [[type IIA string theory]] on [[O8-planes]], which under the [[duality between M-theory and type IIA string theory]] is M-theory [[KK-compactification|KK-compactified]] on the [[orientifold]] $S^1 \times S^1 \sslash \mathbb{Z}_2$ (see also _[[M-theory on S1/G_HW times H/G_ADE]]_):

$$
  \array{
    M
    \\
    {}^{ \mathllap{S^1 \times S^1/\mathbb{Z}_2 }}\big\downarrow
    \\
    I' &\underset{T}{\leftrightarrow}& I
  }
$$

> table from [BLT 13](#BlumenhagenLustTheisen13)

## Properties

### Tadpole cancellation and $SO(32)$-GUT in Type I
 {#TadpoleCancellationAndSO32GUT}

For type I string theory on [[flat orbifold|flat]] ([[toroidal orbifold|toroidal]]) [[target spacetime]] [[orientifolds]] $\mathbb{R}^{9,1}$
(i.e. for [[type IIB string theory]] on flat toroidal [[O9-planes]])
[[RR-field tadpole cancellation]] requires 32 [[D-branes]] (see [this Remark](orientifold+plane#ContingOfDBranesOnOrientifolds) for counting D-branes in [[orientifolds]]) to cancel the [[O-plane charge]] of -32 ([here](orientifold+plane#OPlaneChargeForFlatOrientifolds)).

Under the [[duality between type I and heterotic string theory]] this translates to the [[semi-spin group|semi-spin]] [[gauge group]] [[SemiSpin(32)]] of [[heterotic string theory]].

Discussion of type-I [[string phenomenology]] and [[grand unified theory]] based on [[SO(32)]] type-I strings: ([MMRB 86](#MMRB86), [Ibanez-Munoz-Rigolin 98](#IbanezMunozRigolin98), [Yamatsu 17](#Yamatsu17)).

### Tadpole cancellation and $SO(16) \times SO(16)$-GUT in Type I'

For type I' string theory on [[flat orbifold|flat]] ([[toroidal orbifold|toroidal]]) [[target spacetime]] [[orientifolds]] $X^{8,1} \times \mathbb{S}^1/\mathbb{Z}_2$
(i.e. for [[type IIA string theory]] on two flat toroidal [[O8-planes]])
[[RR-field tadpole cancellation]] requires 16 [[D-branes]] (see [this Remark](orientifold+plane#ContingOfDBranesOnOrientifolds) for counting D-branes in [[orientifolds]]) on each of the two [[O8-planes]] to cancel the total [[O-plane charge]] of $-32 = 2 \cdot (-16)$ ([here](orientifold+plane#OPlaneChargeForFlatOrientifolds)).

Discussion of [[Spin(16)]]-[[GUT]] phenomenology:

(...)

### Orbifolds of type I

Type I' on [[toroidal orbifold|toroidal]] [[orientifolds]] with [[ADE-singularities]] (e.g. [Bergman&Rodriguez-Gomez 12, Sec. 3](#BergmanRodriguezGomez12))

[[duality in string theory|dual]] to [[heterotic M-theory on ADE-orbifolds]].

(...)

### Dualities

#### String-string dualities

See at _[[duality between type I and heterotic string theory]]_

#### Horava-Witten theory

One considers the [[KK-compactification]] of [[M-theory]] on a [[cyclic group of order 2|Z/2]]-[[orbifold]] of a [[torus]], hence of the [[Cartesian product]] of two [[circles]]

$$
  \array{
    & S^1_A &\times& S^1_B
    \\
    \text{radius}: &  R_{11} && R_{10}
  }
$$

such that the reduction on the first factor $S^1_A$ corresponds to the [[duality between M-theory and type IIA string theory]], hence so that subsequent [[T-duality]] along the second factor yields [[type IIB string theory]] (in its [[F-theory]]-incarnation). Now the diffeomorphism which exchanges the two circle factors and hence should be a symmetry of M-theory is interpreted as [[S-duality]] in [[type II string theory]]:

$$
  IIB \overset{S}{\leftrightarrow} IIB
$$

> graphics taken from [Horava-Witten 95, p. 15](#HoravaWitten95)

If one considers this situation additionally with a $\mathbb{Z}/2\mathbb{Z}$-[[orbifold]] quotient of the first circle factor, one obtains the [[duality between M-theory and heterotic string theory]] ([[Horava-Witten theory]]). If instead one performs it on the second circle factor, one obtains [[type I string theory]].

Here in both cases the [[involution]] [[action]] is by [[reflection]] of the circle at a line through its center. Hence if we identify $S^1 \simeq \mathbb{R} / \mathbb{Z}$ then the action is by multiplication by /1 on the [[real line]].

In summary:

M-theory on

* $(S^1_A \sslash \mathbb{Z}_2 ) \times S^1_B$ yields [[heterotic string theory]]

* $S^1_A \times \left( S^1_B \sslash \mathbb{Z}_2 \right)$ yields [[type I' string theory]]

Hence the [[S-duality]] that swaps the two circle factors corresponds to a duality between heterotic E and type I' string theory. And T-dualizing turns this into a _[[duality between type I and heterotic string theory]]_.

$$
  \array{
     HE
       &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}&
     M
       &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}&
     I'
     \\
     \mathllap{T}\updownarrow
     &&  &&
     \updownarrow \mathrlap{T}
     \\
     HO
     && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} &&
     I
  }
$$

> graphics taken from [Horava-Witten 95, p. 16](#HoravaWitten95)

## Related concepts

[[!include string theory and cohomology theory -- table]]

## References

### General

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 4.4.3 of: _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Section 9.4 and 10.6 of: _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics, Springer 2013

Relation to [[M-theory]] (via [[Horava-Witten theory]]):

* {#HoravaWitten95} [[Petr Hořava]], [[Edward Witten]], _Heterotic and Type I string dynamics from eleven dimensions_, Nucl. Phys. B460 (1996) 506 ([arXiv:hep-th/9510209](http://arxiv.org/abs/hep-th/9510209))

* {#HoravaWitten96} [[Petr Hořava]], [[Edward Witten]],  _Eleven dimensional supergravity on a manifold with boundary_, Nucl. Phys. B475 (1996) 94 ([arXiv:hep-th/9603142](http://arxiv.org/abs/hep-th/9603142))

A comprehensive discussion of the ([[differential cohomology|differential]]) [[cohomology|cohomological]] nature of general type II/type I [[orientifold]] backgrounds is in

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

with details in

* {#Freed} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_ ([[FreedESI2012.pdf:file]])

* {#DistlerFreedMooreII} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_, Surveys in Differential Geometry, Volume 15 (2010) ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581), [doi:10.4310/SDG.2010.v15.n1.a4](http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4))

Related lecture notes / slides include

* [[Jacques Distler]], _Orientifolds and Twisted KR-Theory_ (2008) ([pdf](http://www.perimeterinstitute.ca/pdf/files/731c5f3a-928f-453a-b569-db5c574d2a6c.pdf))

* [[Daniel Freed]], _Dirac charge quantiation, K-theory, and orientifolds_, talk at a workshop _Mathematical methods in general relativity and quantum field theories_, November, 2009 ([pdf](http://www.ma.utexas.edu/users/dafr/paris_nt.pdf))

* {#Moore10} [[Greg Moore]], _The RR-charge of an orientifold_, Oberwolfach talk 2010 ([pdf](https://www.physics.rutgers.edu/~gmoore/Oberwolfach_June2010_FINAL.pdf), [[MooreOrientifold2010.pdf:file]], [ppt](http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt))

### Type I'

Original articles on [[type I' string theory]]:

* {#Schwarz00} [[John Schwarz]], _Some Properties of Type I' String Theory_, in: [[Mikhail Shifman]] (ed.), _[[The Many Faces of the Superworld]]_, pp. 388-397 (2000) ([arXiv:hep-th/9907061](https://arxiv.org/abs/hep-th/9907061), [doi:10.1142/9789812793850_0023](https://doi.org/10.1142/9789812793850_0023))

* Justin R. David, Avinash Dhar, Gautam Mandal, _Probing Type I' String Theory Using D0 and D4-Branes_, Phys. Lett. B415 (1997) 135-143 ([arXiv:hep-th/9707132](https://arxiv.org/abs/hep-th/9707132))

Type I' on [[toroidal orbifold|toroidal]] [[orientifolds]] with [[ADE-singularities]] ([[duality in string theory|dual]] to [[heterotic M-theory on ADE-orbifolds]]):

* {#BergmanRodriguezGomez12} [[Oren Bergman]], Diego Rodriguez-Gomez, _5d quivers and their $AdS_6$ duals_, JHEP07 (2012) 171 ([arxiv:1206.3503](https://arxiv.org/abs/1206.3503))

### Phenomenology
 {#Phenomenology}

Type I [[string phenomenology]] and discussion of [[GUT]]s based on [[SO(32)]] type I strings (see also at [heterotic phenomenology](https://ncatlab.org/nlab/show/heterotic+string+theory#ReferencesPhenomenology)):

* {#MMRB86} H.S. Mani, A. Mukherjee, R. Ramachandran, A.P. Balachandran, _Embedding of $SU(5)$ GUT in $SO(32)$ superstring theories_, Nuclear Physics B Volume 263, Issues 3–4, 27 January 1986, Pages 621-628 (arXiv:10.1016/0550-3213(86)90277-4)

* {#IbanezMunozRigolin98} [[Luis Ibáñez]], C. Muñoz, S. Rigolin, _Aspects of Type I String Phenomenology_, Nucl.Phys. B553 (1999) 43-80 ([arXiv:hep-ph/9812397](https://arxiv.org/abs/hep-ph/9812397))

* {#Dudas00} Emilian Dudas, _Theory and Phenomenology of Type I strings and M-theory_, Class. Quant. Grav.17:R41-R116, 2000 ([arXiv:hep-ph/0006190](https://arxiv.org/abs/hep-ph/0006190))

* {#Yamatsu17} Naoki Yamatsu, _String-Inspired Special Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1  ([arXiv:1708.02078](https://arxiv.org/abs/1708.02078), [doi:10.1093/ptep/ptx135](https://doi.org/10.1093/ptep/ptx135))

### Duality

Discussion of [[duality in string theory|duality]] with [[heterotic string theory]] includes the following.

The original conjecture is due to

* {#Witten05} [[Edward Witten]], section 5 of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126 (1995) ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

More details are then in

* {#PolchinskiWitten96} [[Joseph Polchinski]], [[Edward Witten]], _Evidence for Heterotic - Type I String Duality_, Nucl.Phys.B460:525-540,1996 ([arXiv:hep-th/9510169](http://arxiv.org/abs/hep-th/9510169))

### Geometric engineering of $D=6$ $\mathcal{N}=(1,0)$ SCFT

On [[D=6 N=(1,0) SCFTs]] via [[geometric engineering of QFT|geometric engineering]] on [[M5-branes]]/[[NS5-branes]] at D-, E-type [[ADE-singularities]], notably from [[M-theory on S1/G_HW times H/G_ADE]], hence from [[orbifolds]] of [[type I' string theory]] (see at [half NS5-brane](NS5-brane#NSHalfBranes)):

* Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* Kantaro Ohmori, Hiroyuki Shimizu, _$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs_, J. High Energ. Phys. (2016) 2016: 24 ([arXiv:1509.03195](https://arxiv.org/abs/1509.03195))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_, JHEP05 (2019)203 ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

* Ibrahima Bah, Achilleas Passias, [[Alessandro Tomasiello]], _$AdS_5$ compactifications with punctures in massive IIA supergravity_, JHEP11 (2017)050 ([arXiv:1704.07389](https://arxiv.org/abs/1704.07389))

[[!redirects type I superstring]]
[[!redirects type I superstring theory]]

[[!redirects type I' string theory]]


type II geometry

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

#Contents#
* table of contents
{:toc}

## Idea

_Type II geometry_ (often _doubled geometry_ as in "[[double field theory]]") is to [[Riemannian geometry]] as [[generalized complex geometry]] is to [[complex geometry]].

Where the latter is the [[geometry]] induced by [[reduction of the structure group]] of the [[generalized tangent bundle]] of an even dimensional [[manifold]] along the inclusion $U(d,d) \to O(2d,2d)$ of the indefinite [[unitary group]] into the [[orthogonal group]], type II geometry is the geometry induced by reduction along the inclusion of the product of [[orthogonal group|orthogonal groups]]

$$
  O(n) \times O(n) \to O(n,n)
  \,,
$$

which is the inclusion of the [[maximal compact subgroup]] into the [[Narain group]].

This notion takes its name from the fact that it describes a good bit of the geometry of [[type II supergravity]].

## Definition

The definition of type II geometry proceeds in direct analogy with that of [[Riemannian geometry]] in terms of [[orthogonal structure]]/[[vielbein]] fields on the tangent bundle, generalized here to the [[generalized tangent bundle]]:

1. [As a fiberwise metric on the generalized tangent bundle](#FiberwiseMetric)

1. [By reduction of the structure group of the generalized tangent bundle](#ReductionOfStructureGroup)

### By fiberwise metric on the generalized tangent bundle
 {#FiberwiseMetric}

(...)

### By reduction of the generalized tangent bundle
 {#ReductionOfStructureGroup}

We discuss how a type II geometry is the [[reduction of the structure group]] of the [[generalized tangent bundle]] along the inclusion $O(d) \times O(d) \to O(d,d)$.

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

Consider  the [[Lie group]] inclusion
$$
  \mathrm{O}(d) \times \mathrm{O}(d)
  \to
  \mathrm{O}(d,d)
$$

of those [[orthogonal group|orthogonal transformations]], that preserve the positive definite part
or the negative definite part of the [[bilinear form]] of signature $(d,d)$, respectively.

If $\mathrm{O}(d,d)$ is presented as the group of $2d \times 2d$-[[matrix|matrices]] that
preserve the [[bilinear form]] given by the $2d \times 2d$-matrix
$$
  \eta
  \coloneqq
  \left(
     \array{
       0 & \mathrm{id}_d
       \\
       \mathrm{id}_d & 0
     }
  \right)
$$

then this inclusion sends a pair $(A_+, A_-)$ of [[orthogonal group|orthogonal]] $n \times n$-matrices
to the matrix
$$
  (A_+ , A_-)
    \mapsto
  \frac{1}{\sqrt{2}}
  \left(
    \array{
	   A_+ + A_- & A_+ - A_-
	   \\
	   A_+ - A_- & A_+ + A_-
     }
  \right)
  \,.
$$

=--

This inclusion of [[Lie groups]] induces the corresponding morphism of
[[smooth infinity-groupoid|smooth]] [[moduli stacks]]
of [[principal bundles]]

$$
  \mathbf{TypeII}
  :
  \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d))
  \to
  \mathbf{B} \mathrm{O}(d,d)
  \,.
$$

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

There is a [[fiber sequence]] of [[smooth infinity-groupoid|smooth stacks]]

  $$
      O(d) \backslash O(d,d) / O(d)
	 \to
  \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d))
     \stackrel{\mathbf{TypeII}}{\to}
    \mathbf{B} \mathrm{O}(d,d)
  \,,
  $$

where the fiber on the left is the [[coset space]] of the [[action]] of $O(d) \times O(d)$ on $O(d,d)$.

=--

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

There is a canonical embedding
$$
  \mathrm{GL}(d) \hookrightarrow \mathrm{O}(d,d)
$$

of the [[general linear group]].

In the above matrix presentation this is given by sending

$$
  a
  \mapsto
  \left(
     \array{
	    a & 0
		\\
		0 & a^{-T}
      }
  \right)
  \,,
$$

where in the bottom right corner we have the [[transpose matrix|transpose]] of the inverse matrix of the invertble
matrix $a$.

=--

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

Under inclusion of def. \ref{InclusionForTypeIIGeometry}, the
[[tangent bundle]] of a $d$-[[dimension|dimensional]]
[[manifold]] $X$ defines an $\mathrm{O}(d,d)$-[[cocycle]]

$$
  T X \oplus T^* X
  :

    X
     \stackrel{T X}{\to}
    \mathbf{B}\mathrm{GL}(d)
      \stackrel{}{\to}
    \mathbf{B} \mathrm{O}(d,d)
  \,.
$$

The [[vector bundle]] canonically associated to this composite cocycles may
canonically be identified with
the [[direct sum]] vector bundle $T X \oplus T^* X$, and so we will
refer to this cocycle by these symbols, as indicated.
This is also called the **[[generalized tangent bundle]]** of $X$.

=--

Therefore we may canonically consider the groupoid of
$T X \oplus T^* X$-twisted $\mathbf{TypeII}$-structures,
according to the general notion of [[twisted differential c-structures]].

More generally, instead of $E = T X \oplus T^* X$ one considers bundle [[extensions]] $E$ of the form

$$
  T^* X \to E \to T X
  \,.
$$

These may have structure froups in $O(n,n)$ but not in the inclusion $GL(n) \hookrightarrow O(n,n)$. For more on this see the section _[Geometric and non-geometric type II geometries](#GeometricAndNonGeometric)_ below. Accordingly, in all of the following $T X \oplus T^* X$ could be replaced by a more general extension $E$.

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

A **type II generalized vielbein** on a [[smooth manifold]] $X$ is a diagram

$$
  \array{
    X &&\stackrel{\widetilde(T X \oplus T^* X)}{\to}&& \mathbf{B}(O(n) \times O(n))
    \\
    & {}_{\mathllap{T X \oplus T^* X}}\searrow &\swArrow_{E}& \swarrow_{\mathrlap{\mathbf{TypeII}}}
    \\
    && \mathbf{B} O(n,n)
  }
$$

in $\mathbf{H} = $ [[Smooth∞Grpd]], hence a cocycle in the smooth [[twisted cohomology]]

$$
  E
   \in
  \mathbf{TypeII}Struc(X)
  \coloneqq
  \mathbf{H}_{/\mathbf{B} O(n,n)}(T X \oplus T^* X, \mathbf{TypeII})
  \,.
$$

=--

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

  The [[groupoid]] $\mathbf{TypeII}\mathrm{Struc}(X)$
  is that of "generalized vielbein fields" on $X$, as considered for instance
  around equation (2.24) of ([GMPW](#GMPW))
  (there only locally, but the globalization is evident).

  In particular, its set of equivalence classes is the set of
  type-II generalized geometry structures on $X$.

=--

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

Over a local [[coordinate chart]] $\mathbb{R}^d \simeq U_i \hookrightarrow X$, the most general such generalized vielbein
(hence the most general $\mathrm{O}(d,d)$-valued function)
may be parameterized as
$$
  E =
  \frac{1}{2}
  \left(
    \array{
	  (e_+ + e_-) + (e_+^{-T} - e_-^{-T})B & (e_+^{-T} - e_-^{-T})
	  \\
	  (e_+ - e_-) - (e_+^{-T} + e_-^{-T})B & (e_+^{-T} + e_-^{-T})
     }
  \right)
  \,,
$$

where $e_+, e_- \in C^\infty(U_i, \mathrm{O}(d))$ are thought of as two
[[vielbein|ordinary vielbein]] fields, and where $B$ is any smooth skew-symmetric $n \times n$-matrix valued function on
$\mathbb{R}^d \simeq U_i$.

By an $\mathrm{O}(d) \times \mathrm{O}(d)$-[[gauge transformation]] this can always be brought
into a form where $e_+ = e_- =: \tfrac{1}{2}e$ such that
$$
  E =
  \left(
    \array{
	  e & 0
	  \\
	  - e^{-T}B  & e^{-T}
	}
  \right)
  \,.
$$

The corresponding "generalized metric" over $U_i$ is

$$
  E^T
  E
  =
  \left(
    \array{
	  e^T & B e^{-1}
	  \\
	  0  & e^{-1}
    }
  \right)
  \left(
    \array{
	  e & 0
	  \\
	  - e^{-T}B  & e^{-T}
    }
  \right)
  =
  \left(
    \array{
	  g - B g^{-1} B & B g^{-1}
	  \\
	  - g^{-1} B  & g^{-1}
     }
  \right)
  \,,
$$

where

$$
  g \coloneqq e^T e
$$

is the [[metric]] (over $\mathbb{R}^q \simeq U_i$ a smooth function with values in symmetric $n \times n$-matrices)
given by the [[vielbein|ordinary vielbein]] $e$.

=--

### Geometric and "non-geometric" type II geometries
 {#GeometricAndNonGeometric}

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

An element in $O(d,d)$ which in the canonical matrix presentation is of the block form

$$
  e^\omega \coloneqq \left(
    \array{
      1_d & 0
      \\
      \omega &  1_d
    }
  \right)
$$

is called a **$B$-transform**. An element of the block form

$$
  e^\beta \coloneqq \left(
    \array{
      1_d & \beta
      \\
      0 &  1_d
    }
  \right)
$$

is called a **$\beta$-transform**. The [[subgroup]]

$$
  G_{geom}(d) \hookrightarrow O(d,d)
$$

generated by $Gl(d) \hookrightarrow O(d,d)$ and the B-transforms, hence that of matrices with vaishing top right block is called the **geometric subgroup** (e.g. [GMPW, p.5](#GMPW)).

A type II background where the structure group of the [[generalized tangent bundle]] is not in the inclusion of the geometric subgroup is often called a **non-geometric background** (e.g. [GMPW, section 5](#GMPW)).

=--

## Application in type II supergravity

The [[target space]] geometry for [[type II superstrings]]
in the NS-NS sector , [[type II supergravity]], is naturally
encoded by type II geometry.

...

## Related concepts

* [[exceptional generalized geometry]]

* [[non-geometric string theory vacua]]

* [[T-folds]]

* [[twisted smooth cohomology in string theory]]

## References

### General

The appearance of type II geometry in [[type II supergravity]]/[[type II string theory]] is discussed for instance in

* Ian Ellwood, _NS-NS fluxes in Hitchin's generalized geometry_ ([arXiv:hep-th/0612100](http://arxiv.org/abs/hep-th/0612100))

* {#GMPW} [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Daniel Waldram]], _T-duality, Generalized Geometry and Non-Geometric Backgrounds_ ([arXiv:0807.4527](http://arxiv.org/abs/0807.4527))

The genuine reformulation of type II supergravity as a $(O(d)\times O(d) \hookrightarrow O(d,d))$-gauge/gravity theory is in

* André Coimbra, Charles Strickland-Constable, [[Daniel Waldram]], _Supergravity as Generalised Geometry I: Type II Theories_ ([arXiv:1107.1733](http://arxiv.org/abs/1107.1733))

In

* [[Chris Hull]], _Generalised Geometry for M-Theory_, JHEP 0707:079,2007 ([arXiv:hep-th/0701203](http://arxiv.org/abs/hep-th/0701203))

the geometry of the reduction $O(d) \times O(d) \to O(d,d)$ was referred to as "type I geometry", with "type II geometry" instead referring to further [[U-duality]] group extensions, discussed at _[[exceptional generalized geometry]]_.

See also for [[para-Hermitian manifolds]]:

* Vincenzo Emilio Marotta, [[Richard J. Szabo]], *Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality*,  Reviews in Mathematical Physics **33** 09 (2021)  2150031 [[arXiv:1910.09997](https://arxiv.org/abs/1910.09997), [doi:10.1142/S0129055X21500318](https://doi.org/10.1142/S0129055X21500318)]

### Doubled super-geometry
  {#ReferencesDoubledSupergeometry}

Much of the discussion of type II geometry has in fact been purely bosonic, ignoring the [[super-geometry]] of [[supergravity]].

In contrast, discussion combining [[doubled geometry]] with [[super-geometry]] includes the following:

* Machiko Hatsuda, Kiyoshi Kamimura, [[Warren Siegel]], _Superspace with manifest T-duality from type II superstring_, J. High Energ. Phys. (2014) 2014: 39 ([arXiv:1403.3887](https://arxiv.org/abs/1403.3887))

* {#Bandos15} [[Igor Bandos]], _Superstring in doubled superspace_, Physics Letters B Volume 751, 17 December 2015, Pages 408-412 ([arXiv:1507.07779](https://arxiv.org/abs/1507.07779))

* {#Cederwall16} [[Martin Cederwall]], _Double supergeometry_, J. High Energ. Phys. (2016) 155 ([arXiv:1603.04684](https://arxiv.org/abs/1603.04684))

* Bojan Nikolić, Branislav Sazdović, _T-dualization of type II superstring theory in double space_, Eur. Phys. J. C77 (2017) no.3, 197 ([arXiv:1505.06044](https://arxiv.org/abs/1505.06044))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 6 of: _[[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))

* Bojan Nikolić, Branislav Sazdović, _Advantage of the second-order formalism in double space T-dualization of type II superstring_ ([arXiv:1907.03571](https://arxiv.org/abs/1907.03571))

See also the references on the corresponding [[super-geometry]]-enhancement of [[exceptional generalized geometry]]: _[Super-exceptional geometry -- References](exceptional+generalized+geometry#SuperExceptionalGeometryReferences)_

[[!redirects type II geometries]]

[[!redirects double geometry]]
[[!redirects doubled geometry]]

[[!redirects double geometries]]
[[!redirects doubled geometries]]

[[!redirects double supergeometry]]
[[!redirects double supergeometries]]
[[!redirects double super-geometry]]
[[!redirects double super-geometries]]

[[!redirects doubled supergeometry]]
[[!redirects doubled supergeometries]]
[[!redirects doubled super-geometry]]
[[!redirects doubled super-geometries]]


type II string theory

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

#Contents#
* table of contents
{:toc}

## Idea

Perturbative [[string theory]] is defined in terms of certain classes of 2d [[CFT]]s. Depending on which class that is, one speaks of different _types_ of string theory.

* In _type II string theory_ the CFTs in question are $(1,1)$-supersymmetric and defined on [[orientation|oriented]] [[worldsheet]]s;

* In _[[heterotic string theory]]_ the CFTs in question are $(0,1)$-supersymmetric and defined on [[orientation|oriented]] [[worldsheet]]s;

* In ...

The Type II string theory referred to here is defined on spaces in [[Lorentzian geometry]]. It is thought that Type II (and the other string theories) can also be defined in spaces with other metric signatures, such as [[type II* string theory]]. See [[non-Lorentzian type II string theory]].

## Properties

### Effective QFT

The [[effective quantum field theory]] of type II string theory containts --besides [[type II supergravity]] -- the [[self-dual higher gauge theory]] of [[RR-fields]] and [[Kalb-Ramond field]]s.

### Anomaly cancellation

Apart from the [[Weyl anomaly]], which cancels for 10-dimensional [[target space]]s, the [[action functional]] of the [[string]]-[[sigma-model]] also in general has an _[anomalous action functional](http://ncatlab.org/nlab/show/quantum+anomaly#AnomalousActionFunctional)_ , for two reasons:

1. The [[holonomy|higher holonomy]] of the higher [[background gauge field]]s is in general not a function, but a [[section]] of a [[line bundle]];

1. The fermionic [[path integral]] over the [[worldsheet]]-[[spinor]]s of the [[superstring]] produces as section of a [[Pfaffian line bundle]].

In order for the action functional to be well-defined, the [[tensor product]] of these different anomaly [[line bundle]]s over the bosonic [[configuration space]] must have trivial class (as [[connection on a bundle|bundles with connection]], even). This gives rise to various further anomaly cancellation conditions:

For the open [[type II string]] the condition is known as the [[Freed-Witten anomaly cancellation]] condition: it says that the restriction of the [[B-field]] to any [[D-brane]] must consistute the twist of a [[twisted spin^c structure]] on the brane.

A more detailed analysis of these type II anomalies is in ([DFMI](#DFMI)) and ([DFMII](#DFMII)).

### Background fields and orientifolding

[[!include string theory and cohomology theory -- table]]

* [[intersecting D-brane models]]

### Dualities

Some [[dualities in string theory]] involving [[type II string theory]]:

#### Duality with heterotic string theory

See _[[duality between type II and heterotic string theory]]_.

#### Duality with M-theory

See _[[duality between type IIA string theory and M-theory]]_.

#### Duality with F-theory

See _[[F-theory]]_.

### Holographic dual

By a [[holographic principle]] realized in this case as [[AdS/CFT correspondence]] (see the references there), type II string theory is supposed to be dual to 4-dimensional [[super Yang-Mills theory]].

### Partition function and elliptic genus

[[!include genera and partition functions - table]]

## Related concepts

* [[string theory]]

  * [[heterotic string theory]]

    * [[Green-Schwarz mechanism]]

    * [[dual heterotic string theory]]

    * [[Witten genus]], [[string orientation of tmf]]

  * **type II string theory**

    * [[massive type IIA string theory]]

    [[Diaconescu-Moore-Witten anomaly]]

    [[type II geometry]]

    [[elliptic genus]]

    4d [[super Yang-Mills theory]]

  * **[[type I string theory]]**

  * [[type 0 string theory]]

* [[landscape of string theory vacua]]

## References

### General

Physics textbook accounts include

* [[Joseph Polchinski]], _[[String theory]]_, volume II, section 10

* [[Eric D'Hoker]], _String theory -- lecture 7: Free superstrings_ , in part 3 of

  [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,
[[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

A comprehensive discussion of the ([[differential cohomology|differential]]) [[cohomology|cohomological]] nature of general type II/type I [[orientifold]] backgrounds is in

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

with details in

* {#Freed} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_ ([[FreedESI2012.pdf:file]])

* {#DistlerFreedMooreII} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_, Surveys in Differential Geometry, Volume 15 (2010) ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581), [doi:10.4310/SDG.2010.v15.n1.a4](http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4))

Related lecture notes / slides include

* [[Jacques Distler]], _Orientifolds and Twisted KR-Theory_ (2008) ([pdf](http://www.perimeterinstitute.ca/pdf/files/731c5f3a-928f-453a-b569-db5c574d2a6c.pdf))

* [[Daniel Freed]], _Dirac charge quantiation, K-theory, and orientifolds_, talk at a workshop _Mathematical methods in general relativity and quantum field theories_, November, 2009 ([pdf](http://www.ma.utexas.edu/users/dafr/paris_nt.pdf))

* [[Greg Moore]], _The RR-charge of an orientifold_ ([ppt](http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt))

### $(p,q)$-Strings and S-duality

* [[John Schwarz]], opening talk at [Strings 2013](http://strings2013.sogang.ac.kr/), ([pdf](http://strings2013.sogang.ac.kr/design/default/data/john_schwarz.pdf))

### Quantum anomalies

Discussion of type II [[quantum anomalies]] is in

* [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))
 {#DFMI}

* [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_ ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581))
  {#DFMII}

An exposition is at

* [[Dan Freed]], _Lectures on K-theory and orientifolds_ (2012) ([pdf](http://www.ma.utexas.edu/users/dafr/ESI.pdf))

### Classical solutions / vacua

Description of type II backgrounds in terms of [[generalized complex geometry]]/[[Courant Lie 2-algebroids]] is in

* [[Mariana Grana]], Francesco Orsi, _$N=1$ vacua in Exceptional Generalized Geometry_ ([arXiv:1105.4855](http://arxiv.org/abs/1105.4855))

### Holography

A [[holographic principle|holographic]] description of type II by [[higher dimensional Chern-Simons theory]] is discussed in

* Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))

[[!redirects type II superstring]]
[[!redirects type II string]]
[[!redirects type II superstrings]]
[[!redirects type II strings]]

[[!redirects type II superstring theory]]

[[!redirects type II String Theory]]

[[!redirects type IIA string theory]]
[[!redirects type IIB string theory]]
[[!redirects Type IIB string theory]]

[[!redirects type IIA superstring theory]]
[[!redirects type IIB superstring theory]]

[[!redirects type IIA superstring]]
[[!redirects type IIB superstring]]
[[!redirects type IIA superstrings]]
[[!redirects type IIB superstrings]]


type II supergravity

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

#Contents#
* table of contents
{:toc}

## Idea

_Type II supergravity_ is a [[supergravity]] in [[dimension]] 10 which is the low-energy [[effective quantum field theory]] underlying [[type II string theory]].

## Related concepts

* [[massive type IIA supergravity]]

* [[type II supergravity Lie 2-algebra]]

* [[type II geometry]], [[generalized complex geometry]]

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

* [[11-dimensional supergravity]]

* [[heterotic supergravity]],

* [[7-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

* [[duality between M-theory and type IIA string theory]]

* [string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

## References
 {#References}

### General

Original constructions (mostly of [[type IIA supergravity|type IIA]]):

* F. Giani, M. Pernici, *$N=2$ supergravity in ten dimensions*, Phys. Rev. D **30** (1984) 325 [[doi:10.1103/PhysRevD.30.325](https://doi.org/10.1103/PhysRevD.30.325)]

* M. Huq, M. A. Namazie, *Kaluza-Klein supergravity in ten dimensions*, Classical and Quantum Gravity **2** 3 (1985) [[doi:10.1088/0264-9381/2/3/007](https://iopscience.iop.org/article/10.1088/0264-9381/2/3/007)]

* I. C. G. Campbell, [[Peter C. West]] *$N = 2$, $D = 10$ non-chiral supergravity and its spontaneous compactification*, Nuclear Physics B
**243** 1 (1984) 112-124 [doi:10.1016/0550-3213(84)90388-2]

See also the general references at _[[supergravity]]_.

* [[Joseph Polchinski]], chapter 12.1 of *[[String Theory]]* Vol 2: Superstring and beyond_, Cambridge Monographs on Mathematical Physics (1998)

Construction of [[type IIA supergravity]] via [[KK-compactification]] from [[11d supergravity]]:

* M. Huq, M. A. Namazie, _Kaluza-Klein Supergravity In Ten Dimensions_, Class. Quantum Grav. 2 (1985) 293 ([spire:196711](https://inspirehep.net/literature/196711))

* {#DFGT08} [[Riccardo D'Auria]], [[Pietro Fré]], [[Pietro Grassi]], Trigiante, _Pure Spinor Superstrings on Generic type IIA Supergravity Backgrounds_ [arXiv:0803.1703](https://arxiv.org/abs/0803.1703)

### Explicit self-dual formulations

[[!include self-duality for pregeometric RR-fields -- references]]

### Solutions and BPS states

Disucssion of [[black hole]] solutions (see also at _[[black holes in string theory]]_) includes

* U. Gran, [[Jan Gutowski]], [[George Papadopoulos]], _IIB black hole horizons with five-form flux and extended supersymmetry_, JHEP 1109:047,2011 ([arXiv:1104.2908](http://arxiv.org/abs/1104.2908))

Discussion of [[black branes]] and [[BPS states]] for type II supergravity includes

* Andrew Callister, [[Douglas Smith]], _Topological BPS charges in 10 and 11-dimensional supergravity_, Phys. Rev. D78:065042,2008 ([arXiv:0712.3235](http://arxiv.org/abs/0712.3235))

* Andrew Callister, [[Douglas Smith]], _Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA_, Phys.Rev.D80:125035,2009 ([arXiv:0907.3614](http://arxiv.org/abs/0907.3614))

* Andrew Callister, _Topological BPS charges in 10- and 11-dimensional supergravity_, thesis 2010 ([spire](http://inspirehep.net/record/1221591?ln=en))

* A. A. Golubtsova, V.D. Ivashchuk, _BPS branes in 10 and 11 dimensional supergravity_, talk at DIAS 2013 ([pdf slides](http://theor.jinr.ru/~diastp/summer13/lectures/Golubtsova.pdf))

Discussion of *asymptotic* [[de Sitter spacetimes]] from time-dependent [[KK-compactification]] of [[type II supergravity]]:

* Paul Marconnet, [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*,  J. High Energ. Phys. **2023** 33 (2023) [[arXiv:2210.10813](https://arxiv.org/abs/2210.10813), doi:10.1007/JHEP01(2023)033]

reviewed in:

* [[Dimitrios Tsimpis]], *Universal accelerating cosmologies from 10d supergravity*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Tsimpis2023)]

### In terms of (exceptional) generalized complex geometry
 {#InTermsOfGeneralizedComplexGeometryReferences}

A relation of the [[U-duality]] symmetry to [[generalized complex geometry]] is discussed in

* André Coimbra, Charles Strickland-Constable, [[Daniel Waldram]], _Supergravity as Generalised Geometry I: Type II Theories_ ([arXiv:1107.1733](http://arxiv.org/abs/1107.1733))

* Paulo Pires Pacheco, Daniel Waldram, _M-theory, exceptional generalised geometry and superpotentials_ ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

A thesis reviewing some aspects is

* Nicholas Houston, _Supergravity and Generalized Geometry_ Thesis (2010) ([pdf](https://workspace.imperial.ac.uk/theoreticalphysics/Public/MSc/Dissertations/2010/Nicholas%20Houston%20Dissertation.pdf))

### Higher curvature corrections

On [[higher curvature corrections]]:

* [[James Liu]], [[Ruben Minasian]], Raffaele Savelli, Andreas Schachner, *Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles* $[$[arXiv:2205.11530](https://arxiv.org/abs/2205.11530)$]$

### Via double field theory

Discusdion of type IIA and IIB supergravities via [[double field theory]]:

* Olaf Hohm, Seung Ki Kwak, Barton Zwiebach, _Unification of Type II Strings and T-duality_ ([arXiv:1106.5452](https://arxiv.org/abs/1106.5452))

and,  to the full order in fermions, in

* Imtak Jeon, Kanghoon Lee, Jeong-Hyuck Park, Yoonji Suh, _Stringy Unification of Type IIA and IIB Supergravities under N=2 D=10 Supersymmetric Double Field Theory_ ([arxiv:1210.5078](https://arxiv.org/abs/1210.5078))

Comprehensive discussion 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))

* [[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))

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

[[!redirects type II supergravities]]

[[!redirects D=10 type II supergravity]]
[[!redirects D=10 type II supergravities]]

[[!redirects type IIA supergravity]]
[[!redirects type IIB supergravity]]

[[!redirects 10d type IIA supergravity]]
[[!redirects 10d type IIB supergravity]]


type II supergravity Lie 2-algebra

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## Idea

A [[super L-infinity algebra|super Lie 2-algebra]] [[infinity-Lie algebra cohomology|extension]] of the [[super Poincare Lie algebra]] in $D = 10$ for $N=2$ supersymmetry (as in [[type II supergravity]]).

The [[Chevalley-Eilenberg algebra]] of the $\mathfrak{sugra}_{typeII}$ (defining it) is that of the $D = 10$ $N = 2$ [[super Poincaré Lie algebra]] generated from $\{E, \Psi\}$ with one further generator $B$ in degree 2 and [[differential]] given schematically by

$$
  d B = \big( \overline{\Psi} \Gamma^a \Gamma_{11} \Psi\big) \wedge  E_a
  \,.
$$

This is ([CAIB 99, equation (6.3)](#CAIB99)) for type IIA with $N = (1,1)$ and in ([Sakaguchi 99 (2.4) and (2.25)](#Sakaguchi99)) for type IIB with $N = (2,0)$.

It also makes sense to write $\mathfrak{string}_{IIA}$ and $\mathfrak{string}_{IIB}$ for these. See also at _[[string Lie 2-algebra]]_.

## Properties

The [[cocycles]] of the exceptional [[∞-Lie algebra cohomology]] of $\mathfrak{sugra}_{typeII}$ induce the [[Green-Schwarz action functional]] [[schreiber:infinity-Wess-Zumino-Witten theory]]-terms for the [[D-branes]] of [[type II superstring theory]] ([CAIB 99, section 6.1](#CAIB99) for IIA and ([Sakaguchi 99, section 2](#Sakaguchi99)) for IIB):

[[!include brane scan]]

## Related concepts

* [[string Lie 2-algebra]]

[[supergravity Lie 6-algebra]] $\to$ [[supergravity Lie 3-algebra]] $\to$ [[super Poincaré Lie algebra]]

* [[type II supersymmetry algebra]]

## References

The [[type IIA supergravity]] Lie 2-algebra and its [[D-brane]]-[[Green-Schwarz action functional]]-type cocycles are discussed in section 6 of

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

The [[type IIB supergravity]] Lie 2-algebra and its [[D-brane]]-[[Green-Schwarz action functional]]-type cocycles are discussed in section 2 of

* Makoto Sakaguchi, _IIB-Branes and New Spacetime Superalgebras_,  JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](http://arxiv.org/abs/hep-th/9909143))
 {#Sakaguchi99}

The formulation in [[super L-infinity algebra]] theory is in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_


type II supersymmetry algebra

[[!redirects type II super Lie algebra]]

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#Contents#
* table of contents
{:toc}

## Idea

A [[super Lie algebra]] which is a [polyvector](super+Poincare+Lie+algebra#PolyvectorExtensions) [[Lie algebra extension|extension]] of the [[super Poincaré Lie algebra]] ([[supersymmetry]]) in $D = 9+1$ for  $N=(2,0)$ or $N = (1,1)$ [[supersymmetry]] by [[charges]] corresponding to the [[type II superstring]], the [[D-branes]] and the [[NS5-brane]] ("[[extended supersymmetry]]").

## Related concepts

* [[M-theory super Lie algebra]]

* [[type II supergravity Lie 2-algebra]]

* [[BPS state]]

## References

The original articles include

* [[Paul Townsend]], _$p$-Brane Democracy_ ([arXiv:hep-th/9507048](http://arxiv.org/abs/hep-th/9507048))

* [[Chris Hull]], section 2 of _Gravitational Duality, Branes and Charges_, Nucl.Phys. B509 (1998) 216-251 ([arXiv:hep-th/9705162](http://arxiv.org/abs/hep-th/9705162))

Detailed derivation from [[conserved currents]] of the [[Green-Schwarz super p-brane sigma models]] for the D-branes is in

* Hanno Hammer, _Topological Extensions of Noether Charge Algebras carried by D-p-branes_, Nucl.Phys. B521 (1998) 503-546 ([arXiv:hep-th/9711009](http://arxiv.org/abs/hep-th/9711009))

Relation to the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$ is discussed in

* {#BergshoessProeyen00} [[Eric Bergshoeff]], [[Antoine Van Proeyen]], _The many faces of $OSp(1|32)$_, Class. Quantum Grav. 17 (2000) 3277–3303 ([arXiv:hep-th/0003261](http://arxiv.org/abs/hep-th/0003261))

See also

* Tatsuya Ueno, _BPS States in 10+2 Dimensions_, JHEP 0012:006,2000 ([arXiv:hep-th/9909007](http://arxiv.org/abs/hep-th/9909007))

[[!redirects type II supersymmetry algebras]]


type II* string theory

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## Idea

Type $II^\ast$ string theory is supposed to be the [[T-duality|T-dual]] to [[type II string theory]] under [[KK-compactification]] on a [[timelike curve|timelike]] [[circle]].

## References

General:

* [[Christopher Hull]]. *Timelike T-duality, de Sitter space, large N gauge theories and topological field theory*, JHEP 07 (1998) 021 [[hep-th/9806146](https://arxiv.org/abs/hep-th/9806146)\]

* [[Christopher Hull]]. *Duality and the signature of space-time*, JHEP 9811 (1998) 017 [[hep-th/9807127](https://arxiv.org/abs/hep-th/9807127)\]

* [[Christopher Hull]] and [[Ramzi Khuri]], *Branes, times and dualities*, Nucl.Phys. B536 (1998) 219 [[hep-th/9808069](https://arxiv.org/abs/hep-th/9808069)\]

* [[Christopher Hull]]. *Duality and Strings, Space and Time*. (1999) [[hep-th/9911080](https://arxiv.org/abs/hep-th/9911080)\]

* [[Hitoshi Nishino]], [[Jim Gates]]. *The $*$-Report*. (1999). [[hep-th/9908136](https://arxiv.org/abs/hep-th/9908136)\]

On the relation between the [[super Lie algebra|super Lie algebras]] of II and $II^\ast$ and the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$:

* [[Eric Bergshoeff]], [[Antoine Van Proeyen]]. *The many faces of $OSp(1\vert 32)$*. (2000). [[hep-th/0003261](https://arxiv.org/abs/hep-th/0003261)\]

* [[Eric Bergshoeff]], [[Antoine Van Proeyen]]. *The unifying superalgebra $OSp(1\vert 32)$*. (2000). [[hep-th/0010194](https://arxiv.org/abs/hep-th/0010194)\]

The supposed branes in $II^\ast$, [[E-branes]], are discussed in:

* Somdatta Bhattacharya, Sudipta Mukherji, Shibaji Roy. *On the effective action of a space-like brane*. (2003). [[hep-th/0308069](https://arxiv.org/abs/hep-th/0308069)\]

That the boundary state for the Dirichlet [[S-brane]] constructed is a solution of type $II^\ast$ rather than type II string theory is in:

* Bruno Durin, [[Boris Pioline]]. *Aspects of Dirichlet S-branes*. (2005) [[hep-th/0507059](https://arxiv.org/abs/hep-th/0507059)\]

The suggestion that $II^\ast$ and its would-be [[M-theory]] lift may play a role in interpreting M-theory as defined on the base space of an [[octonionic projective space]] $\mathbb{O}P^2$ bundle with structure group the [[exceptional Lie group]] $F_4$ is in Remark 5, Section 2.3 of:

* [[Hisham Sati]]. *On the geometry of the supermultiplet in M-theory*. (2009). [[arXiv:0909.4737](https://arxiv.org/abs/0909.4737)\]

On the relation of twisted $\mathcal{N}=2$ algebra, [Para-quaternionic Kähler manifolds](https://ncatlab.org/nlab/show/para-quaternionic+structure), and compactifications of $II^\ast$ on [[Calabi-Yau variety|Calabi-Yau's]]:

* Maxime Médevielle, [[Thomas Mohaupt]], and G. Pope. *Type-II Calabi-Yau compactifications, T-duality and special geometry in general spacetime signature*. J. High Energ. Phys. 2022, 48 (2022). [[doi:10.1007/JHEP02\(2022\)048](https://doi.org/10.1007/JHEP02(2022)048)\]


type of classes

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#### Type theory
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\tableofcontents

## Idea

In [[dependent type theory]], the [[type]] of *[[classes]]* relative to a [[Tarski universe]] $U$ is the [[function type]] $V_U \to \mathrm{Prop}_U$ from the [[material set theory|material]] [[cumulative hierarchy]] [[higher inductive type]] $V_U$ constructed from $U$ to the [[type of propositions|type $\mathrm{Prop}_U$ of]] $U$-small [[h-propositions]].

## Definition

Let $(U, T)$ be a [[Tarski universe]]. The [[cumulative hierarchy]] relative to $U$ is the [[higher inductive type]] $V_U$ generated by the following:

* A [[function]]
$$\mathrm{set}:\left(\sum_{A:U} T(A) \to V_U\right) \to V_U$$

* An [[equivalence in homotopy type theory|equivalence]]
$$\mathrm{idVchar}:\prod_{A:U} \prod_{B:U} \prod_{f:T(A) \to V_U} \prod_{g:T(B) \to V_U} (\mathrm{set}(A, f) =_{V_U} \mathrm{set}(B, g)) \simeq \left(\left(\prod_{a:T(A)} \left| \sum_{b:T(B)} f(a) =_{V_U} g(b)  \right|\right) \times \left(\prod_{b:T(B)} \left| \sum_{a:T(A)} f(a) =_{V_U} g(b)  \right|\right) \right)$$

* A set-truncator
$$\tau_0:\mathrm{isSet}(V_U)$$

The type of all $U$-small propositions is defined as

$\mathrm{Prop}_U \coloneqq \sum_{A:U} \mathrm{isProp}(A)$

The **type of classes** relative to $U$ is the function type $V_U \to \mathrm{Prop}_U$, and a **[[class]]** relative to $U$ is just a function $C:V_U \to \mathrm{Prop}_U$.

The type of classes provides a general model of [[material class theory]].

## See also

* [[class]]

* [[class theory]]

* [[category of classes]]

## References

The type of classes is defined in section 10.5.3 of:

* [[Univalent Foundations Project]], *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]*, Institute for Advanced Study (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]


type of entire relations

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\tableofcontents

## Idea

In [[predicative]] [[constructive mathematics|constructive]] [[set theory]], there is a hierarchy of possible axioms, starting from the existence of [[function sets]] and ending with the existence of [[power sets]] representing impredicativity. Intermediate is the existence of the set of [[entire relations]], which is given by the [[axiom of fullness]] in set theory.

In [[predicative]] [[constructive mathematics|constructive]] [[dependent type theory]], there is a similar hierarchy of possible axioms, which begins with the existence of [[dependent product types]] (corresponding to function sets in set theory) and ends with the existence of both [[dependent product types]] and the [[type of all propositions]] (corresponding to power sets in set theory), representing impredicativity. Intermediate to both that is the existence of the type of all entire relations between two types $A$ and $B$, which is the type theoretic equivalent of the axiom of fullness.

Types of entire relations are important in [[dependent type theory]] because, assuming [[propositional truncations]], [[quotient sets]], and [[natural numbers]], they allow the user to define the type of [[multivalued Cauchy real numbers]] - which are equivalent to [[Dedekind real numbers]] - predicatively (i.e. without [[power sets]]).

## Definition

Recall that an [[entire relation]] between types $A$ and $B$ in type theory is a [[family]] $x:A, y:B \vdash R(x, y)$ such that for all $x:A$

* for all $y:A$, $R(x, y)$ is an [[h-proposition]], and

* there exists an $y:A$ such that $R(x, y)$.

$$\mathrm{isEntire}(x:A.y:B.R(x, y)) \equiv \prod_{x:A} \left(\prod_{u:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)$$

If there exists a [[type of propositions]] $\mathrm{Prop}$, then the type of entire relations is given by the type

$$\sum_{R:A \times B \to \mathrm{Prop}} \mathrm{isEntire}(R)$$

where

$$\mathrm{isEntire}(R) \equiv \prod_{x:A} \exists y:B.R(x, y)$$

for Russell types of propositions and

$$\mathrm{isEntire}(R) \equiv \prod_{x:A} \exists y:B.\mathrm{El}(R(x, y))$$

for Tarski types of propositions. Otherwise, we could use [[inference rules]] to directly define the type of all entire relations between types $A$ and $B$:

### A la Russell

A la Russell, the type of all entire relations is given by the following rules:

Formation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}$$

Elimination rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash R(x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.R(x, y))}$$

Computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash (\mathrm{toElem}_{A, B, R}(p))(x, y) \equiv R(x, y) \; \mathrm{type}}$$

* Judgmental computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}$$

* Typal computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}$$

Uniqueness rules:

* Judgmental uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}$$

* Typal uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, R}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}$$

Extensionality rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} R(x, y) \simeq S(x, y)}$$

### Strictly a la Tarski

Strictly a la Tarski, the type of all entire relations is given by the following rules:

Formation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}$$

Elimination rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash \mathrm{El}(R, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y))}$$

Computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash \mathrm{El}(\mathrm{toElem}_{A, B, R}(p), x, y) \equiv R(x, y) \; \mathrm{type}}$$

* Judgmental computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}$$

* Typal computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p)) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}$$

Uniqueness rules:

* Judgmental uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}$$

* Typal uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}$$

Extensionality rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} \mathrm{El}(R, x, y) \simeq \mathrm{El}(S, x, y)}$$

### Weakly a la Tarski

Weakly a la Tarski, the type of all entire relations is given by the following rules:

Formation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{EntRel}(A, B) \; \mathrm{type}}$$

Introduction rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y)}{\Gamma \vdash \mathrm{toElem}_{A, B, R}:\mathrm{isEntire}(x:A.y:B.R(x, y)) \to \mathrm{EntRel}(A, B)}$$

Elimination rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma, x:A, y:B \vdash \mathrm{El}(R, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelwitn}(R):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y))}$$

Computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma, x:A, y:B \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p, x, y):\mathrm{El}(\mathrm{toElem}_{A, B, R}(p), x, y) \simeq R(x, y)}$$

* Judgmental computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash (\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)))(\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p))) \equiv p:\mathrm{isEntire}(x:A.y:B.R(x, y))}$$

* Typal computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A, y:B \vdash R(x, y) \quad \Gamma \vdash p:\mathrm{isEntire}(x:A.y:B.R(x, y))}{\Gamma \vdash \beta_{\mathrm{EntRel}(A, B)}^{\mathrm{entrelwitn}, R}:(\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)))(\mathrm{entrelwitn}(\mathrm{toElem}_{A, B, R}(p))) =_{\mathrm{isEntire}(x:A.y:B.R(x, y))} p}$$

where the equivalence
$$\mathrm{congform}_{\mathrm{isEntire}}(\beta_{\mathrm{EntRel}(A, B)}^{\mathrm{El}, R}(p)):\mathrm{isEntire}(x:A.y:B.\mathrm{El}(R, x, y)) \simeq \mathrm{isEntire}(x:A.y:B.R(x, y))$$
is provided from the typal computation rules for $\mathrm{isEntire}(x:A.y:B.R(x, y))$.

Uniqueness rules:

* Judgmental uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) \equiv R:\mathrm{EntRel}(A, B)}$$

* Typal uniqueness rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B)}{\Gamma \vdash \eta_{\mathrm{EntRel}(A, B)}(R):\mathrm{toElem}_{A, B, \mathrm{El}(R)}(\mathrm{entrelwitn}(R)) =_{\mathrm{EntRel}(A, B)} R}$$

Extensionality rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash R:\mathrm{EntRel}(A, B) \quad \Gamma \vdash S:\mathrm{EntRel}(A, B)}{\Gamma \vdash \mathrm{entrelext}_{A, B}(R, S):(R =_{\mathrm{EntRel}(A, B)} S) \simeq \prod_{x:A} \prod_{y:B} \mathrm{El}(R, x, y) \simeq \mathrm{El}(S, x, y)}$$

## Related concepts

* [[axiom of fullness]]

* [[type of propositions]]

[[!redirects type of entire relations]]
[[!redirects types of entire relations]]

[[!redirects weak type of entire relations]]
[[!redirects weak types of entire relations]]

[[!redirects strict type of entire relations]]
[[!redirects strict types of entire relations]]

[[!redirects type of all entire relations]]
[[!redirects types of all entire relations]]

[[!redirects weak type of all entire relations]]
[[!redirects weak types of all entire relations]]

[[!redirects strict type of all entire relations]]
[[!redirects strict types of all entire relations]]