Tarski universe

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[intensional type theory]] and [[homotopy type theory]], there are typically two ways to construct a "type of small types". One way is by [[Russell universes]], where an [[element]] $A:U$ of a [[type universe]] $U$ is literally a [[type]] $A \; \mathrm{type}$. The alternative is by Tarski universes, where elements $A:U$ are not literally types, but rather the universe type $U$ comes with the additional structure of a [[type family]] $T$, such that the [[dependent type]] $T(A)$ represents the actual type.
Usually universes are defined in [[intensional type theory]] and [[homotopy type theory]] using [[derivations]] and [[inference rules]], such as the type reflection rule
$$\frac{\Gamma \vdash A:U}{\Gamma \vdash T(A) \; \mathrm{type}}$$
for Tarski universes. Furthermore, universes are usually defined as being closed under all type formers inside the type theory, regardless of the type theory.

However, this approach of defining Tarski universes means that the notion of "Tarski universe" differs from type theory to type theory. By this approach, a Tarski universe in the bare intensional [[Martin-Löf type theory]], which has [[identity types]], [[dependent product types]], [[dependent sum types]], an [[empty type]], a [[unit type]], a [[booleans type]], and a [[natural numbers type]], would not be considered a Tarski universe in [[homotopy type theory]], since it is missing important type formers present in homotopy type theory, such as internal [[pushout types]], a [[torus type]], [[propositional truncations]], [[W-types]], and [[localizations]]. On the other hand, one could consider an even more spartan type theory which doesn't even have the [[unit type]], let alone the [[empty type]], the [[natural numbers type]], and the [[booleans type]], whose Tarski universes are thus consisting of only internal [[identity types]], [[dependent product types]], [[dependent sum types]].

All three notions of Tarski universe described above are in fact definable in all three versions of intensional type theory mentioned above, but Tarski universes are usually described as having, apart from addiitonal internal Tarski universes, exactly the internal type formers that the external type theory has. One usually doesn't consider Tarski universes with an internal [[James construction type]] in bare intensional Martin-Löf type theory, even though a Tarski universe closed under James constructions is definable in Martin-Löf type theory. On the other hand, in [[homotopy type theory]] the name "Tarski universe" isn't usually used for a universe only closed under [[identity types]], [[dependent product types]], [[dependent sum types]].

From a global perspective, however, all of these types should be considered Tarski universes, since they model some notion of type of small types, even though the small types in the universe do not behave as the external types in the type theory do. And that is how we approach Tarski universes in this article: Tarski universes are a very general notion encompassing all of the above notions of Tarski universe and more, and one could describe more specific versions of Tarski universes by adding additional structure and axioms to Tarski universes, in the same way one adds additional structure and axioms to an [[abelian group]] to get a [[ring]], a [[commutative ring]], a $R$-[[module]], a $R$-algebra, and an $R$-[[commutative algebra]]. We could then study in general Tarski universes and their properties as we do for abelian groups and rings. This requires a shift in the point of view of what Tarski universes are: Tarski universes are actual [[mathematical structure]] in intensional type theory, rather than being something in the [[foundations of mathematics]] to merely address [[size]] issues or do mathematics in.

## Definition

### With a family of types

A **Tarski universe** or **universe à la Tarski** is simply a type $U$ with a type family $T$ whose dependent types $T(A)$ are indexed by elements $A:U$. The elements $A:U$ are usually called **$U$-small types**, or **small types** for short, and $T(A)$ is the **type of elements of $A$**.

### With a type and a function

A **Tarski universe** or **universe à la Tarski** is a type $U$ of all $U$-small types with a type $T'$ of all elements, and a function $\mathrm{typeOf}:T' \to U$ which gets the associated type for every element $a:T'$.

### Equivalence of definitions

From one direction, the individual dependent types $T(A)$ could be defined as the [[fiber type]] of $\mathrm{typeOf}$ at $A$
$$T(A) \coloneqq \mathrm{fiber}_{T', U}(\mathrm{typeOf},A) \equiv \sum_{t:T'} \mathrm{typeOf}(t) =_U A$$

From the other direction, the entire type of elements $T'$ is just the [[dependent sum type]] of all the type reflections of small types
$$T' \coloneqq \sum_{A:U} T(A)$$

We shall be using the first definition throughout this article, but everything could be translated into the second definition of a Tarski universe.

## Tarski universes as models of type theory

The definitions above are already enough to build an internal model of dependent type theory inside the Tarski universe:

* The type $A \; \mathrm{type}$ is represented by the small type $A:U$
* The element $a:A$ is represented by the element $a:T(A)$ for small type $A:U$.
* The type family $B$ indexed by the type $A \; \mathrm{type}$ is represented by the function $B:T(A) \to U$ for small type $A:U$
* The dependent type $a:A \vdash B \; \mathrm{type}$ is represented by the section $B(a):U$ for element $a:T(A)$ and small type $A:U$.
* The section $a:A \vdash b:B$ is represented by the element $b:T(B(a))$ for element $a:T(A)$ and small type $A:U$.

## Properties

### Univalence

There are two notions of equivalence in a Tarski universe $(U, T)$, equality $A =_U B$ and equivalence $T(A) \simeq T(B)$. By the properties of the [[identity type]], there is a canonical [[transport]] dependent function
$$\mathrm{transport}^T:\prod_{A:U} \prod_{B:U} (A =_U B) \to T(A) \simeq T(B)$$
A Tarski universe is a **univalent Tarski universe** if for all elements $A:U$ and $B:U$ the function $\mathrm{transport}^T(A, B)$ is an [[equivalence of types]]
$$\prod_{A:U} \prod_{B:U} \mathrm{isEquiv}(\mathrm{transport}^T(A, B))$$

This is the extensionality principle for any Tarski universe $(U, T)$.

Equivalently, by the [[fundamental theorem of identity types]], a Tarski universe is a **univalent Tarski universe** if it comes with a dependent function
$$t:\prod_{X:U} C(X, X, \mathrm{id}_X) \vdash J(t):\prod_{A:U} \prod_{B:U} \prod_{R:T(A) \simeq T(B)} C(A, B, R)$$
and a dependent function
$$t:\prod_{X:U} C(X, X, \mathrm{id}_X) \vdash \beta(t):\prod_{A:U} J(t, A, A, \mathrm{id}_A) =_{C(A, A, \mathrm{id}_A)} t(A)$$

When the Tarski universe is closed under [[identity types]], [[dependent sum types]], and [[dependent product types]] (see below for more on closure of Tarski universes under type formers), one is able to internalize the [[type of equivalences]] in the universe as $A \simeq_U B$. It can be proven that $T(A \simeq_U B)$ is equivalent to $T(A) \simeq T(B)$, and that the definition of univalence using transport is equivalent to the usual definition given by the internal [[type of equivalences]] and the canonical function $\mathrm{idtoequiv}$. See at [Univalence axiom#For Tarski universes](https://ncatlab.org/nlab/show/univalence+axiom#for_tarski_universes) for more details on this.

#### Construction of univalent Tarski universes via a higher inductive type

Let $(U, T)$ be a Tarski universe. Then one could construct a univalent Tarski universe as the [[higher inductive type]] $(U', T')$ generated by

* a function $F:U \to U'$
* a function $G(A):T(A) \to T'(A)$ for all elements $a:U$
* a dependent function
$$\mathrm{ua}_{U'}:\prod_{A:U} \prod_{B:U} \mathrm{isEquiv}(\mathrm{transport}^{T'}(A, B))$$

### Regularity and product-regularity

A univalent Tarski universe $(U, T)$ is **regular** if it is closed under [[dependent sum types]]: namely, for all $U$-small types $A:U$ and $U$-small type families $B:T(A) \to U$, the dependent sum type $\sum_{x:T(A)} T(B(x))$ is [[essentially small type|essentially $U$-small]].

$$\mathrm{ax}_\Sigma:\sum_{\Sigma:\prod_{A:U} (T(A) \to U) \to U} \prod_{A:U} \prod_{B:T(A) \to U} T(\Sigma(A, B)) \simeq \sum_{x:T(A)} T(B(x))$$

Regular univalent Tarski universes are the type theoretic equivalent of [[regular cardinals]] in [[set theory]].

A univalent Tarski universe $(U, T)$ is **product-regular** if it is closed under [[dependent product types]]: namely, for all $U$-small types $A:U$ and $U$-small type families $B:T(A) \to U$, the dependent product type $\sum_{x:T(A)} T(B(x))$ is [[essentially small type|essentially $U$-small]].

$$\mathrm{ax}_\Pi:\sum_{\Pi:\prod_{A:U} (T(A) \to U) \to U} \prod_{A:U} \prod_{B:T(A) \to U} T(\Pi(A, B)) \simeq \prod_{x:T(A)} T(B(x))$$

Product-regular univalent Tarski universes are the type theoretic equivalent of [[product-regular cardinals]] in [[set theory]].

Since product-regularity implies that [[function types]] between $U$-small types are essentially $U$-small, and regularity implies that [[subtypes]] of essentially $U$-small types are essentially $U$-small, this implies that in a regular and product-regular universe, [[equivalence types]] between $U$-small types are essentially $U$-small, because equivalence types are [[subtypes]] of function types. In combination with [[univalence]], regularity and product-regularity implies that the [[identity types]] between $U$-small types are also essentially $U$-small.

### Closure under identity types

A univalent Tarski universe $(U, T)$ is **closed under identity types** if for all $U$-small types $A:U$ and elements $x:T(A)$ and $y:T(A)$, the [[identity type]] $x =_{T(A)} y$ is [[essentially small type|essentially $U$-small]].

$$\mathrm{ax}_\mathrm{Id}:\sum_{\mathrm{Id}:\prod_{A:U} (T(A) \times T(A)) \to U} \prod_{A:U} \prod_{x:T(A)} \prod_{y:T(A)} T(\mathrm{Id}(A, x, y)) \simeq (x =_{T(A)} y)$$

Regularity and closure under identity types implies that the universe is closed under [[pullbacks]].

### Singletons

A [[univalent Tarski universe]] satisfies the **axiom of singletons** if it is closed under the (weak) [[unit type]]: if the unit type $\mathbb{1}$ is essentially $U$-small. The axiom of singletons comes from the [[formation rule]], the [[introduction rules]], and the [[dependent universal property of the natural numbers]], which are represented by the following elements:

* formation rules: $\mathbb{1}:U$

* introduction rules: $0:T(\mathbb{1})$

and either an universal property or a dependent universal property

* universal property:
$$\mathrm{up}_\mathbb{1}^U:\prod_{C:U} \prod_{c_0:T(C)} \exists!c:T(\mathbb{1}) \to T(C).(c(0) =_{T(C)} c_0)$$

* dependent universal property:
$$\mathrm{dup}_\mathbb{1}^U:\prod_{C:T(\mathbb{1}) \to U} \prod_{c_0:T(C(0))} \exists!c:\prod_{x:T(\mathbb{1})} T(C(x)).(c(0) =_{T(C(0))} c_0)$$
where $\exists!x:A.B(x)$ is the [[uniqueness quantifier]] for the [[type family]] $x:A \vdash B(x)$.

By using [[dependent sum types]], these can be combined into a single element of the following types

$$\mathrm{axsingl}_U:\sum_{\mathbb{1}:U} \sum_{0:T(\mathbb{1})} \prod_{C:U} \prod_{c_0:T(C)} \exists!c:T(\mathbb{1}) \to T(C).(c(0) =_{T(C)} c_0)$$

$$\mathrm{axsingl}_U:\sum_{\mathbb{1}:U} \sum_{0:T(\mathbb{1})} \prod_{C:\mathbb{1} \to U} \prod_{c_0:T(C(0))} \exists!c:\prod_{x:T(\mathbb{1})} T(C(x)).(c(0) =_{T(C(0))} c_0)$$

Alternatively, since the notion of [[truth]] could be defined using the type of $U$-small [[contractible types]] $\sum_{A:U} \mathrm{isContr}(T(A))$, the axiom of singletons could be represented as a resizing axiom, similar to propositional resizing.

A [[univalent Tarski universe]] $(U, T)$ satisfies the **axiom of singletons** if it is closed under the type of all $U$-small [[contractible types]]: namely, the dependent sum type $\sum_{A:U} \mathrm{isContr}(T(A))$ is [[essentially small type|essentially $U$-small]]:

$$\mathrm{axSingl}_U:\sum_{\mathbb{1}:U} T(\mathbb{1}) \simeq \sum_{A:U} \mathrm{isContr}(T(A))$$

### Empty type

A [[univalent Tarski universe]] satisfies the **[[axiom of empty set|axiom of empty type]]** if it is closed under the (weak) [[empty type]]: if the empty type $\emptyset$ is essentially $U$-small. The axiom of empty type comes from the [[formation rule]] and the dependent universal property of the empty type, which are represented by the following elements:

* formation rules: $\emptyset:U$

and either an universal property or a dependent universal property

$$\mathrm{up}_\emptyset^U:\prod_{C:U} \mathrm{isContr}\left(T(\emptyset) \to T(C)\right)$$

* dependent universal property:
$$\mathrm{up}_\emptyset^U:\prod_{C:T(\emptyset) \to U} \mathrm{isContr}\left(\prod_{x:T(\emptyset)} T(C(x))\right)$$

By using [[dependent sum types]], these can be combined into a single element of the following types

$$\mathrm{axempty}_U:\sum_{\emptyset:U} \prod_{C:U} \mathrm{isContr}\left(T(\emptyset) \to T(C)\right)$$

$$\mathrm{axempty}_U:\sum_{\emptyset:U} \prod_{C:T(\emptyset) \to U} \mathrm{isContr}\left(\prod_{x:T(\emptyset)} T(C(x))\right)$$

Alternatively, since [[falsehood]] or the empty proposition could be defined using the type of $U$-small propositions $\sum_{A:U} \mathrm{isProp}(T(A))$, the axiom of empty set could be represented as a resizing axiom, similar to propositional resizing.

A [[univalent Tarski universe]] $(U, T)$ satisfies the **[[axiom of empty set|axiom of empty type]]** if it is closed under the empty proposition or falsehood: namely, the [[dependent product type]] $\prod_{A:\mathrm{Prop}}_U T(A)$ is [[essentially small type|essentially $U$-small]]:
$$\mathrm{axempty}_U:\sum_{\emptyset:U} T(\emptyset) \simeq \prod_{A:\mathrm{Prop}_U} T(A)$$
where
$$\mathrm{Prop}_U \equiv \sum_{A:U} \mathrm{isProp}(T(A))$$

Product regularity and the axiom of empty type imply the axiom of singletons, because the dependent universal property of the empty set states that for every type family $C:T(\mathbb{0}) \to U$ the dependent function type $\prod_{x:T(\mathbb{0})} T(C(x))$ is a singleton, and product regularity implies that $\prod_{x:T(\mathbb{0})} T(C(x))$ is essentially $U$-small.

### Propositional resizing

A [[univalent Tarski universe]] $(U, T)$ satisfies **[[propositional resizing]]** if it is closed under the [[type of propositions|type of all $U$-small propositions]]: namely, the dependent sum type $\sum_{A:U} \mathrm{isProp}(T(A))$ is [[essentially small type|essentially $U$-small]]:
$$\mathrm{propresize}_U:\sum_{\Omega:U} T(\Omega) \simeq \sum_{A:U} \mathrm{isProp}(T(A))$$

This is a version internal to the universe $U$ of having a [[type of all propositions]] in the type theory itself. While propositional resizing implies that the universe is [[impredicative]], it does not imply that the type theory as a whole is impredicative; the latter requires an actual [[type of all propositions]] in the type theory.

Product regularity and propositional resizing imply the axiom of empty set, because propositional resizing implies that $\mathrm{Prop}_U$ is essentially $U$-small, and then product regularity implies that $\prod_{A:\mathrm{Prop}_U} T(A)$ is essentially $U$-small.

### Infinity

A [[univalent Tarski universe]] satisfies the **[[axiom of infinity]]** if it is closed under the (weak) [[natural numbers type]]: if the natural numbers type $\mathbb{N}$ is essentially $U$-small. The axiom of infinity comes from the [[formation rule]], the [[introduction rules]], and the [[dependent universal property of the natural numbers]], which are represented by the following elements:

* formation rules: $\mathbb{N}:U$

* introduction rules: $0:T(\mathbb{N})$ and $s:T(\mathbb{N}) \to T(\mathbb{N})$

and either an universal property or a dependent universal property

* universal property:
$$\mathrm{up}_\mathbb{N}^U::\prod_{C:U} \prod_{c_0:T(C)} \prod_{c_s:T(C) \to T(C)} \exists!c:T(\mathbb{N}) \to T(C).(f(0) =_{T(C)} c_0) \times \prod_{n:T(\mathbb{N})} c(s(n)) =_{T(C)} c_s(c(n))$$

* dependent universal property:
$$\mathrm{dup}_\mathbb{N}^U:\prod_{C:\mathbb{N} \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:\mathbb{N}} T(C(x)) \to T(C(s(x)))} \exists!c:\prod_{x:\mathbb{N}} T(C(x)).(c(0) =_{T(C(0))} c_0) \times \prod_{x:\mathbb{N}} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$
where $\exists!x:A.B(x)$ is the [[uniqueness quantifier]] for the [[type family]] $x:A \vdash B(x)$.

By using [[dependent sum types]], these can be combined into a single element of the following types:

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:U} \prod_{c_0:T(C)} \prod_{c_s:T(C) \to T(C)} \exists!c:T(\mathbb{N}) \to T(C).(f(0) =_{T(C)} c_0) \times \prod_{n:T(\mathbb{N})} c(s(n)) =_{T(C)} c_s(c(n))$$

or

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:T(\mathbb{N}) \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:T(\mathbb{N})} T(C(x)) \to T(C(s(x)))} \exists!c:\prod_{x:T(\mathbb{N})} T(C(x)).(c(0) =_{T(C(0))} c_0) \times \prod_{x:T(\mathbb{N})} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$

Alternatively, since the [[type of finite types|type of all $U$-small finite types]] and [[set truncations]] could be defined using the type of $U$-small propositions $\sum_{A:U} \mathrm{isProp}(T(A))$, the axiom of infinity could be represented as a resizing axiom, similar to propositional resizing.

A [[univalent Tarski universe]] $(U, T)$ satisfies the **[[axiom of infinity]]** if it is closed under the [[set truncation]] of the [[type of finite types|type of all $U$-small finite types]]: namely, the type $\left[\sum_{A:U} \mathrm{isFinite}(T(A))\right]_0$ is [[essentially small type|essentially $U$-small]]:
$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} T(\mathbb{N}) \simeq \left[\sum_{A:U} \mathrm{isFinite}(T(A))\right]_0$$

where

$$
\mathrm{isFinite}(A) \equiv
\begin{array}{c}
\prod_{S:(A \to \mathrm{Prop}_U) \to \mathrm{Prop}_U} (((\lambda x:A.\bot) \in S) \times \prod_{P:A \to \mathrm{Prop}_U} \prod_{Q:A \to \mathrm{Prop}_U} (P \in S) \\
\times (\exists!x:A.x \in Q) \times (P \cap Q =_{A \to \mathrm{Prop}_U} \lambda x:A.\bot) \to (P \cup Q \in S)) \to ((\lambda x:A.\top) \in S)
\end{array}
$$

and

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

The membership relation and the subtype operations used above are defined in the nLab article on [[subtypes]].

### Replacement

A [[univalent Tarski universe]] satisfies the __[[axiom of replacement]]__ if for every [[essentially small type|essentially $U$-small]] type $A$, every [[locally small type|locally $U$-small]] type $B$, and every function $f:A \to B$, the [[image]] of $f$, $\mathrm{im}(f)$, is essentially $U$-small.

Equivalently, given types $A$ and $B$, there is a function

$$\mathrm{axrepl}_U^{A, B}:\left(\sum_{C:U} T(C) \simeq A\right) \times \left(\sum_{R:B \times B \to U} \prod_{a:B} \prod_{b:B} T(R(a, b)) \simeq (a =_B b)\right) \to \left(\sum_{I:(A \to B) \to U} \prod_{f:A \to B} T(I(f)) \simeq \mathrm{im}(f)\right)$$

### Choice

A [[univalent Tarski universe]] satisfies the __[[axiom of choice]]__ if for every $U$-small type $A:U$ and every $U$-small type family $B:T(A) \to U$, and for every every $U$-small type family $C:\prod_{x:T(A)} T(B(x)) \to U$, if $T(A)$ is a set and each $T(B(x))$ is a set for all $x:T(A)$, and each $T(C(x, y))$ is a proposition for all $x:T(A)$ and $y:T(B(x))$, then if for all $x:A$ there merely exists a $y:T(B(x))$ such that $T(C(x, y))$, then there merely exists a dependent function $g:\prod_{x:T(A)} T(B(x))$ such that for all $x:T(A)$, $T(C(x, g(x)))$.

$$\mathrm{choice}_U:
\begin{array}{c}
\prod_{A:U} \prod_{B:T(A) \to U} \prod_{C:\prod_{x:T(A)} T(B(x)) \to U} \\
\left(\mathrm{isSet}(T(A)) \times \prod_{x:T(A)} \left(\mathrm{isSet}(T(B(x))) \times \prod_{y:T(B(x))} \mathrm{isProp}(T(C(x, y)))\right)\right) \\
\to \forall x:T(A).\exists y:T(B(x)).T(C(x, y)) \to \exists g:\prod_{x:T(A)} T(B(x)).\forall x:T(A).T(C(x, g(x)))
\end{array}
$$

### Choice operators and existence

A [[choice operator]] on a type is a function from its [[propositional truncation]] to the type itself, and represents the concept that if there exists an element of the set (i.e. the propositional truncation has an element), then the set itself has an element chosen by the choice operator. A Tarski universe satisfies the **[[type-theoretic axiom of existence]]** if every $U$-small type in the Tarski universe has a choice operator, represented by the following dependent function
$$\varepsilon_U:\prod_{A:U} \vert T(A) \vert \to T(A)$$
$U$ satisfying the type-theoretic axiom of existence implies that $U$ satisfies [[axiom K]] or [[UIP]]. If $U$ is also univalent, then it is an [[h-groupoid]].

### Closure of Tarski universes under other type formers

There are many ways of defining type formers internally in a universe:

* by an equivalence or definitional equality with an existing global type former for the entire type theory.

* by translating the rules for the type former into internal structure on the universe - the [[formation rules]], the [[introduction rules]], and the dependent [[universal property]] of the structure.

* by using the [[universal properties]] corresponding to the [[categorical semantics]] of the types

Using a definitional equality with an existing global type former for each type former results in a **strict Tarski universe**, while using equivalences for each type former results in a **weak Tarski universe**. There are also various Tarski universes where some type formers are strictly closed, and some type formers are only weakly closed, resulting in Tarski universes which are intermediate between strict Tarski universes and weak Tarski universes.

### Tarski universes with all propositions

A Tarski universe $(U, T)$ **has all propositions** if given a type $A$ with a family of identities $x:A, y:A \vdash \tau_{-1}(x, y):x =_A y$, the universe $U$ comes with the structure of an element $A_U:U$ and an equivalence $\delta_A:T(A_U) \simeq A$.

The rules for this condition on Tarski universes is as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash \tau_{-1}(x, y):x =_A y}{\Gamma \vdash A_U:U} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash \tau_{-1}(x, y):x =_A y}{\Gamma \vdash \delta_A:T(A_U) \simeq A}$$

Given a Tarski universe which has all propositions, the type $\sum_{A:U} \mathrm{isProp}(T(A))$ is the [[type of all propositions]]. This is a form of strong [[impredicativity]] in the type theory, as given a type $A$, the type $A \to \sum_{A:U} \mathrm{isProp}(T(A))$ is the [[power set]] on $A$.

## Tarski subuniverses

Let $(U, T_U)$ be a Tarski universe. A Tarski universe $(V, T_V)$ is a **Tarski subuniverse** of $(U, T_U)$ if it comes with an [[embedding]] $t:V \hookrightarrow U$ and a dependent function
$$p:\prod_{a:V} T_U(t(a)) \simeq T_V(a)$$

This is equivalent to a family of Tarski universes $(U(i), T(i))$ indexed by elements $i:\mathbb{2}$ of the [[interval preorder]] $(\mathbb{2}, \leq_\mathbb{2})$, which comes with

* a dependent function
$$t:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} (a \leq_\mathbb{2} b) \to (U(a) \hookrightarrow U(b))$$

* a dependent function
$$q:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \prod_{A:U(a)} T(a)(A) \simeq T(b)(t(a, b)(p)(A))$$
indicating that lifting each small type results in a type equivalent to the original small type.

## Modalities and comodalities

A **[[modal operator]]** on a Tarski universe $(U, T)$ is just an [[endofunction]] $L:U \to U$. Given a Tarski universe, the **modal unit family** is a family of functions
$$\eta:\prod_{L:U \to U}\prod_{A:U} T(A) \to T(L(A))$$
and the **comodal counit family** is a family of functions
$$\epsilon:\prod_{L:U \to U}\prod_{A:U} T(L(A)) \to T(A)$$
Given a modal operator $L:U \to U$, $\eta(L)$ is called the **modal unit** of $L$ and $\epsilon(L)$ is called the **comodal counit** of $L$. A small type $A:U$ is **$L$-modal** if the function
$$\eta(L)(A):T(A) \to T(L(A))$$
is an [[equivalence of types]], and **$L$-comodal** if the function
$$\epsilon(L)(A):T(L(A)) \to T(A)$$
is an equivalence of types.

A **[[modality]]** is...

...and a **[[comodality]]** is...

### Reflective and coreflective Tarski subuniverses

(Section under construction see [[reflective subuniverse]] for the [[Russell universe]] variant for the time being)

## Essentially small Tarski universes

A Tarski universe $(V, T_V)$ inside of a Tarski universe $(U, T_U)$ consists of an element $V:U$ and a function $T_V:T_U(V) \to U$.

Given a Tarski universe $(U, T_U)$, a Tarski universe $(V, T_V)$ is essentially $U$-small if it comes with

* an element $V':U$

* a function $T_{V'}:T_U(V') \to U$

* an equivalence
$$\mathrm{smallType}:V \simeq T_U(V')$$

* a dependent function
$$\mathrm{smallFamily}:\prod_{A:V} T_V(A) \simeq T_U(T_{V'}(\mathrm{smallType}(A)))$$

Equivalently, the entire structure could be regarded as a family of Tarski universes $(U(a), T(a))$ indexed by elements $a:\mathbb{2}$ of the [[interval preorder]] $(\mathbb{2}, \leq_\mathbb{2})$, which comes with

* a dependent function
$$U':\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to U(b)$$

* a dependent function
$$T':\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \left(T(b)(U'(a, b, p)) \to U(b)\right)$$

* an dependent function
$$\mathrm{smallType}:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \left(U(a) \simeq T(b)(U'(a, b, p))\right)$$

* a dependent function
$$\mathrm{smallFamily}:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \prod_{A:U(a)} T(a)(A) \simeq T(b)(T'(a, b, p)(\mathrm{smallType}(a, b, p)(A)))$$

## Hierarchy of Tarski universes

Let $P$ be a [[preorder]]. Then, a $P$-indexed hierarchy of Tarski universes is a type family $U$ such that for all indices $a:P$, there is a Tarski universe $(U(a), T(a))$, such that for all indices $a:P$ and $b:P$ and witnesses $p:a \leq_P b$, the Tarski universe $(U(a), T(a))$ is an essentially $U(b)$-small Tarski subuniverse of the Tarski universe $(U(b), T(b))$.

Expanded out, the family of Tarski universes $(U(a), T(a))$ indexed by elements $a:P$ additionally come with

* a dependent function
$$U':\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to U(b)$$

* a dependent function
$$T':\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \left(T(b)(U'(a, b, p)) \to U(b)\right)$$

* an dependent function
$$\mathrm{smallType}:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \left(U(a) \simeq T(b)(U'(a, b, p))\right)$$

* a dependent function
$$\mathrm{smallFamily}:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \prod_{A:U(a)} T(a)(A) \simeq T(b)(T'(a, b, p)(\mathrm{smallType}(a, b, p)(A)))$$

* a dependent function
$$t:\prod_{a:P} \prod_{b:P} (a \leq_P b) \to (U(a) \hookrightarrow U(b))$$

* a dependent function
$$q:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \prod_{A:U(a)} T(a)(A) \simeq T(b)(t(a, b)(p)(A))$$

Usually, hierarchies of Tarski universes are indexed by the type of [[natural numbers]]. A type theory may have multiple hierarchies of type universes.

## Examples

### Cumulative hierarchy

Any [[cumulative hierarchy]] $V$ with a [[membership relation]] $\in$ is a Tarski universe with universal type family

$$x:V \vdash \mathrm{El}(x) \coloneqq \sum_{y:V} y \in x$$

This is not a univalent universe if the von Neumann universe satisfies the set-theoretic [[axiom of extensionality]], which states that

$$\prod_{x:V} \prod_{y:V} (x =_V y) \simeq \prod_{z:V} (z \in x) \simeq (z \in y)$$

However, the axiom of extensionality implies that $V$ is an [[h-sets]], since the membership relation is an [[h-proposition]], which in turn implies that each $\mathrm{El}(x)$ is an [[h-set]].

Examples include [[ZFC]] and its large cardinal, choice-less, constructive, predicative, and non-foundational/anti-foundational variants, such as [[ZF]], [[Zermelo set theory]], [[CZF]], [[IZF]], [[Mac Lane set theory]], [[Kripke–Platek set theory]], [[Tarski-Grothendieck set theory]], [[New Foundations]], [[Mostowski set theory]], and so forth.

### Well-pointed categories

Any [[well pointed category]] $\mathcal{E}$ with [[hom-set]] type family $\mathrm{hom}(x, y)$ is a Tarski universe with universal type family

$$x:V \vdash \mathrm{El}(x) \coloneqq \mathrm{hom}(1, y)$$

where $1:\mathcal{E}$ is the [[terminal object]] and [[indecomposable object|indecomposable]] [[projective object|projective]] [[separator]] of the category $\mathcal{E}$.

Examples of this include [[ETCS]].

### Concrete categories

Every [[concrete category]] and every [[concrete dagger category]] is a Tarski universe, as given a concrete category or concrete dagger category $C$ by definition for each object $A:Ob(C)$ there is an [[h-set]] $El(A)$. This includes models of [[Mike Shulman]]'s [[SEAR]], which is a concrete [[power allegory]], as well as [[ETCS with elements]], which is a concrete [[elementary topos]].

### Other examples

The [[empty type]], [[unit type]], [[boolean domain]], and [[FinSet]] are all regular univalent Tarski universes. The [[types of propositions]] in a type theory are univalent Tarski universes: they model a dependent type theoretic model of [[propositional logic]] with [[function types]], [[product types]], [[disjunction]] [[higher inductive types]], [[empty type]], and [[unit type]].

## See also

For universes as a replacement for type judgments, see

* [[Russell universe]]

Regarding the general notion of types of small types, see

* [[type universe]]

Other mathematical structures and their univalent counterparts:

* [[preorder]], [[partial order]]
* [[precategory]], [[univalent category]]
* [[dagger precategory]], [[univalent dagger category]]
* [[locally univalent bicategory]], [[univalent bicategory]]

## References

Historical review of [[Alfred Tarski]]'s original notion in [[set theory]]:

* [[Ralf Krömer]], §6.4.5 in: *Tool and object: A history and philosophy of category theory*, Science Networks. Historical Studies **32** (2007) [[doi:10.1007/978-3-7643-7524-9](https://doi.org/10.1007/978-3-7643-7524-9)]

The terminology "universe à la Tarski" --- and now in the context of [[type universes]] for [[Martin-Löf dependent type theory]] --- is due to:

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]): p. 48 in: _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

Further discussion:

* {#Hofmann95} [[Martin Hofmann]], *Universes*, section 2.3.5 in: *Syntax and semantics of dependent types*, Chapter 2 in: _Extensional concepts in intensional type theory_, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [[ECS-LFCS-95-327](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/), [[HofmannExtensionalIntensionalTypeTheory.pdf:file]], [doi:10.1007/978-1-4471-0963-1](https://doi.org/10.1007/978-1-4471-0963-1)]

See also:

* {#Luo12} [[Zhaohui Luo]], *Notes on Universes in Type Theory*, 2012 ([pdf](http://www.cs.rhul.ac.uk/home/zhaohui/universes.pdf))

* {#Gallozzi14} [[Cesare Gallozzi]], *Constructive Set Theory from a Weak Tarski Universe*, MSc thesis (2014) ([arXiv:1411.5591](http://xxx.tau.ac.il/abs/1411.5591))

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

* {#UFP13} *Homotopy Type Theory: Univalent Foundations of Mathematics*,
The Univalent Foundations Program, Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* {#RSS17} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], _[[Modalities in homotopy type theory]]_ ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526))

[[!redirects a la Tarski]]
[[!redirects Tarski universe]]
[[!redirects Tarski universes]]
[[!redirects universe a la Tarski]]
[[!redirects universes a la Tarski]]

[[!redirects univalent Tarski universe]]
[[!redirects univalent Tarski universes]]
[[!redirects univalent universe a la Tarski]]
[[!redirects univalent universes a la Tarski]]

[[!redirects regular Tarski universe]]
[[!redirects regular Tarski universes]]
[[!redirects regular universe a la Tarski]]
[[!redirects regular universes a la Tarski]]

[[!redirects regular univalent Tarski universe]]
[[!redirects regular univalent Tarski universes]]
[[!redirects regular univalent universe a la Tarski]]
[[!redirects regular univalent universes a la Tarski]]

[[!redirects product-regular Tarski universe]]
[[!redirects product-regular Tarski universes]]
[[!redirects product-regular universe a la Tarski]]
[[!redirects product-regular universes a la Tarski]]

[[!redirects product-regular univalent Tarski universe]]
[[!redirects product-regular univalent Tarski universes]]
[[!redirects product-regular univalent universe a la Tarski]]
[[!redirects product-regular univalent universes a la Tarski]]

[[!redirects inaccessible Tarski universe]]
[[!redirects inaccessible Tarski universes]]
[[!redirects inaccessible universe a la Tarski]]
[[!redirects inaccessible universes a la Tarski]]

[[!redirects inaccessible univalent Tarski universe]]
[[!redirects inaccessible univalent Tarski universes]]
[[!redirects inaccessible univalent universe a la Tarski]]
[[!redirects inaccessible univalent universes a la Tarski]]

[[!redirects weakly Tarski]]
[[!redirects weakly a la Tarski]]
[[!redirects weak Tarski universe]]
[[!redirects weak Tarski universes]]
[[!redirects weakly Tarski universe]]
[[!redirects weakly Tarski universes]]
[[!redirects weakly a la Tarski universe]]
[[!redirects weakly a la Tarski universes]]

[[!redirects strictly Tarski]]
[[!redirects strictly a la Tarski]]
[[!redirects strict Tarski universe]]
[[!redirects strict Tarski universes]]
[[!redirects strictly Tarski universe]]
[[!redirects strictly Tarski universes]]
[[!redirects strictly a la Tarski universe]]
[[!redirects strictly a la Tarski universes]]

[[!redirects judgmentally Tarski]]
[[!redirects judgmentally a la Tarski]]
[[!redirects judgmental Tarski universe]]
[[!redirects judgmental Tarski universes]]
[[!redirects judgmentally Tarski universe]]
[[!redirects judgmentally Tarski universes]]
[[!redirects judgmentally a la Tarski universe]]
[[!redirects judgmentally a la Tarski universes]]

[[!redirects judgmentally strict Tarski universe]]
[[!redirects judgmentally strict Tarski universes]]

[[!redirects propositionally Tarski]]
[[!redirects propositionally a la Tarski]]
[[!redirects propositional Tarski universe]]
[[!redirects propositional Tarski universes]]
[[!redirects propositionally Tarski universe]]
[[!redirects propositionally Tarski universes]]
[[!redirects propositionally a la Tarski universe]]
[[!redirects propositionally a la Tarski universes]]

[[!redirects propositionally strict Tarski universe]]
[[!redirects propositionally strict Tarski universes]]

[[!redirects typally Tarski]]
[[!redirects typally a la Tarski]]
[[!redirects typal Tarski universe]]
[[!redirects typal Tarski universes]]
[[!redirects typally Tarski universe]]
[[!redirects typally Tarski universes]]
[[!redirects typally a la Tarski universe]]
[[!redirects typally a la Tarski universes]]

[[!redirects typally strict Tarski universe]]
[[!redirects typally strict Tarski universes]]

[[!redirects type with a type family]]
[[!redirects univalent type with a type family]]

[[!redirects Tarski universe hierarchy]]
[[!redirects hierarchy of Tarski universes]]

[[!redirects weak Tarski universe hierarchy]]
[[!redirects hierarchy of weak Tarski universes]]
[[!redirects weakly Tarski universe hierarchy]]
[[!redirects hierarchy of weakly Tarski universes]]

[[!redirects strict Tarski universe hierarchy]]
[[!redirects hierarchy of strict Tarski universes]]
[[!redirects strictly Tarski universe hierarchy]]
[[!redirects hierarchy of strictly Tarski universes]]


Tarski-Grothendieck set theory

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

#Contents#
* table of contents
{:toc}

## Idea

__Tarski-Grothendieck set theory__ is an axiomatic [[set theory]] whose [[axioms]] are thos of [[ZFC]] plus the axiom that every set belongs to some [[Grothendieck universe]].

This framework is standard in the works of [[Alexander Grothendieck]] and his school.

## References

The notion first appeared in:

* [[Alfred Tarski]], _On well-ordered subsets of any set_, Fundamenta Mathematicae 32 (1939) 176--183,[pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm32/fm32115.pdf)

See also

* Wikipedia, *[Tarski-Grothendieck set theory](https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory)*

[[!redirects Tarski-Grothendieck set theories]]

[[!redirects Tarski-Grothendieck set theory]]
[[!redirects Tarski-Grothendieck set theories]]

[[!redirects Tarski–Grothendieck set theory]]
[[!redirects Tarski–Grothendieck set theories]]

[[!redirects Tarski--Grothendieck set theory]]
[[!redirects Tarski--Grothendieck set theories]]

[[!redirects Tarski-Grothendieck]]
[[!redirects Tarski–Grothendieck]]
[[!redirects Tarski--Grothendieck]]


Tarski-Seidenberg theorem

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

# Contents
* table of contents
{: toc}

## Statement

+-- {: .num_theorem}
######Theorem
**(Tarski-Seidenberg)**

The [[image]] of a [[semialgebraic set]] in $R^{n+1}$ under the [[projection]] [[map]]

$$
 \pi \colon
  R^{n+1} \to R^n: (x_1, \ldots, x_{n+1}) \mapsto (x_1, \ldots, x_n)
$$

is also a semi-algebraic set.

=--

## Commentary

This powerful theorem is usually understood in the context of [[formal logic]] and [[model theory]], where one considers subsets of [[Euclidean space]] $\mathbb{R}^n$ that are definable by [[first-order logic|first-order logical]] [[formulas]] in the [[language]] of ordered rings. It says that the [[definable set|definable]] subsets are precisely semi-algebraic sets, i.e., sets defined by taking Boolean combinations of sets of the form

$$\{(x_1, \ldots, x_n) \in \mathbb{R}^n: p(x_1, \ldots, x_n) \geq 0\}$$

where $p$ is a [[polynomial]].

In logic, the result is understood as connected to a [[quantifier elimination]] result. By definition, semi-algebraic sets $X$ are definable by formulas $P$ without any quantifiers. Taking an image under a projection amounts to applying an existential quantifier:

$$\pi(X) = \pi(\{(x_1, \ldots, x_{n+1}) \in \mathbb{R}^{n+1}: P(x_1, \ldots, x_n, x_{n+1})\} = \{(x_1, \ldots, x_n) \in \mathbb{R}^n: \exists_{x_{n+1}} P(x_1, \ldots, x_n, x_{n+1})\}$$

and Tarski-Seidenberg allows us to replace $\exists_{x_{n+1}} P$ by some quantifier-free formula. By induction, it follows that any quantified formula may be replaced by a quantifier-free one.

There is a host of consequences of this result, including the fact that the theory $Th(\mathbb{R})$ in this language is decidable (in particular, [[Euclidean geometry]] is decidable), and that all [[models]] of the theory of [[real-closed field]]s are [[elementary equivalence|elementarily equivalent]].

## Extensions

The entire theory of [[o-minimal structure]]s can be viewed as a vast extrapolation stemming from the Tarski-Seidenberg theorem and its consequences.

Not all the extrapolations are based on quantifier elimination, but practically all do involve some reduction of logical complexity of formulas. An example is Wilkie's theorem on the structure of the ordered exponential field $\mathbb{R}$ (in the language $(0, 1, +, \cdot, \exp, \lt)$ where $\exp$ is interpreted as the usual [[exponential function]] $\exp(x) = e^x$):

+-- {: .num_theorem}
###### Theorem
**(Wilkie)**
Any subset of $\mathbb{R}^m$ definable in the language of ordered exponential fields can be defined by means of an existential formula

$$\exists x_{m+1}\ldots\exists x_n \, f_1(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})=\cdots= f_r(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})=0$$

where the $f_i$ are polynomials.
=--

Here the critical point is that the _complement_ of a set defined by such an existential formula is definable by another such existential formula. This is called a [[model completeness]] result, and it implies that an alternating string of quantifiers $\forall = \neg \exists \neg, \exists$ can be eliminated in favor of just existential quantifiers at the head of a defining formula.

+-- {: .num_remark}
###### Remark
It is not known whether the theory of ordered exponential fields is decidable, but it is known that such decidability follows if one assumes [[Schanuel's conjecture]].
=--


Tarun Grover

* [institute page](https://physics.ucsd.edu/Directory/Person/492)

## Selected writings

On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:

* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$

* [[Yi Zhang]], [[Tarun Grover]], [[Ashvin Vishwanath]], *General procedure for determining braiding and statistics of anyons using entanglement interferometry*, Phys. Rev. B **91** (2015) 035127 $[$[arXiv:1412.0677](https://arxiv.org/abs/1412.0677), [doi:10.1103/PhysRevB.91.035127](https://doi.org/10.1103/PhysRevB.91.035127)$]$

* [[Tarun Grover]], *Entanglement entropy and strongly correlated topological matter*,  Modern Physics Letters A **28** 05 (2013)  1330001  $[$[doi:10.1142/S0217732313300012](https://doi.org/10.1142/S0217732313300012)$]$

category: people


Tate algebra

#Contents#
* table of contents
{:toc}

## Idea

In [[analytic geometry]], the [[algebra of functions]] on the $n$-dimensional [[polydisc]] is called the _Tate algebra_ $T_n$.

The Tate algebra is a completion of the ring of [[polynomials]] on $n$ [[variables]] under the [[norm]] called the [[Gauss norm]]. Equivalently it is a subalgebra of the ring of [[formal power series]] on those whose [[coefficients]] tend to 0 in a suitable [[limit of a sequence|limit]].

## References

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

[[!redirects Tate algebras]]


Tate conjecture

See [[Hodge conjecture]] for now.

## Related concepts

* [[Hodge conjecture]]
* [[algebraic cycles]]
* [[etale cohomology]]

## References

* [[Benjamin Antieau]], _On the integral Tate conjecture for finite fields and representation  theory_, [arXiv:1504.04879](http://arxiv.org/abs/1504.04879).

* [[Gereon Quick]], _Torsion algebraic cycles and etale cobordism_, [arXiv:0911.0584](http://arxiv.org/abs/0911.0584v3).


Tate curve

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

#Contents#
* table of contents
{:toc}

## Idea

The _Tate curve_ is the [[formal neighbourhood]] of the [[nodal curve]] inside the  [[compactification|compactified]] [[moduli stack of elliptic curves]] $\mathcal{M}_{\overline{ell}}$.

Over a given base ring $R$ it is an elliptic curve over the [[power series]] $R [ [ q ] ]$ which may be thought of as the quotient of the [[multiplicative group]] by $q^{\mathbb{Z}}$.

## Properties

### Relation to complex K-theory
 {#RelationToComplexKTheory}

The Tate curve is naturally an [[oriented elliptic curve]] and as such classified by a map

$$
  Spec \left(  KU ( ( q ) ) \right) \longrightarrow \mathcal{M}^{Der}
$$

in [[derived algebraic geometry]], from the spectrum of the [[power series]] over the [[KU]] [[E-∞ ring]] to the [[derived moduli stack of derived elliptic curves]].

### Relation to Tate K-theory
 {#RelationToTateKTheory}

The [[generalized cohomology theory]] over the Tate curve by the [[Goerss-Hopkins-Miller theorem]] is [[Tate K-theory]] $KO[ [q] ]$.

This is the [[homotopy limit]] over copies $KO \wedge \{1, q, \cdots, q^n\}_+$ of [[KO]] with formal parameters adjoined.

### Relation to the Witten genus

The [[elliptic genus]] associated with the Tate curve is, according to the [above](#RelationToComplexKTheory), a [[formal power series]] of [[K-theory]] classes. This is the _[[Witten genus]]_ ([Hopkins 94](#Hopkins94)).

Physically, Witten obtained this as the [[partition function]] of the [[heterotic string]] in [[perturbation theory]] about configurations where all the [[worldsheet]] sits in a single point of [[target space]]. This limiting case of the [[worldsheet]] is the Tate curve.

Formally this is given by the homotopy groups of the [[String orientation of tmf]] $M String \to tmf$ followed by the map to [[Tate K-theory]] [above](#RelationToTateKTheory), discussed [here](tmf#MapToTateKTheory).

[[!include moduli stack of curves -- table]]

## Related concepts

* [[spin orientation of Tate K-theory]]

## References

Conceptual discussion is in

* [[Jacob Lurie]], pages 32, 33 of _[[A Survey of Elliptic Cohomology]]_.

Details are spelled out in

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], section 3.4 of _Topological modular forms with level structure_ ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394))

Basic reviews include

* Wikipedia, _[Tate curve](https://en.wikipedia.org/wiki/Tate_curve)_

See also

*  Antonios Broumas, _Congruences between the coefficients
of the Tate curve via formal groups_, Proc. Amer. Math. Soc. 128 (2000), 677-681 ([doi:10.1090/S0002-9939-99-05133-3](https://doi.org/10.1090/S0002-9939-99-05133-3))

The relation to the [[Witten genus]] originates around

* {#Hopkins94} [[Michael Hopkins]], _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

More details on this are in ([Hill-Lawson 13, appendix A](#HillLawson13)).

[[!redirects Tate curves]]


Tate diagonal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

While the [[symmetric monoidal (∞,1)-category of spectra]] is not a [[cartesian monoidal (∞,1)-category]], hence while the [[smash product of spectra]] does not admit [[diagonal]] morphisms, it turns out that for each [[prime number]] $p$ there is a analogue of the diagonal into  the $p$-fold [[smash product]]. This is the _Tate diagonal_ (def. \ref{TateDiagonal}). One way that this really does behave like a [[diagonal]] map is the construction of the [[Frobenius morphism]] on [[E-∞ rings]] (see [this def.](Frobenius+morphism#EInfinityFrobenius) and [this example](Frobenius+morphism#EInfinityFrobeniusReducesToOrdinaryFrobenius))

## Definition

For $n \in \mathbb{N}$ a [[natural number]], write $C_p \coloneqq \mathbb{Z}/n\mathbb{Z}$ for the [[cyclic group]] of order $n$.

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

For $X \in Spectra$ a [[spectrum]] and $n$ a [[natural number]], consider the [[Tate spectrum]]

$$
  (X \wedge \cdots \wedge X)^{t C_n} \in Spectra
$$

where the $n$-fold [[smash product of spectra]] is regarded as equipped with the [[∞-action]] by the [[cyclic group]]  given by cyclic [[permutation]] of smash factors. This construction canonically extends to an [[(∞,1)-functor]]

$$
  T_n
    \;\colon\;
  Spectra
    \longrightarrow
  Spectra
$$

on the [[(∞,1)-category of spectra]].

=--

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

For $p$ a [[prime number]], the [[(infinity,1)-category of (infinity,1)-functors|∞-groupoid of natural transformations]]

$$
  id_{spectra} \longrightarrow T_p
$$

from the [[identity functor]] on the [[(∞,1)-category of spectra]] to the [[(∞,1)-functor]] $T_p$ from def. \ref{TatedSmashProductOfSpectra} is equivalent to the [[stabilization|underlying space]] of $T_p$ applied to the sphere spectrum:

$$
  Hom( id_{Specta}, T_p )
  \;\simeq\;
  \Omega^\infty T_p(\mathbb{S})
    \simeq
   Hom_{Specta}(\mathbb{S}, T_p(\mathbb{S}))
  \,.
$$

([Nikolaus-Scholze 17, corollary III.1.3](#NikolausScholze17))

=--

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

For $p$ a [[prime number]], the _Tate diagonal_ is the [[natural transformation]]

$$
  \array{
     id_{Specta} &\overset{\Delta_p}{\longrightarrow}& T_p
     \\
     X &\mapsto& (X \wedge \cdots X)^{t C_p}
  }
$$

is the one which under the equivalence of prop. \ref{SpaceOfTransformationsFromIdentityToTatedSmashProduct} corresponds to the [[composition|composite]] morphism

$$
  \mathbb{S}
    \overset{}{\longrightarrow}
  \mathbb{S}^{C_p}
    \overset{}{\longrightarrow}
  \mathbb{S}^{t C_p}
$$

where the first is the [[adjunction unit|unit]] map into the [[homotopy fixed point]] spectrum, and the second is the defining one of the [[Tate spectrum]].

=--

([Nikolaus-Scholze 17, def. III.1.4](#NikolausScholze17))

## References

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_ ([arXiv:1707.01799](https://arxiv.org/abs/1707.01799))

[[!redirects Tate diagonals]]


Tate K-theory

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

#Contents#
* table of contents
{:toc}

## Idea

Tate K-theory is the [[elliptic cohomology theory]] associated with the [[Tate curve]] (the Tate [[elliptic curve]] over the [[Laurent series]] ring $\mathbb{Z}((q))$ )  ([AHS 01, 2.7](#AHS01), [Lurie 09, 4.3](#Lurie09))

The corresponding [[elliptic genus]] is the [[Witten genus]] ([AHS 01, Sec. 2.7](#AHS01)).

### Plain version

The underlying [[generalized (Eilenberg-Steenrod) cohomology|cohomology theory]] is given by [[Laurent series]] in [[topological K-theory]], and equivalently by completion of [[circle group]]-[[equivariant K-theory]] of [[free loop spaces]] $\mathcal{L}(-)$ ([KM 04, Section 5](#KM04), see also [Lurie 09, Section 5.2](#Lurie09)):

$$
  \begin{aligned}
  Ell_{Tate}(X)
  &
  \simeq
  \;
  K(X)((q))
  \\
  &
  \simeq
  \;
  K_{S^1}(\mathcal{L}X)
  \otimes_{K_{S^1}(\ast)}
  \mathbb{Z}((q))
  \,.
  \end{aligned}
$$

(For more exposition see also [Dove 19, Sec. 6.2](#Dove19)).

### Equivariant version
 {#IdeaEquivariantVersion}

The [[equivariant cohomology|equivariant]] version of Tate K-theory is a form of [[equivariant elliptic cohomology]]. For $G$ a [[finite group]] and $X$ a [[topological G-space]] it comes down ([Ganter 07, Def. 3.1](#Ganter07), [Ganter 13, Def. 2.6](#Ganter13), review in [Dove 19, Def. 6.16](#Dove19)) to the sub-ring

$$
  Ell_{Tate}
  \big(
    \prec (X \sslash G)
  \big)
  \;\subset\;
  \underset{[g]}{\bigoplus}
  K_{C_g}(X^g)(( q^{1/\left\vert g\right\vert} ))
$$

of the [[direct sum]], over [[conjugacy classes]] of group elements $g$, of [[Laurent polynomials]] with [[coefficients]] in [[equivariant K-theory]]-groups on the $g$-[[fixed loci]] for equivariance group the [[centralizer]] of $g$

on those elements which satisfy the _rotation condition_:

**{#RotationCondition}Rotation condition**. _The $C_g$-[[equivariant vector bundles]] $V_j$ which form the coefficient of $q^{j/\left\vert g \right\vert }$ are such that $g$ acts on them by multiplication with $\exp\big( 2 \pi i \frac{j}{\left\vert g \right\vert}  \big)$_.

{#RotationConditionImpliedByHuanInertiaEquivariance} This rotation condition may be understood more intrinsically
(this is made explicit on [p. 63](https://arxiv.org/pdf/1912.02374.pdf#page=69) of [Dove 19](#Dove19))
as that in implied by the [[orbifold K-theory]] on [[Huan's inertia orbifold]] ([Huan 18](#Huan18)), induced by the nature of the quotient groups
$$
  \Lambda_g \;\coloneqq\; \frac{C_g \times \mathbb{R}}{ \langle (g^{-1},1) \rangle }
$$
appearing [there](Huan's+inertia+orbifold#HuanCentralizer). This quotient exactly equates the action of $g \in G$ with that of a rotation of $S^1$.

Hence, in generalization of [[twisted ad-equivariant K-theory]] there is [[twisted ad-equivariant Tate K-theory]] (an [[equivariant elliptic cohomology]] theory) relating to the [[Verlinde ring]] of [[positive energy representation|positive energy]] [[loop group representations]]
([Lurie 09, Sec. 5.2](#Lurie09),
[Luecke 19, Cor. 3.2.5](#Luecke19), [Dove 19](#Dove19)).

## Properties

### Relation to elliptic genus

The [[Ochanine genus]] lifts to a homomorphism of [[ring spectra]] $M Spin \to KO((q))$ from [[spin structure]] [[cobordism cohomology theory]] to [[Tate K-theory]] ([Kreck-Stolz 93, lemma 5.8, lemma 5.4](spin-orientation+of+elliptic+cohomology#KreckStolz93)). This is the _[[spin-orientation of elliptic cohomology]]_

## Related concepts

* [[elliptic genus]]

  * [[Ochanine genus]]

  * [[Witten genus]]

* [[elliptic Chern character]]

* [[topological K-theory]]

* [[twisted ad-equivariant K-theory]]

## References

As [[elliptic cohomology]] over the [[Tate sphere]]:

* {#AHS01}  [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], Section 2.7 of: _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. math. 146, 595–687 (2001) [[doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175)]

* {#Lurie09} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_, in: _Algebraic Topology_ Abel Symposia **4** (2009) 219-277 [[pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf), [doi:10.1007/978-3-642-01200-6_9](https://doi.org/10.1007/978-3-642-01200-6_9)]

As completed $S^1$-[[equivariant K-theory]] of [[free loop space]]

* {#KM04}  [[Nitu Kitchloo]], [[Jack Morava]], Section 5 of: _Thom Prospectra for Loopgroup representations_ [[arXiv:math/0404541](https://arxiv.org/abs/math/0404541)]

As a form of [[equivariant elliptic cohomology]] ([[twisted ad-equivariant Tate K-theory]]):

* {#Ganter07} [[Nora Ganter]], Section 3.1 in: _Stringy power operations in Tate K-theory_ [[arXiv:math/0701565](https://arxiv.org/abs/math/0701565)]

* {#Ganter13} [[Nora Ganter]], _Power operations in orbifold Tate K-theory_, Homology Homotopy Appl. **15** 1 (2013)  313-342 [[arXiv:1301.2754](https://arxiv.org/abs/1301.2754), [euclid:hha/1383943680](https://projecteuclid.org/euclid.hha/1383943680)]

* {#Rezk14} [[Charles Rezk]], *Quasi-Elliptic Cohomology* (2014) [[[Rezk-QuasiEllipticCohomology.pdf:file]]]

* [[Zhen Huan]], _Quasi-elliptic cohomology_ (2017) [[hdl:2142/97268](http://hdl.handle.net/2142/97268)]

* [[Zhen Huan]], *Quasi-Elliptic Cohomology and its Power Operations*, J. Homotopy and Related Structures **13** (2018) 715–767 [[arXiv:1612.00930](https://arxiv.org/abs/1612.00930), [doi:10.1007/s40062-018-0201-y](https://doi.org/10.1007/s40062-018-0201-y)]

* [[Zhen Huan]], *Quasi-elliptic cohomology and its Spectrum* [[arXiv:1703.06562](https://arxiv.org/abs/1703.06562)]

* {#Huan18} [[Zhen Huan]], _Quasi-Elliptic Cohomology I_, Advances in Mathematics, **337** (2018) 107-138 [[arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007)]

with a streamlined account in

* [[Zhen Huan]], *Quasi-theories* [[arXiv:1809.06651](https://arxiv.org/abs/1809.06651)]

* [[Zhen Huan]], *Quasi-theories and their equivariant orthogonal spectra* [[arXiv:1809.07622](https://arxiv.org/abs/1809.07622)]

and some general abstract clarification on [[Huan's inertia orbifold]] in:

* [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Cyclification of Orbifolds]]*, Comm. Math. Phys. (2023) [[arXiv:2212.13836](https://arxiv.org/abs/2212.13836), [[schreiber:cyclic loop spaces 2022|talk]]]

Review:

* [[Zhen Huan]], *Quasi-elliptic cohomology theory and the twisted, twisted Real theories*, talk at Perimeter Institute (2020) [video: [doi:10.48660/20050059](https://doi.org/10.48660/20050059)]

For [[simply-connected topological space|simply-connected]] [[compact Lie groups]]:

* {#Luecke19}  [[Kiran Luecke]], *Completed K-theory and Equivariant Elliptic Cohomology*, Advances in Mathematics **410** B (2022) 108754 [[arXiv:1904.00085](https://arxiv.org/abs/1904.00085), [doi:10.1016/j.aim.2022.108754](https://doi.org/10.1016/j.aim.2022.108754)]

For [[finite groups]]:

* {#Dove19} [[Thomas Dove]], _Twisted Equivariant Tate K-Theory_ [[arXiv:1912.02374](https://arxiv.org/abs/1912.02374)]

Generalization to [[twisted equivariant K-theory]]

* [[Zhen Huan]], [[Matthew Spong]], *Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology* ([arXiv:2006.00554](https://arxiv.org/abs/2006.00554))

and further to [[KR-theory]]:

* [[Zhen Huan]], [[Matthew B. Young]], *Twisted Real quasi-elliptic cohomology* [[arXiv:2210.07511](https://arxiv.org/abs/2210.07511)]

  [[Zhen Huan]], *Twisted Real quasi-elliptic cohomology*, talk at [[CQTS]] (Nov 2022) [[video](https://zoom.us/rec/play/_2MIenxSff2vBZeFPO8lV6g_lyLgrUEBDP5Au7tySNyeJpeLeXfZ96ZBkgKs7wNYAnBDauB4atzVJAt-.McANEtFgQHFACmbr?_x_zm_rhtaid=966&_x_zm_rtaid=F4VKJYOLRDu8ouQdfmDz3w.1668757869896.554b5189e89fa9af0846d97599b6bf49&autoplay=true&continueMode=true&iet=6_ddkV5u8UGa9M9U4dDvH2SN55OyZgmZiCrQZ6wbo4E.AG.Gp56JL6y5Z3naAtSVPOGAHkLGEPMDtxPB370GxyQz_EIJOEq1ylk0bXAowqSDnPSYr62xiX92tLdNLdbchq-w5Vhavx5UCPbjfaCIIObE183OI8.142oNpQmC8JiJOzcA4eIXw.O0nEkHAMvD5awvgi&startTime=1667999236000)]

[[!redirects twisted ad-equivariant Tate K-theory]]

[[!redirects quasi-elliptic cohomology]]


Tate module

## Definition

Let $A$ be an abelian group and let $p$ be a prime number.

The *$p$-adic Tate Module* $T_p(A)$ of $A$ is defined as the limit

$$T_p(A):=lim_n A[p^n]:=lim_n ker(A\stackrel{\cdot p^n }{\to}A)$$

of the directed diagram given by kernels of the endomorphisms of $A$ defined by multiplication by $p^n$ with transition maps $d_i:=(\cdot p):A[p^{n+1}]\to A[p^n]$.

The Tate module can equivalently be described in terms of the endomorphism ring of the [[Prüfer group]].

## References

* Faltings, Gerd (1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern", Inventiones Mathematicae 73 (3): 349–366, doi:10.1007/BF01388432

* Hazewinkel, Michiel, ed. (2001), "Tate module", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

* Murty, V. Kumar (2000), Introduction to abelian varieties, CRM Monograph Series, 3, American Mathematical Society, ISBN 978-0-8218-1179-5

* Section 13 of Rohrlich, David (1994), "Elliptic curves and the Weil–Deligne group", in Kisilevsky, Hershey; Murty, M. Ram, Elliptic curves and related topics, CRM Proceedings and Lecture Notes, 4, American Mathematical Society, ISBN 978-0-8218-6994-9

* Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae 2: 134–144, MR 0206004John Tate, endomorphisms of abelian varieties over finite fields, [web](http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=166676)

* Tate module. Encyclopedia of Mathematics. [web](http://www.encyclopediaofmath.org/index.php/Tate_module)


Tate spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For [[∞-actions]] of [[finite groups]] $G$ on objects $E$ in [[stable (∞,1)-categories]], then the [[homotopy cofiber]] $X^{t G}$ of the [[norm map]] is called the _[[Tate construction]]_, sitting in a [[homotopy fiber sequence]]

$$
  X_G \stackrel{norm}{\longrightarrow} X^G \longrightarrow X^{t G}
  \,.
$$

(e.g [Lurie, def. 6.1.6.24](#Lurie))

+-- {: .standout }
Depending on which edition you have, chapter 6 may be chapter 7.
=--

For the [[stable (∞,1)-category of spectra]] this is accordingly called the _Tate spectrum_.

## Examples

* Tate spectra for [[cyclic groups]] play a role in the characterization of [[cyclotomic spectra]].

* For $N$ a $G$-[[representation]] and $H N$ its [[Eilenberg-MacLane spectrum]], then the Tate spectrum $H N^{t G}$ represents what is classically called _[[Tate cohomology]]_

  $$
    \hat H^{-n}(G,N) \;\simeq\; \pi_n( H N^{t G} )
  $$

  ([Nikolaus-Scholze 17, p. 13](#NikolausScholze17)) This may be generalized to possibly infinite [[discrete groups]], see at _[[Farrell-Tate cohomology]]_.

## References

The general abstract discussion is due to

* [[Jacob Lurie]], section 6.1 of _[[Higher Algebra]]_

Review with an eye towards [[topological cyclic homology]] is in

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], section I of _On topological cyclic homology_ ([arXiv:1707.01799](https://arxiv.org/abs/1707.01799))

[[!redirects Tate spectra]]

[[!redirects Tate construction]]
[[!redirects Tate constructions]]


Tate sphere

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Tate sphere_, in the strict sense, is the [[object]] in a [[sheaf topos]] (or [[(∞,1)-sheaf (∞,1)-topos]], or its [[stabilization]]) over a [[gros site]] of [[algebraic varieties]], which is one of the following: $\mathbb{P}^{1}$ (the [[projective line]]); the quotient

$$
  S^1_{Tate}
  \;\coloneqq\;
  \mathbb{A}^1 / (\mathbb{A}^1 \setminus \{0\})
  \,,
$$

which denotes the ([[homotopy cofiber|homotopy]]-)[[cofiber]] of the inclusion morphism into the [[sheaf]] [[representable functor|represented]] by the [[affine line]] $\mathbb{A}^1$ from the [[subobject]] $\mathbb{A}^1 \setminus \{0\}$ [[representable functor|represented]] by the affine line with origin removed (equivalently the object underlying the [[multiplicative group]] $\mathbb{G}_m$); or $S^{1} \wedge \mathbb{G}_{m}$.

These constructions are understood by default in the context of [[motivic homotopy theory]] working over the [[Nisnevich site]] ([VRO 07, Remark 2.22](#VRO07)), where they are all $\mathbb{A}^{1}$-homotopy equivalent.
But the construction principle, especially that of the homotopy cofibre definition, is clearly more general.

In formal constructions of categories of [[motive|motives]], one typically 'inverts the Tate sphere' in some sense in order to represent [[Tate twist|Tate twists]] correctly in [[Weil cohomology theory|Weil cohomology theories]] such as [[étale cohomology]], or in [[motivic cohomology|motivic cohomology theories]]. This is the case, for example, both for the [[stable motivic homotopy category|stable motivic homotopy categories]] à la [[Vladimir Voevodsky|Voevodsky]], where one more precisely inverts the operation of smashing with the Tate sphere, and for the category of [[pure motive|pure motives]], where one inverts the [[Lefschetz motive]].

## Related concepts

* [[motivic homotopy theory]]

* [[motivic sphere]]

* [[n-sphere]]

* [[Tate twist]]

## References

* {#VRO07} [[Vladimir Voevodsky]], [[Oliver Röndigs]], [[Paul Arne Østvær]], _Voevodsky’s Nordfjordeid Lectures: Motivic Homotopy Theory_, In:  [[Bjørn Ian Dundas]], [[Marc Levine]], [[Paul Arne Østvær]], [[Oliver Röndigs]], [[Vladimir Voevodsky]] (eds.), _Motivic Homotopy Theory_,  Springer 2007 ([doi:10.1007/978-3-540-45897-5_7](https://doi.org/10.1007/978-3-540-45897-5_7), [pdf](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Nordfjordeid_lectures_published.pdf))

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

[[!redirects Tate spheres]]


Tate twist

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

#Contents#
* table of contents
{:toc}

## Idea

[[cohomology|Cohomology theories]] often have two aspects, which one might refer to as _geometric_ and _arithmetic_. A prototypical example is the [[l-adic cohomology]] $H^{i}(X, \mathbb{Z}_{l})$ of a (sufficiently nice) [[scheme]] $X$ over a field $k$ of [[positive characteristic|characteristic]] $p$, with $l$ coprime to $p$. This is not only an abelian group (the geometric aspect), but a representation of the [[absolute Galois group]] of $k$ (the arithmetic aspect).

In the case of the [[singular cohomology]] of a complex manifold, the 'arithmetic' aspect arises as the [[Hodge structure]] on the cohomology groups. It has been speculated (for example by Manin?) that there should be some kind of '[[Galois group]]' whose representations are Hodge structures, (and similarly for mixed Hodge modules vs [[perverse sheaves]], and so on) but this remains mysterious; it may be that a good theory of algebraic geometry over $\mathbb{F}_{1}$ (the would-be "[[field with one element]]") would provide an explanation.

Tate twists play an important role in cohomology theories with this dual geometric and arithmetic aspect, allowing one to express [[Poincaré duality]] canonically, that is, without choosing an orientation of one's geometric object (scheme, complex manifold, ...).

In [[étale cohomology]] in [[positive characteristic|characteristic]] $p$, the Tate twist of a $\mathbb{Z} / p\mathbb{Z}$-[[module]], or [[sheaf of modules|sheaf of such modules]], $A$ is, concretely, $A \otimes_{\mathbb{Z} / p\mathbb{Z}} \mathbb{Z}(1)$, where $\mathbb{Z}(1)$ is the abelian group $\mu_{p}$ of $p^{th}$ roots of unity with respect to the algebraic closure of $\mathbb{Z} / p\mathbb{Z}$. For $n \geq 0$, the $n^{th}$ Tate twist of $A$, often denoted $A(n)$, is defined to be the result of carrying out the above construction $n$ times. One defines $\mathbb{Z}(-n)$ to be the dual of $\mathbb{Z}(n)$, that is to say $\mathsf{Hom}_{\mathbb{Z} / p\mathbb{Z} -\mathsf{mod}}(\mathbb{Z}(n), \mathbb{Z} / p\mathbb{Z})$, and defines $A(-n)$ to be $A \otimes_{\mathbb{Z} / p\mathbb{Z}} \mathbb{Z}(-n)$.

The $l$-adic Tate twist $\mathbb{Z}_{l}(1)$ is defined by means of the inverse system consisting of the groups $\mu_{l^{i}}$ along with the morphisms $\mu_{l^{i+1}} \rightarrow \mu_{l^{i}}$ given by $a \mapsto a^{l}$, for $i \geq 1$, and one can then define $\mathbb{Z}_{l}(n)$ for any integer $n$.

The significance of $\mathbb{Z}_{l}(1)$ is that it is the first $l$-adic homology (or equivalently the étale fundamental group) of $\mathbb{G}_{m}$, the multiplicative group of $\mathbb{Z} / p\mathbb{Z}$ as a scheme (i.e. $\mathbb{A}^{1} \setminus \{ 0 \}$, the affine line over $\mathbb{Z} / p\mathbb{Z}$ minus the origin). It is a free $\mathbb{Z}_{l}$-module of rank $1$, and one therefore deduces that the first $l$-adic cohomology group of $\mathbb{G}_{m}$ is the dual of $\mathbb{Z}_{l}(1)$, that is to say, $\mathbb{Z}_{l}(-1)$. Some basic considerations in $l$-adic cohomology show that $H^{2}(\mathbb{P}^{1}, \mathbb{Z}_{l}(1))$ is canonically isomorphic to $H^{1}(\mathbb{G}_{m}, \mathbb{Z}_{1}(1))$, and thus that $H^{2}(\mathbb{P}^{1}, \mathbb{Z}_{l}(1))$ is canonically isomorphic to $\mathbb{Z}_{l}$ (all of the cohomology groups here are $l$-adic cohomology).

The point of this is that it shows that $\mathbb{Z}_{l}(1)$, that is to say, the Tate twist of $\mathbb{Z}_{l}$, is the correct choice of orientation sheaf in $l$-adic cohomology. And from this follows (with effort!) many things: Tate twists are involved in the canonical formulation of Poincaré duality, as already mentioned, of the Thom isomorphism, of the cycle map, and so on.

There are variations on how to tell the above story (Deligne's discussion in 1.6 of SGA 4$\frac{1}{2}$, via étale cohomology with compact support, can be recommended for instance), but they all come down ultimately to the fact that $\mathbb{Z}_{l}(1)$ is the correct choice of orientation sheaf in an $l$-adic setting.

To return to our starting point, a crucial remark here is that whilst $\mathbb{Z}_{l}(1)$ is non-canonically isomorphic to $\mathbb{Z}_{l}$ as a $\mathbb{Z}_{l}$-module, the two are not isomorphic as Galois representations. Thus the presence of Tate twists is indispensable to the arithmetic aspect of cohomology.

The analogue of this story goes through for singular cohomology of a complex manifold. The roots of unity in this case are a choice of a square root of $-1$, namely either $i$ or $-i$. The choice is invisible in the geometric part of singular cohomology, namely as an abelian group, but it can be seen in the Hodge structure. The analogue of the computations involving $\mathbb{G}_{m}$ and $\mathbb{P}^{1}$ are that the kernel of the exponential map $\mathbb{C} \rightarrow \mathbb{C}^{\times}$ is $\mathbb{Z}$, and the inclusion of $\mathbb{Z}$ in $\mathbb{C}$ is via $1 \mapsto 2 \pi i$. Thus the Tate twist in singular cohomology is tensoring with $2 \pi i \mathbb{Z}$.

Tate twists are so fundamental that they are built into Grothendieck's definition of the category of [[pure motives]]: one formally inverts (this is the analogue of taking the dual in the above story) the Lefschetz motive, namely the motive of a pointed $\mathbb{P}^{1}$.

Tate twists are also fundamental in the various approaches to mixed motives, including Voevodsky's. Indeed, in the definition of motivic cohomology via hypercohomology or via the derived category of mixed motives, 'all the interesting part' comes from a certain complex of presheaves with transfers built from $\mathbb{G}_{m}$ in a way which is formally entirely analogous to the story we told above. In the definition of motivic cohomology via motivic homotopy theory, 'all the interesting part' comes from the inversion of smashing with the 'algebraic circle', namely a pointed $\mathbb{G}_{m}$.

As a final remark, rational Tate twists can also be made sense of, and are used for instance in geometric representation theory.

## Definition

For $n \in \mathbb{N}$ write $\mu_n \to \mathbb{G}_m \stackrel{(-)^n}{\longrightarrow}\mathbb{G}_m$ for the $n$th [[Kummer sequence]]. Then for $k \in \mathbb{N}$ the $k$-fold [[tensor power]] of the group of $n$th [[roots of unity]]

$$
  \mathbb{Z}/n\mathbb{Z}(k) \coloneqq \mu_n^{\otimes k}
$$

is called the $k$th _Tate twist_.

(...)

## References

* Kanetomo Sato, _$p$-adic étale Tate twists and arithmetic duality_ ([arXiv:0610426](http://arxiv.org/abs/math/0610426))

[[!redirects Tate twists]]


Tate's acyclicity theorem

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

#Contents#
* table of contents
{:toc}

## Statement

Let $X = Spec_{an}(A)$ be an [[affinoid space]] with [[affinoid algebra]] $A$.

Then for every [[finite cover]] of $X$ by [[affinoid domains]], the corresponding [[Cech cohomology]] with [[coefficients]] in the [[structure sheaf]] $A$, or with coefficients in any finite [[Banach module]] over $A$, is concentrated in degree 0.

([Berkovich 09, fact 2.2.6](#Berkovich09))

## References

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], section 8.2 of _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

* [pdf](http://www.math.polytechnique.fr/~chenevier/coursIHP/chenevier_lecture5.pdf)

* [[Doosung Park]], _Affinoid domains_, lecture notes ([pdf](http://math.berkeley.edu/~sstich/MAT_274/03-07.pdf))

[[!redirects Tate acyclicity]]


Tate–Shafarevich group

## Related concepts

* [[Picard group]]

* [[Tamagawa measure]]

## References

* Wikipedia, _[Tate-Shafarevich group](http://en.wikipedia.org/wiki/Tate-Shafarevich_group)_

[[!redirects Tate–Shafarevich groups]]

[[!redirects Tate-Shafarevich group]]
[[!redirects Tate-Shafarevich groups]]


Tatiana Ivanova

## Selected writings

A derivation of 4d [[Skyrmions]] by [[KK-compactification]] of [[D=5 Yang-Mills theory]] on a [[closed interval]], motivated by [[M5-branes]] instead of by [[D4/D8-brane intersections]] as in the [[Sakai-Sugimoto model]]:

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Skyrme model from 6d $\mathcal{N}= (2,0)$ theory_, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 ([arXiv:1805.07241](https://arxiv.org/abs/1805.07241))

following

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Non-Abelian sigma models from Yang-Mills theory compactified on a circle_, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 ([arXiv:1803.07322](https://arxiv.org/abs/1803.07322))

category:people


Tatsuya Hagino

* [website](https://web.sfc.keio.ac.jp/~hagino/index.html.en)
* [publications](https://keio.pure.elsevier.com/en/persons/tatsuya-hagino)

## Related pages

* [[dialgebra]]
* [[lambda-calculus]]

category: people


tau cohomology

## Idea

Sometimes the [[super Lie algebra|super]]-[[Lie algebra cohomology]] of [[super Minkowski spacetimes]] is referred to as _$\tau$-cohomology_.

## References

Review includes

* Bertrand Souères, Dimitrios Tsimpis, section 2.2 of _The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity_ ([arXiv:1612.02021](https://arxiv.org/abs/1612.02021))

[[!redirects tau cohomologies]]

[[!redirects tau-cohomology]]
[[!redirects tau-cohomologies]]

[[!redirects ∞-cohomology]]
[[!redirects ∞-cohomologies]]


tau-additive measure

[[!redirects tau-smoothness]]

***

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

#Contents#
* table of contents
{:toc}

## Idea

A **$\tau$-additive measure** (alias τ-regular, τ-smooth) is a [[measure]] on a [[topological space]] which interacts well with the topology. In particular, it has a well-defined and well-behaved [[support]].

$\tau$-additive measures can be thought of as a point of contact between [[measure theory]], and the theory of [[valuation (measure theory)|valuations]]. More on this in [[correspondence between measure and valuation theory]].

## Definition

Let $X$ be a [[topological space]]. A positive [[Borel subset|Borel measure]] $\mu$ on $X$ is called **$\tau$-additive**, or **$\tau$-smooth**, or **$\tau$-regular**, if for every directed [[net]] $\{U_\lambda\}_{\lambda\in\Lambda}$ of [[open subsets]] of $X$,
$$
  \mu \big( \sup_{\lambda} U_\lambda \big)
  =
  \sup_\lambda \mu(U_\lambda) .
$$

This property can be considered an instance of [[Scott topology|Scott continuity]], and is equivalent to continuity of the [[valuation (measure theory)|valuation]] defined by $\mu$ on the open sets.

## Examples

Every [[Radon measure]] on a [[Hausdorff space]] is $\tau$-additive. This includes most regular measures of common use, such as

* the [[Dirac measures]],
* the [[Lebesgue measure]] on the real line,
* the measure induced by the [[volume form]] of a Riemannian manifold,
* the [[Haar measure]] on a Lie group (or more generally a locally compact Hausdorff topological group).

## Properties

### Pushforward, products and marginals

* The [[pushforward measure]] of a $\tau$-additive measure along a [[continuous map]] is itself $\tau$-additive. In particular, the [[marginals]] of a $\tau$-additive measure are $\tau$-additive. This permits to construct a [[functor]], even a [[monad]] (see [below](#monad_structure)).

* The [[product measure|product]] of two $\tau$-additive measures on a product space can be extended to a $\tau$-additive measure.

### Monad structure

One can form a [[measure monad]] analogous to the [[Giry monad]], whose functor part assigns to a topological space the space of $\tau$-additive measures over it. This monad is a submonad of the [[extended probabilistic powerdomain]]. See [[extended probabilistic powerdomain#the_measure_monad_on_top|the measure monad on Top]] for more details.

### Null sets and support

Given a tau-additive measure $\mu$ on a topological space $X$, an open set $U\subseteq X$ is called **null** for $\mu$ if $\mu(U)=0$. A set has **full measure** if its complement is null.

By $\tau$-additivity, the union of all null open sets is null. Its [[complement]], which is a closed set, is called the **[[support]]** of $\mu$.
It can be seen as the smallest closed set of full measure.

This definition can be extended to [[valuation (measure theory)#null_sets_and_support|continuous valuations]].

### Relationship with other measure-theoretical notions

* Every [[Radon measure]] on a [[Hausdorff space]] is $\tau$-additive.
* Every $\tau$-additive measure on a [[compactum|compact Hausdorff space]] is [[Radon measure|Radon]].
* Every $\tau$-additive measure can be restricted to a [[continuous valuation]].
* Every [[continuous valuation]] on a [[regular space|regular]] [[Hausdorff space]] or on a [[locally compact space|locally compact]] [[sober space]] can be extended to a $\tau$-additive measure.

See also: [[valuation (measure theory)#extending_valuations_to_measures|Extending valuations to measures]].

## Related notions

* [[Borel measure]]

* [[Radon measure]]

* [[continuous valuation]]

* [[extended probabilistic powerdomain]]

* [[correspondence between measure and valuation theory]]

## References

* V. Bogachev, _Measure Theory_, vol. 2 (2007).

[[!redirects tau-additive measures]]
[[!redirects tau-regular measure]]
[[!redirects tau-regular measures]]
[[!redirects tau-smooth measure]]
[[!redirects tau-smooth measures]]
[[!redirects τ-additive measure]]
[[!redirects τ-additive measures]]
[[!redirects τ-smooth measure]]
[[!redirects τ-smooth measures]]
[[!redirects τ-regular measure]]
[[!redirects τ-regular measures]]
[[!redirects tau-additive]]
[[!redirects tau-smooth]]
[[!redirects tau-regular]]
[[!redirects τ-additive]]
[[!redirects τ-smooth]]
[[!redirects τ-regular]]
[[!redirects $\tau$-additive measure]]
[[!redirects $\tau$-additive measures]]
[[!redirects $\tau$-additive]]
[[!redirects $\tau$-smooth measure]]
[[!redirects $\tau$-smooth measures]]
[[!redirects $\tau$-smooth]]
[[!redirects $\tau$-regular measure]]
[[!redirects $\tau$-regular measures]]
[[!redirects $\tau$-regular]]
[[!redirects tau-additivity]]
[[!redirects tau-smoothness]]
[[!redirects tau-regularity]]
[[!redirects τ-additivity]]
[[!redirects τ-smoothness]]
[[!redirects τ-regularity]]
[[!redirects $\tau$-additivity]]
[[!redirects $\tau$-smoothness]]
[[!redirects $\tau$-regularity]]


tau-function

This page is about the $\tau$-functions in the study of [[soliton]]s and [[integrable model]]s, as invented by Kyoto school: isomonodromic $\tau$-function and Sato-Segal-Wilson $\tau$-function for integrable hierarchies. There will be a separate entry for a completely different notion of [[Ramanujan tau function]].

## Motivation

Quoting Zabrodin, [arxiv/1211.4428](http://arxiv.org/abs/1211.4428),

> "In nature", $\tau$-functions or their logarithms appear as partition functions, different kinds of correlators and their generating functions, and effective actions as functions of coupling constants.

## Formal idea

Sato and [[Graeme Segal]] have formulated the $\tau$-function as the [[Fredholm determinant]] of certain 1-parameter family of operators on separable Hilbert space.
See also closely related page [[Sato Grassmannian]].

#### Other formal interpretations

$\tau$-functions appear as dependent variables in [[Hirota bilinear equation]]s
associated to integrable hierarchies.

Tau functions can also be associated to the [[isomonodromic problem]]s.

## References

The $\tau$-function has first been introduced in the formalism of [[Clifford group]]
of free fermions in study of [[holonomic quantum field]]s
and in relation to isomonodromic problems

* M. Jimbo M, T. Miwa, K. Ueno, _Monodromy preserving deformation of linear ordinary differential equations with rational coefficients_, I.. Physica D 2, 306–352 (1981)
* Mikio Sato, Tetsuji Miwa, Michio Jimbo

* E. Date, [[Michio Jimbo]], [[Masaki Kashiwara]], T. Miwa, _Transformation groups for soliton equations,IV A new hierarchy of soliton equations of KP-type_,Physica D4, 343-365 (1982)
* M. Jimbo, T. Miwa, _Solitons and infinite dimensional Lie algebras_,Publ. RIMS Kyoto Univ. 19, 943-1001 (1983)

Wikis: wikipedia

The $\tau$-function for [[KP hierarchy]] has originally being studied in

* [[Masaki Kashiwara]], [[Tetsuji Miwa]], _The $\tau$-function of the Kadomtsev-Petviashvili equation transformation groups for soliton equations, I_, Proc. Japan Acad. Ser. A Math. Sci. __57__, N. 7 (1981), 337-386 [euclid](http://projecteuclid.org/euclid.pja/1195516327)

The background on the context of [[Sato-Segal-Wilson Grassmanian]]s is in

* Andrew Pressley, Graeme Segal, _Loop groups_, Oxford University Press (1988)
* G. Segal, G. Wilson, _Loop groups and equations of KdV type_, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5–65.

Reviews include

* A. Alexandrov, A. Zabrodin, _Free fermions and tau-functions_, J.Geom.Phys. __67__ (2013) 37-80 [arxiv/1212.6049](http://arxiv.org/abs/1212.6049) [doi](http://dx.doi.org/10.1016/j.geomphys.2013.01.007)
* A. Zabrodin, _Bethe ansatz and Hirota equation in integrable models_, [arxiv/1211.4428](http://arxiv.org/abs/1211.4428)

The connection to 2D [[gravity]] is elucidated in

* [[Victor Kac]], [[Albert Schwarz]], _Geometric interpretation of the partition function of 2D gravity_ Phys. Lett. B 257:3-4 (1991) 329–334 pdf

Some special cases of $\tau$-functions include

* J. Palmer, _Determinants of Cauchy–Riemann operators as τ-functions_, Acta Appl. Math. 18 (1990), 199-223.

Isomonodromic interpretation is stemming already from the works of holonomic fields, and isomonodromic $\tau$-functions are studied also in

* Bertola, Eynard, Harnad, _Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions_, [nlin.SI/0410043](http://arxiv.org/abs/nlin.SI/0410043)

[[Witten conjecture]] is about the equivalence of two approaches to gravity, and boils down to a connection of the $\tau$-function for the KP hierarchy and theory of Riemann surfaces. It has been proved by Kontsevich who also introduced related family of matrix models.

* [[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_, Comm. Math. Phys. __147__ (1992), no. 1, 1--23, [euclid](http://projecteuclid.org/euclid.cmp/1104250524), [author's pdf](http://193.51.104.7/~maxim/TEXTS/intersection_theory_6.pdf)

* A.Alexandrov, _Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra_, [arxiv/1404.3402](http://arxiv.org/abs/1404.3402)

> In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac-Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.

Connection of $\tau$-functions to [[dessins d'enfants]] is discussed in

* Jan Ambjørn, Leonid Chekhov, _The matrix model for dessins d'enfants_, [arxiv/1404.4240](http://arxiv.org/abs/1404.4240)

A connection to elementary geometry is studied (with nice reference list) in

* Adam Doliwa, _Desargues maps and the Hirota-Miwa equation_,
[pdf](https://www.newton.ac.uk/preprints/NI09049.pdf)

Recent works on $\tau$-function include

* J. W. van de Leur, A. Yu. Orlov, _Pfaffian and determinantal tau functions I_, [arxiv/1404.6076](http://arxiv.org/abs/1404.6076)

* [[Mattia Cafasso]], Chao-Zhong Wu, _Tau functions and the limit of block Toeplitz determinants_, [arxiv/1404.5149](http://arxiv.org/abs/1404.5149)
* [[Mattia Cafasso]], P. Gavrylenko, O. Lisovyy, _Tau functions as Widom constants_, Commun. Math. Phys. __365__, 741–772 (2019) [doi](https://doi.org/10.1007/s00220-018-3230-9)
* J. Harnad, Eunghyun Lee, _Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions_, [arxiv/1304.0020](http://arxiv.org/abs/1304.0020)

* S. Natanzon, A. Zabrodin, _Formal solution to the KP hierarchy_, [arxiv/1509.04472](http://arxiv.org/abs/1509.04472)

There is now a book

* John Harnad, Ferenc Balogh, _Tau functions and their applications_, Cambridge Studies in Math. Physics 2021

category: physics, representation theory, Lie theory

[[!redirects tau function]]


Taub-NUT space

[[!redirects Taub-NUT spacetime]]

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[spacetime]]. For the moment see at _[[KK-monopole]]_ for more.

<img src="https://ncatlab.org/nlab/files/TaubNUT.png" width="300">

> graphics grabbed from [Acharya-Gukov 04](#AcharyaGukov04)

Consider the [[left invariant 1-forms]] on the [[3-sphere]] $\simeq$
[[special unitary group|SU(2)]], which in terms of [[Euler angles]] are

$$
  \begin{aligned}
     \sigma_1 &= \sin \psi \, d \theta - \cos \psi \sin \theta \, d \phi
     \\
     \sigma_2 & = \cos \psi \, d \theta + \sin \psi \sin \theta \, d \phi
     \\
     \sigma_3 &= d \psi + \cos \theta \, d \phi
  \end{aligned}
$$

Then the [[pseudo-Riemannian metric]] defining the Taub-NUT geometry is

$$
  (d s)^2 = \frac{1}{4} \frac{r+n}{r-n} (d r)^2 + \frac{r-n}{r+n} n^2 {\sigma_3}^2 + \frac{1}{4}(r^2 - n^2)({\sigma_1}^2 + {\sigma_2}^2)
$$

## Properties

### Relation to KK-monopoles

[[!include KK-monopole geometries -- table]]

### Relation to D6-branes

[[!include F-branes -- table]]

## Related concepts

* [[KK monopole]]

* [[gravitational instanton]]

## References

* Wikipedia, _[Taub-NUT space](http://en.wikipedia.org/wiki/Taub%E2%80%93NUT_space)_

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

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 6.3.3 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

Discussion in the context of [[M-theory on G2-manifolds]] and [[gauge enhancement]]

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], section 5.2 of _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

[[!redirects Taub-NUT spacetime]]
[[!redirects Taub-NUT spacetimes]]

[[!redirects Taub-NUT space]]
[[!redirects Taub-NUT spaces]]

[[!redirects multi-centered Taub-NUT space]]
[[!redirects multi-centered Taub-NUT spaces]]

[[!redirects NUT charge]]


tauon

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

#Contents#
* table of contents
{:toc}

## Idea

The _tauon_ is one of the fundamental [[particles]]/[[matter]] [[field (physics)|fields]] in the [[standard model of particle physics]], one of the [[leptons]].

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

* Wikipedia, _[Tau (particle)](https://en.wikipedia.org/wiki/Tau_(particle))_

[[!redirects tauons]]

[[!redirects tau (particle)]]


taut functor

# Contents
* automatic table of contents goes here
{: toc}

## Definitions

+-- {: .num_defn}
###### Definition
Let $\mathcal{C}$ and $\mathcal{D}$ be [[categories]] with [[pullbacks]]. A [[functor]] $T: \mathcal{C} \to \mathcal{D}$ is _taut_ if $T$ preserves [[inverse images]], i.e., [[pullbacks]] of the form

$$\array{
P & \to & B \\
\downarrow & & \downarrow \\
A & \underset{i}{\to} & C
}$$

where $i$ is [[monomorphism|monic]]. (This implies in particular that $T$ preserves monos.)
=--

+-- {: .num_defn}
###### Definition
Let $\mathcal{D}$ be a category with pullbacks, and let $\eta: S \to T$ be a [[natural transformation]] between functors $S, T: \mathcal{C} \to \mathcal{D}$. Then $\eta$ is _taut_ if the naturality square

$$\array{
S A & \stackrel{\eta A}{\to} & T A \\
\mathllap{S i} \downarrow & & \downarrow \mathrlap{T i} \\
S B & \underset{\eta B}{\to} & T B
}$$

is a pullback for every monomorphism $i$ in $\mathcal{C}$.
=--

A [[monad]] $(T, m, u)$ on a category with pullbacks is _taut_ if $T$ is a taut functor and $m, u$ are taut transformations.

## Examples

A number of examples of taut functors can be deduced by applying the following observation.

+-- {: .num_prop}
###### Proposition
Let $T: \mathcal{C} \to \mathcal{D}$ be a functor that preserves weak pullbacks, and assume epis in $D$ are [[regular epimorphism|regular]]. Then $T$ is taut.
=--

+-- {: .proof}
###### Proof
In the first place, $T$ preserves monos. For $i: A \to B$ is monic if and only if

$$\array{
A & \stackrel{1_A}{\to} & A \\
\mathllap{1_A} \downarrow & & \downarrow \mathrlap{i} \\
A & \underset{i}{\to} & B
}$$

is a pullback. By hypothesis, the canonical map $\phi: T(A) \to T(A) \times_{T(B)} T(A)$ is (regular) epic, but it is also monic because its composition with either projection $T(A) \times_{T(B)} T(A) \to T(A)$ is the identity. Therefore $\phi$ is an isomorphism, i.e., applying $T$ to the displayed pullback is a pullback, and this forces $T(i)$ to be monic.

Now if

$$\array{
P & \stackrel{j}{\to} & B \\
\mathllap{q} \downarrow & & \downarrow \mathrlap{p} \\
A & \underset{i}{\to} & C
}$$

is a pullback with $i$ monic, we have that $j$ and therefore $T(j)$ is monic.
The canonical map $\phi: T(P) \to T(A) \times_{T(C)} T(B)$ is (regular) epic, but also monic, since the mono $T(j)$ factors through it. Thus $\phi$ is an isomorphism, which completes the proof.
=--

A similar proof shows that weakly cartesian natural transformations are also taut.

As a result of this proposition,

* Any [[cartesian functor]] is (trivially) taut.

* The [[ultrafilter|ultrafilter endofunctor]] on $Set$ is taut. (See [here](/nlab/show/relational+beta-module#functor) for a proof that the ultrafilter functor preserves weak pullbacks.) In fact, the ultrafilter monad is taut.

* Similarly, the [[filter|filter monad]] on $Set$ is taut.

* The covariant power set monad, whose algebras are [[sup-lattices]], is taut.

* An [analytic endofunctor](/nlab/show/operad#analytic) induced by a [[species]] is taut. Furthermore, a morphism of species induces a weakly cartesian transformation between the corresponding analytic functors, thus _a fortiori_ a taut transformation. In particular, an analytic monad is taut.

As an exception, we have

* The double (contravariant) power set functor $P \circ P^{op}: Set \to Set$ is not taut.

## Applications

* Paul Taylor has made tautness of $T$ a central assumption in his account of induction via [[well-founded coalgebras]] over $T$. See chapter VI of his [book](#Taylor).

* Tautness assumptions play a role in viewing relational $T$-algebras and related structures as generalized multicategories in the sense of [Cruttwell-Shulman](#CS). In the prototypical case of [[relational beta-modules]], there is a [[virtual double category]] of relations. A taut monad $T$ on $Set$ (such as the ultrafilter monad) induces a monad $\bar{T}$ on this virtual double category (that is, a monad in an appropriate 2-category of virtual double categories). From there, one can define a horizontal Kleisli construction which is another virtual double category $HKl(Rel, \bar{T})$, and a $\bar{T}$-multicategory in $Rel$ is by definition a monoid in $HKl(Rel, \bar{T})$. In the special case $T = \beta$, the ultrafilter monad, this concept recapitulates Barr's notion of relational $\beta$-module as synonym of "[[topological space]]". This can be generalized further by working with a virtual double category of "$V$-matrices" where $V$ is a completely distributive quantale ($Rel$ being the case $V = \mathbf{2}$). Again with $T$ a taut monad, one can define a virtual double category $HKl(V\text{-}Mat, \bar{T})$ and then define generalized multicategories as before. (These were studied in a series of articles by [Clementino, Hofmann, Tholen](#CHT2), [Seal](#Seal) and others under the name "$(T, V)$-algebras".)

## References

* [[Paul Taylor]], *Practical Foundations of Mathematics*, Cambridge University Press (1999).
{#Taylor}

* [[Geoff Cruttwell]] and [[Michael Shulman]], _A unified framework for generalized multicategories_, [arxiv/0907.2460](http://arxiv.org/abs/0907.2460)
{#CS}

* Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, _One Setting for All: Metric, Topology, Uniformity, Approach Structure._ ([pdf](http://www.math.yorku.ca/~tholen/ProMat_V_.pdf))
{#CHT2}

* Gavin J. Seal, _Canonical and op-canonical lax algebras_, Theory and
Applications of Categories, 14 (2005), 221–243. ([web](http://www.tac.mta.ca/tac/volumes/14/10/14-10abs.html))
{#Seal}

[[!redirects taut functors]]
[[!redirects taut monad]]
[[!redirects taut monads]]


tautological equivariant line bundle

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

#Contents#
* table of contents
{:toc}

## Idea

A _tautological equivariant line bundle_ is an [[equivariant vector bundle|equivariant]] [[tautological line bundle]] over a [[projective G-space]].

## Definition

Let $G$ be a [[finite group]] (or maybe a [[compact Lie group]]) and let $V$ be a $G$-[[linear representation]] over some [[topological field|topological]] [[ground field]] $k$, with $k P(V)$ its [[projective G-space]].

Then the corresponding _tautological equivariant line bundle_ $\mathcal{L}_V$ is the $k^\ast$-[[fiber bundle]] which is [[associated bundle|associated]] to the canonical $k^\times$-[[principal bundle]] over [[projective G-space]]:

$$
  \array{
    \mathcal{L}_V
    & \coloneqq &
    \frac{
      (V \setminus \{0\}) \times k^\ast
    }{
      k^\times
    }
    &
    \overset{
      [v,z]
      \mapsto
      \big(
        [v], v \cdot z
      \big)
    }{\hookrightarrow}
    &
    \frac{
      V \setminus \{0\}
    }{
      k^\times
    }
    \times
    V
    \\
    \big\downarrow
    &&
    \big\downarrow {}^{\mathrlap{
       \frac{id \times pt}{ k^\times }
    }}
    \\
    k P(V)
    &=&
    \frac{
      (V \setminus \{0\}) \times \ast
    }{
      k^\times
    }
  }
$$

and equipped with the induced $G$-[[action]] through $V$ (which passes to the [[quotient spaces]] since the $k$-multiplication action commutes with it, by linearity).

Here

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

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

* and $k^\ast$ is $k$ equipped with the dual $k^\times$-action

  $$
    \array{
      k^\times \times k^\ast
      &\longrightarrow&
      k^\ast
      \\
      (g,z) &\mapsto& z/g
    }
  $$

* so that, for $z \neq 0$,

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

## Related concepts

* [[resolution of singularities]]

* [[equivariant complex-oriented cohomology theory]]

[[!redirects tautological equivariant line bundle]]

[[!redirects equivariant tautological line bundle]]
[[!redirects equivariant tautological line bundles]]