type universe

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### Context
#### Universes
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[[!include universe - contents]]
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#### Type theory
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[[!include type theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[type theory]], a _type universe_ -- usually written $\mathcal{U}$ or $Type$ -- is a [[type]] whose [[terms]] are either themselves [[type|types]] ([[Russell universes]]), or representations of types in an internal model of the type theory ([[Tarski universes]]). Either way, it is a [[universe]] of (small) [[type|types]], a _universe in type theory_, and sometimes called a _type of types_.

{#TypeUniverseAsReflection} One also speaks of $\mathcal{U}$ as being a _reflection_ of the type system in itself (e.g. [MartinLöf 74, p. 6](#MartinLoef74), [Palmgren, pp. 2-3](#Palmgren), [Rathjen, p. 1](#Rathjen), [Luo 11, section 2.5](#Luo11), [Luo 12, p. 2](#Luo12), [Stanf. Enc. Phil.](#SEP)), following the _[[reflection principle]]_ in [[set theory]].

In [[homotopy type theory]] a type of (small) types is what in [[(∞,1)-category theory|higher]] [[categorical semantics]] is interpreted as a (small) _[[object classifier]]_.  Thus, the type of types is a refinement of the [[type of propositions]] which only contains the [[(-1)-truncated]]/[[h-level]]-1 types (and is semantically a [[subobject classifier]]).

In the presence of a type universe a [[judgement]] of the form

$$
  \vdash A : \mathcal{U}
$$

says that $A$ is a [[term]] of [[type]] $\mathcal{U}$, hence is either a (small) [[type]] itself (for Russell universes), or a representation of types in an internal model of the type theory (for Tarski universes).

More generally, in Russell universes a [[hypothetical judgement]] of the form

$$
  x : X \vdash A(x) : \mathcal{U}
$$

says that $A$ is an $X$-[[dependent type]].

In Tarski universes the corresponding hypothetical judgment would be of the form

$$
  x : X \vdash El(A)(x)\; type
$$

Type universes are important in [[dependent type theory]] for numerous reasons:

1. Dependent type theory with a separate [[type]] [[judgment]] serves as a background for a variety of [[foundations of mathematics]] via the [[propositions as some types]] interpretation of dependent type theory: One can add a [[von Neumann universe]] to the dependent type theory and work entirely in the von Neumann universe for [[material set theory]]; one can add a [[category of sets]] to the dependent type theory and work entirely in the category of sets for [[structural set theory]]; and one can add a [[type of all propositions]] and a [[natural numbers type]] and work in the dependent type theory itself for [[higher-order logic]]. For [[univalent type theory]] and [[univalent foundations]], on the other hand, one needs to add a type universe satisfying certain [[axioms]] and [[axiom schemata]], such as [[universe extensionality]], closure under identity types, closure under dependent sum types, closure under dependent product types, [[propositional resizing]], [[axiom of replacement|replacement]], [[axiom of infinity|infinity]], and [[axiom of choice|choice]], to the dependent type theory.

1. In [[set theory]], one could usually quantify over sets using the [[universal quantifier]] and the [[existential quantifier]]. However, in [[dependent type theory]], one could only quantify over elements of types. Without type universes, types are not elements of type universes, but rather [[judgment|judged]] separately as a type, and thus it is impossible to quantify over types. Quantification over types is necessary for proving [[universal properties]] of various [[mathematical structures]], such as that the [[integers]] are the [[initial object|initial]] [[commutative ring]].

1. Having a [[hierarchy of universes]] where every type is an element of the hierarchy, instead of a single separate type judgment has its own benefits. These includes the lack of annotations in defining types, since all types are elements of universes $A:U_i$ and all family of types are elements of function types $B:A \to U_i$; the lack of additional rules for judgmental equality in type theories with judgmental equality, since they are all admissible; the ease of definitions of types in [[objective type theory]] which does not have [[judgmental equality]]; and the lack of separate type judgments for modes in [[split context]] [[modal type theory]]. However, this comes at the cost of having to formalize the theory of universe levels of the hierarchy of universes before formalizing the type theory.

In [[homotopy type theory]] the type universe $\mathcal{U}$ is often assumed to satisfy the [[univalence]] [[axiom]]. This is a reflection of the fact that in its [[categorical semantics]] as an [[object classifier]] is part of an [[internal (∞,1)-category]] in the ambient [[(∞,1)-topos]]: the one that as an [[indexed category]] is the small [[codomain fibration]].

[[Per Martin-Lof]]'s original type theory contained a Russell universe which contained *all* types, which therefore in particular contained itself, i.e. one had $Type : Type$. But it was pointed out by [[Jean-Yves Girard]] that this was inconsistent; see [[Girard's paradox]]. Thus, modern type theories generally contain a [[hierarchy of universes]].

## Formalizations

### Russell universe
  {#RussellStyle}

A **[[Russell universe]]** or **universe à la Russell** is a [[type]] whose terms *are* types. In the presence of a separate [[judgment]] "$A \;type$", this can be formulated as a [[natural deduction|deduction rule]] of the form

$$\frac{A:\mathcal{U}}{A \;type}$$

Thus, the [type formers](natural+deduction#IntroductionAndElimination) have rules saying which universe they belong to, such as:

$$\frac{A:\mathcal{U}\quad B:A\to \mathcal{U}}{\Pi\, A\, B : \mathcal{U}}$$

With Russell universes, we can also omit the judgment "$A\; type$" and replace it everywhere by a judgment that A is a term of some universe. This is the approach taken in [UFP13](#UFP13).

### Tarski universes
 {#TarskiStyle}

A **[[Tarski universe]]** or a **universe à la Tarski** ([Hofmann, section 2.1.6](#Hofmann), [Luo 12](#Luo12), [Gallozzi 14, p. 40](#Gallozzi14)) is a type $U$ together with an "interpretation" operation allowing us to regard its [[terms]] as _codes_ or _names_ for actual types. Thus we have a rule such as

$$\frac{A:\mathcal{U}}{El(A)\;type}$$

saying that for each term $A$ of the type universe $U$ there is an actual type $El(A)$. This is the approach taken by [[Egbert Rijke]]'s [[Introduction to Homotopy Type Theory]].

Conversely, with notation as used at _[[object classifier in an (infinity,1)-topos]]_, one might write $A = 'El(A)'$ to indicate that $A$ is the _name_ of the type $El(A)$ in the type universe.

We usually also have operations on the universe corresponding to (but not identical to) type formers, such as

$$\frac{A:\mathcal{U}\quad B:A\to \mathcal{U}}{\pi(A, B) : \mathcal{U}}$$

with an [[equality]] $El(\pi(A,B))=\Pi \, El(A)\, El(B)$. Usually this latter equality (and those for other type formers) is a [[judgmental equality]]. If it is only an [[equivalence in homotopy type theory|equivalence]] (i.e. we have a rule which gives us a canonical term of the equivalence type), we may speak of a **[[weakly Tarski universe]]** ([Gallozzi 14, p. 49-50](#Gallozzi14)).

We can give a slightly different definition of weakly Tarski universe using [[propositional equality]] and a larger universe. More precisely, we can consider two (or many) universes $\mathcal{U}$ and $\mathcal{U}'$ with the usual rules for the relative reflection $el(a):\mathcal{U}'$ for any $a:U$, a choice of weakly or strongly Tarski [[computation rules]] for the reflections $El$ and $El'$, and a computation rule for the relative reflection el of $\mathcal{U}$ inside $\mathcal{U}'$ based on propositional equality, which gives us canonical elements of the identity types $Id_{\mathcal{U}'}(\pi'(el(a),el(b)),el(\pi(a,b)))$ and similarly for the other type formers.

If the containing universe is univalent the two definitions turn out to coincide.

The three notions of Tarski universe can be ordered by generality: Every Tarski universe is weakly Tarski by equivalences, because both definitional equality of types $A \equiv B$ and the identity types between two types $A = B$ imply that the two types of equivalent $A \simeq_\mathcal{U} B$. Similarly, every strict Tarski universe is weakly Tarski by the identity type, because definitional equality of types $A \equiv B$ implies that two types are identified $A = B$.

Since each type former is independent of each other, one could also have mixed versions of Tarski universes, where some of the type formers are strictly Tarski and some are weakly Tarski. This leads to $2^n$ possible Tarski universes, where $n$ is the number of type formers in a type theory, and the $2^n$ type theories are only [[partially ordered]] by generality. Internally in an ambient universe, that number becomes $3^n$.

Universes defined internally via [[induction-recursion]] are stricty Tarski. Weakly Tarski universes are easier to obtain in [[semantics]] (see [below](#CategoricalSemantics)): they are somewhat more annoying to use, but probably suffice for most purposes. In [[objective type theory]], there is no [[definitional equality]], so every Tarski universe is weakly Tarski.

### Other universes

There are other universes which one could use. For example, any model of [[material set theory]] $V$ or [[structural set theory]] $\mathcal{E}$ would suffice. Material set theories become Tarski universes by the type family

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

and structural set theories become Tarski universes by the type family

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

where $1:\mathcal{E}$ is the [[terminal object|terminal]] [[separator]].

Thus, in a [[dependent type theory]] with a separate [[type]] [[judgment]], one could simply include inference rules for [[ZFC]] or [[ETCS]].

## Properties

### Universe enlargement
 {#UniverseEnlargement}

Both [[Coq]] and [[Agda]] support [[universe polymorphism]] to deal with the issue of universe enlargement. Moreover, Coq supports [[typical ambiguity]].

### Extensionality principle of type universes

The extensionality principle of type universes is given by the [[univalence axiom]], which for [[Tarski universes]] $(U, T)$ states that [[transport]] across the universal type family $T$ is an equivalence of types for all small types $A:U$ and $B:U$; and for [[Russell universes]] $U$ states that the recursively defined function $\mathrm{idToEquiv}$, which takes identifications between small types $A:U$ and $B:U$ to equivalences between $A$ and $B$, is an equivalence of types for all small types $A:U$ and $B:U$.

### Cumulativity

A tower of universes is **cumulative** if $A:\mathcal{U}_i$ implies $A:\mathcal{U}_{i+1}$ (rather than, say, $Lift(A):\mathcal{U}_{i+1}$).

Cumulative Russell universes have some issues; see for instance [Luo 12](#Luo12).

* [[Coq]] uses Russell style universes. For practical purposes, it also has cumulativity, although there is some question (perhaps mainly semantic) of whether this is true internally or whether it uses casts that are simply hidden from the user.

* [[Agda]] uses non-cumulative Russell style universes.

* [UFP13](#UFP13) (first edition) uses cumulative Russell style universes.

### Categorical semantics
 {#CategoricalSemantics}

[[univalence|Univalent]] [Russell universes](#RussellStyle) have been shown to be interpreted in [[type-theoretic model categories]] presenting the base [[(∞,1)-topos]] [[∞Grpd]] ([Kapulkin-Lumsdaine-Voevodsky 12](#KapulkinLumsdaineVoevodsky12))
and more generally presenting [[(∞,1)-toposes]] of [[(∞,1)-presheaves]] over [[elegant Reedy categories]] ([Shulman 13](#Shulman13)).

Discussion for general [[(∞,1)-toposes]] (of [[(∞,1)-sheaves]]) that should have implementation [weakly Tarski](#TarskiStyle) ([Gallozzi 14, p. 49-50](#Gallozzi14)) is in ([Gepner-Kock 12](#GepnerKock12)).

For more on this see the respective sections at _[[relation between type theory and category theory]]_.

## Other types of types

There is other notions of type of types. Suppose we have a Russell universe $U$ or a Tarski universe $(U, T)$ and a function $P:U \to \mathrm{Prop}_U$. Then the type of $U$-small types which satisfy $P$ is given by the [[dependent sum type]] $\sum_{A:U} P(A)$ for Russell universes and $\sum_{A:U} T(P(A))$ for Tarski universes.

Since universes are internal models of type theory, in a dependent type theory with a separate [[type]] [[judgment]], one could generalize the above notion of "type of $U$-small types which satisfy $P$" to the notion of "type of all types which satisfy $P$". Types $A$, previously elements of the universe $A:U$, are now judged to be types $A \; \mathrm{type}$. The function $P:U \to \mathrm{Prop}_U$ which takes types to propositons is now an operator on types, which is defined using [[inference rules]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash P(A) \; \mathrm{type}} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{proptrunc}_{P(A)}:\mathrm{isProp}(P(A)) \; \mathrm{type}}$$

Examples of such $P$ definable from the existing type formers in [[dependent type theory]] include

* $\mathrm{isProp}$, which results in the [[type of all propositions]],
* $\mathrm{isFinite}$, which results in the [[type of all finite types]],
* given a type $T$, $[(-) \simeq T]$, which results in the type of all types merely [[equivalence of types|equivalent]] to $T$.
* given a type $T$, $[(-) \hookrightarrow T]$, which results in the [[subobject preorder|subtype preorder]] of $T$, the type of all types which merely [[embedding of types|embed into]] $T$.

The resulting **type of types which satisfy $P$** is denoted by $\mathrm{Type}_P$.

Care must be taken for which $P$ one could use to define the type of all types which satisfy $P$. For example, given [[unit type]] $\mathbb{1}$, for $P(A) \equiv A \to \mathbb{1}$ and $P(A) \equiv \mathrm{isSet}(A)$, the resulting $\mathrm{Type}_P$ always contains itself or its [[set truncation]] in addition to the [[empty type]], resulting in [[Girard's paradox]]; thus, one cannot form such types of types in the type theory. Similarly, for
$$P(A) \equiv \prod_{x:A} \prod_{y:A} (x =_A y) \vee \neg (x =_A y)$$
the resulting $\mathrm{Type}_P$ contains its set truncation in addition to the empty type if the [[principle of excluded middle]] holds for all propositions, resulting in [[Girard's paradox]]; thus, one could only form $\mathrm{Type}_P$ for
$$P(A) \equiv \prod_{x:A} \prod_{y:A} (x =_A y) \vee \neg (x =_A y)$$
if one doesn't have excluded middle in the type theory.

Now, supposing that the [[dependent type theory]] has the above rules for $P(A)$, $\mathrm{Type}_P$ could be presented either as a [[Russell universe]] or a [[Tarski universe]] in a [[dependent type theory]]. The difference between the two is that in the former, every type which satisfies $P$ in the type theory is literally an element of the type of types which satisfy $P$, while in the latter, elements of $\mathrm{Type}_P$ are only indices of a type family $\mathrm{El}$; every [[finite type]] in the type theory is only [[essentially small type|essentially $\mathrm{Type}_P$-small]] for [[weak Tarski universes]] or [[judgmentally equal]] to an $P(A)$ for $A:\mathrm{Type}_P$ for [[strict Tarski universes]].

For weak Tarski type of types with satisfy $P$, one also needs the typal congruence rules for forming $P(A)$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{congform}_P(e):P(A) \simeq P(B) \; \mathrm{type}}$$

#### As a strict Tarski universe

As a [[strict Tarski universe]], the type of all types which satisfy $P$ is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Type}_P \; \mathrm{type}}$$

Introduction rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toElem}_A:P(A) \to \mathrm{Type}_P}$$

Elimination rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{witn}_P(A):P(\mathrm{El}(A))}$$

Computation rules for the type of all types which satisfy $P$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \mathrm{El}(\mathrm{toElem}_A(p)) \equiv A \; \mathrm{type}}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \mathrm{witn}_P(\mathrm{toElem}_A(p)) \equiv p:P(A)}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \beta_{\mathrm{Type}_P}^{\mathrm{Witn}_P,A}(p):\mathrm{witn}_P(\mathrm{toElem}_A(p)) =_{P(A)} p}$$

Uniqueness rules for the type of all types which satisfy $P$:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{toElem}_{\mathrm{El}(A)}(\mathrm{witn}_P(A)) \equiv A:\mathrm{Type}_P}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \eta_{\mathrm{Type}_P}(A):\mathrm{toElem}_{\mathrm{El}(A)}(\mathrm{witn}_P(A)) =_{\mathrm{Type}_P} A}$$

[[univalence axiom|Extensionality principle]] of the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P \quad \Gamma \vdash B:\mathrm{Type}_P} {\Gamma \vdash \mathrm{ext}_{\mathrm{Type}_P}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

#### As a weak Tarski universe

As a [[weak Tarski universe]], the type of all types which satisfy $P$ is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Type}_P \; \mathrm{type}}$$

Introduction rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toElem}_A:P(A) \to \mathrm{Type}_P}$$

Elimination rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{witn}_P(A):P(\mathrm{El}(A))}$$

Computation rules for the type of all types which satisfy $P$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \beta_{\mathrm{Type}_P}^{\mathrm{El}, A}(p):\mathrm{El}(\mathrm{toElem}_A(p)) \simeq A \; \mathrm{type}}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \mathrm{congform}_P(\beta_{\mathrm{Type}_P}^{\mathrm{El}, A}(p))(\mathrm{witn}_P(\mathrm{toElem}_A(p))) \equiv p:P(A)}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \beta_{\mathrm{Type}_P}^{\mathrm{witn}_P,A}(p):\mathrm{congform}_P(\beta_{\mathrm{Type}_P}^{\mathrm{El}, A}(p))(\mathrm{witn}_P(\mathrm{toElem}_A(p))) =_{P(A)} p}$$

where the equivalence
$$\mathrm{congform}_P(\beta_{\mathrm{Type}_P}^{\mathrm{El}, A}(p)):P(\mathrm{El}(\mathrm{toElem}_A(p))) \simeq P(A)$$
is provided from the typal computation rules for $P(A)$.

Uniqueness rules for the type of all types which satisfy $P$:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash A \equiv \mathrm{toElem}_{\mathrm{El}(A)}(\mathrm{witn}_P(A)):\mathrm{Type}_P}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \eta_{\mathrm{Type}_P}(A):A =_{\mathrm{Type}_P} \mathrm{toElem}_{\mathrm{El}(A)}(\mathrm{witn}_P(A))}$$

[[univalence axiom|Extensionality principle]] of the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P \quad \Gamma \vdash B:\mathrm{Type}_P} {\Gamma \vdash \mathrm{ext}_{\mathrm{Type}_P}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

#### As a Russell universe

As a [[Russell universe]], the type of all types which satisfy $P$ is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Type}_P \; \mathrm{type}}$$

Introduction rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toElem}_A:P(A) \to \mathrm{Type}_P}$$

Elimination rules for the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{witn}_P(A):P(A)}$$

Computation rules for the type of all types which satisfy $P$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \mathrm{toElem}_A(p) \equiv A \; \mathrm{type}}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \mathrm{witn}_P(\mathrm{toElem}_A(p)) \equiv p:P(A)}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:P(A)}{\Gamma \vdash \beta_{\mathrm{Type}_P}^{\mathrm{finWitn},A}(p):\mathrm{witn}_P(\mathrm{toElem}_A(p)) =_{P(A)} p}$$

Uniqueness rules for the type of all types which satisfy $P$:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \mathrm{toElem}_{A}(\mathrm{witn}_P(A)) \equiv A:\mathrm{Type}_P}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P}{\Gamma \vdash \eta_{\mathrm{Type}_P}(A):\mathrm{toElem}_{A}(\mathrm{witn}_P(A)) =_{\mathrm{Type}_P} A}$$

[[univalence axiom|Extensionality principle]] of the type of all types which satisfy $P$:
$$\frac{\Gamma \vdash A:\mathrm{Type}_P \quad \Gamma \vdash B:\mathrm{Type}_P} {\Gamma \vdash \mathrm{ext}_{\mathrm{Type}_P}(A, B):\mathrm{isEquiv}(\mathrm{idToEquiv}(A, B))}$$

## Related concepts

* [[resizing axiom]]

* [[universe polymorphism]]

* [[univalence]]

* [[Prop]], the [[type of propositions]]

  * [[subobject classifier]]

* [[object classifier]]

* [[type of classes]]

* [[parametric polymorphism]]

* [[Girard's paradox]]

* [[Awodey's conjecture]]

* [[univalent type theory]]

* [[hierarchy of universes]]

## References

Type universes in [[Martin-Löf type theory]] originate in un-stratified and hence inconsistent form with

* [[Per Martin-Löf]], §2.7.1 in: *A Theory of Types*, unpublished note (1971) [[pdf](https://raw.githubusercontent.com/michaelt/martin-lof/master/pdfs/martin-loef1971%20-%20A%20Theory%20of%20Types.pdf), [[MartinLoef1971-ATheoryOfTypes.pdf:file]]]

and then in stratified and consistent form in

* {#MartinLof75} [[Per Martin-Löf]], §1.10 in: _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80**, Elsevier (1975) 73-118 [<a href="https://doi.org/10.1016/S0049-237X(08)71945-1">doi:10.1016/S0049-237X(08)71945-1</a>, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926)]

elaborated on in

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]): *Universes*, pp. 47 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]]]

which also introduces (p. 48) the distinction into notions of *[[Russell universes]]* and *[[Tarski universes]]*.

Further discussion in

* {#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)]

Review and further discussion:

* {#Palmgren} [[Erik Palmgren]], _On Universes in Type Theory_, in *Twenty-Five Years of Constructive Type Theory*, Oxford University Press (1998) 191–204 [[pdf](http://www2.math.uu.se/~palmgren/universe.pdf), [doi:10.1093/oso/9780198501275.003.0012](https://doi.org/10.1093/oso/9780198501275.003.0012)]

Introduction with an eye towards [[homotopy type theory]]:

* {#UFP13} [[Univalent Foundations Project]], §1.3 in *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

Definition of weakly Tarski universes:

* {#Hofmann} [[Martin Hofmann]], section 2.1.6 of _Syntax and semantics of dependent types_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8985))

* {#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))

Detailed discussion of the type of types in [[Coq]] is in

* [[Adam Chlipala]], _[Certified programming with dependent types](http://adam.chlipala.net/cpdt/)_ -- _[Library Universes](http://adam.chlipala.net/cpdt/html/Universes.html)_

See also around slide 8 of the survey

* Frade, _Calculus of inductive constructions_ (2008/2009) ([pdf](http://www3.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf))

A [[formal proof]] in [[homotopy type theory]] that the type of [[homotopy n-types]] is not itself a homotopy $n$-type (it is an $(n+1)$-type) is in

* [[Nicolai Kraus]], [[Christian Sattler]], _The universe $\mathcal{U}_n$ is not an $n$-type_ May 2013 ([pdf](http://red.cs.nott.ac.uk/~ngk/universes.pdf))

[[(∞,1)-category theory|(∞,1)-Categorical]] semantics for [[univalence|univalent]] type universes is discussed in

* {#KapulkinLumsdaineVoevodsky12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], _The Simplicial Model of Univalent Foundations_ ([arXiv:1211.2851](http://arxiv.org/abs/1211.2851))

* {#Shulman13} [[Michael Shulman]], _The univalence axiom for elegant Reedy presheaves_ ([arXiv:1307.6248](http://arxiv.org/abs/1307.6248))

* {#GepnerKock12} [[David Gepner]], [[Joachim Kock]], _Univalence in locally cartesian closed ∞-categories_ ([arXiv:1208.1749](http://arxiv.org/abs/1208.1749))

Relation to [[injective types]]:

* {#Escardo19} [[Martín Escardó]], _Injectives types in univalent mathematics_ ([arXiv:1903.01211](https://arxiv.org/abs/1903.01211))

See also

* {#Rathjen} [[Michael Rathjen]], _The strength of Martin-Löf type theory with superuniverse. Part I_ [pdf](https://www1.maths.leeds.ac.uk/~rathjen/Super.pdf)

* {#SEP} Stanford Encyclopedia of Philosophy,  _[Type theory -- Extensions of type systems, Polymorphism, Paradoxes](http://plato.stanford.edu/entries/type-theory/#6)_

* {#Luo11} [[Zhaohui Luo]], _Contextual analysis of word meanings in type-theoretical semantics_, in Pogodalla, Prost (eds.) _Logical Aspects of Computational Linguistics_, 2011 ([pdf](http://www.cs.rhul.ac.uk/home/zhaohui/LACL11.pdf))

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type-theoretic definition of category

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### Context
#### Category theory
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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
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# Type-theoretic definition of category
* table of contents
{: toc}

## Motivation

There are two main styles of definition of [[category]] in the literature: one which immediately generalises to the usual definition of [[internal category]] and one which immediately generalises to the usual definition of [[enriched category]].  Here we consider the latter definition, and show how it may naturally be expressed in [[dependent type theory]].

Note that in a [[type theory]] without [[identity types]], where [[types]] are [[presets]] (without an inherent [[equality predicate]]), it is not manifestly possible to express concepts that violate the [[principle of equivalence]]; in that categories are not necessarily [[strict category|strict]].  In [[homotopy type theory]], we have identity types, but they are not [[extensional type theory|extensional]]; thus it is also possible to define non-strict categories there (but there is a subtlety; see below).

## Definition

In a [[dependent type theory]] with [[dependent record type|dependent record types]] we can define a type of categories as follows.

We use a two-dimensional syntax, which is convenient to allow inference of [[implicit parameter|implicit parameters]], and to signify notation. We read the horizontal line as a rule, so for instance the second line means that whenever $a$ and $b$ have type $\mathrm{Obj}$, then we have a type $\hom(a,b)$ (with equality).

The notation $p\coloneq P$ signifies that $p$ is a proof of the [[proposition]] $P$ (under [[propositions as types]] or [[propositions as some types]], this may be the same as $p\colon P$, but many type theories treat propositions as distinct from types).

\[\begin{aligned}
  \biggl\{&\mathrm{Obj}\colon\mathrm{Type}, \\
    &\frac{a,b\colon\mathrm{Obj}}{\hom(a,b)\colon\mathrm{Type}_=}, \\
    &\frac{a\colon\mathrm{Obj}}{1_a\colon\hom(a,a)}, \\
    &\frac{f\colon\hom(b,c) \quad g\colon\hom(a,b)}{f\circ g\colon\hom(a,c)}, \\
    &\frac{g\colon\hom(a,b)}{\mathrm{left}\text{-}\mathrm{unit}\coloneq 1_b\circ g=g}, \\
    &\frac{f\colon\hom(a,b)}{\mathrm{right}\text{-}\mathrm{unit}\coloneq f\circ 1_a=f}, \\
    &\frac{f\colon\hom(c,d) \quad
        g\colon\hom(b,c) \quad h\colon\hom(a,b)}{\circ\text{-}\mathrm{assoc}\coloneq f\circ(g\circ h) =
        (f\circ g)\circ h} \\
  &\!\biggr\}
\end{aligned}\]
Here, $\mathrm{Type}$ is a type of types, and $\mathrm{Type}_=$ is a type of types with equality predicates (we may or may not have $\mathrm{Type}=\mathrm{Type}_=$).  Specifically:

* In [[extensional type theory]] (with extensional [[identity types]]), we have $\mathrm{Type} = \mathrm{Type}_=$, and the above definition simply makes no use of the equality predicate on the type of objects.  In this case we obtain [[strict categories]], although that is not immediately visible from the definition.

* In dependent type theory without identity types, basically the only option for $\mathrm{Type}_=$ is the type of [[setoids]].  In this case we obtain a notion of non-strict category, since the type of objects has no equality predicate at all.

* In [[homotopy type theory]], it is natural to take $\mathrm{Type}_=$ to be the type of [[h-sets]]: types whose identity/path types behave extensionally.  We should also restrict the [[homotopy level]] of the type of objects, however, since a true 1-category should have no more than a 1-groupoid of objects; thus $\mathrm{Type}$ in the definition above is really the type of [[h-groupoids]] instead of the type of *all* small types.  That is, we should take $\mathrm{Obj}$ to be $1$-truncated in addition to taking each $\hom(a,b)$ to be $0$-truncated.

  This gives a notion of non-strict category (since there is no equality predicate on a $1$-truncated type other than isomorphism).  However, it is not quite the right definition of "$1$-category" in homotopy type theory, because nothing requires that the paths in $\mathrm{Obj}$ are the same as the isomorphisms defined categorically.  We need to impose a version of the "completeness" condition on a [[complete Segal space]]; in other words, we require that the [[core]] of the category be the [[equivalence of groupoids|equivalent]] as a [[groupoid]] to the original type of objects.

\begin{remark}
The defined type of categories cannot itself be a member of $\mathrm{Type}$, otherwise we run into [[Girard's paradox]]. This is related to the size issues for categories.
\end{remark}

## Related concepts

* [[fully formal definition of ETCS]]

* [[internal category in homotopy type theory]]

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type-theoretic model category

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#### Type theory
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#### Model category theory
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# Type-theoretic model categories

* table of contents
{: toc}

## Idea

[[homotopy type theory|Homotopy type theory]] has [[categorical semantics]] in suitable [[homotopical categories]] which in turn present certain [[(∞,1)-categories]].  The additional structure of type theory corresponds to structure on these homotopical categories that makes them into a certain kind of [[fibration category]], known as a [[type-theoretic fibration category]] or a [[tribe]].

In practice, however, semantic examples of tribes naturally sit inside certain [[Quillen model categories]].  The concept of _type-theoretic model category_ refers to a model category with additional structure that in particular ensures that its subcategory of fibrant objects is a tribe, but also includes additional conditions that make it easier to use model-categorical tools to prove things about the type-theoretic behavior of that tribe.  At present it is not clear whether there is a unique "correct" notion of "type-theoretic model category"; instead there is a range of stronger or weaker hypotheses that are often useful in proofs of this sort.

Regardless, one purpose of the notion(s) is to ensure that all [[(∞,1)-categories]] with sufficient structure can be presented by a type-theoretic model category, and hence provide higher [[categorical semantics]] for [[homotopy type theory]] (without possibly [[univalence]]).  Specifically, every [[locally presentable (∞,1)-category|locally presentable]] [[locally cartesian closed (∞,1)-category]] has a presentation by a type-theoretic model category.  For more on this see also the respective sections at _[[relation between type theory and category theory]]_.

## Definitions

Some of the additional assumptions on a model category $M$ that are often useful to include when constructing semantics of type theory are:

* $M$ is a [[locally cartesian closed category]], or at least pullbacks along [[fibrations]] have a right adjoint.

* $M$ is [[combinatorial model category|combinatorial]],

  or at least [[accessible model category|accessible]]

  or [[cofibrantly generated model category|cofibrantly generated]].

* $M$ is [[right proper model category|right proper]].

* The cofibrations in $M$ are the monomorphisms;

  or at least: they are closed under [[limits]], preserved by [[pullbacks]], and all [[monomorphisms]] are [[cofibrations]],

  or all [[fibrant objects]] are also [[cofibrant objects|cofibrant]].

* Acyclic cofibrations are preserved by pullback along fibrations (implied by the previous two requirements.)

* $M$ is a [[simplicial model category]].

* $M$ is a [[simplicially locally cartesian closed category]].

* The underlying category of $M$ is a [[Grothendieck topos]], or even a [[presheaf topos]].

Definitions in the literature include:

* In [Arndt-Kapulkin](#AK) a "logical model category" was defined to be a model category in which pullback along any fibration has a right adjoint and acyclic cofibrations are preserved by pullback along fibrations.

* In [Shulman 15](#Shulman15) a "type-theoretic model category" was defined to be a right proper model category in which pullback along any fibration has a right adjoint and cofibrations are closed under limits.

* In [Gepner-Kock](#GK) a "combinatorial type-theoretic model category" was defined to be a right proper combinatorial locally cartesian closed model category whose cofibrations are the monomorphisms.

* In [Lumsdaine-Shulman](#LS) a "good model category" was defined to be a simplicial, right proper, simplicially locally cartesian closed model category in which every monomorphism is a cofibration and cofibrations are closed under limits, while an "excellent model category" (no relation to the similarly-named [[excellent model category]]) was defined to be a good model category that is additionally combinatorial.

## Examples

* The [[classical model structure on simplicial sets]] satisfies *all* the above properties, as does the [[injective model structure on simplicial presheaves]].

* Every [[presentable (∞,1)-category|locally presentable]] [[locally Cartesian closed (∞,1)-category]] (by the discussion there) has a presentation by a [[right proper model category|right proper]] [[Cisinski model category]] (a combinatorial model structure on a Grothendieck topos whose cofibrations are the monomorphisms), indeed one whose underlying category is a presheaf topos.  Thus very strong conditions may be assumed without significantly restricting the class of (∞,1)-categories that can be presented.

* Another (counter-)example to keep in mind, however, is the [[canonical model structure on groupoids]], which is combinatorial and simplicial, all objects are fibrant and cofibrant, pullback along fibrations has a right adjoint, cofcibrations are closed under limits, and all monomorphisms are cofibrations; but the category $Gpd$ is not locally cartesian closed (hence not a Grothendieck topos), and not every cofibration is a monomorphism.

* The [[type-theoretic model structure]] on the presheaves on a small category with an [[atomic object|atomic]] [[interval object]]. This gives examples on various type of cartesian [[cubical sets]].

## Properties

### Modeling type theory

(cf. [[categorical semantics of dependent types]])

* Since fibrations are closed under composition, $M$ always models [[Σ-types]].

* If cofibrations are preserved by pullback (such as if they are the monomorphisms), then acyclic cofibrations are preserved by pullback along fibrations if and only if $M$ is right proper.  And by adjointness, if pullback $f^*$ along any fibration $f$ has a right adjoint $f_*$, then acyclic cofibrations are preserved by pullback along fibrations if and only if the functors $f_*$ (for $f$ a fibration) preserve fibrations.  This implies that $M$ models [[Π-types]].

* These $\Pi$-types satisfy [[function extensionality]] if and only if $f_*$ also preserves acyclic fibrations, or equivalently if $f^*$ preserves cofibrations ([Shulman 15, lemma 5.9](#Shulman15)).  In particular, if $M$ is right proper, cofibrations are preserved by pullback, and pullback along any fibration has a right adjoint, then $M$ models $\Pi$-types with function exensionality.

* If acyclic cofibrations are preserved by pullback along fibrations, then for any map $f:x\to y$ between fibrant objects, the functor $f^*: M/y \to M/x$ preserves acyclic cofibrations between fibrant objects of $M/y$ (i.e. fibrations over $y$); see e.g. [Shulman 17, lemma 7.2](#Shulman17) (originally due to Joyal).  This "Frobenius condition" implies that the [[path objects]] of $M$ model [[identity types]].

* [Lumsdaine-Shulman](#LS) shows that an excellent model category (in their sense, see above) models a wide class of [[higher inductive types]].

### Model-categorical constructions

* By [Gepner-Kock, Proposition 7.8](#GK), a [[semi-left-exact reflection|semi-left-exact]] [[left Bousfield localization]] of a combinatorial type-theoretic model category (in their sense, see above) is again a combinatorial type-theoretic model category.

### A warning

In general one wants to think of the interpretation of type theory in the underlying [[tribe]] of a type-theoretic model category as "living in" the $(\infty,1)$-category presented by the model category.  However, this is not automatic merely from the fact that the subcategory of fibrant objects in a model category is a tribe; one needs some stronger conditions such as those above to ensure that the 1-categorical constructions present the relevant $\infty$-categorical ones.

For instance, in [Bordg 17](#Bordg17) it is shown that the category of fibrant objects in the [[projective model structure]] on the category of [[groupoids]] with $\mathbb{Z}/2$-action is a tribe with $\Pi$-types and a universe, but that the universe fails to be [[univalence axiom|univalent]] and indeed that [[function extensionality]] fails to hold, even though the $(\infty,1)$-category presented by this model structure is locally cartesian closed and has an [[object classifier]] for discrete objects.

Generally one wants at least to require that all fibrant objects are cofibrant, in order that the underlying tribe of fibrant objects has the same [[simplicial localization]] as the model category itself.

## References

* {#AK} [[Peter Arndt]] and [[Krzysztof Kapulkin]], *Homotopy-Theoretic Models of Type Theory*, In: Ong L. (eds) Typed Lambda Calculi and Applications. TLCA 2011. Lecture Notes in Computer Science, vol 6690. Springer, Berlin, Heidelberg, [doi](https://doi.org/10.1007/978-3-642-21691-6_7)

* {#Shulman15} [[Michael Shulman]], *Univalence for inverse diagrams and homotopy canonicity*, Mathematical Structures in Computer Science **25** 5 "From type theory and homotopy theory to Univalent Foundations of Mathematics" (2015)  1203-1277 [[arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565)]

* {#Shulman12a} [[Mike Shulman]], _[Minicourse on Homotopy Type Theory](https://home.sandiego.edu/~shulman/hottminicourse2012/)_ part 3, _Categorical models of homotopy type theory_, April 2012 ([pdf](https://home.sandiego.edu/~shulman/hottminicourse2012/03models.pdf))

* {#Cisinski14} [[Denis-Charles Cisinski]], _Univalent universes for elegant models of homotopy types_ ([arXiv:1406.0058](http://arxiv.org/abs/1406.0058))

* {#Bordg17} [[Anthony Bordg]], _On the inadequacy of the projective structure with respect to the Univalence Axiom_, [arxiv:1712.02652](https://arxiv.org/abs/1712.02652)

* {#Shulman17} [[Mike Shulman]], _Univalence for inverse EI diagrams_.  Homology, Homotopy and Applications, 19:2 (2017), p219–249, [DOI](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a12), [arXiv:1508.02410](https://arxiv.org/abs/1508.02410).

* {#LS} [[Peter LeFanu Lumsdaine]] and [[Mike Shulman]], _Semantics of higher inductive types_, [arxiv:1705.07088](https://arxiv.org/abs/1705.07088).

* {#GK} [[David Gepner]] and [[Joachim Kock]], *Univalence in locally cartesian closed categories*, [arxiv:1208.1749](https://arxiv.org/abs/1208.1749)

Combination with the terminology of *[[model topos]]* to *type-theoretic model topos*:

* {#Shulman19} [[Michael Shulman]], §1.3 in: _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004)).

reviewed in

* {#Riehl22} [[Emily Riehl]], §6.1 of: *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) [[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]]

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typed predicate logic

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### Context
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\tableofcontents

## Idea

**Typed predicate logic** or **logic over type theory** can be represented in the language of [[natural deduction]] as a two-[[layered type theory]] consisting of a layer of [[propositions]] represented by the judgment $P \; \mathrm{prop}$, over a layer of [[types]] represented by the judgment $A \; \mathrm{type}$. A [[predicate]] $P$ on a type $A$ in typed predicate logic is simply a [[proposition]] $P$ in the context of a variable of $A$, $x:A \vdash P \; \mathrm{prop}$.

One could add additional logical operations to the proposition layer, such as rules for [[false]], [[true]], [[conjunction]], [[disjunction]], [[implication]], [[universal quantifier]], and [[existential quantifier]], as well as additional rules to the type layer, such as rules for [[propositional equality]].

The type layer could either be a [[simple type theory]] or [[dependent type theory]], examples of the former include [[ZFC]] and [[fully formal ETCS]], while the latter includes the [[SEAR]], as well as presentations of [[dependent type theory]] which use [[propositional equality]] instead of [[judgmental equality]] for [[definitional equality]] and [[conversional equality]].

Typed predicate logic is used in many type theories, including [[simplicial type theory]] and [[cubical type theory]], where the main [[dependent type theory]] is built over a typed predicate logic.

## Presentation

### Judgments and contexts

Typed predicate logic is a type theory which consists of two layers, a layer for types and a layer for propositions.

We have the basic judgement forms of the type layer:

* $\Gamma \vdash A\; \mathrm{type}$ - $A$ is a well-typed type in context $\Gamma$.
* $\Gamma \vdash a : A$ - $a$ is a well-typed term of type $A$ in context $\Gamma$.

And the basic judgement forms of the proposition layer:

* $\Gamma \vdash \phi \; \mathrm{prop}$ - $\phi$ is a well-formed proposition in context $\Gamma$
* $\Gamma \vdash \phi \; \mathrm{true}$ - $\phi$ is a well-formed true proposition in context $\Gamma$

As well as context judgments:

* $\Gamma \; \mathrm{ctx}$ - $\Gamma$ is a well-formed context.

Contexts are defined by the following rules:

$$\frac{}{() \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash A \; \mathrm{type}}{(\Gamma, a:A) \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash \phi \; \mathrm{prop}}{(\Gamma, \phi \; \mathrm{true}) \; \mathrm{ctx}}$$

### Structural rules

The structural rules in logic over type theory include the variable rules, the [[weakening rules]], and the [[substitution rule]].

The variable rules states that we may derive a typing judgment or a true proposition judgment if the typing judgment or true proposition judgment is in the context already:

$$\frac{\Gamma, a \in A, \Delta \; \mathrm{ctx}}{\Gamma, a \in A, \Delta \vdash a \in A} \qquad \frac{\Gamma, P \; \mathrm{true}, \Delta \; \mathrm{ctx}}{\Gamma, P \; \mathrm{true}, \Delta \vdash P \; \mathrm{true}}$$

Let $\mathcal{J}$ be any arbitrary judgment. Then we have the following rules:

The weakening rules:

$$\frac{\Gamma, \Delta \vdash \mathcal{J} \quad \Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, \Delta \vdash \mathcal{J}} \qquad \frac{\Gamma, \Delta \vdash \mathcal{J} \quad \Gamma \vdash P \; \mathrm{prop}}{\Gamma, P \; \mathrm{true}, \Delta \vdash \mathcal{J}}$$

The substitution rule:

$$\frac{\Gamma \vdash a:A \quad \Gamma, b:A, \Delta\vdash \mathcal{J}}{\Gamma, \Delta[a/b] \vdash \mathcal{J}[a/b]}$$

The weakening and substitution rules are admissible rules: they do not need to be explicitly included in the type theory as they could be proven by induction on the structure of all possible derivations.

## Typed predicate logic with equality

Sometimes, typed predicate logic comes with a notion of equality on all the types, [[propositional equality]], and the theory then becomes **typed predicate logic with equality**.

Propositional equality of types and terms is formed by the following rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A = B \; \mathrm{prop}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash a =_A b \; \mathrm{prop}}$$

Propositional equality has its own structural rules:  reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

* Reflexivity of propositional equality

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A = A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a =_A a \; \mathrm{true}}$$

* Symmetry of propositional equality

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true}}{\Gamma \vdash B = A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash a =_A b \; \mathrm{true}}{\Gamma \vdash b =_A a \; \mathrm{true}}$$

* Transitivity of propositional equality

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true} \quad \Gamma \vdash B = C \; \mathrm{true}}{\Gamma \vdash A = C \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash c:A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma \vdash b =_A c \; \mathrm{true}}{\Gamma \vdash a =_A c \; \mathrm{true}}$$

* Principle of substitution of propositionally equal terms:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] = B[b/x] \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] =_{B[b/x]} c[b/x] \; \mathrm{true}}$$

* Variable conversion rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

### Definitions

In any typed predicate logic with equality, we could formally define the [[single assignment operator]] $\coloneqq$ used in [[definitions]] in the theory itself.

We add the following judgments to the theory:

* $\Gamma \vdash A' \coloneqq A \; \mathrm{type}$ - $A'$ is defined to be the type $A$ in context $\Gamma$.
* $\Gamma \vdash a' \coloneqq a : A$ - $a'$ is defined to be the term $a:A$ of type $A$ in context $\Gamma$.

The single assignment operator has its own structural rules: type formation, term introduction, and equality reflection.

* Formation and equality reflection rules for type definition:
$$\frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B \; \mathrm{type}} \qquad \frac{\Gamma \vdash B \coloneqq A \; \mathrm{type}}{\Gamma \vdash B = A \; \mathrm{true}}$$

* Introduction and equality reflection rules for term definition:
$$\frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b:A} \qquad \frac{\Gamma \vdash b \coloneqq a:A}{\Gamma \vdash b =_A a \; \mathrm{true}}$$

## Examples

* [[ZFC]]
* [[ETCS]]
* [[SEAR]]
* the first two layers of [[cubical type theory]] representing the theory of an [[interval]] primitive.
* Some presentations of [[book HoTT]], where the first two layers of the presentation represent the theory of the [[natural numbers]] primitive. This additionally requires the theory to have [[split contexts]].

## See also

* [[predicate logic]]

* [[layered type theory]]

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types and calculus

# Types and calculus
* table of contents
{: toc}

## References

* John Baez, this week's finds in mathematical physics, week 202, [web](http://math.ucr.edu/home/baez/week202.html)

* Conor McBride, The Derivative of a Regular Type is its Type of One-Hole Contexts, [pdf](http://strictlypositive.org/diff.pdf)

* Conor McBride, clowns to the left of me, jokers to the right, [pdf](http://strictlypositive.org/CJ.pdf)

* Conor McBride, differential calculus with data types, [web](http://strictlypositive.org/calculus/)

* derivatives and dissections of data types, lambda the ultimate, [blog](http://lambda-the-ultimate.org/node/1957)

### Differential lambda calculus

* Oleksandr Manzyuk, Tangent bundles in differential lambda-categories, [arXiv:1202.0411v5](http://arxiv.org/abs/1202.0411)

* Thomas Ehrhard, Laurent Regnier, the differential lambda-calculus, [CiteSeerX](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.3283)

## Related entries

* [[seven trees in one]]

* [[Goodwillie calculus]]

category: type theory, formal calculus


types and logic - table

**[[computational trinitarianism]]** =
\linebreak
**[[propositions as types]]**
+**[[programs as proofs]]**
+**[[relation between type theory and category theory|relation type theory/category theory]]**

| [[logic]] | [[set theory]] ([[internal logic of set theory|internal logic]] of) | [[category theory]] | [[type theory]] |
|-----------|---------------------|-----------------|-----------------|
| [[proposition]] | [[set]] | [[object]] | [[type]] |
| [[predicate]] | [[family of sets]] | [[display morphism]] | [[dependent type]] |
| [[proof]] | [[element]] | [[generalized element]] | [[term]]/[[program]] |
| [[cut rule]] |  | [[composition]] of [[classifying morphisms]] / [[pullback]] of [[display maps]] | [[substitution]] |
| [[introduction rule]] for [[implication]] | | [[counit]] for hom-tensor adjunction | lambda |
| [[elimination rule]] for [[implication]] | | [[unit]] for hom-tensor adjunction | application |
| [[cut elimination]] for [[implication]] | | one of the [[zigzag identities]] for hom-tensor adjunction | [[beta reduction]] |
| identity elimination for [[implication]] | | the other [[zigzag identity]] for hom-tensor adjunction | [[eta conversion]] |
| [[true]]  | [[singleton]] | [[terminal object]]/[[(-2)-truncated object]] | [[h-level 0]]-[[type]]/[[unit type]] |
| [[false]] | [[empty set]] | [[initial object]] | [[empty type]] |
| [[proposition]], [[truth value]] | [[subsingleton]] | [[subterminal object]]/[[(-1)-truncated object]] | [[h-proposition]], [[mere proposition]] |
| [[logical conjunction]] | [[cartesian product]] |[[product]] | [[product type]] |
| [[disjunction]] |  [[disjoint union]] ([[support]] of) |[[coproduct]] ([[(-1)-truncation]] of) | [[sum type]] ([[bracket type]] of) |
| [[implication]] | [[function set]] (into [[subsingleton]]) | [[internal hom]] (into [[subterminal object]]) | [[function type]] (into [[h-proposition]]) |
| [[negation]] | [[function set]] into [[empty set]] | [[internal hom]] into [[initial object]] | [[function type]] into [[empty type]] |
| [[universal quantification]] | indexed [[cartesian product]] (of family of [[subsingletons]]) | [[dependent product]] (of family of [[subterminal objects]]) | [[dependent product type]] (of family of [[h-propositions]]) |
| [[existential quantification]] | indexed [[disjoint union]] ([[support]] of) | [[dependent sum]] ([[(-1)-truncation]] of) | [[dependent sum type]] ([[bracket type]] of) |
| [[logical equivalence]] | [[bijection set]] | [[object of isomorphisms]] | [[equivalence type]] |
|  | [[support set]] | [[support object]]/[[(-1)-truncation]] | [[propositional truncation]]/[[bracket type]]
| | | [[n-image]] of [[morphism]] into [[terminal object]]/[[n-truncation]] | [[n-truncation modality]]
| [[equality]] | [[diagonal function]]/[[diagonal subset]]/[[diagonal relation]] | [[path space object]] | [[identity type]]/[[path type]] |
| [[completely presented set]] | [[set]] | [[discrete object]]/[[0-truncated object]]  | [[h-level 2]]-[[type]]/[[set]]/[[h-set]] |
| [[set]] | [[set]] with [[equivalence relation]] | [[groupoid object in an (infinity,1)-category|internal 0-groupoid]] | [[Bishop set]]/[[setoid]] with its [[pseudo-equivalence relation]] an actual [[equivalence relation]] |
| | [[equivalence class]]/[[quotient set]] | [[quotient]] | [[quotient type]] |
| [[induction]] | |[[colimit]] | [[inductive type]], [[W-type]], [[M-type]] |
| higher [[induction]] | | [[(infinity,1)-colimit|higher colimit]] | [[higher inductive type]] |
| - | |  [[0-truncated]] [[(infinity,1)-colimit|higher colimit]] | [[quotient inductive type]] |
| [[coinduction]] | |[[limit]] | [[coinductive type]] |
| | [[preset]] | | [[type]] without [[identity types]] |
| | [[set]] of [[truth values]] | [[subobject classifier]] | [[type of propositions]] |
| [[domain of discourse]] | [[universe]] | [[object classifier]] | [[type universe]] |
| [[modality]] |  |[[closure operator]], ([[idempotent monad|idempotent]]) [[monad]] | [[modal type theory]], [[monad (in computer science)]] |
| [[linear logic]] |  |([[symmetric monoidal category|symmetric]], [[closed monoidal category|closed]]) [[monoidal category]] | [[linear type theory]]/[[quantum computation]] |
| [[proof net]] |  |[[string diagram]] | [[quantum circuit]] |
| (absence of) [[contraction rule]] | | (absence of) [[diagonal]] | [[no-cloning theorem]] |
| | | [[synthetic mathematics]]  |  [[domain specific embedded programming language]] |


typical ambiguity

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

#Contents#
* table of contents
{:toc}

## Ideas

In ([[homotopy type theory|homotopy]]) [[type theory]] with [[type universes]], by the *typical universe ambiguity* or just *typical ambiguity*, for short, one refers to the universe level remaining implicit, as long as a [[proof assistant]] is or would be able to deduce it from context.

Both Russell and Tarski style universes can be typically ambiguous or not.

## Examples

* [[Coq]] uses Russell style universes with typical ambiguity.

* [[Agda]] uses Russell style universes without typical ambiguity: you always have to explicitly indicate the universe level.

* The [[HoTT Book]] (first edition) uses Russell style universes with (except for a few places) typical ambiguity.

## Related concepts

* [[universe polymorphism]]

## References

* Solomon Feferman, _Typical ambiguity: Trying to have your cake and eat it too_

* [Blog Post](http://golem.ph.utexas.edu/category/2012/12/universe_polymorphism_and_typi.html)


Tyrone Cutler

* [Institute page](https://ekvv.uni-bielefeld.de/pers_publ/publ/PersonDetail.jsp?personId=116818871&lang=EN)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=249799)

## Selected writings

On [[fibrations]], such as

on [[Hurewicz fibrations]]:

* [[Tyrone Cutler]], _Fibrations III -- Locally trivial maps and bundles_ (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Fibrations%20III.pdf), [[CutlerFibrationsIII.pdf:file]]]

on [[Hopf fibrations]]:

* [[Tyrone Cutler]], *Fibrations IV* (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Fibrations%20IV.pdf), [[Cutler-FibrationsIV.pdf:file]]]

On [[pointed topological spaces]] and [[one-point compactification]]:

* [[Tyrone Cutler]], _The category of pointed topological spaces_ (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), [[CutlerPointedTopologicalSpaces.pdf:file]]]

On the [[Pontryagin product]] on the [[ordinary homology]] of [[based loop spaces]]:

* [[Tyrone Cutler]], *The Bott-Samelson Theorem* (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Week%2011%20-%20The%20Bott-Samelson%20Theorem.pdf)]

See also

category: people


Tzu-Chieh Wei

* [personal page](http://insti.physics.sunysb.edu/~twei/)

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

## Selected writings

On [[measurement-based quantum computation]]:

* [[Tzu-Chieh Wei]], *Measurement-Based Quantum Computation*, Oxford Research Encyclopedia of Physics (2021) [[arXiv:2109.10111](https://arxiv.org/abs/2109.10111), [doi:10.1093/acrefore/9780190871994.013.31](https://doi.org/10.1093/acrefore/9780190871994.013.31)]

category: people


Tôhoku

[[!redirects Tohoku]]

[[Alexander Grothendieck]] wrote in 1955 a revolutionary article on [[homological algebra]], which was, after almost 3 years in redaction, published in 1957 in Tôhoku Mathematical Journal:

* A. GROTHENDIECK, Sur quelques points d'algèbre homologique, Tôhoku Math. J. vol 9, n.2, 3, 1957, The [Tohoku university](http://www.tohoku.ac.jp/english/), Sendai, Japan; [MR21♯1328](http://www.ams.org/mathscinet-getitem?mr=0102537). Project Euclid open access pdf scans of the French original: [part 1](http://projecteuclid.org/euclid.tmj/1178244839),
[part 2](http://projecteuclid.org/euclid.tmj/1178244774). Russian translation as a separate booklet (Izdatel'stvo inostrannoj literatury, Biblioteka Sbornika Matematika, Moskva 1961): [free djvu scan](http://ncatlab.org/nlab/files/GrotendikTohoku.djvu). Michael Barr's English translation: [pdf](http://www.math.mcgill.ca/barr/papers/gk.pdf) ([review](http://inference-review.com/article/tohoku) by [[Rick Jardine]])

In _Tôhoku_, as it is nowadays called, Grothendieck observes that [[modules]] over [[rings]], and [[sheaves]] of [[abelian groups]] have similar behaviour and that one can develop their [[homological algebra]] in a unified way; this includes the axiomatics of what is for the first time called [[abelian category|abelian categories]]. Essentially, they were defined in an earlier paper by Buchsbaum as "exact categories", with different motivation

* D. A. Buchsbaum, Exact categories and duality, Trans. Amer. Math. Soc. **8** (1955), 1–34 ([MR74407](https://www.ams.org/mathscinet-getitem?mr=74407)) doi:[10.1090/S0002-9947-1955-0074407-6](https://doi.org/10.1090/S0002-9947-1955-0074407-6)

[[Saunders MacLane]] had rudiments of the definition of abelian category, around 1950, but it was a bit different and less invariant notion (and under a different name "bicategory"). The Tôhoku paper also introduces the new weaker notion of an _[[additive category]]_ (in which he also postulates the existence of finite products), as well as some additional axioms (including AB5) to abelian categories ensuring existence of sufficiently many injective objects, what is now called a [[Grothendieck category]]. See _[[additive and abelian categories]]_ for more.

The Tôhoku paper is the place where the notion of an [[equivalence of categories]] is introduced for the first time. In fact the definition in question is a definition of an [[adjoint equivalence]] (unit and counit isomorphisms and the corresponding triangle identities are a part of the definition). This was predating just a little bit Kan's introduction of [[adjoint functors]] in general.

Grothendieck defined universal (co)homological functors and studied special properties of [[resolutions]], including showing that the [[Godement resolution]] of sheaves is really an [[injective resolution]]. There is also a section on [[sheaf cohomology]] of spaces with [[group]] [[action]]. In sheaf theory part of _Tôhoku_, Grothendieck partly continues in spirit of his work from Kansas

* A. Grothendieck, A general theory of fibre spaces with structure sheaf, University of Kansas 1955. ([Grothendieck circle pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf))

During his work on the Tôhoku article in Kansas, Grothendieck did not have access to the manuscript of the 1956 book of Cartan–Eilenberg, about which he heard from his correspondence with [[Serre]]. Thus some of the constructions are overlapping with Cartan–Eilenberg, while being independent.

One of the most important discoveries in _Tôhoku_ is the [[spectral sequence]] for the [[derived functor]] of the composition of two functors (the _[[Grothendieck spectral sequence]]_, which is now more naturally treated in terms of [[triangulated categories]] which Grothendieck invented later with [[Verdier]]).

Chapters:

1. Généralités sur les catégories abéliennes
1. Algebre homologique dans les catégories abéliennes
1. Cohomologie à coefficients dans un faisceau
1. Les Ext de faisceaux de modules
1. Étude cohomologique des espaces a opérateurs

category: reference

## References

The original paper is

* [[Alexandre Grothendieck]], *Sur quelques points d'algèbre homologique*, Tohoku Math. J. (2) 9(2) (1957) 119-221  [[doi:10.2748/tmj/1178244839](https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-9/issue-2/Sur-quelques-points-dalg%c3%a8bre-homologique-I/10.2748/tmj/1178244839.full), [euclid:1178244839](http://projecteuclid.org/euclid.tmj/1178244839)]

English translation:

* by Marcia L. Barr & [[Michael Barr]] (2011): *Some aspects of homological algebra*  [[pdf](https://www.math.mcgill.ca/barr/papers/gk.pdf), [[BarrTranslOf-GrothedieckTohoku.pdf:file]]]

A Russian translation:

* А. Гротендик, _О некоторых вопросах гомологической алгебры_, Москва, Издательство иностранной литературы, 1961.  Перевод с французского Б. Б. Венкова, А. В. Руколайне, Б. В. Степанова.  [DjVu](http://ncatlab.org/nlab/files/GrotendikTohoku.djvu).


U(ℋ)

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

#Contents#
* table of contents
{:toc}

## Idea

The [[unitary group]] of [[unitary operators]] on an [[infinity|infinite]]-[[dimension of a vector space|dimensional]] [[separable Hilbert space|separable]] [[complex vector space|complex]] [[Hilbert space]] $\mathcal{H}$ is traditionally denoted $U(\mathcal{H})$. It does not quite have an established prose name, but is often referred to by some combination of the words "unitary group" and "strong topology".

Its [[quotient]] by the [[circle group|circle]] [[subgroup]] [[U(1)]] is the corresponding [[projective unitary group]] [[PU(ℋ)]].

## Definition
 {#Definition}

Throughout, let $\mathcal{H}$ be any [[separable Hilbert space|separably]] infinite-dimensional [[complex numbers|complex]] [[Hilbert space]].

\begin{definition}\label{TheTopologies}
The set $B(\mathcal{H})$ of [[bounded operators]] carries the following [[topological space|topologies]], characterized by the conditions under which a [[sequence]] $\{ T_k \,\in\, B(\mathcal{H}) \}_{k \in \mathbb{N}}$ [[convergence|converges]] to a fixed operator $T \,\in\, B(\mathcal{H})$:

The [[limit of a sequence|limit]] of this sequence is $\underset{k \to \infty}{lim} T_k \;=\; T$

* {#NormTopology} ...in the **[[norm topology]]** or *[[topology of uniform convergence]]* if

  $$
    \underset{
      { x \in \mathcal{H} }
      \atop
      { \vert x \vert \leq 1 }
    }{sup}
    \big(
      \vert T_k(x) - T(x)
    \big)
    \;\to\; 0
    \,;
  $$

* {#StrongOperatorTopology} ...in the **[[strong operator topology]]** or *[[topology of pointwise convergence]]* if

  $$
    \underset{ x \in \mathcal{H} }{\forall}  \; T_k(x) \to T(x)
    \,;
  $$

* {#WeakOperatorTopology} ...in the **[[weak operator topology]]** if

  $$
    \underset{ x, y \in \mathcal{H}  }{\forall}
    \;
    \big\langle
      T_k(x), \, y
    \big\rangle
    \to
    \big\langle
      T_k(x), \, y
    \big\rangle
    \,;
  $$

* {#CompactOpenTopology} ...in the **[[compact-open topology]]** if

  uniform convergence [above](#NormTopology) holds for the restrictions $T_k\vert_{C}$ and $T\vert_C$ to any [[compact subset]] $C \subset \mathcal{H}$.

\end{definition}
(This list follows [Espinoza & Uribe 2014, p. 2](#EspinozaUribe14).)

## Properties

\begin{prop}\label{WithNormTopology}
  The [norm topology](#NormTopology) makes $U(\mathcal{H})$ a [[topological group]], in fact a [[Banach Lie group]]. Further, it is [[metrizable]], hence [[paracompact]].
\end{prop}
(That it is a Banach Lie group, see e.g. [Schottenloher 2013 p. 4/Sec. 3](#Schottenloher13). Paracompactness/metrizability is [Nikolaus-Sachse-Wockel, Sec. 3](#NSW))

\begin{prop}\label{WithOperatorTopology}
  On $U(\mathcal{H})$ the [weak operator topology](#WeakOperatorTopology) and [strong operator topology](#StrongOperatorTopology) agree and make it a [[topological group]].
\end{prop}
([Hilgert & Neeb 1993, Cor. 94](#HilgertNeeb93); [Schottenloher 2013, Prop. 1](#Schottenloher13); [Espinoza & Uribe 2014, Lem. 1.5](#EspinozaUribe14))

\begin{prop}\label{WithNormTopology}
  The [compact-open topology](#CompactOpenTopology) coincides with the [strong operator topology](#StrongOperatorTopology)
\end{prop}
([Schottenloher 2013 Prop. 2](#Schottenloher13); [Espinoza & Uribe 2014, Lem. 1.8](#EspinozaUribe14))

\begin{prop}
  The [norm topology](#NormTopology) on $U(\mathcal{H})$ (from Prop. \ref{WithNormTopology}) is strictly finer than the (weak or strong) [[operator topology]] (from Prop. \ref{WithOperatorTopology}).
\end{prop}
(e.g. [Espinoza & Uribe 2014, p. 5-6](#EspinozaUribe14), see also [Schottenloher 2013, p. 4](#Schottenloher13))

In **summary**:

\[
  \label{DiagramOfAllTheTopologies}
  U(\mathcal{H})_{norm}
    \xrightarrow{ \;\;\; \neq \;\;\; }
  U(\mathcal{H})_{strong}
    \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow}
  U(\mathcal{H})_{weak}
    \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow}
  U(\mathcal{H})_{comp.open}
\]

(also [Espinoza & Uribe 2014, Thm. 1.2](#EspinozaUribe14))

\begin{prop}
  Equipped with the strong topology (Prop. \ref{WithOperatorTopology}), $U(\mathcal{H})$
is [[complete metric space|completely metrizable]].
\end{prop}
([Neeb 1997, Prop. II.1](#Neeb97), [Schottenloher 2013 Prop. 3](#Schottenloher13), [Espinoza & Uribe 2014, Lem. 1.6](#EspinozaUribe14))

\begin{prop}
  Equipped with the strong topology (Prop. \ref{WithOperatorTopology}), $U(\mathcal{H})$
is *not* [[locally compact topological space|locally compact]].
\end{prop}
([Grigorchuk & de la Harpe 2014, Sec. 5](#GrigorchukHarpe2014))

\begin{theorem}\label{KuiperTheorem}
**(Kuiper's theorem)**
\linebreak
The [[topological group|topological]] [[unitary group]] [[U(ℋ)|$\mathrm{U}(\mathcal{H})$]] in either the

* [norm topology](#NormTopology)

or the

* [strong operator topology](#StrongOperatorTopology)

* [weak operator topology](#WeakOperatorTopology)

* [compact-open topology](#CompactOpenTopology)

is [[contractible]] in that there is a [[left homotopy]] between the [[identity]] $id \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$  and the [[constant function]] $const_{\mathrm{e}} \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$.
\end{theorem}
See the [references](Kuiper's+theorem#References) at *[[Kuiper's theorem]]*.

\begin{proposition}\label{UHCircleFiberBundle}
  The [[U(1)]]-[[quotient space]] [[coprojection]] of [[U(ℋ)]]  over [[PU(ℋ)]] -- both in their strong [[operator topology]] -- is a [[circle group|circle]]-[[principal bundle]]:
$$
  \array{
    S^1 &\hookrightarrow& \mathrm{U}(\mathcal{H})
    \\
    && \big\downarrow
    \\
    && PU(\mathcal{H})
    \mathrlap{ \; \simeq \; \mathrm{U}(\mathcal{H})/S^1 }
  }
$$
\end{proposition}
([Simms 1970, Thm. 1](#Simms70))
\begin{remark}
  Prop. \ref{UHCircleFiberBundle} means in particular that $\mathrm{U}(\mathcal{H}) \xrightarrow{\;} PU(\mathcal{H})$ is [[locally trivial bundle|locally trivial]], hence that the [[coset space]] [[coprojection]] $\mathrm{U}(\mathcal{H}) \xrightarrow{\;} \mathrm{U}(\mathcal{H})/S^1$ admits [[local sections]]. See also at *[[coset space coprojection admitting local sections]]*.
\end{remark}

## Related concepts

* [[stable unitary group]]

* [[PU(ℋ)]]

* [[one-parameter semigroup]]

* [[universal equivariant PU(ℋ)-bundle]]

## References

Early discussion in relation to [[PU(ℋ)]]:

* {#Simms70} [[David John Simms]], *Topological aspects of the projective unitary group*, Math. Proc. Camb. Phil. Soc. **68** 1 (1970) 57-60  ([doi:10.1017/S0305004100001043](https://doi.org/10.1017/S0305004100001043))

Proof that the the weak- and strong- [[operator topologies]] agree on $U(\mathcal{H})$:

* {#HilgertNeeb93} [[Joachim Hilgert]], [[Karl-Hermann Neeb]], Cor. 9.4 in: *Lie Semigroups and their Applications*. Lecture Notes in Mathematics **1552**, Springer 1993 ([doi:10.1007/BFb0084640](https://link.springer.com/book/10.1007/BFb0084640))

Proof that weak-, strong- as well as the [[compact-open topology]] all agree on $U(\mathcal{H})$:

* {#Schottenloher13} [[Martin Schottenloher]], _The Unitary Group In Its Strong Topology_ ([arXiv:1309.5891](https://arxiv.org/abs/1309.5891)), Advances in Pure Mathematics **08** 05 (2018) ([doi:10.4236/apm.2018.85029](https://www.scirp.org/html/4-5301487_84967.htm))

* {#EspinozaUribe14} [[Jesus Espinoza]], [[Bernardo Uribe]], _Topological properties of the unitary group_, JP Journal of Geometry and Topology **16** 1 (2014) 45-55 ([arXiv:1407.1869](https://arxiv.org/abs/1407.1869), [journal](http://www.pphmj.com/abstract/8730.htm))

Previous influential but wrong claim that they do not:

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], Appendix of: _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

  > This paper made the mistake of assuming that because various topologies were distinct on $GL(\mathcal{H})$, they remain distinct on $U(\mathcal{H})$.

Further properties:

Proof of complete metrizability for the strong (and hence weak) operator topology:

* {#Neeb97} [[Karl-Hermann Neeb]], *On a Theorem of S. Banach*, Journal of Lie Theory **8**  (1997) 293–300 ([pdf](https://www.heldermann-verlag.de/jlt/jlt07/NEEBPL.PDF))

Proof of failure of local compactness:

* {#GrigorchukHarpe2014} Rostislav Grigorchuk, Pierre de la Harpe, *Amenability and ergodic properties of topological groups: from Bogolyubov onwards*, in: *Groups, Graphs and Random Walks*, Cambridge University Press 2017 ([arXiv:1404.7030](https://arxiv.org/abs/1404.7030), [doi:10.1017/9781316576571.011]( https://doi.org/10.1017/9781316576571.011))

Proof of [[Kazhdan's property (T)]] for $U(\mathcal{H})$:

* [[Bachir Bekka]], *Kazhdan's Property (T) for the unitary group of a separable Hilbert space*, Geom. funct. anal. 13, 509–520 (2003) [[doi:10.1007/s00039-003-0420-0](https://doi.org/10.1007/s00039-003-0420-0)]

* {#NSW} [[Thomas Nikolaus]], [[Christoph Sachse]], [[Christoph Wockel]], _A Smooth Model for the String Group_, Int. Math. Res. Not. IMRN **16** (2013) 3678-3721 [[doi:10.1093/imrn/rns154](https://doi.org/10.1093/imrn/rns154), [arXiv:1104.4288](https://arxiv.org/abs/1104.4288)]

[[!redirects U(H)]]
[[!redirects UH]]

[[!redirects infinite unitary group with strong topology]]


U-duality

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

#Contents#
* table of contents
{:toc}

## Idea

U-duality is a kind of [[duality in string theory]].

The [[KK-compactifications]] of [[11-dimensional supergravity]] to lower dimensional [[gauged supergravity]] theories have global/local [[gauge groups]] given by [[split real forms]] of the $E$-series of the [[exceptional Lie groups]].

Here the [[compact space|compact]] [[exceptional Lie groups]] form a series [[E8]],[[E7]], [[E6]]

$$
  E_8, E_7, E_6
$$

which is usefully thought of to continue as

$$
  E_5 \coloneqq Spin(10), E_4 \coloneqq SU(5), E_3 \coloneqq SU(3) \times SU(2)
  \,.
$$

(Notice that $E_4$, $E_5$ and $E_6$ are also the traditional choices for [[phenomenology|phenomenologically]] realistic [[grand unified theories]], see there for more.)

The [[split real forms]] of this are traditionally written

$$
  E_{8(8)}, E_{7(7)}, E_{6(6)}
$$

and one sets

$$
  E_{5(5)} \coloneqq Spin(5,5), E_{4(4)} \coloneqq SL(5, \mathbb{R}),
  E_{3(3)} \coloneqq SL(3, \mathbb{R}) \times SL(2, \mathbb{R})
  \,.
$$

For instance the [[scalar fields]] in the field [[supermultiplet]] of $3 \leq d \leq 11$-dimensional supergravity have [[moduli spaces]] parameterized by the [[homogeneous spaces]]

$$
  E_{n(n)}/ K_n
$$

for

$$
  n = 11 - d
  \,,
$$

where $K_n$ is the [[maximal compact subgroup]] of $E_{n(n)}$:

$$
  K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)
$$

$$
  K_5 \simeq Spin(5) \times Spin(5),
  K_4 \simeq Spin(5),
  K_3 \simeq SU(2) \times SO(2)
  \,.
$$

Therefore $E_{n(n)}$ acts as a [[global symmetry]] on the supergravity fields and more generally certain [[subgroups]] of it are "gauged" (have [[gauge fields]]) in [[gauged supergravity]] version.

So for instance maximal [[3d supergravity]] has global (and in fact also local, see there) gauge group given by (the [[split real form]] of) [[E8]].

This is no longer verbatim true for their [[UV-completion]] by the corresponding [[Kaluza-Klein mechanism|compactifications]] of [[string theory]] (e.g. [[type II string theory]] for [[type II supergravity]], etc.). Instead, on these a [[discrete group|discrete subgroup]]

$$
  E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)}
$$

acts as global symmetry. This is called the **U-duality** group of the supergravity theory.

It has been argued that this pattern should continue in some way further to the remaining values $0 \leq d \lt 3$,
with "[[Kac-Moody groups]]" [[E9]], [[E10]], [[E11]] corresponding to the [[Kac-Moody algebras]]

$$
  \mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11}
  \,.
$$

Continuing in the other direction to $d = 10$ ($n = 1$) connects to the [[T-duality]] group $O(d,d,\mathbb{Z})$ of [[type II string theory]].

[[!include U-duality -- table]]

More generally, there is a "[[magic pyramid]]" of super-[[Einstein-Yang-Mills theories]]  and their U-duality groups.

## Properties

### Relation to T-duality and S-duality

U-duality may be understood as being the combination of [[T-duality]] for the compactification torus and [[S-duality]] of [[type IIB superstring theory]]. see ([West 12, section 17.5.4](#West12)).

## Related concepts

* [[embedding tensor]], [[tensor hierarchy]]

* [[exceptional field theory]]

* [[duality in string theory]]

  * [[T-duality]]

  * [[S-duality]]

  * **U-duality**

* [[M-theory on G2-manifolds]]

* [[magic pyramid]]

* [[mysterious duality]]

## References
 {#References}

### General

The hidden [[E7]]-symmetry of the [[KK-compactification]] of [[11-dimensional supergravity]] on a 7-dimensional fiber to maximal $N = 8$ [[4d supergravity]] was first noticed in

* {#CremmerJulia78} [[Eugene Cremmer]], [[Bernard Julia]], _The $N = 8$ supergravity theory. I. The Lagrangian_, Phys. Lett. 80B (1978) 48

* {#CremmerJulia79} [[Eugene Cremmer]], [[Bernard Julia]], _The $SO(8)$ Supergravity_, Nucl. Phys. B 159 (1979) 141 ([spire](http://inspirehep.net/record/140465?ln=en))

and more generally in

* [[Bernard de Wit]], [[Hermann Nicolai]], _D = 11 Supergravity With Local SU(8) Invariance_, Nucl. Phys. B 274, 363 (1986) ([spire](http://inspirehep.net/record/227409?ln=en)),  _Local SU(8) invariance in $d = 11$ supergravity_ ([spire](http://inspirehep.net/record/218601?ln=en))

The further U-duality groups for compactifications of 11d SuGra including [[E6]], [[E7]], [[E8]] were identified in

* [[Eugene Cremmer]], _Supergravities in 5 dimensions_, in Hawking, Rocek (eds.) _Superspace and Supergravity_, Cambridge University Press 1981

* [[Bernard Julia]], _Group Disintegrations_, in Hawking, Rocek (eds.) _Superspace and Supergravity_, Cambridge University Press 1981

Review includes ([Obers-Pioline 98, section 4.2](#ObersPioline98)), see also

* [[Pietro Fré]], Floriana Gargiulo, Ksenya Rulik, Mario Trigiante, _The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards_, Nucl.Phys. B741 (2006) 42-82 ([arXiv:hep-th/0507249](http://arxiv.org/abs/hep-th/0507249))

The concept and terminology of U-duality in string theory/M-theory originates in

* {#HullTownsend94} [[Chris Hull]], [[Paul Townsend]], _Unity of Superstring Dualities_ Nucl.Phys.B438:109-137 (1995) ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

A textbook account is in

* {#West12} [[Peter West]], section 17.3 of _[[Introduction to Strings and Branes]]_, Cambridge University Press 2012

Systematization of U-duality via the relation between [[supersymmetry and division algebras]] and the [[Freudenthal magic square]] is due to

* {#BorstenDuffHughesNagy13} [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic square from Yang-Mills squared*, Phys. Rev. Lett. **112** (2014) 131601  [[arXiv:1301.4176](http://arxiv.org/abs/1301.4176)]

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

Quick surveys include

* [[Jacques Distler]], _Split real forms_ ([blog post](http://golem.ph.utexas.edu/~distler/blog/archives/001213.html)).

Reviews focusing on [[gauged supergravity]] and the non-discrete duality groups include

* {#Samtleben08} [[Henning Samtleben]], _Lectures on Gauged Supergravity and Flux Compactifications_ ([arXiv:0808.4076](http://arxiv.org/abs/0808.4076))

with slides in

* {#Samtleben07} [[Henning Samtleben]], _Gauged supergravity and U-duality_, 2007  ([pdf](http://www.desy.de/uni-th/stringth/ggfl/talks/Samtleben.pdf))

Reviews with more [[M-theory]] lore include

* {#ObersPioline98} [[Niels Obers]], [[Boris Pioline]], *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](https://arxiv.org/abs/hep-th/9809039), doi:10.1016/S0370-1573(99)00004-6]

* Shun'ya Mizoguchi, Germar Schroeder, _On Discrete U-duality in M-theory_, Class.Quant.Grav. 17 (2000) 835-870 ([arXiv:hep-th/9909150](http://arxiv.org/abs/hep-th/9909150))

* {#Roest04} Diederik Roest, _M-theory and Gauged Supergravities_, Fortsch.Phys.53:119-230,2005 ([arXiv:hep-th/0408175](http://arxiv.org/abs/hep-th/0408175))

Discussion in line with the [[F-theory]] perspective on the $SL(2,\mathbb{Z})$-[[S-duality]] -- namely "[[F'-theory]]" -- is in

* {#KumarVafa96} Alok Kumar, [[Cumrun Vafa]], _U-Manifolds_, Phys.Lett. B396 (1997) 85-90 ([arXiv:hep-th/9611007](http://arxiv.org/abs/hep-th/9611007))

Discussion of [[11-dimensional supergravity]] in a form that exhibits the higher U-duality groups already before [[KK-compactification]], via a kind of [[exceptional generalized geometry]],is in

* {#HohmSamtleben13} [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Form of $D=11$ Supergravity_, Phys. Rev. Lett. 111, 231601 (2013) ([arXiv:1308.1673](http://arxiv.org/abs/1308.1673))

Review of [[U-duality]] and [[exceptional generalized geometry]]  in [[KK-compactification]] of [[D=11 supergravity]]:

* [[Henning Samtleben]], *11D Supergravity and Hidden Symmetries*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2303.12682](https://arxiv.org/abs/2303.12682)]

On U-duality (and possibly [[mysterious duality]]) via [[Hypothesis H]] as [[automorphisms]] of [[iterated loop space|iterated]] ([[Sullivan model of loop space|rational]]) [[cyclic loop spaces]] of the ([[rational n-sphere|rational]]) [[4-sphere]]:

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious triality* ([arXiv:2111.14810](https://arxiv.org/abs/2111.14810))

review in:

* {#Voronov21} [[Alexander Voronov]] (joint with [[Hisham Sati]]), *Mysterious Duality*, talk at Texas Tech 2021 ([abstract pdf](https://www.math.ttu.edu/~lhoang/PureMath/abstract-2021-03-15-Voronov.pdf) [[Voronov_MysteriousAbstract.pdf:file]], [slides pdf](https://www.math.ttu.edu/~lhoang/PureMath/Slides-2021-03-15-Voronov.pdf), [[Voronov_MysteriousSlides.pdf:file]])

* [[Alexander Voronov]]: *The $E_k$ symmetry of dimensional reductions of M-theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Voronov2023)]

### $n=3$

The case of $SL(3,\mathbb{Z}) \times SL(2,\mathbb{Z})$ in [[8d supergravity]] is discussed in

* {#LiuMinasian97} [[James Liu]], [[Ruben Minasian]], _U-branes and $T^3$ fibrations_, Nucl.Phys. B510 (1998) 538-554 ([arXiv:hep-th/9707125](http://arxiv.org/abs/hep-th/9707125))

### $n=4$

The case of $SL(5,\mathbb{Z})$ in [[7d supergravity]] from [[M-theory]] is discussed in

* {#Rozali97} [[Moshe Rozali]], _Matrix Theory and U-Duality in Seven Dimensions_, Phys.Lett. B400 (1997) 260-264 ([arXiv:hep-th/9702136](http://arxiv.org/abs/hep-th/9702136))

### $n=7$

The $E_{7(7)}$-symmetry was first discussed in

* [[Bernard de Wit]], [[Hermann Nicolai]], _$D = 11$ Supergravity With Local $SU(8)$ Invariance_, Nucl. Phys. B 274, 363 (1986)

### $n=8$

The case of $E_{8(8)}$ is discussed in

* [[Hermann Nicolai]], _$D = 11$ Supergravity with Local $SO(16)$ Invariance_ , Phys. Lett. B 187, 316 (1987).

* K. Koepsell, [[Hermann Nicolai]], [[Henning Samtleben]], _An exceptional geometry for $d = 11$
supergravity?_, Class. Quant. Grav. 17, 3689 (2000) ([arXiv:hep-th/0006034](http://arxiv.org/abs/hep-th/0006034)).

### $n=9$

The case of [[E9]] is discussed in

* {#EHKNT07} [[François Englert]], Laurent Houart, [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, _An E9 multiplet of BPS states_, JHEP 0705:065,2007 ([arXiv:hep-th/0703285](http://arxiv.org/abs/hep-th/0703285))

### $n=10$
 {#Referencesn10}

The case of [[E10]] is discussed for [[boson|bosonic]] degrees of freedom in

* [[Thibault Damour]], [[Marc Henneaux]], [[Hermann Nicolai]], _$E(10)$ and a 'small tension expansion' of M
theory_, Phys. Rev. Lett. 89, 221601 (2002) ([arXiv:hep-th/0207267](http://arxiv.org/abs/hep-th/0207267));

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _$E(10)$ and $SO(9,9)$ invariant supergravity_, JHEP 0407,
041 (2004) ([arXiv:hep-th/0407101](http://arxiv.org/abs/hep-th/0407101))

and for fermionic degrees of freedom in [[supergravity|supersymmetric]] [[quantum cosmology]] in

* {#Damour14} [[Thibault Damour]], _Quantum supersymmetric Bianchi IX  Cosmology and its hidden Kac-Moody Structure_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[DamourSQC14.pdf:file]])

Review includes

* {#Nicolai08} [[Hermann Nicolai]], _Wonders of $E_{10}$ and $K(E_{10})$_ (2008) ([pdf](http://ipht.cea.fr/Pisp/pierre.vanhove/Paris08/talk_PDF/nicolai.pdf))

* {#Nicolai14} [[Hermann Nicolai]], _On Exceptional Geometry and Supergravity_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[NicolaiTalk14.pdf:file]])

Discussion of [[phenomenology]]:

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _Standard model fermions and $K(E_{10})$_ ([arXiv:1504.01586](https://arxiv.org/abs/1504.01586))

* {#MeissnerNicolai18a} Krzysztof A. Meissner, [[Hermann Nicolai]], _Standard Model Fermions and Infinite-Dimensional R-Symmetries_, Phys. Rev. Lett. 121, 091601 (2018) ([arXiv:1804.09606](https://arxiv.org/abs/1804.09606))

* {#MeissnerNicolai18b} Krzysztof A. Meissner, [[Hermann Nicolai]], _Planck Mass Charged Gravitino Dark Matter_, Phys. Rev. D 100, 035001 (2019) ([arXiv:1809.01441](https://arxiv.org/abs/1809.01441))

### $n=11$

The case of of [[E11]] is discussed in

* [[Peter West]], _$E_{11}$ and M-theory_, Class. Quant. Grav. 18, 4443 (2001) ([arXiv:hep-th/0104081](http://arxiv.org/abs/hep-th/0104081)).

* {#West16} [[Peter West]], _A brief review of E theory_ ([arXiv:1609.06863](http://arxiv.org/abs/1609.06863))

### Further details

A careful discussion of the [[topology]] of the [[Kac-Moody group|Kac-Moody]] U-duality groups is in

* [[Arjan Keurentjes]], _The topology of U-duality (sub-)groups_ ([arXiv:hep-th/0309106](http://arxiv.org/abs/hep-th/0309106))

* [[Arjan Keurentjes]], _U-duality (sub-)groups and their topology_ ([arXiv:hep-th/0312134](http://arxiv.org/abs/hep-th/0312134))

A discussion in the context of [[generalized complex geometry]] / [[exceptional generalized complex geometry]] is in

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

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

General discussion of the [[Kac-Moody groups]] arising in this context is for instance in

* Philipp Fleig, [[Axel Kleinschmidt]], _Eisenstein series for infinite-dimensional U-duality groups_ ([arXiv:1204.3043](http://arxiv.org/abs/1204.3043), [[Kleinschmidt12.pdf:file]])

### Relation to automorphic forms

String theory [[partition functions]] as [[automorphic forms]] for U-duality groups are discussed in

* {#GRV10} [[Michael Green]], Jorge G. Russo, Pierre Vanhove, _Automorphic properties of low energy string amplitudes in various dimensions_ ([arXiv:1001.2535](http://arxiv.org/abs/1001.2535))

[[!redirects U-duality]]
[[!redirects U-duality groups]]


U-duality -- table

|   | [[supergravity]] [[gauge group]] ([[split real form]]) | [[T-duality]] group (via toroidal [[KK-compactification]])  |  [[U-duality]] | maximal [[gauged supergravity]] |  | |
|--------|-----|----|---------------|---------------------|---|
|   | $SL(2,\mathbb{R})$  | 1  |  [[SL(2,Z)|$SL(2,\mathbb{Z})$]] [[S-duality]]   |   10d [[type IIB supergravity]]  |  |
|   |    [[special linear group|SL]]$(2,\mathbb{R}) \times$ [[orthogonal group|O]](1,1) | $\mathbb{Z}_2$  |  [[SL(2,Z)|$SL(2,\mathbb{Z})$]] $\times \mathbb{Z}_2$ |  [[9d supergravity]] |  |
| [[special unitary group|SU]](3)$\times$ SU(2) |    [[special linear group|SL]]$(3,\mathbb{R}) \times SL(2,\mathbb{R})$ | $O(2,2;\mathbb{Z})$  | $SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$ |  [[8d supergravity]] |  |
| [[special unitary group|SU]](5)  | $SL(5,\mathbb{R})$ | $O(3,3;\mathbb{Z})$ | $SL(5,\mathbb{Z})$  | [[7d supergravity]] |    |
| [[spin group|Spin]](10) | $Spin(5,5)$  | $O(4,4;\mathbb{Z})$  | $O(5,5,\mathbb{Z})$ |  [[6d supergravity]] |  |
| [[E6]]  |  $E_{6(6)}$ | $O(5,5;\mathbb{Z})$  | $E_{6(6)}(\mathbb{Z})$  |  [[5d supergravity]] | |
| [[E7]]  |  $E_{7(7)}$ | $O(6,6;\mathbb{Z})$  | $E_{7(7)}(\mathbb{Z})$ |  [[4d supergravity]] | |
| [[E8]]  |  $E_{8(8)}$ | $O(7,7;\mathbb{Z})$ | $E_{8(8)}(\mathbb{Z})$ |  [[3d supergravity]] | |
| [[E9]]  |  $E_{9(9)}$ | $O(8,8;\mathbb{Z})$  | $E_{9(9)}(\mathbb{Z})$ |  [[2d supergravity]] | [[E8]]-[[equivariant elliptic cohomology]] |
| [[E10]]  |  $E_{10(10)}$ | $O(9,9;\mathbb{Z})$  | $E_{10(10)}(\mathbb{Z})$ |  | |
| [[E11]]  |  $E_{11(11)}$ | $O(10,10;\mathbb{Z})$  | $E_{11(11)}(\mathbb{Z})$ |  | |

([Hull-Townsend 94, table 1, table 2](U-duality#HullTownsend94))


ubugnu

## Ubugnu ##
Post--graduate student in theoretical physics, Laboratoire de Physique Théorique, Université d'Oran, Algeria(31100).
email: <ubugnu@gmail.com>
category: people


UC Riverside Seminar on Cobordism and Topological Field Theories

In the last quarter of 2009, there is a seminar at UC Riverside on Cobordism and Topological Field Theories run by [[Julie Bergner]]. The seminar webpage is [here](http://www.math.ucr.edu/~jbergner/cobordism.htm). The goal is to work through recent notes of [[Jacob Lurie|Lurie]] giving an outline of his proof of the [[cobordism hypothesis|Cobordism Hypothesis]], relating [[cobordism]] classes of [[manifolds]] and [[TQFT|topological field theories]].  This work brings together several areas of recent mathematical interest: topological field theories, cobordisms of manifolds, and [[homotopy theory|homotopical approaches]] to [[higher category theory|higher categories]].  The basic definitions and examples of all of the above will be covered and then we'll work towards understanding Lurie's proof.

The main reference is

[[Jacob Lurie|J. Lurie]], _[[On the Classification of Topological Field Theories]]_

##### Friday, September 25 --- John Baez, Introduction to the cobordism hypothesis

[[John Baez]] kicked off the seminar with an introduction to the [[cobordism hypothesis]], how it began its life, and what it means in dimension $n=1$. Already here an issue of "framing" comes into play concerning the first [[Reidemeister move]].

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture1.pdf) by [[Alex Hoffnung]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/10/seminar_on_cobordism_and_topol.html)

##### Friday, October 2 --- Julie Bergner, Manifolds and cobordism

The definition of cobordism, and the category nCob of closed oriented $(n-1)$-manifolds and diffeomorphism classes of oriented cobordisms between them.

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture2.pdf) by [[Alex Hoffnung]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/10/cobordism_and_topological_fiel.html)

##### Friday, October 9 --- Julie Bergner, Topological field theories

The basics of symmetric monoidal categories and functors and the definition of topological field theory.

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture3.pdf) by [[Alex Hoffnung]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/11/cobordism_and_topological_fiel_1.html)

##### Friday, October 16 --- Chris Carlson, Topological field theories in low dimensions

Introduces 2-dimensional topological field theories and commutative Frobenius algebras.

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture4.pdf) by [[Alex Hoffnung]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/12/cobordism_and_topological_fiel_2.html)

##### Friday, October 23 --- John Huerta, A short history of the interaction between quantum field theory and topology

Today's story should involve lots of people, but the stars are Sir Michael Atiyah and Edward Witten. It begins with a paper Witten wrote in 1982, called "Supersymmetry and Morse Theory". In this paper, Witten shows how to use 'supersymmetric quantum mechanics' to compute the de Rham cohomology of a compact manifold, $M$, via Morse theory. This was perhaps the first instance of using quantum theory to find topological invariants.

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture5.pdf) by [[John Huerta]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/12/cobordism_and_topological_fiel_3.html)

##### Friday, October 30 --- Julie Bergner, 2-Extended topological field theories

A sketch of a $2$-extended TFT illustrated by chopping up manifolds into pieces and considering the problems encountered when one considers TFT's of dimension higher than $2$.

* [Notes](http://math.ucr.edu/~alex/cobordism_lecture6_7.pdf) by [[Alex Hoffnung]].
* [Blog entry](http://golem.ph.utexas.edu/category/2009/12/cobordism_and_topological_fiel_4.html)
* (These are notes for both October 30 and November 6)

##### Friday, November 6 --- Julie Bergner, 2-Extended topological field theories and 2-categories

Introduces $2$-categories of cobordisms and suggests the need for $(\infty,2)$-categories and gives a definition for extended  $\mathit{C}$-valued TFT's.

##### Friday, November 13 --- John Baez, Cobordism bicategories

##### Friday, November 20 --- Julie Bergner, Framings, dualizable objects, and the statement of the cobordism hypothesis

##### Friday, December 4 --- Julie Bergner, $(\infty,n)$-categories and diffeomorphisms

All pictures were drawn by [[Christopher Walker]].  Otherwise these notes are mostly the unpolished version from seminar.

[[!redirects UC Riverside Seminar on Cobordisms and Topological Field Theories]]


UC Riverside Seminar on Cobordisms and Topological Field Theories > history

< [[UC Riverside Seminar on Cobordisms and Topological Field Theories]]


Uday Reddy

* [webpage](http://www.cs.bham.ac.uk/~udr/)

category: people


UF-IAS-2012

A research program at [IAS](http://www.ias.edu/) in 2012 on [[homotopy type theory]].

* [Webpage](http://uf-ias-2012.wikispaces.com/)

## Related pages

* Textbook: _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

[[!redirects Univalent Foundations Project]]


Uffe Haagerup

Uffe Haagerup was a Danish mathematician.
He received a Master's degree in 1973 from the University of Copenhagen, advised by [[Gert Pedersen]],
and went on to become a professor at the University of Odense in 1981 without a doctorate.

His work is concentrated around the field of [[von Neumann algebras]]
(and [[noncommutative measure theory]]).
He is responsible for developing many fundamental concepts in this area,
including the [[standard form of a von Neumann algebra]] (analogous
to the [[L^2-space]]),
[[noncommutative L^p-spaces]], [[weights]] and [[operator valued weights]] on [[von Neumann algebras]] (analogous to (fiberwise) [[measures]]), and many others.


Ugo Bruzzo

* [webpage](https://people.sissa.it/~bruzzo/webpage/)

## Selected writings

On [[Fourier-Mukai transforms]]:

* C. Bartocci, [[Ugo Bruzzo]], D. Hernandez Ruiperez, _Fourier-Mukai and Nahm transforms in geometry and mathematical physics_, Progress in Mathematics 276, Birkhauser 2009.

On [[Lie algebroid cohomology]] via [[derived functors]]:

* [[Ugo Bruzzo]], _Lie algebroid cohomology as a derived functor_ ([arxiv/1606.02487](http://arxiv.org/abs/1606.02487))

On [[super Riemann surfaces]]:

* [[Ugo Bruzzo]], Daniel Hernández Ruipérez, *The supermoduli of SUSY curves with Ramond punctures* ([arXiv:1910.12236](https://arxiv.org/abs/1910.12236))

Discussion of traditional [[algebraic geometry]]  for [[super-schemes]]:

* [[Ugo Bruzzo]], Daniel Hernandez Ruiperez, [[Alexander Polishchuk]], _Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes_ ([arXiv:2008.00700](https://arxiv.org/abs/2008.00700))

On [[derived functors]] and [[abelian sheaf cohomology]]:

* [[Ugo Bruzzo]], _Derived Functors and Sheaf Cohomology_, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2, 2020 ([doi:10.1142/11473](https://doi.org/10.1142/11473))

category: people