objective type theory

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Deduction and Induction
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[[!include deduction and induction - contents]]
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#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
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#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
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## Idea

**Objective type theory** is a [[dependent type theory]] without [[judgmental equality]].

This means that in additon to being a [[weak type theory]], where the computation and uniqueness rules ([[beta-reduction]] and [[eta-conversion]]) for each type in the type theory are all propositional computational and uniqueness rules using the [[identity type]], objective type theory is similar to other non-type theory foundations such as the various flavors of set theory, since it also only has one notion of equality, propositional equality, and uses propositional equality in [[definitions]].

Objective type theory has [[decidable]] [[type checking]], and the type checking can be done in quadratic time.

## Formalizations and syntax

### Substitutions

There are two ways one could define [[substitutions]] in [[dependent type theory]], either by [[explicit substitution]] or by showing that the [[substitution rule]] is an [[admissible rule]]. However, for objective type theory, only the latter approach is possible.

The problem with explicit substitutions is that with explicit substitution approaches, the contexts are usually formalized with [[judgmental equality]] of [[morphisms]] of contexts, such that the [[categorical semantics]] correspond to a functor from the category of contexts $C$ to the [[2-category]] $\mathrm{Cat}$ which takes a context $\Gamma:C$ to the slice category $C/\Gamma$. However, without judgmental equality, $C$ doesn't form a category anymore, since the morphisms in $C$ don't form a [[set]] anymore, and one can no longer express the unit and associatvity laws for composition of morphisms of contexts, which are needed for the unit and associativity laws of explicit substitution. The same holds for explicit substitution via [[telescopes]]; here, substitutions are elements of telescopes, and one has judgmental equalities of the substitutions themselves.

Alternatively, one can try to define structure indicating that the collection of contexts $C$ forms an [[(infinity,1)-category|$(\infty, 1)$-category]], avoiding the need for judgmental equality entirely, but one quickly runs into [[coherence]] issues, since the structure of an $(\infty, 1)$-category cannot be expressed from nothing using a finite amount of structure. This is similar to the issues in defining an $(\infty, 1)$-category in dependent type theory without using pre-existing [[simplicial set|simplicial]] or [[cubical set|cubical]] structure or [[modal type theory|modalities]]. Thus, it is currently unknown how to avoid this use of judgmental equality for explicit substitution and/or telescopes in dependent type theory, rendering it impossible for those to formulate objective type theory.

The other approach involving an admissible substitution rule doesn't suffer from this issue because the analogue in admissible substitution approaches of the judgmental equality in explicit substitution approaches resides in the metatheory rather than the actual theory itself, so the type theory itself can genuinely be said to not contain any judgmental equalities.

### Definitions in objective type theory

In section 9 of [Winterhalter2020](#Winterhalter20), [[Theo Winterhalter]] indicates that there are many ways to formalize [[dependent type theory]]. One important aspect is whether to use [[Russell universes]] or a separate [[type]] [[judgment]] to denote types: [van der Berg & den Besten 2021](#BB21) and [UPF 2013](#UPF13) both use [[Russell universes]] in their formal presentation of dependent type theory, while [Rijke 2022](#Rijke22) uses a separate type judgment to denote types.

In the context of objective type theory, there is no [[judgmental equality]] in the type theory in the traditional sense, so there is the question of how to define aliases of terms and types in the type theory. For the case of terms, that is easily resolved by using identity types. For example, to define $2$ as the successor of the succcessor of zero in the [[natural numbers type]], one could postulate an [[identification]] $\mathrm{def}2:2 =_{\mathrm{N}} s(s(0))$. On the other hand, the situation for types is a bit more complicated. For example, the [[isContr]] [[modality]] $\mathrm{isContr}(A)$ in [[dependent type theory]] indicating whether the type $A$ is a [[contractible type]] is usually defined to be $\sum_{x:A} \prod_{y:A} \mathrm{Id}_A(x, y)$. But without [[judgmental equality]], one cannot simply write
$$\mathrm{isContr}(A) \equiv \sum_{x:A} \prod_{y:A} \mathrm{Id}_A(x, y)$$
When presenting [[dependent type theory]] using [[Russell universes]], the answer is as simple as that for terms: one simply uses [[typal equality]] instead, because every type is an element of a Russell universe, and so one could write
$$\mathrm{defisContr}_A:\mathrm{isContr}(A) =_{\mathrm{Type}_i} \sum_{x:A} \prod_{y:A} \mathrm{Id}_A(x, y)$$
for types $\mathrm{isContr}(A):\mathrm{Type}_i$ and $\sum_{x:A} \prod_{y:A} \mathrm{Id}_A(x, y):\mathrm{Type}_i$. On the other hand, when using a separate type judgment, types are not elements of other types, and thus one cannot compare them for typal equality. Instead, one has to use [[equivalences of types]] instead:
$$\mathrm{defisContr}_A:\mathrm{isContr}(A) \simeq \sum_{a:A} \prod_{b:A} \mathrm{Id}_A(a, b)$$
This poses another problem: the [[equivalence type]] $A \simeq B$ has not yet been defined yet.

When expanded out using [[dependent sum types]] and [[function types]], one gets
$$\sum_{f:A \to B} \mathrm{isEquiv}(f)$$

[UPF 2013](#UPF13) says that the type family $f:A \to B \vdash \mathrm{isEquiv}(f)$ comes with

* a family of functions
$$f:A \to B \vdash \mathrm{qinvtoisEquiv}(f):\sum_{g:B \to A} \left(\prod_{y:B} \mathrm{Id}_B(f(g(y)), y)\right) \times \left(\prod_{x:A} \mathrm{Id}_A(g(f(x)), x)\right) \to \mathrm{isEquiv}(f)$$

* a family of functions
$$f:A \to B \vdash \mathrm{isEquivtoqinv}(f):\mathrm{isEquiv}(f) \to \sum_{g:B \to A} \left(\prod_{y:B} \mathrm{Id}_B(f(g(y)), y)\right) \times \left(\prod_{x:A} \mathrm{Id}_A(g(f(x)), x)\right)$$

* a family of identifications
$$f:A \to B, p:\mathrm{isEquiv}(f), q:\mathrm{isEquiv}(f) \vdash \mathrm{proptruncisEquiv}(f, p, q):\mathrm{Id}_{\mathrm{isEquiv}(f)}(p, q)$$

This could be made into inference rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B \vdash \mathrm{isEquiv}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B \vdash \mathrm{qinvtoisEquiv}(f):\sum_{g:B \to A} \left(\prod_{y:B} \mathrm{Id}_B(f(g(y)), y)\right) \times \left(\prod_{x:A} \mathrm{Id}_A(g(f(x)), x)\right) \to \mathrm{isEquiv}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B \vdash \mathrm{isEquivtoqinv}(f):\mathrm{isEquiv}(f) \to \sum_{g:B \to A} \left(\prod_{y:B} \mathrm{Id}_B(f(g(y)), y)\right) \times \left(\prod_{x:A} \mathrm{Id}_A(g(f(x)), x)\right)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, p:\mathrm{isEquiv}(f), q:\mathrm{isEquiv}(f) \vdash \mathrm{proptruncisEquiv}(f, p, q):\mathrm{Id}_{\mathrm{isEquiv}(f)}(p, q)}$$

Breaking the functions down, the inference rules become

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B \vdash \mathrm{isEquiv}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, g:B \to A, G:\prod_{y:B} \mathrm{Id}_B(f(g(y)), y), H:\prod_{x:A} \mathrm{Id}_A(g(f(x)), x) \vdash \mathrm{qinvtoisEquiv}(f, g, G, H):\mathrm{isEquiv}(f)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, p:\mathrm{isEquiv}(f) \vdash \mathrm{isEquivtoqinv}(f, p):\sum_{g:B \to A} \left(\prod_{y:B} \mathrm{Id}_B(f(g(y)), y)\right) \times \left(\prod_{x:A} \mathrm{Id}_A(g(f(x)), x)\right)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, p:\mathrm{isEquiv}(f), q:\mathrm{isEquiv}(f) \vdash \mathrm{proptruncisEquiv}(f, p, q):\mathrm{Id}_{\mathrm{isEquiv}(f)}(p, q)}$$

One could postulate a single equality judgment between types $A \equiv B \; \mathrm{type}$ which reflects into an equivalence:
$$\frac{\Gamma \vdash B \equiv A \; \mathrm{type}}{\Gamma \vdash \mathrm{def}_{A, B}:\sum_{f:A \to B} \mathrm{isEquiv}(f)}$$
A similar equality judgment could be made for terms, with a similar rule to reflect it into an identification:
$$\frac{\Gamma \vdash b \equiv a:A}{\Gamma \vdash \mathrm{def}_{A, a, b}:\mathrm{Id}_A(a, b)}$$
These equalities, while judgmental, are different from the judgmental equality found in other dependent type theories like [[Martin-Löf type theory]] and [[cubical type theory]] in that they have neither the [[structural rules]] nor the [[congruence rules]] found in those theories: they are only a formalization of a notational representation of definitional equality and definitional equivalence otherwise represented by [[identity types]] and [[equivalence types]].

### Without a separate type judgment

In this section, we describe a formalization of objective type theory using an infinite hierarchy of Russell universes with cumulativity, in the style of [UPF 2013](#UPF13).

#### Judgments, contexts, and universes

This presentation of objective type theory consists of the following judgments:

* Element judgments, where we judge $a$ to be an element of $A$, $a:A$

* Context judgments, where we judge $\Gamma$ to be a context, $\Gamma \; \mathrm{ctx}$.

In addition, we have a [[countable set|countably]] [[infinite set|infinite]] number of [[inference rules]] for the countably infinite number of [[Russell universes]] $\mathrm{Type}_0, \mathrm{Type}_1, \mathrm{Type}_2, \ldots$ in the theory, here represented by a [[natural numbers]] [[metavariable]] for conciseness:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Type}_i:\mathrm{Type}_{i + 1}}$$

as well as inference rules for either [[cumulativity]], that any type in a universe is also in the next universe of the hierarchy, or [[lifting]], that any type in a universe can be lifted to the next universe of the hierarchy:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma \vdash A:\mathrm{Type}_{i + 1}}\mathrm{cumul} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma \vdash \mathrm{Lift}(A):\mathrm{Type}_{i + 1}}\mathrm{lifting}$$

Contexts are lists of element judgments $a:A$, $b:B$, $c:C$, et cetera, and are formalized by the inference rules for the empty context and extending the context by a element judgment

$$\frac{}{() \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash A:\mathrm{Type}_i}{(\Gamma, a:A) \; \mathrm{ctx}}$$

#### Variable rule

There is one additional [[structural rule]] in objective type theory, the [[variable rule]].

The variable rule states that we may derive a element judgment if the element judgment is in the context already:

$$\frac{\Gamma, a:A, \Delta \; \mathrm{ctx}}{\Gamma, a:A, \Delta \vdash a:A}$$

#### Admissible structural rules

The [[weakening rule]] and the [[substitution rule]] 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.

Let $\mathcal{J}$ be any arbitrary judgment. Then the weakening rule is
$$\frac{\Gamma, \Delta \vdash \mathcal{J} \quad \Gamma \vdash A:\mathrm{Type}_i}{\Gamma, a:A, \Delta \vdash \mathcal{J}}$$
and the substitution rule is
$$\frac{\Gamma \vdash a:A \quad \Gamma, b:A, \Delta(b) \vdash \mathcal{J}(b)}{\Gamma, \Delta(a) \vdash \mathcal{J}(a)}$$

#### Families of types and elements

A family of elements is an element $b:B$ in the context of the variable judgment $x:A$, $x:A \vdash b:B$. They are usually written as $b(x)$ to indicate its dependence upon $x$. Given a particular element $a:A$, the element $b(a)$ is an element dependent upon $a:A$.

Since types are elements of universe, a family of types is simply a family of elements of universes.

#### Identity types

Formation rules for identity types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma, a:A, b:A \vdash a =_A b:\mathrm{Type}_i}$$

Introduction rules for identity types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma, a:A \vdash \mathrm{refl}_A(a) : a =_A a}$$

Elimination rule for identity types:
$$\frac{\Gamma, x:A, y:A, p:x =_A y, \Delta(x, y, p) \vdash C(x, y, p):\mathrm{Type}_i}{\Gamma, x:A, t:C(x, x, \mathrm{refl}_A(x), y:A, p:x =_A y, \Delta(x, t, y, p) \vdash \mathrm{ind}_{=_A}(x, t, y, p):C(x, y, p)}$$

Computation rules for identity types:
$$\frac{\Gamma, x:A, y:A, p:x =_A y, \Delta(x, y, p) \vdash C(x, y, p):\mathrm{Type}_i}{\Gamma, x:A, t:C(x, x, \mathrm{refl}_A(x), \Delta(x, t, x, \mathrm{refl}_A(x)) \vdash \beta_{=_A}(x, t):\mathrm{ind}_{=_A}(x, t, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t}$$

#### Definitions

Definitions of a symbol $b$ for the element $a:A$ are made by using [[identity types]] between the symbol and element: $\mathrm{def}_{a, b}:a =_A b$. Definitions of a symbol $B$ for the type $A:\mathrm{Type}_i$ are made in the same way, as $\mathrm{def}_{A, B}:A =_{\mathrm{Type}_i} B$. Thus, the [[identity type]] behaves very similarly to [[explicit conversion]] as discussed in section 9.2 of [Winterhalter2020](#Winterhalter20).

#### Dependent function types

Formation rules for dependent function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma \vdash \prod_{x:A} B(x):\mathrm{Type}_i}$$

Introduction rules for dependent function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

Elimination rules for dependent function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, f:\prod_{x:A} B(x), a:A \vdash \mathrm{ind}_{\prod_{x:A} B(x)}(f, a):B(a)}$$

Computation rules for dependent function types
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma, a:A \vdash \beta_{\prod_{x:A} B(x)}(a):\mathrm{ind}_{\prod_{x:A} B(x)}(\lambda(x:A).b(x), a) =_{B(a)} b(a)}$$

Uniqueness rules for dependent function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, f:\prod_{x:A} B(x) \vdash \eta_{\prod_{x:A} B(x)}(f):f =_{\prod_{x:A} B(x)} \lambda(x:A).\mathrm{ind}_{\prod_{x:A} B(x)}(f, x)}$$

#### Function types

Formation rules for function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma \vdash A \to B:\mathrm{Type}_i}$$

Introduction rules for function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i \quad \Gamma, x:A \vdash b(x):B}{\Gamma \vdash \lambda(x:A).b(x):A \to B}$$

Elimination rules for function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, f:A \to B, a:A \vdash \mathrm{ind}_{A \to B}(f, a):B}$$

Computation rules for function types
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i \quad \Gamma, x:A \vdash b(x):B}{\Gamma, a:A \vdash \beta_{A \to B}(a):\mathrm{ind}_{A \to B}(\lambda(x:A).b(x), a) =_{B} b(a)}$$

Uniqueness rules for function types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, f:A \to B \vdash \eta_{A \to B}(f):f =_{A \to B} \lambda(x:A).\mathrm{ind}_{A \to B}(f, x)}$$

#### Pair types

Formation rules for pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma \vdash A \times B:\mathrm{Type}_i}$$

Introduction rules for pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, x:A, y:B \vdash \mathrm{in}(x, y):A \times B}$$

Elimination rules for pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, z:A \times B \vdash \mathrm{ind}_{A \times B}^A(z):A} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, z:A \times B \vdash \mathrm{ind}_{A \times B}^B(z):B}$$

Computation rules for pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, x:A, y:B \vdash \beta_{A \times B}^A(x, y):\mathrm{ind}_{A \times B}^A(\mathrm{in}(x, y)) =_A x}$$

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, x:A, y:B \vdash \beta_{A \times B}^B(x, y):\mathrm{ind}_{A \times B}^B(\mathrm{in}(x, y)) =_B y}$$

Uniqueness rules for pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, z:A \times B \vdash \eta_{A \times B}(z):z =_{A \times B} \mathrm{in}(\mathrm{ind}_{A \times B}^A(z), \mathrm{ind}_{A \times B}^B(z))}$$

#### Dependent pair types

Formation rules for dependent pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma \vdash \sum_{x:A} B(x):\mathrm{Type}_i}$$

Introduction rules for dependent pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, x:A, y:B(x) \vdash \mathrm{in}(x, y):\sum_{x:A} B(x)}$$

Elimination rules for dependent pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma \vdash \mathrm{ind}_{\sum_{x:A} B(x)}^A:\left(\sum_{x:A} B(x)\right) \to A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_{\sum_{x:A} B(x)}^B:\prod_{z:\sum_{x:A} B(x)} B(\mathrm{ind}_{\sum_{x:A} B(x)}^A(z))}$$

Computation rules for dependent pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, x:A, y:B(x) \vdash \beta_{\sum_{x:A} B(x)}^A(x, y):\mathrm{ind}_{\sum_{x:A} B(x)}^A(\mathrm{in}(x, y)) =_A x}$$

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, x:A, y:B(x) \vdash \beta_{\Sigma(x:A).B(x)}^B(x, y):\mathrm{ind}_{\sum_{x:A} B(x)}^B(\mathrm{in}(x, y)) =_{B(\mathrm{ind}_{\sum_{x:A} B(x)}^A(\mathrm{in}(x, y)))} y}$$

Uniqueness rules for dependent pair types:
$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma, z:\sum_{x:A} B(x) \vdash \eta_{\sum_{x:A} B(x)}(z):z =_{\sum_{x:A} B(x)} \mathrm{in}(\mathrm{ind}_{\sum_{x:A} B(x))}^A(z), \mathrm{ind}_{\sum_{x:A} B(x)}^B(z))}$$

#### isContr, uniqueness quantifiers, isEquiv, and equivalence types

Now that we have [[identity types]], [[dependent sum types]], and [[dependent product types]], we can use that to define

* the [[isContr]] [[modality]]:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma \vdash \mathrm{isContr}(A):\mathrm{Type}_i} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i}{\Gamma \vdash \mathrm{defisContr}(A):\mathrm{isContr}(A) =_{\mathrm{Type}_i} \sum_{y:A} \prod_{z:A} y =_{A} z}$$

* the [[uniqueness quantifier]]:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma \vdash \exists!x:A.B(x):\mathrm{Type}_i} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma, x:A \vdash B(x):\mathrm{Type}_i}{\Gamma \vdash \mathrm{def}\exists!_{x:A.B(x)}:\exists!x:A.B(x) =_{\mathrm{Type}_i} \mathrm{isContr}\left(\sum_{x:A} B(x)\right)}$$

* the [[isEquiv]] [[type family]]:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, f:A \to B \vdash \mathrm{isEquiv}(f):\mathrm{Type}_i} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma, f:A \to B \vdash \mathrm{defisEquiv}_{A, B}(f):\mathrm{isEquiv}(f) =_{\mathrm{Type}_i} \prod_{y:B} \exists!x:A.f(x) =_{B} y}$$

* the [[equivalence type]]:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma \vdash A \simeq B:\mathrm{Type}_i} \qquad \frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma \vdash \mathrm{def}_{A \simeq B}:(A \simeq B) =_{\mathrm{Type}_i} \sum_{f:A \to B} \mathrm{isEquiv}(f)}$$

#### Univalence

The [[univalence axiom]] states that the identity type between two types of a universe is equivalent to the equivalence type between said types:

$$\frac{\Gamma \vdash A:\mathrm{Type}_i \quad \Gamma \vdash B:\mathrm{Type}_i}{\Gamma \vdash \mathrm{ua}_i(A, B):(A =_{\mathrm{Type}_i} B) \simeq (A \simeq B)}$$

#### Positive types

Now that we have the [[uniqueness quantifier]] we can combine the [[elimination rule]], the [[computation rule]], and the [[uniqueness rule]] for any [[positive type]] into one rule, the [[induction]] rule.

##### Unit type

Formation rules for the unit type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{1}:\mathrm{Type}_i}$$

Introduction rules for the unit type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{pt}:\mathbb{1}}$$

Induction rule for the unit type:
$$\frac{\Gamma, x:\mathbb{1} \vdash C(x):\mathrm{Type}_i \quad \Gamma \vdash c_\mathrm{pt}:C(\mathrm{pt})}{\Gamma \vdash \mathrm{up}_\mathbb{1}^C(c_\mathrm{pt}):\exists!c:\prod_{x:\mathbb{1}} C(x).(c(\mathrm{pt}) =_{C(\mathrm{pt})} c_\mathrm{pt})}$$

##### Empty type

Formation rules for the empty type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{0}:\mathrm{Type}_i}$$

Induction rule for the empty type:
$$\frac{\Gamma, x:\mathbb{0} \vdash C(x):\mathrm{Type}_i}{\Gamma \vdash \mathrm{up}_\mathbb{0}^C:\mathrm{isContr}\left(\prod_{x:\mathbb{0}} C(x)\right)}$$

##### Booleans type

Formation rules for the booleans type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{2}:\mathrm{Type}_i}$$

Introduction rules for the booleans type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 1:\mathbb{2}}$$

Induction rule for the booleans type:
$$\frac{\Gamma, x:\mathbb{2} \vdash C(x):\mathrm{Type}_i \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_1:C(1)}{\Gamma \vdash \mathrm{up}_\mathbb{2}^C(c_0, c_1):\exists!c:\prod_{x:\mathbb{2}} C(x).(c(0) =_{C(0)} c_0) \times (c(1) =_{C(1)} c_1)}$$

##### Natural numbers type

Formation rules for the natural numbers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N}:\mathrm{Type}_i}$$

Introduction rules for the natural numbers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{N} \to \mathbb{N}}$$

Induction rule for the natural numbers type:
$$\frac{\Gamma, x:\mathbb{N} \vdash C(x):\mathrm{Type}_i \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))}{\Gamma \vdash \mathrm{up}_\mathbb{N}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{N}} C(x).(c(0) =_{C(0)} c_0) \times \prod_{x:\mathbb{N}} c(s(x)) =_{C(s(x))} c_s(c(x))}$$

##### Integers type

Formation rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{Z}:\mathrm{Type}_i}$$

Introduction rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{Z}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{Z} \simeq \mathbb{Z}}$$

Induction rule for the integers type:
$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x):\mathrm{Type}_i \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{Z}} C(x).(c(0) =_{C(0)} c_0) \times \prod_{x:\mathbb{Z}} c(s(x)) =_{C(s(x))} c_s(c(x))}$$

### With a separate type judgment

In this section, we describe a formalization of objective type theory using a type judgment, in the style of [Rijke 2022](#Rijke22).

#### Judgments and contexts

This presentation of objective type theory consists of the following judgments:

* Type judgments, where we judge $A$ to be a type, $A \; \mathrm{type}$

* Element judgments, where we judge $a$ to be an element of $A$, $a:A$

* Context judgments, where we judge $\Gamma$ to be a context, $\Gamma \; \mathrm{ctx}$.

Contexts are lists of element judgments $a:A$, $b:B$, $c:C$, et cetera, and are formalized by the rules for the empty context and extending the context by a element judgment

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

#### Variable rule

There is one additional [[structural rule]] in objective type theory, the [[variable rule]].

The variable rule states that we may derive a element judgment if the element judgment is in the context already:

$$\frac{\Gamma, a:A, \Delta \; \mathrm{ctx}}{\Gamma, a:A, \Delta \vdash a:A}$$

#### Admissible structural rules

The [[weakening rule]] and the [[substitution rule]] 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.

Let $\mathcal{J}$ be any arbitrary judgment. Then the weakening rule is
$$\frac{\Gamma, \Delta \vdash \mathcal{J} \quad \Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, \Delta \vdash \mathcal{J}}$$
and the substitution rule is
$$\frac{\Gamma \vdash a:A \quad \Gamma, b:A, \Delta(b) \vdash \mathcal{J}(b)}{\Gamma, \Delta(a) \vdash \mathcal{J}(a)}$$

#### Families of types and elements

A family of types is a type $B$ in the context of the element judgment $x:A$, $x:A \vdash B \; \mathrm{type}$, they are usually written as $B(x)$ to indicate its dependence upon $x$. Given a particular element $a:A$, the type $B(a)$ is a type dependent upon $a:A$.

A family of terms is a term $b:B$ in the context of the variable judgment $x:A$, $x:A \vdash b:B$. They are likewise usually written as $b(x)$ to indicate its dependence upon $x$. Given a particular element $a:A$, the element $b(a)$ is an element dependent upon $a:A$.

#### Identity types

Formation rules for identity types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, b:A \vdash a =_A b \; \mathrm{type}}$$

Introduction rules for identity types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash \mathrm{refl}_A(x) : x =_A x}$$

Elimination rule for identity types:
$$\frac{\Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \; \mathrm{type}}{\Gamma, x:A, t:C(x, x, \mathrm{refl}_A(x), y:A, p:x =_A y, \Delta(x, t, y, p) \vdash \mathrm{ind}_{=_A}(x, t, y, p):C(x, y, p)}$$

Computation rules for identity types:
$$\frac{\Gamma, x:A, y:A, p:x =_A y, \Delta(x, y, p) \vdash C(x, y, p) \; \mathrm{type}}{\Gamma, x:A, t:C(x, x, \mathrm{refl}_A(x) \vdash \beta_{=_A}(x, t):\mathrm{ind}_{=_A}(x, t, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t}$$

#### Dependent function types

Formation rules for dependent function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \prod_{x:A} B(x) \; \mathrm{type}}$$

Introduction rules for dependent function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

Elimination rules for dependent function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x), a:A \vdash \mathrm{ind}_{\prod_{x:A} B(x)}(f, a):B(a)}$$

Computation rules for dependent function types
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \beta_{\prod_{x:A} B(x)}^{x:A.b(x)}:\prod_{a:A} \mathrm{ind}_{\prod_{x:A} B(x)}(\lambda(x:A).b(x), a) =_{B(a)} b(a)}$$

Uniqueness rules for dependent function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, f:\prod_{x:A} B(x) \vdash \eta_{\prod_{x:A} B(x)}(f):f =_{\prod_{x:A} B(x)} \lambda(x:A).\mathrm{ind}_{\prod_{x:A} B(x)}(f, x)}$$

#### Equivalence types

Formation rules for equivalence types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \simeq B \; \mathrm{type}}$$

Introduction rules for equivalence types:
$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}{c}
}{\Gamma \vdash \mathrm{toEquiv}(f, g, G, H, K):A \simeq B}$$

Elimination rules for equivalence types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{func}(e):\prod_{x:A} B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{finv}(e):\prod_{y:B} A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{sec}(e):\prod_{x:A} \mathrm{finv}(e)(\mathrm{func}(e)(x)) =_A x}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{ret}(e):\prod_{y:B} \mathrm{func}(e)(\mathrm{finv}(e)(y) =_{B} y}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \mathrm{coh}(e):\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))}$$

Computation rules for equivalence types:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}
}{\Gamma \vdash \beta_{A \simeq B}^\mathrm{func}(f, g, G, H, K):\mathrm{func}(\mathrm{toEquiv}(f, g, G, H, K)) =_{\prod_{x:A} B} f}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}
}{\Gamma \vdash \beta_{A \simeq B}^\mathrm{finv}(f, g, G, H, K):\mathrm{finv}(\mathrm{toEquiv}(f, g, G, H, K)) =_{\prod_{y:B} A} g}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}
}{\Gamma \vdash \beta_{A \simeq B}^\mathrm{sec}(f, g, G, H, K):\mathrm{sec}(\mathrm{toEquiv}(f, g, G, H, K)) =_{\prod_{y:B} f(g(y)) =_{B} y} G}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}
}{\Gamma \vdash \beta_{A \simeq B}^\mathrm{ret}(f, g, G, H, K):\mathrm{ret}(\mathrm{toEquiv}(f, g, G, H, K)) =_{\prod_{x:A} g(f(x)) =_{A} x} H}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B \quad \Gamma \vdash g:\prod_{y:B} A \\
\Gamma \vdash G:\prod_{x:A} g(f(x)) =_{A} x \quad \Gamma \vdash H:\prod_{y:B} f(g(y)) =_{B} y \\
\Gamma \vdash K:\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))
\end{array}
}{\Gamma \vdash \beta_{A \simeq B}^\mathrm{coh}(f, g, G, H, K):\mathrm{coh}(\mathrm{toEquiv}(f, g, G, H, K)) =_{\prod_{x:A} H(f(x)) =_{f(g(f(x)) =_{B} f(x)} \mathrm{ap}_f(f(g(x)), x, G(x))} K}$$

Uniqueness rules for equivalence types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B}{\Gamma \vdash \eta_{A \simeq B}(e):\mathrm{toEquiv}(\mathrm{func}(e), \mathrm{finv}(e), \mathrm{sec}(e), \mathrm{ret}(e), \mathrm{coh}(e)) =_{A \simeq B} e}$$

#### Definitions

Definitions of a symbol $b$ for the element $a:A$ are made by using [[identity types]] between the symbol and element: $\mathrm{def}_{a, b}:a =_A b$. Definitions of a symbol $B$ for the type $A \; \mathrm{type}$ are made by using [[equivalence types]] between the symbol and the type: $\mathrm{def}_{A, B}:A \simeq B$.

#### Function types

Formation rules for function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \to B \; \mathrm{type}}$$

Introduction rules for function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B}{\Gamma \vdash \lambda(x:A).b(x):A \to B}$$

Elimination rules for function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \to B, a:A \vdash \mathrm{ind}_{A \to B}(f, a):B}$$

Computation rules for function types
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B}{\Gamma \vdash \beta_{A \to B}^{x:A.b(x)}:\prod_{a:A} \mathrm{ind}_{A \to B}(\lambda(x:A).b(x), a) =_{B} b(a)}$$

Uniqueness rules for function types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \eta_{A \to B}:\prod_{f:A \to B} f =_{A \to B} \lambda(x:A).\mathrm{ind}_{A \to B}(f, x)}$$

#### Pair types

Formation rules for pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \times B \; \mathrm{type}}$$

Introduction rules for pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash \mathrm{in}(x, y):A \times B}$$

Elimination rules for pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \mathrm{ind}_{A \times B}^A(z):A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \mathrm{ind}_{A \times B}^B(z):B}$$

Computation rules for pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash \beta_{A \times B}^A(x, y):\mathrm{ind}_{A \times B}^A(\mathrm{in}(x, y)) =_A x}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, x:A, y:B \vdash \beta_{A \times B}^B(x, y):\mathrm{ind}_{A \times B}^B(\mathrm{in}(x, y)) =_B y}$$

Uniqueness rules for pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, z:A \times B \vdash \eta_{A \times B}(z):z =_{A \times B} \mathrm{in}(\mathrm{ind}_{A \times B}^A(z), \mathrm{ind}_{A \times B}^B(z))}$$

#### Dependent pair types

Formation rules for dependent pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \sum_{x:A} B(x) \; \mathrm{type}}$$

Introduction rules for dependent pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, x:A, y:B(x) \vdash \mathrm{in}(x, y):\sum_{x:A} B(x)}$$

Elimination rules for dependent pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_{\sum_{x:A} B(x)}^A:\left(\sum_{x:A} B(x)\right) \to A} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_{\sum_{x:A} B(x)}^B:\prod_{z:\sum_{x:A} B(x)} B(\mathrm{ind}_{\sum_{x:A} B(x)}^A(z))}$$

Computation rules for dependent pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, x:A, y:B(x) \vdash \beta_{\sum_{x:A} B(x)}^A(x, y):\mathrm{ind}_{\sum_{x:A} B(x)}^A(\mathrm{in}(x, y)) =_A x}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, x:A, y:B(x) \vdash \beta_{\Sigma(x:A).B(x)}^B(x, y):\mathrm{ind}_{\sum_{x:A} B(x)}^B(\mathrm{in}(x, y)) =_{B(\mathrm{ind}_{\sum_{x:A} B(x)}^A(\mathrm{in}(x, y)))} y}$$

Uniqueness rules for dependent pair types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma, z:\sum_{x:A} B(x) \vdash \eta_{\sum_{x:A} B(x)}(z):z =_{\sum_{x:A} B(x)} \mathrm{in}(\mathrm{ind}_{\sum_{x:A} B(x))}^A(z), \mathrm{ind}_{\sum_{x:A} B(x)}^B(z))}$$

#### isContr and uniqueness quantifiers

Now that we have [[equivalence types]], [[identity types]], [[dependent sum types]], and [[dependent product types]], we can use that to define

* the [[isContr]] [[modality]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isContr}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{defisContr}(A):\mathrm{isContr}(A) \simeq \sum_{y:A} \prod_{z:A} y =_{A} z}$$

* the [[uniqueness quantifier]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \exists!x:A.B(x) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{def}\exists!_{x:A.B(x)}:\exists!x:A.B(x) \simeq \mathrm{isContr}\left(\sum_{x:A} B(x)\right)}$$

#### Positive types

Now that we have the [[uniqueness quantifier]] we can combine the [[elimination rule]], the [[computation rule]], and the [[uniqueness rule]] for any [[positive type]] into one rule, the [[induction]] rule.

##### Empty type

Formation rules for the empty type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{0} \; \mathrm{type}}$$

Recursion rule for the empty type:
$$\frac{\Gamma \vdash C \; \mathrm{type}}{\Gamma \vdash \mathrm{up}_\mathbb{0}^C:\mathrm{isContr}\left(\mathbb{0} \to C\right)}$$

Induction rule for the empty type:
$$\frac{\Gamma, x:\mathbb{0} \vdash C(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{dup}_\mathbb{0}^C:\mathrm{isContr}\left(\prod_{x:\mathbb{0}} C(x)\right)}$$

##### Unit type

Formation rules for the unit type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{1} \; \mathrm{type}}$$

Introduction rules for the unit type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{pt}:\mathbb{1}}$$

Recursion rule for the unit type:
$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{pt}:C}{\Gamma \vdash \mathrm{up}_\mathbb{1}^C(c_\mathrm{pt}):\exists!c:\mathbb{1} \to C.(c(\mathrm{pt}) =_{C} c_\mathrm{pt})}$$

Induction rule for the unit type:
$$\frac{\Gamma, x:\mathbb{1} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{pt}:C(\mathrm{pt})}{\Gamma \vdash \mathrm{dup}_\mathbb{1}^C(c_\mathrm{pt}):\exists!c:\prod_{x:\mathbb{1}} C(x).(c(\mathrm{pt}) =_{C(\mathrm{pt})} c_\mathrm{pt})}$$

##### Booleans type

Formation rules for the booleans type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{2} \; \mathrm{type}}$$

Introduction rules for the booleans type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 1:\mathbb{2}}$$

Recursion rule for the booleans type:
$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash c_1:C}{\Gamma \vdash \mathrm{up}_\mathbb{2}^C(c_0, c_1):\exists!c:\mathbb{2} \to C.(c(0) =_{C} c_0) \times (c(1) =_{C} c_1)}$$

Induction rule for the booleans type:
$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_1:C(1)}{\Gamma \vdash \mathrm{dup}_\mathbb{2}^C(c_0, c_1):\exists!c:\prod_{x:\mathbb{2}} C(x).(c(0) =_{C(0)} c_0) \times (c(1) =_{C(1)} c_1)}$$

##### Natural numbers type

Formation rules for the natural numbers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N} \; \mathrm{type}}$$

Introduction rules for the natural numbers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{N} \to \mathbb{N}}$$

Recursion rule for the natural numbers type:
$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash c_s:C \to C}{\Gamma \vdash \mathrm{up}_\mathbb{N}^C(c_0, c_s):\exists!c:\mathbb{N} \to C.(c(0) =_{C} c_0) \times \prod_{x:\mathbb{N}} c(s(x)) =_{C} c_s(c(x))}$$

Induction rule for the natural numbers type:
$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))}{\Gamma \vdash \mathrm{dup}_\mathbb{N}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{N}} C(x).(c(0) =_{C(0)} c_0) \times \prod_{x:\mathbb{N}} c(s(x)) =_{C(s(x))} c_s(c(x))}$$

##### Integers type

Formation rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{Z} \; \mathrm{type}}$$

Introduction rules for the integers type:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{Z}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash s:\mathbb{Z} \simeq \mathbb{Z}}$$

Recursion rule for the integers type:
$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash c_s:C \simeq C}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_0, c_s):\exists!c:\mathbb{Z} \to C.(c(0) =_{C} c_0) \times \prod_{x:\mathbb{Z}} c(s(x)) =_{C} c_s(c(x))}$$

Induction rule for the integers type:
$$\frac{\Gamma, x:\mathbb{Z} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{Z}} C(x) \simeq C(s(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_0, c_s):\exists!c:\prod_{x:\mathbb{Z}} C(x).(c(0) =_{C(0)} c_0) \times \prod_{x:\mathbb{Z}} c(s(x)) =_{C(s(x))} c_s(c(x))}$$

## Categorical semantics

The fragment of objective type theory consisting of only [[identity types]] and [[dependent product types]] can be interpreted in any [[path category]] with weak homotopy $\Pi$-types.

## See also

* [[weak type theory]]

* [[judgmental equality]], [[identity type]]

* [[dependent type theory]], [[Martin-Löf type theory]], [[homotopy type theory]]

## References

The original paper on this topic:

* {#BB21} [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

General dependent type theory references used for the formalizations of objective type theory in this article:

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

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

* {#Winterhalter20} [[Theo Winterhalter]], *Formalisation and Meta-Theory of Type Theory* ([github](https://github.com/TheoWinterhalter/phd-thesis/releases/tag/v1.2.1))

[[!redirects objective type theory]]
[[!redirects objective type theories]]


observable

> See also at *[[quantum observable]]*.

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=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In [[physics]] and in the theory of [[dynamical system]]s (deterministic, stochastic, quantum, autonomous, nonautonomous, open, closed, discrete, continuous, with finite or infinite number of degrees of freedom...), an **observable** is a quantity in some theoretical framework whose value can be measured and observed in principle. Any good theoretical framework of physical phenomena should come with carefully established notion of an observable.

## In classical physics

In [[classical mechanics]] an **observable** is any smooth function on the [[phase space]] of the system, and of time. The value of the observable is just the value of the function for fixed argument.

## In quantum physics

In [[quantum mechanics]] an **observable** is a Hermitean operator on the physical [[Hilbert space]] of the theory. See [[quantum observable]] for more details.

In this case, one distinguishes the concepts of the expectation value of the observable and the concept of the measured value; they are evaluated in some state of the system. The expectation value can be taken in any state of the system, while the measured value is always in some eigenstate of the observable operator. The process of measurement results in the quantum mechanical collapse or reduction, in which the system passes to an eigenstate of the measured operator. The probability of taking a given eigenstate depends on the the transition matrix element from the previously prepared state to the given eigenstate.

## In quantum field theory

In relativistic quantum mechanics and relativistic [[quantum field theory]] the question of observables is more complicated: issues like causality and superselection sectors are involved.

In the [[AQFT]] approach to [[quantum field theory]] the system of [[quantum observables]] localized in given spacetime regions are the very foundation of the theory, called a _[[local net of observables]]_ (the [[Haag-Kastler axioms]] for QFT).

In [[non-perturbative quantum field theory]] the [[algebras of observables]] are meant to be [[C*-algebras]], while in [[perturbative quantum field theory]] ([[perturbative AQFT]]) they are [[formal power series algebras]].

[[!include perturbative observables -- table]]

## Related concepts

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

\linebreak

[[!include Isbell duality - table]]

## References

In the [[AQFT|algebraic]] formulation of [[quantum mechanics]]:

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, *Algebra of Observables and State Space* [[doi:10.1007/978-94-009-0491-0_6](https://doi.org/10.1007/978-94-009-0491-0_6)],  Chapter in: *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

* {#Gleason09} [[Jonathan Gleason]], *The $C^*$-algebraic formalism of quantum mechanics* (2009) [[[Gleason09.pdf:file]], [[GleasonAlgebraic.pdf:file]]]

* {#Gleason11} [[Jonathan Gleason]], *From Classical to Quantum: The $F^\ast$-Algebraic Approach*, contribution to *[VIGRE REU 2011](http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html)*, Chicago (2011) [[pdf](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf), [[GleasonFAlgebraic.pdf:file]]]

Careful discussion of local gauge invariant observables in [[gravity]]/[[general relativity]] is in

* {#Khavkine14} [[Igor Khavkine]],  _Local and gauge invariant observables in gravity_, Class. Quantum Grav. 32 185019, 2015 ([arXiv:1503.03754](https://arxiv.org/abs/1503.03754), [CQG+](http://cqgplus.com/2015/09/16/local-and-gauge-invariant-observables-in-gravity/), [pdf slides](http://www.science.unitn.it/~moretti/convegno/khavkine.pdf) for talk at [Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)
Levico Terme, Italy, September 2014)

showing that, while there are no globally defined local gauge invariant observables, they do exist on an [[open cover]] of the space of field configuration and form something like a [[sheaf]] of observables (but, hence, one without [[global sections]]).

Careful discussion of observables in abelian [[gauge theory]] ([[electromagnetism]]) is in

* {#BeniniSchenkelSzabo15} [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Homotopy colimits and global observables in Abelian gauge theory_ ([arXiv:1503.08839](http://arxiv.org/abs/1503.08839))

[[!redirects observable]]
[[!redirects observables]]
[[!redirects algebra of observables]]
[[!redirects algebras of observables]]

[[!redirects algebra of quantum observables]]
[[!redirects algebras of quantum observables]]


observable universe

> under construction

> This entry is about the notion in [[physics]]. For a different (but at least roughly analogous) notion in [[mathematics]]/[[logic]] see at _[[universe]]_.

***

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### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On the very largest scales observable in astrophysical [[experiment]], the cosmos is well described by an [[FRW model]] with [[cosmological constant]] $\Lambda$ (but see at _[[inhomogeneous cosmology]]_), with plenty of [[dark matter]] and with primordial [[cosmic inflation]] (called the $\Lambda$-CDM "[[standard model of cosmology|concordance model of cosmology]]"). A [[horizon]] in there and everything on smaller scales is the _observable universe_.

There has never been a reason to assume that beyond this _cosmic horizon_ visible to us today, the cosmos would not extend further. In fact in the simple [[FRW models]] with positive cosmological constant, the [[spacetime]] [[manifold]] is not a [[closed manifold]] and extends indefinitely beyond our observable horizon. This is directly analogous to the ancient mariner who would stand on the deck of his ship and see a few miles of ocean around him. That was the world within his horizon, but there was no telling what lay beyond.

Therefore it is important to distinguish the _observable universe_ from the _universe_ as such. Unfortunately, intellectual laziness tended to ignore this distinction and at some point some people who said "universe" to mean just the stretch of our cosmic horizon felt the need to have a new term for whatever may lie beyond. For better or worse, that new term has become wide-spread these days and is "[[multiverse]]".

The main scientific observation that goes with this is the observation that [[cosmic inflation]] -- for which there is by now excellent and ever increasing [[experiment|experimental]] evidence that indeed it happened -- does, at least by the simple method which is currently used to model it, naturally predict a large ambient space in which local regions stochastically undergo inflationary expansion. This scenario of [[chaotic inflation]] has in fact been singled out as the most likely one (within the standard model) by the [[bicep2]]-[[experiment]] in 2014.

In any case, as far as the _observable_ universe is concerned, two facts are worth noticing:

1. the observable universe alone is already pretty darn large, see the various discussion of its scales [below](#References);

1. [[Giordano Bruno]]'s old intuition remains plausible even at the largest scales: possibly the full xyz-verse beyond our present horizon is vastly larger still... and we may never know.

> graphics taken from [Primack Chapter 6](#PrimackCh6)

> ([color version](http://viewfromthecenter.com/images/Cosmic_Uroboros.jpg))

> Recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know that the universe is flat with only a 2% margin of error

[[energy]] and [[length scales]] in the [[observable universe]] (from [[cosmological constant|cosmic]] scales, over [[fundamental particle]]-[[masses]] around the [[electroweak symmetry breaking]] to [[GUT scale]] and [[Planck scale]]):

> graphics grabbed from [Zupan 19](flavour+in+particle+physics#Zupan19)

\begin{center}

\end{center}

> graphics grabbed from [here](http://sten.astronomycafe.net/the-particle-desert/)

## Related entries

* [[experimental observation]]

  * [[hydrogen line]]

* [[cosmology]],

  * [[standard model of cosmology]],

    * [[FRW model]]

    * [[big bang]]

    * [[dark energy]]

    * [[dark matter]]

  * [[structure formation]]

  * [[inhomogeneous cosmology]]

    [[KBC void]]

* [[quantum cosmology]]

* [[particle physics]], [[standard model of particle physics]]

## References
 {#References}

* Wikipedia _[Observable universe](http://en.wikipedia.org/wiki/Observable_universe)_

Popular introduction:

* [[Carl Sagan]], *Cosmos*, Ballantine Books (1985) [Wikipedia entry, [GoogleBooks entry](https://books.google.de/books/about/Cosmos.html?id=_-XhL6_xsVkC&redir_esc=y)]

A visualization of the scales in the [[solar system]] are in

* {#VisSolarSystem} [A tediously accurate scale model of the solar system](http://joshworth.com/dev/pixelspace/pixelspace_solarsystem.html)

Accurate visualization of many astronomical phenomena is at

* [en.spaceengine.org](#http://en.spaceengine.org)

Zoom through the full range of scales:

* _[Cosmic Eye](https://www.youtube.com/watch?v=8Are9dDbW24)_

Visualization of the [[speed of light]] in relation to distances in the solar system:

* [www.techexplorist.com/3-animations-provided-nasa-scientist-suggests-the-speed-of-light-frustratingly-slow/20215/](https://www.techexplorist.com/3-animations-provided-nasa-scientist-suggests-the-speed-of-light-frustratingly-slow/20215/)

A visualization of the big observable structures in the universe, drawn to scale, is in

* {#VisObservableUniverse} American Museum of Natural History, _The known
universe_ ([video](https://www.youtube.com/watch?v=17jymDn0W6U#t=374))

Visualization of local gakaxy densities and voids

https://phys.org/news/2019-07-astronomers-vast-void-cosmic-neighborhood.amp

A simulation of the [[structure formation]] (see there for more) of these is at

* _[The Millennium Simulation Project](http://www.mpa-garching.mpg.de/galform/virgo/millennium/)_

On scales

* {#PrimackCh6} [[Joel Primack]], _[The View from the Center of the Universe](http://viewfromthecenter.com/)_ Chapter 6: _What Size is the Universe? The Cosmic Uroboros_ ([pdf](http://physics.ucsc.edu/~joel/Chapter6.pdf), [[PrimackSizeOfUniverse.pdf:file]])

Pointers to the past and future timeline of the observable universe are collected in

* [[John Baez]], _[A Brief History of the Universe](http://math.ucr.edu/home/baez/timeline.html)_

Survey and exposition of aspects of organizing the large-scale structure of the universe are at

* Wikipedia, _[The cosmic distance ladder](http://en.wikipedia.org/wiki/Cosmic_distance_ladder)_

* [[Terence Tao]], _[The cosmic distance ladder](http://terrytao.wordpress.com/2007/05/31/the-cosmic-distance-ladder/)_

More technical discussion of cosmic horizons contains

* [[George Ellis]], T. Rothman: _Lost Horizons_, 1993 ([scribd](http://de.scribd.com/doc/235971566/G-F-R-Ellis-T-Rothman-Lost-Horizons))

A sketch of a comprehensive picture had been proposed in

* [[Erich Jantsch]], _[[The Self-Organizing Universe]]_ (1980)

[[!redirects cosmic horizon]]
[[!redirects cosmic horizons]]


observational equality

+-- {: .rightHandSide}
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### Context
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[dependent type theory]], sometimes one could [[inductive definition|inductively define]] an equality type directly on an [[inductive type]], rather than using Martin-Loef's [[identity types]] using the j-rule. This equality type is called **observational equality**.

## Examples
 {#Examples}

### Empty type
 {#EmptyType}

Observational equality for the [[empty type]] $\mathrm{Eq}_\mathbb{0}(x, y)$ is an indexed [[inductive type]] on the empty type $\mathbb{0}$ inductively defined by no constructors

### Unit type

Observational equality for the [[unit type]] $\mathrm{Eq}_\mathbb{1}(x, y)$ is an indexed inductive type on the unit type $\mathbb{1}$ inductively defined by the following constructor

$$\mathrm{eq}_*: \mathrm{Eq}_\mathbb{1}(*, *)$$

### Booleans type

Observational equality for the [[booleans type]] $\mathrm{Eq}_\mathbb{2}(x, y)$ is an indexed inductive type on the booleans type $\mathbb{2}$ inductively defined by the following constructors

$$\mathrm{eq}_0: \mathrm{Eq}_\mathbb{2}(0, 0)$$
$$\mathrm{eq}_1: \mathrm{Eq}_\mathbb{2}(1, 1)$$

### Natural numbers

Observational equality for the [[natural numbers]] $\mathrm{Eq}_\mathbb{N}(x, y)$ is an indexed inductive type on the natural numbers $\mathbb{N}$ inductively defined by the following constructors

$$\mathrm{eq}_0: \mathrm{Eq}_\mathbb{N}(0, 0)$$
$$\mathrm{eq}_s: \prod_{x:\mathbb{N}} \prod_{y:\mathbb{N}} \mathrm{Eq}_\mathbb{N}(x, y) \to \mathrm{Eq}_\mathbb{N}(s(x), s(y))$$

## See also

* [[identity type]]

## References

* {#Rijke19} [[Egbert Rijke]], _[[Introduction to Homotopy Type Theory]]_ (2019)  ([web](http://www.andrew.cmu.edu/user/erijke/hott/), [pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [GitHub](https://github.com/EgbertRijke/HoTT-Intro))

* [Univalent mathematics in Agda](https://github.com/UniMath/agda-unimath)

[[!redirects observational equalities]]


observational type theory

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

# Contents
* table of contents
{: toc}

## Idea

What is called _observational type theory_ (OTT) is a flavor of [[type theory]] in between [[extensional type theory]] and [[intensional type theory]].

This may be regarded as a first-stage approximation to
[[homotopy type theory]] (HoTT, and for more motivation see at *[[higher observational type theory]]*):

The [[propositions]] of OTT correspond to the [[h-propositions]] of HoTT, and the [[types]] in OTT correspond to [[h-sets]] in HoTT.  The notion of [[equality]] on OTT is based on [[symmetric prosets]], which is a special case of [[internal category in an (infinity,1)-category|higher internal groupoids]].  Since equality is defined type-by-type, OTT enjoys [[function extensionality]], and a form of [[propositional extensionality]] at least for a specified universe of propositions (not necessarily including all [[h-propositions]]).

There are a few technical differences (e.g. proofs of propositions are [[definitional equality|definitionally equal]] in OTT, whereas proofs of hprops are only [[propositional equality|propositionally equal]] in HoTT) but on the whole HoTT looks a lot like a [[higher homotopy]] version of OTT.

## See also

* [[higher observational type theory]]

* [[structure identity principle]]

## References

Observational type theory was introduced in:

* [[Thorsten Altenkirch]], [[Conor McBride]], _Towards observational type theory_, draft (2006) [[pdf](http://strictlypositive.org/ott.pdf), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/ott-conf.pdf), [[AltenkirchMcBrided-TowardsOTT.pdf:file]]]

* {#AltenkirchMcBrideSwierstra07} [[Thorsten Altenkirch]], [[Conor McBride]], [[Wouter Swierstra]], *Observational Equality, Now!*, PLPV '07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, [doi:10.1145/1292597.1292608](http://doi.org/10.1145/1292597.1292608), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/obseqnow.pdf)]

A blog post about an [[Agda]] implementation, including [[propositional extensionality]] (which is not mentioned in [ABS07](#AltenkirchMcBrideSwierstra07)):

* [[Conor McBride]], [Hier Soir, an OTT Hierarchy](http://mazzo.li/epilogue/index.html%3Fp=1098.html)

The above comparison between OTT and HoTT is taken from

* [[Dan Licata]], _[comment](http://homotopytypetheory.org/2012/11/12/abstract-types-with-isomorphic-types/#comment-2476)_ (2012)

[[!redirects observational type theory]]
[[!redirects observational type theories]]
[[!redirects OTT]]


observer

## Idea

In [[general relativity]]: a [[timelike curve]] in [[spacetime]].

## Related concepts

* [[observable]]

[[!redirects observers]]


observing black holes

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

## Idea

The typical ways of -- indirectly -- observing [[black holes]] in Nature -- are

* <nowiki>1.</nowiki> via X-ray emission. The nearby gas or nearby stars get acreted about a black hole and accelerate in the vicinity producing radiation emission;
* <nowiki>2.</nowiki> in star clusters with no gas, such as globular clusters, the only way is to see how the black hole affects the motion of nearby stars.

## X-ray emission

Consider first stars. Radiation pressure outwards, gravity inwards. If the radiation stronger the star dissolves. __Eddington's luminosity limit__

$$L_{Edd} = \frac{G M m_0 4\pi c}{\sigma}$$

In astrophysical units $L_{Edd} = \frac{M}{M_{sun}}\cdot 1.26 \cdot 10^38 erg/s$. For about ten solar masses this is about $10^{39} erg/s$. So something radiating above this limit is heavier of ten solar masses, under the assumption that the emission is isotropic and there are no other mechanisms which can violate the Eddington's limit (e.g. no beaming).
So the sources above $10^{39} erg/s$ (and under the assmptions above) are called the ultraluminous X-ray sources. Candidates for IMBH.

Spectrum observation in the acretion disk. Get temperature as a function of $R$ by various means, this now into black body radiation formulas, and then model the spectrum. Hoter toward the inside, fit to the mass of the black hole supposed around a center.

This method for fitting with the spectrum is called **multicolor** disk model.

## Effects on nearby stars

Effects of a [[black hole]] on the motion of nearby stars can be observed directly via black hole influence sphere or as global evidence (the whole cluster is affected). If the black hole has mass $M$, we can look at the escape velocity $\sqrt{\frac{2GM}{R}_{inf}}\sim \langle v \rangle$.

$R_{inf}$ -- is the influence sphere radius. $GM_{BH}/R_{inf} = GM_{GC}/R_{GC}$. Thus

Typically, for $R_{GC} = 1 pc$, $M_{BH} \sim 1 \% M_{GC}$, $R_{inf}\sim 1/100 pc\sim 1 arc sec$. $D_{sun GC} \sim Kpc$.

Very good estimates of black hole mass. But there is a problem here. The number of stars which is in the black hole influence region, it depends on the volume of the black hole influence region. The ratio of volumes

$$\frac{V_{inf}}{V_{GC}} \cong 10^{-6}$$

while the number of stars in the globular cluster is $10^4 - 10^6$.

so even less than one star in the influence sphere unless there is much bigger density of stars. In the center of the globular cluster there is $10^2$-factor of higher density. That is the place where the black holes would typically occur (heavier than the average star, they tend in the thermalization limit to be found about in the center of the star).

Crowding is the issue here, sometimes lucky to see the individual star.
How angularly distant from the center you find the star. At least 100 solar masses black hole. Size around square root of the mass. Another problem is low completeness -- we will observe at least two stars, usually $02-0.5$ sun masses each star in the cluster. All this is about the observation of globular clusters in our galaxy (more difficult for other galaxies of course) -- about 200 globular clusters in the [[halo of a galaxy|halo]] of Milky Way.

It is important to compare the properties like density and velocity dispersion in the cluster (the corresponding graphs are in the [[Croatian black hole school|school]] notes, versus $log R$). Velocity dispersion bigger in the center; actually a peak is near the center: Cusp phenomenon in the presence of [[intermediate mass black hole]].

* see Noyola and Gebherdt 2006, 2007, 2008 for observations

For example in the cluster $\Omega$-centauri.

Small number of stars then also lots of fluctuations. So one needs to be careful about the statistics. The presence of the cusp has been predicted by Bahcall and Wolf in 1970-s (analytic calculation, gives the shape of the density cusp as well; later simulations have been found).

More recently it has been shown by Trenti, Vesperini, Pasquato 2010, even if you do not put a black hole in the cluster after a while when the relaxation takes place the cusp cn appear anyway. So other phenomena possible.

 If we look outside of the black hole influence sphere, it has been shown by simulations that if you take the ratio between the radius $R_c$ at which the projected density declines and compare with $R_h$, then we can see the energy loss in the cluster. It can be supported by the presence of energy sources in the cluster  like binary stars or by a black hole. This can be seen on the profile of the projected density -- the larger density when energy source.

IMBH --  hosting GCs has a large core.

Another signature of the presence of black hole, acceleration of the  global shrinking the cluster puff the core and mass segregation accelerates in the cluster (infall of heavy stars).

Heavier main sequence stars (0.8 solar mass) compare with smaller ones (0.2 solar masses). Distribution tells if there is an energy source near the core. If you know how many binary stars there in principle one could be able to tell if there is a black hole there or not.

## References

The above is in parts taken from [[Mario Pasquato|Mario]]'s 3rd lecture at the [[Croatian black hole school]], 2010.

Resolved image of the direct vicinity of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* {#EHT19} [[Event Horizon Telescope Collaboration]], _First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole_,  The Astrophysical Journal Letters, Volume 875, Number 1, 2019 ([doi:10.3847/2041-8213/ab0ec7](https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7))


obstruction

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

Given a [[fiber sequence]] $F \to A \stackrel{\mathbf{c}}{\to} B$ of [[classifying spaces]]/[[moduli stacks]], hence $[\mathbf{c}]$ a [[universal characteristic class]],  and given an "$A$-structure" in the form of a morphism ([[cocycle]]) $f : X \to A$, then a lift $\hat f$ through $F \to A$ to an "$F$-structure" exists precisely if the induced $B$-structure $\mathbf{c}(f) : X \to B$ is trivializable in $B$-[[cohomology]]. One says that $[\mathbf{c}(f)]$ it is the **obstruction** to lifting the $A$-structure to an $F$-structure.

$$
  \array{
    && F &\to& *
    \\
    & {}^{\mathllap{\hat f}}\nearrow & \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{pt_B}}
    \\
    X &\stackrel{f}{\to}& A &\stackrel{\mathbf{c}}{\to}& B
  }
$$

### Twisted cohomology

Conversely, by the [[universal property]] of [[fiber sequences]], $A$-cocycles are equivalent to $B$-cocycles whose obstruction class under $f$ is trivial.

Therefore it makes sense to ask for the [[infinity-groupoid]] of $B$-cocycles whose class under $f$ has some other fixed value $\chi$. This gives $\chi$-_[[twisted cohomology]]_ with coefficients in $A$.

### In terms of homotopy type theory

Formulated in [[homotopy type theory]], obstruction theory reduces to a rather simple statement about factorization, or not, of [[functions]] through [[kernels]] of other functions. We spell out some details.

Let $\mathbf{c} : A \to B$ be a [[term]] of [[function type]] and let $pt_B : * \to B$ be a global point of $B$. The [[homotopy fiber|fiber]] of $\mathbf{c}$ over $pt_B$

$$
  F \coloneqq \sum_{a : A} (\mathbf{c}(a) = pt_B)
$$

comes with a canonical "inclusion" function

$$
  i : F \to A
$$

given by

$$
  i : (a, \mathbf{c}(a) \stackrel{\simeq}{\to} pt_B) \mapsto a
  \,.
$$

Now let

$$
  f : X \to A
$$

be any other function. We are asking for the _obstruction_ to lift it to a function $\hat f : X \to F$ such that

$$
  i \circ \hat f \simeq f
  \,.
$$

This exists precisely if there is an equivalence

$$
  \phi : \mathbf{c}\circ f \stackrel{\simeq}{\to} pt_B
$$

hence if the obstruction class

$$
  f^* \mathbf{c} \coloneqq \mathbf{c} \circ f
$$

is trivial.

## Examples

### Lift through Postnikov stages

If $F \to A$ in the above is a stage $\tau_{\leq n+1} B \to \tau_{\leq n}B$ in the [[Postnikov tower]] of an object $B$, then the lifting problem is that of lifting through the Postnikov tower of $A$ and the universal obstruction class is that which classified $\tau_{\leq n+1} B \to \tau_{\leq n}B$ as a $\pi_{n+1} B$-[[principal infinity-bundle]].

### Obstruction to extension

The formal dual of the lift obstruction problem discussed above is the following extension problem:

we start with a [[universal characteristic class|universal characteristic map]]

$$
  \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A
$$

representing a class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ in the $A$-cohomology of $\mathbf{B}G$. Then given a morphism  $\phi : \mathbf{B}G \to \mathbf{B}H$ we may ask for the obstruction to extending $\mathbf{c}$ along it.

Now the statement is: if  $\phi$ is a [[homotopy cofiber|homotopy cofiber]], then there is a good obstruction theory to answer this question. Namely in that situation we are looking at a [[diagram]] of the form

$$
  \array{
    \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A
    \\
    \downarrow && \downarrow^{\mathrlap{\phi}} & \nearrow_{\mathrlap{\hat \mathbf{c}}}
    \\
    * &\to& \mathbf{B}H
  }
$$

where the left square is an [[homotopy pushout]]. By its universal property, the extension $\hat {\mathbf{c}}$ of $\mathbf{c}$ exists as indicated precisely if the class

$$
  [f^* \mathbf{c}] \in H^n(\mathbf{B}Q, A)
$$

is trivial.

One class of examples for this sort of situation is where one considers refined [[Lie group cohomology]] on simply connected [[Lie groups]] and is asking for ways to push it down to discrete [[quotients]], hence to non-simply connected Lie groups integrating the same [[Lie algebra]]. This is often phrased in terms of "multiplicative bundle gerbes" over these Lie groups, but that is just another way of talking about the corresponding cohomology of the [[smooth infinity-groupoid|smooth]] [[moduli stack]] $\mathbf{B}G$.

### Obstruction to quantization: Quantum anomaly
 {#QuantumAnomaly}

There are various formalizations of the notion of [[quantization]] in [[physics]], or at least various aspects of that formalization. This involves various steps, some of which may have obstructions to being carried out. In [[physics]] such an obstruction in the process of quantization is often called a [[quantum anomaly]].

For instance for many [[theory (physics)|theories in physics]] the [[action functional]] is a priori not a [[function]] on the [[field (physics)|fields]] but a [[section]] of a circle-[[principal bundle]]. For this to qualify as an action functional therefore one needs a trivialization of that bundle and so the [[Chern class]] of the bundle is the obstruction and hence an _[[quantum anomaly|anomaly]]_ of the system. See there for more.

(...)

[[!redirects obstructions]]
[[!redirects obstruction theory]]


Ocneanu Rigidity

#Contents#
* table of contents
{:toc}

## Idea ##

Every [[fusion category]] has an associated [[fusion ring]]. Ocneanu rigidity states that, up to [[monoidal category|monoidal]] [[equivalence of categories | equivalence]], there are finitely many fusion categories with any given fusion ring.

This can be interpreted as saying that, when one fixes a set of fusion rules, you cannot continuously deform your category between monoidal equivalence classes.

Ocneanu rigidity is exemplary among many finiteness results for fusion categories. It is useful because it can translate many questions about fusion categories into questions about fusion rings, which are much simpler algebraic structures. For instance, it is an open problem whether there are finitely many equivalence classes of fusion categories of a given rank. Ocneanu rigidity reduces this problem to asking whether there are finitely many fusion rings of a given rank.

## References ##

A modern statement and proof can be found here:

* Azat Gainutdinov, Jonas Haferkamp, and [[Christoph Schweigert]]. "Davydov-Yetter cohomology, comonads and Ocneanu rigidity." Advances in Mathematics 414 (2023): 108853.

A good reference is here:

* [[Pavel Etingof]], [[Dmitri Nikshych]], and [[Viktor Ostrik]]. "On fusion categories." Annals of mathematics (2005): 581-642.

The original discussion of rank-finiteness can be found here:

* [[Viktor Ostrik]]. "Fusion categories of rank 2." arXiv preprint math/0203255 (2002).


octahedral group

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

#Contents#
* table of contents
{:toc}

## Idea

The _octahedral group_, a [[finite group]], is the group of [[symmetries]] of an [[octahedron]].

As a symmetry group of one of the [[Platonic solids]], the octahedral group participates in one of the three exceptional entries cases of the _ADE pattern_:

[[!include ADE -- table]]

More in detail, there are variants of the octahedral group corresponding to the stages of the [Whitehead tower of O(3)](Whitehead+tower#OfTheOrthogonalGroup):

* the full octahedral group is the [[subgroup]] of [[orthogonal group|O(3)]]

  $$
    O_h \hookrightarrow O(3)
  $$

  which is the [[stabilizer subgroup|stabilizer]] of the standard embedding of the [[octahedron]] into [[Cartesian space]] $\mathbb{R}^3$;

* the _rotational octahedral group_ $O \hookrightarrow SO(3)$ is the restriction to [[orientation]]-preserving symmetries, hence to [[special orthogonal group|SO(3)]]; this is [[isomorphism|isomorphic]] to the [[symmetric group]] $S_4$;

* the _binary octahedral group_ is the [[double cover]], hence the lift of $O$ to [[spin group|Spin(3)]]$\simeq$ [[special unitary group|SU(2)]];

* next there is a [[string 2-group]] lift $String_{2O} \hookrightarrow String_{SU(2)}$ of the octahedral group ([Epa 10](#Epa10), [Epa & Ganter 16](#EpaGanter16))

$$
  \array{
     String_{2O} &\hookrightarrow& String_{SU(2)}
     \\
     \downarrow && \downarrow
     \\
     2 O &\hookrightarrow & Spin(3) \simeq SU(2)
     \\
     \downarrow && \downarrow
     \\
     O \simeq S_4 &\hookrightarrow&  SO(3)
     \\
     \downarrow && \downarrow
     \\
     O_h \simeq S_4 \times \mathbb{Z}/2 &\hookrightarrow & O(3)
  }
$$

## Properties

### Basic properties

The [[order of a group|group order]] is:

$\vert O_h\vert = 48$

$\vert O \vert = 24$

$\vert 2O \vert = 48$

The [[subgroup]] of the octahedral group on the [[orientation]]-preserving symmetries is [[isomorphism|isomorphic]] to the [[symmetric group]] $S_4$. This also happens to be the full [[tetrahedral group]].

+-- {: .num_prop #QuaternionSubgroup}
###### Proposition
**([[quaternion group]] inside [[binary tetrahedral group]])**

The [[binary octahedral group]] contains the [[quaternion group]] of [[order]] 8, hence the [[binary dihedral group]] of [[order of a group|order]] 8, as a [[subgroup]] ([[normal subgroup|normal]]):

$$
  2 D_4 = Q_8 \subset 2 O
  \,.
$$

In fact the only [[finite subgroups of SU(2)]] which contain $2 D_4 =Q_8$ as a proper subgroup are the exceptional ones, hence the [[binary tetrahedral group]], the [[binary octahedral group]] and the [[binary icosahedral group]].

=--

See [this Prop](quaternion+group#InclusionInLargerFininteSubgroupsOfSU2) at _[[quaternion group]]_.

### Character table

[[!include character table of 2O]]

### Group cohomology

The [[group cohomology]] of the orientation-preserving octahedral group is discussed in [Groupprops](#Groupprops), [Tomoda & Zvengrowski 08, Sec. 4.2](#TomodaZvengrowski08), [Kirdar 13](#Kirdar13), [Epa & Ganter 16, p. 12](#EpaGanter16).

## References

* {#Klein1884} [[Felix Klein]], chapter I.7 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

Aspects of the linear [[representation theory]] of the binary octahedral group ([[irreducible representations]], [[character table]]) is spelled out at

* [[Groupprops]], _[Linear representation theory of binary octahedral group](https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_binary_octahedral_group)_

See also

* Wikipedia, _[Octahedral symmetry](https://en.wikipedia.org/wiki/Octahedral_symmetry)_

Discussion of the group cohomology:

* {#Groupprops} Groupprops, _[Group cohomology of symmetric group:S4](http://groupprops.subwiki.org/wiki/Group_cohomology_of_symmetric_group:S4)_

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

* {#Kirdar13} Mehmet Kirdar, _On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters_ [[arXiv:1309.4238](http://arxiv.org/abs/1309.4238)]

Discussion of [[Platonic 2-group]]-extensions:

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

[[!redirects octahedral groups]]

[[!redirects binary octahedral group]]
[[!redirects binary octahedral groups]]

[[!redirects octahedral symmetry]]
[[!redirects octahedral symmetry group]]

[[!redirects 2O]]


octahedron

#Contents#
* table of contents
{:toc}

## Idea

 (image source)

One of the [[Platonic solids]]

$\,$

$\,$

$\,$

## In the ADE patter

[[!include ADE -- table]]

## References

* {#Klein1884} [[Felix Klein]], chapter I.7 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

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

[[!redirects octahedra]]


octonion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Arithmetic
+-- {: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
=--
=--

# Octonions
* table of contents
{: toc}

## Idea

The _octonions_ or _Cayley numbers_ ([Cayley 1845](#Cayley1845)) form a [[non-associative algebra|non-associative]] [[real numbers|real]] [[star-algebra]] $\mathbb{O}$ similar to the [[complex numbers]] and the [[quaternions]] but with seven [[imaginary number|imaginary]] [[units]] adjoined.

The octonions arise from the [[quaternions]] in analogy -- namely: by the [[Dickson double construction]] ([Dickson 1919, (6)](#Dickson1919)) -- of how the [[quaternions]] arise from the [[complex numbers]], and the [[complex numbers]] from the [[real numbers]]. These are precisely the [[normed division algebras]] over the [[real numbers]], the octonions being the largest of the four. While the further [[Dickson double]] of the quaternions exists, called the _[[sedenions]]_, it is no longer a [[normed division algebra]].

In continuation of how the [[complex numbers]] and [[quaternions]] control [[spin groups]], [[real spin representations]] and [[supersymmetry]] up to [[dimension]] 7, the octonions control these up to the maximal dimension 11:

[[!include exceptional spinors and division algebras -- table]]

For more on this see at _[[supersymmetry and division algebras]]_.

Generally, the algebra of octonions shows up, in one way or another, behind most, if not all, [[exceptional structures]] in [[group theory]], [[Lie theory]] and [[differential geometry]]. See also at _[[universal exceptionalism]]_ for more on this.

## Definition

The following definition is in the style of [Dickson 1919](#Dickson1919), [Baez 02, second half of Section 2.2](#Baez02):

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

The _octonions_ $\mathbb{O}$ are the elements of the [[nonassociative algebra|non-associative]] [[star-algebra]] over the [[real numbers]] which is the [[Cayley-Dickson double]] of the [[star-algebra]] of [[quaternions]] $\mathbb{H}$ (with $\overline{(-)}$ denoting the conjugation-operation).

This means (see [there](Cayley-Dickson+construction#DefinitionByGeneratorsAndRelations)) that if $i, j, k \in \mathbb{H}$ denote an [[orthonormal basis]] of [[imaginary number|imaginary]] unit-[[quaternions]]

\begin{imagefromfile}
    "file_name": "QuaternionOctonionMultiplicationTableII.jpg",
    "float": "right",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "graphics from [SS21](#SS21)"
\end{imagefromfile}

$$
  \begin{aligned}
     & i^2 = j^2 = k^2 = -1
     \\
     &
     i j = k, \, j i = - k \;\;\;\text{and cyclic}
  \end{aligned}
$$

then the algebra $\mathbb{O}$ of octonions is [[generators and relations|generated]] from these $i, j , k$ and from one more element $\ell$, subject to these [[generators and relations|relations]]:

$$
  \begin{aligned}
  \ell^2 \;=\; -1
  \,,
  \phantom{AAA}
  \overline{\ell}
  \;=\;
  - \ell
  \end{aligned}
$$

and

$$
  \begin{aligned}
  q (\ell q')
  =
  \ell (\overline{q} q')
  \\
  (q \ell) q'
  =
  (q \overline{q}') \ell
  \\
  (\ell q) (q' \ell)
  =
  - \overline{q q'}
  \end{aligned}
$$

for all [[quaternions]] $q_1, q_2 \in \mathbb{H}$.

This gives the multiplication table on the right, where any two consecutive arrows $a \to b \to c$ mean that $a b = c$, $c a = b$, $b c = a$ and $b a = -c$.

=--

+-- {: .num_example #FixedLocusOfe4e5e6e7}
###### Example

The following computation shows the operation of consecutive left multiplication by the generators $\mathrm{e}_4$, $\mathrm{e}_5$, $\mathrm{e}_6$ $\mathrm{e}_7$ (according to Def. \ref{QuaternionCompatibleComponentDefinition}) on any octonion
$x = q + p \ell$ ($q,p \in \mathbb{H}$) is by reversal of the sign of the $\ell$-component, hence has as [[fixed point|fixed]] [[linear subspace]] the [[quaternions]] (see [HSS 18, Lemma 4.13](#HSS18) for application of this fact to [[M-branes]]):

$$
  \begin{aligned}
    \mathrm{e}_4
    \Big(
    \mathrm{e}_5
    \big(
      \mathrm{e}_6
      (\mathrm{e}_7 x)
    \big)
    \Big)
    & =
    \ell
    \bigg(
    (i \ell)
    \Big(
      (j \ell)
      \big(
        (k \ell)
        x
      \big)
    \Big)
    \bigg)
    \\
    & =
    \ell
    \bigg(
    (i \ell)
    \Big(
      (j \ell)
      \big(
        (k \overline{x})
        \ell
      \big)
    \Big)
    \bigg)
    \\
    & =
    \ell
    \Big(
      (i \ell)
      \big(
        (x k) j
      \big)
    \Big)
    \\
    & =
    \ell
    \bigg(
      \Big(
        i
        \big(
          j
          (k \overline{x}
        \big)
      \Big)
    \ell
    \bigg)
    \\
    & =
    \big(
      (
       \underset{
         \mathclap{
           q + p \ell
         }
       }{
         \underbrace{
           x
         }
       }
      k)
      j
    \big)
    i
    \\
    & =
    q k j i
    +
    \Big(
      \big(
        (q \ell) k
      \big)
      j
    \Big)
    i
    \\
    & =
    q
    \underset{
      = 1
    }{
    \underbrace{
      k j i
    }
    }
    -
    (p
      \underset{
        = 1
      }{
        \underbrace{
          k j i
        }
      }
    ) \ell
    \\
    & =
    q - p \ell
    \,.
  \end{aligned}
$$

=--

\linebreak

\linebreak

Of course the labels of the generators is not fixed. Here is another version:

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

The _octonions_ $\mathbb{O}$ is the [[nonassociative algebra]] over the [[real numbers]] which is [[generators and relations|generated]] from seven generators $\{e_1, \cdots, e_7\}$ subject to the [[generators and relations|relations]]

1. for all $i$

   $e_i^2 = -1$

1. for $e_i \to e_j \to e_k$ an edge or circle in the following diagram (a labeled version of the [[Fano plane]]) the relations

   1. $e_i e_j  = e_k$

   1. $e_j e_i  = -e_k$

{#FanoPlaneDiagram} $\,$

This becomes a [[star-algebra]] with star [[involution]]

\[
  \label{StarInvolution}
  \overline{(-)} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}
\]

which is the [[antihomomorphism]] $\overline{a b} = \overline{b} \overline{a}$ that is  given on the above generators by

$$
  \overline{e_i} \coloneqq - e_i
  \phantom{AAAA}
  i \in \{1, \cdots, 7\}
 \,.
$$

=--

+-- {: .num_example #ProductOfImaginaryOctonions}
###### Example

The product of all the generators with each other, bracketed to the right, is

$$
  e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 ))))))
  \;=\;
  + 1
$$

=--

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

By iteratively using the multiplication table in def. \ref{ComponentDefinition} we compute as follows:

$$
  \begin{aligned}
   & e_7 (e_6 (e_5 (e_4 (e_3 (\underset{-e_4}{\underbrace{e_2 e_1}})))))
   \\
   & =
   - e_7 (e_6 (e_5 (e_4 (\underset{e_6}{\underbrace{e_3 e_4}}))))
   \\
   & =
   - e_7 (e_6 (e_5 (\underset{e_3}{\underbrace{e_4 e_6}})))
   \\
   & =
   - e_7 (e_6 (\underset{-e_2}{\underbrace{e_5 e_3}}))
   \\
   & =
   + e_7 (\underset{-e_7}{\underbrace{e_6 e_2}})
   \\
   & =
   - \underset{= -1}{\underbrace{e_7 e_7}}
   \\
   & =
   + 1
  \end{aligned}
$$

=--

+-- {: .num_defn #RealAndImaginary}
###### Definition
**(real and imaginary octonions)**

As for the [[complex numbers]] one says that

* an _imaginary octonion_ is an $a \in \mathbb{O}$ shuch that under the star involution (eq:StarInvolution) it is sent to its negative:

  $$
    \overline{a} = -a
  $$

* a _real octonions_ is an $a \in \mathcal{O}$ shuch that under the star involution (eq:StarInvolution) it is sent to itself

  $$
    \overline{a} = a
  $$

Accordingly every octonion decomposes into a [[real part]] and an [[imaginary part]]:

$$
  Re(a) \coloneqq \tfrac{1}{2}(a + \overline{a})
  \phantom{AA}
  Im(a) \coloneqq \tfrac{1}{2}(a - \overline{a})
  \,.
$$

=--

## Properties

### General

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

The octonions are _not_ an [[associative algebra]]. But the _non-zero_ octonions and the _unit_ octonions form [[Moufang loops]].

=--

+-- {: .num_prop #Alternativity}
###### Proposition
**(octonions are alternative)**

The octonions form an [[alternative algebra]]

=--

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

By linearity it is sufficient to check this on generators.
So let $e_i \to e_j \to e_k$ be a circle or a cyclic permutation of an edge in the [[Fano plane]] as in Def. \ref{ComponentDefinition}.
Then by definition of the octonion multiplication we have

$$
  \begin{aligned}
    (e_i e_j) e_j
    &=
    e_k e_j
    \\
    &=
    - e_j e_k
    \\
    & =
    -e_i
    \\
    & =
    e_i (e_j e_j)
  \end{aligned}
$$

and similarly

$$
  \begin{aligned}
    (e_i e_i ) e_j
    &=
    - e_j
    \\
    &=
    - e_k e_i
    \\
    &=
    e_i e_k
    \\
    &=
    e_i (e_i e_j)
  \end{aligned}
  \,.
$$

=--

### Automorphisms

+-- {: .num_prop #AutomorphismsOfOctonionAlgebraIsG2}
###### Proposition

The [[automorphism group]] of the octonions, as a real algebra, is the [[exceptional Lie group]] [[G2]].

=--

See also at _[normed division algebra -- automorphism](normed+division+algebra#Automorphisms)_

### Left multiplication by imaginary octonions

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

Given any octonion $o$, then the operation of left multiplication by $o$

$$
  \array{
    \mathbb{O} &\overset{L_o}{\longrightarrow}& \mathbb{O}
    \\
    a &\mapsto& o a
  }
$$

is an $\mathbb{R}-$[[linear map]]. Under [[composition]] of linear maps, this defines an _[[associativity|associative]]_ [[monoid]] [[action|acting]] linearly on $\mathbb{O}$.

=--

+-- {: .num_prop #CliffordActionOfImaginaryOctonions}
###### Proposition
**(Clifford action of imaginary octonions)**

Consider the [[Clifford algebra]]

$$
  Cl(Im(\mathbb{O}), -{\vert -\vert}^2)
$$

on the underlying [[real vector space]] of that of the imaginary octonions (Def. \ref{RealAndImaginary}) regarded as an [[inner product space]] via the [[quadratic form]] given by the _negative_ square norm.

Then the operation of left multiplication on $\mathbb{O}$ (def. \ref{LeftMultiplicationAction}) induces a [[representation]] of this Clifford algebra on $\mathbb{R}^8 \simeq_{\mathbb{R}} \mathbb{O}$.

=--

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

By [[alternative algebra|alternativity]] (Prop. \ref{Alternativity}) we have for every $v \in Im(\mathbb{O})$ and every $a \in \mathbb{O}$

$$
  \begin{aligned}
    L_v L_v (a)
    & \coloneqq
    v (v a)
    \\
    & =
    (v v) a
    \\
    & =
    - {\vert v\vert}^2 a
    \\
    & =
    L_{- {\vert v\vert}^2} (a)
  \end{aligned}
$$

=--

+-- {: .num_prop #ConsecutiveLeftActionByImaginaryGenerators}
###### Proposition
**(consecutive left action by imaginary generators is unity)**

The consecutive left multiplication action (Def. \ref{LeftMultiplicationAction}) by all the imaginary octonion generators $e_i$ (Def. \ref{ComponentDefinition}) is $\pm$ the [[identity function]] on the octonions. Specifically, if one acts in increasing order of the labels in Def. \ref{ComponentDefinition}, then it is +1:

$$
  L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1}
  \;=\;
  + Id_{\mathbb{O}}
$$

=--

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

All the generators $e_i$ are imaginary octonions (Def. \ref{RealAndImaginary}). By Prop. \ref{CliffordActionOfImaginaryOctonions} their left action on $\mathbb{O}$ [[representation|represents]] a [[Clifford algebra]]-action of $Cl(Im(\mathbb{O}), -{\vert-\vert}^2) \simeq Cl_{0,7}$ on $\mathbb{R}^{8} \simeq_{\mathbb{R}} \mathbb{O}$.

By the [classification of real Clifford algebras](Clifford+algebra#ClassificationOverTheRealNumbers), $Cl_{0,7}$ has, up to [[isomorphism]], two different [[irreducible representation|irreducible]] [[modules]]. Their underlying vector space is $\mathbb{R}^8$ in both cases, and so the left action of imaginary octonions we have must be one of the two. The two irreps may be distinguished by the action of the "volume element" $\Gamma_7 \Gamma_6 \cdots \Gamma_1$: On one of the two it acts as the identity, on the other as minus the identity.

Hence we may check the remaining sign by acting on any one octonion, for instance on the unit $1 \in \mathbb{O}$. Then the claim follows with the computation in Example \ref{ProductOfImaginaryOctonions}:

$$
  \begin{aligned}
    L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} (1)
    & =
    e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 ))))))
    \\
    & =
    + 1
    \,.
  \end{aligned}
$$

=--

### Basic triples
 {#ElementaryTriples}

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

A _special triple_ or _basic triple_ is a [[triple]] $(e_1, e_2, e_3) \in \mathbb{O}^3$ of three octonions such that

* $e_i^2 = -1$

* $e_i e_j = - e_j e_i$.

=--

([Whitehead 71, p. 691](#Whitehead71))

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

The choice of $e_1$ identifies an inclusion of the [[complex numbers]] $\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{O}$.

Then the choice of $e_2$ on top of that identifies a compatible inclusion of the [[quaternions]] $\mathbb{R} \hookrightarrow \mathbb{C}\hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}$.

Finally the choice of $e_3$ on top of that induces a basis for all of $\mathbb{O}$.

=--

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

The set of basic triples, def. \ref{BasicTriple}, forms a [[torsor]] over the [[automorphism group]] [[G2]] $= Aut(\mathbb{O})$.

=--

(e.g. [Baez 02, 4.1](#Baez02))

### Relation to quaternions

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

Let

$$
  \mathbb{H} = \langle 1, i, j, k\rangle
$$

be the [[quaternions]] equipped with canonical [[linear basis|basis]] elements, and let

\[
  \label{ForProjectionStatementOcotonionsDirectSumUnderCayleyDickson}
  \mathbb{O} = \mathbb{H} \oplus \ell \mathbb{H}
\]

be the octonions equipped with the linear basis induced by the [[Cayley-Dickson construction]] (via [this def.](Cayley-Dickson+construction#CayleyDicksonDoubleByAdjoiningFurtherGenerator)).

Then the linear map

$$
  L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k}
  \;\colon\;
  \mathbb{O} \longrightarrow \mathbb{O}
$$

is an [[involution]] whose +1 [[eigenspace]] is $\ell \mathbb{H}$ and whose -1 eigenspace is $\mathbb{H}$, under the above identification (eq:ForProjectionStatementOcotonionsDirectSumUnderCayleyDickson).

(Here $L_{(-)}$ denotes the linear map on $\mathbb{O}$ given by left multiplication in $\mathbb{O}$.)

=--

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

We use the Cayley-Dickson relations ([this def.](Cayley-Dickson+construction#CayleyDicksonDoubleByAdjoiningFurtherGenerator))

$$
  a (\ell b) = \ell (\overline{a} b)
  \,,
  \phantom{AA}
  a(\ell b) = (a \overline{b}) \ell
  \,,
  \phantom{AA}
  (\ell a) (b \ell^{-1}) = \overline{a b}
$$

that hold in $\mathbb{O}$ for all $a,b \in \mathbb{H}$, as well as

$$
  \ell e = - e \ell
$$

for all imaginary elements $e \in \mathbb{H}$.

With this we compute

$$
  \begin{aligned}
    & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( a)
    \\
    & =  \ell( (\ell i) ( (\ell j) ( (\ell k) a ) ) )
    \\
    & = - \ell( (\ell i) ( (\ell j) ( (k \ell) a ) ) )
    \\
    & = - \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell ) ) )
    \\
    & = + \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell^{-1} ) ) )
    \\
    & = + \ell( (\ell i) ( \overline{j (k \overline{a})} ) )
    \\
    & = + \ell( (\ell i) ( \overline{i \overline{a}} ) )
    \\
    & = - \ell( (i \ell) ( \overline{i \overline{a}} ) )
    \\
    & = - \ell( ( i i \overline{a}  ) \ell )
    \\
    & = + \ell( \overline{a}  \ell )
    \\
    & = - \ell( \overline{a} \ell^{-1} )
    \\
    & = - a
  \end{aligned}
$$

and

$$
  \begin{aligned}
    & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( \ell a)
    \\
    & = \ell( (\ell i) ( (\ell j) ( (\ell k) (\ell a) ) ) )
    \\
    & =
      \ell( (\ell i) ( (\ell j) ( (\ell k) ( \overline{a} \ell) ) ) )
    \\
    & =
      -
      \ell( (\ell i) ( (\ell j) ( \overline{k {\overline{a}}} ) ) )
    \\
    & =
      + \ell( (\ell i) ( (j \ell) ( \overline{k {\overline{a}}} ) ) )
    \\
    & =
      + \ell( (\ell i) ( (j k {\overline{a}}) \ell ) )
    \\
    & =
      - \ell( (\ell i) ( (j k {\overline{a}}) \ell^{-1} ) )
    \\
    & =
      - \ell( \overline{ i j k {\overline{a}} } )
    \\
    & =
      + \ell( \overline{ {\overline{a}} } )
    \\
    & =
      + \ell a
  \end{aligned}
$$

=--

## Related concepts

* [[associative 3-form]]

* [[Fano plane]]

* [[Cayley plane]]

* [[octonionic Hopf fibration]]

* [[octonionic projective space]]

* [[split octonion]]

[[!include exceptional spinors and division algebras -- table]]

## References

### General

The definition is originally due to

* {#Cayley1845} [[Arthur Cayley]], _On certain results relating to quaternions_,  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 ([doi:10.1080/14786444508562684](https://doi.org/10.1080/14786444508562684))

The formulation as the [[Dickson double construction]] is due to

* {#Dickson1919} [[Leonard Dickson]], _On Quaternions and Their Generalization and the History of the Eight Square Theorem_,
Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 ([jstor:1967865](https://www.jstor.org/stable/1967865))

Review:

* {#Baez02} [[John Baez]], _The Octonions_,  Bull. Amer. Math. Soc. 39 (2002), 145-205. ([web](http://math.ucr.edu/home/baez/octonions/octonions.html), [pdf](http://math.ucr.edu/home/baez/octonions/octonions.pdf) [doi:10.1090/S0273-0979-01-00934-X](https://doi.org/10.1090/S0273-0979-01-00934-X))

Textbook accounts:

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000 ([doi:10.1007/978-3-662-12622-6](https://doi.org/10.1007/978-3-662-12622-6))

* [[Tevian Dray]], [[Corinne Manogue]], _The Geomety of Octonions_, World Scientific 2015 ([doi:10.1142/8456](https://doi.org/10.1142/8456))

The concept of "special triples" or ("basic triples") used above seems to go back to

* {#Whitehead71} [[George Whitehead]], appendix A in _Homotopy Theory_, MIT press 1971

Relation to the [[Leech lattice]]:

* {#Wilson09} [[Robert A. Wilson]], _Octonions and the Leech lattice_, Journal of Algebra
Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, ([pdf](http://www.maths.qmul.ac.uk/~raw/pubs_files/octoLeech1.pdf), [slides](http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf))

### Relation to 10d/11d spin geometry

Application of [[octonion]]-algebra to analysis of [[spin representations]] and [[spin geometry]] specifically in 11d (for general discussion in other dimensions see at _[[supersymmetry and division algebras]]_):

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. 371: 425. (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

* {#SS21} [[Hisham Sati]], [[Urs Schreiber]], *Super-exceptional M5-brane model -- Emergence of $SU(2)$-flavor sector*, Journal of Geometry and Physics **170** (2021) 104349 [[arXiv:2006.00012](https://arxiv.org/abs/2006.00012), [doi:10.1016/j.geomphys.2021.104349](https://doi.org/10.1016/j.geomphys.2021.104349)]

[[!redirects octonion]]
[[!redirects octonions]]
[[!redirects octonion algebra]]
[[!redirects algebra of octonions]]

[[!redirects Cayley number]]
[[!redirects Cayley numbers]]
[[!redirects Cayley algebra]]
[[!redirects algebra of Cayley numbers]]