boolean domain

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

The __boolean domain__ or __boolean field__ (often just: $Bool$) is a $2$-element [[set]], say $\mathbb{B} = \{ 0, 1 \}$ ("[[bits]]") or $\mathbb{B} = \{ \bot, \top \}$ ("[[bottom]]", "[[top]]"), whose [[elements]] may be interpreted as [[truth values]].

Note that $\mathbb{B}$ is the set of *all* [[truth values]] in [[classical logic]], but this cannot be assumed in non-classical logic such as [[intuitionistic logic]].

The Boolean domain plays the role of the [[subobject classifier]] in the [[Boolean topos]] of [[Sets]].

If we think of the classical $\mathbb{B}$ as a [[pointed set]] equipped with the true element, then there is an [[generalized the|effectively unique]] boolean domain.

A __boolean variable__ $x$ is a variable that takes its value in a boolean domain, as $x \in \mathbb{B}$.  If this variable depends on parameters, then it is (or defines) a [[Boolean-valued function]], that is a [[function]] whose target is $\mathbb{B}$.

An [[element]] of $\mathbb{B}$ is a __[[binary digit]]__, or __bit__.

The boolean domain is the initial set with two elements. It is also the initial set with an element and an [[involution]]. It is also the [[tensor unit]] for the [[smash product]] in the [[monoidal category]] of [[pointed sets]].

\begin{remark}
**(relation to boolean algebra)**
\linebreak
The term 'boolean field' (or just 'field', depending on the context) is sometimes used more generally for any *[[boolean algebra]]*. In fact, the boolean domain is the [[initial object|initial]] boolean algebra.  If we interpret a boolean algebra as a [[boolean ring]], then the boolean domain is the [[finite field]] with $2$ elements.
\end{remark}

## Definition

### In type theory
 {#InTypeTheory}

As an [[inductive type]], the boolean domain is given by

    Inductive Bool : Type
      | zero : Bool
      | one : Bool

This says that the type is inductively constructed from two [[terms]] in the type Bool, whose [[semantics|interpretation]] is as the two points of the type.

#### As a positive type

Assuming that [[identification types]] and [[dependent product types]] exist in the type theory, the boolean domain is the [[inductive type]] generated by two elements, and is defined by the following [[inference rules]]:

**[[type formation rules]]** for the boolean domain
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Bool} \; \mathrm{type}}$$

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**[[term introduction rules]]** for the boolean domain:
$$\frac{\Gamma \; \mathrm{ctx}}{\vdash 0:\mathrm{Bool}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 1:\mathrm{Bool}}$$

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**[[term elimination rules]]** for the boolean domain:
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1), p:\mathrm{Bool} \vdash \mathrm{ind}_\mathrm{Bool}^C(c_0, c_1, p):C(p)}$$

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**[[computation rules]]** for the boolean domain:

* **judgmental computation rules**
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1) \vdash \mathrm{ind}_\mathrm{Bool}^C(c_0, c_1, 0) \equiv c_0:C(0)}$$
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1) \vdash \mathrm{ind}_\mathrm{Bool}^C(c_0, c_1, 1) \equiv c_1:C(1)}$$

* **typal computation rules**
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1) \vdash \beta_\mathrm{Bool}^0(c_0, c_1):\mathrm{Id}_{C(0)}(\mathrm{ind}_\mathrm{Bool}^C(c_0, c_1, 0), c_0)}$$
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1) \vdash \beta_\mathrm{Bool}^1(c_0, c_1, 1):\mathrm{Id}_{C(1)}(\mathrm{ind}_\mathrm{Bool}^C(c_0, c_1, 1), c_1)}$$

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**[[uniqueness rules]]** for the boolean domain:

* **judgmental uniqueness rules**
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c:\prod_{x:\mathrm{Bool}} C(x), p:\mathrm{Bool} \vdash \mathrm{ind}_\mathrm{Bool}^C(c(0), c(1), p) \equiv c(p):C(p)}$$
* **typal uniqueness rules**
$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c:\prod_{x:\mathrm{Bool}} C(x), p:\mathrm{Bool} \vdash \eta_\mathrm{Bool}(c, p):\mathrm{Id}_{C(p)}(\mathrm{ind}_\mathrm{Bool}^C(c(0), c(1), p), c(p))}$$

\linebreak

The elimination, typal computation, and typal uniqueness rules for the boolean domain state that the boolean domain satisfies the **dependent universal property of the boolean domain**. If the dependent type theory also has [[dependent sum types]] and [[product types]], allowing one to define the [[uniqueness quantifier]], the dependent universal property of the boolean domain can be simplified to the following rule:

$$\frac{\Gamma, x:\mathrm{Bool} \vdash C(x) \; \mathrm{type}}{\Gamma, c_0:C(0), c_1:C(1)\vdash \mathrm{up}_\mathrm{Bool}^C(c_0, c_1):\exists!c:\prod_{x:\mathrm{Bool}} C(x).\mathrm{Id}_{C(0)}(c(0), c_0) \times \mathrm{Id}_{C(1)}(c(1), c_1)}$$

The judgmental computation and uniqueness rules imply the typal computation and uniqueness rules and thus imply the dependent universal property of the boolean domain.

In type theories with a separate type judgment where not all types are elements of universes, one has to additionally add the following elimination and computation rules:

Elimination rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, x:\mathrm{Bool} \vdash \mathrm{typerec}_{\mathrm{Bool}}^{A, B}(x) \; \mathrm{type}}$$

Computation rules:

* judgmental computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{typerec}_{\mathrm{Bool}}^{A, B}(0) \equiv A \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{typerec}_{\mathrm{Bool}}^{A, B}(1) \equiv B \; \mathrm{type}}$$

* typal computation rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \beta_{\mathrm{Bool}}^{0, A, B}:\mathrm{typerec}_{\mathrm{Bool}}^{A, B}(0) \simeq A}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \beta_{\mathrm{Bool}}^{1, A, B}:\mathrm{typerec}_{\mathrm{Bool}}^{A, B}(1) \simeq B}$$

#### As the type of decidable propositions

The boolean domain can also be defined as the type of decidable propositions. Suppose that we have a [[type of propositions]] $\mathrm{Prop}$. Then, the type of booleans is defined as

$$\mathrm{Bool} \equiv \sum_{P:\mathrm{Prop}} P \vee \neg P$$

where $A \vee B$ is the [[disjunction]] of two types $A$ and $B$ and $\neg A \equiv A \to \emptyset$ is the [[negation]] of the type $A$. Both disjunctions and the empty set can be directly defined from $\mathrm{Prop}$, [[dependent function types]], and [[product types]].

One can also define directly the boolean domain as the *type of all decidable propositions*, provided one already has [[disjunctions]] defined in the type theory. Similarly to the [[type of all propositions]], the boolean domain can be presented either as a [[Russell universe]] or a [[Tarski universe]]. The difference between the two is that in the former, every [[decidable proposition]] in the type theory is literally an element of the boolean domain, while in the latter, elements of $\mathrm{Bool}$ are only indices of a (-1)-truncated type family $\mathrm{El}$; every [[decidable proposition]] in the type theory is only [[essentially small type|essentially $\mathrm{Bool}$-small]] for [[weak Tarski universes]] or [[judgmentally equal]] to an $\mathrm{El}(P)$ for $P:\mathrm{Bool}$ for [[strict Tarski universes]].

##### As a strict Tarski universe

As a strict [[Tarski universe]], the boolean domain is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the boolean domain:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Bool} \; \mathrm{type}}$$

Introduction rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toBool}_A:\mathrm{isProp}(A) \to (A \vee \neg A) \to \mathrm{Bool}}$$

Elimination rules for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(\mathrm{El}(A))} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{lem}(A):\mathrm{El}(A) \vee \neg \mathrm{El}(A)}$$

Computation rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{El}(\mathrm{toBool}_A(p)(q)) \equiv A \; \mathrm{type}}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{proptrunc}(\mathrm{toBool}_A(p)(q)) \equiv p:\mathrm{isProp}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{lem}(\mathrm{toBool}_A(p)(q)) \equiv q:A \vee \neg A}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{proptrunc}, A}(p, q):\mathrm{Id}_{\mathrm{isProp}(A)}(\mathrm{proptrunc}(\mathrm{toBool}_A(p)(q)), p)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{lem}, A}(p, q):\mathrm{Id}_{A \vee \neg A}(\mathrm{lem}(\mathrm{toBool}_A(p)(q)), q)}$$

Uniqueness rules for the boolean domain:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{toBool}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)) \equiv A:\mathrm{Bool}}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \eta_{\mathrm{Bool}}(A):\mathrm{Id}_{\mathrm{Bool}}(\mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)), A)}$$

Extensionality principle for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool} \quad \Gamma \vdash B:\mathrm{Bool}} {\Gamma \vdash \mathrm{ext}_\mathrm{Bool}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

##### As a weak Tarski universe

As a weak [[Tarski universe]], the boolean domain is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the boolean domain:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Bool} \; \mathrm{type}}$$

Introduction rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toBool}_A:\mathrm{isProp}(A) \to (A \vee \neg A) \to \mathrm{Bool}}$$

Elimination rules for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{El}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(\mathrm{El}(A))} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{lem}(A):\mathrm{El}(A) \vee \neg \mathrm{El}(A)}$$

Computation rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q):\mathrm{El}(\mathrm{toBool}_A(p)(q)) \simeq A}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q))(\mathrm{proptrunc}(\mathrm{toBool}_A(p)(q))) \equiv p:\mathrm{isProp}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{congform}_{(-) \vee \neg (-)}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q))(\mathrm{lem}(\mathrm{toBool}_A(p)(q))) \equiv q:A \vee \neg A}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{proptrunc}, A}(p, q):\mathrm{Id}_{\mathrm{isProp}(A)}(\mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q))(\mathrm{proptrunc}(\mathrm{toBool}_A(p)(q))), p)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{proptrunc}, A}(p, q):\mathrm{Id}_{A \vee \neg A}(\mathrm{congform}_{(-) \vee \neg (-)}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q))(\mathrm{lem}(\mathrm{toBool}_A(p)(q))), q)}$$

where the equivalences
$$\mathrm{congform}_\mathrm{isProp}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q)):\mathrm{isProp}(\mathrm{El}(\mathrm{toProp}_A(p)(q))) \simeq \mathrm{isProp}(A)$$
$$\mathrm{congform}_{(-) \vee \neg (-)}(\beta_\mathrm{Bool}^{\mathrm{El}, A}(p, q)):(\mathrm{El}(\mathrm{toProp}_A(p)(q)) \vee \neg \mathrm{El}(\mathrm{toProp}_A(p)(q))) \simeq (A \vee \neg A)$$
can always be constructed in a type theory with [[dependent product types]], [[dependent sum types]], [[identity types]], [[disjunctions]], and [[negations]], as given types $A$ and $B$ and an equivalence $e:A \simeq B$, it is possible to form the equivalences
$$\mathrm{congform}_\mathrm{isProp}(e):\mathrm{isProp}(A) \simeq \mathrm{isProp}(B)$$
$$\mathrm{congform}_{(-) \vee \neg (-)}(e):(A \vee \neg A) \simeq (B \vee \neg B)$$
through [[function application to identifications|application of equivalences to identifications]] and the typal congruence rules of [[function types]], [[dependent product types]], [[product types]], [[dependent sum types]], [[disjunctions]], and [[negations]].

Uniqueness rules for the boolean domain:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash\mathrm{toBool}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)) \equiv A:\mathrm{Bool}}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \eta_{\mathrm{Bool}}(A):\mathrm{Id}_{\mathrm{Bool}}(\mathrm{toProp}_{\mathrm{El}(A)}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)), A)}$$

Extensionality principle for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool} \quad \Gamma \vdash B:\mathrm{Bool}} {\Gamma \vdash \mathrm{ext}_\mathrm{Bool}(A, B):\mathrm{isEquiv}(\mathrm{transport}^\mathrm{El}(A, B))}$$

##### As a Russell universe

As a [[Russell universe]], the boolean domain is given by the following [[natural deduction]] [[inference rules]]:

Formation rules for the boolean domain:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Bool} \; \mathrm{type}}$$

Introduction rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{toBool}_A:\mathrm{isProp}(A) \to (A \vee \neg A) \to \mathrm{Bool}}$$

Elimination rules for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash A \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{proptrunc}(A):\mathrm{isProp}(A)} \qquad \frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{lem}(A):A \vee \neg A}$$

Computation rules for the boolean domain:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{toBool}_A(p)(q) \equiv A \; \mathrm{type}}$$

* Judgmental computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{proptrunc}(\mathrm{toBool}_A(p)(q)) \equiv p:\mathrm{isProp}(A)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \mathrm{lem}(\mathrm{toBool}_A(p)(q)) \equiv q:A \vee \neg A}$$

* Typal computation rules:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{proptrunc}, A}:\mathrm{Id}_{\mathrm{isProp}(A)}(\mathrm{proptrunc}(\mathrm{toBool}_A(p)(q)), p)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash q:A \vee \neg A}{\Gamma \vdash \beta_{\mathrm{Bool}}^{\mathrm{lem}, A}:\mathrm{Id}_{A \vee \neg A}(\mathrm{lem}(\mathrm{toBool}_A(p)(q)), q)}$$

Uniqueness rules for the boolean domain:

* Judgmental computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \mathrm{toBool}_{A}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)) \equiv A:\mathrm{Bool}}$$

* Typal computation rules:
$$\frac{\Gamma \vdash A:\mathrm{Bool}}{\Gamma \vdash \eta_{\mathrm{Bool}}(A):\mathrm{Id}_{\mathrm{Bool}}(\mathrm{toBool}_{A}(\mathrm{proptrunc}(A))(\mathrm{lem}(A)), A)}$$

Extensionality principle for the boolean domain:
$$\frac{\Gamma \vdash A:\mathrm{Bool} \quad \Gamma \vdash B:\mathrm{Bool}} {\Gamma \vdash \mathrm{ext}_\mathrm{Bool}(A, B):\mathrm{isEquiv}(\mathrm{idToEquiv}(A, B))}$$

### In homotopy type theory
 {#InHomotopyTypeTheory}

In [[homotopy type theory]] the type of booleans / bits looks as [above](#InTypeTheory) (using [[judgemental equality]], [[propositional equality]], or [[typal equality]] for the [[computation rule]] and [[uniqueness rule]]) but now it may equivalently be thought of as the [[sphere type]] of the *[[0-sphere]]* and as such as the beginning of the [[suspension type]]-tower of types of "higher homotopy bits" --- the [[n-sphere|$n$-]][[sphere types]]:

\linebreak

## Properties

### Descent and large elimination

The [[descent]] for the boolean domain states that given any types $A$ and $B$ one can construct a type family $x:\mathbb{2} \vdash \mathrm{descFam}_\mathbb{2}^{A, B}(x)$ with [[equivalences of types]]
$$\mathrm{descEquiv}_A:\mathrm{descFam}_\mathbb{2}^{A, B}(0) \simeq A \quad \mathrm{and} \quad \mathrm{descEquiv}_B:\mathrm{descFam}_\mathbb{2}^{A, B}(1) \simeq B$$
Large elimination for the boolean domain strengthens the equivalences of types in descent to [[judgmental equality of types]]
$$\mathrm{descFam}_\mathbb{2}^{A, B}(0) \equiv A \quad \mathrm{and} \quad \mathrm{descFam}_\mathbb{2}^{A, B}(1) \equiv B$$

### Extensionality principle of the boolean domain

The elements of the boolean domain represent certain truth values or propositions, namely, [[true]] and [[false]]. By the principle of [[propositions as some types]], truth values or propositions are represented as certain types: specifically, [[true]] or $1$ is represented by the [[unit type]] $\mathbb{1}$, and [[false]] or $0$ is represented by the [[empty type]] $\mathbb{0}$. The boolean domain is a [[Tarski universe]] through the following type family:

$$x:\mathrm{Bool} \vdash \mathrm{El}(x) \coloneqq (x =_\mathrm{Bool} 1) \times \neg (x =_\mathrm{Bool} 0)$$

The extensionality principle of the boolean domain is then given by the [[univalence axiom]]:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathrm{Bool}, y:\mathrm{Bool} \vdash \mathrm{ua}(x, y):\mathrm{Id}_\mathrm{Bool}(x, y) \simeq (\mathrm{El}(x) \simeq \mathrm{El}(y))}$$

Since the empty type is not equivalent to the unit type, this automatically implies that $0$ is not equal to $1$. The extensionality principle of the boolean domain can be proven from descent of the boolean domain, see [Sattler 2023](#Sattler23) for a proof.

### Relation to sum types

The [[sum types]] can be defined in terms of the boolean domain and the [[dependent sum type]]. Given types $A$ and $B$, the sum type $A + B$ is defined as

$$A + B \coloneqq \sum_{x:\mathrm{Bool}} ((x =_\mathrm{Bool} 1) \to A) \times ((x =_\mathrm{Bool} 0) \to B)$$

### Boolean logic

One could recursively define the logical functions on $\mathrm{Bool}$ as follows

* For negation $\neg$
  * $\neg 0 := 1$
  * $\neg 1 := 0$
* For conjunction $\wedge$
  * $0 \wedge a := 0$
  * $1 \wedge a := a$
* For disjunction $\vee$
  * $0 \vee a := a$
  * $1 \vee a := 1$
* For implication $\implies$
  * $0 \implies a := 1$
  * $1 \implies a := a$
* For the biconditional $\iff$
  * $0 \iff a := \neg a$
  * $1 \iff a := a$

One could prove that $(\mathrm{Bool}, 0, 1, \neg, \wedge, \vee, \implies)$ form a [[Boolean algebra]]. The [[poset]] structure is given by implication.

One could also inductively define [[observational equality]] on the booleans $\mathrm{Eq}_\mathrm{Bool}(x, y)$ as an indexed inductive type on the boolean type $\mathrm{Bool}$ with the following constructors

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

A _boolean predicate_ valued in a type $T$ is a function $P: T \rightarrow \mathrm{Bool}$, and the type $T \to \mathrm{Bool}$ is a boolean [[function algebra]] for finite types $T$, and if [[path types]] exist, for all types $T$. Thus the [[functor]] $F: U \to BoolAlg$, $F(T) = T \to \mathrm{Bool}$ for a [[type universe]] $U$ is a [[Boolean hyperdoctrine]], and one could do [[classical logic|classical]] [[first-order logic]] inside $U$ if $\mathrm{Bool}$ and path types exist in $U$.

In fact, just with [[dependent sum types]], [[dependent product types]], [[empty type]], [[unit type]], and the two-valued type in a type universe $U$, any [[two-valued logic]] could be done inside $U$. Furthermore, since binary [[disjoint coproducts]] exist when $\mathrm{Bool}$ exists, all finite types exist in $U$, and any [[finitely-valued logic]], such as the [[internal logic]] of a [[finite]] [[cartesian power]] of [[Set]], could be done inside $U$.

For finite types, one could also inductively define specific functions

$$\forall a \in A.(-)(a):(A \to \mathrm{Bool}) \to \mathrm{Bool}$$

$$\exists a \in A.(-)(a):(A \to \mathbf{2}) \to \mathrm{Bool}$$

from the type of boolean predicates on $A$ and $\mathrm{Bool}$ such that they behave like [[existential quantification]] and [[universal quantification]].

### Other properties

* The boolean domain is the [[suspension type]] of the [[empty type]], and the suspension of the boolean domain is the [[circle type]]. Geometrically, the boolean domain is a zero-dimensional sphere.

* The boolean domain is [[homotopy equivalent]] to the type of [[decidable propositions]] in a universe $\mathcal{U}$.

$$\mathrm{Bool} \cong \sum_{P:\mathcal{U}} isProp(P) \times isDecidable(P)$$

As a result, sometimes the boolean domain is called a __decidable subset classifier__.

## Related concepts

* [[decidable equality]]

* [[type of propositions]]

* [[boolean domain object]]

* [[two-valued logic]]

## References

See also:

* Wikipedia, *[Boolean domain](https://en.wikipedia.org/wiki/Boolean_domain)*

Discussion in [[type theory]] as a simple example of an [[inductive type]]:

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), [p. 37](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=43) of _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

Discussion in [[homotopy type theory]]:

* {#AwodeyGambinoSojakova} [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], §3.1 in: *Inductive types in homotopy type theory*, LICS'12 (2012) 95–104 [[arXiv:1201.3898](http://arxiv.org/abs/1201.3898), [doi:10.1109/LICS.2012.21](https://doi.org/10.1109/LICS.2012.21)]

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* [[Egbert Rijke]], Exc. 4.2 (pp. 35) in: *[[Introduction to Homotopy Type Theory]]* (2023) [[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)]

...

* {#LumsdaineShulman16} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Abstracting away from cell complexes*, [talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf)]

Large elimination and descent of the boolean domain can be found in

* {#Sattler23} [[Christian Sattler]], *Natural numbers from integers* ([pdf](https://www.cse.chalmers.se/~sattler/docs/naturals.pdf))

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[[!redirects Russell types of decidable propositions]]
[[!redirects Tarski type of decidable propositions]]
[[!redirects Tarski types of decidable propositions]]
[[!redirects weak Tarski type of decidable propositions]]
[[!redirects weak Tarski types of decidable propositions]]
[[!redirects strict Tarski type of decidable propositions]]
[[!redirects strict Tarski types of decidable propositions]]

[[!redirects type of all decidable propositions]]
[[!redirects types of all decidable propositions]]

[[!redirects Russell type of all decidable propositions]]
[[!redirects Russell types of all decidable propositions]]
[[!redirects Tarski type of all decidable propositions]]
[[!redirects Tarski types of all decidable propositions]]
[[!redirects weak Tarski type of all decidable propositions]]
[[!redirects weak Tarski types of all decidable propositions]]
[[!redirects strict Tarski type of all decidable propositions]]
[[!redirects strict Tarski types of all decidable propositions]]

[[!redirects booleans type]]
[[!redirects type of booleans]]

[[!redirects decidable subset classifier]]

[[!redirects two-valued type]]
[[!redirects two-valued types]]


boolean domain object

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

# Boolean domain object
* table of contents
{: toc}

## Idea

Recall that a [[topos]] is a [[category]] that behaves likes the category [[Set]] of [[sets]].

A boolean domain object internal to a topos is an object that behaves in that topos like the [[boolean domain]] $\mathrm{Bool}$ does in Set.

## Definition

### In a topos or cartesian closed category

A **boolean domain object** in a [[topos]] (or any [[cartesian closed category]]) $E$ with [[terminal object]] $1$ is

* an [[object]] $\mathbf{2}$ in $E$

* equipped with

  * a [[morphism]] $zero:1 \rightarrow \mathbf{2}$ from the [[terminal object]] $1$;

  * a [[morphism]] $one:1 \rightarrow \mathbf{2}$ from the [[terminal object]] $1$;

* such that for every other tuple $(A, t, f)$ there is a morphism $u : \mathbf{2} \to A$ such that

\begin{center}
  \begin{tikzcd}
    \mathbf{2} \ar[d, "u"] &
    1 \ar[l, "zero"] \ar[ld, "f"] \ar[r, "one"] \ar[rd, "t"]        &
    \mathbf{2} \ar[d, "u"] \\
    A & & A
  \end{tikzcd}
\end{center}

By the [[universal property]], the boolean domain object is unique up to [[isomorphism]].

## Properties

### As an initial object

The boolean domain object $\mathbf{2}$ is an [[initial object]] in the category of [[bi-pointed object]]s.

### Coproducts

The boolean domain object $\mathbf{2}$ is the [[disjoint coproduct]] of the terminal object $1$ with itself.

... (One should be able to define binary [[coproduct]]s using the [[dependent sum]] functor and the boolean domain object, as dependent sums exist in topoi and cartesian closed categories.)

### Products

... (Same thing as above but with binary [[products]] and [[dependent product]]s.)

### Subobject classifier

A topos with a [[subobject classifier]] that is also a boolean domain object is a [[two-valued topos]]. The [[internal logic]] of such a topos is [[two-valued logic]].

### Type theory

Boolean domain objects are the [[categorical semantics]] of the [[boolean domain]] in type theory. The [[inductive type|inductive property]] of the boolean domain type, case analysis or if/else expressions, corresponds to the [[initial object|initiality]] of the boolean domain object in the subcategory of triples $(A, t:1\rightarrow A, f:1\rightarrow A)$ representing [[bi-pointed object]]s, similar to how the principle of [[induction]] over [[natural numbers]] corresponds to the initiality of the [[natural numbers object]] in the subcategory of triples $(A, q:1\rightarrow A, f:A\rightarrow A)$ representing infinite sequences.

## See also

* [[natural numbers object]]

* [[bi-pointed object]]

* [[subobject classifier]]

* [[boolean domain]]

* [[two-valued topos]], [[two-valued logic]]

[[!redirects two-valued object]]


boolean function

A __boolean function__ of $n$ variables is a [[function]] from $\mathbb{B}^n$ to $\mathbb{B}$, where $\mathbb{B}$ is the [[boolean domain]].  Compare a [[boolean-valued function]], where $\mathbb{B}^n$ may be replaced by any set $X$.

The logical operations in [[classical logic]] may all be interpreted as boolean functions, given by their truth tables.

[[!redirects Boolean function]]
[[!redirects Boolean functions]]
[[!redirects boolean functions]]


Boolean hyperdoctrine

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Boolean hyperdoctrine_ $P$ is a [[hyperdoctrine]] for [[lattices]] which are [[Boolean algebras]]

$$
  P \;\colon\; T^{op} \to BooleanAlgebra
  \,.
$$

## Definition

Let $C$ be a [[category]] with [[finite limits]]. A **[[classical logic|Boolean]] [[hyperdoctrine]]** over $C$ is a [[functor]]

$$
  P \;\colon\; C^{op} \to BoolAlg
$$

from the [[opposite category]] of $C$ to the category of [[Boolean algebras]], such that for every [[morphism]] $f : A \to B$ in $C$, the functor $P(B) \to P(A)$ has a [[left adjoint]] $\exists_f$ and a [[right adjoint]] $\forall_f$ satisfying

1. [[Frobenius reciprocity]];

1. [[Beck-Chevalley condition]].

## Examples

Every [[Boolean category]] $\mathcal{C}$ is a Boolean hyperdoctrine given by the [[subobject poset]] [[functor]] $\mathrm{sub}:\mathcal{C}^{op} \to BoolAlg$.

## Related concepts

[[!include categories and logic - table]]

## References

A discussion of the free Boolean hyperdoctrine generated from a [[signature (in logic)|signature]] is in

* [[Todd Trimble]], _[[toddtrimble:Notes on predicate logic]]_

[[!redirects Boolean hyperdoctrines]]


Boolean locale

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Boolean locale
* tables of contents
{: toc}

## Definition

A [[locale]] $L$ is __Boolean__ if all of its opens (i.e. elements of the corresponding [[frame]]) are [[complemented]],
i.e., for any $a\in O(L)$ we have $a\vee\neg a=1$.
Equivalently, we could say that for all $a\in A$ we have $a=\neg\neg a$.

## Properties

The underlying [[frames]] of Boolean locales
are precisely [[complete Boolean algebras]].

Maps of Boolean locales are automatically [[open map|open]].
Their underlying [[morphisms of frames]]
are precisely [[complete Boolean homomorphisms]],
i.e., suprema-preserving homomorphisms of [[Boolean algebras]].

Thus, the opposite category of Boolean locales
is precisely the category of [[complete Boolean algebras]]
and complete homomorphisms thereof.

By [[Stonean duality]] the category of Boolean locales
is equivalent to the category of [[Stonean locales]]
and open maps thereof.
(Not every [[map of locales]] between [[Stonean locales]] is open, unlike for Boolean locales.)

Any [[locale]] $L$ has a maximal dense [[sublocale]],
whose opens are precisely the [[regular elements]] of $O(L)$,
i.e., elements $a\in O(L)$ such that $a=\neg\neg a$.
This sublocale is Boolean and is also known
as the [[double negation sublocale]].

## Related concepts

* [[complete Boolean algebra]]
* [[Stonean locale]]
* [[Stonean space]]

[[!redirects Boolean locales]]


Boolean prealgebra object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A version of a [[Boolean algebra]] without the preorder being an [[antisymmetric relation]], [[internal relation|internal]] to a [[finitely complete category]].

## Definition

In a [[finitely complete category]] $C$, a **Boolean prealgebra object** is a [[bicartesian closed preordered object]] $X$ with a function
$$\zeta:((* \to X) \times (* \to X)) \to (* \to R)$$
such that for all global elements $a:* \to X$ and $b:* \to X$,

* $s \circ \zeta(a, b) = a \Rightarrow b$
* $t \circ \zeta(a, b) = (a \Rightarrow \bot) \vee b$

## See also

* [[bicartesian closed preordered object]]

* [[Boolean algebra object]]

[[!redirects Boolean prealgebra objects]]


Boolean rig

# Boolean rigs
* table of contents
{: toc}

## Idea

As a [[Boolean ring]] is a [[ring]] in which multiplication is [[idempotent]], so a Boolean rig is a [[rig]] in which multiplication is idempotent; that is, a rig in which $x^2 = x$ always holds.

While many further properties follow from the idempotence of multiplication in a ring, most of these require subtraction (or at least cancellation of addition), so these will not necessarily hold for a Boolean rig.  As such, it might be proper to add some of them as axioms for a Boolean rig.

## Properties

Even without subtraction, we may define the Boolean join operation $x \vee y \coloneqq x + x y + y$; however, it also seems to have no nice properties without additional axioms.  In a *commutative* Boolean rig, however, we can prove that multiplication distributes over join, meaning that multiplication and join together form another rig structure.  But we cannot prove that this rig is also Boolean, or that join is a [[semilattice]] operation.

## Examples

The main examples are probably [[distributive lattices]].  In this case, the join operation called $\vee$ above will match the original join/addition operation called $+$ above.

Of course, a [[Boolean ring]] is a Boolean rig, since any ring is a rig.  However, since it\'s also a distributive lattice, a Boolean ring is actually a Boolean rig in two different ways.

Let see what are the smallest commutative boolean rigs.

- The only boolean rig of cardinal $1$ is the zero ring $0$.

There are exactly two boolean rings of cardinal $2$. In such a boolean ring, we necessarily have $0 \neq 1$, because $0=1$ would imply that $x=1.x=0.x=0$ for every $x$ and then the boolean ring would be the zero ring $0$. The two elements of the boolean rings are thus $0$ and $1$. From the axioms of a rig, we have $0+0=0$, $1+0=0+1=1$, $0.0=0$, $1.0=0.1=0$ and $1.1=1$. We then have two possibilities for $1+1$, either $0$ or $1$. The two possibilities give a boolean rig. The multiplications being commutative, one has only to check the distributivity $(a+b).c = a.c +b.c$.

- One of the two boolean rigs of cardinal $2$ is $\mathbb{Z}/2\mathbb{Z}$ ($0$ can be interpreted as "False", $1$ as "True", $+$ as the [[exclusive disjunction]] and $.$ as the [[conjunction]]).
- One of the two boolean rigs of cardinal $2$ is $\mathbb{B}=\{0,1\}$ where $1+1=1$ ($0$ can be interpreted as "False", $1$ as "True", $+$ as the [[disjunction]] and $.$ as the [[conjunction]]).

## Related concepts

* [[Boolean algebra]]
* [[Boolean ring]]

## References

There seem to be none.  The 'Boolean semirings' in the literature (by which I mean, what Google found for me that wasn\'t cloaked) are something much more complicated than what we\'re looking at here.

[[!redirects Boolean rig]]
[[!redirects Boolean rigs]]
[[!redirects boolean rig]]
[[!redirects boolean rigs]]

[[!redirects Boolean semiring]]
[[!redirects Boolean semirings]]
[[!redirects boolean semiring]]
[[!redirects boolean semirings]]


Boolean ring

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

# Boolean rings
* table of contents
{: toc}

## Idea

A [[boolean algebra]] is an algebraic structure that models the fragment of the classical propositional calculus that deals with the connectives “and”, “or”, “implies”, and “not”.  In some approaches the definition of boolean algebra is rather lengthy, but boolean algebras are equivalent to **boolean rings**, which are simply rings obeying the identity $x^2 = x$.

In a boolean ring, the multiplication can be interpreted as the conjunction, the multiplicative unit $1$ as the truth value "true", the additive unit $0$ as the truth value "false". The unary operation $x \mapsto 1 + x$ provides the negation. Compared to [[boolean algebras]], which are also [[semiring| semirings]] but not rings (except the trivial boolean algebra), the addition expresses the exclusive disjunction and not the inclusive disjunction. The fact that the exclusive disjunction of $x$ and $x$ is the truth value "false" makes the commutative additive monoid an abelian group where $-x = x$.

## Definitions

A [[ring with unit]] $R$ is __boolean__ if the operation of multiplication is [[idempotent]]; that is, $x^2 = x$ for every element $x$. Although the terminology would make sense for rings without unit, the common usage assumes a unit.

Boolean rings and the [[ring]] [[homomorphisms]] between them form a [[category]] $Bool Ring$.

## Properties

*  A boolean ring is an [[algebra]] over the [[field]] [[prime field|$\mathbb{F}_2$]] with two elements, since
   $$ 2 x = 4 x - 2 x = 4 x^2 - 2 x = (2 x)^2 - 2 x = 2 x - 2 x = 0 .$$

*  $R$ is commutative (meaning that $x y = y x$ for all $x, y$):
   $$ y x = x + y - x - y + y x = (x + y)^2 - x^2 - y^2 + y x = x^2 + x y + y x + y^2 - x^2 - y^2 + y x = x y + 2 y x = x y .$$

Thus, multiplication in a boolean ring makes it into a [[semilattice]], while addition makes it into a [[vector space]] over the field with two elements, $\mathbb{F}_2$.

Define $x \vee y$ to mean $x + x y + y$.  Then:

*  $\vee$ is commutative (as it would be in any commutative ring) and idempotent:
   $$ x \vee x = x + x^2 + x = 3 x = x .$$
*  The absorption law ($x \vee x y = x$) also holds:
   $$ x \vee x y = x + x^2 y + x y = x + 2 x y = x .$$
   We could now prove the other absorption law to conclude that $R$ is a [[lattice]] using multiplication as [[meet]] and $\vee$ as [[join]].
*  But in fact, we can skip that step since it follows the distributive law ($x (y \vee z) = x y \vee x z$):
   $$ x (y \vee z) = x (y + y z + z) = x y + x y z + x z = x y + x^2 y z + x z = x y + (x y) (x z) + x z = x y \vee x z .$$
   Thus $R$ is a [[distributive lattice]].

Next define $\neg{x}$ to be $x + 1$.  Then:

*  $\neg{x}$ is a pseudocomplement of $x$ (meaning that $x (\neg{x}) = 0$):
   $$ x (\neg{x}) = x (x + 1) = x^2 + x = 2 x = 0 .$$
   By relativising from $x + 1$ to $x y + x + 1$, we can show that $R$ is a [[Heyting algebra]].
*  But don\'t bother, because $\neg{x}$ is also an op-pseudocomplement of $x$:
   $$ x \vee \neg{x} = x + x (x + 1) + (x + 1) = x^2 + 3 x + 1 = x + x + 1 = 1 .$$
   Therefore, $\neg{x}$ is a [[complement]] of $x$, and $R$ is a [[Boolean algebra]].

Conversely, starting with a boolean algebra $R$ (with the meet written multiplicatively), let $x + y$ be $x (\neg{y}) \vee (\neg{x}) y$ (which is called [[exclusive disjunction]] in $\{\top,\bot\}$ and [[symmetric difference]] in $2^X$).  Then $R$ is a boolean ring.

In fact, we have:
\begin{proposition}
The categories of Boolean rings and Boolean algebras are equivalent.
\end{proposition}

+-- {: .proof}
###### Proof
Since the Boolean algebra operations $\wedge, \vee, \neg, 0, 1$ are definable from the Boolean ring operations $+, -, \cdot, 0, 1$, and conversely, it follows that a function $f: {|A|} \to {|B|}$ between the underlying sets of Boolean rings is a Boolean ring homomorphism (i.e., preserves the Boolean ring operations) if and only if it is a Boolean algebra homomorphism (between the Boolean algebra structures defined on the same sets). In other words, the subsets

$$BoolRing(A, B) \hookrightarrow Set({|A|}, {|B|}), \qquad BoolAlg(A, B) \hookrightarrow Set({|A|}, {|B|})$$

coincide. Therefore the categories of Boolean rings and of Boolean algebras are equivalent as [[concrete categories]].
=--

The category of Boolean algebras is discussed further in [[BoolAlg]], but some of the results about this category are  proved there by working with the equivalent category of Boolean rings.

Here is a very convenient result: although a boolean ring $R$ is a [[rig]] in two different ways (as a ring or as a distributive lattice), these have the same concept of [[ideal]]!

## Examples

The most familiar example is the [[power set]] $\mathcal{P}S$ of any set $S$. This is a boolean ring with [[symmetric difference]] as the addition and the intersection of sets as the multiplication. In [[constructive mathematics]], one would use the set of [[decidable subsets]] $2^S$ instead of the set of all subsets $\mathcal{P}S$ to get the corresponding boolean ring.

More generally, given any Boolean ring $R$ and a set $S$, the [[function ring]] $R^S$ is a Boolean ring.

In classical mathematics the free boolean ring on a set $X$ can be identified with $\mathcal{P}_f \mathcal{P}_f X$, where $\mathcal{P}_f X$ is  the set of all finite subsets of $X$.  In fact $\mathcal{P}_f$ can be extended to a functor in two different ways, which agree on objects but differ on morphisms, and each of these gives a monad:

* the monad for [[semilattices]], $M \colon Set \to Set$ (which we use to describe multiplication in a boolean ring)

* the monad for [[vector spaces]] over the [[field]] with 2 elements, $S \colon Set \to Set$ (which we use to describe addition in a boolean ring).

These two monads are related by a distributive law which expresses the distributivity of multiplication over addition.   Their composite $S \circ M$ is then the monad for Boolean rings.

## Terminology

Back in the day, the term 'ring' meant (more often than now is the case) a possibly *non*unital ring; that is a [[semigroup]], rather than a [[monoid]], in [[Ab]].  This terminology applied also to boolean rings, and it changed even more slowly.  Thus older books will make a distinction between 'boolean ring' (meaning an idempotent semigroup in $Ab$) and 'boolean algebra' (meaning an idempotent monoid in $Ab$), in addition to (or even instead of) the difference between $+$ and $\vee$ as fundamental operation.  This distinction survives most in the terminology of $\sigma$-[[sigma-ring|rings]] and $\sigma$-[[sigma-algebra|algebras]].

+-- {: .num_remark}
###### Remark
We pause to note that "idempotent monoid" doesn't make sense a priori in a general monoidal category: generally speaking the idempotency axiom would be expressed by an equation

$$1_M = \left(M \stackrel{\delta_M}{\to} M \otimes M \stackrel{mult}{\to} M \right)$$

where $\delta_M$ is an appropriate diagonal map, not generally available for monoidal categories. But in a *[[concrete category|concrete]]* monoidal category $\mathbf{M}$ where the underlying-set functor $U: \mathbf{M} \to Set$ is [[lax monoidal functor|lax monoidal]], the meaning is that the evident equation holds:

$$1_{U(M)} = \left(U(M) \stackrel{\delta}{\to} U(M) \times U(M) \stackrel{\lambda}{\to} U(M \otimes M) \stackrel{U(mult)}{\to} U(M) \right)$$
where $\lambda$ is a lax monoidal constraint.
=--

## Analogues

Inasmuch as a [[semilattice]] is a commutative idempotent monoid, a boolean ring may be defined as a semilattice in $Ab$.  However, with boolean rings, we do not need to hypothesize commutativity; it follows.  That is, any idempotent monoid in $Ab$ is commutative; indeed, any idempotent [[magma]] in $Ab$ is commutative.

+-- {: .proof}
###### Proof
Write the magma operation as $(x, y) \mapsto x y$. Then for any element $x$, idempotence and bilinearity imply

$$x + x = (x + x)(x + x) = (x + x)x + (x + x)x = (x x + x x) + (x x + x x) = (x + x) + (x + x)$$

which by cancellation gives $x + x = 0$, or $x = -x$. Similarly, for any elements $x, y$,

$$x + y = (x + y)(x + y) = (x + y)x + (x + y)y = (x x + y x) + (x y + y y) = x + y x + x y + y$$

which by cancellation gives $y x + x y = 0$, or $y x = -(x y) = x y$.
=--

## Related concepts

* [[Boolean algebra]]
* [[BoolAlg]] - the category of boolean algebras
* [[Boolean rig]]

[[!redirects Boolean ring]]
[[!redirects boolean ring]]
[[!redirects Boolean rings]]
[[!redirects boolean rings]]
[[!redirects Boo Rng]]
[[!redirects BooRng]]
[[!redirects Boo Ring]]
[[!redirects BooRing]]
[[!redirects Bool Ring]]
[[!redirects BoolRing]]


boolean space

# Boolean spaces
* table of contents
{: toc}

## Definitions

A __boolean space__ is ...

Boolean spaces and their [[homomorphisms]] form a [[category]] $Boo Sp$.

## References

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

[[!redirects Boo Spa]]
[[!redirects BooSpa]]
[[!redirects Boo Sp]]
[[!redirects BooSp]]
[[!redirects boolean space]]
[[!redirects boolean spaces]]
[[!redirects Boolean space]]
[[!redirects Boolean spaces]]


Boolean topical dagger 2-poset

[[!redirects Boolean topical dagger 2-posets]]

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

## Contents ##
* table of contents
{:toc}

## Idea ##

A Boolean topical dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[Boolean topos]].

## Definition ##

A **Boolean topical dagger 2-poset** $C$ is a [[Boolean dagger 2-poset|Boolean]] [[power dagger 2-poset|power]] [[dagger 2-poset]].

## Examples ##

The dagger 2-poset of [[sets]] with [[decidable equality]] and [[relations]] is a Boolean topical dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


Boolean topos

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

# Contents
* table of contents
{:toc}

## Definition

A **Boolean topos** is a [[topos]] that is also a [[Boolean category]].

There are several conditions on a topos that are necessary and sufficient to be Boolean:

* Every [[subobject]] has a complement (the general definition of Boolean category).
* Every [[subobject poset|subobject lattice]] is a [[Boolean algebra]].
* The [[subobject classifier]] $\Omega$ is an [[internalization|internal]] Boolean algebra.
* The maps $\top, \bot: 1 \to \Omega$ are a [[coproduct]] cone (so in particular, $\Omega \cong 1 + 1$, and in fact this is enough, because the map $[\top, \bot]: 1 + 1 \to \Omega$ is always a monomorphism, and any monic endomorphism of $\Omega$ is an automorphism).

### Warning

Let $C$ be a Boolean pretopos, i.e. a [[pretopos]] in which is also a [[Boolean category]], and let $Sh(C)$ be the [[classifying topos]] for $C$. Then it does *not* follow that $Sh(C)$ is Boolean. In fact, this is rarely the case, even if $C$ is the classifying pretopos for a theory in classical first-order logic. See [Blass and Scedrov](#BlassScedrov75) for a characterization of which classical first-order theories have Boolean classifying topoi -- in particular, any such theory is $\aleph_0$-[[categoricity|categorical]]. Nevertheless, as discussed below, [[Barr's theorem]] shows that any topos admits a surjection from a Boolean topos.

## Properties

### As a context for foundations

The [[internal logic]] of a Boolean topos with [[natural numbers object]] can serve as [[foundations]] for "ordinary" mathematics, except for that which relies on the [[axiom of choice]].  If you add the axiom of choice, then you get (an internal version of) [[ETCS]]; conversely, if you use an arbitrary topos, then you get [[constructive mathematics]].  (For some high-powered work, you may also need to add a version of the [[axiom of replacement]] or an axiom of [[Grothendieck universe]]s.)

Every [[cartesian closed category|cartesian closed]] Boolean [[pretopos]] is in fact a topos.  This is why 'generalised [[predicative mathematics|predicativism]]' (with function types but not power types) is necessarily a feature of [[constructive mathematics]] only.

### Some characterisations

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

For $\mathcal{E}$ a [[topos]], then the following are equivalent:

1. $\mathcal{E}$ is a Boolean topos;

1. Every [[subtopos]] of $\mathcal{E}$ is Boolean.

1. Every [[subtopos]] of $\mathcal{E}$ is an [[open subtopos]].

1. Every [[subtopos]] of $\mathcal{E}$ is a [[closed subtopos]].

=--

([Johnstone, prop. A 4.5.22](#Johnstone))

+--{: .num_prop}
###### Proposition
Let $j$ be a [[Lawvere-Tierney topology]] on $\mathcal{E}$. Then $\mathcal{E}_j$ is Boolean iff there exists a [[subterminal object]] $U$ such that $j$ is the largest topology such that $U\rightarrowtail 1$ is $j$-closed.
=--

[(Johnstone 2002, p.220)](#Johnstone)

+--{: .num_prop}
###### Proposition
$\mathcal{E}$ is Boolean iff the only dense subtopos of $\mathcal{E}$ is $\mathcal{E}$ itself.
=--

**Proof**.  Suppose $\mathcal{E}$ is Boolean. $\mathcal{E}_{\neg\neg}=\mathcal{E}$ is the smallest dense subtopos (cf. [[double negation]]{#boolean_subtopos}).
Conservely, suppose $\mathcal{E}$ is not Boolean then $\mathcal{E}_{\neg\neg}$ is a second dense subtopos.

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

In a [[lattice of subtoposes]] the 2-valued Boolean toposes are the [[atoms]].

=--

See [this proposition](subtopos#BooleantoposesAreAtoms).

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

Let $\mathcal{E}$ be a topos. Then automorphisms of $\Omega$ correspond bijectively to [[closed subtopos|closed]] Boolean subtoposes. The group operation on $Aut(\Omega)$ corresponds to symmetric difference of subtoposes.

=--

This result appears in [Johnstone (1979)](#Johnstone79). (See also [Johnstone (2002)](#Johnstone), A1.6.11 pp.66-67.)

## Examples

With [[excluded middle]] in the meta-logic, every [[well-pointed topos]] is a Boolean topos. This includes [[Set]] and models of [[ETCS]].

The topos of sheaves on a [[Boolean algebra]] $B$ for the [[coherent category|coherent topology]] is Boolean if and only if the Boolean algebra is finite. Indeed, in $Sh(B)$ the sections of $\Omega$ over $b \in B$ are the ideals of $\downarrow b \leqslant B$, while those of $1+1$ are the principal ideals, and these coincide just when $B$ is finite.

The topos of [[canonical topology|canonical]] sheaves on a complete [[Boolean algebra]] is Boolean (note this includes the finite case above).

If $\mathcal{E}$ is any topos, the category of [[sheaf|sheaves]] for the [[double negation|double-negation topology]] is a Boolean [[subtopos]] of $\mathcal{E}$.

Any topos satisfying the [[axiom of choice]] is Boolean. This result is due to R. Diaconescu ([1975](#Diaconescu75)); see [[excluded middle]] for a brief discussion.

[[Barr's theorem]] implies that any topos $\mathcal{E}$ can be covered by a Boolean topos $\mathcal{F}$, in the sense of there being a [[surjective geometric morphism]] $f \colon \mathcal{F} \to \mathcal{E}$.

## Relation to measure theory

Boolean toposes are closely related to [[measurable spaces]] (e.g [Jackson 06](#Jackson06), [Henry 14](#Henry14)).

## Related entries

* [[Boolean category]]

* [[Boolean domain]]

* [[axiom of choice]]

* [[double negation]]

* [[de Morgan topos]]

## References

### General

* {#BlassScedrov75} [[Andreas Blass]], Andrej Scedrov, _Boolean Classifying Topoi_ , JPAA **28** (1983) pp.15-30.

* {#Diaconescu75}Radu Diaconescu, _Axiom of Choice and Complementation_ , Trans. AMS **51** no.1 (1975) pp.176-178. ([pdf](http://www.ams.org/journals/proc/1975-051-01/S0002-9939-1975-0373893-X/S0002-9939-1975-0373893-X.pdf))

* {#Johnstone79}[[Peter Johnstone]], _Automorphisms of $\Omega$_ , Algebra Universalis **9** (1979) pp.1-7.

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vols. I,II_, Oxford UP 2002. (A4.5.22, D3.4, D4.5)

### Relation to measure theory
 {#ReferencesRelationToMeasureTheory}

On relation of Boolean toposes to [[measure theory]]:

* {#Jackson06} Matthew Jackson, _A sheaf-theoretic approach to measure theory_, 2006. ([pdf](http://www.andrew.cmu.edu/~awodey/students/jackson.pdf), [d-scholarship:7348](http://d-scholarship.pitt.edu/7348))

* {#Henry14} [[Simon Henry]], _Measure theory over boolean toposes_, Mathematical Proceedings of the Cambridge Philosophical Society, 2016 ([arXiv:1411.1605](https://arxiv.org/abs/1411.1605))

* [[Asgar Jamneshan]], [[Terence Tao]], _Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration_ ([arXiv:2010.00681](arXiv:2010.00681))

[[!redirects Boolean topos]]
[[!redirects boolean topos]]
[[!redirects Boolean topoi]]
[[!redirects boolean topoi]]
[[!redirects Boolean toposes]]
[[!redirects boolean toposes]]


boolean-valued function

\tableofcontents

## Definition

A __boolean-valued function__ is a [[function]] from any [[set]] to the [[boolean domain]] $\{\bot, \top\}$.

In [[classical logic]], the definable boolean-valued functions on a [[type]] $X$ correspond precisely to [[predicates]] on $X$.  Assuming the law of [[excluded middle]], the boolean-valued functions on $X$ correspond precisely to the [[subsets]] of $X$; even in [[constructive mathematics]], they corresond to the [[decidable subsets]] of $X$.

## Related concepts

* [[predicate]], [[subset]]

[[!redirects Boolean-valued function]]
[[!redirects Boolean-valued functions]]
[[!redirects boolean-valued functions]]


Boone conjecture

[[!redirects empty 155]]
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# Contents
* table of contents
{:toc}

## Idea
A [[Higman's embedding theorem|famous result by G. Higman]] in [[group theory]] says that a finitely generated group can be [[subgroup|embedded]] in a finitely presented group precisely if it has a presentation whose defining relations are a recursively enumerable set of words. It has been conjectured by the group theorist W. Boone that the same result holds more generally for every single-sorted algebraic theory.

While in fact the conjecture is not true for all single-sorted algebraic theories (see [below](#FailureForSingleSorted)),
as was known by Soviet mathematicians, the general question "Which algebraic categories have the [[Higman's embedding theorem|Higman property]]?" is still interesting (and potentially something [[category theory|category-theorists]] could study).

## Statement

From [Lawvere 2002](#Lawvere02):

>There is an issue involving recursivity that categorists should settle: How general is Higman's theorem? In group theory the [[word problem]] (whether
a given finitely generated group is recursively related) is equivalent to the purely algebraic one of whether the given group can be embedded as a subgroup of a finitely presentable one.  For which other algebraic categories is the same statement true? or is it possibly true for the category of single-sorted [[algebraic theory|algebraic theories]]?
The latter problem was posed by Bill Boone, but as far as I know, is  not yet resolved.

>For the category of [[predicate logic|first-order]] [[theory|theories]], a theorem analogous to Higman's was proved by Craig and Vaught; for that case, they gave a kind of
intuitive argument: using a few additional predicates one can express enough of number theory to encode a fragmentary satisfaction relation and any given recursive set of axioms, so that one can then add the one additional axiom that says "all those axioms are true". Of course, it is non-trivial that this argument actually works.

## Partial results and failure of the conjecture for all single-sorted algebraic theories
  {#FailureForSingleSorted}

A category of models of an [[algebraic theory]] is said to satisfy the Higman property whenever any recursively presentable model embeds into a finitely presented model. The Higman property has been shown to hold for models of several algebraic theories such as those of [[rings]], [[semigroups]], [[inverse semigroups]] and [[Lie algebras]] (see [Kukin 1989](#Kukin89) for rings and references therein for the others).

However, there are single-sorted algebraic theories whose models do not have the Higman property as remarked in the introduction of ([Kukin 1989](#Kukin89)). This is because having the Higman property implies an undecidable word problem under suitable conditions (see connection 2.8. in ([Kharlampovich 1995](#Kharlampovich95)) but for instance (non-associative) commutative algebras over a fixed field have a decidable word-problem ([Shirshov 2009](#Shirshov09)).

## Related entries

* [[Higman's embedding theorem]]

* [[Lawvere theory]]

## References

* W. Craig, R. L. Vaught, _Finite axiomatizability using additional predicates_ , JSL **23** (1958) pp.289-308.

* M. Hébert, _Finitely presentable morphisms in exact sequences_ , TAC **24** no.9 (2010) pp.209-220.([pdf](http://www.tac.mta.ca/tac/volumes/24/9/24-09.pdf))

* G. Higman, _Subgroups of finitely presented groups_ , Proc. Royal. Soc. London Ser. A (1961) pp.455-475.

* {#Kharlampovich95}O. G. Kharlampovich, &  M. V. Sapir, _Algorithmic problems in varieties._ International Journal of Algebra and Computation 5(04n05) (1995) pp.379-602.

* {#Kukin89}G. Kukin, _The variety of all rings has Higman's property._
Third Siberian School: Algebra and Analysis (1989); American Mathematical Society translations. Series 2 v. **163** (1995) pp.91-101.

* {#Lawvere02}[[William Lawvere|W. F. Lawvere]], _On the [[effective topos]] - message to catlist_, January 2002. ([link](http://facultypages.ecc.edu/alsani/ct02%281-2%29/msg00016.html))

* {#Shirshov09}Shirshov, Anatolii Illarionovich. _Some algorithmic problems for ε-algebras._ Selected Works of AI Shirshov. Birkhäuser Basel (2009) pp. 119-124.


BooRng > history

< [[BooRng]]


boost

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#### Group Theory
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{:toc}

## Idea

The elements in the [[Lorentz group]] $O(d-1,1)$ in the image of the [[special orthogonal group]] $SO(d-1) \hookrightarrow O(d-1,1)$ are _[[rotations]]_ in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are "hyperbolic rotations" in a space-time plane, are called _boosts_ in [[physics]].

## Related concepts

* [[translation]], [[reflection]], [[rotation]]

## References

* [Wikipedia](https://en.wikipedia.org/wiki/Lorentz_transformation#Physical_formulation_of_Lorentz_boosts)

[[!redirects boosts]]

[[!redirects Lorentz boost]]
[[!redirects Lorentz boosts]]


Borcherds algebra

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* table of contents
{:toc}

## Idea

The concept of **Borcherds algebra** or **Borcherds-Kac-Moody algebra** is a generalization of that of [[Kac-Moody algebra]] (hence also called **generalized Kac-Moody algebra**) obtained by allowing imaginary [[root (in representation theory)|simple roots]].

## References

Original articles:

* [[Richard Borcherds]], _Generalized Kac-Moody algebras_, J. Algebra 115 (1988), 501–512.

* [[Richard Borcherds]], _Central extensions of generalized Kac-Moody algebras_, J. Algebra.140, 330-335 (1991).

* [[Victor Kac]], _Infinite dimensional Lie algebras_, third edition, Cambridge University Press, 1990.

See also:

* Wikipedia, _[Generaralized Kac-Moody algebra](https://en.wikipedia.org/wiki/Generalized_Kac–Moody_algebra)_

Relation to [[U-duality]] and [[E11]] (via  [[mysterious duality]]) is discussed in

* Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, _Borcherds symmetries in M-theory_, JHEP 0204 (2002) 049 ([arXiv:http://arxiv.org/abs/hep-th/0203070](http://arxiv.org/abs/hep-th/0203070))

* [[Marc Henneaux]], [[Bernard Julia]], Jérôme Levie, _$E_{11}$, Borcherds algebras and maximal supergravity_ ([arxiv:1007.5241](http://arxiv.org/abs/1007.5241))

and specifically to [[tensor hierarchies]] in [[gauged supergravity]]

* {#Palmkvist12} [[Jakob Palmkvist]], _Tensor hierarchies, Borcherds algebras and $E_{11}$_, JHEP 1202 (2012) 066 ([arXiv:1110.4892](http://arxiv.org/abs/1110.4892))

* [[Jakob Palmkvist]], _The tensor hierarchy algebra_, J. Math. Phys. 55, 011701 (2014) ([arXiv:1305.0018](https://arxiv.org/abs/1305.0018))

and to [[exceptional generalized geometry]]

* [[Jakob Palmkvist]], _Exceptional geometry and Borcherds superalgebras_ ([arXiv:1507.08828](http://arxiv.org/abs/1507.08828))

[[!redirects Borcherds algebras]]

[[!redirects Borcherds' algebra]]
[[!redirects Borcherds' algebras]]

[[!redirects Borcherds' algebra]]
[[!redirects Borcherds' algebras]]

[[!redirects Borcherds superalgebra]]
[[!redirects Borcherds superalgebras]]

[[!redirects generalized Kac-Moody algebra]]
[[!redirects generalized Kac-Moody algebras]]


Borchers property

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* table of contents
{:toc}

## Idea ##

In [[algebraic quantum field theory]], the *Borchers property* is a property of [[local nets]] in the [[Haag-Kastler approach]] to [[quantum field theory]] that follows, by a theorem of Borchers, from three physically motivated axioms on [[local nets]].
If every local algebra of the net is a type III factor, then the Borchers property is also a direct consequence.

The Borchers property is therefore often used as an "intermediate technical assumption".

Sometimes this property is also abbreviated as "property B".

## Definition ##

Let $\mathcal{M}(\mathcal{O})$ be a net of [[von Neumann algebras]] indexed by bounded open subsets of [[Minkowski spacetime]], for more details see [[Haag-Kastler vacuum representation]],

\begin{definition}

The net $\mathcal{M}(\mathcal{O})$ satisfies the **Borchers property** if for every double cones $K_1, K_2$ with $\bar K_1 \subseteq K_2$ and $E \in \mathcal{M}(\mathcal{K}_1)$ a nonzero projection, $E$ is (Murray-von Neumann) equivalent to the identity with respect to $\mathcal{M}(\mathcal{K}_2)$. That is, there is a partial isometry $V \in \mathcal{M}(\mathcal{K}_2)$ such that $VV^* = E, V^*V = \mathbb{1}$.

\end{definition}

## Properties ##

\begin{proposition}
A net satisfying causality, the spectrum condition and weak additivity (see [[Haag-Kastler vacuum representation]] for definitions) satisfies the Borchers property.
\end{proposition}

\begin{remark}
In particular. the local algebras of nets that satisfy the Borchers property cannot be finite where finite is meant in the sense of the Murray-von Neumann classification of [[von Neumann algebra factor|factors]].
\end{remark}

## References ##

* [[Hans-Jürgen Borchers]], _A remark on a theorem of B. Misra_,  Comm. Math. Phys. **4** 5 (1967) 315-323 [[euclid](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-4/issue-5/A-remark-on-a-theorem-of-B-Misra/cmp/1103839939.full)]

[[!redirects B property]]


bordism categories following Stolz-Teichner

<div class="rightHandSide toc">

[[!include functorial quantum field theory - contents]]

</div>

+-- {: .standout}

This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]

See there for background and context.

This entry here is about the definition of [[cobordism category|cobordism categories]] for [[Riemannian cobordism]]s.

=--

> **raw material**: this are notes taken more or less verbatim in a seminar -- needs polishing

Previous:

* [[Axiomatic field theories and their motivation from topology]].

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

* [[(2,1)-dimensional Euclidean field theories and tmf]]

Next:

* [[(2,1)-dimensional Euclidean field theories]]

see also

* [[cobordism]]

* [[Riemannian cobordism]]

* [[cobordism category]]

* [[(infinity,n)-category of cobordisms]]

***

# Contents #

* tic
{:toc}

#Idea {#Idea}

The goal here is to define a category of [[cobordism]]s that carry the structure of [[Riemannian manifold]]s. Where a functor on an ordinary [[cobordism category]] defines a [[TQFT]], the assignments of a functor on a category of [[Riemannian cobordism]]s do not only depend on the topology of a given cobordism, but also on its Riemannian structure. In physics terms such a functor is a _Euclidean quantum field theory_ .

>Notice however that the physicist's use of the word "Euclidean" is different from the way Stolz-Teichner use it: for a physicist it means that the Riemannian structure is not _pseudo_-Riemannian. For Stolz-Teichner it means (later on) that the Riemannian metric is _flat_ .

One central technical difference between plain topological cobordisms and those with Riemannian structure is that we want the functors on these to smoothly depend on variations of the Riemannian structure. This requires refining the bordism category to a _smooth_ category. By the logic of [[space and quantity]], one way to do this is to realize it as a [[stack]] on [[Diff]] with values in categories. This realization will be described here.

# Part 1 (topological) bordism category  {#topological}

**definition sketch**

the [[category]] $Bord_d$  has

* as [[object]]s closed $(d-1)$-dimensional _smooth_ [[manifold]]s

* and the [[morphism]]s are [[compact space|compact]] $d$-dimensional smooth [[manifold]]s with boundary, modulo [[diffeomorphism]] "rel boundaries" (i.e. those that restrict to the identy on the boundary)

The composition of [[morphism]]s is given by gluing of manifolds along their boundary

# Part 2 Riemannian bordism category {#Riemannian}

in all of the following

* the symbol $Y$ denotes $d$-dimensional a [[Riemannian manifold]]

  without boundary.

* **note on boundaries** technically it is convenient to never ever work with manifolds with Riemannian or other structure with boundary. Instead, we always just mention manifolds without boundary and encoded the way in which they are still to be thouhgt of as [[cobordism]]s by injecting _[[collars]]_ into them. The manifolds with boundary could be obtained by cutting of at the _core_ of these collars (see the definition below) but, while this is morally the idea, in the construction this is never explicitly considered.

  Also, later when we generalize [[manifold]]s to [[supermanifold]]s it will be very convenient not to have to talk about boundaries

$R Bord_d$ is defined using bicollars from the beginning

an object in $R Bord_d$ is a quintuple

consisting of

* a $d$-dimensional [[Riemannian manifold]] $Y$;

* a **core** $(d-1)$-manifold $Y^c$ sitting $Y^c \hookrightarrow Y$ in a thickening $Y^d$ -- being a $d$-manifold --

* $Y^+, Y^- \hookrightarrow Y$ two disjointly embedded open $d$-dimensional manifolds such that

  * $Y^c$ is in the closure of both $Y^+$ and $Y^-$

  * that $Y^+ \coprod Y^- = Y \backslash Y^C$

> so the _picture_ of an object, which is missing in this writeup here for the moment, is a $d-1$-dimensional Riemannian manifold that is thickened a bit in one further othogonal direction

**definition** A **Riemannian bordism** from $(Y_0,Y_0^c, Y_0^{\pm})$ to $(Y_1,Y_1^c, Y_1^{\pm})$ is a triple $(\Sigma, i_0, i_1)$ where

* $\Sigma$ is a $d$-dimensional [[Riemannian manifold]] without boundary

* for $i=0,1$ an open neighbourhood of the core $Y_i^c \hookrightarrow W_i \stackrel{open}{\hookrightarrow} Y_i$

  this defines the intersections $W^\pm_k := W_k \cap Y^\pm_k$ with the two collars for each $k = 0,1$.

* a smooth map $i_k : W_k \to \Sigma$

  such that

  * $i_k : W^+_k \cup Y_k^c \to Z$ is a proper map;

  * (+) for $i^+_k := i_k/W^+_k$ are [[isometry|isometric]] embeddings into $\Sigma \backslash i_1(W^-_1 \cup Y^c_1)$

    i.e. restricted to the (+)-collar the embedding of the thickened object into the would-be cobordisms is isomertric

  * the core $\Sigma^c := \Sigma \backslash (i_0(W^+_0) \cup i_1(W^-_1))$ is compact

    i.e. cutting of the (+)-collar of the incoming object and the (-)-collar of the outgoing object yields a compact manifold

**Remark**. Notice that this builds in an asymmetry: the (+)-side is preferred. This is intentionally: also the category $TV$ of [[topological vector space]]s will have a similar asymmetry (from the fact that for $\infty$-dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for $V \otimes V^*$), similarly, with the above asymmetric definition we have a cobordims $Y \coprod Y^* \to \emptyset$ (where $Y^*$ is obtained from $Y$ by reversing orientation) but _not_ one going the other way round.

A big difference between [[TQFT]]s and the Riemannian QFTs is that for [[TQFT]]s the vector spaces assigned to objects are necessarily finite-dimensional. So this issue here with infinite-dimensional vector spaces and the asymmetry that this introduces is crucial for Riemannian QFTs.

**example** Given any [[isometry]]

$$
  \phi : W_0 \to W_1
$$

such that $\phi$ preserves the decomposition $W_k^\pm, Y_k^c$ we get a [[Riemannian cobordism]] using

$$
  \Sigma := W_1
$$

and

$$
  i_1 = Id_{W_1}\,,\;\;\;\;\; i_0 = \phi
$$

**definition (morphisms in $R Bord_d$)** morphisms from $Y_0$ to $Y$ in $R Bord_d$ (or $d-RB$ or whatever the notation is) are [[isometry]] classes _rel. boundary_ (see below) of [[Riemannian cobordism]]s from $Y_0$ to $Y_1$.

We require the commutativity of the following diagram

$$
  \array{
    V_1 &\stackrel{i_1}{\to}&
    X
    &\stackrel{i_0}{\leftarrow}&
     V_0
    \\
    \downarrow^{f_1} &&
    \downarrow &&
    \downarrow^{f_0}
    \\
    V'_1 &\stackrel{i'_1}{\to}&
    X'
    &\stackrel{i'_0}{\leftarrow}&
    V'_0
  }
$$

The [[isometry]] $(F,f_0, f_1)$ is "rel. boundary" if $f_0 = Id$ and $f_1 = Id$

so an isomorphism "rel boundary" in the sense here (more "rel collars", really) is an [[isometry]] $F$ sitting in a diagram

$$
  \array{
    V_1 &\stackrel{i_1}{\to}&
    X
    &\stackrel{i_0}{\leftarrow}&
     V_0
    \\
    \downarrow^{Id} &&
    \downarrow &&
    \downarrow^{Id}
    \\
    V'_1 &\stackrel{i'_1}{\to}&
    X'
    &\stackrel{i'_0}{\leftarrow}&
    V'_0
  }
$$

## description for $d=1$ {#onedim}

we decribe $R Bord_1$ explicitly

it has at least the object

$$
 pt =
  \left(
    \array{
        pt^- & pt^c & pt^+
        \\
        -- & \bullet & --
    }
  \right)
  =
  (\mathbb{R}, \{0\}, \mathbb{R}_\pm)
$$

which is a point with collar all of $\mathbb{R}$.

**Lemma** every object in $R Bord_1$ which is _connected_ and not the empty set is [[isomorphism|isomorphic]] to this $pt$

now for $t \in \mathbb{R}_+$ consider the morphism

$$
  I_t \in R Bord_1(pt,pt)
$$

defined as the triple $(\mathbb{R}, i_0, i_1)$ where $i_0 : \mathbb{R} \to \mathbb{R}$ is the identity map, and where $i_1 : \mathbb{R} \to \mathbb{R}$ is translation by $t$.

This means that $i_0$ takes the core of in the incoming point to $0 \in \mathbb{R}$ while $i_1$ takes the core of the outgoing point to $t \in \mathbb{R}$. Everything in $\mathbb{R}$ outside of $[0,1]$ is hence "collar" and this describes what naively one would think of as just the [[interval]] $[0,1]$ regarded as a [[Riemannian cobordism]].

**Lemma** The composition of these cobordisms is given by

$$
  I_t \circ I_{t'} = I_{t+t'}
$$

There are also morphisms

$$
  L_+ : pt \coprod pt \to \emptyset
$$

and

$$
  R_+ : \emptyset \to pt \coprod pt
$$

which describe morally the same cobordisms as $I_t$ does, but where both boundary components are regarded as incoming or noth as outgoing, respectively.

Here $L_t$ is formall given exactly as $I_t$ only that the map $i_0 : \mathbb{R} \to \mathbb{R}$ is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.

This is defined for $t \gt 0$. For $t= 0$ the morphism $L_0$ is still defined, but $R_0$ is not!! Exercise: check carefully with the above definition, keeping the asymmetry mentioned there in mind, to show that the obvious definition of $R_0$ does not satisfy the axioms above.

So this means that we have a cobordism of length 0 going $\emptyset \to pt \coprod pt$, but all cobordisms going the other way round $pt \coprod pt \to \emptyset$ will have to have non-vanishing length.

Another morphism in $R Bord_1$ is the morphism

$$
  \sigma : pt \coprod pt \to pt \coprod pt
$$

which just interchanges the two points, without having any length.

**Lemma** We have the following composition laws:

* $L_t \circ \sigma = L_t$

* $R_t = \sigma \circ R_t$

* $R_t \circ_{L_0} R_{t'} = R_{t+t'}$

where in the last line we have the composition that is obvious once you draw the corresponding picture, which in full beuaty is

$$
  (Id_{pt} \otimes L_0 \otimes Id_{pt}) \circ (R_t \otimes R_{t'})
$$

where the [[tensor product]] $\otimes$ is given by disjoint union.

**theorem** the [[symmetric monoidal category]] $R Bord_1$ is _generated_ as a symmetric monoidal category by

* the object $pt$

* the morphisms $L_0$, $\{R_t\}_{t \gt 0}$

subject to the relations

$$
  L_0 \circ \sigma = L_0
$$

$$
  \sigma \circ R_t = R_t
$$

$$
  \forall t,t' \gt 0 : R_t \circ_{L_0} R_{t'} = R_{t + t'}
$$

**corollary** symmetric monoidal functors

$$
  E \in Fun^\otimes(R Bord_1, TV)
$$

to the category $TV_\mathbb{R}$ of [[topological vector space]]s are specified by their imagges of these generators. We have

* $E : pt \mapsto V$

* $E : L_0 \mapsto (\lambda :  V\otimes V \to \mathbb{R})$

* $E : R_t \mapsto \rho_t \in V \otimes V$

The map $\lambda : V \otimes V \to \mathbb{R}$ is necessarily a _nondegenerate_ and _symmetric_ bilinear form and thus may be used to produce and fix an isomorphism $V \simeq V^*$.

This isomorphism is used to get an embedding

$$
  V \otimes V to V \otimes V^* \hookrightarrow End(V)
  \,.
$$

The image of this embedding is the set of what in this context will be called "trace class" operators.

With respect to this identification the map $\rho$ is to be understood. For varying $t$ the
$\rho_t$ form a [[semigroup]] (for instance a typical example would be $V = \Gamma(E)$ a space of sections of a vector bundle and $\rho_t = e^{-t \Delta}$ for $\Delta$ a [[Laplace operator]] on $E$).

**note** for $\lambda : V \otimes V \to \mathbb{R}$ to be continuous, one cannot use the Hilbert tensor product $\otimes_H$

the reason is that we have the folloing possible mpas out of the following possible tensor products

$$
  \mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes_{algebraic} V \stackrel{finite rank}{\hookrightarrow}
  End(V)
$$

$$
  \mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes V \stackrel{trace class}{\hookrightarrow}
  End(V)
$$

$$
  \mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes_H V \stackrel{Hilbert Schmitdt}{\hookrightarrow}
  End(V)
$$

(so here the middle is the projective tensor product, the one that we are actually using)

## smooth version / families version {#smoothversion}

We now refine the definition of the categories $R Bord_d$ and $TV$ such that they remember smooth stucture.

>Effectively, what the following implicitly does is to refine these categories to [[stack]]s with values in categories over [[Diff]]. The [[fibered category|fibred categories]] that appear in the following, $R Bord_d^{fam} \to Diff$ and $TV^{fam} \to Diff$ are the [[Grothendieck
construction]] of these stacks.

**definition of $TV^{fam}$**

recall that $TV$ denotes the category of locally convex Hausdorff [[topological vector space]]

now let $TV^{fam}$ be the [[fibred category]] over [[Diff]] whose fiber over $X \in Diff$ is the category of topological [[vector bundle]]s over $X$. This has as objects [[vector bundle]]s of [[topological vector space]]s, and the morphisms are fiberwise linear $C^\infty$-morphisms of bundles in the following sense:

let $V, W \in TV$

Then a linear map $F : V \to W$ is -- for any inclusion $U \hookrightarrow V$

$$
  \array{
    V &\stackrel{f}{\to}& W
    \\
    \uparrow^\subset & \nearrow
    \\
    U
  }
$$

-- called $C^1$ at $u \in U$ in the direction $v \in V$ if

$$
  \lim_{t \to 0} \frac{F(u+t v) - F(u)}{t}
$$

exists in $W$ and

$$
  U \times V \to W
$$

$$
  (u,v) \mapsto d F_u(v)
$$

is continuous.

Iteratively one defines $C^n$ and then $C^\infty$. The morphsims of $TV$-bundles are supposed to be $C^\infty$ maps in this sense (linear in the fibers, of course)

$$
  \array{
    V' &\stackrel{\tilde f}{\to}& V
    \\
    \downarrow &&\downarrow
    \\
    S' &\stackrel{f}{\to}& S
  }
$$

**definition of $R Bord_d^{fam}$**

Similarly $R Bord_d^{fam}$ has as objects submersions $Y \to S$ and $Y^c \to S$ (not necessarily surjective) with a smooth rank-2 tensor on $Y$ that fiberwise induces the structure of a [[Riemannian manifold]] (so these are $S$-families of [[Riemannian manifold]]s) such that

$$
  \array{
    Y &\leftarrow^\subset& Y^c
    \\
    \downarrow^{submersion} & \swarrow_{proper subm.}
    \\
    S
  }
$$

recall that a map is a [[proper map]] if inverse images of compact sets are compact.

**remark** Notice that if we fix the topology of the fibers in $Y \to S$, then what varies as we vary the fibers is the [[Riemannian metric]] on the fibers, so here each $S$ can be thought of as a (subspace of a) moduli space of Riemannian metrics on a given topological space. Don't confuse this with the role the space always called $X$ here will play as a kind of "moduli space of field theories".

a morphism in $R Bord_d^{fam}$  in

$$
  R Bord_d^{fam}\left(
    \array{
       Y_0 \\ \downarrow \\ S_0
    },
    \;\;
    \array{
       Y_1 \\ \downarrow \\ S_1
    }
  \right)
$$

are [[isometry|isometric]] rel boundary classes
of submersions $\Sigma \to S_0$ such that

$$
  \array{
    \Sigma
    &\stackrel{i_1}{\leftarrow}&f^* Y_1 &\to&  Y_1
    \\
    \uparrow^{i_0} &&\downarrow && \downarrow
    \\
    Y_0& \to&S_0 &\stackrel{f}{\to}&
    S_1
  }
$$

so here $\Sigma$ is an $S_0$-family of cobordisms.

## Riemannian field theories {#fieldtheories}

**definition**

A **$d$-dimensional Riemannian quantum field theory** is a [[symmetric monoidal functor]]

$$
  E \in Fun^\otimes_{Diff}(R Bord_d^{fam}, TV^{fam})
$$

such that

$$
  \array{
      R Bord_d^{fam} &&\stackrel{}{\to}&&
      TV^{fam}
      \\
      & \searrow && \swarrow
      \\
      && Diff
  }
$$

and such that it preserves [[pullback]]

(so its a [[cartesian functor]] between these [[fibered category|fibered categories]] that is also symmetric monoidal)

## Invertible Field Theories

In the study of invertible field theories, let $\mid \mid Bord^G_n \mid \mid$ be the fundamental groupoid of the classifying space of $Bord^G_n$. An [[invertible field theory]] factors through this and the Picard subgroupoid $\mathcal{C}^*$ of the target.

Kreck, S. Stolz, and P. Teichner.Invertible topological field theories are SKK invariants.


bordism homology theory

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

#Contents#
* table of contents
{:toc}

## Idea

A [[generalized homology theory]] [[Brown representability theorem|represented]] by a [[Thom spectrum]]. The dual concept of [[cobordism cohomology theory]].

## Examples

[[!include flavours of cobordism cohomology theories -- table]]

## Geometric model

See Section 2 in [Atiyah](#Atiyah).
Atiyah's geometric model for bordism homology groups
defines them as equivalence classes of maps of manifolds.

We concentrate on the oriented case, corresponding to the [[Thom spectrum]]
$\mathrm{M} \mathrm{SO}$.

Specifically, the $k$-dimensional oriented bordism group $\mathrm{M} \mathrm{SO}_k(X)$ of a smooth manifold $X$
(more generally, a [[paracompact Hausdorff topological space]],
even more generally, an arbitrary [[topological space]] provided
we use [[numerable open covers]] for trivializations)
is defined as the quotient of the [[commutative monoid]] $C_k(X)$ by
the equivalence relation $\sim$ of bordism, defined below.

Elements of $C_k(X)$ are smooth (or continuous) maps $F\colon M\to X$,
where $M$ is a [[compact]] $k$-dimensional
[[oriented]] [[smooth manifold]] (without boundary).

Two such maps $F$ and $F'$ are equivalent
if there is a smooth (or continuous) map $G\colon N\to X$,
where $N$ is a [[compact]] $(k+1)$-dimensional
[[oriented]] [[smooth manifold]] with boundary
$$\partial N = M\sqcup M'$$
such that $G|_M=F$ and $G|_{M'}=F'$.

One can also defined [[twisted homology groups]] in the same manner.
Twists are [[principal bundles]] $\alpha$ over $X$ with [[structure group]] $\mathbf{Z}/2$.
Elements of $C_k(X,\alpha)$
are maps $F\colon (M,\tau)\to (X,\alpha)$,
where $M$ is a [[compact]] $k$-dimensional [[smooth manifold]]
equipped with a principal $\mathbf{Z}/2$-bundle $\tau$,
and the map $F$ is a morphism of such [[principal bundles]].
Likewise, $F\sim F'$ if there is a map $G\colon (N,\sigma)\to(X,\alpha)$
with the same properties as above.

## Related concepts

* [[cobordism cohomology theory]]

* [[bordism]]

* [[ordinary homology]]

* [[equivariant bordism homology theory]]

* [[stable homotopy homology theory]]

## References

* {#Thom54} [[René Thom]], _Quelques propriétés globales des variétés différentiables_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

The original article introducing bordism as a [[Whitehead-generalized cohomology theory]]:

* {#Atiyah} [[Michael Atiyah]], _Bordism and Cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 ([doi:10.1017/S0305004100035064](https://doi.org/10.1017/S0305004100035064), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahb.pdf))

Surveys:

* [[Peter Landweber]], _A survey of bordism and cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 2 September 1986 , pp. 207-223 ([doi:10.1017/S0305004100066032](https://doi.org/10.1017/S0305004100066032))

* Max Hopkins, _The Extraordinary Bordism Homology_, 2016  ([pdf](http://cseweb.ucsd.edu/~nmhopkin/papers/Bordism.pdf), [[HopkinsBordism.pdf:file]])

For more, see the references at _[[cobordism cohomology theory]]_.

[[!redirects bordism homology theories]]


bordism ring

[[!redirects cobordism ring]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **(co)bordism ring** $\Omega_*=\oplus_{n\geq 0}\Omega_n$ is the [[graded ring]] whose

* degree $n$ elements are classes of $n$-[[dimension|dimensional]] [[smooth manifolds]] modulo [[cobordism]];

* product operation is given by the [[Cartesian product]] of manifolds;

* addition operation is given by the [[disjoint union]] of manifolds.

Instead of bare manifolds one may consider manifolds with extra structure, such as [[orientation]], [[spin structure]], [[string structure]], etc. and accordingly there is

* the _oriented cobordism ring_ $\Omega^{SO}_*$,

* the _spin cobordism ring_ $\Omega^{Spin}_*$,

etc.

In this general context the bare cobordism ring is also denoted $\Omega^O_*$ or $\Omega^{un}_*$, for emphasis.

A ring [[homomorphism]] out of the cobordism ring is a (multiplicative) _[[genus]]_.

More generally, for $X$ a fixed [[manifold]] there is a relative cobordism ring $\Omega_\bullet(X)$ whose

* elements are classes modulo cobordism over $X$ of manifolds equipped with [[continuous functions]] to $X$ ("singular manifolds");

* multiplication of $[f_1 \colon \Sigma_1 \to X]$ with $[f_2 \colon \Sigma_2 \to X]$ is given by _transversal intersection_ $\Sigma_1 \cap_X \Sigma_2$ over $X$: perturb $f_1$ such $(f_1',f_2)$ becomes a [[transversal maps]] and then form the [[pullback]] $\Sigma_1 \times_{(f_1',f_2)} \Sigma_2$ in [[Diff]].

This product is graded in that it satisfies the **dimension formula**

$$
  (dim X - dim \Sigma_1) + (dim X - dim \Sigma_2) = dim X - dim (\Sigma_1 \cap_X \Sigma_2)
$$

hence

$$
  dim (\Sigma_1 \cap_X \Sigma_2 ) = (dim \Sigma_1 + dim \Sigma_2) - dim X
  \,.
$$

Still more generally, this may be considered for $\Sigma$ being [[manifolds with boundary]]. Then $\Omega(X,A)$ for $(X,A)$ a [[CW pair]] is the ring of cobordism classes, relative boundary, of singular manifolds $\Sigma \to X$ such that the [[boundary]] of $\Sigma$ lands in in $A$.

The resulting [[functor]]

$$
  (X,A) \mapsto \Omega^G_\bullet(X,A)
$$

constitutes a [[generalized homology theory]] (see e.g. [Buchstaber, II.8](#Buchstaber)). Accordingly this is called _[[bordism homology theory]]_.

The [[spectrum]] that represents this under the [[Brown representability theorem]] is the universal [[Thom spectra]] $M G$ (e.g. [[MO]] for $G=O$ or [[MU]] for $G = U$), which canonically is a [[ring spectrum]] under [[Whitney sum]] of [[universal vector bundles]]. Accordingly the (co-bordism ring) itself is equivalently the bordism homology groups of the point, hence the [[stable homotopy groups]] of the [[Thom spectrum]] (this is _[[Thom's theorem]]_)

$$
  \Omega_\bullet^G \simeq \M G_\bullet(\ast) \simeq \pi_\bullet(M G)
  \,.
$$

This remarkable relation between [[manifolds]] and [[stable homotopy theory]] is known as _[[cobordism theory]]_ (or "Thom theory").

On general grounds this is equivalently the $M G$-[[generalized cohomology]] of the point ([[cobordism cohomology theory]])

$$
  \Omega_\bullet^G \simeq M G^\bullet(\ast)
$$

which justifies calling $\Omega_\bullet^G$ both the "bordism ring" as well as the "cobordism ring".

## Properties

### Relation to cohomotopy
 {#RelationToCohomotopy}

Let $X$ be a [[smooth manifold]] of [[dimension]] $n \in \mathbb{N}$ and let $k \leq n$. Then the [[Pontryagin-Thom construction]] induces a [[bijection]]

$$
  [X, S^k]
  \overset{\simeq}{\longrightarrow}
  \Omega^{n-k}(X)
$$

from the [[cohomotopy]] sets of $X$ to the [[cobordism group]] of $(n-k)$-dimensional [[submanifolds]] with [[normal bundle|normal]] [[framed manifold|framing]] up to normally framed [[cobordism]].

In particular, the natural group structure on [[cobordism group]] (essentially given by [[disjoint union]] of submanifolds) this way induces a group structure on the cohomotopy sets.

This is made explicit for instance in [Kosinski 93, chapter IX](#Kosinski93).

## Examples

### Framed cobordism
 {#FramedCobordism}

By [[Thom's theorem]], for any [[(B,f)-structure]] $\mathcal{B}$, there is an [[isomorphism]] (of [[commutative rings]])

$$
  \Omega^{\mathcal{B}}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(M\mathcal{B})
$$

from the [[cobordism ring]] of manifolds with stable normal $\mathcal{B}$-structure to the [[homotopy groups of a spectrum|homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].

Now for $\mathcal{B} = Fr$ [[framing]] structure, then

$$
  M Fr \simeq \mathbb{S}
$$

is equivalently the [[sphere spectrum]]. Hence in this case [[Thom's theorem]] states that there is an isomorphism

$$
  \Omega^{fr}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(\mathbb{S})
$$

between the framed cobordism ring and the [[stable homotopy groups of spheres]].

For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance ([Wang-Xu 10, section 2](#WangXu10)).

For instance

* $\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;

* $\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.

### Oriented cobordism
 {#OrientedCobordismRing}

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

The cobordism ring over the point for [[orientation|oriented]] manifolds starts out as

| $k$             | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | $\geq 9$ |
|-----------------|---|---|---|---|---|---|---|---|---|---|
| $\Omega^{SO}_k$ | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | 0 | 0 | $\mathbb{Z}\oplus \mathbb{Z}$ | $\neq 0$ |

=--

see e.g. ([ManifoldAtlas](#ManifoldAtlas))

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

For $X$ a [[CW-complex]] (for instance a [[manifold]]), then  the oriented cobordism ring is expressed in terms of the [[ordinary homology]] $H_q(X,\Omega^{SO}_{p-q})$ of $X$ with [[coefficients]] in the cobordism ring over the point, prop. \ref{OrientedCobordismOverPoint}, as

$$
  \Omega_p^{SO}(X)
  =
  \oplus_{q = 0}^p
  H_q(X,\Omega_{p-q}^{SO})
  \;
  mod\; odd \; torsion
  \,.
$$

=--

e.g. [Connor-Floyd 62, theorem 14.2](#ConnorFloyd62)

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

Original articles:

* {#Thom54} [[René Thom]], _Quelques propriétés globales des variétés différentiables_ Comment. Math. Helv. 28, (1954). 17-86 ([digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259))

* {#Milnor60} [[John Milnor]], _On the cobordism ring $\Omega^\bullet$ and a complex analogue_, Amer. J. Math. 82 (1960), 505–521 ([jstor:2372970](https://www.jstor.org/stable/2372970))

* {#Novikov60} [[Sergei Novikov]], _Some problems in the topology of manifolds connected with the theory of Thom spaces_, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

* [[Daniel Quillen]], *Elementary proofs of some results of cobordism theory using Steenrod operations*, Advances in Mathematics **7** 1 (1971) 29-56 [doi:10.1016/0001-8708(71)90041-7]

  > (using [[Steenrod operations]])

Textbook accounts:

* {#Stong68} [[Robert Stong]], _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

* {#Kochmann96} [[Stanley Kochmann]], section 1.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Lecture notes:

* {#Buchstaber} [[Victor Buchstaber]], _Geometric cobordism theory_ ([pdf](http://www.maths.manchester.ac.uk/~peter/MATH41101notes07.pdf))

* {#Francis} [[John Francis]] (notes by [[Owen Gwilliam]]), _[Topology of manifolds](http://math.northwestern.edu/~jnkf/classes/mflds/)_, _Lecture 2: Cobordism_ ([pdf](http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf))

* [[Gerald Höhn]], _Komplexe elliptische Geschlechter und $S^1$-äquivariante Kobordismustheorie_ (german) ([pdf](http://arxiv.org/PS_cache/math/pdf/0405/0405232v1.pdf))

* [[Sander Kupers]], _Oriented bordism: Calculation and application_ ([pdf](http://web.stanford.edu/~kupers/orientedbordism.pdf))

Details for [[framed cobordism]]:

* {#WangXu10} [[Guozhen Wang]], Zhouli Xu, section 2 of _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([pdf](http://math.mit.edu/~guozhen/homotopy%20groups.pdf))

* {#Putnam} [[Andrew Putman]], _Homotopy groups of spheres and low-dimensional topology_ ([pdf](http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf))

The relation to [[cohomotopy]] is made explicit in

* {#Kosinski93} [[Antoni Kosinski]], chapter IX of _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

Further discussion of oriented cobordism includes

* {#ManifoldAtlas} Manifold Atlas, _[Oriented bordism](http://www.map.mpim-bonn.mpg.de/Oriented_bordism)_

* {#ConnorFloyd62} P. E. Conner, E. E. Floyd, _Differentiable periodic maps_, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. ([Euclid](http://projecteuclid.org/euclid.bams/1183524501), [pdf](http://www.maths.ed.ac.uk/~aar/papers/cf1.pdf))

A historical review in the context of [[complex cobordism cohomology theory]]/[[Brown-Peterson theory]] is in

* [[Doug Ravenel]], chapter 4, section 2 of _[[Complex cobordism and stable homotopy groups of spheres]]_

On fibered cobordism groups:

* Astey, Greenberg, Micha, Pastor, _Some fibered cobordisms groups are not finitely generated_ ([pdf](http://www.ams.org/journals/proc/1988-104-01/S0002-9939-1988-0958093-3/S0002-9939-1988-0958093-3.pdf))

Discussion of the $G$-equivariant complex coborism ring includes

* G. Comezana and [[Peter May]], _A completion theorem in complex cobordism_, in Equivariant Homotopy and Cohomology Theory, CBMS Regional conference series in Mathematics, American Mathematical Society Publications, Volume 91, Providence, 1996.

* [[Igor Kriz]], _The $\mathbb{Z}/p$–equivariant complex cobordism ring_, from: "Homotopy invariant algebraic structures (Baltimore, MD, 1998)", Amer. Math. Soc. Providence, RI (1999) 217–223

* {#Strickland01} [[Neil Strickland]], _Complex cobordism of involutions_, Geom. Topol. 5 (2001) 335-345 ([arXiv:math/0105020](http://arxiv.org/abs/math/0105020))

* William Abrams, _Equivariant complex cobordism_, 2013 ([pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf))

[[!redirects bordism ring]]

[[!redirects cobordism rings]]

[[!redirects cobordism group]]
[[!redirects cobordism groups]]

[[!redirects bordism ring]]
[[!redirects bordism rings]]
[[!redirects bordism group]]
[[!redirects bordism groups]]