heterogeneous identity type

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### Context
#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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#### Type theory
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[[!include type theory - contents]]
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#### Induction
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[[!include induction - contents]]
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#Contents#
* table of contents
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## Idea

In [[dependent type theory]], a **heterogeneous identity type** or **dependent identity type** is a version of the [[identity type]] where given a [[type]] $A$, a [[type family]] $x:A \vdash B(x)$, [[terms]] $a:A$, $b:A$, and an [[identification]] $p:a =_A b$, the heterogeneous identity type between two elements $y:B(a)$ and $z:B(b)$ is a type whose elements witness that $y$ and $z$ are "equal" over or modulo the identification $p$.

### Note on terminology

There are many different names used for this particular dependent type, as well as many different names used for the terms of this dependent type. These include the following:

| name of type | name of terms |
|--------------|---------------|
| heterogeneous identity type | heterogeneous identities |
| heterogeneous path type | heterogeneous paths |
| heterogeneous identification type | heterogeneous identifications |
| heterogeneous equality type | heterogeneous equalities |

These four names have different reasons behind the use of the name:

* The name "heterogeneous identity type" comes from the fact that the dependent identity type is the canonical [[one-to-one correspondence]] $x:B(a), y:B(b) \vdash \mathrm{Id}_B^p(x, y)$ of the [[transport]] [[equivalence in type theory|equivalence]] $\mathrm{tr}_B^p:B(a) \simeq B(b)$ on a type family $x:A \vdash B(x)$ and an identification $p:a =_A b$, which is the dependent/heterogeneous version of the identity type for the identity equivalence $\mathrm{id}_A:A \simeq A$

* The name "heterogeneous path type" comes from the fact that, in a context where terms of ordinary identity types can be represented as functions from an [[interval type]], the terms of the heterogeneous identity type can similarly be represented as [[dependent functions]] from the [[interval type]].  For instance, this is true in some categorical semantics and in [[cubical type theory]].

## Definitions

### Using identity types

There are many different ways to define heterogeneous identity types. The simplest definition states that given a [[type family]]  $x:A \vdash B(x)$, heterogeneous identity types are an [[inductive family]] of [[types]]

$$a:A, b:A, p:\mathrm{Id}_A(a, b), y:B(a), z:B(b) \vdash \mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$$

generated by the family of elements called "heterogeneous reflexivity"

$$a:A, y:B(a) \vdash \mathrm{hrefl}_{x:A.B(x)}(a, y):\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)$$

#### Inference rules

The inference rules for forming heterogeneous identity types and terms are as follows.  First the inference rule that defines the heterogeneous identity type itself, as a [[dependent type]], in some [[context]] $\Gamma$.

Formation rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b)
    \end{array}
  }{\Gamma \vdash \mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \; \mathrm{type}}$$

Now the basic "introduction" rule, which says that everything is equal to itself in a canonical way.

Introduction rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{hrefl}_{x:A.B(x)}(a, y):\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)}$$

Next we have the "elimination" rule:

Elimination rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), y:B(a), z:B(b), q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \vdash C(a, b, p, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b) \quad \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, a, b, p, y, z, q):C(a, b, p, y, z, q)}$$

The elimination rule then says that if:

1. for any elements $a:A$ and $b:A$, any identification $p:\mathrm{Id}_A(a,b)$, any elements $y:B(a)$ and $z:B(b)$, and any heterogeneous identification $q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$ we have a type $C(a, b, p, y, z, q)$, and
1. we have a specified dependent function
$$t:\prod_{a:A} \prod_{y:B(a)} C(a,a,\mathrm{refl}_A(x),y,y,\mathrm{hrefl}_{x:A.B(x)}(a, y))$$

then we can construct a canonically defined term called the *eliminator*
$$\mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t,a,b,p,y,z,q):C(a,b,p,y,z,q)$$
for *any* $a$, $b$, $p$, $y$, $z$, and $q$.

Then, we have the [[computation rule]] or [[beta-reduction]] rule. This says that for all elements $a:A$ and $y:B(a)$, substituting the dependent function $t:\prod_{a:A} \prod_{y:B(a)} C(a,a,\mathrm{refl}_A(x),y,y,\mathrm{hrefl}_{x:A.B(x)}(a, y))$ into the eliminator along heterogeneous [[reflexive relation|reflexivity]] for $a$ and $y$ yields an element equal to $t(a, y)$ itself.  Normally "equal" here means [[judgmental equality]] (a.k.a. definitional equality), but it is also possible for it to mean [[propositional equality]] (a.k.a. typal equality), so there are two possible computation rules.

Computation rules for heterogeneous identity types:

* Judgmental computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), y:B(a), z:B(b), q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \vdash C(a, b, p, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \equiv t(a, y):C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))}$$

* Propositional computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), y:B(a), z:B(b), q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \vdash C(a, b, p, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{hId}_{x:A.B(x)}}(t, a, y):\mathrm{Id}_{C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))}(\mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y), t(a, y))}$$

Finally, one might consider a [[uniqueness rule]] or [[eta-conversion]] rule.  But similar to the case for [[identity types]], a judgmental uniqueness rule for heterogeneous identity types implies that the type theory is an [[extensional type theory]], in which case there is not much need for heterogeneous identity types, so such a rule is almost never written down.  And as for identity types and other inductive types, the propositional/typal uniqueness rule is provable from the other four inference rules, so we don't write it out explicitly.

#### Dependent universal property

The elimination, typal computation, and typal uniqueness rules for heterogeneous identity types state that heterogeneous identity types satisfy the **dependent universal property of heterogeneous identity types**. 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 heterogeneous identity types could be simplified to the following rule:

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), y:B(a), z:B(b), q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \vdash C(a, b, p, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{up}_{\mathrm{hId}_{x:A.B(x)}}(t, f, a, b, p):
\begin{array}{c}
\exists!J:\left(\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))\right) \\
\to \left(\prod_{c:A} \prod_{d:A} \prod_{r:\mathrm{Id}_A(c, d)} \prod_{e:B(c)} \prod_{f:B(c)} \prod_{s:\mathrm{hId}_{x:A.B(x)}(c, d, r, e, f)} C(c, d, r, e, f, s)\right) \\
.\mathrm{Id}_{C(a, a, \mathrm{refl}_A(a), y, y)}(J(t, a, a, \mathrm{refl}_A, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)), t(a, y))
\end{array}
}$$

In natural language this states that given a type $A$, a type family $B(x)$ indexed by elements $x:A$, and given a type family $C(a, b, p, y, z, q)$ indexed by elements $a:A$, $b:A$, $p:\mathrm{Id}_A(a, b)$, $y:B(a)$, $z:B(b)$, and $q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$, for all elements $t:\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))$, $a:A$, and $y:B(a)$, there exists a unique function $J$ with domain
$$\prod_{a:A} \prod_{y:B(a)} C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))$$
and codomain
$$\prod_{c:A} \prod_{d:A} \prod_{r:\mathrm{Id}_A(c, d)} \prod_{e:B(c)} \prod_{f:B(c)} \prod_{s:\mathrm{hId}_{x:A.B(x)}(c, d, r, e, f)} C(c, d, r, e, f, s)$$
such that $J(t, a, a, \mathrm{refl}_A, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))$ is equal to $t(a, y)$ in the type $C(a, a, \mathrm{refl}_A(a), y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))$.

### Using function types from the interval type

There is another notion of heterogeneous identity type, which uses [[functions]] from the [[interval type]] $f:\mathbb{I} \to A$, rather than elements $a:A$, $b:A$, and $p:a =_A b$:

Given a [[type family]] $x:A \vdash B(x)$, heterogeneous identity types are an [[inductive family]] of [[types]]

$$f:\mathbb{I} \to A, y:B(f(0)), z:B(f(1)) \vdash \mathrm{hId}_{x:A.B(x)}(f, y, z)$$

generated by the family of elements called "heterogeneous reflexivity"

$$a:A, y:B(a) \vdash \mathrm{hrefl}_{x:A.B(x)}(a, y):\mathrm{hId}_{x:A.B(x)}(\lambda i:\mathbb{I}.a, y, y)$$

since by definition of [[constant function]] we have that

$$(\lambda i:\mathbb{I}.a)(0) \equiv a \qquad (\lambda i:\mathbb{I}.a)(1) \equiv a$$

#### Inference rules

The inference rules for forming heterogeneous identity types and terms are as follows.  First the inference rule that defines the heterogeneous identity type itself, as a [[dependent type]], in some [[context]] $\Gamma$.

Formation rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash f:\mathbb{I} \to A \quad \Gamma \vdash y:B(f(0)) \quad \Gamma \vdash z:B(f(1))
    \end{array}
  }{\Gamma \vdash \mathrm{hId}_{x:A.B(x)}(f, y, z) \; \mathrm{type}}$$

Now the basic "introduction" rule, which says that everything is equal to itself in a canonical way.

Introduction rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{hrefl}_{x:A.B(x)}(a, y):\mathrm{hId}_{x:A.B(x)}(\lambda i:\mathbb{I}.a, y, y)}$$

Next we have the "elimination" rule:

Elimination rule for heterogeneous identity types:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, f:\mathbb{I} \to A, y:B(f(0)), z:B(f(1)), q:\mathrm{hId}_{x:A.B(x)}(f, y, z) \vdash C(f, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(\lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \\
      \Gamma \vdash f:\mathbb{I} \to A \quad \Gamma \vdash y:B(f(0)) \quad \Gamma \vdash z:B(f(1)) \quad \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(f, y, z)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, f, y, z, q):C(f, y, z, q)}$$

The elimination rule then says that if:

1. for any function $f:\mathbb{I} \to A$, any elements $y:B(f(0))$ and $z:B(f(1))$, and any heterogeneous identification $q:\mathrm{hId}_{x:A.B(x)}(f, y, z)$ we have a type $C(f, y, z, q)$, and
1. we have a specified dependent function
$$t:\prod_{a:A} \prod_{y:B(a)} C(\lambda i:\mathbb{I}.a,y,y,\mathrm{hrefl}_{x:A.B(x)}(a, y))$$

then we can construct a canonically defined term called the *eliminator*
$$\mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t,f,y,z,q):C(a,b,p,y,z,q)$$
for *any* $f$, $y$, $z$, and $q$.

Then, we have the [[computation rule]] or [[beta-reduction]] rule. This says that for all elements $a:A$ and $y:B(a)$, substituting the dependent function $t:\prod_{a:A} \prod_{y:B(a)} C(\lambda i:\mathbb{I}.a,y,y,\mathrm{hrefl}_{x:A.B(x)}(a, y))$ into the eliminator along heterogeneous [[reflexive relation|reflexivity]] for $a$ and $y$ yields an element equal to $t(a, y)$ itself.  Normally "equal" here means [[judgmental equality]] (a.k.a. definitional equality), but it is also possible for it to mean [[propositional equality]] (a.k.a. typal equality), so there are two possible computation rules.

Computation rules for heterogeneous identity types:

* Judgmental computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, f:\mathbb{I} \to A, y:B(f(0)), z:B(f(1)), q:\mathrm{hId}_{x:A.B(x)}(f, y, z) \vdash C(f, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(\lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, \lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \equiv t(a, y):C(\lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))}$$

* Propositional computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, f:\mathbb{I} \to A, y:B(f(0)), z:B(f(1)), q:\mathrm{hId}_{x:A.B(x)}(f, y, z) \vdash C(f, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(\lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{hId}_{x:A.B(x)}}(t, a, y):\mathrm{Id}_{C(\lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))}(\mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t, \lambda i:\mathbb{I}.a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y), t(a, y))}$$

Finally, one might consider a [[uniqueness rule]] or [[eta-conversion]] rule.  But similar to the case for [[identity types]], a judgmental uniqueness rule for heterogeneous identity types implies that the type theory is an [[extensional type theory]], in which case there is not much need for heterogeneous identity types, so such a rule is almost never written down.  And as for identity types and other inductive types, the propositional/typal uniqueness rule is provable from the other four inference rules, so we don't write it out explicitly.

### From two-type identity types

Given [[two-type identity types]] in the type theory, heterogeneous identity types between $y:B(a)$ and $z:B(b)$ across the path $p:a =_A b$ could be defined as the two-type identity type between $y:B(a)$ and $z:B(b)$ across the [[transport]] equivalence $\mathrm{transport}_{x:A.B(x)}(a, b, p)$

$$\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \coloneqq \mathrm{Id}_{B(a), B(b)}(\mathrm{transport}_{x:A.B(x)}(a, b, p), y, z)$$

## Properties

### Groupoidal structure of heterogeneous identity types

Reflexivity of heterogeneous identifications is already given in the [[introduction rule]] of heterogeneous identity types.

\begin{definition}
Given

* a type family $x:A \vdash B(x)$,

* elements $a_1:A$, $a_2:A$, $a_3:A$,

* identifications $p_1:\mathrm{Id}_A(a_1, a_2)$ and $p_2:\mathrm{Id}_A(a_2, a_3)$,

* elements $y_1:B(a_1)$, $y_2:B(a_2)$, and $y_3:B(a_3)$,

there is a function called the **concatenation of heterogeneous identifications**

$$(-) \bullet (-):\mathrm{hId}_{x:A.B(x)}(a_1, a_2, p_1, y_1, y_2) \times \mathrm{hId}_{x:A.B(x)}(a_2, a_3, p_2, y_2, y_3) \to \mathrm{hId}_{x:A.B(x)}(a_1, a_3, p_1 \bullet p_2, y_1, y_3)$$

defined by double induction on heterogeneous reflexivity for $a:A$ and $y:B(a)$

$$\mathrm{hrefl}_A(a, y) \bullet \mathrm{hrefl}_A(a, y) \equiv \mathrm{hrefl}_A(a, y)$$

in the type

$$\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)$$
\end{definition}

\begin{definition}
Given

* a type family $x:A \vdash B(x)$,

* elements $a:A$, $b:A$

* identifications $p:\mathrm{Id}_A(a, b)$

* elements $y:B(a)$, $z:B(b)$,

there is a dependent function called the **left unitor of heterogeneous identifications**

$$\mathrm{hlunitor}:\prod_{q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)} \mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)}(\mathrm{hrefl}_A(a, y) \bullet q, q)$$

defined by induction on heterogeneous reflexivity for $a:A$ and $y:B(a)$:

$$\mathrm{hlunitor}(\mathrm{hrefl}_A(a, y)) \equiv \mathrm{refl}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)}(\mathrm{hrefl}_A(a, y))$$

in the type

$$\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)}(\mathrm{hrefl}_A(a, y) \bullet \mathrm{hrefl}_A(a, y), \mathrm{hrefl}_A(a, y))$$
\end{definition}

\begin{definition}
Given

* a type family $x:A \vdash B(x)$,

* elements $a:A$, $b:A$

* identifications $p:\mathrm{Id}_A(a, b)$

* elements $y:B(a)$, $z:B(b)$,

there is a dependent function called the **right unitor of heterogeneous identifications**

$$\mathrm{hrunitor}:\prod_{q:\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)} \mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)}(q \bullet \mathrm{hrefl}_A(b, z), q)$$

defined by induction on heterogeneous reflexivity for $b:B$ and $z:B(b)$

$$\mathrm{hrunitor}(\mathrm{hrefl}_A(b, z)) \equiv \mathrm{refl}_{\mathrm{hId}_{x:A.B(x)}(b, b, \mathrm{refl}_A(b), z, z)}(\mathrm{hrefl}_A(b, z))$$

in the type

$$\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(b, b, \mathrm{refl}_A(b), z, z)}(\mathrm{hrefl}_A(b, z) \bullet \mathrm{hrefl}_A(b, z), \mathrm{hrefl}_A(b, z))$$
\end{definition}

\begin{definition}
Given

* a type family $x:A \vdash B(x)$,

* elements $a_1:A$, $a_2:A$, $a_3:A$, $a_4:A$

* identifications $p_1:\mathrm{Id}_A(a_1, a_2)$, $p_2:\mathrm{Id}_A(a_2, a_3)$, $p_3:\mathrm{Id}_A(a_3, a_4)$,

* elements $y_1:B(a_1)$, $y_2:B(a_2)$, $y_3:B(a_3)$, $y_4:B(a_4)$,

there is a dependent function called the **associator of heterogeneous identifications**

$$\mathrm{hassoc}:\prod_{q_1:\mathrm{hId}_{x:A.B(x)}(a_1, a_2, p_1, y_1, y_2)} \prod_{q_2:\mathrm{hId}_{x:A.B(x)}(a_2, a_3, p_2, y_2, y_3)} \prod_{q_3:\mathrm{hId}_{x:A.B(x)}(a_3, a_4, p_3, y_3, y_4)} \mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a_1, a_4, \mathrm{assoc}(p_1, p_2, p_3), y_1, y_4)}(q_1 \bullet (q_2 \bullet q_3), (q_1 \bullet q_2) \bullet q_3)$$

defined by triple induction on heterogeneous reflexivity for $a:A$ and $y:B(a)$

$$\mathrm{hassoc}(\mathrm{hrefl}_A(a, y),  \mathrm{hrefl}_A(a, y), \mathrm{hrefl}_A(a, y)) \equiv \mathrm{refl}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)}(\mathrm{hrefl}_A(a, y))$$

in the type

$$\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)}(\mathrm{hrefl}_A(a, y) \bullet (\mathrm{hrefl}_A(a, y) \bullet \mathrm{hrefl}_A(a, y)), (\mathrm{hrefl}_A(a, y) \bullet \mathrm{hrefl}_A(a, y)) \bullet \mathrm{hrefl}_A(a, y))$$
\end{definition}

\begin{definition}
Given

* a type family $x:A \vdash B(x)$,

* elements $a_1:A$, $a_2:A$

* identifications $p_1:\mathrm{Id}_A(a_1, a_2)$

* elements $y_1:B(a_1)$, $y_2:B(a_2)$,

there is a function called the **inverse of heterogeneous identifications**

$$\mathrm{inv}(a_1, a_2, p_1, y_1, y_2):\mathrm{hId}_{x:A.B(x)}(a_1, a_2, p_1, y_1, y_2) \to \mathrm{hId}_{x:A.B(x)}(a_2, a_1, p_1^{-1}, y_2, y_1)$$

defined by induction on heterogeneous reflexivity:

$$a:A, y:B(a) \vdash \mathrm{inv}(a, a, \mathrm{refl}_A(a), y, y)(\mathrm{hrefl}_A(a, y)) \equiv \mathrm{hrefl}_A(a, y):\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y) \to \mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), y, y)$$
\end{definition}

### Relation between identity types and heterogeneous identity types

There are equivalences between identity types and heterogeneous identity types

$$\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \simeq \mathrm{Id}_{B(b)}(\mathrm{transport}_{x:A.B(x)}(a, b, p)(y), z)$$

and

$$\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z) \simeq \mathrm{Id}_{B(a)}(y, \mathrm{transport}_{x:A.B(x)}(a, b, p)^{-1}(z))$$

for all $a:A$, $b:A$, $p:\mathrm{Id}_A(a, b)$, $y:B(a)$, and $z:B(b)$. For constant families $x:A \vdash B$, $\mathrm{transport}_{x:A.B}(a, b, p)(y) \coloneqq y$ and $\mathrm{transport}_{x:A.B}(a, b, p)^{-1}(z) \coloneqq z$, so the above two equivalences simplify down to

$$\mathrm{hId}_{x:A.B}(a, b, p, y, z) \simeq \mathrm{Id}_{B}(y, z)$$

for constant type families $x:A \vdash B$.

### One-to-one correspondence structure

Heterogeneous identity types are [[one-to-one correspondences]]. Given elements $a:A$, $b:A$, and $p:\mathrm{Id}_A(a, b)$,

* for all elements $z:B(b)$, there exists a unique element $y:B(a)$ such that $\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$; the witness is given by the right projection function of [[transport]] across $p$, $\mathrm{transport}_{x:A.B(x)}(a, b, p):B(a) \simeq B(b)$

$$\pi_2(\mathrm{transport}_{x:A.B(x)}(a, b, p)):\prod_{z:B(b)} \exists!y:B(a).\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$$

* for all elements $y:B(a)$, there exists a unique element $z:B(b)$ such that $\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$; the witness is given by the right projection function of the [[inverse]] [[transport]] across $p$, $\mathrm{transport}_{x:A.B(x)}(a, b, p)^{-1}:B(b) \simeq B(a)$

$$\pi_2(\mathrm{transport}_{x:A.B(x)}(a, b, p)^{-1}):\prod_{y:B(a)} \exists!z:B(b).\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$$

### Function application to identifications

The [[dependent function application to identifications]] is defined in the usual way, by induction on the [[identity type]] of the index type:

* given a type $A$, a type family $x:A \vdash B(x)$, and a [[dependent function]] $f:\prod_{x:A} B(x)$, there are dependent functions

$$\mathrm{apd}_{x:A.B(x)}(f):\prod_{a:A} \prod_{b:A} \prod_{p:\mathrm{Id}_A(a, b)} \mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))$$

inductively defined respectively by:

$$\mathrm{apd}_{x:A.B(x)}(f, a, a, \mathrm{refl}_A(a)) \equiv \mathrm{hrefl}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), f(a), f(a)):\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), f(a), f(a))$$

### Extensionality principles

Heterogeneous identity types are used in many extensionality principles.

* The extensionality principle for [[function types]] (non-dependent [[function extensionality]]) states that there is an equivalence

$$\mathrm{ext}_{\to}:\prod_{f:A \to B} \prod_{g:A \to B} \mathrm{Id}_{A \to B}(f, g) \simeq \prod_{a:A} \prod_{b:A} \prod_{p:\mathrm{Id}_A(a, b)} \mathrm{hId}_{B}(a, b, p, f(a), g(b))$$

* The extensionality principle for [[dependent product types]] (dependent [[function extensionality]]) states that there is an equivalence

$$\mathrm{ext}_\Pi:\prod_{f:\prod_{x:A} B(x)} \prod_{g:\prod_{x:A} B(x)} \mathrm{Id}_{\prod_{x:A} B(x)}(f, g) \simeq \prod_{a:A} \prod_{b:A} \prod_{p:\mathrm{Id}_A(a, b)} \mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), g(b))$$

* The extensionality principle for [[product types]] states that there is an equivalence

$$\mathrm{ext}_{\times}:\prod_{s:A \times B} \prod_{t:A \times B} \mathrm{Id}_{A \times B}(s, t) \simeq \sum_{p:\mathrm{Id}_A(\pi_1(s), \pi_1(t))} \mathrm{hId}_{B}(\pi_1(s), \pi_1(t), p, \pi_2(s), \pi_2(t))$$

* The extensionality principle for [[dependent sum types]] states that there is an equivalence

$$\mathrm{ext}_\Sigma:\prod_{s:\sum_{x:A} B(x)} \prod_{t:\sum_{x:A} B(x)} \mathrm{Id}_{\sum_{x:A} B(x)}(s, t) \simeq \sum_{p:\mathrm{Id}_A(\pi_1(s), \pi_1(t))} \mathrm{hId}_{x:A.B(x)}(\pi_1(s), \pi_1(t), p, \pi_2(s), \pi_2(t))$$

### Inference rules for higher inductive types

Whenever there is an [[identification]] given in an [[introduction rule]] for a [[higher inductive type]], one of the hypothesis in the elimination and computation rules for the higher inductive types is a judgment of a term of the corresponding heterogeneous identification type along the given identification in the introduction rule.

For example, the introduction rules for the [[circle type]] $S^1$ state that one could construct an element $\mathcal{b}:S^1$ and an identification $\mathcal{l}:\mathrm{Id}_{S^1}(\mathcal{b}, \mathcal{b})$. The elimination rule for the circle type states that given the hypotheses of

* a type family $x:S^1 \vdash C(x)$,

* an element $c_\mathcal{b}:C(\mathcal{b})$,

* a heterogeneous identification $c_\mathcal{l}:\mathrm{hId}_{x:S^1.C(x)}(\mathcal{b}, \mathcal{b}, \mathcal{l}, c_\mathcal{b}, c_\mathcal{b})$,

one could construct the family of elements $p:S^1 \vdash \mathrm{ind}_{S^1}(c_\mathcal{b}, c_\mathcal{l})(p):C(p)$.

## Categorical semantics

needs to be written

## In cubical type theory

The [[path types]] in cubical type theory are heterogeneous path types. See [[cubical path type]] for more details.

## In higher observational type theory

In [[higher observational type theory]], the heterogeneous identity type is a primitive type former (although depending on the presentation, it can also be obtained using $ap$ into the universe).  In its general form, the type family can depend not just on a single type but on a [[type telescope]] $\Delta$. The resulting heterogeneous identity type then depends on an "identification in that telescope", which is defined by mutual recursion as a telescope of heterogeneous identity types.  The [[type formation|formation rule]] is then

$$\frac{\varsigma:\delta =_\Delta \delta^{'} \quad \delta \vdash A\; \mathrm{type} \quad a:A[\delta] \quad a^{'}:A[\delta^{'}]}{a =_{\Delta.A}^\varsigma a^{'}\; \mathrm{type}}$$

... needs to be finished

### Heterogeneous identity types in universes

Given a term of a universe $A:\mathcal{U}$, a judgment $z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U}$, terms $x:\mathcal{T}_\mathcal{U}(A)$ and $y:\mathcal{T}_\mathcal{U}(A)$, and an identity $p:\mathrm{id}_{\mathcal{T}_\mathcal{U}(A)}(x,y)$, we have

$$\mathrm{ap}_{z.B}(p):\mathrm{id}_\mathcal{U}(B(x),B(y))$$

and

$$(u,v):\mathcal{T}_\mathcal{U}(B(x)) \times \mathcal{T}_\mathcal{U}(B(y)) \vdash \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u,v):\mathcal{U}$$

We could define a heterogeneous identity type as

$$\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \coloneqq \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u, v)$$

There is a rule

$$\frac{A:\mathcal{U} \quad z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U} \quad a:\mathcal{T}_\mathcal{U}(A)}{\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{\refl_{a}}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B[a/z])}(u, v)}$$

and for constant families $B:\mathcal{U}$

$$\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B)}(u, v)$$

## A variant using dependent function types

There is a variant of heterogeneous identity types which uses a dependent function $f:\prod_{x:A} B(x)$ instead of two elements $y:B(a)$ and $z:B(b)$ in the constructor. This variant is useful for defining [[dependent function application to identifications]].

These are given by the following inference rules:

* Formation rules
  $$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b)
    \end{array}
  }{\Gamma \vdash \mathrm{fhId}_{x:A.B(x)}(f, a, b, p) \; \mathrm{type}}$$

* Introduction rules
  $$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \\
      \Gamma \vdash a:A
    \end{array}
  }{\Gamma \vdash \mathrm{hrefl}_{x:A.B(x)}(f, a):\mathrm{fhId}_{x:A.B(x)}(f, a, b, p)}$$

* Elimination rules
  $$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), q:\mathrm{fhId}_{x:A.B(x)}(f, a, b, p) \vdash C(a, b, p, q) \; \mathrm{type} \\
      \Gamma \vdash \vdash t:\prod_{x:A} C(a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_A(a, b) \quad \Gamma \vdash q:\mathrm{hId}_{x:A.B(x)}(f, a, b, p)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{fhId}_{x:A.B(x)}}(t, f, a, b, p, q):C(a, b, p, q)}$$

* Judgmental computational rule
  $$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), q:\mathrm{fhId}_{x:A.B(x)}(f, a, b, p) \vdash C(a, b, p, q) \; \mathrm{type} \\
      \Gamma \vdash \vdash t:\prod_{x:A} C(a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b \quad \Gamma \vdash q:\mathrm{fhId}_{x:A.B(x)}(f, a, b, p)
    \end{array}
  }{\Gamma, a:A \vdash \mathrm{ind}_{\mathrm{fhId}_{x:A.B(x)}}(t, f, a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a)) \equiv t(a):C(a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a))}$$

* Propositional computational rule
  $$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \\
      \Gamma, a:A, b:A, p:\mathrm{Id}_A(a, b), q:\mathrm{fhId}_{x:A.B(x)}(f, a, b, p) \vdash C(a, b, p, q) \; \mathrm{type} \\
      \Gamma \vdash \vdash t:\prod_{x:A} C(a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b \quad \Gamma \vdash q:\mathrm{fhId}_{x:A.B(x)}(f, a, b, p)
    \end{array}
  }{\Gamma, a:A \vdash \beta_{\mathrm{hId}_{x:A.B(x)}}(t, f, a):\mathrm{Id}_{C(a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a))}(\mathrm{ind}_{\mathrm{fhId}_{x:A.B(x)}}(t, f, a, a, \mathrm{refl}_A(a), \mathrm{hrefl}_{x:A.B(x)}(f, a)), t(a))}$$

Given a dependent function $f:\prod_{x:A} B(x)$, we have

$$\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b)) \simeq \mathrm{fhId}_{x:A.B(x)}(f, a, b, p)$$

since by induction we have functions

$$\mathrm{hIdTofhId}(f, a, b, p):\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b)) \to \mathrm{fhId}_{x:A.B(x)}(f, a, b, p)$$

and

$$\mathrm{fhIdTohId}(f, a, b, p):\mathrm{fhId}_{x:A.B(x)}(f, a, b, p) \to \mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))$$

inductively defined by

$$\mathrm{hIdTofhId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{hrefl}_{x:A.B(x)}(a, f(a))) \equiv \mathrm{hrefl}_{x:A.B(x)}(f, a)$$

and

$$\mathrm{fhIdTohId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{hrefl}_{x:A.B(x)}(f, a)) \equiv \mathrm{hrefl}_{x:A.B(x)}(a, f(a))$$

Thus, we have identifications

$$H(f, a, b, p):\mathrm{Id}_{\mathrm{fhId}_{x:A.B(x)}(f, a, b, p)}(\mathrm{hIdTofhId}(f, a, b, p)(\mathrm{fhIdTohId}(f, a, b, p)(q), q)$$

and

$$K(f, a, b, p):\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, b, p, f(a), f(b))}(\mathrm{fhIdTohId}(f, a, b, p)(\mathrm{hIdTofhId}(f, a, b, p)(q)), q)$$

inductively defined by

$$H(f, a, a, \mathrm{refl}_A(a)) \equiv \mathrm{refl}_{\mathrm{fhId}_{x:A.B(x)}(f, a, a, \mathrm{refl}_A(a))}(\mathrm{hrefl}_{x:A.B(x)}(f, a))$$

and

$$K(f, a, a, \mathrm{refl}_A(a)) \equiv \mathrm{refl}_{\mathrm{hId}_{x:A.B(x)}(f, a, a, \mathrm{refl}_A(a))}(\mathrm{hrefl}_{x:A.B(x)}(a, f(a))$$

since by the properties of [[judgmental equality]]

$$\mathrm{Id}_{\mathrm{fhId}_{x:A.B(x)}(f, a, a, \mathrm{refl}_A(a))}(\mathrm{fhIdTofhId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{hIdTofhId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{hrefl}_{x:A.B(x)}(f, a))), \mathrm{hrefl}_{x:A.B(x)}(f, a)) \equiv \mathrm{Id}_{\mathrm{fhId}_{x:A.B(x)}(f, a, a, \mathrm{refl}_A(a))}(\mathrm{hrefl}_{x:A.B(x)}(f, a)), \mathrm{hrefl}_{x:A.B(x)}(f, a))$$

and

$$\mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), f(a), f(a))}(\mathrm{hIdTofhId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{fhIdTohId}(f, a, a, \mathrm{refl}_A(a))(\mathrm{hrefl}_{x:A.B(x)}(a, f(a))), \mathrm{hrefl}_{x:A.B(x)}(a, f(a))) \equiv \mathrm{Id}_{\mathrm{hId}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a), f(a), f(a))}(\mathrm{hrefl}_{x:A.B(x)}(a, f(a)), \mathrm{hrefl}_{x:A.B(x)}(a, f(a)))$$

Then by the properties of [[quasi-inverse functions]] and [[equivalences of types]] one could prove that both $\mathrm{hIdTofhId}(f, a, b, p)$ and $\mathrm{fhIdTohId}(f, a, b, p)$ are equivalences of types.

## Related concepts

* [[two-type identity type]]

* [[fibered heterogeneous identity type]]

[[!include notions of type]]

## References

Inference rules for heterogeneous identity types (referred to as "dependent identity types" in the article) could be found in section 6 of:

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Semantics of higher inductive types*, Math. Proc. Camb. Phil. Soc. **169** (2020) 159-208 [[arXiv:1705.07088](https://arxiv.org/abs/1705.07088), talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf), [doi:10.1017/S030500411900015X](https://doi.org/10.1017/S030500411900015X)]

For heterogeneous identity types in the context of [[higher observational type theory]], see:

* [[Mike Shulman]], *Towards Third-Generation HOTT -- Part 1* ([slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-04-28.pdf), [video](https://www.youtube.com/watch?v=FrxkVzItMzA))

* [[Mike Shulman]], *Towards Third-Generation HOTT -- Part 2* ([slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-05-05.pdf), [video](https://www.youtube.com/watch?v=5ciDNfmvMdU))

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heterogeneous path type

## Disambiguation

There are a few different types which can be called *heterogeneous path types* or *dependent path types* in [[dependent type theory]]:

* The [[heterogeneous identity type]] $u =_{B(-)}^{p} v$ for type family $B(x)$ indexed by $x:A$, and elements $x:A$, $y:A$, $u:B(x)$, $v:B(y)$, and identification $p:x =_A y$, which is the type of paths between $u:B(x)$ and $v:B(y)$ across $p:x =_A y$;

* The [[dependent sum type]] $\sum_{u:B(x)} \sum_{v:B(y)} u =_{B(-)}^{p} v$, which is the type of all heterogeneous paths in the type family $B(x)$ across the identification $p:x =_A y$;

* The [[dependent function type]] $\prod_{i:\mathbb{I}} B(x)$ from the [[interval type]] $\mathbb{I}$ to the type family $B(x)$.

* In [[cubical type theory]], the [[cubical path type]] $\mathrm{path}_{i.A}(x, y)$ for a line of types $A(i)$ indexed by interval variable $i:\mathbb{I}$, given interval terms $i:\mathbb{I}$ and $j:\mathbb{I}$ and elements $x:A(i)$ and $y:A(j)$.

See also [[path type]] for the homogeneous version of path types.

category: disambiguation

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[[!redirects dependent path types]]
[[!redirects heterogeneous path type]]
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heteromorphism

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## Idea

The term _heteromorphism_ ([Ellerman 2006](#Ellerman06), [Ellerman 2007](#Ellerman07)) refers to concept of a [[morphism]] not (necessarily) between two [[objects]] in the same [[category]], but between objects in two different categories that are related by a [[functor]], and typically by an [[adjoint functor]], in which case the notion is first made explicit in [Pareigis 1970, §2.2](#Pareigis70). Indeed, sets of heteromorphism may be used to characterize [[adjoint functors]]. Generally, the set of heteromorphisms is that assigned by the corresponding [[profunctor]] to the pair of objects.

The concept is also known as the *[[cograph of a functor]]*. While in traditional [[category theory]] literature this is maybe somewhat neglected, it serves for instance as the very definition of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functor]] in the context of [[quasi-categories]] in [Lurie 2009](#Lurie09)) (without using the term "heteromorphism" there).

## Definition

Given two [[categories]] $\mathcal{C}$ and $\mathcal{D}$ and a [[functor]] $L \colon \mathcal{C} \to \mathcal{D}$, then for $c\in \mathcal{C}$ and $d \in \mathcal{D}$ two [[objects]], the set of _heteromorphisms_ between them is the [[hom set]]

$$
  Het(c,d) \coloneqq \mathcal{D}\big(L(c),d\big)
  \,.
$$

When $L$ has a [[right adjoint]] $R \colon \mathcal{D}\to \mathcal{C}$ then this is of course equivalent to

$$
  Het(c,d) \cong \mathcal{C}\big(c,R(d)\big)
  \,.
$$

More generally, for $Het \colon \mathcal{C}^{op}\times \mathcal{D}\to Set$ a [[profunctor]] from $\mathcal{C}$ to $\mathcal{D}$, then $Het(c,d)$ may be called its set of heteromorphisms from $c$ to $d$.

A pair of [[adjoint functors]] arise when a Het profunctor is a [[representable functor]] on both the left and right. The [[left adjoint]] is representing functor on the left:
$$
  \mathcal{D}\big(L(c),d\big) \cong  Het(c,d)
  \,
$$
and symmetrically the [[right adjoint]] is the representing functor on the right:
$$
  Het(c,d) \cong \mathcal{C}\big(c,R(d)\big)
  \,.
$$

Putting the two representations together gives the usual natural isomorphism characterization of a [[adjunction]] but with the het middle term:
$$
\mathcal{D}\big(L(c),d\big) \cong  Het(c,d) \cong \mathcal{C}\big(c,R(d)\big)
\,.
$$

## Properties

### Characterization of adjunctions

Heteromorphisms may be used to express/characterize [[adjunctions]]. For more on this see at

* [adjoint functor -- In terms of cographs/heteromorphisms](adjoint+functor#InTermsOfCographsHeteromorphisms)

* _[adjoint (infinity,1)-functor -- In terms of cographs/Heteromorphisms](adjoint+%28infinity%2C1%29-functor#InTermsOfCographsHeteromorphisms)_

## References

* {#Pareigis70} [[Bodo Pareigis]], section 2.2 in: *Categories and Functors*, Pure and Applied Mathematics **39**, Academic Press (1970) [[doi:10.5282/ubm/epub.7244](https://doi.org/10.5282/ubm/epub.7244)]

* {#Ellerman06} [[David Ellerman]], _A Theory of Adjoint Functors --- with Some Thoughts on Their Philosophical Significance_, in: G. Sica (e.) _What Is Category Theory?_, 12783. Milan: Polimetrica. (2006)

* {#Ellerman07} [[David Ellerman]], _Adjoint Functors and Heteromorphisms_ [[arXiv:0704.2207](http://arxiv.org/abs/0704.2207)]

* [[David Ellerman]], *Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective* (2015) [[pdf](http://www.ellerman.org/wp-content/uploads/2015/06/Maclane-Bourbaki-Redux.pdf), [doi:10.2139/ssrn.2620333](https://dx.doi.org/10.2139/ssrn.2620333)]

* {#Lurie09} [[Jacob Lurie]], _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [arXiv:0608040](http://arxiv.org/abs/math/0608040)&rbrack

[[!redirects heteromorphisms]]


heterotic line bundle

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## Idea

If the [[gauge field|gauge]]-[[complex vector bundle]] in a [[heterotic string theory]] [[perturbative string theory vacuum|vacuum]] has [[reduction of the structure group]] to an [[abelian group]] of the form

$$
  S\big( U(1)^n \big)
  \;\subset\;
  SU(n)
  \; \subset\;
  E_8
  \;\;\;\;\;
  2 \leq n \leq 5
$$

(the [[direct product group]] of $(n-1)$-copies of the [[circle group]], regarded as a [[diagonal morphism|diagonal]] [[subgroup]] of [[SU(n)]] and thus of [[E8]])

it is called a _heterotic line bundle_ in [Anderson-Gray-Lukas-Palti 11](#AndersonGrayLukasPalti11).

Considering these models has led to a little revolution in [[heterotic string theory|heterotic]] [[string phenomenology]] ([Anderson-Gray-Lukas-Palti 12](#AndersonGrayLukasPalti12)).

In the observable sector of [[heterotic M-theory]] the values $n = 4,5$ lead to good phenomenology, while for the [[hidden sector]] the value $n = 2$ is used (in [ADO 20a, Sec. 4.2](#ADO20a), [ADO 20a, Sec. 2.2](#ADO20a)).

## References

Heterotic line bundle models were first considered in

* {#AndersonGrayLukasPalti11} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_, Phys. Rev. D 84, 106005 (2011) ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* {#AndersonGrayLukasPalti12} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], *Heterotic Line Bundle Standard Models*, JHEP **2012** 113 (2012) 06 [[arXiv:1202.1757](https://arxiv.org/abs/1202.1757), doi:10.1007/JHEP06(2012)113]

The resulting scan of [[SU(5)]] [[GUT]] [[perturbative string theory vacuum|vacua]] among [[heterotic line bundle models]]:

* [[Lara Anderson]], [[Andrei Constantin]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic $SU(5)$ GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic standard model database_ ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html))

* [[Yang-Hui He]], Seung-Joo Lee, [[Andre Lukas]], Chuang Sun, _Heterotic Model Building: 16 Special Manifolds_, J. High Energ. Phys. 2014, 77 (2014) ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223))

* Stefan Groot Nibbelink, Orestis Loukas, Fabian Ruehle, Patrick K.S. Vaudrevange, _Infinite number of MSSMs from heterotic line bundles?_, Phys. Rev. D 92, 046002 (2015) ([arXiv:1506.00879](https://arxiv.org/abs/1506.00879))

* [[Andreas Braun]], Callum R. Brodie, [[Andre Lukas]], _Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds_, JHEP04 (2018) 087 ([arXiv:1706.07688](https://arxiv.org/abs/1706.07688))

* [[James Gray]], Juntao Wang, _Jumping Spectra and Vanishing Couplings in Heterotic Line Bundle Standard Models_, JHEP11 (2019) 073 ([arXiv:1906.09373](https://arxiv.org/abs/1906.09373))

* Magdalena Larfors, Robin Schneider, _Explore and Exploit with Heterotic Line Bundle Models_, Fortschritte der Physik Vol 86 Nr. 5 ([arXiv:2003.04817](https://arxiv.org/abs/2003.04817))

Review:

* [[Andrei Constantin]], _Heterotic line bundle models_, talk at _[New Developments in Gravity, Cosmology and Strings 2013](https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_13/ppcsst/)_, Munich 2013 ([pdf](https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_13/ppcsst/constantin.pdf), [[ConstantiHeteroticLineBundles.pdf:file]])

* [[Andre Lukas]], _Line Bundles in String Model Building_, talk at _[StringPhenomenology 2019](https://indico.cern.ch/event/782251/)_ ([indico:782251/contributions/3441850](https://indico.cern.ch/event/782251/contributions/3441850/), [pdf](https://indico.cern.ch/event/782251/contributions/3441850/attachments/1868056/3072624/Lukas.pdf), [[LukasHeteroticLineBundles.pdf:file]])

On [[heterotic line bundles]] in the [[hidden sector]] of [[heterotic M-theory]]:

* {#ADO20a} [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], Section 4.2 of: _Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models_, Fortsch. Phys. **69** 7 (2021) 2100052 [[arXiv:2003.05455](https://arxiv.org/abs/2003.05455), [doi:10.1002/prop.202100052](https://doi.org/10.1002/prop.202100052)]

* {#ADO20b} [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], Section 2.2 of: _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_, J. High Energ. Phys. 2021, 33 (2021). ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029), doi:10.1007/JHEP08(2021)033)

* Anthony Ashmore, [[Sebastian Dumitru]], [[Burt Ovrut]], *Hidden Sectors from Multiple Line Bundles for the B−L MSSM* ([arXiv:2106.09087](https://arxiv.org/abs/2106.09087))

* [[Sebastian Dumitru]], [[Burt A. Ovrut]], *Heterotic M-Theory Hidden Sectors with an Anomalous $U(1)$ Gauge Symmetry* ([arXiv:2109.13781](https://arxiv.org/abs/2109.13781))

* [[Sebastian Dumitru]], Sec. 2.4 of: *The strongly coupled $E_8 \times E_8$ heterotic string: Geometry & Phenomenology* [[arXiv:2206.12310](https://arxiv.org/abs/2206.12310)]

On [[heterotic line bundles]] seen in [[F-theory]] under [[duality between M/F-theory and heterotic string theory]]:

* [[Andreas Braun]], Callum Brodie, [[Andre Lukas]], [[Fabian Ruehle]], _NS5-Branes and Line Bundles in Heterotic/F-Theory Duality_, Phys. Rev. D 98, 126004 (2018) ([arXiv:1803.06190](https://arxiv.org/abs/1803.06190), [doi:10.1103/PhysRevD.98.126004](https://doi.org/10.1103/PhysRevD.98.126004))

See also:

* [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], _The Moduli Space of Heterotic Line Bundle Models: a Case Study for the Tetra-Quadric_, 	JHEP03 (2014) 025 ([arXiv:1311.1941](https://arxiv.org/abs/1311.1941))

* Magdalena Larfors, Davide Passaro, Robin Schneider, _Heterotic Line Bundle Models on Generalized Complete Intersection Calabi Yau Manifolds_ ([arXiv:2010.09763](https://arxiv.org/abs/2010.09763))

Similar discussion in [[SemiSpin(32)]]-[[heterotic string theory]]:

* Hajime Otsuka, _$SO(32)$ heterotic line bundle models_, JHEP 05 (2018) 045 ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684))

Discussion via [[machine learning]] of [[connection on a bundle|connections]] on heterotic line bundles over [[Calabi-Yau manifold|Calabi-Yau 3-folds]]:

* [[Anthony Ashmore]], Rehan Deen, [[Yang-Hui He]], [[Burt Ovrut]], *Machine Learning Line Bundle Connections* ([arXiv:2110.12483](https://arxiv.org/abs/2110.12483))

Appearance of heterotic line bundles via [[Hypothesis H]]:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation]]*, Reviews in Mathematical Physics **34** 05 (2022) 2250013 [[arXiv:2008.08544](https://arxiv.org/abs/2008.08544), [doi:10.1142/S0129055X22500131](https://doi.org/10.1142/S0129055X22500131)]

(see commentary on p. 5).

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heterotic M-theory on ADE-orbifolds

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## Idea

The [[KK-compactification]] of [[M-theory]] on [[fibers]]

$$
  S^1 \!\sslash\! G_{HW} \times \mathbb{H}\!\sslash\! G_{ADE}
$$

which are, locally, the [[Cartesian product]] of

1. the [[circle]] [[orientifold|orientifolded]] by $G_{HW} \simeq \mathbb{Z}_2$ as in [[Horava-Witten theory]];

1. the [[quaternions]] [[orbifold|orbifolded]] by a [[finite subgroup of SU(2)]] $G_{ADE}$.

([Sen 97, Sec 3](#Sen97), [Faux-Lüst-Ovrut 99](#FLO99), [Kaplunovsky-Sonnenschein-Theisen-Yankielowicz 99](#KSTY99), [Faux-Lüst-Ovrut 00a](#FLO00a), [00b](#FLO00b), [00c](#FLO00c))

> graphics grabbed from [HSS18, Example 2.2.7](#HSS18)

For $G_{ADE} = \mathbb{Z}_2$ this subsumes M-theory on [[K3]] times $S^1 \sslash G_{HW}$ ([Seiberg-Witten 96](#SeibergWitten96))

\linebreak

## Dual string theory perspectives
 {#DualStringTheoryPerspectives}

Under [[duality in string theory]] (specifically: [[duality between M-theory and type IIA string theory]] and [[duality between M-theory and heterotic string theory]]) M-theory on $\mathbb{S}^{1}\sslash \mathbb{Z}_2^{HW} \times \mathbb{T}^{4}\sslash G^{ADE}$ appears through the following [[string theory]]-perspectives

1. [$HET_{E}$-Theory on ADE-singularities](#HetEWithADESingularities)

1. [$I'$-Theory with D6-branes on O8-planes](#TypeIPrimeWithD6OnO8)

1. [$I'$-Theory on ADE-singularities intersecting O8-planes](#IPrimeTheoryOnADESingularitiesIntersectingO8Planes)

The following graphics shows how the three perspectives arise from [[KK-compactification]] on three different choices of [[circle]]-[[fibers]]. Indicated also are the [[M5-branes]] and their string theoretic images at [[NS5-branes]]/[[D4-branes]] with [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]] (see [further below](#GeometricEngineeringOfDIs6NIs1SCFTs)).

> graphics grabbed from [SS19](#SatiSchreiber19)

### $HET_{E}$-Theory on ADE-singularities
 {#HetEWithADESingularities}

(...)

[[Horava-Witten theory]], hence [[heterotic string theory]], on [[ADE-singularities]] $\mathbb{H} \sslash G_{ADE}$

([Witten 99](#Witten99),...)

(...)

### $I'$-Theory with D6-branes on O8-planes
 {#TypeIPrimeWithD6OnO8}

> from [GKSTY 02](#GKSTY02)

If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a "[[half-brane]]" -- the [[half M5-brane]] -- with only one copy of [[D6-branes]] ending on it:

> graphics grabbed from [GKSTY 02](#GKSTY02)

(In [Hanany-Zaffaroni 99](#HananyZaffaroni99) this is interpreted in terms of the [['t Hooft-Polyakov monopole]].)

{#LiftToMTheortyOfNS5Halfbrane} The lift to [[M-theory]] of this situation is an [[M5-brane]] [[brane intersection|intersecting]] an [[M9-brane]] (see at _[[M-theory on S1/G_HW times H/G_ADE]]_):

> from [GKSTY 02](#GKSTY02)

Alternatively the [[O8-plane]] may [[brane intersection|intersect]] the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:

> from [HKLY 15](#HKLY15)

In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:

> from [Hanany-Zaffaroni 98](#HananyZaffaroni98)

### $I'$-theory on ADE-singularities intersecting O8-planes
{#IPrimeTheoryOnADESingularitiesIntersectingO8Planes}

([Bergman & Rodriguez-Gomez 12, Sec. 3](#BergmanRodriguezGomez12))

(...)

## Geometric engineering of $D = 6$ $N=(1,0)$ SCFTs on M5-branes
 {#GeometricEngineeringOfDIs6NIs1SCFTs}

The [[M5-branes]]-configurations  as [above](#DualStringTheoryPerspectives) are supposed to [[geometric engineering of QFTs|geometrically engineer]] [[D=6 N=(1,0) SCFTs]].

See the references [below](#ReferencesGeometricEngineeringOfDIs6NIs1SCFT), for example [DHTV 14, Section 6](#DHTV14), [Gaiotto-Tomasiello 14](#GaiottoTomasiello14), [HKLY 15](#HKLY15).

## Related concepts

[[!include KK-compactifications of M-theory -- table]]

\linebreak

[[!include Dp-D(p+4)-brane bound states -- contents]]

## References

### General

The first discussion of this compactification is possibly

* {#Sen97} [[Ashoke Sen]], Section 3 of: _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

in the context of the [[M-theory lift of gauge enhancement on D6-branes]].

The original articles focusing on this situation:

* {#FLO99} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))

* {#FLO00a} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions_, Nucl. Phys. B589: 269-291, 2000 ([arXiv:hep-th/0005251](https://arxiv.org/abs/hep-th/0005251))

* {#FLO00b} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _An M-Theory Perspective on Heterotic K3 Orbifold Compactifications_, Int. J. Mod. Phys. A18:3273-3314, 2003 ([arXiv:hep-th/0010087](https://arxiv.org/abs/hep-th/0010087))

* {#FLO00c} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds_, Int. J. Mod. Phys. A18: 2995-3014, 2003 ([arXiv:hep-th/0011031](https://arxiv.org/abs/hep-th/0011031))

* {#KSTY99} [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B ([arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144), doi:10.1016/S0550-3213(00)00460-0)

* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

Discussion of [[heterotic M-theory]] on smooth [[K3]] originates around

* {#SeibergWitten96} [[Nathan Seiberg]], [[Edward Witten]], _Comments on String Dynamics in Six Dimensions_, Nucl. Phys. B471:121-134, 1996 ([arXiv:hep-th/9603003](https://arxiv.org/abs/hep-th/9603003))

* Zygmunt Lalak, Steven Thomas, _Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models_, Nuclear Physics B
Volume 515, Issues 1–2, 30 March 1998, Pages 55-72
Nuclear Physics B ([hep-th/9707223](https://arxiv.org/abs/hep-th/9707223), doi:10.1016/S0550-3213(97)00784-0)

  (on [[gaugino condensation]])

See also

* Jacob Cole Claussen, _The deconstruction of orbifold fixed points in heterotic M-theory_, 2016 ([hdl:2152/41748](http://hdl.handle.net/2152/41748))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], Section 4.1 of: _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

### As $HET_E$-theory with ADE-singularities

As [[heterotic string theory]] on [[orbifold]] [[ADE-singularities]]:

* {#Witten99} [[Edward Witten]], _Heterotic String Conformal Field Theory And A-D-E Singularities_, JHEP 0002:025, 2000 ([arXiv:hep-th/9909229](https://arxiv.org/abs/hep-th/9909229))

### As $I'$-theory with D6-branes

As [[type I' string theory]] with [[D6-branes]]:

* {#BrunnerKarch97} [[Ilka Brunner]], [[Andreas Karch]], _Branes at Orbifolds versus Hanany Witten in Six Dimensions_, JHEP 9803:003, 1998 ([arXiv:hep-th/9712143](https://arxiv.org/abs/hep-th/9712143))

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

* Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, _A new 5d description of 6d D-type minimal conformal matter_, JHEP 1508:097, 2015 ([arXiv:1505.04439](https://arxiv.org/abs/1505.04439))

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

and in [[massive type IIA string theory]] with [[D6-D8 brane bound states]]:

* Fabio Apruzzi, [[Marco Fazzi]], _AdS7/CFT6 with orientifolds_, JHEP 01 (2018) 124 ([arXiv:1712.03235](https://arxiv.org/abs/1712.03235))

### As $I'$-theory with ADE-singularities

As [[type I' string theory]] at [[orbifold]] [[ADE-singularities]]:

* {#BergmanRodriguezGomez12} [[Oren Bergman]], Diego Rodriguez-Gomez, Section 3 of: _5d quivers and their $AdS_6$ duals_, JHEP07 (2012) 171 ([arxiv:1206.3503](https://arxiv.org/abs/1206.3503))

* Chiung Hwang, Joonho Kim, Seok Kim, [[Jaemo Park]], Section 3.4.2 of: _General instanton counting and 5d SCFT_, JHEP07 (2015) 063 ([arxiv:1406.6793](https://arxiv.org/abs/1406.6793))

### F-theory perspective

The [[F-theory]] perspective:

* [[Monika Marquart]], [[Daniel Waldram]], _F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$_ ([arXiv:hep-th/0204228](https://arxiv.org/abs/hep-th/0204228))

* Christoph Lüdeling, Fabian Ruehle, _F-theory duals of singular heterotic K3 models_, Phys. Rev. D 91, 026010 (2015) ([arXiv:1405.2928](https://arxiv.org/abs/1405.2928))

### Geometric engineering of $D=6, \mathcal{N}=(1,0)$ SCFT
 {#ReferencesGeometricEngineeringOfDIs6NIs1SCFT}

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

* {#DHTV14} Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], Section 6 of: _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

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

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

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

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

* Santiago Cabrera, [[Amihay Hanany]], [[Marcus Sperling]], _Magnetic Quivers, Higgs Branches, and 6d $\mathcal{N}=(1,0)$ Theories_, J. High Energ. Phys. (2019) 2019: 71 ([arXiv:1904.12293](https://arxiv.org/abs/1904.12293))

[[!include flavor on heterotic M5-branes -- references]]

[[!redirects heterotic M-theory on ADE-singularities]]

[[!redirects M-theory on S1/G_HW times H/G_ADE]]


heterotic S-duality -- table

**[[S-duality]] in [[string theory]]**

| reduction from 11d | [[electric charge|electric]] [[∞-model]] | weak/strong coupling duality   |  [[magnetic charge|magnetic]] [[∞-model]]  |
|--|-----|------|-----|
| | [[M2-brane]]  in [[11-dimensional supergravity|11d sugra]] [[effective field theory|EFT]]    |  $\leftarrow$[[electric-magnetic duality]]$\rightarrow$   | [[M5-brane]] in  [[11-dimensional supergravity|11d sugra]]  [[effective field theory|EFT]]  |
| [[Horava-Witten theory|HW reduction]]  |  | | |
| | $\downarrow$ on [[orientifold]] [[K3]]$\times S^1//\mathbb{Z}_2$ |  | $\downarrow$ on [[orientifold]] [[K3]]$\times S^1//\mathbb{Z}_2$ |
|  | [[F1-brane]] in [[heterotic supergravity]] | $\leftarrow$[[S-duality]]$\rightarrow$ | [[black string]] in [[heterotic supergravity|heterotic sugra]] |
| [[Horava-Witten theory|HW reduction]]  |  | | |
| | $\downarrow$ on [[orientifold]] [[torus|T4]]$ \times S^1//\mathbb{Z}_2$ | | $\downarrow$ on [[orientifold]] [[torus|T4]]$ \times S^1//\mathbb{Z}_2$ |
| | [[F1-brane]] in [[heterotic supergravity]] | $\leftarrow$[[S-duality]]$\rightarrow$ | [[black string]] in [[type IIA supergravity|type IIA sugra]] |
| [[Kaluza-Klein mechanism|KK reduction]]  |  | | |
| | $\downarrow$ on [[K3]]$\times S^1$ | |  $\downarrow$ on [[K3]]$\times S^1$ |
| | [[F1-brane]] in [[type IIA supergravity|IIA sugra]] | $\leftarrow$[[S-duality]]$\rightarrow$ | [[black string]] in [[heterotic supergravity|heterotic sugra]] |
| [[Kaluza-Klein mechanism|KK reduction]]  |  | | |
| | $\downarrow$ on $T^4\times S^1$ | |  $\downarrow$ on $T^4 \times S^1$ |
| | [[F1-brane]] in [[type IIA supergravity|IIA sugra]] | $\leftarrow$[[S-duality]]$\rightarrow$ | [[black string]] in [[type IIA supergravity|type IIA sugra]] |
| [[F-theory|F-reduction]] | $\updownarrow$ [[T-duality]] on $S^1$ |  |  |
| | [[F1-brane]] in [[type IIB supergravity|IIB sugra]] | $\leftarrow$[[S-duality]]$\rightarrow$ | [[D1-brane]] in [[type IIB supergravity|10d IIB sugra]] |
| | | [[!include SVG short nwsearrow]] [[U-duality]] | $\updownarrow$ [[T-duality]] on $T^2$ |
| | [[D3-brane]] in [[type II supergravity|IIB sugra]] |  $\leftarrow$[[S-duality]]$\rightarrow$ | [[D3-brane]] in [[type II supergravity|IIB sugra]] |


heterotic string phenomenology -- references

### Heterotic string phenomenology
 {#HeteroticStringPhenomenologyReferences}

The historical origin of all [[string phenomenology]] is the [[top-down model building|top-down]] [[GUT]]-model building in [[heterotic string theory]] due to

* {#CHSW85} [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], [[Edward Witten]], _Vacuum configurations for superstrings_, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) \[doi:10.1016/0550-3213(85)90602-9\]

Review and exposition:

* {#Witten02} [[Edward Witten]], _Quest For Unification_, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg \[arXiv:hep-ph/0207124\]

* {#Nilles11} [[Hans-Peter Nilles]], _Strings, Exceptional Groups and Grand Unification_, talk at _[Planck 2011](https://indico.cern.ch/event/112851/)_ \[pdf, [[Nilles11GUT.pdf:file]]\]

* [[Saul Ramos-Sanchez]], [[Michael Ratz]], *Heterotic Orbifold Models*, in *[[Handbook of Quantum Gravity]]*, Springer (2024) \[doi:10.1007/978-981-19-3079-9,
[arXiv:2401.03125](https://arxiv.org/abs/2401.03125)\]

#### The $E_8 \times E_8$-heterotic string
 {#HeteroticStringPhenomenologyReferencesForE8}

The following articles claim the existence of exact realization of the [[gauge group]] and [[matter]]-content of the [[MSSM]] in [[heterotic string theory]] on [[orbifolds]] (not yet checking [[Yukawa couplings]]):

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _A Heterotic Standard Model_, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th/0501070](http://arxiv.org/abs/hep-th/0501070))

* Wilfried Buchmuller, Koichi Hamaguchi, Oleg Lebedev, [[Michael Ratz]], _Supersymmetric Standard Model from the Heterotic String_,  Phys. Rev. Lett. **96** 121602 (2006) (\[doi:v[arXiv:hep-ph/0511035](https://arxiv.org/abs/hep-ph/0511035)\]

* {#BraunHeOvrutPantev06} [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _The Exact MSSM Spectrum from String Theory_, JHEP 0605:043, 2006 ([arXiv:hep-th/0512177](http://arxiv.org/abs/hep-th/0512177))

* [[Vincent Bouchard]], [[Ron Donagi]], _An SU(5) Heterotic Standard Model_, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th/0512149](http://arxiv.org/abs/hep-th/0512149))

A computer search through the "[[landscape of string theory vacua|landscape]]" of [[Calabi-Yau varieties]] showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as [[SU(5)]]-[[GUTs]]):

general computational theory:

* [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], _Heterotic Compactification, An Algorithmic Approach_, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210))

using [[heterotic line bundle models]]:

* {#AndersonGrayLukasPalti11} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_, Phys. Rev. D 84, 106005 (2011) ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

* [[Lara Anderson]], [[Andrei Constantin]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic $SU(5)$ GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[Yang-Hui He]], Seung-Joo Lee, [[Andre Lukas]], Chuang Sun, _Heterotic Model Building: 16 Special Manifolds_, J. High Energ. Phys. 2014, 77 (2014) ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223))

* Stefan Groot Nibbelink, Orestis Loukas, Fabian Ruehle, Patrick K.S. Vaudrevange, _Infinite number of MSSMs from heterotic line bundles?_, Phys. Rev. D 92, 046002 (2015) ([arXiv:1506.00879](https://arxiv.org/abs/1506.00879))

* [[Andreas Braun]], Callum R. Brodie, [[Andre Lukas]], _Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds_, JHEP04 (2018) 087 ([arXiv:1706.07688](https://arxiv.org/abs/1706.07688))

* {#CHE18} [[Andrei Constantin]], [[Yang-Hui He]], [[Andre Lukas]], _Counting String Theory Standard Models_, Physics Letters B
Volume 792, 10 May 2019, Pages 258-262 ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444))

* Alon E. Faraggi, Glyn Harries, Benjamin Percival, John Rizos, _Towards machine learning in the classification of $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold compactifications ([arXiv:1901.04448](https://arxiv.org/abs/1901.04448))_

* Magdalena Larfors, Robin Schneider, _Explore and Exploit with Heterotic Line Bundle Models_, Fortschritte der Physik Vol 86 Nr. 5 ([arXiv:2003.04817](https://arxiv.org/abs/2003.04817))

The resulting database of [[heterotic line bundle models]] is here:

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic standard model database_ ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.))

Review includes

* [[Lara Anderson]], _New aspects of heterotic geometry and phenomenology_, talk at [Strings2012](http://wwwth.mpp.mpg.de/conf/strings2012/), Munich 2012 ([pdf](http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf))

* {#He18} [[Yang-Hui He]], _Deep-learning the landscape_, talk at _[String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php)_ ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE))

* [[Yang-Hui He]], _Calabi-Yau Spaces in the String Landscape_ ([arXiv:2006.16623](https://arxiv.org/abs/2006.16623))

Computation of [[metrics]] on these Calabi-Yau compactifications (eventually needed for computing their induced [[Yukawa couplings]]) is started in

* [[Volker Braun]], Tamaz Brelidze, [[Michael Douglas]], [[Burt Ovrut]], _Calabi-Yau Metrics for Quotients and Complete Intersections_, JHEP 0805:080, 2008 ([arXiv:0712.3563](https://arxiv.org/abs/0712.3563))

and via [[machine learning]]:

* Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, [[Andre Lukas]], [[Burt Ovrut]], *Computation of Quark Masses from String Theory* \[arXiv:2402.01615\]

This "heterotic standard model" has a "hidden sector" copy of the actual standard model, more details of which are discussed here:

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], _Supersymmetric Hidden Sectors for Heterotic Standard Models_ ([arXiv:1301.6767](http://arxiv.org/abs/1301.6767))

The issue of [[moduli stabilization]] in these kinds of models is discussed in

* Michele Cicoli, Senarath de Alwis, Alexander Westphal, _Heterotic Moduli Stabilization_ ([arXiv:1304.1809](http://arxiv.org/abs/1304.1809))

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

Principles singling out heterotic models with three generations of fundamental particles are discussed in:

* [[Philip Candelas]], [[Xenia de la Ossa]], [[Yang-Hui He]], Balazs Szendroi, _Triadophilia: A Special Corner in the Landscape_, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http://arxiv.org/abs/0706.3134))

Discussion of non-[[supersymmetry|supersymmetric]]: [[GUT]] models:

* Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, _Classification of Non-Supersymmetric Pati-Salam Heterotic String Models_ ([arXiv:2011.04113](https://arxiv.org/abs/2011.04113))

See also:

* {#AngelantonjFlorakis18} Carlo Angelantonj, Ioannis Florakis, _GUT Scale Unification in Heterotic Strings_ ([arXiv:1812.06915](https://arxiv.org/abs/1812.06915))

#### The $SemiSpin(32)$-heterotic string
 {#HeteroticStringPhenomenologyReferencesForSemiSpin32}

Discussion of [[string phenomenology]] for the [[SemiSpin(32)]]-[[heterotic string]] (see also at [type I phenomenology](type+I+string+theory#Phenomenology)):

* Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, _Heterotic $SO(32)$ model building in four dimensions_, JHEP 0412:063, 2004 ([arXiv:hep-th/0410232](https://arxiv.org/abs/hep-th/0410232))

* [[Hans-Peter Nilles]], [[Saul Ramos-Sanchez]], Patrick K.S. Vaudrevange, Akin Wingerter, _Exploring the $SO(32)$ Heterotic String_, JHEP 0604:050 (2006) \[doi:10.1088/1126-6708/2006/04/050[arXiv:hep-th/0603086](https://arxiv.org/abs/hep-th/0603086)\]

* [[Saul Ramos-Sanchez]], _Towards Low Energy Physics from the Heterotic String_,  Fortsch. Phys. **10** (2009) 907-1036 \[doi:10.1002/prop.200900073[arXiv:0812.3560](https://arxiv.org/abs/0812.3560)\]

  ([[heterotic string|heterotic]] [[SemiSpin(32)]] [[R-parity]] [[MSSM]] [[string theory vacua|vacua]])

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

* Jihn E. Kim, _Grand unfication models from $SO(32)$ heterotic string_ ([arXov:2008.00367](https://arxiv.org/abs/2008.00367))

On [[heterotic line bundle models]]:

* Hajime Otsuka, _$SO(32)$ heterotic line bundle models_, ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684))

#### The "$SO(16)\times SO(16)$"-heterotic string
 {#HeteroticStringPhenomenologyReferencesForSO16}

This non-[[supersymmetry|supersymmetric]] string theory was first described in:

* {#AGMV86het} [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], [[Gregory Moore]] and [[Cumrun Vafa]], *An $O(16)\times O(16)$ heterotic string*, Phys. Lett. B **171** (1986) 155 \[doi:10.1016/0370-2693(86)91524-8\]

* {#DH86het} [[Lance Dixon]] and [[Jeffrey Harvey]], *String Theories in Ten-Dimensions Without Space-Time Supersymmetry*, Nucl. Phys. B **274** (1986) 93 \[doi:10.1016/0550-3213(86)90619-X\]

A proposal on what the correct global character of the [[gauge group]] is appears in:

* {#McIn99spin} [[Brett McInnes]], _The Semispin Groups in String Theory_, J. Math. Phys. **40** (1999) 4699-4712 \[arXiv:hep-th/9906059, [doi:10.1063/1.532999](https://doi.org/10.1063/1.532999)\]

A suggestion that the $SO(16)\times SO(16)$ heterotic string is a the $E_8\times E_8$ string "minus" the [[semispin group]] [[SemiSpin(32)|$\text{Spin}(32)/\mathbb{Z}_2$]]:

* {#SW87anom} [[Adrian Norbert Schellekens]], and [[Nicholas Warner]] *Anomalies, characters and strings*, Nuclear Physics B **287** (1987) 317-361 \[doi:10.1016/0550-3213(87)90108-8\]