transport

> This page is about the notion in [[homotopy type theory|homotopy]] [[type theory]]. For *[[parallel transport]]* via [[connections]] in [[differential geometry]] see there. For the relation see [below](#RelationToParallelTransport).

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

[[Gottfried Leibniz]] said (in translation from [Lewis 1918, p. 373](#Lewis18) with original Latin terms in parenthesis; see also [Cartwright 1971, p. 119](#Cartwright71) and [Gries & Schneider 1998](#GriesSchneider98))

> Two terms are the same (eadem) if one can be substituted for the otherwithout altering the truth of any statement (salva veritate). If we have $P$ and $Q$, and $P$ enters into some true proposition, and the substitution of $Q$ for $P$ wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then $P$ and $Q$ are said to be the same; conversely, if $P$ and $Q$ are the same, they can be substituted for one another.

The converse mentioned at the end of the last sentence is known as the **principle of substitution** or the **indiscernibility of identicals**. In any simply typed first-order theory, given a predicate $P$ on $T$, this principle is formalized as the following

$$
  \forall_{x,y\colon T} (x =_T y) \implies (P(x) \iff P(y))
$$

In the interpretation of [[propositions as types]] in [[type theory]], [[propositions]] are interpreted as [[types]], and the above statement has a generalization from the types which are propositions to all types: the [[universal quantifier]] becomes a [[dependent product type]], the [[predicate]] becomes a [[type family]], [[implication]] becomes an a [[function type]], and [[logical equivalence]] of [[propositions]] becomes an [[equivalence type]]. This results in what is known as *transport* in type theory, which for the type $T$ and the type family $x:T \vdash P(x)$ results in the [[dependent function]]:

\[
  \label{TheTransportFunctions}
  \mathrm{tr}_P
  \;\colon\;
  \prod_{x,y:T} (x =_T y) \to (P(x) \simeq P(y))
  \,.
\]

## Definitions

There are actually two notions of transport in dependent type theory. One notion of transport is transport across the identification $p:a =_A b$ for elements $a:A$ and $b:A$. The other notion of transport is transport across the path $p:\mathbb{I} \to A$. These two are interchangeable through the [[recursion principle]] and uniqueness principle of the [[interval type]].

Similarly, there are multiple notions of [[equivalence types]] in [[dependent type theory]], which can be used for a definition of univalence for a [[type universe]] $U$; these include

* [[judgmentally strict equivalence types]]
* [[propositionally strict equivalence types]]
* various notions of [[weak equivalence types]]
  * the type of [[functions]] with [[contractible type|contractible]] [[fiber type|fibers]]
  * the type of [[spans]] with contractible fibers
  * the type of [[multivalued partial functions]] which are single-valued and [[total function|total]] and have contractible fibers
  * the type of [[one-to-one correspondences]]
* type of $U$-small equivalences, given a type universe $U$ and a definition of equivalence above

The transport equivalence is called **judgmentally strict transport** if the equivalence type is a [[judgmentally strict equivalence type]], and similarly it is called **propositionally strict transport** if the equivalence type is a [[propositionally strict equivalence type]]. It is called **weak transport** if the equivalence type is a [[weak equivalence type]].

### Transport across paths

#### Inductive definition

Given any notion of [[equivalence type]] $A \simeq B$ between types $A$ and $B$, let $\mathrm{id}_A^\simeq$ denote the [[identity equivalence]] on $A$.

Transport across paths is inductively defined using [[path induction]] for function types $\mathbb{I} \to A$ out of the interval type. Path induction states that

\begin{theorem}
Given any type $A$, and any type family $C(p)$ indexed by paths $p:\mathbb{I} \to A$ in $A$, and given any dependent function $t:\prod_{x:A} C(\lambda i:\mathbb{I}.x)$ which says that for all elements $x:A$, there is an element of the type defined by substituting the constant path of $x:A$ into $C$, $C(\lambda i:\mathbb{I}.x)$, one can construct a dependent function $\mathrm{ind}_{\mathbb{I} \to A}(t):\prod_{z:\mathbb{I} \to A} C(z)$ such that for all $x:A$, $\mathrm{ind}_{\mathbb{I} \to A}(t, \lambda i:\mathbb{I}.x) \equiv t(x):C(\lambda i:\mathbb{I}.x)$.
\end{theorem}

Now, given a type family $B(x)$ indexed by $x:A$, there exists a dependent function called **transport**

$$\mathrm{tr}_{\underline{ }.B(-)}^{(-)}:\prod_{p:\mathbb{I} \to A} B(p(0)) \simeq B(p(1))$$

This is inductively defined by path induction, where for all $x:A$,

$$\mathrm{tr}_{\underline{ }.B(-)}^{\lambda i:\mathbb{I}.x} \equiv \mathrm{id}_{B(x)}^{\simeq}:B(x) \simeq B(x)$$

#### Using function application and Russell universes

Suppose that the dependent type theory uses a sequence of [[Russell universes]] $U_i$ to define types, instead of a separate type [[judgment]]. Then, given a path $p:\mathbb{I} \to A$, [[transport]] across the type family $B:A \to U_i$ can be defined as the evaluation of the inductively defined function
$$\mathrm{idtoequiv}(B(p(0)), B(p(1))):(B(p(0)) =_{U_i} B(p(1))) \to (B(p(0)) \simeq B(p(1)))$$
at the function application of $B$ to the path $p$, $\mathrm{ap}_{U_i}(B, p):B(p(0)) =_{U_i} B(p(1))$:

$$\mathrm{transport}(B, p) \equiv \mathrm{idtoequiv}(B(p(0)), B(p(1)))(\mathrm{ap}_{U_i}(B,p)):B(p(0)) \simeq B(p(1))$$

### Transport across identifications

#### Inductive definition

Given any notion of [[equivalence type]] $A \simeq B$ between types $A$ and $B$, let $\mathrm{id}_A^\simeq$ denote the [[identity equivalence]] on $A$.

Transport across identification is inductively defined using the [[J-rule]]. The J-rule states that

\begin{theorem}
Given a type $A$ and given a type family $C(x, y, p)$ indexed by $x:A$, $y:A$, and $p:x =_A y$, and a dependent function $t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))$, one can construct a dependent function

$$\mathrm{ind}_{=, A}(t):\prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} C(x, y, p)$$

such that for all $x:A$,

$$J(t, x, x, \mathrm{refl}_A(x)) \equiv t(x)$$
\end{theorem}

Now, given a type family $B(x)$ indexed by $x:A$, there exists a dependent function called **transport**

$$\mathrm{tr}_{\underline{ }.B(-)}^{(-, -, -)}:\prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} B(x) \simeq B(y)$$

This is inductively defined by path induction, where for all $x:A$,

$$\mathrm{tr}_{\underline{ }.B(-)}^{(x, x, \mathrm{refl}_A(x))} \equiv \mathrm{id}_{B(x)}^{\simeq}:B(x) \simeq B(x)$$

#### Using function application and Russell universes

Suppose that the dependent type theory uses a sequence of [[Russell universes]] $U_i$ to define types, instead of a separate type [[judgment]]. Then, given elements $a:A$, $b:A$, and identification $p:a =_A b$, [[transport]] across the type family $B:A \to U_i$ can be defined as the evaluation of the inductively defined function
$$\mathrm{idtoequiv}(B(a), B(b)):(B(a) =_{U_i} B(b)) \to (B(a) \simeq B(b))$$
at the function application of $B$ to the identification $p$, $\mathrm{ap}_{U_i}(B, a, b, p):B(a) =_{U_i} B(b)$:

$$\mathrm{transport}(B, a, b, p) \equiv \mathrm{idtoequiv}(B(a), B(b))(\mathrm{ap}_{U_i}(B, a, b, p)):B(a) \simeq B(b)$$

## Properties

### Transport across paths

The type of families of transport functions on a [[type family]] $x:A \vdash B(x)$ is given by the type

$$\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

where $x =_{x:A.B(x)}^{p} y$ is a [[heterogeneous identity type]] defined using paths.

In general, a family of transport functions is an element of this type. However, any element of the type of families of transport functions is unique up to identification:

\begin{theorem}
The type of families of transport functions

$$\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

is a [[contractible type]].
\end{theorem}

\begin{proof}
We first show that the type

$$\sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

is contractible for all $p:\mathbb{I} \to A$. By [[path induction]], it suffices to show that the type

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{\lambda i:\mathbb{I}.a} f(x)$$

is contractible for all $a:A$. There is an equivalence of types

$$(x =_{x:A.B(x)}^{\lambda i:\mathbb{I}.a} f(x)) \simeq (x =_{B(a)} f(x))$$

and thus by the typal congruence rules for dependent function types and dependent pair types, there is an equivalence of types

$$\left(\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{\lambda i:\mathbb{I}.a} f(x)\right) \simeq \left(\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{B(a)} f(x)\right)$$

Now,

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{B(a)} f(x)$$

is just the [[type of identity functions]] on $B(a)$, and is thus always contractible, which implies that

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{\lambda i:\mathbb{I}.a} f(x)$$

is contractible for all $a:A$. By induction on identity types, the type

$$\sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

is contractible for all paths $p:\mathbb{I} \to A$. By [[weak function extensionality]], this implies that

$$\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

is contractible.
\end{proof}

Thus, it makes sense to refer to the canonical family of transport functions

$$\mathrm{transport}_{x:A.B(x)}:\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$

inductively defined by

$$\mathrm{transport}_{x:A.B(x)}(\lambda i:\mathbb{I}.a) \equiv \left(\lambda x:B(a).x, \lambda x:B(a).\mathrm{idTohId}(x, (\lambda x:B(a).x)(x), \beta_{B(a) \to B(a)}^{x:B(a).x}(x))\right)$$

for all $a:A$ as *[[generalized the|the]]* transport function on the type family $x:A \vdash B(x)$.

\begin{theorem}
Given a family of transport functions
$$t:\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$
for all $p:\mathbb{I} \to A$, the function $\pi_1(t(p^{-1}))$ is a [[quasi-inverse function]] of $\pi_1(t(p))$, where $p^{-1}$ is the inverse path of $p$.
$$p^{-1} \coloneqq \mathrm{rec}_\mathbb{I}^A(p(1), p(0), \mathrm{ap}_p(\mathcal{p}^{-1}))$$
\end{theorem}

\begin{proof}
For all paths $p:\mathbb{I} \to A$ and for all $x:B(p(0))$, we have
$$\pi_2(t(p))(x):x =_{x:A.B(x)}^{p} \pi_1(t(p))(x)$$
and for all $y:B(p(1))$ we have
$$\pi_2(t(p^{-1}))(x):x =_{x:A.B(x)}^{p^{-1}} \pi_1(t(p^{-1}))(x)$$

Thus, we have
$$\pi_2(t(p))(\pi_1(t(p^{-1}))(y)):\pi_1(t(p^{-1}))(y) =_{x:A.B(x)}^{p} \pi_1(t(p))(\pi_1(t(p^{-1}))(y))$$
and
$$\pi_2(t(p^{-1}))(\pi_1(t(p))(x)):\pi_1(t(p)(x)) =_{x:A.B(x)}^{p^{-1}} \pi_1(t(p^{-1}))(\pi_1(t(p))(x))$$

By concatenating heterogeneous identifications, we get
$$\pi_2(t(p))(x) \bullet \pi_2(t(p^{-1}))(\pi_1(t(p))(x)):x =_{x:A.B(x)}^{\mathrm{rec}_\mathbb{I}^A(p(0), p(0), \mathrm{fill}(p,  p^{-1}, \mathrm{refl}_{A}(p(1))} \pi_1(t(p^{-1}))(\pi_1(t(p))(x))$$
and
$$\pi_2(t(p^{-1}))(y) \bullet \pi_2(t(p))(\pi_1(t(p^{-1}))(y)):y =_{x:A.B(x)}^{\mathrm{rec}_\mathbb{I}^A(p(1), p(1), \mathrm{fill}(p^{-1},  p, \mathrm{refl}_{A}(p(0))} \pi_1(t(p))(\pi_1(t(p^{-1}))(y))$$

where

$$p \bullet p^{-1} \coloneqq \mathrm{rec}_\mathbb{I}^A(p(0), p(0), \mathrm{ap}_p(\mathcal{p}) \bullet \mathrm{ap}_p(\mathcal{p}^{-1}))$$

$$p^{-1} \bullet p \coloneqq \mathrm{rec}_\mathbb{I}^A(p(1), p(1), \mathrm{ap}_p(\mathcal{p}^{-1}) \bullet \mathrm{ap}_p(\mathcal{p}))$$

Since $p \bullet p^{-1} \equiv \lamba i:\mathbb{I}.a$ and $p^{-1} \bullet p \equiv \lamba i:\mathbb{I}.b$, we thus have
$$\pi_2(t(p))(x) \bullet \pi_2(t(p^{-1}))(\pi_1(t(p))(x)):x =_{x:A.B(x)}^{p \bullet p^{-1}} \pi_1(t(p^{-1}))(\pi_1(t(p))(x))$$
and
$$\pi_2(t(p^{-1}))(y) \bullet \pi_2(t(p))(\pi_1(t(p^{-1}))(y)):y =_{x:A.B(x)}^{p^{-1} \bullet p} \pi_1(t(p))(\pi_1(t(p^{-1}))(y))$$

By converting heterogeneous identifications over constant paths, one gets
$$\mathrm{hIdToId}(x, \pi_1(t(p^{-1}))(\pi_1(t(p))(x)), \pi_2(t(p))(x) \bullet \pi_2(t(p^{-1}))(\pi_1(t(p)(x))))):x =_{B(p(0))} \pi_1(t(p^{-1}))(\pi_1(t(p))(x))$$
and
$$\mathrm{hIdToId}(y, \pi_1(t(p))(\pi_1(t(p^{-1}))(y)), \pi_2(t(p^{-1}))(y) \bullet \pi_2(t(p))(\pi_1(t(p^{-1}))(y))):y =_{B(b)} \pi_1(t(p))(\pi_1(t(p^{-1}))(y))$$

Thus, we have shown that $\pi_1(t(p^{-1}))$ is a quasi-inverse function of $\pi_1(t(p))$ for all paths $p:\mathbb{I} \to A$.
\end{proof}

We define the dependent functions

$$\mathrm{linv}(t, p) \equiv \lambda x:B(p(0)).\mathrm{hIdToId}(x, \pi_1(t(p^{-1}))(\pi_1(t(p))(x)), \pi_2(t(p))(x) \bullet \pi_2(t(p^{-1}))(\pi_1(t(p)(x)))))$$

in type $\prod_{x:B(p(0))} x =_{B(p(0))} \pi_1(t(p^{-1}))(\pi_1(t(p))(x))$

and

$$\mathrm{rinv}(t, p) \equiv \lambda y:B(p(1)).\mathrm{hIdToId}(y, \pi_1(t(p))(\pi_1(t(p^{-1}))(y)), \pi_2(t(p^{-1}))(y) \bullet \pi_2(t(p))(\pi_1(t(p^{-1}))(y)))$$

in type $\prod_{y:B(p(1))} y =_{B(p(1))} \pi_1(t(p))(\pi_1(t(p^{-1}))(y))$

for all $p:\mathbb{I} \to A$.

\begin{theorem}
Given a family of transport functions
$$t:\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$
for all $p:\mathbb{I} \to A$, the function $\pi_1(t(p))$ is bi-invertible.
\end{theorem}

\begin{proof}
We construct the element

$$((\pi_1(t(p^{-1})), \mathrm{linv}(t, p)), (\pi_1(t(p^{-1})), \mathrm{rinv}(t, p))$$

in the type

$$\left(\sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{B(p(0))} f(\pi_1(t(p))(x))\right) \times \left(\sum_{g:B(p(1)) \to B(p(0))} \prod_{y:B(p(1))} y =_{B(p(1))} \pi_1(t(p))(g(y))\right)$$

indicating that $\pi_1(t(p))$ is bi-invertible for all $p:\mathbb{I} \to A$.
\end{proof}

By definition, it follows that:

\begin{corollary}
Given a family of transport functions
$$t:\prod_{p:\mathbb{I} \to A} \sum_{f:B(p(0)) \to B(p(1))} \prod_{x:B(p(0))} x =_{x:A.B(x)}^{p} f(x)$$
for all $p:\mathbb{I} \to A$, the function $\pi_1(t(p))$ is an [[equivalence of types]].
\end{corollary}

Hence, in [[dependent type theory]], transport functions are also called **transport equivalences**.

### Transport across identifications

The type of families of transport functions on a [[type family]] $x:A \vdash B(x)$ is given by the type

$$\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

where $x =_{x:A.B(x)}^{(a, b, p)} y$ is a [[heterogeneous identity type]] defined using identifications.

In general, a family of transport functions is an element of this type. However, any element of the type of families of transport functions is unique up to identification:

\begin{theorem}
The type of families of transport functions

$$\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

is a [[contractible type]].
\end{theorem}

\begin{proof}
We first show that the type

$$\sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

is contractible for all $a:A$, $b:A$, and $p:a =_A b$. By induction on identity types, it suffices to show that the type

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, a, \mathrm{refl}_A(a))} f(x)$$

is contractible for all $a:A$. There is an equivalence of types

$$(x =_{x:A.B(x)}^{(a, a, \mathrm{refl}_A(a))} f(x)) \simeq (x =_{B(a)} f(x))$$

and thus by the typal congruence rules for dependent function types and dependent pair types, there is an equivalence of types

$$\left(\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, a, \mathrm{refl}_A(a))} f(x)\right) \simeq \left(\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{B(a)} f(x)\right)$$

Now,

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{B(a)} f(x)$$

is just the [[type of identity functions]] on $B(a)$, and is thus always contractible, which implies that

$$\sum_{f:B(a) \to B(a)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, a, \mathrm{refl}_A(a))} f(x)$$

is contractible for all $a:A$. By induction on identity types, the type

$$\sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

is contractible for all $a:A$, $b:A$, and $p:a =_A b$. By [[weak function extensionality]], this implies that

$$\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

is contractible.
\end{proof}

Thus, it makes sense to refer to the canonical family of transport functions

$$\mathrm{transport}_{x:A.B(x)}:\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$

inductively defined by

$$\mathrm{transport}_{x:A.B(x)}(a, a, \mathrm{refl}_A(a)) \equiv \left(\lambda x:B(a).x, \lambda x:B(a).\mathrm{idTohId}(x, (\lambda x:B(a).x)(x), \beta_{B(a) \to B(a)}^{x:B(a).x}(x))\right)$$

for all $a:A$ as *[[generalized the|the]]* transport function on the type family $x:A \vdash B(x)$.

\begin{theorem}
Given a family of transport functions
$$t:\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$
for all $a:A$, $b:A$, and $p:a =_A b$, the function $\pi_1(t(b, a, p^{-1}))$ is a [[quasi-inverse function]] of $\pi_1(t(a, b, p))$.
\end{theorem}

\begin{proof}
For all $a:A$, $b:A$, and $p:a =_A b$ and for all $x:B(a)$, we have
$$\pi_2(t(a, b, p))(x):x =_{x:A.B(x)}^{(a, b, p)} \pi_1(t(a, b, p))(x)$$
and for all $y:B(b)$ we have
$$\pi_2(t(b, a, p^{-1}))(x):x =_{x:A.B(x)}^{(b, a, p^{-1})} \pi_1(t(b, a, p^{-1}))(x)$$

Thus, we have
$$\pi_2(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)):\pi_1(t(b, a, p^{-1}))(y) =_{x:A.B(x)}^{(a, b, p)} \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))$$
and
$$\pi_2(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x)):\pi_1(t(a, b, p)(x)) =_{x:A.B(x)}^{(b, a, p^{-1})} \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x))$$

By concatenating heterogeneous identifications, we get
$$\pi_2(t(a, b, p))(x) \bullet \pi_2(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x)):x =_{x:A.B(x)}^{(a, a, p \bullet p^{-1})} \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x))$$
and
$$\pi_2(t(b, a, p^{-1}))(y) \bullet \pi_2(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)):y =_{x:A.B(x)}^{(b, b, p^{-1} \bullet p)} \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))$$

Since $p \bullet p^{-1} \equiv \mathrm{refl}_{A}(a)$ and $p^{-1} \bullet p \equiv \mathrm{refl}_{A}(b)$, we thus have
$$\pi_2(t(a, b, p))(x) \bullet \pi_2(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x)):x =_{x:A.B(x)}^{(a, a, p \bullet p^{-1})} \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x))$$
and
$$\pi_2(t(b, a, p^{-1}))(y) \bullet \pi_2(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)):y =_{x:A.B(x)}^{(b, b, p^{-1} \bullet p)} \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))$$

By converting heterogeneous identifications over reflexivity to homogeneous identifications, one gets
$$\mathrm{hIdToId}(x, \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x)), \pi_2(t(a, b, p))(x) \bullet \pi_2(t(b, a, p^{-1}))(\pi_1(t(a, b, p)(x))))):x =_{B(a)} \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x))$$
and
$$\mathrm{hIdToId}(y, \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)), \pi_2(t(b, a, p^{-1}))(y) \bullet \pi_2(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))):y =_{B(b)} \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))$$

Thus, we have shown that $\pi_1(t(b, a, p^{-1}))$ is a quasi-inverse function of $\pi_1(t(a, b, p))$ for all $a:A$, $b:A$, and $p:a =_A b$.
\end{proof}

We define the dependent functions

$$\mathrm{linv}(t, a, b, p) \equiv \lambda x:B(a).\mathrm{hIdToId}(x, \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x)), \pi_2(t(a, b, p))(x) \bullet \pi_2(t(b, a, p^{-1}))(\pi_1(t(a, b, p)(x)))))$$

in type $\prod_{x:B(a)} x =_{B(a)} \pi_1(t(b, a, p^{-1}))(\pi_1(t(a, b, p))(x))$

and

$$\mathrm{rinv}(t, a, b, p) \equiv \lambda y:B(b).\mathrm{hIdToId}(y, \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)), \pi_2(t(b, a, p^{-1}))(y) \bullet \pi_2(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y)))$$

in type $\prod_{y:B(b)} y =_{B(b)} \pi_1(t(a, b, p))(\pi_1(t(b, a, p^{-1}))(y))$

for all $a:A$, $b:A$, and $p:a =_A b$.

\begin{theorem}
Given a family of transport functions
$$t:\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$
for all $a:A$, $b:A$, and $p:a =_A b$, the function $\pi_1(t(a, b, p))$ is bi-invertible.
\end{theorem}

\begin{proof}
We construct the element

$$((\pi_1(t(b, a, p^{-1})), \mathrm{linv}(t, a, b, p)), (\pi_1(t(b, a, p^{-1})), \mathrm{rinv}(t, a, b, p))$$

in the type

$$\left(\sum_{f:B(b) \to B(a)} \prod_{x:B(a)} x =_{B(a)} f(\pi_1(t(a, b, p))(x))\right) \times \left(\sum_{g:B(b) \to B(a)} \prod_{y:B(b)} y =_{B(b)} \pi_1(t(a, b, p))(g(y))\right)$$

indicating that $\pi_1(t(a, b, p))$ is bi-invertible for all $a:A$, $b:A$, and $p:a =_A b$.
\end{proof}

By definition, it follows that:

\begin{corollary}
Given a family of transport functions
$$t:\prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \sum_{f:B(a) \to B(b)} \prod_{x:B(a)} x =_{x:A.B(x)}^{(a, b, p)} f(x)$$
for all $a:A$, $b:A$, and $p:a =_A b$, the transport function $\pi_1(t(a, b, p))$ is an [[equivalence of types]].
\end{corollary}

Hence, in [[dependent type theory]], transport functions are also called **transport equivalences**.

## Examples and applications

### Univalent Tarski universes

Tarski universes are types which behave like internal models of [[dependent type theory]] inside of the type theory itself. We use a very general notion of Tarski universe: a Tarski universe merely consists of a type of encodings for types $\mathcal{U}$ and a universal type family $\mathcal{T}_\mathcal{U}$. Given an encoding $A:\mathcal{U}$, an internal type family indexed by $A$ is a function $B:\mathcal{T}_\mathcal{U}(A) \to \mathcal{U}$.

There are two ways to say that $A:\mathcal{U}$ and $B:\mathcal{U}$ are the same, by way of the [[identity type]] of the Tarski universe $A =_\mathcal{U} B$, and by way of the [[equivalence in homotopy type theory|type of equivalences]] between the type reflections of $A:\mathcal{U}$ and $B:\mathcal{U}$, $\mathcal{T}_\mathcal{U}(A) \simeq \mathcal{T}_\mathcal{U}(B)$. By [[transport]], there is a canonical function
$$\mathrm{trans}^{\mathcal{T}_\mathcal{U}}(A, B):(A =_\mathcal{U} B) \to (\mathcal{T}_\mathcal{U}(A) \simeq \mathcal{T}_\mathcal{U}(B))$$
The Tarski universe $\mathcal{U}$ is then said to be a [[univalent universe]] if the transport function $\mathrm{trans}^{\mathcal{T}_\mathcal{U}}(A, B)$ is an equivalence of types
$$\mathrm{trans}^{\mathcal{T}_\mathcal{U}}(A, B):(A =_\mathcal{U} B) \simeq (\mathcal{T}_\mathcal{U}(A) \simeq \mathcal{T}_\mathcal{U}(B))$$
for all encodings $A:\mathcal{U}$ and $B:\mathcal{U}$.

One could also internalize the type of equivalences inside of the Tarski universe $\mathcal{U}$, but that requires the Tarski universe be closed under [[identity types]],  [[dependent product types]], and [[dependent sum types]].

A Tarski universe $\mathcal{U}$ is weakly closed under identity types if it has an internal encoding of identity types in the Tarski universe: Given an element $A:\mathcal{U}$, there is a function $\mathrm{Id}_A^\mathcal{U}:\mathcal{T}_\mathcal{U}(A) \times \mathcal{T}_\mathcal{U}(A) \to \mathcal{U}$ and for all elemments $a:\mathcal{T}_\mathcal{U}(A)$ and $b:\mathcal{T}_\mathcal{U}(A)$ an equivalence
$$\mathrm{canonical}_{\mathrm{Id}_A}(a, b):\mathcal{T}_\mathcal{U}(\mathrm{Id}_A^\mathcal{U}(a, b)) \simeq (a =_{\mathcal{T}_\mathcal{U}(A)} b)$$

A Tarski universe $\mathcal{U}$ is weakly closed under dependent product types if it has an internal encoding of dependent product types in the Tarski universe: given an element $A:\mathcal{U}$ and a function $B:\mathcal{T}_\mathcal{U}(A) \to \mathcal{U}$, there is an element $\Pi_\mathcal{U}(x:A).B(x):\mathcal{U}$, and an equivalence
$$\mathrm{canonical}_\Pi(A, B):\mathcal{T}_\mathcal{U}(\Pi_\mathcal{U}(x:A).B(x)) \simeq \prod_{x:\mathcal{T}_\mathcal{U}(A)} \mathcal{T}_\mathcal{U}(B(x))$$

A Tarski universe $\mathcal{U}$ is weakly closed under dependent sum types if it has an internal encoding of dependent sum types in the Tarski universe: given an element $A:\mathcal{U}$ and a function $B:\mathcal{T}_\mathcal{U}(A) \to \mathcal{U}$, there is an element $\Sigma_\mathcal{U}(x:A).B(x):\mathcal{U}$, and an equivalence
$$\mathrm{canonical}_\Sigma(A, B):\mathcal{T}_\mathcal{U}(\Sigma_\mathcal{U}(x:A).B(x)) \simeq \sum_{x:\mathcal{T}_\mathcal{U}(A)} \mathcal{T}_\mathcal{U}(B(x))$$

Every Tarski universe $\mathcal{U}$ which is weakly closed under identity types, dependent product types, and dependent sum types types has an internal encoding of type of equivalences $A \simeq_\mathcal{U} B$, defined for all $A:\mathcal{U}$ and $B:\mathcal{U}$ as
$$A \simeq_\mathcal{U} B \coloneqq \Sigma_\mathcal{U}(f:A \to_\mathcal{U} B).\mathrm{isEquiv}_\mathcal{U}(f)$$
with $\mathrm{isEquiv}_\mathcal{U}$ defined by whatever suitable definition of [[equivalence in homotopy type theory]].

Thus, there is a third way to say that $A:\mathcal{U}$ and $B:\mathcal{U}$ are the same, by way of the type reflection of the internal encoding of the type of equivalences: $\mathcal{T}_\mathcal{U}(A \simeq_\mathcal{U} B)$. It could be proven, from the closure of the universe under identity types, dependent product types, and dependent sum types, that there is an equivalence
$$\mathrm{canonical}_\simeq(A, B):\mathcal{T}_\mathcal{U}(A \simeq_\mathcal{U} B) \simeq (\mathcal{T}_\mathcal{U}(A) \simeq \mathcal{T}_\mathcal{U}(B))$$

There is a more common definition of a univalent Tarski universe: a  Tarski universe $\mathcal{U}$ is a [[univalent universe]] if the canonical function
$$\mathrm{idtoequiv}(A, B):(A =_\mathcal{U} B) \to \mathcal{T}_\mathcal{U}(A \simeq_\mathcal{U} B)$$
is an equivalence
$$\mathrm{idtoequiv}(A, B):(A =_\mathcal{U} B) \simeq \mathcal{T}_\mathcal{U}(A \simeq_\mathcal{U} B)$$
These two definitions of univalent universes are the same. In order to show that the two definitions are the same, we need to show that there is an identification
$$i(p):\mathrm{canonical}_\simeq(A, B)(\mathrm{idtoequiv}(A, B)(p)) =_{\mathcal{T}_\mathcal{U}(A) \simeq \mathcal{T}_\mathcal{U}(B)}  \mathrm{trans}^{\mathcal{T}_\mathcal{U}}(A, B)(p)$$
for all identifications $p:A =_\mathcal{U} B$. By the J rule it is enough to show that $\mathrm{canonical}_\simeq(A, A)$ maps the identity function of $\mathcal{T}_\mathcal{U}(A \to_\mathcal{U} A)$ to the identity function of $\mathcal{T}_\mathcal{U}(A) \to \mathcal{T}_\mathcal{U}(A)$. Since the identity function in $\mathcal{T}_\mathcal{U}(A \to_\mathcal{U} A)$ is just $\mathrm{canonical}_\simeq^{-1}(A, A)(id_A)$ the above statement is always true. Thus, the second definition of a univalent universe is equivalent to the definition by transport.

These results apply to strictly Tarski universes as well, which are strictly closed rather than weakly closed (the equivalences become [[definitional equality]]): since $A \equiv B$ implies that $A \simeq B$ for types $A$ and $B$, every strictly Tarski universe is weakly Tarski.

### Relation to parallel transport

\begin{remark}\label{RelationToParallelTransport}
**(relation to parallel transport -- [[schreiber:differential cohomology in a cohesive topos|dcct §3.8.5]], [ScSh12 §3.1.2](cohesive+homotopy+type+theory#SchreiberShulman2012))**
\linebreak
  In [[cohesive homotopy type theory]] the [[shape modality]] $\esh$ has the [[shape via cohesive path ∞-groupoid|interpretation]] of turning any cohesive type $X$ into its [[path infinity-groupoid|path $\infty$-groupoid]] $\esh X$: The [[1-morphisms]] $p \colon (x =_{\esh X} y)$ of $\esh X$ have the [[shape via cohesive path ∞-groupoid|interpretation]] of being (whatever identities existed in $X$ composed with) cohesive (e.g. [[continuous function|continuous]] or [[smooth function|smooth]]) *[[paths]]* in $X$, and similarly for the higher order paths-of-paths.

Accordingly, an [[shape modality|$\esh X$]]-[[dependent type]] $B$ has the interpretation of being a "[[local system]]" of $B$-[[coefficients]] over $X$, namely a $B(x)$-[[fiber infinity-bundle|fiber $\infty$-bundle]] equipped with a [[flat infinity-connection|flat $\infty$-connection]].

In this case, the identity transport (eq:TheTransportFunctions) along paths in [[shape modality|$\esh X$]] has the interpretation of being the *[[parallel transport]]* (in the original sense of [[differential geometry]]) with respect to this [[flat infinity-connection|flat $\infty$-connection]] (and the [[higher parallel transport]] when applied to paths-of-paths).
\end{remark}

## See also ##

* [[identity type]]

* [[dependent identity type]]

* [[identity of indiscernibles]]

* [[identity function]]

## References ##

The traditional notion of the "transport" in [[algebraic topology]] (via the [[fundamental theorem of covering spaces]] and more generally via the notion of [[Serre fibrations]]):

* {#tomDieck2008} [[Tammo tom Dieck]],  §3.2 and §5.6 in: _Algebraic topology_, European Mathematical Society, Zürich (2008) [[doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf)]

The notion in [[homotopy type theory]]:

* [[Univalent Foundations Project|UVP]], [§2.3 "Type families and fibrations"](https://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf#page=81) in: *[[Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Egbert Rijke]]: §5.4 in: *[[Introduction to Homotopy Type Theory]]* Cambridge University Press (2023) [[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)&rbrack

* [[Egbert Rijke]], pp. 18 in: *Homotopy Type Theory* (2012) [[pdf](https://studenttheses.uu.nl/bitstream/handle/20.500.12932/11704/hott.pdf?sequence=1)]

Implementation in [[Agda]]:

* [[Martín Hötzel Escardó]], *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/)]

Leibniz's original paragraph (from an unpublished manuscript probably written after 1683) is reproduced in the Latin Original in

* {#Gerhard1890} K. Gerhard (ed.), Section XIX, [p. 228](https://archive.org/details/diephilosophisc00gerhgoog/page/228/mode/2up) in: *Die philosophischen Schriften von Gottfried Wilhelm Leibniz*, Siebenter Band, Weidmannsche Buchhandlung (1890) [[archive.org](https://archive.org/details/diephilosophisc00gerhgoog/page/n7/mode/2up)]

and in English translation in:

* {#Lewis18} [[Clarence I. Lewis]], Appendix (p. 373) of: *A Survey of Symbolic Logic*, University of California (1918) [[[LewisAppendix-LeibnizIndiscernibles.pdf:file]]]

and further discussed in:

* {#Cartwright71} [[Richard Cartwright]], *Identity and Substitutivity*, p. 119-133 of: Milton Munitz (ed.) *Identity and Individuation*, New York University Press (1971) [[[Cartwright-IdentityAndSubstitutivity.pdf:file]]]

* {#GriesSchneider98} David Gries, Fred Schneider, *Formalizations Of Substitution Of Equals For Equals* (1998) [[pdf](https://ecommons.cornell.edu/bitstream/handle/1813/7340/98-1686.pdf?sequence=1&isAllowed=y), [ecommons:1813/7340](https://ecommons.cornell.edu/handle/1813/7340)

The understanding of transport in HoTT as expressing [[Leibniz]]'s principle of "[[indiscernibility of identicals]]" (aka "substitution of equals", "substitutivity") has been made explicit in:

* {#LadymanPresnell15} [[James Ladyman]], [[Stuart Presnell]], §6.3 in: *Identity in Homotopy Type Theory, Part I: The Justification of Path Induction*, Philosophia Mathematica **23** 3 (2015) 386–406 [[doi:10.1093/philmat/nkv014](https://doi.org/10.1093/philmat/nkv014), [pdf](http://philsci-archive.pitt.edu/11079/1/Identity_in_HTT_public.pdf)]

* [[Benedikt Ahrens]], [[Paige Randall North]], §3.1 in: *Univalent foundations and the equivalence principle*, in: *Reflections on the Foundations of Mathematics*, Synthese Library **407** Springer (2019)  [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

The converse implication of [[path induction]] from transport (together with the uniqueness principle for id-types) is made explicit in:

* [Ladyman & Presnell 2015, §6.3](#LadymanPresnell15)

* Lennard Götz, §4.2 of: *Martin-Löf's J-rule*, LMU (2018) [[pdf](https://www.math.lmu.de/~petrakis/Goetz.pdf)]

For the role of transport in defining an equivalent notion of [[univalence]] in [[weakly Tarski universes]], see:

* Madeleine Birchfield, Valery Isaev, *Univalence for weakly Tarski universes*, MathOverflow, ([web](https://mathoverflow.net/q/431723))

[[!redirects transport]]
[[!redirects transports]]

[[!redirects weak transport]]
[[!redirects weak transports]]

[[!redirects strict transport]]
[[!redirects strict transports]]

[[!redirects judgmentally strict transport]]
[[!redirects judgmentally strict transports]]

[[!redirects propositionally strict transport]]
[[!redirects propositionally strict transports]]

[[!redirects transporting]]

[[!redirects identity transport]]
[[!redirects identity transports]]

[[!redirects type transport]]
[[!redirects type transports]]

[[!redirects transport equivalence]]
[[!redirects transport equivalences]]

[[!redirects transport function]]
[[!redirects transport functions]]

[[!redirects transport functor]]
[[!redirects transport functors]]

[[!redirects type of transport functions]]
[[!redirects types of transport functions]]

[[!redirects indiscernibility of identicals]]

[[!redirects principle of substitutivity]]
[[!redirects principle of substitution]]
[[!redirects principle of substitution of equals for equals]]
[[!redirects principle of substituting equals for equals]]


transport plan

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### Context
#### Measure and probability theory
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#Contents#
* table of contents
{:toc}

## Idea

One can view a [[probability measure]] $p$ on a space $(X,\mathcal{A})$ as a "pile of mass", for example, of sand, on the space $X$.
Using this picture, given two probability spaces $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, there could be many ways of moving the mass from $X$ to $Y$ in such a way that the sand from the pile $p$ is arranged to form the pile $q$. (The mass from which point goes to which point, or points?)
This "way of moving the mass" is called a *transport plan*, and it is usually encoded by a joint distribution or by a [[Markov kernel]] (see below).

It is useful to keep track of *in which way* we are rearranging the mass $p$ to form $q$, and we can see these different ways as different morphisms, between the objects $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, in a  [[category of couplings]].

## Definition

Let $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ be [[probability spaces]]. A **coupling** or **transport plan** between $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ is a probability space $(X\times Y, \mathcal{A}\otimes\mathcal{B},r)$ where

* $\mathcal{A}\otimes\mathcal{B}$ is the [[tensor product]] [[sigma-algebra]] on the [[topological product space|product space]] $X\times Y$ (generated by the sets $A\times B$ with $A\in\mathcal{A}$ and $B\in\mathcal{B}$);

* the [[measure]] $r$ has $p$ and $q$ as marginals, in the sense that for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$,

   $$
     r(A\times Y) \,=\, p(A)
     \;\;
     \text{and}
     \;\;
     r(X\times B) \,=\, q(B)
     \,.
   $$

## Main constructions

### Identity coupling

Given a probability space $(X,\mathcal{A},p)$, the **identity coupling** or **diagonal coupling** is given by the following measure on $\mathcal{A}\otimes\mathcal{A}$:
$$
\Delta_p (A\times A') = p(A\cap A')
$$
for all $A,A'\in\mathcal{A}$.

Intuitively, this is a copy of $p$ on $X$ concentrated on the diagonal subset $\{(x,x):x\in X\}\subseteq X\times X$.
(Whenever $(X,\mathcal{A})$ is standard Borel, the diagonal subset is measurable, and so this intuition can be made precise.)

This coupling gives the [[identity]] in the [[category of couplings]].
In terms of transport plans, this corresponds to not moving any mass (almost surely).

### Independent coupling

Given probability spaces $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ the **independent coupling** or **product coupling** or **constant coupling** is given by the product measure $p\otimes q$, i.e.
$$
(p\otimes q)(A\times B) = p(A)\,q(B)
$$
for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$.

In terms of transport plans, this arranges the mass from almost all points of $X$ to a distribution proportional to $q$, (almost surely) independently of the point of origin.

### Composition of couplings

Let $(X,\mathcal{A},p)$, $(Y,\mathcal{B},q)$, $(Z,\mathcal{C},r)$ be [[standard Borel]] probability spaces, and consider transport plans $s$ from $p$ to $q$ and $t$ from $q$ to $r$.
The **composite transport plan** $t\circ s$ from $p$ to $r$ is defined as follows:
$$
(t\circ s)(A\times C) = \int_Y s'(A|y)\,t'(C|y) q(dy)
$$
for all $A\in\mathcal{A}$ and $C\in\mathcal{C}$, and where $s'$ and $t'$ are the [[Markov kernel#regular_conditionals_from_a_joint_distribution|regular conditional distributions]] associated to $s$ and $t$ given $Y$.
The interpretation is that the mass in moved according to the plan $s$ and then according to the plan $t$, and in case the transport is stochastic, the two transitions are taken independently.

This construction gives composition in the [[category of couplings]]. When the transport plans are induced by functions or kernels (see below), the composition of transport plans is given by the composition of functions or kernels.

In [Kozen-Silva-Voogd'23](#non_borel), this construction was extended beyond the standard Borel case. (See there for the details.)

### Couplings induced by functions

Let $f:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$ be a [[measure-preserving function]].
One can define the "deterministic" transport plan $r_f$ as follows,
$$
r_f(A\times B) = p\big(A\cap f^{-1}(B)\big)
$$
for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$.
Intuitively, this maps all the mass at $x$ to the point $f(x)$, for every $x\in X$.

Note that in general there may exist no measure-preserving function between two probability spaces, for example, on the real line, if $p$ is a Dirac delta and $q$ is not.
A construction that always exists is in terms of [[Markov kernels]], see below.

### Couplings induced by kernels

Let $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$ be a [[Markov kernel#measurepreserving_kernels|measure-preserving Markov kernel]].
One can define a transport plan $r_k$ as follows,
$$
r_k(A\times B) = \int_A k(B|x)\,p(dx)
$$
for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$.
Intuitively, this maps all the mass at $x$ to a measure on $Y$ proportional to the measure $B\mapsto k(B|x)$.

Note that in the formula above, the measure $B\mapsto k(B|x)$ is invoked only for almost all $x$, and so it is insensitive to changes in $k$ on a $p$-measure-zero set.
In a certain sense, this transporting the *mass* of $p$, more than the single points $x$.

In many cases, such as if $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are [[standard Borel]], every transport plan is in the form $r_k$ for some $k$. See also [[category of couplings#equivalence_of_the_definitions|the discussion at "categories of couplings"]].

### Bayesian inversion

Couplings are in some sense undirected, meaning that every transport plan from $X$ to $Y$ can also be seen as (and canonically induces) a transport plan from $Y$ to $X$.

This makes the [[category of couplings]] canonically a [[dagger category]].

For transport plans specified by kernels, this symmetry corresponds exactly to [[Bayesian inversion]] of kernels.

## Related concepts

* [[category of couplings]]

* [[probability space]]

* [[Markov kernel]]

* [[categorical probability]]

* [[optimal transport]]

## References

* {#villani} [[Cedric Villani]], _Optimal transport: old and new_, Springer, 2008.

* Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Alexandra Silva, _Borel kernels and
their approximation, categorically_, MFPS 2018.  [arXiv](https://arxiv.org/abs/1803.02651).

* {#non_borel} Dexter Kozen, Alexandra Silva, Erik Voogd, _Joint Distributions in Probabilistic Semantics_, MFPS 2023. ([arXiv](https://arxiv.org/abs/2309.06913))

* [[Paolo Perrone]], _Lifting couplings in Wasserstein spaces_, 2021. ([arXiv:2110.06591](https://arxiv.org/abs/2110.06591))

category: probability

[[!redirects transport plans]]
[[!redirects coupling (probability)]]
[[!redirects couplings (probability)]]
[[!redirects coupling (probability theory)]]
[[!redirects couplings (probability) theory]]


transpose matrix

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#### Linear algebra
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## Definition

If $A = (a_{i j})$ is a [[matrix]], then its _transpose_ $A^T$ is the matrix $A^T \coloneqq (a_{j i})$.

A square matrix which is its own transpose, $A =A^T$, is called a [[symmetric matrix]].

## Related concepts

* [[conjugate transpose matrix]]

* [[hermitian matrix]]

* [[bilinear form]]

* [[dagger-compact category]]

[[!redirects transpose matrices]]
[[!redirects matrix transposition]]

[[!redirects transposed matrix]]


transposition permutation

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## Idea

In [[group theory]] a  *transposition* is a [[permutation]] that exchanges two elements and leaves all others fixed.

## Definition

For $n \in \mathbb{N}$ and $1 \leq i \neq j \leq n$, the *transposition* of $i$ with $j$ among $\{1,\cdots, n\}$ is the [[permutation]]

$$
  \array{
    \{1, \cdots, n\}
    &
    \underoverset{\simeq}{\;\;\; \sigma_{i,j} \;\;\; }{\longrightarrow}
    &
    \{1, \cdots, n\}
    \\
    k
    &\mapsto&
    \left\{
    \array{
       k &\vert& k \notin \{i,j\}
       \\
       j &\vert& k = i
       \\
       i &\vert& k = j
    }
    \right.
  }
$$

## Properties

* Transpositions are [[generators]] for the [[symmetric group]], exhibiting it as a [[finitely generated group]].

* In fact already just the *adjacent transpositions* $t_{i , i+1}$ for $1 \leq i \leq n$ generate the symmetric group, see at *[[Kendall tau distance]]*.

## Related concepts

* [[permutation representation]], [[permutation matrix]]

* [[cyclic permutation]], [[rotation permutation]]

[[!redirects transposition permutations]]

[[!redirects transposition]]
[[!redirects transpositions]]


transseries

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#### Analysis
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## Idea

A _transseries_ is an expansion of a [[function]] $f(x)$ as a [[power series]] in a [[vector]] $z$ whose components are fixed functions of $x$.

In application to [[perturbative quantum field theory]] one is interested in functions of the [[coupling constant]] $g$ and the relevant expansions are in terms of $x=g$ or $x=g^2$ and $z_1=x$ for the contributions of the [[Feynman perturbation series]] and  $z_2=e^{-c/x}$ for capturing [[non-perturbative effects]]. Also $z_3=\log x/x_0$.

## Related concepts

* [[asymptotic series]]

* [[resurgence theory]]

## Reference

* Daniele Dorigoni, _An Introduction to Resurgence, Trans-Series and Alien Calculus_ ([arXiv:1411.3585](https://arxiv.org/abs/1411.3585))

[[!redirects trans-series]]


transvectant

Certain invariant bilinear differential operators used to obtain new invariants of bilinear forms from old, discovered by Gordan,

* P. Gordan, _Das Zerfallen der Curven in gerade Linien_, Math. Ann., 45 (1894) 411-427

Appearance of transvectants is related to certain transformations on symmetric spaces, [[transvection]]s.

Transvectants in the context of modular forms are [[Rankin-Cohen bracket]]s.

* P.J. Olver, J.A. Sanders, _Transvectants, modular forms and the Heisenberg algebra_,  Adv. Appl. Math., 25 (2000) 252-283 [doi](https://doi.org/10.1006/aama.2000.0700)
* V. Ovsienko, P. Redou, _Generalized transvectants-Rankin–Cohen brackets_ Lett. Math. Phys. 63, 19–28 (2003). https://doi.org/10.1023/A:1022956710255
* P.J. Olver, M. Petitot, P. Solé, _Generalized transvectants and Siegel modular forms_, Advances in Applied Mathematics __38__:3 (2007) 404-418 [doi](https://doi.org/10.1016/j.aam.2005.11.001) (preprint [pdf](https://www-users.cse.umn.edu/~olver/a_/gtv.pdf))
* Sofiane Bouarroudj, [[Dimitry Leites]], _Invariant differential operators in positive characteristic_, J. Algebra __499__ (2018) 281-297 [doi](https://doi.org/10.1016/j.jalgebra.2017.11.048)
* G. Mari Beffa, Jan A. Sanders, _Transvection and differential invariants of parametrized curves_, J. Lie Theory __18__ (2008) 93–123


transvection

### Idea

Transvection is a name for a number of concepts of involutive transformations in algebra and geometry which use shift in a direction of a fixed vector, along geodesic line (say, through another fixed point) or alike. They leave some nondegenerate hypersurface (in some contexts, hyperplane) fixed. In the geometry of symmetric spaces, one defines transvections using a parallel transport along geodesic lines; at any point these transvections induce [[Killing vector]]s, or equivalently, elements of the tangent [[Lie triple system]].

> Warning: In the usual Euclidean geometry transvection is sometimes also used as a name for _shear mapping_,  cf. wikipedia [shear mapping](http://en.wikipedia.org/wiki/Shear_mapping).

### For vector spaces

__Definition.__ For a left vector space $V$ over a [[skewfield]] $F$, a __transvection__ is a left $F$-linear map $t:V\to V$ such that the map $v\mapsto t(v)-v$ has 1-dimensional image (called the center of $t$) and this image is a subset of the fixed space of $V$ (the set of fixed points of $t$, i.e. the kernel of $v\mapsto t(v)-v$).

For finite dimensional vector spaces over a (commutative) field, $t\in GL(V)$ is a transvection if it is either the identity or $dim(Im(t-Id))=1$ and $det(t) = 1$. For any $s\in GL(V)$ the image $Im(s-Id)$ is also called the residual space of $s$ and $res(s) = dim(Im(s-Id))$ the residue of $s$. Residue is $0$ iff $s=Id$; residue of a product is smaller or equal the sum of residues. The fixed space and the residue space of $s$ are both fixed by $s$ and their dimensions add to the dimension of $V$; both spaces for $s^{-1}$ are equal to those of $s$.

In the geometry of symplectic vector spaces (of positive dimension $\geq 2$ over a commutative field) special role is played by
the transvections within the symplectic group. The fixed points of a
transvection $t\in Sp(V)$ are precisely those points $v\in V$ for which $\omega(v,t(v))=0$.

Theorem. $Sp(V)$ is generated by symplectic transvections (a result of [[Jean Dieudonne]] for fields other than $\mathbb{F}_2$ and of D. Callan for $\mathbb{F}_2$). This is theorem 2.1.9 in

* O. T. O'Meara, _Lectures on symplectic groups_, Amer. Math. Soc. Mathematical Monographs __16__ (1978)

There are similar generation results for alternating spaces (i.e. when the form $\omega$ is possibly degenerate, hence not symplectic) of U. Spengler and H. Wolff. For the symplectic groups over commutative rings the validity of the generation result depends on the ring, cf.

* Yu. I. Merzlyakov, _Linear groups_, J. Soviet Math. __1__ (1973) 571--593

The study of transvections is often more complicated in the characteristic 2 case.

There is a natural projection from the general linear to the projective general linear group (quotienting out by the subgroup of diagonal matrices). A __projective transvection__ is an element of $PGL(V)$ which is an image (under this natural projection) of a transvection in
$GL(V)$.

### In symmetric spaces

For transvections defined via geodesics in a [[symmetric space]], the relation to Killing vectors and the infinitesimal transvections see

* J.-H. Eschenburg, _Lecture notes on symmetric spaces_, 24 pages, [pdf](http://myweb.rz.uni-augsburg.de/~eschenbu/symspace.pdf)

The reflections define the Coxeter quandle what motivates the definition of transvections in the [[quandle]] context.
The operations $R_a R_b^{-1}$ (where $R_a: x\mapsto x\triangleright a$) for $a,b$ in a quandle $X$ form the transvection group
$Transv(X)$ which is a normal subgroup of the inner automorphism group of a quandle. For Coxeter quandle, their quotient is a cyclic group (result of D. Joyce, JPAA 1982).

There are also quandles which are secretly entirely defined in terms of transvections; these are the transvection quandles, also called
symplectic quandles; the operation is defined in terms of an alternative form on the underlying $R$-module:

* Esteban Adam Navas, Sam Nelson, _On symplectic quandles_, Osaka J. Math. __45__:4 (2008) 973--985 [MR2493966](https://mathscinet.ams.org/mathscinet-getitem?mr=2493966) [euclid](https://projecteuclid.org/euclid.ojm/1227708829)

### Other literature

Transvections in other contexts

* J.I Hall, _Graphs, geometry, 3-transpositions, and symplectic-transvection groups_, Proc. London Math. Soc. __58__ (1989) 89--111 (3)

* Hans Cuypers, _Symplectic geometries, transvection groups, and modules_, J. Combin. Theory A __65__:1, January 1994, 39--59 doi

* Hans Cuypers. Anja Steinbach, _Linear transvection groups_, [pdf](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1063-9.pdf); _Linear transvection groups and embedded polar spaces_, Invent. math. __137__, 169--198 (1999) [doi](https://doi.org/10.1007/s002220050328)

* P.J Cameron, J.I Hall, _Some groups generated by transvection subgroups_, J. Algebra __140__ (1991) 184--209

* F. Timmesfeld, _Groups generated by k-transvections_, Invent. Math. __100__ (1990) 167--206

category: geometry, algebra

[[!redirects transvections]]


transversal maps

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

# Contents
* table of contents
{: toc}

## Idea

Two [[differentiable functions]] $f \colon X \to Z$ and $g \colon Y \to Z$ between [[differentiable manifolds]] are called _transversal_ if, roughly, the [[images]] of $X$ and $Y$ in $Z$ do not "touch tangentially".

If $f$ or $g$ are inclusions of (possibly [[immersion|immersed]]) [[submanifolds]] one speaks of these submanifolds being transversal to each other.

## Definition

Two [[differentiable functions]] $f \colon X \to Z$ and $g \colon Y \to Z$ between  [[differentiable manifold]] (for instance [[smooth functions]] between [[smooth manifolds]]) are __transversal__ if for all pairs of points $x \in X$ and $y \in Y$ with $f(x) = z = g(y)$ the [[differentials]] of $f$ and $g$ in these points [[linear span|span]] the entire [[tangent space]] at $z$ in the sense that

$$
  im(d f) + im(d g) \simeq T_z Z
  \,.
$$

(Notice that this is not required to be a [[direct sum]].)

## Examples

* A [[submersion]] is transversal to all differentiable functions into its [[codomain]].

## Properties

### Pullbacks along transversal maps

Various constructions involving [[pullbacks]] of [[differentiable manifolds]] work as expected only for pullbacks involving transversal maps.

For example, two [[differentiable functions]] with a common [[codomain]] are transversal only if their [[pullback]] exists and is preserved by the [[tangent bundle]] [[functor]]; that is, $T(X \times_Z Y) = T X \times_{T Z} T Y$.  (However, the pullback may exist and be preserved without transversality; for example if $X$ and $Y$ are both abstract [[points]], $Z$ is *not* a point, and the maps $X, Y \to Z$ are equal as concrete points of $Z$.)

This is to be regarded as the dual of the possibly more familiar statement that various constructions involving [[quotient object|quotient]]s only work as expected for _free_ [[actions]].

Both of these "problems" are solved by passing from the ordinary [[1-category]] [[Diff]] of manifolds to a suitable [[higher category theory|higher category]] of [[generalized smooth spaces]].

More precisely:

* the problem with the [[pushouts]] (quotients) is resolved by passing to [[stacks]] and [[smooth infinity-stacks]].

* the problem with the [[pullbacks]] is resolved by passing to [[derived stacks]]. Concretely for the case of manifolds this is discussed at _[[derived smooth manifold]]_.

### Thom's transversality theorem

See at _[[Thom's transversality theorem]]_.

## Related concepts

* [[regular value]]

## References

* [[Theodor Bröcker]], [[Klaus Jänich]], *Introduction to differentiable topology* (1982) [[ISBN:9780521284707](https://www.cambridge.org/ae/universitypress/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-differential-topology?format=PB&isbn=9780521284707)]

  > (translated from the German 1973 edition)

* [[Morris Hirsch]], _Differential topology_, GTM **33**, Springer (1976) [[doi:10.1007/978-1-4684-9449-5](https://link.springer.com/book/10.1007/978-1-4684-9449-5), [gBooks](http://books.google.com/books/about/?id=iSvnvOodWl8C)]

* {#Kosinski93} [[Antoni Kosinski]], chapter IV (pp. 59) of: _Differential manifolds_, Academic Press (1993) \[pdf, [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C)\]

* [[Ieke Moerdijk]], [[Gonzalo E. Reyes]], p. 27 in: *[[Models for Smooth Infinitesimal Analysis]]*, Springer (1991) [[doi:10.1007/978-1-4757-4143-8](https://link.springer.com/book/10.1007/978-1-4757-4143-8)]

[[!redirects transversal map]]
[[!redirects transversal maps]]
[[!redirects transverse map]]
[[!redirects transverse maps]]

[[!redirects transversality]]


Travis Schedler

__Travis Schedler__ is a mathematician at  Imperial College London ([webpage](https://www.imperial.ac.uk/people/t.schedler)).

* [[V. Ginzburg]], T. Schedler, _Differential operators and BV structures in noncommutative geometry_, Sel. Math. New Ser. 16, 673–730 (2010) [doi](https://doi.org/10.1007/s00029-010-0029-8) [arXiv:0710.3392](https://arxiv.org/abs/0710.3392)
* [[V. Ginzburg]], T. Schedler, _A new construction of cyclic homology_, Proc. Lon. Math. Soc 112 (2016), no. 3, 549-587 [doi](https://doi.org/10.1112/plms/pdw001)  [arXiv:1201.6635](https://arxiv.org/abs/1201.6635)
* Travis Schedler, _Zeroth Hochschild homology of preprojective algebras over the integers_, Adv. Math. __299__ (2016) 451--542 [doi](https://doi.org/10.1016/j.aim.2016.02.015)
* Gwyn Bellamy, Travis Schedler, _Kostka polynomials from nilpotent cones and Springer fiber cohomology_, [arxiv/1509.02520](https://arxiv.org/abs/1509.02520)
* [[Andrey Lazarev]], Travis Schedler, _Curved infinity-algebras and their characteristic classes_, J Topology (2012) 5 (3): 503-528 [arXiv:1009.6203](https://arxiv.org/abs/1009.6203)
* Raf Bocklandt, Travis Schedler, Michael Wemyss, _Superpotentials and higher order derivations_, J. Pure and Appl. Alg. __214__:9 (2010) 1501-1522 [doi](https://doi.org/10.1016/j.jpaa.2009.07.013)
* Travis Schedler, _Poisson algebras and Yang-Baxter equations_, in: Advances in quantum computation (Tyler, TX, 2007) (Providence, RI) (K. Mahdavi and D. Koslover, eds.), Contemp. Math. __482__ (2009) 91--106
arXiv:[math.QA/0612493](https://arxiv.org/abs/math.QA/0612493)
* [[Pavel Etingof]], [[Travis Schedler]], Alexandre Soloviev, _Set-theoretical solutions to the quantum Yang-Baxter equation_,  Duke Math. J. 100(2) (1999) 169--209 [doi](10.1215/S0012-7094-99-10007-X)
* Daniel Kaplan, Travis Schedler, _Multiplicative preprojective algebras are 2-Calabi-Yau_, [arXiv:1905.12025](https://arxiv.org/abs/1905.12025)

category: people

[[!redirects T. Schedler]]


tree

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

# Contents
* table of contents
{:toc}

## Idea

A **tree** is a connected [[graph]] without cycles. Some notions of tree (such as planar tree) consider extra structure or properties as well, but the baseline notion is tree as acyclic connected graph.

Trees are fundamental combinatorial objects which appear in a wide variety of guises. Some are given below.

## Definitions

### As undirected graphs

Recall that an [[undirected graph]] $G$ consists of a set $V$ of _vertices_ and a set $E$ of unordered pairs of vertices. A **path** in $G$ is a finite sequence of vertices $x_0, \ldots, x_n$ such that each pair $x_i x_{i+1}$ belongs to $E$. If $x_n = x_0$, then the path is called a **cycle**. A graph is **connected** if it is nonempty and if for every distinct pair of vertices $x, y$, there is a path for which $x_0 = x$ and $x_n = y$. A graph is **acyclic** if the only cycles are paths

$$x_0, \ldots, x_{n-1}, x_n, x_{n-1}, \ldots, x_0$$

where steps are retraced; an [[acyclic graph]] is also called a _[[forest]]_. A **tree** is a connected forest.

Each forest is a [[disjoint union|disjoint sum]] (a [[coproduct]] in the category of graphs) of trees. Removal of a vertex of a tree (and any edges incident to it) results in a forest.

Not all authors include nonemptiness as part of the notion of connectedness. See *[[connected space]]* for commentary on this. Forests on the other hand may be empty.

### As spaces

A _topological tree_ is a 1-dimensional [[CW-complex]] which is contractible, i.e., homotopy equivalent to a point.

In this language, a topological graph is simply a 1-dimensional CW-complex; more precisely, we mean a _CW-presentation_ and not merely a space that can be so presented. A topological forest is then a 1-dimensional CW-complex with vanishing first [[homology group]] with integral coefficients, and a tree is a connected forest.

For finite graphs $G$ as CW-complexes, one has the useful [[Euler characteristic|Euler formula]]:

$$h_0 - h_1 = c_0 - c_1$$

where $c_i$ is the number of $i$-cells and $h_i$ is the rank of the homology group $H_i(G; \mathbb{Z})$. This gives a useful criterion for a graph to be a tree: that it be connected ($h_0 = 1$) and that the number of vertices $c_0$ be 1 greater than the number of edges $c_1$.

### As digraphs

A **rooted tree** is a [[directed graph]] (or [[quiver]]) $G$ such that the [[free category on a quiver|free category]] generated by $G$ has a [[terminal object]], called the **root** of $G$. (Thus for every vertex $x$ in $G$ there is exactly one directed path from $x$ to the root. It follows easily that the free category is a poset. There is no significant change in content if "terminal" here is replaced by "initial"; we can distinguish the conventions by a tree directed _towards_ the root rather than _away from_ the root.)

A **rooted forest** is a coproduct of rooted trees, and again we have the fundamental fact that there is a natural [[equivalence of categories]]

$$Tree \overset{\to}{\leftarrow} Forest$$

which in one direction is the functor which sends a tree to the forest which results by excising the root and all edges incident to it, and in the other sends a forest to the tree gotten by adjoining a new root and joining the roots of the forest to that new root by new edges.

This equivalence can be exploited to relate the [[exponential generating function]] for finite rooted trees to the [[Lambert W-function]] (q.v.).

A tree in the form of an _undirected_ graph together with a chosen vertex can be regarded as a rooted tree in digraph form (with root the chosen vertex), by orienting all edges in the direction of paths going to the root. The category of rooted trees has very nice categorical properties not shared by the category of unrooted trees; see the following section.

### As ordered sets

In [[order theory]], a tree is described by a [[partial order]] with the added property that it is downward [[well ordering|well ordered]]. Let $S$ be a [[partially ordered set]], then $S$ is a tree when for each $x \in S$ the downward closure or history $x^- = \{ y \mid y \leq x \}$ is a [[well ordered]] subset of $S$. If instead of a well-order we find that $x^-$ is only totally ordered (meaning that points in the tree do not have a distinguished successor) we say that $S$ is [[prefix ordered]].

### As functors

Let $\mathbb{N}$ be the [[totally ordered set]] of natural numbers $0 \leq 1 \leq 2 \leq \ldots$, viewed as a [[category]]. A **rooted forest** is a [[presheaf]] on $\mathbb{N}$, i.e., a functor
$$F: \mathbb{N}^{op} \to Set$$
to "the" [[category of sets]]. A **rooted tree** is a forest for which $F(0)$ is terminal (a [[point]]).

In this case, we get from a tree as functor $F$ to a tree as digraph by passing to the [[category of elements]] $El(F)$. The vertices of the digraph are the elements, and the edges are those morphisms of $El(F)$ which map to morphisms $i + 1 \to i$ in $\mathbb{N}^{op}$. Notice that $\mathbb{N}^{op}$ is itself (the free category on) a tree as digraph, one that is terminal in the category of forests.

In the other direction, starting with a tree as digraph, we define a functor $F \colon \mathbb{N}^{op} \to Set$ by letting $F(n)$ be the set of vertices $v$ such that the path from $v$ to the root has $n$ edges. The map $F(n+1) \to F(n)$ moves a vertex $v \in F(n+1)$ to the vertex that is one step closer to the root (along the unique path from $v$ to the root).

The functor description makes it clear that the category of forests is a (presheaf) [[topos]], indeed a [[Grothendieck topos]]; therefore the [[category of trees]], which is equivalent, is also a [[topos]].

If we replace $Set$ here by the category of [[totally ordered set]]s $Tos$, then we may define a **planar forest** as a functor

$$F: \mathbb{N}^{op} \to Tos$$

### As terminal coalgebra

The category of trees as described in the section immediately above can be described universally in 2-universal terms, as the 2-terminal coalgebra for the endofunctor on $Cat$ (locally small categories) which takes a category to its small-coproduct cocompletion. See [[terminal coalgebra]] for an extended discussion.

### As operations

Let $\mathbf{F}$ denote the free (strict) monoidal category generated by $n$-ary operations $\theta_n: x^n \to x$, one for each arity $n \geq 0$. A (finite) **planar forest with ($n$) live leaves** is a morphism $\phi: x^n \to x^m$ in $\mathbf{F}$; a (finite) **planar tree with ($n$) live leaves** is a morphism $\phi: x^n \to x$.

The previous description is more a theorem than definition, but it points to the recursive construction of trees based on the fundamental equivalence between forests and trees:

* The identity $x \to x$ corresponds to a root considered as a live leaf;

* Given trees with live leaves $f_j \colon x^{n_j} \to x$, where $j \in \{1 \ldots, k\}$, there is a forest
$$f_1 \otimes \ldots \otimes f_k \colon x^{n_1 + \ldots + n_k} = x^{n_1} \otimes \ldots \otimes x^{n_k} \to x \otimes \ldots \otimes x = x^k$$
whose set of live leaves is the disjoint union of sets of live leaves of the $f_j$.

* Given a forest with live leaves
$$f_1 \otimes \ldots \otimes f_k \colon x^{n_1 + \ldots + n_k} = x^{n_1} \otimes \ldots \otimes x^{n_k} \to x \otimes \ldots \otimes x = x^k$$
there is a tree
$$\theta_n \circ (f_1 \otimes \ldots \otimes f_k) \colon x^{n_1 + \ldots + n_k} \to x$$
with the same set of live leaves.

In more straightforward combinatorial language: if a finite planar tree is a functor $F \colon [n]^{op} \to \Delta_a$ from a finite ordinal to the augmented [[simplex category]], and a **leaf** is an element without predecessors in the poset of elements of $F$, then we can consider as extra structure some subset of the leaves whose elements are designated as **live**. The principle is that one is permitted to graft roots of trees onto live leaves, but not onto dead ones. (Dead leaves arise by plugging in the constant = $0$-ary operation $\theta_0 \colon I = x^0 \to x$.)

Fancier versions of this basic recursive construction may involve "colored trees", "cherry trees", and so on, and figure prominently in the theory of [[operads]], especially in the construction of [[free operads]].

In fact, this type of recursivity is at the root (pun perhaps intended) of the [[cumulative hierarchy]] conception of sets in a material set theory like [[ZFC|ZFC set theory]]: all sets are collections of sets $\{x_1, \ldots, x_n\}$, and all sets are formed in this way. In order to found this conception on trees in a way that accepts the [[axiom of foundation]], one must restrict to _well-founded_ trees. A tree (pictured as a free category on a digraph, i.e., as a certain poset $P$) is **well-founded** if every object of $P^{op}$ is bounded above by a maximal element, or equivalently if there are no infinite chains $x_0 \lt x_1 \lt \ldots$ in $P^{op}$ starting from the root $x_0$. (Chains starting from the root are also called **branches**.)  See [[well-founded relation]] for other formulations of this definition (including those suitable for [[constructive mathematics]]; see [[pure set]] for details on this construction of the cumulative hierarchy, including using arbitrary trees for _ill-founded_ sets.

The category of well-founded trees admits a (2-)universal description, as an [[initial algebra]] for a certain endofunctor on [[Cat]]. See [[well-founded forest]] for details.

[To be continued...]

+-- {: .query}
Very nice! Is there a similar story worth telling for possibly infinite trees, corecursion, etc.

_Toby_:  Well-founded trees (and pure sets) may be defined recursively; ill-founded trees (and pure sets) may be defined corecursively; there is stuff about this at [[pure set]].  (Note that even a well-founded tree may be infinite, as long as some vertex has infinitely many branches out of it.)
=--

### As relational structures

A _rooted tree_ can be defined as a [[relational structure]] and as such serves as a useful [[geometric models for modal logics|model]] in the context of [[modal logics]].  In this form it consists of a set of _nodes_, _vertices_ or whatever your preferred terminology is, and a binary relation, $S$, on $T$.  These are to satisfy the following:

* the set $T$ contains a unique element $r$ called the **root**, such that, for all $t\in T$, $S^*r t$ holds, where $S^*$ is the reflexive-[[transitive closure]] of $S$;

* if $t\in T$, and $t\neq r$, then there is a unique $t'\in T$ such that $S t' t$;

* $S$ is **acyclic**, so for all $t\in T$, $\neg S^+ t t$, where $S^+$ is the [[transitive closure]] of $S$.

This last property, of course, implies that $S$ is irreflexive.

### As polynomial endofunctors (after Kock)

A new way of thinking about (rooted, usually assumed finite) trees has been introduced by [Joachim Kock](#Kock).

+-- {: .num_defn}
###### Definition
A *tree* $T$ consists of a set $T_0$ of edges, a set $T_1$ of nodes, and a set $T_2$ of *input flags*, all assumed finite, together with functions

$$T_0 \stackrel{s}{\leftarrow} T_2 \stackrel{p}{\to} T_1 \stackrel{t}{\to} T_0$$

such that

* $t$ is injective,
* $s$ is injective and its image is complementary to a singleton $\{r\}$ (called the _root_), and
* for all $e \in T_0$ there exists a natural number $k$ such that $\sigma^k(e) = r$, where $\sigma : T_0 \to T_0$ is the "walk-to-root" function defined by $e \mapsto t(p(s^{-1}(e)))$ if $e \neq r$, else $r \mapsto r$.
=--

Some commentary is in order. The subscript $0$ for edges and $1$ for nodes is consonant with standard usage in [[string diagrams]], where edges are labeled by $0$-cells in a monoidal category and nodes by $1$-cells. Each node is pictured as having a multiplicity of (and possibly zero) input edges and exactly one outgoing edge; the map $t: T_1 \to T_0$ maps a node to its outgoing edge. Only the root edge is not an input edge of any node, i.e., the unique inhabitant of the complement of $im(s)$ is what we call the root $r$. Now: the input "function" $T_1 \to T_0$ is actually a multivalued function, taking a node to the set of its input edges, and is therefore represented as a [[relation]] or [[span]] from $T_0$ to $T_1$. The domain of this relation $T_2$ is the set of ordered pairs $(e, v)$ where $e \in T_0$ is an input edge of $v \in T_1$; this may be called a "flag" (thinking of a general flag as a chain of incidence relations, here a 1-step chain with $e$ input-incident to $v$).

To each tree $T$ described in this form, there is an associated [[polynomial endofunctor]] $p_T: Set/T_0 \to Set/T_0$ controlled by the data

$$T_0 \stackrel{s}{\leftarrow} T_2 \stackrel{p}{\to} T_1 \stackrel{t}{\to} T_0$$

and defined by the composition

$$Set/T_0 \stackrel{s^\ast}{\to} Set/T_2 \stackrel{\Pi_p}{\to} Set/T_1 \stackrel{\Sigma_t}{\to} Set/T_0$$

where as usual $\Sigma_f \dashv f^\ast \dashv \Pi_f$. Notice that the functors on display preserve pullbacks, so $p_T$ preserves pullbacks.

There is a [[double category]] whose $0$-cells are sets $I$, whose horizontal $1$-cells are polynomial endofunctors $p: Set/I \to Set/J$, whose vertical $1$-cells are pullback functors $f^\ast: Set/I \to Set/I'$ induced by functions $f: I' \to I$, and whose $2$-cells are [[cartesian natural transformations]] of the form

$$\array{
Set/I & \stackrel{p}{\to} & Set/J \\
\mathllap{f^\ast} \downarrow & \swArrow & \downarrow \mathrlap{g^\ast} \\
Set/I' & \underset{p'}{\to} & Set/J'
}$$

The category consisting of horizontal arrows and 2-cells between them is called the *category of polynomial endofunctors*, $PolyEnd$. Kock defines a category of trees where objects are trees $T$ and whose morphisms are morphisms between the corresponding endofunctors $p_T$ as objects in $PolyEnd$.

As Kock shows, this description of trees is well-adapted to the usual sorts of combinatorics that emerge in the study of [[operads]], such as free operads.

## Related concepts

* [[category of trees]]

* in [[perturbative quantum field theory]] [[Feynman graphs]] of vanishing "[[loop order]]" are trees, and their contribution to the [[Feynman perturbation series]] is hence called the _tree level_ contribution

* [[shuffle of trees]]

* [[Bruhat-Tits tree]]

* [[tree tensor network state]]

* [[abstract syntax tree]]

## References

* {#Kock} [[Joachim Kock]], *Polynomial functors and trees*, arXiv:[0807.2874](http://arxiv.org/abs/0807.2874)

[[!redirects tree]]
[[!redirects trees]]

[[!redirects planar tree]]
[[!redirects planar trees]]

[[!redirects rooted tree]]
[[!redirects rooted trees]]

[[!redirects binary tree]]
[[!redirects binary trees]]


tree category

# Contents
* table of contents
{: toc}

## Idea

The [[simplex category]] may be regarded as the category of all linear [[directed graphs]]. The _tree category_ generalizes this to directed rooted [[trees]].

## Definition

### Finite planar level tree

(... ) see Berger

### Finite symmetric rooted trees

We define the category $\Omega$ _finite symmetric rooted trees_.

The objects of $\Omega$ are non-empty non-planar [[trees]] with specified root.

Each such tree may naturally be regarded as specifying an (colored) [[symmetric operad]] with one color per edge of the tree. A morphism of trees in $\Omega$ is a morphism of the corresponding operads.

As such, $\Omega$ is by construction a full subcategory of that of [[symmetric operads]] enriched over [[Set]].

### Operadic structure

$\Omega$ can be turned into a (symmetric) [[operad]]
by grafting trees.

## Dendroidal sets

A [[presheaf]] on $\Omega$ is a [[dendroidal set]], a generalization of a [[simplicial set]].

## References

* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))

* See the references at [[dendroidal set]].

[[!redirects tree categories]]

[[!redirects dendrex category]]
[[!redirects dendrex categories]]

[[!redirects category of trees]]
[[!redirects categories of trees]]

[[!redirects operad of trees]]
[[!redirects operads of trees]]


tree tensor network state

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What are called _tree tensor network states_ (TTN states) in [[quantum physics]] (specifically in [[solid state physics]] and in [[AdS/CFT]]) are those [[tensor network states]] of the form of [[tree]], often considered of fixed valence.

For example, given a [[metric Lie algebra]] $\mathfrak{g}$ (with [[string diagram]]-notation as discussed [there](metric+Lie+algebra#Definition)) its [[Lie bracket]]-[[tensor]] $f \coloneqq [-,-] \in \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast \otimes \mathfrak{g}$ gives rise to tree tensor network of the following form:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The [[tree tensor network states]] in the form of [[Bruhat-Tits trees]] play a special role in the [[AdS/CFT correspondence]], either as

1. a kind of [[lattice QFT]]-approximation to an actual [[boundary field theory|boundary]] [[CFT]] [[quantum state]],

1. as the [[p-adic geometry|p-adic geometric]] incarnation of [[anti de Sitter spacetime]] in the [[p-adic AdS/CFT correspondence]],

1. as a reflection of actual [[crystal]]-site quantum states in [[AdS/CFT in solid state physics]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

(As in [HMSS 16](p-adic+AdS/CFT+correspondence#HMSS16), [HLM 19](p-adic+AdS/CFT+correspondence#HLM19). But  maybe one wants the Poincaré-dual networks, instead, as in [HMPS 18](p-adic+AdS/CFT+correspondence#HMPS18)?)

## Related concepts

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

## References

### General

For more see the references at _[[tensor network state]]_.

* Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein, Section 2.3.3 of: _Tensor Network Contractions_, Lecture Notes in Physics, Springer (2020) ([arXiv:1708.09213](https://arxiv.org/abs/1708.09213), [doi:10.1007/978-3-030-34489-4](https://link.springer.com/book/10.1007/978-3-030-34489-4))

### Application to entanglement entropy

Application to [[entanglement entropy]];

* [[Luca Tagliacozzo]], G. Evenbly, [[Guifre Vidal]],
_Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law_, Phys. Rev. B 80, 235127 (2009) ([arXiv:0903.5017](https://arxiv.org/abs/0903.5017))

### Application in solid state physics

Application in [[solid state physics]]:

* Valentin Murg, Örs Legeza, Reinhard M. Noack, [[Frank Verstraete]], _Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks_, Phys. Rev. B 82, 205105 (2010) ([arXiv:1006.3095](https://arxiv.org/abs/1006.3095))

### Application in quantum chemistry

For application in [[quantum chemistry]]:

* Naoki Nakatani, Garnet Kin-Lic Chan, _Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm_, J. Chem. Phys. 138, 134113 (2013) ([arXiv:1302.2298](https://arxiv.org/abs/1302.2298))

* Klaas Gunst, [[Frank Verstraete]], Sebastian Wouters, Örs Legeza, Dimitri Van Neck, _T3NS: three-legged tree tensor network states_, Chem. Theory Comput. 2018, 14, 4, 2026-2033 ([arXiv:1801.09998](https://arxiv.org/abs/1801.09998))

* Henrik R. Larsson, _Computing vibrational eigenstates with tree tensor network states (TTNS)_, J. Chem. Phys. 151, 204102 (2019) ([arXiv:1909.13831](https://arxiv.org/abs/1909.13831))

### Application to $p$-adic AdS/CFT correspondence:

Discussion of [[Bruhat-Tits tree]]-tensor networks in the context of the [[p-adic AdS/CFT correspondence]]:

* {#BHLL17} Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, _Tensor network and ($p$-adic) AdS/CFT_, JHEP 1801 (2018) 139 ([arXiv:1703.05445](https://arxiv.org/abs/1703.05445))

* {#HLM19} Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, _$p$-adic CFT is a holographic tensor network_ ([arXiv:1902.01411](https://arxiv.org/abs/1902.01411))

### Application in machine learning

Application in [[machine learning]]:

* Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Blázquez García, Gang Su, Maciej Lewenstein, _Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure_, New Journal of Physics, 21, 073059 (2019) ([arXiv:1710.04833](https://arxiv.org/abs/1710.04833))

* Song Cheng, Lei Wang, Tao Xiang, Pan Zhang, _Tree Tensor Networks for Generative Modeling_, Phys. Rev. B 99, 155131 (2019) ([arXiv:1901.02217](https://arxiv.org/abs/1901.02217))

[[!redirects tree tensor network states]]

[[!redirects tree tensor network]]
[[!redirects tree tensor networks]]


trefoil knot

The __trefoil knot__ is a famous [[knot]]. One of the reasons is that in the list of knots, ordered by  [[crossing number]], it is the first 'real' knot one meets, being the simplest non-trivial knot. (The first knot listed is usually the 'unknot', i.e. the unknotted circle.) The trefoil has crossing number 3.

Here is a traditional view:

[[!include trefoil knot - SVG]]

Here is a depiction with [[bridge number]] 2:

[[!include trefoil knot (2 bridge) - SVG]]

The [[knot group]] of the trefoil knot (calculated either by the Dehn or Wirtinger presentations) has two very useful presentations:

* $\langle x,y \mid x y x=y x y\rangle$, which is the [[braid group]], **Br 3**;

*  $\langle a,b | a^2= b^3\rangle$, in which the pair of numbers, $(2,3)$, is apparent.  These reflect the fact that the trefoil is a $(2,3)$-[[torus knot]]. (Of course, it is also a (3,2)-torus knot.)

+-- {: .un_remark}
###### Note
To include one of the above pictures on a page, write <nowiki>[[!include trefoil knot - SVG]]</nowiki> or <nowiki>[[!include trefoil knot (2 bridge) - SVG]]</nowiki>.

category : knot theory
[[!redirects trefoil knot]]
[[!redirects trefoil knots]]
[[!redirects Trefoil knot]]
[[!redirects Trefoil Knot]]
[[!redirects trefoil]]
[[!redirects Trefoil]]


trefoil knot (2 bridge) - SVG

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category: svg


trefoil knot - SVG

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Trevor Hyde

* [personal page](https://math.uchicago.edu/~tghyde/)

## Selected writings

On [[FI-representations]] and [[representation stability]]:

* [[Trevor Hyde]], *$FI$-Modules and Representation Stability* (2016) [[pdf](https://math.uchicago.edu/~tghyde/Hyde%20--%20FI-modules%20and%20representation%20stability.pdf), [[Hyde-FIModules.pdf:file]]]

category: people


triadic relation" > history

< [[triadic relation"]]


trialgebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _trialgebra_ is meant to be a structure at least similar to or involving an [[associative algebra]] but equipped with two further compatible algebra structures, the first additional one making it a [[bialgebra]], the second one making it then a trialgebra.

In the spirit of [[Tannaka duality]], this is to be such that the [[category of modules]] of the associative algebra underlying the trialgebra inherits the structure not just of a [[monoidal category]] (as it does for a bialgebra), but even of a [[Hopf monoidal category]].

Notice that next the [[2-category]] of [[module categories]] over a [[Hopf monoidal category]] inherits the structure of a [[monoidal 2-category]]. This finally has a [[3-category]] of module 2-categories.

In the spirit of _[[n-modules]]_ by iterated internalization, this 3-category may be thought of as the [[4-module]] (over the groun field), presented by the trialgebra.

Accordingly, trialgebras may be thought of as 4-[[bases]] for [[spaces of quantum states]] of [[extended TQFTs|extended]] [[4d TQFTs]] . This has been the main motivation for considering them in ([Pfeiffer 04](#Pfeiffer)) (not quite in the above language), in turn forming a 2-basis for the corresponding [[Hopf monoidal categories]] in ([Crane-Frenkel 04](#CraneFrenkel)). A review of related literature is in ([Baez-Lauda 09, p. 98](#BaezLauda))

## Related concepts

* [[algebra]], [[module]]

  * [[2d TQFT]]

* [[bialgebra]],  [[2-module]]

  * [[quantum group]]

  * [[3d TQFT]]

* **trialgebra**, [[3-module]]

  * [[quantum 2-group]]

  * [[4d TQFT]]

[[!include structure on algebras and their module categories - table]]

## References

A proposal for the definition of trialgebras is in

* [[Hendryk Pfeiffer]], _2-Groups, trialgebras and their Hopf categories of representations_, Adv. Math. 212 No. 1 (2007) 62-108 ([arXiv:math/0411468](http://arxiv.org/abs/math/0411468))
 {#Pfeiffer}

motivated from the definition of [[Hopf monoidal categories]] in

* [[Louis Crane]], [[Igor Frenkel]], _Four dimensional topological quantum field theory, Hopf categories, and the canonical bases_, J.Math.Phys. 35 (1994) 5136-5154, ([arXiv:hep-th/9405183](http://arxiv.org/abs/hep-th/9405183))
 {#CraneFrenkel}

A survey of related references is in p. 98 of

* [[John Baez]], [[Aaron Lauda]], _A prehistory of $n$-categorical physics_, in _Deep beauty_, 13-128, Cambridge Univ. Press, Cambridge, 2011 ([arXiv:0908.2469](http://arxiv.org/abs/0908.2469))
 {#BaezLauda}

[[!redirects trialgebras]]


triality

#Contents#
* table of contents
{:toc}

## Idea

The [[Dynkin diagram]]/[[Dynkin quiver]] [[D4]] has an [[symmetric group|S3]]-[[symmetry group]]

   (graphics grabbed from Wikipedia [here](https://upload.wikimedia.org/wikipedia/commons/5/59/Dynkin_diagram_D4.png))

This manifests itself in "triality" symmetries enjoyed by the objects that are labeled by [[D4]] in an [[ADE-classification]].

Specifically for the [[simple Lie group]] [[SO(8)]] corresponding to [[D4]], triality is the statement that the two [[spin representations]] of the corresponding [[spin group]] $Spin(8)$ and the defining vector representations are all [[isomorphism|isomorphic]], and in a nice way.

## Details

### Triality of subgroups of $Spin(8)$ and $SO(8)$
 {#TrialityOfSubgroupsOfSpin8}

+-- {: .num_prop #Spin7SubgroupsInSpin8}
###### Proposition
**([[Spin(7)]]-[[subgroups]] in [[Spin(8)]])

There are precisely 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[Spin(7)]]-[[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      \mathrm{Spin}(7)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      & \mathrm{Spin}(8)
      &
      \mathrm{Spin}(7)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      \mathrm{Spin}(7)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 6](#Varadarajan01), see also [Kollross 02, Prop. 3.3 (1)](#Kollross02))

+-- {: .num_prop #G2IsIntersectionOfSpin7SubgroupsInSpin8}
###### Proposition
**([[G2]] is [[intersection]] of [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])**

The [[intersection]] inside [[Spin(8)]] of any two [[Spin(7)]]-subgroups from distinct [[conjugacy classes of subgroups]] (according to Prop. \ref{Spin7SubgroupsInSpin8})
is the [[exceptional Lie group]] [[G2]], hence we have [[pullback squares]] of the form

\begin{center}
\begin{xymatrix}
  G_2
   \ar@{^{(}->}[r]
   \ar@{^{(}->}[d]
  &
  \mathrm{Spin}(7)
  \ar@{^{(}->}[d]^-{\iota^\prime}
  \\
  \mathrm{Spin}(7)
   \ar@{^{(}->}[r]_-{\iota}
  &
  \mathrm{Spin}(8)
 \end{xymatrix}
\end{center}

=--

([Varadarajan 01, Theorem 5 on p. 13](#Varadarajan01))

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

We have the following [[commuting diagram]] of [[subgroup]] inclusions, where each [[square]] exhibits a [[pullback]]/[[fiber product]], hence an [[intersection]] of subgroups:

\begin{center}
\begin{xymatrix}
    \mathrm{SU}(2)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{SU}(3)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{G}_2
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[d]^{B \iota'}
    \\
    \mathrm{Spin}(5)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(6)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[r]_-{\iota}
    &
    \mathrm{Spin}(8)
\end{xymatrix}
\end{center}

Here in the bottom row we have the [[Lie groups]]

[[Spin(5)]] $\hookrightarrow$ [[Spin(6)]] $\hookrightarrow$ [[Spin(7)]] $\hookrightarrow$ [[Spin(8)]]

with their canonical [[subgroup]]-inclusions, while in the top row we have

[[SU(2)]] $\hookrightarrow$ [[SU(3)]] $\hookrightarrow$ [[G2]] $\hookrightarrow$ [[Spin(7)]]

and the right vertical inclusion $B \iota'$ is the [[delooping]] of one of the two non-standard inclusions, according to Prop. \ref{Spin7SubgroupsInSpin8}.

=--

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

The square on the right is that from Prop. \ref{G2IsIntersectionOfSpin7SubgroupsInSpin8}.

The square in the middle is [Varadarajan 01, Lemma 9 on p. 10](#Varadarajan01).

The statement also follows with
[Onishchik 93, Table 2, p. 144](#Onishchik93):

\begin{center}
\begin{imagefromfile}
  "file_name": "SO8SubgroupIntersection.jpg",
  "width": 690
\end{imagefromfile}
\end{center}

=--

+-- {: .num_prop #Spin3DotSpin5SubgroupsInSO8}
###### Proposition
**([[Spin(5).Spin(3)]]-[[subgroups]] in [[SO(8)]])

The [[direct product group]] [[SO(3)]] $\times$ [[SO(5)]] together with the groups [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[SO(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[SO(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      Sp(2)\cdot Sp(1)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      &
      SO(8)
      &
      SO(3) \times SO(5)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      Sp(1) \cdot Sp(2)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Kollross 02, Prop. 3.3 (3)](#Kollross02))

Similarly:

+-- {: .num_prop #Spin3DotSpin5SubgroupsInSpin8}
###### Proposition
**([[Spin(5).Spin(3)]]-[[subgroups]] in [[Spin(8)]])

The groups [[Spin(5).Spin(3)]], [[Sp(2).Sp(1)]] and $Sp(1) \cdot Sp(2)$, with their canonical inclusions into [[Spin(8)]], form 3 [[conjugacy classes of subgroups|conjugacy classes]] of [[subgroups]] inside [[Spin(8)]], and the [[triality]] group $Out(Spin(8))$ [[action|acts]] [[transitive action|transitively]] on these three classes.

\begin{center}
  \begin{xymatrix}
      Sp(2)\cdot Sp(1)
      \ar@{^{(}->}[dr]^{\iota'}
      \\
      &
      Spin(8)
      &
      Spin(3) \cdot Spin(5)
      \ar@{_{(}->}[l]^-{\iota}
      \\
      Sp(1) \cdot Sp(2)
      \ar@{^{(}->}[ur]_-{ \iota'' }
  \end{xymatrix}
\end{center}

=--

([Čadek-Vanžura 97, Sec. 2](#CadekVanzura97))

{#Sp2DotSp1SubgroupInclusionsInSummary} In summary we have these subgroup inclusions

\begin{xymatrix}
    \mathrm{Sp}(1) {\cdot} \mathrm{Sp}(2)
    \ar@{=}[dddddr]
    \ar@{_{(}->}[ddrr]|-{ \iota' }
    \\
    \\
    &&
    \mathrm{Spin}(8)
    \ar@{->>}[dddddr]
    &&
    \mathrm{Spin}(3) {\cdot} \mathrm{Spin}(5)
    \ar@{->>}[dddddr]
    \ar@{_{(}->}[ll]|-{\iota}
    \\
    \\
    \mathrm{Sp}(2) {\cdot} \mathrm{Sp}(1)
     \ar@{=}[dddddr]
     \ar@{^{(}->}[uurr]
       |<<<<<<<<<<<<<<<<<<<<<<<<<<{ \phantom{AA \atop AA} }
       |>>>>>>>>>>>>>>>>>>>>>>>{ \iota'' }
    \\
    &
    \mathrm{Sp}(1) {\cdot} \mathrm{Sp}(2)
    \ar@{_{(}->}[ddrr]|-{ \iota' }
    \\
    \\
    &&&
    \mathrm{SO}(8)
    &&
    \mathrm{SO}(3) \times \mathrm{SO}(5)
    \ar@{_{(}->}[ll]|-{\iota}
    \\
    \\
    &
    \mathrm{Sp}(2) {\cdot} \mathrm{Sp}(1)
    \ar@{^{(}->}[uurr]|-{ \iota'' }
\end{xymatrix}

permuted by triality:

\begin{center}
\begin{imagefromfile}
  "file_name": "TrialityOnSp2Sp1.jpg",
  "width": 680
\end{imagefromfile}
\end{center}

> graphics grabbed from [FSS 19, Sec. 3.3](#FSS19)

\linebreak

## References

### General

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 3 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

Discussion of triality of [[subgroups]] of [[Spin(8)]] and [[SO(8)]]:

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, [pdf](https://core.ac.uk/download/pdf/81114499.pdf))

* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures  in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))

* Megan M. Kerr, _New examples of homogeneous Einstein metrics_, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 ([euclid:1030132086](https://projecteuclid.org/euclid.mmj/1030132086))

* {#Kollross02} Andreas Kollross, Prop. 3.3 of _A Classification of Hyperpolar and Cohomogeneity One Actions_, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 ([jstor:2693761](https://www.jstor.org/stable/2693761))

See also

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

* [[John Baez]], _[Spinors and trialities](http://math.ucr.edu/home/baez/octonions/node7.html)_

### In string theory

Exposition with an eye to application in [[string theory]]:

* {#TongTurner19} [[David Tong]], Carl Turner, _Notes on 8 Majorana Fermions_ ([arXiv:1906.07199](https://arxiv.org/abs/1906.07199))

Discussion in the foundations of [[M-theory]]:

* {#FSS19} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3.3 of _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

[[!redirects trialities]]


triangle

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

#Contents#
* table of contents
{:toc}

## Idea

The 2-[[simplex]]. The [[polygon]] with three sides.

## Properties

* [[Pythagorean theorem]]

## Related concepts

* [[trigonometry]]

* [[circle]]

* [[square]]

* [[tetrahedron]]

* [[triangle type]]

[[!redirects triangles]]