natural numbers type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]]: the the _natural numbers type_ is the [[type]] of [[natural numbers]].

## Definition

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

The [[type of natural numbers]] $\mathbb{N}$ is the [[inductive type]] defined by the following [[inference rules]].

1. **[[type formation rule]]**:

   $$
     \frac{}{
       \mathclap{\phantom{\vert^{\vert}}}
       \mathbb{N} \,\colon\, Type
     }
   $$

2. **[[term introduction rules]]**:

   $$
     \frac{}{
       \mathclap{\phantom{\vert^{\vert}}}
       0 \,\colon\, \mathbb{N}
     }
     \;\;\;\;\;\;\;\;
     \frac{
       n \,\colon\, \mathbb{N}
       \mathclap{\phantom{\vert_{\vert}}}
     }{
       \mathclap{\phantom{\vert^{\vert}}}
       succ(n) \,\colon\, \mathbb{N}
     }
   $$

3. **[[term elimination rule]]**:

   $$
     \frac{
       n \,\colon\, \mathbb{N}
         \;\vdash\;
       P(n) \,\colon\, Type
       \;;
       \;\;\;\;
       \vdash\; 0_P \,\colon\, P(0)
       \;;
       \;\;\;\;
       n \,\colon\, \mathbb{N}
       \,,
       \;
       p \,\colon\, P(x)
         \;\vdash\;
       succ_P(n,\,p) \,\colon\, P\big(succ(n)\big)
       \mathclap{\phantom{\vert_{\vert}}}
     }{
       \mathclap{\phantom{\vert^{\vert}}}
       n \,\colon\, \mathbb{N}
       \;\vdash\;
       ind_{(P, 0_P, succ_P)}(n)
        \,\colon\,
       P(n)
     }
   $$

4. **[[computation rules]]**:

   $$
     \frac{
       n \,\colon\, \mathbb{N}
        \;\vdash\;
       P(n) \,\colon\, Type
       \;;
       \;\;\;\;
         \vdash\;
       0_P \,\colon\, P(0)
       \;;
       \;\;\;\;
       n \,\colon\, \mathbb{N}
       \,,
       \;
       p \,\colon\, P(x)
         \;\vdash\;
       succ_P(n,p) \,\colon\, P\big(succ(n)\big)
       \mathclap{\phantom{\vert_{\vert}}}
     }{
       \mathclap{\phantom{\vert^{\vert}}}
       ind_{(P, 0_P, succ_P)}(0)
        \,=\,
       0_P
     }
   $$

   and

   $$
     \frac{
       n \,\colon\, \mathbb{N}
         \;\vdash\;
       P(x) \,\colon\, Type
       \;;
       \;\;\;\;
       \vdash\; 0_P \,\colon\, P(0)
       \;;
       \;\;\;\;
       n \,\colon\, \mathbb{N}
       \,,
       \;
       p \,\colon\, P(x)
         \;\vdash\;
       succ_P(x,p) \,\colon\, P(s x)
       \;;
       \;\;\;\;
       \vdash\; n \,\colon\, \mathbb{N}
       \mathclap{\phantom{\vert_{\vert}}}
     }{
       \mathclap{\phantom{\vert^{\vert}}}
       ind_{(P, 0_P, succ_P)}\big(succ(n)\big)
       \,=\,
       succ_P\big(n,\, ind_{(P, 0_P, succ_P)}(n) \big)
     }
   $$

(In the last line, "$=$" denotes [[judgemental equality]].)

=--

That this is the right definition (and a special case of the general principle of [[inductive types]]) was clearly understood around [Martin-Löf (1984)](#Martin-Löf84), [pp. 38](/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=44); [Coquand & Paulin (1990, p. 52-53)](#CoquandPaulin90); [Paulin-Mohring (1993, §1.3)](#Paulin-Mohring93); [Dybjer (1994, §3)](#Dybjer94). For review see also, e.g., [Pfenning (2009, §2)](#Pfenning); [UFP (2013, §1.9)](#UFP13); [Söhnen (2018, §2.4.5)](#Söhnen18).

In [[Coq]]-[[syntax]] the [[natural numbers]] are the [[inductive type]] defined [cf. [Paulin-Mohring (2014, p. 6)](#Paulin-Mohring14)] by:

    Inductive nat : Type :=
     | zero : nat
     | succ : nat -> nat.

In the [[categorical semantics]] (via the [[categorical model of dependent types]], see [below](#CategoricalSemantics))
this is interpreted as the [[initial algebra for an endofunctor|initial algebra]] for the [[endofunctor]] $F$ that sends an object to its [[coproduct]] with the [[terminal object]]

\[
  \label{TheEndofunctor}
  F(X) \;\coloneqq\; * \sqcup X
  \,,
\]

or in different equivalent notation, which is very suggestive here:

$$
  F(X) \;=\; 1 + X
  \,.
$$

That [[initial algebra for an endofunctor|initial algebra]] is (as also explained [there](#initial+algebra+of+an+endofunctor#NaturalNumbers)) precisely a [[natural number object]] $\mathbb{N}$. The two components of the morphism $F(\mathbb{N}) \to \mathbb{N}$ that defines the algebra structure are the 0-[[generalized element|element]] $0 \,\colon\, * \to \mathbb{N}$ and the [[successor]] [[endomorphism]] $succ \colon \mathbb{N} \to \mathbb{N}$

$$
  (0 ,\, succ)
  \;\colon\;
  * \sqcup \mathbb{N}
  \longrightarrow
  \mathbb{N}
  \,.
$$

### With typal computation and uniqueness rules

Assuming that [[identification types]], [[function types]] and [[dependent sequence types]] exist in the type theory, the natural numbers type is the [[inductive type]] generated by an element and a [[function]]:

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

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

Elimination rules for the natural numbers type:
$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x)) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C(c_0, c_s, n):C(n)}$$

Computation rules for the natural numbers type:

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))}{\Gamma \vdash \beta_\mathbb{N}^0(c_0, c_s):\mathrm{Id}_{C(0)}\mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0), c_0)}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_0:C(0) \quad \Gamma \vdash c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x)) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \beta_\mathbb{N}^s(c_0, c_s, n):\mathrm{Id}_{C(s(n))}(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(n)), c_s(n)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, n)))}$$

Uniqueness rules for the natural numbers type:

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathbb{N}} C(x) \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \eta_\mathbb{N}(c, n):\mathrm{Id}_{C(n)}(\mathrm{ind}_\mathbb{N}^C(c(0), \lambda x:\mathbb{N}.c(s(x)), n), c(n))}$$

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

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

The dependent universal property of the natural numbers is used to characterize the [[dependent product type]] of an type family $C(x)$ dependent on $x:\mathbb{N}$, and states that the [[fibers]] of the function
$$\lambda c.(c(0), \lambda x.c(s(x))):\prod_{x:\mathbb{N}} C(x) \to \left(C(0) \times \prod_{x:\mathbb{N}} C(x) \to C(s(x))\right)$$
are [[contractible types]]. This is equivalent to saying that the above function is an [[equivalence of types]]:

$$\mathrm{isEquiv}(\lambda c.(c(0), \lambda x.c(s(x))))$$

The non-dependent universal property similarly says that given a type $C$ the function
$$\lambda c.(c(0), \lambda x.c(s(x))):(\mathbb{N} \to C) \to \left(C \times (\mathbb{N} \to C \to C)\right)$$
is an equivalence of types
$$\mathrm{isEquiv}(\lambda c.(c(0), \lambda x.c(s(x))))$$

### Generalized induction principle

There is also a generalized induction principle (cf. the talk slides in [LumsdaineShulman17](#LumsdaineShulman17)), which uses a type $C$ and a function $f:C \to \mathbb{N}$ instead of a type family $x:\mathbb{N} \vdash P(x)$, and one uses the [[fiber]] $\sum_{z:C} f(z) =_\mathbb{N} n$ to express the generalized induction principle.

Then the induction principle states that given a type $C$ and a function $f:C \to \mathbb{N}$ along with

* dependent pair

$$c_0:\sum_{z:C} f(z) =_\mathbb{N} 0$$

* dependent function

$$c_s:\prod_{n:\mathbb{N}} \left(\sum_{z:C} f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)$$

* and natural number $n:\mathbb{N}$

one could construct the dependent pair

$$\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, n):\sum_{z:C} f(z) =_\mathbb{N} n$$

such that

$$\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, 0) \equiv c_0$$

and for all $n:\mathbb{N}$

$$\mathrm{ind}_\mathbb{N}^C(f, c_0, c_s, s(n)) \equiv c_s(n, \mathrm{ind}(f, c_0, c_s, n))$$

However, by the rules of dependent pair types, one could instead postulate separate elements and identifications instead of an element of a [[fiber type]] throughout the generalized principle.

Instead of the dependent pair $c_0:\sum_{z:C} f(z) =_\mathbb{N} 0$ we have the element $c_0:C$ and identificaiton $p_0:f(c_0) =_\mathbb{N} 0$, where the original element is given by $(c_0, p_0)$. In addition, given the dependent type

$$c_s:\prod_{n:\mathbb{N}} \left(\sum_{z:C} f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)$$

by currying this is equivalent to

$$c_s:\prod_{n:\mathbb{N}} \prod_{z:C} \left(f(z) =_\mathbb{N} n\right) \to \left(\sum_{z:C} f(z) =_\mathbb{N} s(n)\right)$$

and by the [[type theoretic axiom of choice]] this is equivalent to

$$c_s:\prod_{n:\mathbb{N}} \prod_{y:C} \sum_{g:(f(y) =_\mathbb{N} n) \to C} \prod_{p:f(y) =_\mathbb{N} n} f(g(p)) =_\mathbb{N} s(n)$$

By the rules of dependent pair types, the family of dependent pair types could be split up into

$$c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C$$

$$p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)$$

where the original dependent funciton is given by

$$\lambda n:\mathbb{N}.\lambda y:C.(c_s(n, y), p_s(n, y))$$

Then the induction principle of the natural numbers states that given a type $C$ and a function $f:C \to \mathbb{N}$, along with

* an element $c_0:C$

* an identification $p_0:f(c_0) =_\mathbb{N} 0$

* dependent functions

$$c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C$$

$$p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)$$

we have a function

$$\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s):\mathbb{N} \to C$$

and a homotopy

$$\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s):\prod_{n:\mathbb{N}} f(\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n)) =_\mathbb{N} n$$

indicating that $\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s)$ is a [[section]] of $f$, such that

$$\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, 0) \equiv c_0$$

$$\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, 0) \equiv p_0$$

and for all $n:\mathbb{N}$,

$$\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, s(n)) \equiv c_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))$$

$$\mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, s(n)) \equiv p_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))$$

As inference rules these are given by the following:

**elimination rules**:
$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s):\mathbb{N} \to C}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s):\prod_{n:\mathbb{N}} f(\mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n)) =_\mathbb{N} n}$$

**computation rules**:
$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, 0) \equiv c_0}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n)
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, 0) \equiv p_0}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \quad \Gamma \vdash n:\mathbb{N}
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, s(n)) \equiv c_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash f:C \to \mathbb{N} \quad \Gamma \vdash c_0:C \quad \Gamma \vdash p_0:f(c_0) =_\mathbb{N} 0 \\
\Gamma \vdash c_s:\prod_{n:\mathbb{N}} \prod_{y:C} (f(y) =_\mathbb{N} n) \to C \quad \Gamma \vdash p_s:\prod_{n:\mathbb{N}} \prod_{y:C} \prod_{p:f(y) =_\mathbb{N} n} f(c_s(n, y, p)) =_\mathbb{N} s(n) \quad \Gamma \vdash n:\mathbb{N}
\end{array}
}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, s(n)) \equiv p_s(n, \mathrm{ind}_\mathbb{N}^{C}(f, c_0, p_0, c_s, p_s, n), \mathrm{ind}_\mathbb{N}^{C, \mathrm{sec}}(f, c_0, p_0, c_s, p_s, n))}$$

One gets back the usual induction principle of the natural numbers type when $C \equiv \sum_{n:\mathbb{N}} P(n)$ and $f \equiv \pi_1$ the first projection function of the dependent sum type, and one gets back the recursion principle of the natural numbers type when $C \equiv \mathbb{N} \times X$ and $f \equiv \pi_1$ the first projection function of the product type.

### Extensionality principle of the natural numbers

First we [[inductive definition|inductively define]] a [[binary function]] into the [[boolean domain]] called [[observational equality]] of the natural numbers:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Eq}_\mathbb{N}:\mathbb{N} \times \mathbb{N} \to \mathrm{Bit}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \delta^{0, 0}:\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(0, 0), 1)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \delta^{0, s}(n):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(0, s(n)), 0)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N} \vdash \delta^{s, 0}(m):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(s(m), 0), 0)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta^{s, s}(m, n):\mathrm{Id}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(s(m), s(n)),\mathrm{Eq}_\mathbb{N}(m, n))}$$

The extensionality principle of the natural numbers states that the natural numbers has [[decidable equality]] given by observational equality:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta(m, n):\mathrm{Id}_\mathbb{N}(m, n) \simeq \mathrm{El}_\mathrm{Bit}(\mathrm{Eq}_\mathbb{N}(m, n))}$$

or equivalently

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, m:\mathbb{N}, n:\mathbb{N} \vdash \delta(m, n):\mathrm{Id}_\mathbb{N}(m, n) \simeq \mathrm{Id}_\mathbb{2}(\mathrm{Eq}_\mathbb{N}(m, n), 1)}$$

## Properties

### General

\begin{example}\label{NaturalNumbersAsWType}
**([[natural numbers type]] as a [[W-type|$\mathcal{W}$-type]])**
\linebreak
The [[natural numbers type]] $(\mathbb{N},\, 0,\, succ)$ (Def. \ref{InferenceRules}) is equivalently the [[W-type|$\mathcal{W}$-type]] $\underset{c \colon C}{\mathcal{W}} A(c)$ with:

* $C \,\coloneqq\, \{0, succ\} \,\simeq\, \ast \sqcup \ast$;

* $A_0 \,\coloneqq\, \varnothing$ ([[empty type]]);

  $A_{succ} \,\coloneqq\, \ast$ ([[unit type]])

\end{example}
[[Martin-Löf (1984)](#Martin-Löf84), [pp. 45](/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=51), [Dybjer (1997, p. 330, 333)](W-type#Dybjer97)]

### Relation to the type of finite types

The natural numbers type is equivalent to the [[set truncation]] of the [[type of finite types]]:

$$\mathbb{N} \simeq [\mathrm{FinType}]_0$$

This is the type theoretic analogue of the [[decategorification]] of the [[permutation category]] resulting in the set of [[natural numbers]].

This gives us an alternate definition of the natural numbers as the type of finite types

$$\mathbb{N} \coloneqq [\mathrm{FinType}]_0$$

One has $[-]_0:\mathrm{FinType} \to [\mathrm{FinType}]_0$ by the introduction rules of set truncation.

The arithmetic operations and order relations on the natural numbers type can be defined by induction on [[set truncation]]:

For all finite types $A:\mathrm{FinType}$ and $B:\mathrm{FinType}$ and finite families $C:A \to \mathrm{FinType}$, we have

$$0 =_\mathbb{N} [\emptyset]_0 \quad 1 =_\mathbb{N} [\mathbb{1}]_0$$
$$[A]_0 + [B]_0 =_\mathbb{N} [A + B]_0$$
$$\sum_{x = 1}^{[A]_0} [C]_0(x) =_\mathbb{N} \left[\sum_{x:A} C(x)\right]_0$$
$$[A]_0 \cdot [B]_0 =_\mathbb{N} [A \times B]_0$$
$$\prod_{x = 1}^{[A]_0} [C]_0(x) =_\mathbb{N} \left[\prod_{x:A} C(x)\right]_0$$
$$[B]_0^{[A]_0} =_\mathbb{N} [A \to B]_0$$

$$[A]_0 =_\mathbb{N} [B]_0 \coloneqq [A \simeq B]_{(-1)} \; \mathrm{or} \; [A =_\mathrm{FinType} B]_{(-1)}$$
$$[A]_0 \leq [B]_0 \coloneqq [A \hookrightarrow B]_{(-1)}$$

### Categorical semantics
 {#CategoricalSemantics}

We spell out how under the canonical [[categorical model of dependent types]], the [[categorical semantics]] of the natural numbers types yields a [[natural numbers object]] together with its expected [[recursion]] and [[induction]] principle.

\linebreak

Throughout, we consider an ambient [[category]] $\mathcal{C}$ (e.g. $\mathcal{C} =$ [[Set]]) and write

\[
  \label{CategoryOfFAlgebras}
  F Alg(\mathcal{C}) \xrightarrow{underlying} \mathcal{C}
\]

for the category of [[algebra over an endofunctor|algebras over the endofunctor]] $F$ (eq:TheEndofunctor).

#### Recursion
 {#Recursion}

We spell out how the fact that $\mathbb{N}$ satisfies Def. \ref{InferenceRules} is the classical [[recursion principle]].

\linebreak

We begin with a simple special case of recursion (cf. Rem. \ref{NeedForDependentRecursion}), where not only the underlying type but also its successor-map is independent of $\mathbb{N}$ (we come to the general form of recursion further [below](#RecursionGenerally)).

So consider any $F$-algebra $\big(D, (0_D, succ_D)\big) \,\in\, F Alg(\mathcal{C})$ (eq:CategoryOfFAlgebras), hence an [[object]] $D \in \mathcal{C}$ equipped with a morphism

$$
  0_D \colon * \to D
$$

and a morphism

$$
  succ_D \colon D \to D
  \,.
$$

By [[initial object|initiality]] of the $F$-algebra $\mathbb{N}$, there is then a (unique) morphism

$$
  rec \,\colon\, \mathbb{N} \to D
$$

such that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    *
    &\stackrel{0}{\longrightarrow}&
    \mathbb{N}
    &\stackrel{succ}{\longrightarrow}&
    \mathbb{N}
    \\
    \big\downarrow
    &&
    \big\downarrow\mathrlap{^\mathrlap{rec}}
    &&
    \big\downarrow\mathrlap{^\mathrlap{rec}}
    \\
    *
    &\underset{0_D}{\longrightarrow}&
    D
    &\underset{succ_D}{\longrightarrow}&
    D
  }
$$

This means precisely that $rec$ is the function [[recursive definition|defined recursively]] by

\[
  rec(0) \;=\; 0_D
\]
and
\[
  \label{FormulaForNonDependentRecursion}
  rec\big(succ(n)\big)
   \;=\;
  succ_D\big(rec(n)\big)
  \,.
\]

{#RecursionGenerally} More generally, consider an $F$-algebra in the [[slice category|slice]] over $\big(\mathbb{N}, (0,succ)\big)$, but with the [[underlying]] slice object assumed (dropping also this assumption leads to the fully general notion of induction further [below](#Induction)) to be independent of $\mathbb{N}$, hence of the form

\[
  \label{AssumingUnderlyingScliceObjectToBeIndependent}
  \left[
  \array{
    \mathbb{N} \times D
    \\
    \big\downarrow\mathrlap{^{pr_{\mathbb{N}}}}
    \\
    \mathbb{N}
  }
  \;\;\;
  \right]
  \;\in\;
  \mathcal{C}_{/\mathbb{N}}
  \,,
\]

while the $F$-algebra structure may now depend on $\mathbb{N}$:

\begin{tikzcd}[
  column sep=56pt,
  row sep=30pt
]
    \ast \,\sqcup\, (\mathbb{N} \times D)
    \ar[
      rr,
      "{
        \big(
          (0, \mathrm{succ} \,\circ\, pr_{\mathbb{N}})
          ,\,
          (0_D, \mathrm{succ}_D)
        \big)
      }"
    ]
    \ar[d, "{ \ast \,\sqcup\, \mathrm{pr}_{\mathbb{N}} }"]
    &&
    \mathbb{N} \times D
    \ar[d, "{ \mathrm{pr}_{\mathbb{N}} }"]
    \\
    \ast \,\sqcup\, \mathbb{N}
    \ar[rr, "{(0, \mathrm{succ})}"{swap}]
    &&
    \mathbb{N}
\end{tikzcd}

in that

$$
  succ_D \;\colon\; \mathbb{N} \times D \to D
$$

may depend on $\mathbb{N}$.

Now, since with $\big(\mathbb{N},(0,\mathrm{succ})\big)$ being the [[initial object]] in $F Alg(\mathcal{C})$, the [[identity morphism]] on $\big(\mathbb{N},(0,\mathrm{succ})\big)$ is the [[initial object]] in the [[slice category]] $F Alg(\mathcal{C})_{/\big(\mathbb{N}, (0,succ)\big)}$ (cf. [there](over+category#InitialAndTerminalObjects)), it follows that from such data is induced a unique morphism $f$ in the following [[commuting diagram]]:

\begin{tikzcd}[
    column sep=34pt,
    row sep=25pt
  ]
    &[+34pt]
    &
    \ast \,\sqcup\, \mathbb{N}
    \ar[
      ddll,
      "{
        (0,\mathrm{succ})
      }"{sloped}
    ]
    \ar[
      dr,
      equals,
      shorten >=-2pt
     ]
    \ar[
      r,
      dashed,
      shorten=-1pt,
      "{
        \mathrm{id}_\ast
        \,\sqcup\,
        (\mathrm{id}_{\mathbb{N}},\, \mathrm{rec})
      }"
    ]
    &[+16pt]
    \ast \,\sqcup\, (\mathbb{N} \times D)
    \ar[
      d,
      "{
        \mathrm{id}_\ast
          \,\sqcup\,
        \mathrm{pr}_{\mathbb{N}}
      }"
    ]
    \\
    &
    &&
    \ast \,\sqcup\, \mathbb{N}
    \ar[
      ddll,
      "{
        (0, \mathrm{succ})
      }"{description, sloped}
    ]
    \\[+10pt]
    \mathbb{N}
    \ar[dr, equals]
    \ar[
      r,
      dashed,
      "{
        (\mathrm{Id}_{\mathbb{N}},\, \mathrm{rec})
      }"{description}
    ]
    &
    \mathbb{N} \times D
    \ar[d]
    \ar[
      from=uurr,
      crossing over,
      "{
        \big(
          (
            0,
            \,
            \mathrm{succ}
              \,\circ\,
             \mathrm{pr}_{\mathbb{N}})
          ,\,
          (0_D,\, \mathrm{succ}_D)
        \big)
      }"{description, sloped, pos=.6}
    ]
    \\
    &
    \mathbb{N}
\end{tikzcd}

Here the [[commuting diagram|commutativity]] of the top square means equivalently that

\[
  \mathrm{rec}(0) \,=\, 0_D
\]
and
\[
\label{FormulaForDependentRecursion}
  \mathrm{rec}\big(
    \mathrm{succ}(n)
  \big)
  \;=\;
  \mathrm{succ}_D\big(n,\, \mathrm{rec}(n)\big)
  \,.
\]

\begin{remark}\label{NeedForDependentRecursion}
(**the need for dependent recursion** [[cf. Paulin-Mohring (1993, p. 330)](#Paulin-Mohring93)])
\linebreak
  The appearance of the argument "$n$" on the right of (eq:FormulaForDependentRecursion) -- in contrast to formula (eq:FormulaForNonDependentRecursion) for non-dependent recursion -- means (in view of the argument "$succ(n)$" on the left) that the recursor $succ_D$ has access to the *[[predecessor]]* [[partial function]] $pred \,\colon\,succ(n) \,\mapsto\, n$. This is necessary in order to express all computable functions on the natural numbers inductively and hence explains the need for the [[dependent type|dependently typed]] recursion principle (eq:AssumingUnderlyingScliceObjectToBeIndependent)
\end{remark}

#### Induction
 {#Induction}

Dropping the above constraint (eq:AssumingUnderlyingScliceObjectToBeIndependent) on the dependent $F$-algebra, we spell out in detail how the fact that $\mathbb{N}$ satisfied Def. \ref{InferenceRules} is the classical [[induction principle]].

That principle says informally that if a [[proposition]] $P$ depending on the natural numbers is true at $n = 0$ and such that if it is true for some $n$ then it is true for $n+1$, then it is true for all natural numbers.

Here is how this is formalized in type theory and then [[categorical semantics|interpreted]] in some suitable ambient category $\mathcal{C}$.

First of all, that $P$ is a proposition depending on the natural numbers means that it is a [[dependent type]]

$$
  n \,\colon\, \mathbb{N}
  \;\;\vdash\;\;
  P(n) \,\colon\, Type
  \,.
$$

The categorical interpretation of this is by a [[display map]]

\[
  \label{DisplayMapInterpretingNDependentType}
  \array{
     P
     \\
     \big\downarrow\mathrlap{^{\pi_P}}
     \\
     \mathbb{N}
  }
\]

in the given category $\mathcal{C}$.

{#DiagramInterpretingInductionPrinciple} With this, the [[commuting diagram]] which interprets the induction principle is the following:

\begin{tikzcd}[
    column sep=30pt,
    row sep=25pt
  ]
    &[+34pt]
    &
    \ast \,\sqcup\, \mathbb{N}
    \ar[
      ddll,
      "{
        (0,\mathrm{succ})
      }"{sloped}
    ]
    \ar[
      dr,
      equals,
      shorten >=-2pt
     ]
    \ar[
      r,
      dashed,
      shorten=-1pt,
      "{
        \mathrm{id}_\ast
        \,\sqcup\,
        \mathrm{ind}
      }"
    ]
    &[+16pt]
    \ast \,\sqcup\, P
    \ar[
      d,
      "{
        \mathrm{id}_\ast
          \,\sqcup\,
        \pi_P
      }"
    ]
    \\
    &
    &&
    \ast \,\sqcup\, \mathbb{N}
    \ar[
      ddll,
      "{
        (0, \mathrm{succ})
      }"{description, sloped}
    ]
    \\[+10pt]
    \mathbb{N}
    \ar[dr, equals]
    \ar[
      r,
      dashed,
      "{
        \mathrm{ind}
      }"{description}
    ]
    &
    P
    \ar[
      d,
      "{ \pi_P }"
    ]
    \ar[
      from=uurr,
      crossing over,
      "{
        (0_P,\, \mathrm{succ}_P)
      }"{description, sloped, pos=.6}
    ]
    \\
    &
    \mathbb{N}
\end{tikzcd}

We now unwind again how this comes about and what it all means:

First, the fact that $P$ holds at 0 means that there is a ([[proof]]-)[[term]]

$$
  \vdash \;\; 0_P \,\colon\, P(0)
  \,.
$$

In the [[categorical semantics]] the [[substitution]] of $n$ for 0 that gives $P(0)$ is given by the [[pullback]] of the given display map (eq:DisplayMapInterpretingNDependentType):

$$
  \array{
    0^* P &\longrightarrow& P
    \\
    \big\downarrow && \big\downarrow
    \\
    * &\underset{0}{\longrightarrow}& \mathbb{N}
  }
$$

and the [[term]] $0_P$ is interpreted as a [[section]] of the resulting fibration over the terminal object

$$
  \array{
    *
    &\overset{p_0}{\longrightarrow}&
    0^* P
    & \longrightarrow &
    P
    \\
    &\searrow& \big\downarrow && \big\downarrow
    \\
    &&
    *
    &\underset{0}{\longrightarrow}& \mathbb{N}
  }
  \,.
$$

But by the defining [[universal property]] of the [[pullback]], this is equivalently just a [[commuting diagram]]

$$
  \array{
    * &\stackrel{p_0}{\longrightarrow}& P
    \\
    \big\downarrow && \big\downarrow
    \\
    * &\underset{0}{\longrightarrow}& \mathbb{N}
  }
  \,.
$$

Next the induction step. Formally it says that for all $n \in \mathbb{N}$ there is an [[implication]] $succ_P(n) \,\colon\, P(n) \to P(n+1)$

$$
  n \in \mathbb{N}
  \;\;\;\vdash\;\;\;
  succ_P(n) \,\colon\, P(n) \to   P(n+1)
  \,.
$$

The [[categorical semantics]] of the [[substitution]] of $n+1$ for $n$ is given by the [[pullback]]

$$
  \array{
     P\big(\ast \sqcup (-)\big) \coloneqq
     & s^*P &\longrightarrow& P
     \\
     & \big\downarrow && \big\downarrow
     \\
     & \mathbb{N} &\underset{s}{\longrightarrow}& \mathbb{N}
  }
$$

and the interpretation of the implication term $succ_P(n)$ is as a [[morphism]] $P \to s^* P$ in $\mathcal{C}_{/\mathbb{N}}$

$$
  \array{
     P
     & \overset{p_s}{\longrightarrow} &
     s^*P &\longrightarrow& P
     \\
     &\searrow & \big\downarrow && \big\downarrow
     \\
     && \mathbb{N} &\overset{s}{\longrightarrow}& \mathbb{N}
  }
  \,.
$$

Again by the [[universal property]] of the pullback this is equivalently a [[commuting diagram]]

$$
  \array{
     P &\overset{succ_P}{\longrightarrow}& P
     \\
     \big\downarrow && \big\downarrow
     \\
     \mathbb{N}
     &\underset{s}{\longrightarrow}&
     \mathbb{N}
  }
  \,.
$$

In summary this shows that

* $P$ being a proposition depending on natural numbers which holds at 0 and which holds at $n+1$ if it holds at $n$

is interpreted precisely as an [[algebra for an endofunctor|endofunctor-algebra homomorphism]]

$$
  \array{
     P
     \\
     \downarrow
     \\
     \mathbb{N}
  }
  \,.
$$

for the [[endofunctor]] $F$ (eq:TheEndofunctor).

The [[induction principle]] is supposed to deduce from this that $P$ holds for every $n$, hence that there is a proof $ind(n) \colon P(n)$ for all $n$:

$$
  n \,\colon\, \mathbb{N}
  \;\;\;\vdash\;\;\;
  ind(n) \,\colon\, P(n)
  \,.
$$

The categorical interpretation of this is as a morphism $p \,\colon\, \mathbb{N} \to P$ in $\mathcal{C}_{/\mathbb{N}}$. The existence of this is indeed exactly what the interpretation of the elimination rule (Def. \ref{InferenceRules}) gives exactly what the initiality of the $F$-algebra $\mathbb{N}$ gives.

## Related concepts

* [[natural numbers]], [[natural numbers object]]

* [[decimal numeral representation of the natural numbers]]

* [[dependent sequence type]]

* [[type of finite types]]

## References

Original articles with emphasis on the nature of $\mathbb{N}$ as an [[inductive type]]:

* {#Martin-Löf84} [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), [pp. 38](/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=44) of: _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

* {#CoquandPaulin90} [[Thierry Coquand]], [[Christine Paulin]], p. 52-53 in:  *Inductively defined types*, COLOG-88 Lecture Notes in Computer Science **417**, Springer (1990) 50-66 [[doi:10.1007/3-540-52335-9_47](https://doi.org/10.1007/3-540-52335-9_47)]

* {#Paulin-Mohring93} [[Christine Paulin-Mohring]], §1.3 in: *Inductive definitions in the system Coq -- Rules and Properties*, in: *Typed Lambda Calculi and Applications* TLCA 1993, Lecture Notes in Computer Science **664** Springer (1993) [[doi:10.1007/BFb0037116](https://doi.org/10.1007/BFb0037116)]

* {#Dybjer94} [[Peter Dybjer]], §3 in: *Inductive families*, Formal Aspects of Computing **6** (1994) 440–465 $[$[doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [pdf](http://www.cse.chalmers.se/~peterd/papers/Inductive_Families.pdf)$]$

The syntax in [[Coq]]:

* {#Paulin-Mohring14} [[Christine Paulin-Mohring]], p. 6 in: *Introduction to the Calculus of Inductive Constructions*, contribution to: *Vienna Summer of Logic* (2014) [[hal:01094195](https://hal.inria.fr/hal-01094195), [pdf](https://hal.inria.fr/hal-01094195/document), [pdf slides](https://easychair.org/smart-program/VSL2014/APPA-invited-slides-6.pdf)]

See also:

* {#Pfenning} [[Frank Pfenning]], _Lecture notes on natural numbers_ (2009) [[pdf](http://www.cs.cmu.edu/~fp/courses/15317-f09/lectures/06-nat.pdf), [[Pfenning-NaturalNumbersType.pdf:file]]]

Discussion in a context of [[homotopy type theory]] and in view of [[higher inductive types]]:

* {#UFP13} [[Univalent Foundations Project]], §1.9 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Egbert Rijke]], §3 in: *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

* {#Söhnen18} Kajetan Söhnen, §2.4.5 in: *Higher Inductive Types in Homotopy Type Theory*, Munich (2018) [[pdf](https://www.math.lmu.de/~petrakis/Soehnen.pdf), [[Soehnen-HigherInductiveTypes.pdf:file]]]

* {#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)]

Equivalence to binary presentations:

* [[Nicolas Magaud]], [[Yves Bertot]], *Changing Data Structures in Type Theory: A Study of Natural Numbers*, in *Types for Proofs and Programs. TYPES 2000*, Lecture Notes in Computer Science **2277** [[doi:10.1007/3-540-45842-5_12](https://doi.org/10.1007/3-540-45842-5_12), [pdf](https://dpt-info.u-strasbg.fr/~magaud/papers/types2000-nmagaud.pdf)]

* [[Nicolas Magaud]], *Changing Data Representation within the Coq*, in *Theorem Proving in Higher Order Logics. TPHOLs 2003*, Lecture Notes in Computer Science **2758** [[doi:10.1007/10930755_6](https://doi.org/10.1007/10930755_6)]

That one can construct the [[natural numbers type]] from the integers type can be found in:

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

[[!redirects natural number type]]
[[!redirects natural-number type]]
[[!redirects natural numbers type]]
[[!redirects natural-numbers type]]
[[!redirects type of natural numbers]]

[[!redirects universal property of the natural numbers type]]
[[!redirects dependent universal property of the natural numbers type]]

[[!redirects universal property of the natural numbers]]
[[!redirects dependent universal property of the natural numbers]]


This page collects links related to

* [[Ivan Kolář]], [[Jan Slovák]], [[Peter Michor]]

  \linebreak

  **Natural operations in differential geometry**

  \linebreak

  Springer 1993

  \linebreak

  [book webpage](https://www.emis.de/monographs/KSM/)

  [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3)

  [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)

on [[differential geometry]] with particular emphasis on [[natural bundles]] and [[natural operations]].

## Chapter I. Manifolds and Lie groups

* [[smooth manifold]]

* [[Lie group]]

## Chapter II. Differential forms

* [[differential form]]

## Chapter III Bundles and connections

* [[fiber bundle]], [[principal bundle]]

* [[connection on a bundle]]

## Chapter IV. Jets and natural bundles

* [[jet]], [[jet bundle]]

* [[higher order frame bundle]]

* [[jet group]]

* [[natural bundle]]

## Chapter V. Finite order theorems

* [[Frölicher–Nijenhuis bracket]]

* [[Schouten-Nijenhuis bracket]]

## Chapter VI. Methods for finding natural operators

## Chapter VII. Further operations

## Chapter VIII.  Product preserving functors

* [[embedding of smooth manifolds into formal duals of R-algebras]]

## Chapter IX. Bundle functors on manifolds

## Chapter X. Prolongation of vector fields and connections

## Chapter XI. General theory of Lie derivatives

## Chapter XII. Gauge natural bundles and operators

category: reference

[[!redirects Natural operators in differential geometry]]


natural parametrization

A [[smooth map|smooth]] (or [[rectifiable curve|rectifiable]]) [[curve]] $s: I\to X$, where $I\subset \mathbb{R}$ is an interval and $X$ is a [[Riemannian manifold]], is **naturally parametrized**, or **parametrized by arclength**, if the [[arclength]] of the arc from $s(t)$ to $s(t')$ is $t' - t$, for any $t\lt t'$, $t,t'\in I$. In other words, the parameter is the length of the curve counted from a given fixed point on the curve. Many classical formulas in the geometry of curves (where $X$ is typically $\mathbb{R}^n$ with its usual metric) assume natural parametrization; otherwise the analogues in terms of general parametrizations become far more complicated.

In particular, [[geodesic]]s are generally taken to be naturally parametrized.

[[!redirects natural parametrization]]
[[!redirects natural parametrisation]]
[[!redirects naturally parametrized curve]]
[[!redirects naturally parametrised curve]]

[[!redirects parametrization by arclength]]
[[!redirects parametrisation by arclength]]
[[!redirects parametrization by arc length]]
[[!redirects parametrisation by arc length]]


natural philosophy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Natural philosophy
* table of contents
{: toc}

## Idea

Before [[mathematics]] was found to be 'the language with which God has written the universe' ([[Galilei]], [Il Saggiatore, 1623](Galileo+Galilei#TheBookOfnature)) (eventually via [[Newton's laws of motion]]), the study of the [[phenomenology]] of [[physics]] was pursued by more informal means, and went by the name _natural philosophy_, that is the [[philosophy]] of [[nature]].  (Compare 'natural history' for [[biology]], although that term is still used for observational biology.)

## Related entries

* [[philosophy of physics]]

* [[Georg Hegel]], _[[Philosophy of Nature]]_


natural transformation

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

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[functor|functor]] is a [[morphism]] between [[category|categories]], a natural transformation is a [[2-morphism]] between two functors.

Natural transformations are the [[2-morphism]]s in the [[2-category]] [[Cat]].

## Definition ##

### Explicit definition

\begin{definition}
**(natural transformations)**
\linebreak
Given [[categories]] $C$ and $D$ and [[functors]] $F,G \colon C \to D,$ a __natural transformation__ $\alpha \colon F \Rightarrow G$ between them, denoted

\begin{xymatrix}
  C
  \ar@/^2pc/[rr]^-{ F }_-{\ }="s"
  \ar@/_2pc/[rr]_-{ G }^-{\ }="t"
  &&
  D
  %
  \ar@{=>}^-{\alpha} "s"; "t"
\end{xymatrix}

is an assignment to every [[object]] $x$ in $C$ of a [[morphism]]
$\alpha_x:F(x) \to G(x)$ in $D$ (called the __component__ of $\alpha$ at $x$) such that for any morphism $f:x \to y$ in $C$, the following [[diagram]] [[commuting diagram|commutes]] in $D$:

\begin{tikzcd}
    F(x) \arrow[r, "F(f)"] \arrow[d, "\alpha_x"'] & F(y) \arrow[d, "\alpha_y"] \\
    G(x) \arrow[r, "G(f)"']                        & G(y)
\end{tikzcd}

\end{definition}

\begin{remark}
**([[composition]] of natural transformations)**
Natural transformations between functors $C \to D$ and $D \to E$ compose in the obvious way to natural transformations $C \to E$ (this is their [[vertical composition]] in the [[2-category]] [[Cat]])
and functors $F : C \to D$ with natural transformations between them form the [[functor category]]

\[
  \label{FunctorCategory}
  [C,D] \;\in\; Cat
\]

The notation alludes to the fact that this makes [[Cat]] a [[closed monoidal category]].  (Since $Cat$ is in fact a [[cartesian closed category]], another common notation is $D^C$, cf. *[[exponential objects]]*.)

 In fact, if we want $Cat$ to be [[cartesian closed category|cartesian closed]], the definition of natural transformation is forced (since an [[adjoint functor]] is unique). This is discussed in [a section below](#InTermsOfCartMon).

There is also a [[horizontal composition]] of natural transformations, which makes [[Cat]] a [[2-category]]: the [[Godement product]]. See there for details.

In fact, [[Cat]] is a 2-category (a $Cat$-enriched category) _because_ it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their [[internal hom]].
\end{remark}

\begin{remark}\label{CategoryOfPresheaves}
**([[categories of presheaves]])**
  \linebreak
  The [[functor categories]]  (eq:FunctorCategory) are also called *[[categories of presheaves]]*, in particular if they are of the form
$$
  PSh(C) \,\coloneqq\, [C^{op}, Set]
  \,,
$$
hence if they are categories whose

* [[objects]] are [[functors]] out of the [[opposite category]] of a given category $C$ into the category [[Set]] of [[sets]] and [[functions]]

* [[morphisms]] are natural transformations between these.

Similarly, functor categories of the form
$$
  CPSh(C) \,\coloneqq\, [C, Set]
$$
are also called *[[categories of copresheaves]]*.

\end{remark}

### In terms of morphismwise components
{#morphismwiseDefn}

An alternative but ultimately equivalent way to define a natural transformation $\alpha : F \rightarrow G$ is as an assignment to every morphism $m : x \rightarrow y$ in $C$ of a morphism $\alpha(m) : F(x) \rightarrow G(y)$, in such a way as that $G(m_1)\alpha(m_0) = \alpha(m_1)F(m_0)$ for every binary composition $m_1 m_0$ in $C$ (or equivalently $\alpha(m_2 m_1 m_0) = G(m_2) \alpha(m_1) F(m_0)$ for every ternary composition $m_2m_1m_0$ in $C$).

The relation of this to the previous definition is that the commutative squares in the previous definition for any morphism $f$ give the value $\alpha(f) = G(f) \circ \alpha_x = \alpha_y \circ F(f)$, and the identity morphisms for any object $x$ give the component $\alpha_x = \alpha(id_x)$.

Vertical composition of natural transformations can be specified directly in terms of this account as well: specifically, an $n$-ary composition $\alpha_1 ... \alpha_n$ of natural transformations is uniquely determined by the property that $(\alpha_1 ... \alpha_n)(m_1 ... m_n) = \alpha_1(m_1) ... \alpha_n(m_n)$, for every $n$-ary composition $m_1 ... m_n$ in $C$.

Horizontal composition is even easier, as the horizontal composite of $\alpha_1, ..., \alpha_n$ is just $\alpha_1 ... \alpha_n$.

### In terms of the cartesian closed monoidal structure on $Cat$ {#InTermsOfCartMon}

The definition of the [[functor category]] $[C,D]$ with morphisms being natural transformations is precisely the one that makes $Cat$ a [[cartesian monoidal category|cartesian]] [[closed monoidal category]].

The category [[Cat]] of all categories (regarded for the moment just as an ordinary 1-[[category]]) is a [[cartesian monoidal category]]: for every two categories $C$ and $D$ there is the cartesian [[product]] category $C \times D$, whose objects and morphisms are simply pairs of objects and morphisms in $C$ and $D$: $Mor(C \times D) = Mor(C) \times Mor(D)$.

It therefore makes sense to ask if there is for each category $C \in Cat$ an [[internal hom]] functor $[C,-] : Cat \to Cat$ that would make [[Cat]] into a [[closed monoidal category]] in that for $A,B,C \in Cat$ we have natural isomorphisms of _sets_ of functors

$$
  Funct(A \times C , B) \simeq Funct(A, [C,B])
  \,.
$$

This is precisely the case for $[C,B]$ being the [[functor category]] with functors $C \to B$ as objects and natural transformations, as defined above, as morphisms.

Since $Cat$ here is cartesian closed, one often uses the exponential notation $C^B := [B,C]$ for the functor category.

To derive from this the definition of natural transformations above, it is sufficient to consider the [[interval category]] $A := I := \{a \to b\}$. For any category $E$, a functor $I \to E$ is precisely a choice of morphism in $E$. This means that we can check what a morphism in the [[internal hom]] category $[C,B]$ is by checking what functors $I \to [C,B]$ are. But by the defining property of $[C,B]$ as an [[internal hom]], such functors are in natural bijection to functors $I \times C \to B$.

$$
  Funct(I, [C,B]) \simeq Funct(I \times C, B)
  \,.
$$

But, as mentioned above, we know what the category $I \times C$ is like: its morphisms are pairs of morphisms in $I$ and $C$, subject to the obvious composition law, which says in particular that for $f : c_1 \to c_2$ any morphism in $C$ we have

$$
  \begin{aligned}
    (c_1,a) \stackrel{(f,(a \to b))}{\to} (c_2,b)
    & =
    (c_1,a) \stackrel{(f, Id)}{\to} (c_2,a)
    \stackrel{(Id, (a \to b)}{\to}
    (c_2, b)
    \\
    &=
    (c_1,a) \stackrel{(Id, (a\to b))}{\to} (c_1,b)
    \stackrel{(f,Id}{\to}
    (c_2, b)
  \end{aligned}
  \,.
$$

Here the right side is more conveniently depicted as a commuting square

$$
  \array{
     (c_1,a) &\stackrel{(f,Id)}{\to}& (c_2,a)
     \\
     \downarrow^{\mathrlap{(Id,(a \to b))}}
     &&
     \downarrow^{\mathrlap{(Id, (a \to b))}}
     \\
     (c_1,b) &\stackrel{(f,Id)}{\to}& (c_2,b)
  }
  \,.
$$

So a natural transformation between functors $C \to D$ is given by the images of such squares in $D$. By tracing back the way the hom-isomorphism works, one finds that the image of such a square in $D$ for a natural transformation $\alpha : F \to G$ is the naturality square from above:

$$
  \array{
    F(c_1)
    &
    \stackrel{F(f)}{\to}
    &
    F(c_2)
    \\
    \alpha_x\downarrow
    &&
    \downarrow \alpha_y
    \\ G(c_1)
    &
    \stackrel{G(f)}{\to} & G(c_2)
  }
  \,.
$$

### In terms of double categories

There is a nice way of describing these structures due to [[Charles Ehresmann]]. For a category $D$ let $(\square D,\circ_1,\circ_2)$  be the [[double category]] of commutative squares in $D$. Then the class of natural transformations of functors $C \to D$ can be described as $Cat(C,(\square D,\circ_1))$. But then $\circ_2$ induces a category structure on this and so we get $CAT(C,D)$.

An advantage of this approach is that it applies to the case of topological categories and groupoids (working in a convenient category of spaces).

An analogous approach works for strict cubical $\omega$-categories with connections, using the good properties of [[cube]]s, so leading to a [[closed monoidal category|monoidal closed structure]] for these objects. This yields by an [[equivalence of categories]] a monoidal closed structure on [[strict omega-category|strict globular omega-categories]], where the [[tensor product]] is the [[Crans-Gray tensor product]].

## Examples ##
 {#Examples}

\begin{example}\label{UnitOfDoubleDualization}
**(unit of double-dualization)**
\linebreak
  For any [[ground field]] $\mathbb{K}$,
  the canonical [[linear map]] from a [[vector space]] $V$ to its [double dual](dual+vector+space#DoubleDualVectorSpaces) $\big(V^{\ast}\big)^\ast$
\[\label{UnitMapToDoubleDual}\]
\begin{tikzcd}[sep=0pt]
  V \ar[rr] && \big(V^\ast\big)^\ast
  \\
  v
    &\mapsto&
  \big(
    \omega(-) \mapsto \omega(v)
  \big)
\end{tikzcd}
is a natural transformation
$$
  Id \longrightarrow \big((-)^{\ast}\big)^\ast
$$

from the [[identity functor]] $Id$ on [[Vect|$\mathbb{K} Vect$]] to the [[composition]] $\big((-)^{\ast}\big)^\ast$ of the [[linear dual space|linear dual]]-[[endofunctor]] with itself.

This example was the motivating example of [Eilenberg & MacLane 1945](#EilenbergMacLane45) (right on the first pages) for introducing the notion of natural transformation (and with it [[category theory]]) in the first place.

The conceptual subtlety that these authors sought to resolve here is that for any *[[finite-dimensional vector space|finite-dimensional]]* [[vector space]] there exists also an [[isomorphism]] from $V$ to its *single*-[[dual vector space]]:

$$
  V \in \mathbb{K} Vect^{fdim}
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  \exists
  \;
  V \xrightarrow{\sim} V^\ast
  \,.
$$

But these linear maps are conceptually different from (eq:UnitMapToDoubleDual) in that they are *not natural* in the technical sense that they do not form the components of a natural transformation between the evident functors. Instead they involve an arbitrary choice equivalent to that of an (possibly indefinite) [[inner product]] on $V$, which is not preserved by general [[linear isomorphisms]] (but just by the corresponding [[isometries]]).
\end{example}

\begin{example}
The [[determinant]] is a natural transformation $det\colon GL_n\rightarrow (-)^\times$ from the [[general linear group]] to the [[group of units]] of a [[ring]], which are both functors from [[Ring]] to [[Grp]].
\end{example}

\begin{example}
The [[Frobenius homomorphism]] is a natural transformation $Frob_p\colon Id_{Ring_p}\Rightarrow Id_{Ring_p}$ from the [[identity functor]] on the [[full subcategory]] of [[Ring]] containing all rings with [[characteristic]] $p$ to itself.
\end{example}

\begin{example}
The [[Hurewicz homomorphism]] is a natural transformation $h_n\colon\pi_n\Rightarrow H_n(-;\mathbb{Z})$ from the [[homotopy group]] to [[singular homology]], which are both functors from [[Top]] to [[Grp]].
\end{example}

\begin{example}
The [[inverse element|inversion]] $G \rightarrow G^{op},g\mapsto g^{-1}$ for every [[group]] $G$ yields a natural transformation $\Id_{Grp}\Rightarrow (-)^{op}$ from the [[identity functor]] on [[Grp]] to the [[opposite group]]-assigning functor.
\end{example}

\begin{example}
The [[coprojection]] $G\rightarrow G^{ab},g\mapsto[g]$ for every group $G$ yields a natural transformation $\Id_{Grp}\Rightarrow -^{ab}$ from the [[identity functor]] on $Grp$ to the [[abelianization]] functor.
\end{example}

\begin{example}\label{HomomorphismsOfDiagrams}
**(homomorphisms of diagrams)**
\linebreak
By Remark \ref{CategoryOfPresheaves}, every category identified with a [[category of presheaves]] or [[copresheaves]] has its morphisms identified with natural transformations.

For instance, the category of [[directed graphs]] ([[digraphs]]) may be identified with the category of [[copresheaves]] on the [[diagram]] shape $\big\{ V \rightrightarrows E\big\}$, and under this identification the natural transformations between functors $\big\{ V \rightrightarrows E \big\} \longrightarrow Set$ are identified with [[digraph]] [[homomorphisms]].
\end{example}

## Variations

For functors between [[n-category|higher categories]], see *[[lax natural transformation]]* etc.

A transformation which is natural only relative to [[isomorphisms]] may be called a [[core-natural transformation|canonical transformation]].

For functors with more complicated shapes than $C \rightrightarrows D$, see [[extranatural transformation]] and [[dinatural transformation]].

## Related entries

* [[homotopy]]

* [[transfor]]

  * **natural transformation**

    * [[transformation of adjoints]]

    * [[monad transformation]]

    * [[extranatural transformation]], [[dinatural transformation]]

    * [[cartesian natural transformation]]

    * [[enriched natural transformation]]

    * [[strong natural transformation]]

  * [[pseudonatural transformation]]

  * [[lax natural transformation]]

* See [[natural transformation (discussion)]] for an informal discussion about natural transformations.

## References

The notion of *natural transformations* between functors is due to

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], _[[General Theory of Natural Equivalences]]_,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

where it served as the motivation for the definition of *[[categories]]* and *[[functors]]* in the first place.

Textbook accounts:

* [[Saunders MacLane]], §I.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.3 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

See also *[[category theory#references|category theory - references]]*.

[[!redirects natural transformations]]

[[!redirects naturality square]]
[[!redirects naturality squares]]

[[!redirects natural map]]
[[!redirects natural maps]]


natural transformation (discussion)

#Contents#

* automatic table of contents goes here
{:toc}

##Discussion 2.0##

[[Eric]]: This discussion began when I asked the following questions:

>[[Eric]]: For some reason, I always thought a natural transformation was a more general map between functors $\alpha:F\Rightarrow G$, where
$$F:A\to B\quad\text{and}\quad G:C\to D.$$

>Then $\alpha(F):C\to D$ is a functor and we'd probably also want **2-component** functors $\alpha_C:A\to C$ and $\alpha_D:B\to D$ such that the following diagram commutes

$$
  \array{
    A
    &
    \stackrel{F}{\to}
    &
    B
    \\
    \mathllap{\scriptsize{\alpha_C}}\downarrow
    &\mathllap{\scriptsize{\alpha}}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_D}}
    \\ C
    &
    \stackrel{G}{\to} & D
  }
$$

>If this is not a natural transformation, is there another name for it?

>When $A = C$ and $B = D$, this should coincide with the usual definition. If so, why don't we define natural transformation in this more general way? It seems more "natural" to me.

[[Eric]]: I hope to pick up where we left off because I was making some progress understanding some of this stuff.

I like the above concept, but I'm still trying to find the right label for it. The standard definition of [[natural transformation]] involves a map between functors (sharing the same domain and codomain) together with **component morphisms**. What I'm trying to describe is a map between functors (not necessarily with the same domain and codomain) together with **component functors**.

This seems to be a natural thing to do, especially when thinking of functors as maps between morphisms, together with component morphisms, as put forward at [[functor (discussion)]].

From there, it is a skip away from thinking of morphisms as maps between objects together with component objects. Hmmm...

The whole thing just wants to bumped up incrementally in dimension.

###Functor Revisited###

####Definition (Experimental)####

Given categories $A$, $B$ and inclusion maps $i_A:A\to A\sqcup B$, $i_B:B\to A\sqcup B$, a **functor** is a map $F:A\to B$ together with component morphisms $\alpha_x:i_A(x)\to i_B\circ F(x)$ and $\alpha_y:i_A(y)\to i_B\circ F(y)$ for any morphism $f:x\to y$ in $A$ such that the following diagram commutes:
$$
  \array{
    i_A(x)
    &
    \stackrel{i_A(f)}{\to}
    &
    i_A(y)
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &\mathllap{F}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ i_B\circ F(x)
    &
    \stackrel{i_B\circ F(f)}{\to} & i_B\circ F(y)
  }
$$

For a less cluttered diagram, we can omit the obvious inclusion maps and look at

$$
  \array{
    x
    &
    \stackrel{f}{\to}
    &
    y
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &\mathllap{F}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ F(x)
    &
    \stackrel{F(f)}{\to} & F(y)
  }
$$

<img src="http://ncatlab.org/nlab/files/alt_functor.jpg" width = "250"/>

##Discussion 1.0##

[[Eric]]: For some reason, I always thought a natural transformation was a more general map between functors $\alpha:F\Rightarrow G$, where
$$F:A\to B\quad\text{and}\quad G:C\to D.$$

Then $\alpha(F):C\to D$ is a functor and we'd probably also want **2-component** functors $\alpha_C:A\to C$ and $\alpha_D:B\to D$ such that the following diagram commutes
$$
  \array{
    A
    &
    \stackrel{F}{\to}
    &
    B
    \\
    \mathllap{\scriptsize{\alpha_C}}\downarrow
    &\mathllap{\scriptsize{\alpha}}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_D}}
    \\ C
    &
    \stackrel{G}{\to} & D
  }
$$
If this is not a natural transformation, is there another name for it?

When $A = C$ and $B = D$, this should coincide with the usual definition. If so, why don't we define natural transformation in this more general way? It seems more "natural" to me.

[[Urs Schreiber]]: I suppose your 2-arrow in the middle labeled $\alpha$ is supposed to be a natural transformation (in the standard sense)?

Then this is a square in the 2-category of categories. Such squares appear in turn as the _components_ of [[lax natural transformation]]s of 2-functors into [[Cat]].

[[Eric]]: Thanks. I was hesitant to put that 2-arrow in there. I'm still learning this stuff and am on shaky ground. But no. That 2-arrow is not supposed to be a natural transformation in the traditional sense because it is not a map between functors with the same domain and codomain. It is tempting to say we might want $\alpha_D\circ F\Rightarrow\alpha_C\circ G$ to be a natural transformation in the traditional sense though.

_Toby_:  That arrow certainly *looks* to me like a $2$-arrow from $\alpha_D\circ F$ to $\alpha_C\circ G$.  Unless it has no particular meaning and is just labelling the entire square as $\alpha\colon F \Rightarrow G$.  In any case, there is the question of what, if anything, goes between $\alpha_D\circ F$ and $\alpha_C\circ G$.  If nothing, then I would call such a thing a __square__ of functors.  If something more interesting, see my answer at [[ericforgy:Natural Transformation]] (which I saw first).

_Todd_: Here's one way of considering how the standard notion of natural transformation is the "right" one. Just as in topology we would like to consider the "space of all continuous maps $X \to Y$" from one space to another as a topological space, and the right notion of topology so we get an appropriate universal property, we want the same thing here. So the term analogous to "space" is "category", the term analogous to "continuous map" is "functor", and the term analogous to "space of all continuous maps from one space to another" or "function space" is "category of all functors from one category to another" or "functor category". And there should be a universal property which gives the right such category.

In the end, if $D^C$ denotes the functor category, the universal property provides an equivalence between functors $X \to D^C$ and functors $X \times C \to D$. Pursuing this, let's figure out what morphisms in $D^C$ should look like. In an ordinary category $Y$, a morphism in $Y$ corresponds to a functor $\mathbf{2} \to Y$, where $\mathbf{2}$ is a category with two objects $0, 1$ and exactly one nonidentity arrow $0 \to 1$. So morphisms in $D^C$ should correspond to functors $\mathbf{2} \to D^C$. And the universal property says these are equivalent to functors $\mathbf{2} \times C \to D$. Now: if you carefully work through what functors of that form give you, you get precisely the notion of natural transformation. This is an exercise well worth working out.

This is how you know the notion of natural transformation is morally the "right" one.

[[Urs Schreiber]]: I added above a subsection on this characterization in terms of the cartesian closed monoidal structure on $Cat$.

[[Eric]]: I think I got it.

Given two functors $F:A\to B$ and $G:C\to D$, let $\alpha:F\Rightarrow G$ together with component functors $\alpha_C:A\to C$ and $\alpha_D:B\to D$ be a map such that
$$\alpha_D\circ F\Rightarrow G\circ\alpha_C$$
is a natural transformation (in the standard sense). Does this guy have a name?

The reason why I bring this up is that I do not like bigons. Bigons are not a good shape for doing computational geometry, geometric realization, etc. So before I could finally grok what a [[cone]] was, I had to draw a bunch of diagrams. These diagrams were bigons. Yucky!

Thanks Todd. That was a valiant effort, but didn't manage to convince me (I have a thick skull!). I still like this more general definition. Given topological spaces $W,X,Y,Z$, we could also be interested in studying spaces $(W\to X)\Rightarrow (Y\to Z)$. What you described would then be a special case where $W=Y$ and $X=Z$.

Here is my attempt to formalize an alternative definition:

***

>##Definition##

>Given categories $A$,$B$,$C$,$D$ and functors $F:A\to B$,$G:C\to D$ a __natural transformation__ $\alpha:F \Rightarrow G$

$$
  \array{
    A
    &
    \stackrel{F}{\to}
    &
    B
    \\
    \mathllap{\scriptsize{\alpha_C}}\downarrow
    &\mathllap{\scriptsize{\alpha}}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_D}}
    \\ C
    &
    \stackrel{G}{\to} & D
  }
$$

>assigns functors $\alpha_C:A\to C$ and $\alpha_D:B\to D$ called **component functors** and for any morphism $f:x\to y$ in $A$ assigns morphisms $\alpha_x:\alpha_D\circ F(x)\to G\circ\alpha_C(x)$ and $\alpha_y:\alpha_D\circ F(y)\to G\circ\alpha_C(y)$ in $D$ called **component morphisms** such that the following diagram commutes in $D$:

$$
  \array{
    \alpha_D\circ F(x)
    &
    \stackrel{\alpha_D\circ F(f)}{\to}
    &
    \alpha_D\circ F(y)
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &&
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ G\circ\alpha_C(x)
    &
    \stackrel{G\circ\alpha_C(f)}{\to} & G\circ\alpha_C(y)
  }
$$

***

Again, this reduces to the traditional definition when $\alpha_C$ and $\alpha_D$ are identity functors. What do you think? If this doesn't already have a name (it probably does), what would be a good name for it?

_Toby_:  This is exactly what I was talking about at [[ericforgy:Natural Transformation]].  I would call it a __lax commutative square of functors__.

[[Eric]]: Ok, but this seems to be "rotated $45^\circ$" relative to what you said there. Here $\alpha:F\Rightarrow G$ is a map between functors (not composites functors) with distinct domains and codomains having component functors $\alpha_C$, $\alpha_D$ and component morphisms $\alpha_x$, $\alpha_y$. I'm trying to emphasize the map $\alpha:F\Rightarrow G$. Now, I'm trying to think about how I might reinterpret functors in this way so that we can iterate the process to higher degrees.

_Urs_: no, the $\alpha$ the way you defined it _is_ a transformation between the composite functors: it is precisely a natural transformation from $\alpha_D \circ F$ to $G \circ \alpha_C$

_Toby_:  You are right that it is rotated *conceptually*, but what I wrote there was also supposed to be rotated conceptually.  The thing itself is indifferent to being from $F$ to $G$, but once you declare (conceptually) that it is to be something from $F$ to $G$, then what you have is a morphism in an arrow $2$-category, as I wrote there.

[[Eric]]: Thanks Toby. Here is a prepared remark with a comment below:

***

I wrote the definition down intentionally so that the composite _is_ a natural transformation, but I hope I can convince you that $\alpha:F\Rightarrow G$ is _not_ a natural transformation because the domains and codomains of $F$ and $G$ do not coincide (which they must to be a standard natural transformation). I'll call it something else then. How about "sheet transformations"?

>##Definition##

>Given categories $A$,$B$,$C$,$D$ and functors $F:A\to B$,$G:C\to D$ a __sheet transformation__ $\alpha:F \Rightarrow G$

$$
  \array{
    A
    &
    \stackrel{F}{\to}
    &
    B
    \\
    \mathllap{\scriptsize{\alpha_C}}\downarrow
    &\mathllap{\scriptsize{\alpha}}\Downarrow &
    \downarrow\mathrlap{\scriptsize{\alpha_D}}
    \\ C
    &
    \stackrel{G}{\to} & D
  }
$$

>assigns functors $\alpha_C:A\to C$ and $\alpha_D:B\to D$ called **component functors** such that

$$\alpha_D\circ F\Rightarrow G\circ\alpha_C$$

>is a **natural transformation**.

***

>The thing itself is indifferent to being from $F$ to $G$, but once you declare (conceptually) that it is to be something from $F$ to $G$, then what you have is a morphism in an arrow $2$-category, as I wrote there.

Interesting! That means a morphism in a arrow 2-category is the same thing as a "sheet transformation" (or vice versa :))

Urs: it seems to me what you are getting is is the relation between 2-categories and [[double category|double categories]]: a double category is a gadget whose 2-cells are squares not bigons. Every 2-category yields a double category by precisely the kind of operation that you are describing: take the squares of the double category to be things built from four 1-morphisms and one 2-morphism in the 2-category. Indeed, as a 2-morphism in the 2-category it is something that goes between the composites of the 1-morphisms, but the resulting square itself in the double category is indeed a 2-morphism to be regarded as going from the top horizontal to the bottom horizontal 1-morphism.

_Toby_:  Don\'t you have to talk about a [[double bicategory]] to get something nontrivial to fill the squares?  I may have this wrong, but that\'s how I remember it.

[[Mike Shulman]]: No, a double bicategory is just a double category in which both composition operations are only associative and unital up to isomorphism.  Sometimes I feel like "double categories" should really have been called "double 2-categories," since they have nontrivial 2-cells.  So it is true that what Eric is interested in are just the 2-cells in the double category $Sq(Cat)$ of "squares" or "quintets" in $Cat$.  As such, they have two different composition operations given by pasting.

Another way to describe it is that these are the morphisms in the 2-category $2Cat_{oplax}(\mathbf{2},Cat)$, where for 2-categories $A$ and $B$, $2Cat_{oplax}(A,B)$ denotes the 2-category whose objects are 2-functors $A\to B$ and whose morphisms are [[oplax natural transformations]], and $\mathbf{2}$ denotes the [[walking arrow]].  Thus an object of $2Cat_{oplax}(\mathbf{2},Cat)$ is a 2-functor $\mathbf{2}\to Cat$, i.e. a functor, and a morphism is an oplax natural transformation, which is precisely what you've described.  Composition of morphisms in $2Cat_{oplax}(\mathbf{2},Cat)$ is one of the above compositions in the double category $Sq(Cat)$.


natural units

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

\tableofcontents

## Idea

In [[physics]], by *natural units* one means [[physical units]] of [[measurements]] formed from combinations of fundamental physical constants, such as

* $c$ -- the [[speed of light]]

* $G$ -- [[gravitational constant]],

* $\hbar$ -- [[Planck's constant]],

* $\epsilon_0$ -- [[permittivity of the vacuum]],

etc.

## Examples

Examples of natural units used in physics include

* Planck units ([[Planck mass]], [[Planck length]], ...)

* Stoney units,

* Hartree units,

* particle physics natural units,

* QCD units,

* Schrödinger units.

## Related concepts

* [[physical unit]]

[[!include fundamental scales -- contents]]

## References

* [[Robert Loren Jaffe]], *Natural Units and the Scales of Fundamental Physics*, Course Notes 2007 ([pdf](https://stuff.mit.edu/afs/athena/course/8/8.06/spring08/handouts/units.pdf), [[Jaffe_NaturalUnits.pdf:file]])

See also:

* Wikipedia, *[Natural units](https://en.wikipedia.org/wiki/Natural_units)*

[[!redirects natural unit]]
[[!redirects natural units]]


naturalness

> This entry is about the notion of naturalness in ([[particle physics|particle]]-)[[physics]]. For the notion in [[mathematics]] see at _[[natural transformation]]_.

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In ([[particle physics|particle]]-)[[physics]] the term "naturalness" [['t Hooft 1980](#tHooft1980), [Gell- Mann 1983](#GellMann83)] refers to the vague idea that a [[model (physics)|model]] of physics is expected to work without requiring unlikely-looking ad-hoc coincidences or "fine-tuning" of its parameters (such as the choice of [[renormalization]] constants), and that instead all such "conspiracies" among parameter values are to have some systematic cause, such as some ([[broken symmetry|broken]]) [[symmetry]] of the model.

An archetypical example, which at times has been referred to as _the_ naturalness probem of the [[standard model of particle physics]], is that the [[mass]] of the [[Higgs boson]] is first of all many orders of magnitude below the [[Planck scale]], while at the same time potentially receiving very large potential [[renormalization|quantum corrections]] from the presence of (potentially undetected) heavy particles, which therefore must coincidentally cancel out. This is also called the _[[hierarchy problem]]_.

Accordingly, a popular suggestion has been that there should be a [[symmetry]] in nature which is to naturally explain this otherwise coincidental cancellation, namely low-energy [[supersymmetry]]. The failure of such a symmetry to be observed at the [[LHC]] experiment is leading the high energy physics community to a re-examination of the naturalness paradigm and/or to focus on alternative mechanisms, such as a [[composite Higgs boson]].

The next main example of (lack of) naturalness often considered is the _[[cosmological constant]]_ ([[dark energy]]), which seems tiny when compared to the [[renormalization|quantum corrections]] which it naively receives from the [[vacuum energy]] of all [[field (physics)|fields]] in the [[observable universe]].

## Problems with the concept
 {#ProblemsWithTheConcept}

A key problem is arguably that the principle of naturalness has been used in a very vague and very ambiguous sense. Even apart from the problem of which large or small numbers are to be regarded as "unlikely" (see the [quote below](#WilsonOnNaturalness) from [Wilson 04](#Wilson04)), there is technical fine print.

### 1) Dependence on renormalization scheme
 {#DependenceOnRenormalizationScheme}

Both the [[mass]] of the [[Higgs particle]] as well as the [[cosmological constant]] are subject to [[renormalization]] in [[perturbative quantum field theory]], whence "quantum corrections".

(For more on the [[renormalization]]-freedom in the [[cosmological constant]] see [there](cosmological+constant#InPerturbativeQuantumGravity).)

A key fact, that seems to be mostly overlooked in discussion of naturalness/fine-tuning, is that [[renormalization]] parameters a priori take values in an [[affine space]] (see [this theorem](geometry+of+physics+--+perturbative+quantum+field+theory#ExistenceRenormalization)).

This means that without choosing an arbitrary origin and then a set of [[coordinates]], there is no sense in which a [[renormalization|quantum correction]] is "large" or "small". An origin and a set of coordinates is however provided by a choice of _[[renormalization scheme]]_, and discussions in the literature typically implicitly take such a choice for granted.

### 2) General viability

Even if one assumes that the parameters under consideration are invariantly defined [[numbers]], independent on choices like [[renormalization schemes]] (see [above](#DependenceOnRenormalizationScheme)) it has been questioned whether it is fruitful to consider the "naturalness" of the values they take

{#FromKane17} The following is from [Kane 17](#Kane17) (there in a discussion of [[G2-MSSM]] [[model (physics)|model building]]):

> Until recently there were no theories predicting the values of superpartner masses. The arguments based on ‘naturalness’ are basically like saying the weather tomorrow should be the same as today. The opposite of naturalness is having a theory. $[$...$]$ Claims $[$superpartners$]$ should have been seen would be valid given so called naturalness arguments, but are wrong in actual theories. Many of us think that is a misuse of the idea of naturalness, but it is the fashionable use.

(p. 33 (3-2))

> Some arguments (‘naturalness’) can be used to estimate what values $[$MSSM parameters$]$ might have. If those arguments were correct some superpartners would already have been discovered at the CERN LHC. It would have been nice if the naturalness arguments had worked, but they did not. Since they were not predictions from a theory it is not clear how to interpret that.

(p. 39 (4-3))

> The failure of naïve naturalness to describe the world tells us we should look harder for a theory that does, an ‘ultraviolet completion’. Compactified string/ M-theories appear to be strong candidates for such a theory.

> {#TheAlternative} The alternative to naturalness, often neglected as an alternative, is having a theory.

(p. 57 (6-1))

Similarly [Wilson 04, p. 10](#Wilson04):

> {#WilsonOnNaturalness} $[ \cdots ]$ The claim was that it would be unnatural for such particles to have masses small enough to be detectable soon. But this claim makes no sense when one becomes familiar with the history of physics. There have been a number of cases where numbers arose that were unexpectedly small or large. An early example was the very large distance to the nearest star as compared to the distance to the Sun, as needed by Copernicus, because otherwise the nearest stars would have exhibited measurable parallax as the Earth moved around the Sun. Within elementary particle physics, one has unexpectedly large ratios of masses, such as the large ratio of the muon mass to the electron mass. There is also the very small value of the weak coupling constant. In the time since my paper was written, another set of unexpectedly small masses was discovered: the neutrino masses. There is also the riddle of dark energy in cosmology, with its implication of possibly an extremely small value for the cosmological constant in Einstein's theory of general relativity. This blunder was potentially more serious, if it caused any subsequent researchers to dismiss possibilities for very large or very small values for parameters that now must be taken seriously.

> ([Wilson 04, p. 10](#Wilson04))

## Related concepts

* [[hierarchy problem]]

* [[split supersymmetry]]

## References

Early discussion:

* {#tHooft1980} [[Gerard 't Hooft]], *Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking* In: G. 'Hooft et al., *Recent Developments in Gauge Theories*, NATO Advanced Study Institutes Series **59**, Springer (1980) [[doi:10.1007/978-1-4684-7571-5_9](https://doi.org/10.1007/978-1-4684-7571-5_9)]

  > [pp 135:] "it is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a philosophy which we would like to apply to the unified gauge theories: the effective interactions at a large length scale, corresponding to a low energy scale $\mu_1$ should follow from the properties at a much smaller length scale, or higher energy scale $\mu_2$, without the requirement that various different parameters at the energy scale $\mu_2$ match with an accuracy of the order of $\mu_1/\mu_2$. That would be unnatural."

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_ (1983) [[[Gell-Mann_ShelterIslandII_1983.pdf:file]], [[Gell-Mann-ShelterIslandII_1983.pdf:file]]]

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

  > "we seem to have to dial various renormalized quantities to small values. The situation of having numerous arbitrary dimensional parameters is even more humiliating when some of them are very small. [...] First of all, we'd like to interpret small or nearly symmetric quantities as coming from a slightly broken symmetry; otherwise they don't make sense to us. [...] When we can't avoid dialing some renormalized quantity to a small value [...] that situation has recently been described as a problem of 'naturalness'."

Survey and further discussion:

* {#Clarke17} [[Jackson Clarke]], _[Naturalness: A Pragmatist's Guide](http://syymmetries.blogspot.de/2017/06/naturalness-pragmatists-guide.html)_, 2017

* [[Gian Francesco Giudice]], _Naturally Speaking: The Naturalness Criterion and Physics at the LHC_ ([arXiv:0801.2562](https://arxiv.org/abs/0801.2562))

* [[Porter Williams]], _Naturalness, the autonomy of scales, and the 125 GeV Higgs_, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, Volume 51, August 2015, Pages 82–96 ([publisher](http://www.sciencedirect.com/science/article/pii/S1355219815000416), [PhilSciArchive](http://philsci-archive.pitt.edu/11529/))

* [[Matthew Strassler]], _[Naturalness](http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/naturalness/)_

* [[Simon Friedrich]], _[Fine-Tuning](https://seop.illc.uva.nl/entries/fine-tuning/)_, Stanford Encyclopedia of Philosophy, 2017

* [[Gian Francesco Giudice]], _The Dawn of the Post-Naturalness Era_ ([arXiv:1710.07663](https://arxiv.org/abs/1710.07663))

* Arthur Hebecker,
_Lectures on Naturalness, String Landscape and Multiverse_ ([arXiv:2008.10625](https://arxiv.org/abs/2008.10625))

An attempt to make the notion precise is due to

* Greg Anderson, Diego Castano, _Measures of fine tuning_, Phys. Lett.B 347:300-308, 1995 ([arXiv:hep-ph/9409419](https://arxiv.org/abs/hep-ph/9409419))

* [[James Wells]], *Evaluation and Utility of Wilsonian Naturalness* ([arXiv:2107.06082](https://arxiv.org/abs/2107.06082))

Critical comments are in

* {#Wilson04} [[Kenneth Wilson]], _The Origins of Lattice Gauge Theory_, ([arXiv:hep-lat/0412043](http://arxiv.org/abs/hep-lat/0412043))

* [[Petr Horava]], _Surprises with Nonrelativistic Naturalness_, Int. J. Mod. Phys. D25 (2016) 1645007 ([arXiv:1608.06287](http://arxiv.org/abs/1608.06287))

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

* {#Wells18a} [[James Wells]], _Naturalness, Extra-Empirical Theory Assessments, and the Implications of Skepticism_ ([arXiv:1806.07289](https://arxiv.org/abs/1806.07289))

* {#Wells18} [[James Wells]], _Finetuned Cancellations and Improbable Theories_ ([arXiv:1809.03374](https://arxiv.org/abs/1809.03374))

* Matěj Hudec, [[Michal Malinský]], _Hierarchy and decoupling_ ([arXiv:1902.04470](https://arxiv.org/abs/1902.04470))

* Goran Senjanovic, _Natural Philosophy versus Philosophy of Naturalness_ ([arXiv:2001.10988](https://arxiv.org/abs/2001.10988))

* {#Mooij-Shaposhnikov} [[Sander Mooij]], [[Mikhail Shaposhnikov]], _QFT without infinities and hierarchy problem_ [[arXiv:2110.05175](https://arxiv.org/abs/2110.05175)]

  > The aim of this work has been to stress that there is no physical sense in the [[naturalness]] criterion.

See also

* Wikipedia, _[Naturalness (physics)](http://en.wikipedia.org/wiki/Naturalness_%28physics%29)_

[[!redirects naturality]]

[[!redirects fine-tuning]]
[[!redirects fine-tunings]]


Nature conformable to herself

## Overview

*Nature conformable to herself* is the name of an article written by [[Murray Gell-Mann]] which takes its title from the writings of [[Isaac Newton]]:

> For Nature is very consonant and conformable to her self... For we must learn from the phaenomena of nature what bodies attract one another, and what are the laws and properties of the attraction, before we enquire the cause by which the attraction is perform'd, the attractions of gravity, magnetism and electricity, react to very sensible distances, and so have been observed by vulgar eyes, and there may be others which reach to so small distances as hitherto escape observation; and perhaps electrical attraction may react to such small distances, even without being excited by friction. (Opticks, [Gell-Mann 96, p. 11](#Gell-Mann96))

Gell-Mann comments on this passage:

> What a wealth of wisdom is contained in these words! Newton suggests that at small distances electrical interactions may play an important role, going far beyond the attraction of bits of paper to an amber rod rubbed against a cat's fur. He anticipates the existence of a multitude of short-range forces such as we now know to exist. (Indeed, theoretical considerations now point to an infinite number of such forces.) He points out how empirical laws generally precede detailed dynamical explanations. *And he seems to think of the various "Phaenomena" as exhibiting conformability among themselves as well as within each one.* The last idea is the key to understanding why the criterion of simplicity should be helpful in the search for the fundamental laws of physics. ([Gell-Mann 96, p. 11](#Gell-Mann96))

Conformity between the physical principles operating at neighboring levels allows for their discovery:

> As we peel the skins of the onion, penetrating to deeper and deeper levels of the structure of the elementary particle system, mathematics with which we become familiar because of its utility at one level suggests new mathematics, some of which may be applicable at the next level down or to another phenomenon at the same level. Sometimes even the old mathematics is sufficient. ([Gell-Mann 96, p. 11](#Gell-Mann96))

## Developments

Gell-Mann's ideas are developed by [[James Hartle]] in ([Hartle 19](#Hartle19)) in the context of their joint work on the [[consistent histories|decoherent histories]] approach to quantum mechanics. Layers of what Gell-Mann calls the 'onion' are given as:

> Quantum Spacetime; Classical Spacetime, the Quasiclassical Realm, and Copenhagen Quantum Mechanics; Elementary Particles; Big Bang Nucleosynthesis; Recombination; Large Scale Structure -- Galaxies, Stars, Planets, Black Holes; Approximately Isolated Subsystems and their Frozen Accidents.

Again, we find an explanation for the applicability of mathematics across levels:

>In this model it is not surprising that the mathematics useful at one level is related to that at another level.  There is only one mathematics underlying the unified theory... Different parts of this mathematics may be more useful than others at different levels but they are all partial expressions of the same mathematical structure. The closer the mathematics at one level is to the mathematics of the next, the simpler it is to make connections between them, to augment them, or to invent any new mathematics needed. ([Hartle 19, p. 7](#Hartle19))

## References

* {#Gell-Mann96} [[Murray Gell-Mann]], _Nature conformable to herself: Some arguments for a unified theory of the universe_, Complexity 1(4): 9-12 (1996), ([doi:10.1002/cplx.6130010404](https://doi.org/10.1002/cplx.6130010404)).

* {#Hartle19} [[James Hartle]], _How Nature is Conformable to Herself: A View from Quantum Cosmology_, ([arXiv:1909.08724](https://arxiv.org/abs/1909.08724))

category: reference


Naturerkennen und Logik

* [[David Hilbert]],

  _Naturerkennen und Logik_ (Engl.: _Logic and the knowledge of nature_),

  Lecture at

  _Kongress der Gesellschaft Deutscher Naturforscher und Ärtze_,

  1930

  ([pdf](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf), [audio](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3))

  reprinted in Hilbert's collected works in German (pp. 378-387)

  translated to English in W. Ewald, _From Kant to Hilbert_, 1996, pp. 1157-1165

The famous _finale_ from this lecture, in English translation:

> {#TheInstrumentThatMediates} The  instrument  that  mediates  between  theory  and  practice,  between  thought  and observation, is [[mathematics]];  it builds the connecting bridge and makes it stronger and stronger.  Thus it happens that our entire
present-day culture, insofar as it rests on intellectual insight into and harnessing of nature, is founded on mathematics.  Already, [[Galilei|GALILEO]] said:  Only he can understand nature who has learned the language and signs
by which it speaks to us;  but this language is mathematics and its signs are mathematical figures.  [[Kant|KANT]] [declared](Immanuel+Kant#IchBehaupteAber), “I maintain that in each  particular natural science there is only as much true science as there is mathematics.”  In fact, we do not master a theory in natural science until we have extracted its  mathematical kernel and laid it completely bare.  Without mathematics today’s astronomy  and physics would be impossible;  in their theoretical parts, these sciences unfold directly into mathematics.  These, like numerous other applications, give mathematics whatever authority it enjoys with the general public.

> Nevertheless, all mathematicians have refused to let applications serve as the standard of value for mathematics.  [[Gauss|GAUSS]] spoke of the magical attraction that made number theory the favorite science for the first mathematicians, not to mention its inexhaustible richness,  in  which  it  so  far  surpasses  all  other  parts  of  mathematics. [[Leopold Kronecker|KRONECKER]] compared number theorists with the Lotus Eaters, who, once they had sampled that delicacy, could never do without it.

> With  astonishing  sharpness,  the  great  mathematician  [[Henri Poincaré|POINCARÉ]] once  attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish.  The achievements of industry, for example, would never have seen the light of day had the practical-minded existed alone and had not these advances been pursued by disinterested fools.

> The glory of the human spirit, so said the famous Königsberg mathematician JACOBI, is the single purpose of all science

> We must not believe those, who today with philosophical bearing and a tone  of superiority prophesy the downfall of culture and accept the [ignorabimus](https://en.wikipedia.org/wiki/Ignoramus_et_ignorabimus).  For us there is no  [ignorabimus](https://en.wikipedia.org/wiki/Ignoramus_et_ignorabimus), and in my opinion even none whatever in natural science. In place of the foolish [ignorabimus](https://en.wikipedia.org/wiki/Ignoramus_et_ignorabimus) let stand our slogan:

> We  must know,

> We will know.

## Related entries

* [[Hilbert's sixth problem]]

* [[mathematical physics]]

## Related references

* [[Galileo Galilei]] ([quote](Galileo+Galilei#Quotes)):

  > [[natural philosophy|Philosophy]]$ {} $ [i.e. [[physics]]] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of [[mathematics]], and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. ([Galilei, Il Saggiatore, 1623](http://en.wikipedia.org/wiki/The_Assayer)).

* [[Eugene Wigner]], _[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]_ 1960.

category: reference


Natàlia Castellana

Natàlia Castellana is a homotopy theorist

* [website](http://mat.uab.cat/~natalia/index.html)

### See also

* [[Handbook of Homotopy Theory]]

category: people

[[!redirects Natàlia Castellana Vila]]


Navier-Stokes equation

Cl
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[partial differential equation]] describing viscous [[hydrodynamics]].

## Related concepts

* [[fluid dynamics]]

* [[fluid/gravity correspondence]]

## References

* [[Terence Tao]], _Searching for singularities in the Navier–Stokes equations_, Nature Reviews Physics (2019) ([doi:10.1038/s42254-019-0068-9](https://doi.org/10.1038/s42254-019-0068-9), pdf)

* {#CMI} Clay Mathematical Institute, _[Navier-Stokes equation](https://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation)_

See also

* Wikipedia, _[Navier-Stokes equation](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations)_

* Wikipedia, _[Navier-Stokes existence and smoothness](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness)_

Discussion of the (relativistic) Navier-Stokes equation in view of the [[fluid/gravity correspondence]]:

* {#BKLS12} Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, [[Andrew Strominger]], _From Navier-Stokes To Einstein_, J. High Energ. Phys. (2012) 2012: 146 ([arXiv:1101.2451](https://arxiv.org/abs/1101.2451))

* {#Thillaisundaram17} Ashok Thillaisundaram, _Aspects of fluid dynamics and the fluid/gravity correspondence_, 2017 ([CambridgeRepositor:1810/267097](https://www.repository.cam.ac.uk/handle/1810/267097), [pdf](https://www.repository.cam.ac.uk/bitstream/handle/1810/267097/ThillaisundaramThesis.pdf?sequence=1&isAllowed=y))

[[!redirects Navier-Stokes equations]]


Naṣīr al-Dīn al-Ṭūsī

Persian scholar, 1201 - 1274 AD

* [Wikipedia entry](https://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusi)

## Translated writings

On [[astronomy]]:

* {#Ragep93} F. J. Ragep, II.4 $[10]$ (p. 128) in: *[[Naṣīr al-Dīn al-Ṭūsī]]’s Memoir on Astronomy (al-Tadhkira fī cilm al-hay’a): Volume I: Introduction, Edition, and Translation*, Sources in the History of Mathematics and Physical Sciences 12, Springer 1993 ([doi:10.1007/978-1-4757-2241-3](https://link.springer.com/book/10.1007/978-1-4757-2241-3))

## Related $n$Lab entries

* [[Milky Way]]

category: people

[[!redirects Nasir_al-Din_al-Tusi]]


nCob

As a [[category]], $n Cob$ has $(n-1)$-dimensional [[compact space|compact]] [[orientation|oriented]] [[manifolds]] as [[objects]] and $n$-dimensional oriented [[cobordisms]] between these as [[morphisms]].  It plays a fundamental role in [[TQFT]].

For some purposes, we want $n Cob$ to be an [[n-category]], but this is less well understood.

category: category


near-field

A __near-field__ is an algebraic structure with two associative binary algebraic structures similar to a division ring (skewfield) but with only one sided distributivity law (conventionally, usually right distributivity).