differentiation

> See also _[[differentiable function]]_.

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### Context
#### Analysis
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[[!include analysis - contents]]
=--
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

_Differentiation_ is the process that assigns to a [[function]] $f : X \to Y$ its *derivative*, sometimes denoted $d f$. The derivative is a function that, roughly speaking, assigns to each point $x \in X$ the linear transformation $d f_x$ that maps [[infinitesimal space|infinitesimal]] [[differences]] $y - x$ (for points $y$ infinitesimally close to $x$) to infinitesimal differences $f(y) - f(x)$.

## Definitions

### Classical (analytic) definitions

Classically, if $f: \mathbb{R}^n \to \mathbb{R}$ is a function, the derivative $d f_x$ at a point $x \in \mathbb{R}^n$ is defined by a [[convergence|limiting process]]: it is the unique (when it exists) [[linear transformation]] $L: \mathbb{R}^n \to \mathbb{R}$ such that

$$\lim_{h \to 0} \frac{f(x + h) - f(x) - L h}{{|h|}} = 0$$

or in other words such that

$$f(y) = f(x) + L(y - x) + g(y)$$

where $g(y)$ is $o({|y - x|})$ (see [[o-notation]]). We say in that case that $f$ is *differentiable* at $x$, with *derivative* $d f_x = L$. When $n = 1$, such a linear transformation $L: \mathbb{R} \to \mathbb{R}$ is multiplication by a [[scalar]] $\lambda \in \mathbb{R}$. This is often denoted $f'(x)$, so that the derivative may be regarded as another function $f': \mathbb{R} \to \mathbb{R}$.

We say $f: \mathbb{R}^n \to \mathbb{R}$ is *differentiable* if it is differentiable at every point $x$ of its [[domain]]. In that case, we get a global derivative function $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ which takes a pair $(x, h)$ to $d f_x(h)$. We can then ask whether this function is itself differentiable, i.e., whether the differentiation process can be iterated. The *$n$-th derivative* is the result of differentiating $n$ times (the $0$-th derivative of $f$ is just $f$). We say that $f: \mathbb{R}^n \to \mathbb{R}$ is *infinitely differentiable* (or more briefly, that $f$ is *smooth*) if it has an $n$-th derivative for each [[natural number]] $n$.

Notice that a differentiable function is necessarily [[continuous map|continuous]] (indeed, it is locally a [[Lipschitz function]]). One denotes the class of functions having a *continuous*[^1] $n^{th}$ derivative by $C^n(\mathbb{R}^n)$. Thus continuous functions $f: \mathbb{R}^n \to \mathbb{R}$ form a filtration

$$\ldots \subset C^n(\mathbb{R}^k) \subset \ldots \subset C^1(\mathbb{R}^k) \subset C^0(\mathbb{R}^k)$$

and the intersection $C^\infty(\mathbb{R}^k) \coloneqq \bigcap_{n \geq 0} C^n(\mathbb{R}^k)$ coincides with the class of smooth functions.

[^1]: Without the continuity assumption on derivatives, the class of (once-)differentiable functions exhibits non-trivial [[descriptive set theory|descriptive set-theoretic]] complexity, as may be surmised by [this paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6984080) by Kechris and Woodin. Thus the continuity assumption eliminates troublesome pathologies.

Evidently differentiation gives a function $D: C^{n+1}(\mathbb{R}^k) \to C^n(\mathbb{R}^k \times \mathbb{R}^k)$. The convenient setting for developing the differential calculus is to go directly to the smooth case, where we have $D: C^\infty(\mathbb{R}^k) \to C^\infty(\mathbb{R}^k \times \mathbb{R}^k)$ without having to keep track of the degree of differentiability.

This development carries over more generally to [[smooth manifolds]] $X$, which are locally (smoothly) isomorphic to [[Euclidean spaces]] $\mathbb{R}^k$.

### Differentiation as a functor
 {#AsAFunctor}

_Differentiation_ in more conceptual terms is a [[functor]] $d : Diff \to Diff$ on the [[category]] [[Diff]] of [[smooth manifold]]s that sends

* a manifold $X$ to its [[tangent bundle]] $T X$; recall that the points of $T X$ are [[ordered pairs]] $(x, v)$ where $x \in X$ and $v$ is a [[tangent vector]] at $x$, i.e., an (augmented) [[derivation]] $v: C^\infty(X) \to \mathbb{R}$ on the [[associative unital algebra|algebra]] of smooth functions, augmented by evaluation $ev_x: C^\infty(X) \to \mathbb{R}$ at $x$;

* a [[smooth function]] $f : X \to Y$ to its differential $d f : T X \to T Y$:

  if $\gamma : [-1,1] \to X$ is a path in $X$ that represents a vector $v \in T_x X$, then $(d f)(v) \in T_{f(x)} Y$ is the vector represented by the path
  $[-1,1] \stackrel{\gamma}{\to} X \stackrel{f}{\to} Y$.

Equivalently, a smooth map $f: X \to Y$ induces an algebra map $f^\ast = - \circ f: C^\infty(Y) \to C^\infty(X)$, and hence by composition sends a derivation $v: C^\infty(X) \to \mathbb{R}$ to a derivation $v \circ f^\ast: C^\infty(Y) \to \mathbb{R}$ augmented by $ev_{f(x)}: C^\infty(Y) \to \mathbb{R}$). Concretely: $v \circ f^\ast(\phi) = v(\phi \circ f)$ for $\phi \in C^\infty(Y)$. Then the differential $d f: T X \to T Y$ is defined by $d f(x, v) = (f(x), v \circ f^\ast)$.

According to either of these descriptions, differentiation is manifestly functorial. In the English literature, this fact is known as the [[chain rule]]. In more traditional form:

$$d(f \circ g)_x = d f_{g(x)} \circ d g_x.$$

In a context $H$ of [[synthetic differential geometry]] (SDG) the functor $d$ is simply the [[internal hom]] out of the [[infinitesimal space|infinitesimal interval]] $D$

$$
  d = [D, -] : H \to H
  \,.
$$

where $D \hookrightarrow \mathbb{R}$ is the smooth locus defined by $x^2 = 0$. (Classically, i.e., in the [[topos]] $Set$, this locus consists of just a single point[^2], but in toposes that model SDG it should be regarded as an infinitesimal neighborhood about a point.) This description as internally [[representable functor]] underlies the following classical fact:

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###### Theorem
Differentiation is a [[product-preserving functor]].
=--

For example, the product of two smooth functions $f, g: X \to \mathbb{R}$ is a composition

$$X \stackrel{\Delta}{\to} X \times X \stackrel{f \times g}{\to} \mathbb{R} \times \mathbb{R} \stackrel{mult}{\to} \mathbb{R}$$

and the product rule for differentiation derives by applying the functor $d$ which results in the composite

$$T X \stackrel{\Delta}{\to} T X \times T X \stackrel{d f \times d g}{\to} T \mathbb{R} \times T \mathbb{R} \stackrel{d mult}{\to} T \mathbb{R}$$

which reduces to computing $d mult$.

[^2]: According to John Bell, the fact that classically this is merely a point bears comparison to Bishop Berkeley's objections to the differential calculus he put forth in *The Analyst*.

## Exposition of differentiation via infinitesimals
 {#ExpositionDifferentiationViaInfinitesimal}

We attempt to give an exposition of differentiation not via taking limits in the sense of [[analysis]], but rather, by working with [[infinitesimal space]]s as in [[synthetic differential geometry]].

### Setup

For that purpose we regard objects such as the [[real line]] $\mathbb{R}$ and more generally any [[smooth manifold]] as being objects of a [[Models for Smooth Infinitesimal Analysis|well adapted]] [[smooth topos]] of [[smooth space]]s. But in fact much less is required for most of the discussion. For the basic theory, it is sufficient to work in the canonical [[site]] of such a topos, namely, the category $SmoothLoc$ of [[smooth loci]]. This site is defined via a categorial equivalence with the [[opposite category]] of the category of [[smooth algebra]]s:

$$
  C^\infty: SmoothLoc \stackrel{\sim}{\longrightarrow} SmoothAlg^{op}
$$

Apart from the real line $\mathbb{R}$, the other object involved in our discussion of the basics of differentiation is the first order [[infinitesimal space|infinitesimal line segment]] $D$ (an [[infinitesimally thickened point]]), a [[subobject]] of $\mathbb{R}$. This subobject collects those points of $\mathbb{R}$ that square to zero under the canonical multiplicative structure of $\mathbb{R}$

$$
  D = \{\epsilon \in \mathbb{R} | \epsilon^2 = 0\}
  \,.
$$

Although this is just $\{0\}$ when interpreted in the usual context of [[Set]], here we are to interpret this in the ambient [[smooth topos]], or equivalently just in our [[site]] $SmoothLoc$ for it. That is to say, $D$ satisfies a [[pullback]] square in $SmoothLoc$:

$$
  \array{
    D &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{0}}
    \\
    \mathbb{R} &\stackrel{(-)^2}{\to}& \mathbb{R}
  }
$$

Dually, the categorial equivalence $C^\infty$ from $SmoothLoc$ to the [[opposite category]] of [[smooth algebra]]s carries the above pullback square to a [[pushout]] square of [[smooth algebras]]:

$$
  \array{
     C^\infty(D) &\leftarrow& C^\infty(*) \simeq \mathbb{R}
     \\
     \uparrow && \uparrow^{\mathrlap{f \mapsto f(0)}}
     \\
     C^\infty(\mathbb{R})
      &\stackrel{x \mapsto x^2}{\leftarrow}&
     C^\infty(\mathbb{R})
  }
$$

Here along the bottom and on the right are the ordinary smooth algebras of [[smooth function]]s on the [[smooth manifold]]s $\mathbb{R}^1 = \mathbb{R}$ and $\mathbb{R}^0 = *$, whereas $C^\infty(D)$ is new notation, suggestive and justified, for whatever the algebra of functions on the object $D$ is, where $D$ is _defined_ by the fact that it has this algebra of functions. The bottom morphism sends a smooth function $f : \mathbb{R} \to \mathbb{R}$ to the function $f((-)^2) : \mathbb{R} \to \mathbb{R}$. If we write $x = Id : \mathbb{R} \to \mathbb{R}$ for the canonical [[coordinate]] function, then this sends $x$ to the function $x^2$.

Notice that the commutativity of this pushout square means that the image in $C^\infty(D)$ of the squared coordinate function $x^2 \in C^\infty(\mathbb{R})$  is zero. This is  because the former is the image under the bottom morphism of the coordinate function $x$ and that maps under the right vertical morphism to $0 \in C^\infty(*) \simeq \mathbb{R}$.

In fact one can show that the [[pushout]] $C^\infty(D)$ is _generated_ by the coordinate function $x \in C^\infty(\mathbb{R}))$ subject to the relation $x^2 = 0$: it is the [[ring of dual numbers]]

$$
  \begin{aligned}
    C^\infty(D) & \simeq C^\infty(\mathbb{R})/(x^2)
    \\
    & \simeq \mathbb{R}[x]/(x^2)
  \end{aligned}
  \,.
$$

(Here in the second step we have used [[Hadamard's lemma]].)

This computation of the smooth algebra $C^\infty(D)$ of smooth functions on $D$ expresses the intuitive idea that $D$ is such a small neighbourhood of $0$ in $\mathbb{R}$ that while the canonical coordinate function of $\mathbb{R}$ restricted to $D$ is non-vanishing (it would vanish only when restricted entirely to the point 0) it is "so small that its square is always 0". Readers who like this intuitive statement should keep it in mind, as it is accurate and useful, whereas readers who do not like it could stick with the above precise definition, which unwinds this slogan in the [[internal logic]] of the ambient [[smooth topos]] that we keep alluding to.

In order to remind us about the infinitesimal nature of the generator $x$ in $C^\infty(D)$ we shall usually call this "$dx$" and write

$$
  C^\infty(D) = \mathbb{R}[dx]/((dx)^2) \simeq \mathbb{R} \oplus dx \mathbb{R}
  \,.
$$

### Functions and differential 1-forms
 {#FunctionsAndForms}

We discussion of [[differential 1-forms]] are functions on the space of infinitesimal paths. For more comprehensive such discussion see ([Stel 13](#Stel13)).

Define the [[smooth locus]] $\mathbb{R}^{(\Delta^1_{inf})}$ as the [[product]] $\mathbb{R} \times D$ of the [[real line]] with the first order [[infinitesimal space|infinitesimal line segment]].
We think of this as the space of **infinitesimal paths** in $\mathbb{R}$ (see [Spaces of infinitesimal k-simplicies](infinitesimal+object#SpacOfInfSimpl)). We do so by thinking of a [[generalized element]] of $\mathbb{R}^{(\Delta^1_{inf})}$ in the category $SmoothLoc$, namely a pair $(x, \epsilon) \in \mathbb{R} \times D$, as the linear path in $\mathbb{R}$ stretching from $x$ to $x + \epsilon$ (both regarded as [[generalized element]]s of $\mathbb{R}$).

We say a **smooth [[differential form|differential 1-form]]** on $\mathbb{R}$ is a function (meaning: a [[morphism]] in $SmoothLoc$)

$$
  \omega :
  X^{(\Delta^1_{inf})} \to \mathbb{R}
$$

with the property that its restriction to $\mathbb{R}$ vanishes, hence so that the composite morphism

$$
 0 : \mathbb{R} \hookrightarrow X^{(\Delta^1_{inf})} \stackrel{\omega}{\to} \mathbb{R}
$$

vanishes, where the first morphism is the inclusion of $\mathbb{R}$ as the constant infinitesimal paths in $\mathbb{R}$, hence $(Id, 0) : \mathbb{R} \to \mathbb{R} \times D$.

We call the [[coproduct]] of [[smooth algebra]]s their [[tensor product]] (classically it is the _smoothly completed_ algebraic tensor product). For each smooth loci $X$ and $Y$, we have

$$
  C^\infty(X \times Y) \simeq C^\infty(X) \otimes C^\infty(Y)
  \,.
$$

Using this property, we can dualize the above definition of a diferential 1-form. Namely, in the [[opposite category]] of [[smooth algebra]]s, a 1-form on the line is a morphism

$$
  C^\infty(\mathbb{R}) \otimes \mathbb{R}[dx]/((dx)^2)
   \leftarrow
  C^\infty(\mathbb{R})
  :
  \omega^*
  \,,
$$

hence an assignment $ x \mapsto g(x) +  h(x) dx $ such that $g(x) = 0$. In other words, it is any element in $C^\infty(\mathrm{R}) \otimes \mathbb{R}[dx]/((dx)^2)$ that is proportional to the generator $dx$. We write $ \omega = h dx $ with $h \in C^\infty(\mathbb{R})$.

### Derivatives as infinitesimal differences

We can now "[[synthetic differential geometry|synthetically]]" describe differentiation as the process of taking infinitesimal differences of functions.

Consider the two morphisms of smooth loci

$$
  + : \mathbb{R}^{(\Delta^1_{inf})} = \mathbb{R} \times D \to \mathbb{R}
$$

and

$$
  p_1 : \mathbb{R}^{(\Delta^1_{inf})} = \mathbb{R} \times D \to \mathbb{R}
  \,,
$$

where the first is the restriction of the canonical addition operation $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ of [[real number]]s, and where the second is the [[projection]] on the first factor out of the [[product]].

Given an element $f \in C^\infty(\mathbb{R})$, hence a [[smooth function]] $f : \mathbb{R} \to \mathbb{R}$, we can then form two functions on infinitesimal paths by precomposing with these two morphisms:

first we get

$$
 f : X^{(\Delta^1_{inf})} \stackrel{p_1}{\to} \mathbb{R}
   \stackrel{f}{\to} \mathbb{R}
  \,.
$$

This is the function on infinitesimal paths in the line that only depends on the starting point of a path, not on the length of the path, and which sends that starting point to the value of $f$ on that point. Therefore by slight abuse of notation we just keep writing "$f$" for this function.

The second combination is

$$
 \tilde f : X^{(\Delta^1_{inf})} \stackrel{+}{\to} \mathbb{R}
   \stackrel{f}{\to} \mathbb{R}
  \,.
$$

This is the function on infinitesimal paths that sends any path $(x) \to (x + \epsilon)$ to the value of the function on the _endpoint_ $(x + \epsilon)$ of the paths: in terms of these [[generalized element]]s $\tilde f$ is the assignment

$$
  \tilde f : (x , \epsilon) \mapsto f(x + \epsilon)
 \,.
$$

For instance consider the function $f : x \mapsto x^2$. Then

$$
  \begin{aligned}
   \tilde f : (x, \epsilon) &\mapsto (x + \epsilon)^2
    \\
    & = x^2  + 2 x \epsilon + \epsilon^2
    \\
    & = x^2 + 2 x \epsilon
   \end{aligned}
   \,.
$$

This is evident, if maybe still somewhat mysterious, in the internal language of the ambient [[smooth topos]] that we keep alluding to, but to make it entirely explicit and concrete notice that the functions here are dually morphisms of smooth algebras

$$
  C^\infty(\mathbb{R}) \otimes (\mathbb{R} \oplus dx \mathbb{R})
  \leftarrow
  C^\infty(\mathbb{R})
  :
  (\tilde f)^*
$$

and the equation

$$
  (x + dx)^2 = x^2 + 2 x dx
$$

holds in $C^\infty(\mathbb{R}) \otimes (\mathbb{R} \oplus dx \mathbb{R})$ by the [above discussion](FunctionsAndForms), where now $x \in C^\infty(\mathbb{R})$ and $dx \in C^\infty(D) = \mathbb{R} \oplus dx \mathbb{R}$ are the two canonical generators.

But notice that the expression $x^2 + 2 x dx $ is not a dfifferential 1-form by the above definition, equivalently the function $\tilde f$ does not vanish when restricted to constant infinitesimal paths.

To get such, we form the **[[derivative]]** or **infinitesimal difference**, the function

$$
  \tilde f - f : X^{(\Delta^1_{inf})} \to \mathbb{R}
$$

that acts on [[generalized element]]s of the space of infinitesimal paths in the line as

$$
  (\tilde f - f) : (x, \epsilon) \mapsto
   f(x + \epsilon) - f(x)
  \,.
$$

By construction on the right this now is something linear in the infinitesimal length $\epsilon$ of the path. Write $f' \in C^\infty(\mathbb{R})$ for the coefficient:

$$
  f(x + \epsilon) - f(x) = \epsilon f'(x)
  \,.
$$

This function $f'$ is the **[[differential]]** of $f$. It is the function that gives the change of the function $f$ along a "unit infinitesimal path".

Dually, we have a morphism

$$
  C^\infty(\mathbb{R})\otimes (\mathbb{R} \oplus dx \mathbb{R})
  \leftarrow
  C^\infty(\mathbb{R})
  :
  (\tilde f - f)^*
$$

that sends

$$
  x \mapsto f(x + dx) - f(x) = f'(x) dx
  \,.
$$

The 1-form on the right is the [[de Rham differential]] of $f$, usually written

$$
  (d f)(x) = f'(x) dx \in C^\infty(\mathbb{R} \times D)
  \,.
$$

### As a map of tangent bundles
 {#InfinitesimalDifferentiationAsAMapOfTangentBundles}

Equivalently we may rephrase this differential $d f$ as a morphism of [[tangent bundles]] $d f \colon T \mathbb{R} \longrightarrow T \mathbb{R}$ as follows.

First, by the [[Kock-Lawvere axiom]] of SDG we have the standard fact that $T \mathbb{R} := \mathbb{R}^D \simeq \mathbb{R} \times \mathbb{R}$. Therefore given a function

$$
  \mathbb{R} \times D \longrightarrow \mathbb{R}
$$

this is by the [[internal hom]]-[[adjunction]] equivalently a function of the form

$$
  \mathbb{R} \longrightarrow \mathbb{R}^D = T \mathbb{R}
  \,.
$$

The condition on the original function makes this [[adjunct]] be a [[section]] of the [[tangent bundle]] of $\mathbb{R}$. This section is $x \mapsto f'(x)$, hence is the derivative of $f$ regarded as a [[tangent vector]] on $\mathbb{R}$. Moreover, since $\mathbb{R}$ is a [[microlinear space]], this induces by rescaling a function

$$
   T \mathbb{R} = \mathbb{R} \times \mathbb{R} \longrightarrow T \mathbb{R}
  \,.
$$

This is the differential of $f$ regarded as a map of tangent bundles.

## Related concepts

* [[product law]], [[chain rule]]

* [[Jacobian matrix]], [[Hessian matrix]], [[extremum]]

* [[partial differentiation]]

* [[Hadamard lemma]], [[Fermat theory]]

* [[derivative of distributions]]

* [[pullback of differential forms]]

* [[tangent space]], [[tangent bundle]]

* [[jet space]], [[jet bundle]], [[metric jet]]

[[!include infinitesimal and local - table]]

## References

### General

The [[Kock-Lawvere axiom]] for the [[axiom|axiomatization]] of differentiation in [[synthetic differential geometry]] was introduced in

* {#Kock77} [[Anders Kock]], _A simple axiomatics for differentiation_,  Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 ([JSTOR](http://www.jstor.org/stable/24491223))

Another [[synthetic mathematics|synthetic]] formalization of the chain rule in [[differential cohesion|differential]] [[cohesive homotopy type theory]] is given in

* {#Wellen18} [[Felix Wellen]], section 3 of _[[schreiber:thesis Wellen|Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

Discussion of differential forms as functions on infinitesimal simplices is in

* {#Stel13} [[Herman Stel]], _Cosimplicial $C^\infty$ rings and the de Rham complex of Euclidean space_ ([arXiv:1310.7407](http://arxiv.org/abs/1310.7407))

### History

Discussion of the history of the concept of differentiation, with emphasis on its roots all the way back in [[Zeno's paradoxes of motion]] is in

* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949


diffiety

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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

#Contents#
* tabel of contents
{:toc}

## Idea

The concept of _diffiety_ ([Vinogradov 81](#Vinogradov81)) reflects the concept of _[[partial differential equation]]_ (generally non-linear) in analogy to how the concept of _[[algebraic variety]]_ reflects that of _[[polynomial]] [[equation]]_:

A _diffiety_ is the [[solution]]-locus $\mathcal{E}\hookrightarrow J^\infty_\Sigma(E)$ of a [[partial differential equation]] $D \Phi = 0$  regarded as an ordinary [[equation]]  $\tilde D = 0$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ of some [[bundle]] $E \overset{fb}{\to} \Sigma$.

Here $\Sigma$ is the space of [[free variables]] of the [[PDE]], $E \overset{fb}{\to} \Sigma$ is the bundle of [[dependent variables]], and a [[differential operator]]

$$
  D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)
$$

on the [[space of smooth sections]] of $fb$ is represented by a [[bundle morphism]]

$$
  \tilde D \colon J^\infty_\Sigma(E) \to F
$$

out of the [[jet bundle]] via [[jet prolongation]] $j^\infty_\Sigma \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(J^\infty_\Sigma(E))$ as $D \Phi = \tilde D (j^\infty_\Sigma(\Phi))$:

$\,$

$$
  \array{
    & \text{diffiety} && \text{jet bundle} & \array{\text{differential} \\ \text{operator}} &
    \\
    &
    \mathcal{E}
      &\overset{\tilde D = 0}{\hookrightarrow}&
    J^\infty_\Sigma(E)
      &\overset{\tilde D}{\longrightarrow}&
    F
  }
$$

For instance in [[Lagrangian field theory]] the bundle in question is a [[field bundle]] $E \overset{fb}{\to} \Sigma$, the [[partial differential equation]] is the [[Euler-Lagrange equation]] $\delta_{EL}\mathbf{L} = 0$, and its diffiety solution locus $\mathcal{E}$ inside the [[jet bundle]] $J^\infty_\Sigma(E)$ is called the _[[shell]]_ of the field theory.

In [Marvan 86](#Marvan86) it was observed that Vinogradov's [[formally integrable PDE|formally integrable diffieties]] are equivalently the [[coalgebra over a comonad|coalgebras]] over the [[jet comonad]] acting on [[locally pro-manifold]]-[[bundles]] (over a base space $\Sigma$ of [[free variables]]). This statement generalizes to the [[synthetic differential geometry]] of the [[Cahiers topos]] $\mathbf{H}$ ([Khavkine-Schreiber 17](#KhavkineSchreiber17)), where the [[jet comonad]] is realized as the comonad corresponding to [[base change]] along the [[de Rham shape]] projection $\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma$. By [[comonadic descent]] this implies that over [[formally smooth]] base spaces $\Sigma$ [[formally integrable PDE|formally integrable diffieties]] are equivalently the [[bundles]] over the [[de Rham shape]] $\Im \Sigma$:

$$
  PDEs_\Sigma(\mathbf{H})
  \;=\;
  Diffeties_\Sigma(\mathbf{H})
  \;\simeq\;
  J^\infty_\Sigma CoAlg\left(\mathbf{H}_{/\Sigma} \right)
  \;\simeq\;
  \mathbf{H}_{/\Im(\Sigma)}
$$

([Khavkine-Schreiber 17, thorem 3.52, theorem 3.60](#KhavkineSchreiber17))

This makes manifest how [[diffieties]] are the analog in [[differential geometry]] of concepts in [[algebraic geometry]]: For $\Sigma$ a suitable [[scheme]] then a [[quasicoherent module]] over its [[de Rham shape]] $\Im \Sigma$ ("[[crystal]]") is called a _[[D-module]]_ and represents an algebraic [[linear partial differential equation]], while a [[relative scheme]] over $\Im \Sigma$ is called a _[[D-scheme]]_ and represents a general algebraic [[partial differential equation]]. See also at _[[D-geometry]]_ for more on this.

## Related concepts

* [[variational calculus]]

## References

### General

* {#Vinogradov81} [[Alexandre Vinogradov]], _Geometry of nonlinear differential equations_, Journal of Soviet Mathematics 17 (1981) 1624–1649 ([doi:10.1007/BF01084594](https://doi.org/10.1007/BF01084594))

* {#Vinogradov84} [[Alexandre Vinogradov]], _Local symmetries and conservation laws_, Acta Appl. Math., Vol. 2, 1984, p. 21 ([MR85m:58192](http://www.ams.org/mathscinet-getitem?mr=736872), [doi](http://dx.doi.org/10.1007/BF01405491))

* {#Vinogradov89} [[Alexandre Vinogradov]], _Symmetries and conservation laws of partial differential equations: basic notions and results_, Acta Appl. Math., Vol. 15, 1989, p. 3. [MR91b:58282](http://www.ams.org/mathscinet-getitem?mr=1007340), [doi](http://dx.doi.org/10.1007/BF00131928)

* [[Alexandre Vinogradov]], _Scalar differential invariants, diffieties and characteristic classes_, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, [MR92e:58244](http://www.ams.org/mathscinet-getitem?mr=1098525)

* G. Cicogna, G. Gaeta, _Lie-point symmetries in bifurcation problems_, Annales de l'institut Henri Poincaré (A) Physique théorique, 56 no. 4 (1992), p. 375-414 [numdam](http://www.numdam.org/item?id=AIHPA_1992__56_4_375_0)

* L. Vitagliano, _Hamilton-Jacobi diffieties_, [arxiv/1104.0162](http://arxiv.org/abs/1104.0162)

* [[Alexandre M. Vinogradov]]'s [webpage](http://diffiety.ac.ru/curvita/amv.htm)

* [[Joseph Krasil'shchik]]'s [webpage](http://diffiety.ac.ru/curvita/isk.htm) (with links to some papers) and [wiki list of publications](http://gdeq.org/index.php?title=Joseph_Krasil%27shchik)

### Review

* [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], _Homological methods in equations of mathematical physics_, [arxiv:math.DG/9808130](http://xxx.lanl.gov/abs/math/9808130v2), 150 p.

* [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], Geometry of jet spaces and integrable systems, J. Geom. Phys. (2010), [doi](doi:10.1016/j.geomphys.2010.10.012), [arXiv:1002.0077](http://arxiv.org/abs/1002.0077)

See also

* wikipedia: [diffiety](http://en.wikipedia.org/wiki/Diffiety), [secondary calculus and cohomological physics](http://en.wikipedia.org/wiki/Secondary_calculus_and_cohomological_physics)

### As jet coalgebras
 {#ReferencesAsJetCoalgebras}

Diffieties as [[coalgebra over a comonad|coalgebras]] over the [[jet comonad]] are discussed in

* {#Marvan86} [[Michal Marvan]], _A note on the category of partial differential equations_, in _Differential geometry and its applications_, Proceedings of the Conference August 24-30, 1986, Brno ([[MarvanJetComonadPDE.pdf:file]])

* {#MarvanThesis} [[Michal Marvan]], thesis, 1989 ([[MarvanThesis.pdf:file]], [web](http://www.slu.cz/math/cz/lide/marvan-michal/docs/mar89-ocr.pdf))

* {#Marvan89} [[Michal Marvan]], _On the horizontal cohomology with general coefficients_, 1989 ([web announcement](http://old.math.slu.cz/People/MichalMarvan/Annotations/horizontal.php), [web archive](http://dml.cz/dmlcz/701469))

* {#Marvan93} [[Michal Marvan]], section 1.1 of _On Zero-Curvature Representations of Partial Differential Equations_,  (1993) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5631))

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

### Conferences
 {#Conferences}

* [diffiety school](http://school.diffiety.org), [diffiety institute](http://diffiety.ac.ru), [gdeq.org wiki](http://gdeq.org/index.php?title=Welcome_to_GDEq.org!)

*

 [XXI Summer Diffiety School School on Geometry of PDEs](https://sites.google.com/site/levicivitainstitute/Activities/DiffietySchools/xxi-summer-diffiety-school), July 19 - 31, 2018

[[!redirects diffeity]] [[!redirects Diffiety]] [[!redirects diffieties]]


Digambara

Summary:

Definition of Stochastic Transtheistic Pre-determinism(Popper):
	There exists a spirit who knows now (i.e. at present instance) every event of the future.

Black swan events: Even if two particles or organisms always had exactly same properties till present instance of time, it is possible that they may have different properties in the next instance of time.

Physicists rarely mention the concept of block universe and how it solves the problems of Quantum Mechanics. They do not mention it partly because it conflicts with the Big Bang theory and mechanical framework of universe and partly because it violates free will of experimenter. However, it is explainable with pre-determinism.

Also, Stochastic Pre-determinism makes additional falsifiable predictions:
a.	Substance dualism is true ie consciousness is fundamental not emergent
b.	Metempsychosis is true
c.	Senses are not the source of knowledge but are channels or mediums of knowledge. It is possible to acquire knowledge of future, past, remote and fine through non-sensory channels.
d.	Discrete spacetime

This largely matches with Digambara Axioms and Gödel’s axioms.

Contents:

1.	Possible and popular interpretations
2.	Proposed interpretation
3.	Argument from Physics
4.	Argument from Discrete Mathematics/Computer Science
5.	Argument from Crowdsourcing
6.	Argument from Mainstream Mathematics
7.	Argument from Invention Discovery Equivalence
8.	Consequences
9.	Continuum Hypothesis
10.	Appendix
10a.	Two games in Town: Continuous SpaceTime/Big Bang/Emergent Consciousness vs Discrete SpaceTime/Digambara Eternal Universe/Fundamental Consciousness
10b.	Three outstanding problems, one logically consistent solution
10c.	Why you have not heard of Block Universe before
10d.	UCLA Professor opinion
10e.	Note I by Jeffrey D Long
10f.	Note 2 by Jeffery D Long
10g.	Note 3 by Jeffery D Long
10h.	Life as per Robert Rosen

1. There is no doubt at all that the issue of determinism versus indeterminism was a central, dominating theme of Popper's thought. There are three mutually exclusive and mutually exhaustive frameworks that doesn’t require supernatural cause: Scientific Determinism, Transtheistic Pre-determinism and Indeterminism.

Current two most popular frameworks are:

a.	Scientific Consensus (Scientific Determinism/Closed Universe/Presentism):
Body is “a complex machine” that works according to the deterministic laws of a Newtonian world. Brain is made of matter, which must follow the laws of nature. There is no ghost in the machine. Quantum mechanics play no part in workings of brain.
- Stephen Hawking

b.	Mainstream Consensus (Absolute Indeterminism/Open Universe/Presentism):

My three arguments against scientific determinism are: approximate nature of scientific knowledge, asymmetry of past and future and inability of predicting growth of our knowledge with time (because if we can predict growth of our future knowledge then that knowledge will be immediately available which is a contradiction.) These arguments, while conclusive against scientific determinism, are not conclusive against pre-determinism. But they are sufficient for me to conclude that future is open.
-	Karl Popper

Mathematics assumes human beings can acquire perfect knowledge. Science says perfect knowledge is not possible. Two ideas are in direct conflict. I sided with science because I was not very good with mathematics, so I rather question the assumptions on which mathematics is based.
- George Soros

A determinist model of brain cannot claim a scientific status because it could not be falsified by testing.
- George Soros

2. Proposed Framework (Stochastic Transtheistic Pre-determinism/Block Universe/Eternalism):

Stochastic Transtheistic Pre-determinism asserts that:
a.	All events in this world are fixed, or unalterable, or predetermined
b.	It also asserts that they are known to a spiritual being and
c.	They are not predictable by scientific means i.e., it admits black swan events.
Thus, it primarily asserts that the future is as little changeable as is the past. Everybody knows what we mean when we say that the past cannot be changed. It is in precisely the same sense that the future cannot be changed, according to pre-determinism.

Perceived finite indeterminism is actually synthesized at every instance of time by psycho-chemical interaction between two infinities in opposite directions: the positive infinite intellect of each living entity that is capable of knowing past and future with perfection is being relentlessly interfered by or attenuated by negative infinite fine matter at every instance of time to realize finite indeterministic cognition for each of the living entity. There is indeed ghost in the machine. Pre-determinism does not preclude the appearance of indeterminism and realization of black swan events.

3. Argument from Physics:

..the effort to reduce all concepts and correlations to as few as possible logically independent basic concepts and axioms.

There has always been present the attempt to find a unifying theoretical basis, consisting of a minimum of concepts and fundamental relationships, from which all the concepts and relationships might be derived by logical process. This is what we mean by the search for a foundation of the whole of physics. The confident belief that this ultimate goal may be reached is the chief source of the passionate devotion which has always animated the researcher.

From what has been said it is clear that the word foundations in this connection does not mean something analogous in all respects to the foundations of a building. Logically considered, of course, the various single laws of physics rest upon this foundation. But whereas a building may be seriously damaged by a heavy storm or spring flood, yet its foundations remain intact, in science the logical foundation is always in greater peril from new experiences or new knowledge than are the branch disciplines with their closer experimental contacts. In the connection of the foundation with all the single parts lies its great significance, but likewise its greatest danger in face of any new factor. When we realize this, we are led to wonder why the so-called revolutionary epochs of the science of physics have not more often and more completely changed its foundation than has actually been the case.
- Albert Einstein

Foundation of physics needs to change if we want to account for consciousness.
- Ed Witten

....it seems to be that a 'fundamental' physical theory that lays claim to any kind of completeness at the deepest levels of physical phenomena must also have the potential to accommodate conscious mentality. Some people will try to evade (or belittle) this problem, arguing that consciousness simply 'emerges' as some sort of 'epiphenomenon'. Accordingly, so it would be claimed, there is no importance, for the emerge of consciousness, in the precise type of physics that happens to underlie the relevant (not necessarily biological) processes. A standard position is that of computational functionalism, according to which it is merely computational activity. I have argued strongly against this view (partly using reasoning based on Gödel’s theorem and the notion of Turing computability), and I have indeed suggested that consciousness actually depends upon the missing .. theory.
- Roger Penrose

As alluded by Einstein, A small change in foundation of physics can cause a big change in understanding of universe without altering most of the secondary sciences.

There are two competing foundations or frameworks: Aristotelian Framework and Digambara Framework.

First, the similarities:
•	Abstract objects don’t exist
•	Intellect cannot be subsumed into matter because it is fundamentally different type of object compared to matter

There is only one subtle difference between Aristotelian Framework and Digambara Framework of Universe. However, this very subtle change in framework completely reverses some of the results related to cognition, morality, and happiness.

Aristotelian Framework says law of excluded middle can be applied only to finite sets because human mind cannot comprehend completed infinity. Also, physical universe is finite with very few exceptions.

Digambara Framework says law of excluded middle can be applied to infinite sets because	human mind can comprehend completed infinity. Also, infinities are much more common, widespread, and integral in universe than implied by Aristotelian Framework.

https://plus.maths.org/content/index.php/does-infinity-exist

This page is maintained by Cambridge University
See reference to eastern infinity in comments

Following corollaries automatically follow from Digambara Framework:
•	Eternal recurrence and by extension rebirth is true https://en.wikipedia.org/wiki/Eternal_return
•	Materialism is false. Eternal "individual knowledge field" is the source of knowledge while senses and mind are only channels or medium of knowledge.
•	This "individual knowledge field" or intelligence, with which each individual is endowed including animals, is infinitely more powerful than matter (finite state machine) and is capable of knowing entire universe in a single instance of time with all of its general and special details.

Following corollaries automatically follow from Aristotelian Framework:
•	Eternal recurrence and by extension rebirth is false. https://en.wikipedia.org/wiki/Eternal_return#Christian_response
•	Mind cannot exist independent of body / cognitive process cannot exist independent of manipulative Process. Knowledge is acquired
o	through eyes via action-at-a-distance and
o	through other senses and mind locally.
•	Human knowledge is finite as it is sourced from senses and mind which are material and finite.

Digambaras point to Gödel’s incomplete theorem as proof for their framework. Aristotelians says existential proofs are not admissible in science. You must provide a constructive proof aka a real example of at least one individual who possesses infinite knowledge to convince us. Digambaras say art of realizing infinite knowledge is lost.

https://en.wikipedia.org/wiki/Existence_theorem
https://en.wikipedia.org/wiki/Constructive_proof

Digambara Framework agrees with Digambara logic.

Digambara logic goes to extra-ordinary length to explain workings of the universe by purely logical means without resorting to any supernatural causes. Digambara logic tries to identify and differentiate between life and matter and their origins - both at microscopic level as well as at macroscopic level - without resorting to any supernatural causes. Digambara logic also tries to explain their inter-dependent evolutions - both at microscopic level as well as at macroscopic level - without resorting to any supernatural causes. Digambara logic also tries to explain their transformations from microscopic level to macroscopic level and vice versa - without resorting to any supernatural causes.

Aristotelian Framework is largely in agreement with western religions and science.

Another difference is that Aristotelian foundation limits consciousness and free will, whatever it means, to only human beings. Digambaraism, in contrast, asserts that difference between consciousness of human beings and other biological beings is that of quantity and not of quality.

So, to summarize: The proposal is to add an additional factor, variously known as "life" or "knowledge fields" or "intelligence" to the list of current fundamental concepts like space, time, motion, rest and matter. And add metempsychosis as the fundamental mechanism of interaction between intelligence and matter.

4. Argument from Computer Science:

It is not possible to create random events from deterministic means - Van Neuman. In other words, it is not possible to create random events from purely inert means which implies that random events can only be created by life concepts i.e., by non-inert means which implies that life concepts are also fundamental irreducible concepts like space, time, motion, rest and matter. Each individual life as understood by mainstream biology has evolved from one such individual primitive microscopic life through a chain of births and deaths.

Corollary: As a true random sequence is an infinite sequence (mathematical consensus), each individual life is past-eternal and future-eternal spending most of time in primitive microscopic form because those forms are found to be the most numerous and are the simplest, lowest-energy state of life. https://en.wikipedia.org/wiki/Nigoda

5. Argument from Crowdsourcing:

About a quarter of Americans believe that reincarnation is true.

Liberalism rightfully maintains existence of "individual soul" as axiomatic but doesn't provide formal proof and mismodel the principle due to their use of Aristotelian logic and their sole focus on human beings.

Here, a more comprehensive approach is taken in modelling individual soul. It is done by replacing Aristotelian logic with Digambara logic.

Or in other words it is proved, startlingly, that Digambaraism, which has the strictest interpretation of individualism, is true.

6. Argument from Mainstream Mathematics:

Quote from Wikipedia article on Mathematics:
"There is no general consensus about exact scope or epistemological status of Mathematics."

There are two models of knowledge: Aristotle's model of knowledge and Digambara model of knowledge. Please see Wikipedia's article on infinity to confirm.

Mathematicians don't like contradictions. They like consistency.
Hence, they assume that everything is knowable i.e. knowledge is infinite. Reasoning being that you can't be 100% sure that something is unknowable without first knowing everything, which is a contradiction and mathematicians avoid contradictions even at the expense of uncommon assumptions.

Mainstream Science's model of universe is based on Aristotle's axioms and Aristotle's model of knowledge. Aristotle's model of knowledge is not compatible with mainstream Mathematical model of knowledge. Hence, Mainstream Science's model of Universe is not compatible with fundamental mainstream Mathematical axioms.

Digambara model of universe is based on Digambara model of knowledge. Digambara model of knowledge is compatible with mainstream Mathematical model of knowledge. Hence, Digambara model of Universe is compatible with fundamental mainstream Mathematical axioms.

Digambara model of Universe predicts universe has no beginning and that reincarnation is true which violates Aristotle axioms but are affirmed by mainstream Mathematical axioms. Digambara model of Universe also predicts human mind is potentially infinitely more powerful than Turing machine which violates Aristotle axioms but is affirmed by mainstream Mathematical axioms. Digambara model of Universe also predicts universe is completely comprehensible i.e., there are no "unknown unknowns" which violates Aristotle axioms but is affirmed by mainstream Mathematical axioms. Digambara model of Universe also predicts substance dualism is true i.e., mind can survive without brain which violates Aristotle axioms but is affirmed by mainstream Mathematical axioms.

Mainstream Science's model of Universe predicts there is beginning of the Universe which violates mainstream Mathematical axioms but is compatible with Aristotle axioms. Mainstream Science's model of Universe also predicts human mind cannot be more powerful than Turning machine which violates mainstream Mathematical axioms but is compatible with Aristotle axioms. Mainstream Science's model of Universe also predicts universe is not completely comprehensible which violates mainstream Mathematical axioms but is compatible with Aristotle axioms. Mainstream Science's model of Universe also predicts substance dualism is false i.e., mind is machine which violates mainstream Mathematical axioms but is compatible with Aristotle axioms.

Digambara axioms are not compatible with Aristotle's axioms but are compatible with mainstream Mathematical axioms. Original Mathematics was also based on Aristotle's axioms. However, George Cantor, a German mathematician, improved upon axioms of mathematics by properly modelling "actual infinities". Cantor, being a devout Christian, as well as his contemporaries, were startled by Cantor's results and were well aware of profound philosophical implications of his work (that human beings can know completed infinity).

Christianity is largely compatible with Aristotle's axioms, but not with mainstream Mathematical axioms. Individualism part of Christianity is compatible with both Aristotle's axioms and Digambara/Mathematical axioms.

Aristotle/Christianity: Everything is finite except God. Individual exists.
Current consensus: Everything is finite except for Space and Time. God doesn't exist. Individual exists. Person and Property. John Locke. Karl Popper.
Atheism/Buddhism: Everything is finite except for Space and Time. God doesn't exist. Individual doesn't exist.

Digambara: Many more things are infinite apart from space and time. knowledge/intelligence come in two flavors: potentially infinite and actual absolute infinite. Hence, no requirement of existence of separate absolute indirect creator aka God. Individual exists without requirement of existence of Creator (Independent Origination with Permanence).

Any Physicist/Mathematician who argues along this line is automatically branded as a Platonist. This implicitly implies that mainstream physicists are Aristotelian.

Mainstream viewpoint based on axioms of Aristotle is not consistent with experiments.

CIA has done research in extra sensory perception. Here is their conclusion:

A large body of reliable-experimental evidence points to the inescapable-conclusion that extrasensory perception does exist as a real phenomenon, albeit characterized by rarity and lack of reliability. It appears as a low-capacity, high-noise information channel exhibiting data rates orders of magnitude less than normal perceptive processes. Almost by definition extra-sensory perception must involve in an essential way the operation of the human mind. There exists no satisfactory theoretical understanding of these phenomena.
-	CIA

Wikipedia says all mathematicians are epistemologically Platonists without knowing it.
I would restate that statement, based on above, as: All mathematicians are epistemologically Digambaras without knowing it.

All scientists are epistemologically empiricist.
All mathematicians are epistemologically Digambaras.

"Epistemological status of mainstream mathematics is Digambaraism and scope of mainstream mathematics is observable universe as well as non-observable universe in all of its generalities and specialties"

"Epistemological status of mainstream science is empiricism and scope of mainstream science is observable universe"

7. Argument from Invention Discovery Equivalence:
Our civilization has rigorous differentiation between invention vs discovery.

Laws are eternal but Universe has beginning. So, laws are discovered whereas everything else is man-made. Works of mathematicians, podcasters, theoretical physicists, and computer programmers is eternal/discovered and hence is not patentable, but the work of pharmacists-physician-GMO is created/invented and hence patentable. This results in degradation of works of mathematicians, podcasters, theoretical physicists, computer programmers and technologists (discoverers and hence non-patentable) for the benefit of pharmaceutical companies, physicians and GMO companies (so called inventors and hence patentable). However, according to Digambara universe, everything is discovered. All knowledge already exists.

Sharp distinction between invention and discovery has also been refuted by Church-Turing thesis half a century ago. This implicitly favors Digambara universe and refutes Aristotelian universe.

However, our laws still strongly practice unprincipled distinction between invention and discovery in conformance with Aristotelian universe.

8. Consequences

Aristotelian model implies that "extreme ascetism" is not the source of real happiness. Aristotelians believe "golden mean" of consumption and austerity is the source of real liberty and real happiness. In fact, extreme ascetism is considered a social ill in modern Aristotelian society and is banned. Digambara model, on the other hand, implies that "physical and mental restraint" is a virtue and the source of real liberty and real happiness.

Aristotelian Framework says universe should be exploited by human beings so long as it is done on a sustainable basis. Digambara Framework says each "knowledge field" in the universe is separate and independent. Each one should not be disturbed as far as possible.

Aristotelian Framework says extrovertism is virtue while Digambara Framework says introvertism is virtue.

Above are three complete reversals observed from just a small change in Framework.

9. Continuum Hypothesis
Continuum Hypothesis in mathematics states that there are no sets which are larger than “countably infinite” sets, but smaller than “uncountably infinite” sets. There is no formal proof of it but is widely held to be true. Digambaraism also affirms Continuum Hypothesis and calls these two types of sets as “noncountable sets” and “endless sets” respectively.

10. Appendix:

Mathematics says infinite knowledge can be realized. Science says infinite knowledge cannot be realized.

Potentially infinite knowledge grasps information serially producing vibrations in the process. Realized infinite knowledge grasps all the information in the universe with all its generalities and specialties simultaneously in a single instance of time.

Aristotle/ Humanism / Abrahamism / Buddhism / Science says Humans cannot realize infinite knowledge, either only God can or no one can.

Gödel / Digambaraism / Mathematics says Humans can realize infinite knowledge and can grasp all the information in the universe with all its generalities and specialties simultaneously in a single instance of time.

10a. Two games in Town:

1. Big Bang/String Theory (mainstream scientists)
•	Forced to fit in arbitrary Aristotelian Axioms (this statement actually copied from a comment of the most popular science blog)
•	Property Dualism: All lives including humans have beginning at birth and end at death. Continued exploitation and enslavement of non-human lives.
•	Mind and brain are identical.
•	Not compatible with individualism and interactionism in the absence of God.

2. Digambaraism (supported (partially) by rebel physicists)
•	Compatible with global mathematical Axioms.
•	Universe has no beginning no end (existential proofs are mainstream in mathematics).
•	Substance Dualism: All lives including humans have no beginning and no end and they are transformable from one type to another.
•	Mind is (potentially) infinitely more knowledgeable than finite state machine. Mind has ability to know entire universe in one instance of time with all its generalities and specialties. While this sounds startling, it is in line with fundamental mathematical axiom of excluded middle (that it is self‐evident or knowable a priori, or a presupposition of all knowledge, or that it cannot be denied without self‐contradiction), also aptly known as "principle of omniscience". This is also in agreement with first conjecture in Gödel’s Disjunction, which Gödel believed to be true.
•	Compatible with individualism and interactionism without any additional non-observable dimension/god/God.
•	Has discrete spacetime which is compatible with both causal set theory – a suggested replacement of general relativity and quantum mechanics.
•	Fixed background as speculated by exiled theoretical physicist Lee Smolin (Lorentzian Ether Theory, preferred version of motion and rest). Also considered unsettled by Popper.
•	Compatible with free will theorem of quantum mechanics.

If there is conflict of opinion about potentiality of mind between science and mathematics, which side should you choose? If you choose mathematics, then Digambaraism, substance dualism, individualism and interactionism are true. If you choose science, then materialism is true.

10b. Three problems, one logically consistent solution:

There are three outstanding problems
1. Relation between math and science and unreasonable effectiveness of math in explaining universe
2. Relation between general relativity and quantum mechanics
3. What happens before life and after death

Only one solution that matches with empirical experience, mental experience, and reasoning without resorting to supernatural cause: Digambara Ontology, Epistemology and Soteriology.

Epistemological axioms, Ontological axioms and Soteriological axioms of a candidate model of universe have to be consistent with each other and can be derived from each other using logic, direct observations, direct experiences etc.

Digambaraism is epistemologically, soteriologically, and ontologically consistent.

Constitution of universe should not have contradictions. Otherwise, you can reach to different results depending on the starting point, which is not possible.

10c. Internet Article: Why Haven't you Heard of the Blockworld Before?

Good question! The basic answer is that you haven't been reading books and papers on modern physics or philosophical papers or books dealing with time. Oh, and, even within physics it is kind of an obscure subject and there are six basic reasons for that.

Reason number 1 is that it sounds crazy. Almost everyone, including most physicists who study modern physics are locked into a view of the world that came from the 19th Century, before Einstein's special relativity appeared in 1905. Philosophers call it presentism. In presentism, it looks like we have all these particles here and they were formed at the Big Bang and all the particles are moving forward in time, the past is gone and the future isn't out there yet.

The blockworld view says something else altogether. It says the past, present and future are all equally real and all of it is frozen in place like a sculpture carved in stone, then it is our consciousness that is moving through the blockworld. This is very hard to believe even though it is obviously true. To quote Einstein:

For we convinced physicists the distinction between past, present, and future is only an illusion however persistent.

Reason number 2 is that the blockworld seems to imply that there is no such thing as free will and physicists generally want to believe in free will. It looks like whatever happens in the blockworld, including whatever it is we decide was fixed in advance. Physicists would be right that we have no free will if all people consisted of is matter since it is the matter that is frozen in place. But if there is a spiritual part to a human being, a human soul, that is moving through the blockworld the human soul could be free to make decisions.

Reason number 3 is that the blockworld concept that comes from relativity and it has been widely believed for a long time that relativity and quantum mechanics (the other part of modern physics) don't fit together very well. To physicists, this makes it look like something important is missing and so they've spent an awful long time trying to make relativity and quantum mechanics fit together. But times are changing a bit. One researcher involved in the study is physicist Antoine Suarez and here is a quote from him:

... the ongoing work on nonlocality is helping us to better understand the relationship between quantum theory and relativity. Initial mis-understandings and controversies hid a deep unity which is now appearing.
Relativity and quantum theory share the very same experimental basis, and derive from the same principles. They are two inseparable aspects of one and the same description of the physical reality.

In addition to Suarez there are also a small number of physicists who are proposing the relational blockworld interpretation of quantum mechanics where they argue that relativity and quantum mechanics fit together very nicely.

Reason number 4 is big. Physicists who've been studying the matter all realize that the blockworld implies that there is an omniscient person who knows the future.

One also knows that the experiments violating the Bell inequalities are compatible with the orthodox relativity if one is prepared to embrace "pre-determinism" ... .

However, this view's demanding philosophical underpinnings, such as its denial of (libertarian) free will and evident need for an all-knowing agent might limit its appeal ...

Fifth reason is that the subject is basically very theoretical and at this time it does not appear that study of this subject will produce anything practical like bigger bombs or faster computers. So there is no reason for anyone to invest in it.

Sixth reason is that it implies that big bang theory is false.

Those are the reasons you haven't heard about the blockworld before. The important bottom line from all this is that the blockworld proves there is a spiritual part to human beings, a soul that is moving through the blockworld. It is all pretty hard for scientists to embrace, so they haven't, at least not so far.

10d. Dean Buonomano, a professor at UCLA and author of "Your Brain Is a Time Machine: The Neuroscience and Physics of Time", gives a simple explanation of the theory of eternalism:

In the context of physics, there’s two general views of the nature of time. One we can think of is "presentism," which only the present is real. And the second, we can think of as "eternalism" in which the past, present, future are equally real. And under this view, now is to time as here is to space.

It’s called the "block universe" view in physics — in which everything has, in a sense, a manner of speaking, already happened.

10e. It would be difficult to conceive of a system of thought and practice more radically at odds with the dominant materialistic paradigm of Western modernity than Digambaraism. I find that when I teach my course on the “Religions of South Asia” at Elizabethtown College, the tradition that my students consistently find to be most “other” to their values and way of life is not Hinduism, Buddhism, Sikhism, or Islam, but Digambaraism. Why is this so? Whereas most of my students tend to assume that the material world exists for human consumption, for the gratification of human physical needs and desires, Digambaraism teaches that, for human beings, the material world is primarily that which is to be renounced. Whereas most of my students celebrate their physical existence as something to be enjoyed, the religious among them regarding life as “sacred,” Digambaraism teaches that the most course in the life is one of “mental and physical restraint” pursued in the name of liberation from physical existence. Whereas most inhabitants of the region in which Elizabethtown College is located are voracious meateaters, Digambaraism teaches not only vegetarianism, but even the avoidance of injury, as much as possible, to insects and micro-organisms.
-	Jeffery D Long

10f. Among the numerous intellectual traditions indigenous to South Asia, the Digambara tradition is distinctive in a number of important ways. Not least among these is its view of consciousness as inherent to all living beings. According to Digambara tradition, not only are all living beings possessed of consciousness, but this consciousness is also, in each being, potentially infinite in extent. Indeed, the ultimate soteriological goal of the Digambara path, as a spiritual practice, entails the attainment of a unique, absolute awareness (kevala jñāna) in which the practitioner realizes a state of literal omniscience. The attainment of this state is taken, in the Digambara tradition, to be an extremely difficult achievement that requires many lifetimes of rigorous ascetic practice. For beings who have not attained this state – the vast majority of beings in the cosmos – consciousness is realized only in a relative way that is dependent upon the amount and type of karma to which a given being is bound. Karma is seen in the Digambara tradition as a type of matter that adheres to a living being, or jīva, and obscures the infinite potential inherent in that being, thus giving rise to the great variety of organisms that exist in the world. This Digambara understanding of consciousness gives rise to a set of doctrines – the Digambara doctrines of relativity – that are among the most important contributions of the Digambara tradition to world philosophy. This is an approach which affirms the relativity of perspectives but avoids a skeptical relativism that would deny the possibility of ever grasping the truth at all.
-	Jeffery D Long

10g. In taking the implications of his research seriously, Tucker has raised the question, “What scientific model of reality might explain phenomena of this kind better than materialism is able to do?”

In Tucker’s work, importantly, he does not display an extensive or in-depth knowledge of Indian philosophical traditions such as Vedanta, Digambaraism, or Buddhism. This is important because he cannot be accused of having a communal axe to grind in advancing his theory, one implication of which is that these traditions have been right about many things in the basic account of reality that they give. Tucker’s theory puts forth the idea that consciousness is fundamental to the nature of being. Certain aspects of quantum theory also fit well with Digambara philosophy. The fact that the same entity can be validly described as both a wave and a particle, for example, is consistent with the Digambara teaching of anekantavada, the multi-faceted nature of reality, nayavada, the teaching that reality can be viewed from many perspectives, and syadvada, the teaching that truth can be expressed in a variety of seemingly contradictory ways. And like Digambaraism, of course, it affirms the phenomenon of rebirth itself–including rebirth across species. Interestingly, Tucker’s cases even include one in which a child seems to remember a past life as a snake. This affirms not only the view of consciousness as distinct from the physical body, but also their view that consciousness is not limited to human beings.
-	 Jeffery D Long

10h. Why are living things alive? As a theoretical biologist, Robert Rosen saw this as the most fundamental of all questions-and yet it had never been answered satisfactorily by science. The answers to this question would allow humanity to make an enormous leap forward in our understanding of the principles at work in our world.

For centuries, it was believed that the only scientific approach to the question "What is life?" must proceed from the Cartesian metaphor (organism as machine). Classical approaches in science, which also borrow heavily from Newtonian mechanics, are based on a process called "reductionism." The thinking was that we can better learn about an intricate, complicated system (like an organism) if we take it apart, study the components, and then reconstruct the system-thereby gaining an understanding of the whole.

However, Rosen argues that reductionism does not work in biology and ignores the complexity of organisms. Life Itself, a landmark work, represents the scientific and intellectual journey that led Rosen to question reductionism and develop new scientific approaches to understanding the nature of life. Ultimately, Rosen proposes an answer to the original question about the causal basis of life in organisms. He asserts that renouncing the mechanistic and reductionistic paradigm does not mean abandoning science. Instead, Rosen offers an alternate paradigm for science that takes into account the relational impacts of organization in natural systems and is based on organized matter rather than on particulate matter alone.

Rosen said that organization must be independent from the material particles which seemingly constitute a living system. As he put it:

“The human body completely changes the matter it is made of roughly every 8 weeks, through metabolism, replication, and repair. Yet, you're still you --with all your memories, your personality... If science insists on chasing particles, they will follow them right through an organism and miss the organism entirely.”

Central to Rosen's work is the idea of a "complex system," defined as any system that cannot be fully understood by reducing it to its parts. In this sense, complexity refers to the causal impact of organization on the system as a whole. Since both the atom and the organism can be seen to fit that description, Rosen asserts that complex organization is a general feature not just of the biosphere on Earth-but of the universe itself.

Links:

https://www.cambridge.org/us/academic/subjects/religion/buddhism-and-eastern-religions/pravacana-sara-kunda-kunda-acarya-together-commentary-tattva-dipika-amtacandra-suri
Digambara Bible, Cambridge University Press

Experimental confirmation: Reincarnation research at University of Virginia: https://med.virginia.edu/perceptual-studies/

Experimental confirmation: CIA's Extra Sensory Perception research: https://www.cia.gov/readingroom/document/cia-rdp96-00787r000200190002-0

https://plus.maths.org/content/index.php/does-infinity-exist

https://plus.maths.org/content/index.php/do-infinities-exist-nature-0

https://towardsdatascience.com/when-science-and-philosophy-meet-randomness-determinism-and-chaos-abdb825c3114


digital identity

* wikipedia [digital identity](https://en.wikipedia.org/wiki/Digital_identity)

> A digital identity is information on an entity used by computer systems to represent an external agent. That agent may be a person, organization, application, or device. ISO/IEC 24760-1 defines identity as "set of attributes related to an entity".

* Telegram Passport [manual](https://core.telegram.org/passport), [example](https://core.telegram.org/passport/example), [telepass.me](https://telepass.me), [guide](https://technandu.com)

> Telegram is introducing a new feature called Passport, a single sign-on option for services that need actual real-world IDs or documents. Passport will let users store their files and data in Telegram's cloud and pass it directly on to services that need that information (via Neowin).

* [OAuth 2.0](https://oauth.net/2) -- industry-standard protocol for authorization; There is a controversy, as the earlier standard, OAuth 1.0, is arguably more secure; see wikipedia.
* Microsoft decentralized identity whitepaper [pdf](https://query.prod.cms.rt.microsoft.com/cms/api/am/binary/RE2DjfY)

* The digital identity is used in various online activities and particular care and many efforts in standardization are related to finance (cf. [[zoranskoda:fintech]]).

* Related pages include [[distributed computing]], [[blockchain]]

category: computer science


digital signal

\tableofcontents

\section{Idea}

A _digital signal_ is a [[signal]] which conveys information about digital data, the latter being typically a sequence of [[bit|bits]]. [[IQ modulation]] in one of its several forms is the [[waveform modulation|modulation]] technique typically used to create the signal, that is to say, to construct a [[waveform]] from the digital data.

\section{Sending of a digital signal}

Several steps are typically involved in creating a waveform from a sequence of bits.

* [[scrambling of a bit sequence|Scrambling]]


digraph

{:principle: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

***

This page is about the concept of [[directed graphs]] that is usual in combinatorics. For the notion of directed graph as commonly understood in category theory, see [[quiver]]. For some commentary on how the two formalizations relate to one another, see [[directed graph]].

***

$\,$

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}

#### Graph Theory
+--{: .hide}
[[!include graph theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In combinatorics, a **digraph** (a shortening of *[[directed graph]]*)  consists of a [[set]] and a [[binary relation]] on that set that is [[irreflexive relation|irreflexive]]. The elements of the set are called *nodes* or *vertices*, and elements of the relation are called *edges* or *arcs*; the idea is that whenever an ordered pair $(x, y)$ belongs to the relation, then we depict it as an arrow or directed edge going from node $x$ to node $y$.
The irreflexivity condition means there is never an edge from a node to itself.

## Definition and basic notions

In this section we collect some definitions that are fundamental to digraph theory, but presented largely from the point of view of category theory.

+-- {: .num_defn #Digraph}
###### Definition
**(digraph)**

A *digraph* is a [[pair]] $(V,A)$ of [[sets]], with $A\subseteq (V\times V)\setminus\{ (v,v)\colon v\in V\}$.
Here, $V\times V$ denotes the [[product]], and ${}\setminus{}$ the [[relative complement]] in the [[category of sets]].
The elements of $V$ are called [[vertices]], the elements of $A$ are called [[arcs]]. For $a = (v, w)$ an arc, we call $v$ the *source* of $a$ and $w$ the *target* of $a$, also denoted $s(a)$ and $t(a)$ respectively.

A *morphism* from a digraph $(V, A)$ to a digraph $(V', A')$ is a function $f: V \to V'$ such that $(f(v), f(w)) \in A'$ whenever $(v, w) \in A$. Digraphs and their morphisms form a [[category]] denoted $\mathsf{Dgr}$.

=--

A digraph $(V, A)$ may be regarded as a [[quiver]] in an evident way, with source and target functions $s, t: A \rightrightarrows V$. It is straightforward that $\mathsf{Dgr}$ is (equivalent to) a [[full subcategory]] of the category of quivers [[Quiv]]. Of course most quivers are not digraphs since quivers can have parallel edges: distinct edges $a, b$ where $s(a) = s(b)$ and $t(a) = t(b)$.

Some useful examples of digraphs are based on [[ordinals]]:

+-- {: .num_example #ExampleDigraph}
###### Example
If $\alpha$ is an ordinal, we may form a digraph $(V, A)$ where $V$ is the underlying set of $\alpha$ and $A$ is the relation $\{(i, i+1): i, i+1 \in \alpha\}$. We call this an *ordinal digraph* and denote it by the ordinal itself.
=--

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

**(walk in a digraph)**
Suppose $D=(V,A)$ is a digraph.
A *walk in $D$* is a digraph-morphism $\alpha \to D$ where $\alpha$ is an ordinal digraph.

=--

Notice that for $\alpha = 1$, the hom-set $\mathsf{Dgr}(1, D)$ may be identified with the set of vertices of $D$. The definition of graph morphism implies that the underlying set functor $U = \mathsf{Dgr}(1, -): \mathsf{Dgr} \to Set$ is [[faithful functor|faithful]], and realizes $\mathsf{Dgr}$ as a [[concrete category]].

For $\alpha = 2$, the hom-set $\mathsf{Dgr}(2, D)$ may be identified with the set of arcs of $D$. However, the arc-set functor $\mathsf{Dgr}(2, -): \mathsf{Dgr} \to Set$ is very far from being faithful.

If we let $C$ denote the full subcategory of $\mathsf{Dgr}$ consisting of the ordinal digraphs $1, 2$, then the inclusion $i: C \hookrightarrow \mathsf{Dgr}$ induces a functor $\mathsf{Dgr} \to Set^{C^{op}} = Quiv$ according to the usual "[[nerve]]" construction

$$\mathsf{Dgr} \to Set^{C^{op}}$$

$$D \mapsto (c \mapsto \mathsf{Dgr}(i c, D))$$

and this nerve functor is exactly the full embedding $\mathsf{Dgr} \to Quiv$. Since full embeddings reflect and preserve isomorphisms, a digraph morphism $f: D \to D'$ is an isomorphism iff the vertex-function $\mathsf{Dgr}(1, f)$ and arc-function $\mathsf{Dgr}(2, f)$ are isomorphisms (which is in any case obvious from the definition of digraph morphism, but this underscores the general importance of walks).

Generally we will only be interested in walks $\alpha \to D$ such that $\alpha \leq \omega$ in the sense of ordinals; here $\omega$ is of course the first infinite ordinal. Unwinding the definitions, here is an elementary description of such walks:

+-- {: .num_defn #WalkElementary}
###### Definition
**(walk in a digraph; elementary definition)**

Suppose $D=(V,A)$ is a digraph.
A *walk* in $D$ consists of

* a [[sequence]] $P\colon\alpha\rightarrow V$, with  $\alpha\leq\omega$,

such that

* for all $i\in\alpha$, if $i+1\in\alpha$, then $(P(i),P(i+1))\in A$.

The phrase *$\alpha$-vertex path* means any *path whose domain is $\alpha$*.

=--

Some special cases: If $\alpha=0$, then $P$ is called *the empty walk*. We have already seen that a $1$-walk is tantamount to a vertex, and that a 2-walk is tantamount to an edge. We call a walk $\omega \to D$ a *ray* in $D$.

+-- {: .num_example #cycle}
###### Example
Another useful example of a digraph is a finite [[cyclic order]]. For each $n \geq 2$, there is a digraph $Z_n$ whose vertex set is the underlying set of the finite cyclic group $\mathbb{Z}/n$, and whose arc relation is the set $\{(i, i+1): i \in \mathbb{Z}/n\}$.
=--

## Categorical properties of $\mathsf{Dgr}$

The category $\mathsf{Dgr}$ is not especially well-behaved for a category theorist. For example, it doesn't have a terminal object (no, the ordinal digraph $1$ is not terminal), and it does not admit all coequalizers. In both cases the culprit is the irreflexivity condition.

It does however have products indexed over nonempty sets, and equalizers. These limits are computed as they are in [[Quiv]], e.g., the quiver product of two digraphs is again a digraph and is their product in $\mathsf{Dgr}$. Such limits are easy to compute because $Quiv$, being a presheaf topos $Set^{C^{op}}$, has limits (and colimits) that are computed pointwise/objectwise.

It follows from these observations that $\mathsf{Dgr}$ has pullbacks which are simply $Quiv$-pullbacks; they are computed objectwise at the objects $c = 1, 2$, i.e., one simply takes pullbacks in $Set$ at each of the vertex and edge levels.

In our development, it will be useful to characterize various classes of [[monomorphisms]] in $\mathsf{Dgr}$. We begin with plain monomorphisms.

+-- {: .num_prop #Mono}
###### Proposition
A morphism of digraphs $f: D \to D'$ is monic iff its underlying function $U(f)$ is monic ([[injective function|injective]]).
=--

+-- {: .proof}
###### Proof
It is well-known that faithful functors $U$ reflect monomorphisms and epimorphisms. For suppose $U(f)$ is monic. Given a diagram in $\mathsf{Dgr}$ of the form

$$C \underoverset{h}{g}{\rightrightarrows} D \stackrel{f}{\to} D'$$

if we have $f \circ g = f \circ h$ then $U(f) \circ U(g) = U(f) \circ U(h)$, whence $U(g) = U(h)$ by monicity of $U(f)$. Since $U$ is faithful, it follows that $g = h$; therefore $f$ is monic. The argument for epimorphisms is dual.

For the other direction, use the fact that $U = \mathsf{Dgr}(1, -)$ is representable. It follows immediately from the definition of monomorphism that representable functors preserve monomorphisms, so if $f$ is monic, then so is $U(f)$.
=--

(Still to come: characterization of regular monomorphisms.)

## Paths, trails, and cycles

+-- {: .num_defn #Path}
###### Definition
**(path in a digraph)**

Suppose $D=(V,A)$ is a digraph.
A *path* in $D$ is a walk $P: \alpha \to D$ whose underlying function is [[injection|injective]].

=--

Or what is the same, that $\mathsf{Dgr}(1, P)$ is a [[monomorphism]] in $Set$. In view of Proposition \ref{Mono}, we may also define a path to be a monic walk. (Notice this digraph concept of path is at variance with the concept of [[path]] in topology which is simply a continuous map $[0, 1] \to X$ into a [[topological space]], where there is no assumption that its underlying function is injective.)

+-- {: .num_defn #Trail}
###### Definition
**(trail in a digraph)**
Suppose $D=(V,A)$ is a digraph.
A *trail* in $D$ is a walk $\alpha\overset{P}{\longrightarrow}V$ such that the [[partial function]] $e_P: \alpha\rightarrow A$ with $e_P(i)=(P(i),P(i+1))$ is injective.

=--

The domain of $e_P$ is canonically identified with the arc set of $\alpha$, and the trail condition is equivalent to saying that $\mathsf{Dgr}(2, P)$ (which we again denote as $e_P$), as a function between edge sets $\mathsf{Dgr}(2, \alpha) \to \mathsf{Dgr}(2, D)$, is a monomorphism in $Set$.

We have the following easy result.

+-- {: .num_prop #PathsAreTrails}
###### Proposition
In a digraph $D$, every path is a trail.
=--

+-- {: .proof}
###### Proof
Since we just observed that a path $P$ is the same as a monic walk, and since representable functors like $\mathsf{Dgr}(2, -)$ preserve monomorphisms, it follows that $e_P = \mathsf{Dgr}(2, P)$ is a monomorphism.
=--

Continuing the theme of monic digraph morphisms, we introduce the notion of cycle:

+-- {: .num_defn}
###### Definition
An $n$-*cycle* in a digraph $D$ is a monic morphism $P: Z_n \to D$ (see Example \ref{cycle}). A *cycle* in $D$ is just an $n$-cycle in $D$ for some $n$. A digraph $D$ is *acyclic* if it admits no cycles.
=--

There is a natural bijection between digraph morphisms $C: Z_n \to D$ and walks $P: \{0, \ldots, n\} \to D$ such that $P(0) = P(n)$. Indeed we have a digraph quotient map $q: \{0, \ldots, n\} \to Z_n$ defined as the mapping $i \mapsto i \mod n$, and the walk associated with $C: Z_n \to D$ is $P = C \circ q$. The map $q$ induces a bijection on arcs:

$$\mathsf{Dgr}(2, q): \mathsf{Dgr}(2, \{0, \ldots, n\}) \stackrel{\sim}{\to} \mathsf{Dgr}(2, Z_n).$$

Thus if $C: Z_n \to D$ is *monic*, we have a composite injective map in $Set$,

$$\mathsf{Dgr}(2, \{0, \ldots, n\}) \underoverset{\mathsf{Dgr}(2, q)}{\sim}{\to} \mathsf{Dgr}(2, Z_n) \underset{\mathsf{Dgr}(2, C)}{\to} \mathsf{Dgr}(2, D),$$

which implies that the walk $P$ associated with a cycle $C: Z_n \to D$ is a trail.

## Symmetrizing and "weak" notions

For various digraph (directed graph) notions above, it can be useful to consider parallel notions for *undirected* graphs. A slick method for doing this is by invoking a "symmetrizing" [[functor]] construction.

+-- {: .num_defn}
###### Definition
For a digraph $D$, we define $Symm(D)$ to have the same vertices as $D$, and the arc relation to be the symmetric closure of the arc relation of $D$. In other words, $(v, w)$ is an arc of $Symm(D)$ iff $(v, w)$ or $(w, v)$ is an arc of $D$.
=--

Clearly the symmetric closure of an irreflexive relation remains irreflexive, so $Symm(D)$ is again a digraph, and the assignment $D \mapsto Symm(D)$ is the object part of an obvious functor $Symm: \mathsf{Dgr} \to \mathsf{Dgr}$. Also the monomorphism $D \to Symm(D)$ that is the identity on vertices is the component $u_D$ of a [[natural transformation]]

$$u: 1_\mathsf{Dgr} \to Symm.$$

+-- {: .num_theorem}
###### Theorem
The functor $Symm$ carries an [[idempotent monad]] structure.
=--

+-- {: .proof}
###### Proof
It is clear that $u_{Symm(D)}: Symm(D) \to Symm^2(D)$ is an [[isomorphism]] (and even an identity map) because the operation of taking the symmetric closure of a relation is itself idempotent. Similarly $Symm(u_D)$ is also an identity map: $u_{Symm(D)} = Symm(u_D)$. The multiplication $m: Symm^2 \to Symm$ of the monad is defined by taking the inverse: $m = u_{Symm}^{-1}$. Then $m$ is natural and the unit law $m \circ u_{Symm} = 1$ is automatic; the other unit law $m \circ Symm(u_D)$ follows. The associativity of $m$ follows by inverting a naturality square for $u$.
=--

Morally we may consider the category of undirected graphs to be the category of algebras over the monad $Symm$. In any case, this is a full subcategory of $\mathsf{Dgr}$ consisting of digraphs of the form $Symm(D)$.

We proceed to "weaken" various of the notions above to take into account the undirected graph context, simply by applying the functor $Symm$. For example:

* If $\alpha$ is an ordinal digraph, a *weak walk* $\alpha \to D$ is defined to be a walk $\alpha \to Symm(D)$.

(Obviously, a weak walk $\alpha \to D$ need not be literally a walk $\alpha \to D$; cf. [[red herring principle]].)

Alternatively, we could define a weak walk to be a digraph morphism $Symm(\alpha) \to Symm(D)$ where $\alpha$ is an ordinal digraph: the two ways of defining weak walks are (naturally) equivalent.

Similarly,

* A *weak trail* $\alpha \to D$ is defined to be a trail $\alpha \to Symm(D)$.

* A *weak cycle* $Z_n \to D$ is defined to be a cycle $Z_n \to Symm(D)$.

## Miscellaneous notions

In [contemporary](#MO277288) mathematics, ordered pairs $(x,y)$, $(y,x)$ are called *antiparallel arcs*.

+-- {: .num_defn #Neighbourhoods}
###### Definition
**(in-neighborhood, out-neighbourhood)**

Suppose $D=(V,A)$ is a digraph and $v\in V$.
The *in-neighborhood* of $v$ in $D$, denoted $N^{\rightarrow}_D(v)$, is the set $\{ u\in V\colon (u,v)\in A\}$. The *out-neighborhood* of $v$ in $D$, denoted $N^{\leftarrow}_D(v)$, is the set $\{ u\in V\colon (v,u)\in A\}$.

=--

We note that the rationale for the slightly non-standard notation $N^{\leftarrow}_D(v)$ is to have it be intuitively understandable: the arrow points away from the vertex and points to the letter which stands for the set which contains the out-neighbours.

+-- {: .num_defn #Degrees}
###### Definition
**(in-degree, out-degree)**

Suppose $D=(V,A)$ is a digraph and $v\in V$.
The *in-degree* of $v$ in $D$, denoted $d^{\rightarrow}_D(v)$, is the [[cardinality]] of the in-neighborhood of $v$ in $D$. The *out-degree* is defined analogously.

=--

The axiom implies---digraphs being *irreflexive*--- that always $\neg \, (P(i)=P(i+1))$. Walks are only allowed to be non-injective *every other step*, so to speak.

+-- {: .num_example #TheLeastInjectiveWalk}
###### Example
In a sense the *least injective* walk is obtained by considering any digraph containing a *2-cycle* (see below) $C$, taking $\alpha=\omega$ and defining $P$ to alternate back and forth on $C$ forever.
This walk has a two-element image.
=--

+-- {: .num_defn #StartAndEndVertices}
###### Definition
**(start vertex, $A$-$B$-walk in a digraph)**

Suppose $D=(V,A)$ is a digraph.
Suppose $\alpha\overset{P}{\rightarrow} V$ is a non-empty walk in $D$.
Then $P(0)$ is called the *start vertex* of $P$.
Suppose that $A,B$ are [[subsets]] of $V$, not necessarily disjoint.
Then $P$ is an $A$--$B$-path if and only if $\alpha\gt 0$ is a finite ordinal and $P(0)\in A$, $P(\alpha-1)\in B$ and $0\lt i\lt \alpha-1$ implies $\neg\, (P(i)\in A\cup B)$.
If $A=\{a\}$ and $B=\{b\}$ are singletons, one also writes $a$--$b$-path for *$\{a\}$--$\{b\}$-path*.

=--

Discussion of weak trails from a previous revision mentioned Power's [proof](#Power1990) of unique interpretability of [[pasting schemes]].

The standard term in topology for a path which is *free of self-intersections* is [[arc]] (eg [Kowalksky 1965](#Kowalksky65), Definition 29b).
As a saving grace, the standard way to rigorously define the arcs of *plane* digraphs uses precisely the topological [[arcs]].

Allowing 2-cycles, yet neither loops nor parallel arcs, has become a  standard meaning of *digraph* in combinatorics.

+-- {: .num_defn #Acyclic}
###### Definition
**(acyclic digraph (DAG); classical definition)**

Suppose $D=(V,A)$ is a digraph.
Then $D$ is called *acyclic* if and only if there does not exist any cycle in $D$.

A usual abbreviation in combinatorics for *acyclic directed graph* is *DAG*.

=--

Here, "classical" refers to the negative definition.

The set of all weak cycles of a digraph $D$ is denoted $\mathrm{WeCy}(D)$.

The following non-standard definition is useful in when constructing proofs of Power's pasting theorem:

+-- {: .num_defn #ConstructiveAcyclic}
###### Definition
**(constructive acyclic digraph (cDAG))**

A *constructive acyclic digraph* is a triple $D=(V,A,z)$ of data

* a digraph $(V,A)$
* a function $z\colon\mathrm{WeCy}(D)\rightarrow V$

subject to the axioms

* for each $P\in \mathrm{WeCy}(D)$, $z(P) \in \mathrm{image}(P)$

* for each $P\in \mathrm{WeCy}(D)$, if (        $u\in \mathrm{image}(P)$ [[and]] (  $(u,z(P))\in A$ [[or]] $(z(P),u)\in A$  ) ) , then $(u,z(P))\in A$.

=--

In words, the axioms mean that $z$ gives us one vertex on any given weak cycle $P$ whose out-degree *on $P$* is two.
Then $z(P)$ witnesses $P$'s not being a cycle.
Evidently, a (classical) DAG can usually be *made* a cDAG in many ways: there usually are many functions $z$ one can specify and that satisfy the axioms.
We are here not interested in describing or counting these possibilities, rather in doing things with a given cDAG.

There is an apparently inescapable arbitrariness in whether to favor out-neighbors or in-neighbor when defining *constructive DAG*.
The above definition would to all intents and purposes be equivalent the one obtained by replacing the conclusion in the axiom with "then $(z(P),u)\in A$".

The following concept appears (under another name) in [Power's proof](#Power1990):

+-- {: .num_defn #King}
###### Definition
**(king, coking, $d$-king, $d$-coking, $\omega$-king, $\omega$-coking)**

Suppose $D=(V,A)$ is a digraph.
A vertex $v$ of $D$ is a *king* (resp. *coking*) of $D$ if and only if $(v,w)\in A$ (resp. $(w,v)\in A$) for all $w\in V\setminus\{v\}$.

For any finite [[cardinal number|cardinal]] $d$, a vertex $v$ of $D$ is a *$d$-king* if and only if each vertex of $D$ can be reached along a *path*, starting with $v$, the path having at most $d$ arcs.

For any finite [[cardinal number|cardinal]] $d$, a vertex $v$ of $D$ is a *$d$-coking* if and only if for each $u\in V$ there exists at least one *path* starting with $u$ and ending with $v$ and having at most $d$ arcs.

A vertex $v$ of $D$ is an *$\omega$-king* if and only if each vertex of $D$ can be reached along a path of $D$, starting with $v$, and having *any finite* number of arcs.

A vertex $v$ of $D$ is an *$\omega$-coking* if and only if for each $u\in V$ there is at least one path of $D$, starting with $u$, ending with $v$, and having *any finite* number of arcs.

=--

The term *$\omega$-king* is a natural variant of the usual term *$k$-king* in digraph theory.
By definition, for each $k\in\omega$, each $k$-(co)king is an $\omega$-(co)king.
Of course, a vertex can be a king and a coking simultaneously. (Which is why e.g. in [Power's proof](#Power1990), non-equality of the king and the coking *must be made a separate axiom* in the definition of what Power calls *plane graph with source and sink*).

+-- {: .num_defn #PlaneDigraph}
###### Definition
**(plane digraph)**

A *plane digraph* consists of the data

* a subset $V\subseteq\mathbb{R}^2$ of the [[plane]]
* a [[set]] $A$ of [[arcs]] (in the usual topological sense), each $a\in A$ of the form $[0,1]\overset{a}{\rightarrow}\mathbb{R}^2$

and the axioms

(v.connect) for each $a\in A$,  $a(0)\in V$ and $a(1)\in V$,

(free.interior) if $a\in A$ and $0\lt t\lt 1$, then $\neg\,(a(t)\in V)$,

(not.par) if $a,a'\in A$ and if $a(0) = a'(0)$ and $a(1) = a'(1)$, then $a=a'$,

(disjoint.interiors) if $a,a'\in A$ and ( $\exists t:[0,1]$, $0\lt t \lt 1$ and $\exists t':[0,1]$, $0 \lt t' \lt 1$ and $a(t)=a'(t')$ ), then $a=a'$.

=--

Note that plane digraphs are often introduced into a context like (abstract) digraphs are, i.e. by writing $D=(V,A)$, although

* the type of $D=(V,A)$ is invisible from this notation (it may be a *digraph* or a *plane digraph*)

* for a plane digraph $D=(V,A)$ we do not in general have $A\subseteq V\times V$.

We make use of the treatment of *undirected* plane graphs given in ([Diestel](#DiestelGraphTheoryFourthEd), Chapter 4), with some adaptations made for the present purposes:

* here, *arc* means arc in the usual topological sense, without exception, while, since the author focuses on undirected graphs, in [Diestel](#DiestelGraphTheoryFourthEd), "arc" is defined to be a [[subset]] of the  [[plane]], in particular is not parametrized, in particular does not carry *directional information* (compare   def. \ref{AbstractDigraphOfAPlaneDigraph} where said information is used).

* in ([Diestel](#DiestelGraphTheoryFourthEd), Chapter 4), both $V$ and $A$ in the definition of plane graphs are required to be [[finite sets]]. Here we make no such restriction (cf. \ref{ExamplePlaneDigraph}).

* for ease of reference, we calibrate the point-set topological terminology according to [[Introduction to Topology]] (example: we will say "component" instead of "region") when defining *faces*, and also [Munkres 2000](#Munkres2000).

+-- {: .num_example #ExamplePlaneDigraph}
###### Example
An example of a plane digraph with countably infinite vertex set, and whose satisfying the axioms of plane digraphs is evident, is given by $T:=\{\varphi\in\mathbb{R}\colon 0\le \varphi\lt 2\pi,\quad \varphi\in\mathbb{Q} \}$, $D=(V,A)$ with $V:=\{(0,0)\} \cup \{   (  \cos( \varphi )  , \sin( \varphi  ) ) \colon  \varphi\in M    \}$ and $a(\varphi)\colon [0,1]\rightarrow\mathbb{R}^2$ with $a(\varphi)(t) =   (  t\cdot\cos(\varphi), t\cdot \sin(\varphi)   )$ and $A:= \{  a(\varphi)\colon     \varphi\in M      \}$.
Note that this example (0) is not a [[pasting scheme|pasting scheme in the sense of Power]] (already for the strong reason that in it there does not exist any $\omega$-coking), and (1), while large, it would make for a rather dull and vacuously-uniquely-interpretable [[pasting diagram]]: there are no arrows that could possibly be composed, let alone is there any 2-cell.

In combinatorics, this is sometimes referred to as an *infinite star*.

=--

Applications of the following standard concept are e.g. the study of [[maps]] on [[surfaces]], and Power's ([proof](#Power1990)) of unique interpretability of [[pasting schemes]]:

+-- {: .num_defn #Face}
###### Definition
**(face of a plane digraph)**

Suppose $D=(V,A)$ is a plane digraph.
The *set of faces of $D$*, denoted $\mathrm{Fa}(D)$, is the set of [[connected components]], in the sense of [Section 7](Introduction+to+Topology+--+1#subspaces), of the [[relative complement]] $\mathbb{R}^2\setminus\bigcup\{   \mathrm{im}(a) \colon a\in A        \}$, where $\mathrm{im}(a)=\{   a(t)\colon 0\leq t\leq 1    \}$.

A *face of $D$* is a *member of the set $\mathrm{Fa}(D)$*.
Each face of $D$ is an open connected subspace of the plane.

=--

Of course one needs a map from *plane digraphs* to (abstract) *digraphs*:
+-- {: .num_defn #AbstractDigraphOfAPlaneDigraph}
###### Definition
**(the (abstract) digraph of a plane digraph)**

Suppose $D=(V,A)$ is a plane digraph.
Then $\mathrm{Dig}(D)=(V',A')$ is the digraph defined as follows: it has vertex set $V':=V$ and arc set $A' :=      \{    ( a(0)  ,   a(1) )  \colon    a\in A           \}$.

=--

Note that by axiom (v.connect), $A'$ is a subset of $V\times V$, and because each $a$ is an topological arc, and, as such, injective, $A'$ is moreover disjoint from the diagonal $\{(v,v)\colon v\in V\}$, so $A'$ has the type required by def.  \ref{Digraph}.

+-- {: .num_defn #DigraphTheoreticTermsForPlaneDigraph}
###### Definition
All digraph-theoretic technical terms like *out-neighborhood*, *outdegree* which are defined for *digraphs* are defined for *plane digraphs* $D$ by first applying the map $\mathrm{Dig}$ to $D$, then applying the digraph-theoretic definition, then applying the inverse of $\mathrm{Dig}$.

=--

A central technical tool in  [Power's proof](#Power1990) are *boundary walks*:
boundary walks validate a form of the [[red herring principle]]: each boundary walk is a *weak walk*, yet need not be a walk.

The boundary walks one works with when using A.J. Power's [application](#Power1990) of plane digraphs to category theory are often weak *trails*, "often" in the sense that most [[pasting diagrams]] one encounters in practice do not have any arc-repetitions.
Equivalently, the 1-cells of usual pasting diagrams have an underlying undirected graphs with is *bridgeless*.
However, this is not generally true, and to have both *walks* and *trails* is not futile but necessary: a typical example of a pasting diagram occurring naturally in category theory are [[whiskering]] diagrams: the *boundary trail* of the *exterior* face of the typical whiskering diagram *does* have arc-repetions, hence is not a trail but only a walk.

## Remarks on the definitions

The *convention* to have digraph imply that there be no loops and no parallel arcs, and resort to other terms such as *directed pseudograph* to signal loops or parallel arcs, is widespread in modern combinatorics.
For example, see p. 2 of ([Gutin and Bang-Jensen](#DG2nd)), and Section 1.2 of ([Csaba et al](#DecompositionProof)).

Unsurprisingly, while the convention that digraph implies that there be no parallel arrows has been widely adopted nowadays, there is a generous disregard for whether a digraph may contain loops. In this set-theoretical definition given above, this corresponds to the decision whether to allow $A$ to contain elements of the form $(v,v)$.

The terms *walk*, *trail* and *path* are standard in modern combinatorics. They are in particular compatible with standard usage in the theory of *undirected* graphs, in that upon forgetting (so to speak) the directions in the various definitions, one obtains the standard notions of walk, trail and path in (undirected) graph theory (for example [Figure 1.8](#BondyMurtyFirstEdition) illustrates precisely this convention).

Many (e.g. [p. 11](#DG2nd) and [Chapter 1](#BondyMurtyFirstEdition)) modern reference define walks to be sequences whose codomain conains both vertices and arcs.
For simplicity we do not do so.
Having paths consist of vertices only, and letting the structure of the target space determine the axioms seems cleaner.
We can afford to make this simplification *only* because digraphs cannot have any parallel arcs: for digraphs, our definition and the vertex-arc-alternating definition of [p. 11](#DG2nd) are equivalent to all intents and purposes.
There is, however a good reason why e.g. [Bondy and Murty](#BondyMurtyFirstEdition) use the latter definition: for them, a *directed graph* is (cf. [10.1](#BondyMurtyFirstEdition), though Bondy--Murty's formalization is different from usual [[quivers]] and a rare hybrid between purely functional and purely relation-theoretic formalizations: they use maps with codomain set $V\times V$) not a *digraph* but a [[quiver]], and---evidently---if multiple parallel arcs are permitted, *then a vertex-sequence does not contain enough information to define a "walk"*.
Upon stepping from one vertex to the next, a vertex sequence by itself cannot tell one which parallel arc to pick.

Needless to say (since given by the definitions), *in a trail, arc-repetitions are forbidden*, and in a path, any repetition is forbidden.

The perhaps oddly hybrid concept of *trail* has a crucial role in A. J. Power's ([proof](#Power1990)) of unique interpretability of [[pasting schemes]]: any *face* of a *plane digraph* determines a *boundary trail* which always is a *weak trail*.

 Generically, such a *boundary trail* has many vertex-repetitions, but it never has any arc-repetition.

The terminology "king" above is standard in modern digraph theory; the term "coking" is not, but in tune with category-theoretic usages. The term "$\omega$-king" appears not to be attested but is in tune with a standard notion of $d$-king in digraph-theory, and also with some usages in mathematics which let $\omega$ signal that some parameter is unconstrained (cf. [[omega-category]] [[omega-groupoid]]).

The choice of definitions documented here is biased towards (one of) the *uses* digraphs are put to in category theory, in particular in giving a rigorous justification of [[pasting diagrams]].
The most salient example of this is the emphasis given to *plane digraphs* in this article.

A reason for treating the concept of $\omega$-kings here is A. J. Power's proof ([Power 1990](#Power1990)) of a *pasting theorem*, a rigorous justification of the notational practice of [[pasting diagrams]]: therein, both $\omega$-kings and $\omega$-cokings play an important role (Power calls $\omega$-kings "sources" and $\omega$-cokings "sinks"; both these terms clash with two standard technical terms in, respectively, contemporary digraph theory and flow theory, which is why the alternative terms were chosen).

Moreover, Power makes crucial use of a concept of *addition of $a$--$b$-paths*, which is one of the reasons why $A$--$B$-paths have been  introduced.

## Uses of digraphs in category theory

One example of why digraphs are relevant to category theory are [[pasting schemes]], which are a rigorous justification of the (older) notational practice of [[pasting diagrams]].
Pasting diagrams are fundamental to [[higher category theory]]: already the axioms for various notions of higher categories are formalized in terms of pasting diagrams.
The formal justification of pasting diagrams was achieved in the late 1980.
Note that there are more than one definition of [[pasting scheme]] in the literature.
E.g. there are M. Johnson's pasting schemes (([Johnson 1987](#Johnson1987), Chapter 2), ([Johnson 1989](#Johnson1989), Section 2)) and Power's pasting schemes ([Power 1990](#Power1990)).
The definition which is strictly relevant to the present article is Power's. He defines defines pasting schemes as a special kind of digraph ([Power 1990](#Power1990), Section 3), details will be found at [[pasting scheme]].

## Related concepts

* [[geometric shape for higher structures]]

* [[loop digraph object]]

## References

* A. Bondy, U.S.R. Murty: Graph Theory with Applications. Fifth Printing. North Holland 1982.
{#BondyMurtyFirstEdition}

* B. Csaba, D. Kühn, A. Lo, D. Osthus, A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures. Memoirs of the American Mathematical Society, 244 (2016), monograph 1154, 170 pages
{#DecompositionProof}

* {#DiestelGraphTheoryFourthEd} R. Diestel, Graph Theory, 4. ed., Springer 2010

* [[Gregory Gutin]], [[Jørgen Bang-Jensen]]: _Digraphs: Theory, Algorithms and Applications_. Springer Monographs in Mathematics. Second Edition (2009)
{#DG2nd}

* [[Michael Johnson]]: _Pasting Diagrams in $n$-Categories with Applications to Coherence Theorems and Categories of Paths_, Doctoral Thesis, University of Sydney, 1987
{#Johnson1987}

* [[Michael Johnson]]: _The Combinatorics of $n$-Categorical Pasting_, Journal of Pure and Applied Algebra 62 (1989)
{#Johnson1989}

* Manfred Weis (https://mathoverflow.net/users/31310/manfred-weis), Calculating Cost-Optimal 1-Factors in Digraphs, URL (version: 2017-07-26): https://mathoverflow.net/q/277288
{#MO277288}

* [[James Munkres]]: _Topology_, Prentice Hall. Second Edition (2000)
{#Munkres2000}

* [[John Power]]: _A 2-Categorical Pasting Theorem_, Journal of Algebra 129 (1990)
{#Power1990}

* {#Kowalksky65} H. J. Kowalksky: Topological Spaces. Academic Press. 1965.

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