infinitesimal extension

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#### Algebra
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#### Formal geometry
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[[!include formal geometry -- contents]]
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#Contents#
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## Idea

An _infinitesimal extension_ of a [[ring]] $R$ is [[quotient]] map $\widehat R \to R$ whose [[kernel]] is a [[nilpotent ideal]], hence a nilpotent [[ring extension]].

(Only if the map $\widehat R \to R$ is finite, such as for [[infinitesimally thickened point|Weil  algebras]], then this may be cast as "epimorphism with nilpotent kernel", by [[Stacks Project]] [lemma 10.106.6](http://stacks.math.columbia.edu/tag/04VM).)

If already the product of any two elements in the kernel is zero, this is also called a _square-zero extension_. See also at _[[tangent category]]_ and at _[[Mod]]_ for more on this.

## Examples

For a [[Noetherian ring]] $R$ (for instance a [[finitely generated ring]]), the projection from $R$ to its [[reduced ring]] $R_{red}$ is an example of an infinitesimal extension. In general, however, the kernel of $R\to R_{red}$ is not nilpotent.

## Related concepts

* [[infinitesimal object]], [[infinitesimally thickened point]], [[Artin algebra]]

* [[deformation theory]]

* [[module]], [[tangent category]]

* [[formally étale morphism]]

[[!include infinitesimal and local - table]]

## References

A category of infinitesimal extensions regarded as a [[site]] over which to characterize [[formally etale morphisms]] by means of an [[infinitesimal shape modality]] is considered in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], section 2.6 of _Noncommutative spaces_, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenbergSpaces}

Discussion in the context of [[simplicial algebras]] is in

* Mauro Porta, [[Gabriele Vezzosi]], _Infinitesimal and square-zero extensions of simplicial algebras_ ([arXiv:1310.3573](http://arxiv-web3.library.cornell.edu/abs/1310.3573))

Further discussion in the context of [[higher algebra]] is in

* [[Jacob Lurie]], _[[Deformation Theory]]_

Infinitesimal neighborhoods of subvarieties play role in completion of subvarieties and various definitions of [[formal schemes]], the ring case of course plays a role, cf.

* J. Illusie, Grothendieck theorem [[FGA explained]]

[[!redirects infinitesimal extensions]]

[[!redirects square-0 extension]]
[[!redirects square-0 extensions]]
[[!redirects square-zero extension]]
[[!redirects square-zero extensions]]

[[!redirects square-0 ring extension]]
[[!redirects square-0 ring extensions]]
[[!redirects square-zero ring extension]]
[[!redirects square-zero ring extensions]]

[[!redirects infinitesimal ring extension]]
[[!redirects infinitesimal ring extensions]]

[[!redirects nilpotent ring extension]]
[[!redirects nilpotent ring extensions]]


infinitesimal flat modality

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#### Cohesion
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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#Contents#
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## Idea

In a context of [[synthetic differential geometry]]/[[differential cohesion]] the _infinitesimal flat modality_ is the [[right adjoint]] in an [[adjoint modality]] with the [[infinitesimal shape modality]].

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]] forming two pairs of [[adjoint modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\& $ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]].

Here $\& $ is the **infinitesimal flat modality**.

## Properties

### Relation to crystalline cohomology

For $A$ a [[geometric homotopy type]], $\& A $ is the [[coefficient]]
for [[crystalline cohomology]] with [[coefficients]] in $A$. See there for more.

## Related concepts

[[!include cohesion - table]]


infinitesimal interval

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## Idea

The "infinitesimal interval" is an [[infinitesimally thickened point]]/[[infinitesimal disk]] which is a subspace of the [[line object]]. Usually meant is the first order infinitesimal interval, which is the formal dual to the [[ring of dual numbers]].

## Properties

The [[internal hom]]/[[mapping space]] out of the infinitesimal interval is the [[synthetic tangent bundle]]-construction. See there for more.

## Related concepts

* [[synthetic differential geometry]]

* [[synthetic tangent bundle]]

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

[[!redirects infinitesimal intervals]]


infinitesimal interval object

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#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#

* automatic table of contents goes here
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## Idea

An _infinitesimal interval object_ is like an [[interval object]] that is an [[infinitesimal space]].

Where maps out of an [[interval object]] model paths that may arrange themselves into [[path ∞-groupoid]]s, maps out of infinitesimal interval object model infinitesimal paths that may arrange themselves into [[infinitesimal path ∞-groupoid in a smooth topos|infinitesimal path ∞-groupoid]].

## Definition

In any [[lined topos]] $(\mathcal{T},(R,+,\cdot))$ the line object $R$ is naturally regarded as a [[interval object|cartesian interval object]].

$$
  \array{
    && R
    \\
    &
    {}^{0_{*}}\nearrow && \nwarrow^{1_{*}}
    \\
    {*}
    &&&&
    {*}
  }
  \,.
$$

If the [[lined topos]] $(\mathcal{T}, R)$ is also a [[smooth topos]] in that it satisfies the [[Kock-Lawvere axiom]] it makes sense to consider the [[subobject]] $D$ of $R$ defined as the [[equalizer]]

$$
  D = \lim\left( R \stackrel{\stackrel{(-)^2}{\to}}{\underset{0}{\to}} R \right)
$$

or equivalently expressed in [[topos]] [[logic]] as

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

and think of $D$ as the **infinitesimal interval object** of the [[smooth topos]] $(\mathcal{T},R)$ inside its finite [[interval object]] $R$. By the axioms satisfied by a [[smooth topos]] it is in particular an [[infinitesimal object]].

Various constructions induced by a finite [[interval object]] have their infinitesimal analog for infinitesimal interval objects.

## Infinitesimal path $\infty$-groupoid induced from infinitesimal interval

+-- {: .query}

[[Urs Schreiber]]: the following should be checked

=--

### Introduction

Recall the discussion at [[interval object]] of how the interval ${*}\sqcup {*} \stackrel{\to}{\to} R$ in $\mathcal{T}$ alone gave rise to the [[cosimplicial object]]

$$
  \Delta_R : \Delta \to \mathcal{T}
$$

of (collared) $k$-simplices modeled on $R$, and how that induces for each object $X \in \mathcal{T}$ the [[simplicial object]]

$$
  \Pi(X) := X^{\Delta_R^\bullet}
  \,.
$$

Here as an object in $\mathcal{T}$ the $k$-simplex is actually a $k$-cube $\Delta_R^k := R^k$, but equipped with face and degeneracy maps that identify the boundary of a $k$-simplex inside the $k$-cube thus realizing the interior of that boundary as the $k$-simplex proper and the exterior as its _collar_ .

We want to mimic that construction with the finite interval $R$ replaced by the infinitesimal interval object $D$, to get a simplicial object $\Pi^{inf}(X)$ for every object $X$.

While the infinitesimal situation is formally very similar to the finite situation, one technical diference is that the infinitesimal interval does not fit into a nontrivial [[cospan]] as the finite interval does. This is because $D$ typically has a _unique_ morphism ${*} \to D$ from the terminal object, as a consequence of the fact that all the infinitesimal elements it contains are genuinely [[generalized element]]s.

The natural way to encode an infinitesimal path between two elements in an object $X$ in the [[smooth topos]] $\mathcal{T}$ is therefore not as an element of $X^D$ but of $X^D \times D$, which we may think of as the space of pairs consisting of infinitesimal paths in $X$ and infinitesimal parameter lengths of these paths.

This naturally yields the [[span]]

$$
  \array{
    && X^D \times D
    \\
    & {}^{\mathllap{p_{T X}\circ p_1}}\swarrow && \searrow^{\mathrlap{ev}}
    \\
    X &&&& X
  }
  \,,
$$

where

* the left leg is the projection $X^D \times D \to X^D$ followed by the [[tangent bundle]] projection $p_{T X} = X^{(* \to D)} : X^D \to X$

* the right leg is the evaluation map, i.e. the [[inner hom]]-[[adjunct]] of $Id : X^D \to X^D$.

With this setup a pair of ([[generalized element|generalized]]) elements $x,y \in X$ may be thought of as connected by an infinitesimal path if there is an element $(v,\epsilon) \in X^D \times D$ such that

$$
  \delta_1(v,\epsilon) = v(0) = x
$$

and

$$
  \delta_0(v,\epsilon) = v(\epsilon) = y
  \,.
$$

But not all elements $(v,\epsilon)$ define _different_ pairs of [[infinitesimal neighbour]] elements this way: specifically in the case that $X$ is a [[microlinear space]], the [[tangent bundle]] object $X^D$ is fiberwise $R$-linear, and thus for any $t \in R$ the elements $(v,t \epsilon)$ and $(t v, \epsilon)$ define the same pair of [[infinitesimal neighbour]]s, $x = v(0)$ and $y = (t v)(\epsilon) = v(t \epsilon)$.

We may identify such elements $(t v, \epsilon)$ and $(v, t \epsilon)$ by passing to the [[tensor product]] $X^D \otimes_R D$, i.e. the coequalizer of

$$
  X^D \times R \times D \stackrel{\stackrel{\cdot \times Id}{\to}}{\underset{Id \times \cdot}{\to}}
  X^D \times D
$$

where $\cdot$ here denotes the [[monoid]]-[[action]] of $(R,\cdot)$ on $D$ (by the embedding $D \hookrightarrow R$) and on $X^D$ (by the fact that $X$ is assumed to be [[microlinear space|microlinear]]).

In this same fashion we can then define infinitesimal analogs of the finite higher path object $X^{\Delta_R^k} = X^{R^{\times k}}$.

### The definition

+-- {: .un_defn}
###### Definition
**(infinitesimal path simplicial object)**

Let $X \in \mathcal{T}$ be a [[microlinear space]] in the
[[smooth topos]] $(\mathcal{T},R)$ with infinitesimal
interval object $D$.

Then define the [[simplicial object]]

$$
  \Pi^{inf}(X)
  : \Delta^{op} \to
  \mathcal{T}
$$

as follows:

* in degree $n$ it assigns the object

  $$
    [n] \mapsto
    X^{D(n)} \otimes_{R^n} D(n)
    \hookrightarrow
    (X^D \otimes_R D)^{\times_X^n}
  $$

  whose [[generalized element]]s we write $(\epsilon_i v_i)_x$ with $\vec \epsilon \in D(n)$ or $(\nu_i)_x$ for short; where $x \in X$ indicates the fiber of $X^{D(n)} \to X$ that the element lives in

* the face maps $d_i : (X^D \times_R D)^{\times_X^{n+1}} \to (X^D \times_R D)^{\times_X^{n}}$ are

  * for $0 \lt i \lt n+1$

    given on [[generalized element]]s by

    $$
      d_i : (\nu_i)_x
      \mapsto
      (\nu_0 , \cdots, \nu_{i-2},
      \nu_{i-1} + \nu_i, \nu_{i+1}, \cdots, \nu_{m+1} )_x
    $$

  * for $i = 0$

    given by

    $$
      d_0 : (\nu_i) \mapsto (v_0(\epsilon_0) + \nu_i)
    $$

    where the element on the right denotes the evaluation of the map $(\nu_i) : D(n) \to X$ in its first argument on $\epsilon_0$, regarded as an element in the fiber over $v_0(\epsilon_0)$.

  * for $i = n+1$

    given by

    $d_{n+1} : (\nu_0, \cdots, \nu_{n+1}) \mapsto (\nu_0, \cdots, \nu_n)$

* the degeneracy maps $\sigma_i$ act by inserting the 0-vector in position i:

  $\sigma_i : (\ni_i) \mapsto (\nu_0, \cdots, \nu_{i-1}, 0, \nu_{i+1}, \cdots, \nu_{n})$.

=--

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

These face and degeneracy maps indeed satisfy the
[[simplicial identities]].

=--

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

This is straightforward checking that proceeds entirely analogously as the proof of the simplicial identities for the finite path $\infty$-groupoid $\Pi(X)$ discussed at [[interval object]]. See also the following remark.

=--

+-- {: .un_remark}
###### Remark

By thinking of the $v_i : D  \to X$ as infinitesimal collared curves in $X$ with source $v_i(0)$ and target $v_i(\epsilon_i)$ the above definition is an immediate analog of the definition of the path $\infty$-groupoid $\Pi(X)$ of finite paths as discussed at [[interval object]].

This is made manifest by the following construction that embeds $\Pi^{inf}(X)$ into $\Pi(X)$.

=--

### The inclusion of infinitesimal paths into finite paths ##

Recall the finite path $\infty$-groupoid $\Pi(X)$ induced from the [[interval object]]

$$
  (0_*,1_*) : * \coprod * \to R
$$

as discussed there. On object this assigns

$$
  \Pi(X) : [n] \mapsto X^{R^n}
  \,.
$$

+-- {: .un_defn}
###### Definition
**(inclusion of infinitesimal into finite paths)**

For $n \in \mathbb{N}$ define a morphism

$$
  X^{(D(n))} \otimes_{R^n} D(n)
  \to
  X^{R^n}
$$

on [[generalized element]]s by

$$
  \iota_n :
  (\epsilon_i v_i) \mapsto
  ((t_0, \cdots, t_{n-1})
   \mapsto \sum_{i=0}^{n-1} v_i(t_i \epsilon_i))
   \,.
$$

=--

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

These morphism $(\iota_n)$ respect the face and degeneracy maps on both sides and hence induce an inclusion of simplicial objects

$$
  \Pi^{inf}(X) \hookrightarrow \Pi(X)
$$

=--

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

Straightforward checking on [[generalized element]]s.

=--

### Properties {#Properties}

> under construction

Let  $X$ be a microlinear space.

**Sketch of Proposition**

We want to show that the morphism of simplicial sets

$$
  [\Delta^{op},\mathcal{T}](U \times D, X^{(\Delta_\inf^\bullet)})
  \to
  [\Delta^{op},\mathcal{T}](U , X^{(\Delta_\inf^\bullet)})
$$

induced by pullback along $U \simeq U \times * \to U \times D$
is a [[weak homotopy equivalence]].

**Sketch of proof**

First consider the case that $U$ itself has no infinitesimal
directions in that $Hom(U,D) = *$. Then we claim that the morphism
$
  [\Delta^{op},\mathcal{T}](U \times D, X^{(\Delta_\inf^\bullet)})
  \to
  [\Delta^{op},\mathcal{T}](U , X^{(\Delta_\inf^\bullet)})
$
is an acyclic Kan fibration in that all squares

$$
  \array{
    \partial \Delta[n] &\to& [\Delta^{op},\mathcal{T}](U \times D, X^{(\Delta_\inf^\bullet)})
    \\
    \downarrow && \downarrow
    \\
    \Delta[n]
    &\to&
    [\Delta^{op},\mathcal{T}](U , X^{(\Delta_\inf^\bullet)})
  }
$$

have lifts.

For $n = 0$ this says that the map is surjective on vertices, which it is, since any $U \to X$ is in the image of $U \times D \to U \times * \simeq U \to X$.

For $n=1$ we need to check that every homotopy $U \to X^D \times D$ downstairs with fixed lifts $U \times D \to X$ over the endpoints may be lifted. But by assumption $U \to D$ factors through the point, so that
the homotopy $U \to X^D \times D \stackrel{\to}{\to} X$ has the same source as target $f : U \to X$. This means the two lifts of its endpoints are morphisms $(u,\epsilon) \mapsto (f(u) + \epsilon \nu_i(u)$ for $i=1,2$ and $\nu_i$ tangent vectors.
A homotopy between these is given by a map $U \times D \to X^D \times D$
defined by $(u,\epsilon) \mapsto (f(u)+ (-)\nu_1(u) + (\epsilon-(-))\nu_2(u), \epsilon)$.

Finally, for $n \geq 2$ we have unique flllers, because by construction every morphism $\Delta[n]\cdot Y \to X^{(\Delta^\bullet_{inf})}$ is uniquely fixed by the restriction $\partial \Delta[n] \to \Delta[n]\cdot Y \to X^{(\Delta^\bullet_{inf})}$ to its boundary.

The case for $U \times D$ replaced with $U \times D^n$ works analogously.


infinitesimal neighborhood

> This entry is about what in [[nonstandard analysis]] are called *monads*. For disambiguation see at *[[monad (disambiguation)]]*.

***

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### Context
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Infinitesimal neighbourhoods
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{: toc}

## Idea

> *Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct* ([Fichte 1795, Grundriss §4.IV](Grundriss+des+Eigenthümlichen+der+Wissenschaftslehre#4IVUnendlichKleinsterTeilDesRaumes))

> ("An infinitesimally small part of space is always a space, something with continuity, but not a bare point.")

An infinitesimal neighbourhood is a "[[neighbourhood]] with [[infinitesimal]] diameter". This can be made precise in several formalisms, including [[nonstandard analysis]], [[ringed spaces]], [[synthetic differential geometry]], [[differential cohesion]], ....

{#MonadTerminology} **"Monads".** In [[nonstandard analysis]]   the infinitesimal neighbourhood of a point is traditionally called (see [below](#InNonstandardAnalysis)) its *[[monad (disambiguation)|monad]]*, apparently following the [ancient terminology going back to Euclid](monad+disambiguation#HistoricalOrigins), for more see [there](monad+disambiguation#HistoricalOrigins).

Curiously, in [[differential cohesion]] the construction of all infinitesimal neighbourhoods (the [[infinitesimal disk bundle]]) is indeed a [[monad]] in the sense of [[category theory]] (see [there](infinitesimal+disk+bundle#MonadicityAdjointToJetBundles), namely the [[left adjoint|left]] [[adjoint monad]] to the [[jet comonad]]). Whether this confluence of the terms "monads" is just a happy coincidence seems to be lost to history, see at *[[monad]]* the section *[Etymology](monad#Etymology)*.

## Definition

### In nonstandard analysis
 {#InNonstandardAnalysis}

In [[nonstandard analysis]], the

* "*[[monad (disambiguation)|monad]]*"  ([Robinson 1966, p. 57](#Robinson66) following , see also [Luxemburg 1966](#Luxemburg66), [Keisler 1976, Def. 1.2](#Keisler76), [Kutateladze 2011](#Kutateladze11))

or

* "*halo*" (eg. [Rayo 2015, Def. 4.11](#Rayo15))

of a standard point $p$ in a [[topological space]] (or even in a [[Choquet space]]) is the hyperset of all [[hyperpoints]] infinitely close to $p$. It is  the (external) set constructed as the [[intersection]] of all of the standard [[neighbourhoods]] of $p$, the _infinitesimal neighbourhood_ of $p$.

Some

### In differential cohesion
 {#DifferentialCohesion}

For $\mathbf{H}$ a context of [[differential cohesion]] with [[infinitesimal shape modality]] $\Im$, then for $x\colon \ast \to X$ a [[global point]] in any object $X \in \mathbf{H}$ the infinitesimal disk $\mathbb{D}^X_x \to X$ around that point is the ([[homotopy pullback|homotopy]]) [[pullback]] of the [[unit of a monad|unit]] $i \colon X \to \Im(X)$ of the $\Im$-monad

$$
  \array{
    \mathbb{D}^X_x &\longrightarrow& X
    \\
    \downarrow && \downarrow^{\mathrlap{i}}
    \\
    \ast &\stackrel{x}{\longrightarrow}& \Im(X)
  }
  \,.
$$

The collection of all infinitesimal disks forms the *[[infinitesimal disk bundle]]* over $X$, see there for more.

### For ringed spaces

Consider a morphism $(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X)$ of [[ringed space]]s for which the corresponding map $f^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y$ of sheaves on $Y$ is surjective. Let $\mathcal{I} = \mathcal{I}_f = Ker\,f^\sharp$, then $\mathcal{O}_Y = f^\sharp(\mathcal{O}_X)/\mathcal{I}_f$. The ring $f^*(\mathcal{O}_Y)$ has the $\mathcal{I}$-preadic filtration which has the associated graded ring $Gr_\bullet =\oplus_{n} \mathcal{I}_f^n/\mathcal{I}^{n+1}_f$ which in degree $1$ gives the [[conormal sheaf]] $Gr_1 = \mathcal{I}_f/\mathcal{I}^2_f$ of $f$. The $\mathcal{O}_Y$-augmented ringed space $(Y,f^\sharp(\mathcal{O}_X)/\mathcal{I}^{n+1})$ is called the $n$-th __infinitesimal neighborhood__ of $Y$ along morphism $f$. Its structure sheaf is called the $n$-th normal invariant of $f$.

## Related concepts

* [[jet groupoid]]

[[!include infinitesimal and local - table]]

## References

In [[algebraic geometry]] (via [[infinitesimal shape modality]]):

* [[A. Grothendieck]], _Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie_, Publications Mathématiques de l'IHÉS __32__ (1967), p. 5-361, [numdam](http://www.numdam.org/item?id=PMIHES_1967__32__5_0)

Discussion of infinitesimal neighbourhoods in [[nonstandard analysis]] and use of the "[[monad (disambiguation)|monad-terminology]]":

* {#Robinson66} [[Abraham Robinson]], p. 57 of: *Non-standard analysis*, Studies in Logic and the Foundations of Mathematics **42**, North-Holland (1966), Princeton University Press (1996) [[ISBN:9780691044903](https://press.princeton.edu/books/paperback/9780691044903/non-standard-analysis)]

* {#Luxemburg66} [[Wilhemus A. J. Luxemburg]], *A General Theory of Monads*, in: *Applications of Model Theory to Algebra, Analysis and Probability*, Holt, Rinehart and Minston (1966) 18–86 [[ark](https://archive.org/stream/in.ernet.dli.2015.141486/2015.141486.Applications-Of-Model-Theory-To-Algebra-analysis-And-Probability_djvu.txt)]

* E. I. Gordon, A. G. Kusraev, [[Semën S. Kutateladze]], Chapter 4 in: *Infinitesimal Analysis*, Mathematics and its Applications **544**, Springer (2002) [[doi:10.1007/978-94-017-0063-4](https://doi.org/10.1007/978-94-017-0063-4)]

* {#Keisler76} [[Jerome Keisler]], Def. 1.2 in: *Foundations of Infinitesimal Calculus*, Prindle Weber & Schmidt (1976, 2022) [[pdf](https://people.math.wisc.edu/~hkeisler/foundations.pdf)]

* {#Kutateladze11} [[Semën S. Kutateladze]], *Leibnizian, Robinsonian, and Boolean valued monads*, Journal of Applied and Industrial Mathematics **5** 3 (2011) 365-373 [[arxiv/1106.2755](http://arxiv.org/abs/1106.2755), [doi:10.1134/S1990478911030082](https://doi.org/10.1134/S1990478911030082)]

* [[Sergio Albeverio]], Jens Erik Fenstad, Raphael Hoegh-Krohn, *Nonstandard methods in stochastic analysis and mathematical physics*, Academic Press 1986

and using the "*halo*"-terminology:

* {#Rayo15} Diego Rayo, Def. 4.11 in: *Introduction to non-standard analysis* (2015) [[pdf](http://math.uchicago.edu/~may/REUDOCS/Rayo.pdf)]

See also:

 * Wikipedia, *[Monad (non-standard analysis)](http://en.wikipedia.org/wiki/Monad_%28non-standard_analysis%29)*

More historical commentary on the [[monad (disambiguation)|monad-terminology]]

* {#Kutateladze06} [[Semën S. Kutateladze]], *Leibniz's Definition of Monad*, NeuroQuantology **4** 3 (2006) 249-241 [[arXiv:math/0608298](https://arxiv.org/abs/math/0608298), [pdf](https://www.researchgate.net/profile/Semen-Kutateladze/publication/2130297_Leibniz%27s_Definition_of_Monad/links/0912f5058405405dbb000000/Leibnizs-Definition-of-Monad.pdf)]

Discussion in [[synthetic differential geometry]], also using the "monad"-terminology:

* {#Kock80} [[Anders Kock]], _Formal manifolds and synthetic theory of jet bundles_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **21** 3 (1980)  [[numdam:CTGDC_1980__21_3_227_0](http://www.numdam.org/item?id=CTGDC_1980__21_3_227_0)]

Discussion in [[differential cohesion]] (see at *[[infinitesimal disk bundle]]*):

* [[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))

and its formalization in differentially [[cohesive homotopy type theory]]:

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_ (2017)

[[!redirects infinitesimal neighborhood]]
[[!redirects infinitesimal neighborhoods]]
[[!redirects infinitesimal neighbourhood]]
[[!redirects infinitesimal neighbourhoods]]

[[!redirects monad in nonstandard analysis]]
[[!redirects monads in nonstandard analysis]]
[[!redirects monad in non-standard analysis]]
[[!redirects monads in non-standard analysis]]

[[!redirects halo]]
[[!redirects halos]]
[[!redirects haloes]]

[[!redirects infinitesimal disk]]
[[!redirects infinitesimal disks]]


infinitesimal neighbour > history

< [[infinitesimal neighbour]]


infinitesimal number

# Infinitesimal numbers
* table of contents
{: toc}

## Idea

The term 'infinitesimal' means the same as 'infinitely small in absolute value'; in Latin, it literally means 'infinity-eth' and should be interpreted in the sense of a fraction.  In the ordinary [[analysis]] of [[real numbers]], the only infinitesimal number is [[zero]].  However, the basic intuitions of calculus since its beginnings have dealt with infinitely small (and sometimes also infinitely large) numbers.  There are several different ways to develop a rigorous theory that includes infinitesimal numbers.

## Definition

Briefly we recall the definition of what makes a number infinitesimal, which we give in some generality.

Let $P$ be a [[cancellative monoid|cancellative]] [[commutative monoid]] equipped with a [[strict weak order]] such that $x \lt 0$ does not hold for any element $x$ of $P$; that is, the additive [[identity]] is a [[bottom element]] with respect to the [[partial order]] induced by the [[negation]] of the strict weak order, and an element $1$ such that $0 \lt 1$. Let $M$ be a [[cancellative monoid|cancellative]] [[commutative monoid]] equipped with a function ${|\cdot|}\colon M \to P$, thought of as a measure of [[absolute value]], such that ${|0|} = 0$. For example, $M$ could consist of the [[real numbers]] with $P$ the nonnegative real numbers; similarly, we may consider the [[natural numbers]], the [[complex numbers]], the [[cardinal numbers]], and many other familiar examples of numbers. (In these examples, $\lt$ and ${|\cdot|}$ satisfy additional compatibility properties with respect to the cancellative commutative monoids $M$ and $P$, but these do not seem to enter into the definition. Conversely, we do not use the full cancellative commutative monoid structure of $M$ and $P$, so the definition should work in even greater generality.)

An element $x$ of $M$ is __infinitesimal__ if the sum of $n$ elements of $x$ has absolute value less than $1$: ${|\sum_{i = 1}^n x|} \lt 1$ for every [[natural number]] $n$. The sum of $n$ elements of an element in a cancellative commutative monoid $M$ always exists, since every monoid is power-associative.

According to this definition, $0$ is always infinitesimal.  Traditionally, one adds to the definition the requirement that $i \neq 0$, but this leads to a less useful notion of the space of all infinitesimals.  We allow $0$ to be infinitesimal for some of the same reasons that we allow it to be a [[natural number]].  (Although the argument is even stronger here, since it's always [[decidable set|decidable]] whether a given natural number is zero, but it may not be decidable whether a given infinitesimal is zero.  This is especially important in [[synthetic differential geometry]], where the logic is unavoidably [[constructive mathematics|constructive]].)

In the usual systems of numbers (real, natural, complex, cardinal, etc), $0$ is the *only* infinitesimal number.  So the interesting question is how to get other infinitesimals.

## Free infinitesimals

The simplest way to add an infinitesimal to an [[ordered ring]] (or rig) is [[free object|freely]].  For example, if we start with the field $\mathbb{R}$ of real numbers, then form the [[polynomial ring]] $\mathbb{R}[x]$, we may make $x$ (and every nonconstant polynomial) infinitesimal by defining $x \lt c$ for every positive real number $c$ and generating the rest of $\lt$ by requiring that $\mathbb{R}[x]$ be an ordered ring.  (This amounts to defining $f \lt g$ to mean that $f(x) \lt g(x)$ for sufficiently small positive values of $x$.)

If you want a [[field]], then form the [[field of fractions]] $\mathbb{R}(x)$ of $\mathbb{R}[x]$, which is the field of [[rational function]]s.  This is a very commonly known example, although it is more usual to make $x$ infinitely large (so that $1/x$ is infinitesimal).

Similarly, one can use polynomials to define infinitesimal versions of complex numbers, cardinal numbers, etc.

The downside of this approach is that the resulting infinitesimal numbers will be subject only to those operations that apply to the category in which one forms the free construction.  For example, there is no way to apply transcendental functions to infinitesimals with this approach.  Presumably that could be fixed by working with $C^\infty$-[[C-infinity-ring|rings]], but no algebraic theory covers everything that can be done with real numbers (or whatever numbers you start with).

Alternatively, one could [[formally complete]] the [[polynomial ring]] $R[x]$ to get the ring of [[formal power series]] $R[[x]]$. This results in a [[local integral domain]] if $R$ is a [[field]]. In the case that $R$ is an [[Archimedean ordered field]], then $R[[x]]$ is an [[Archimedean ordered local integral domain]]; and if $R$ is [[sequentially Cauchy complete]], this allows for any [[analytic function]] to be defined on $R[[x]]$. The infinitesimals in $R[[x]]$ are the formal power series whose leading 0th-power term is equal to zero, or equivalently, the product of $x$ and a formal power series.

## Nilpotent infinitesimals

Instead of working with free infinitesimals, we could also work with [[nilpotent]] infintiesimals. This requires working in an [[Archimedean ordered Artinian local algebra|Archimedean ordered Artinian local $\mathbb{R}$-algebras]]. Since these are not ordered fields, the [[strict weak order]] is not a [[strict total order]], the [[preorder]] is not a [[partial order]], and the [[apartness relation]] is not [[tight apartness relation|tight]]. There is an [[equivalence relation]] $a \approx b$ which says that $a$ is approximately equal to $b$, and is defined by $a \approx b$ if and only if the difference $a - b$ is a zero divisor.

Given an Archimedean ordered Artinian local $\mathbb{R}$-algebra $A$, since $A$ is a [[local ring]], the quotient of $A$ by its [[ideal]] $D$ of non-invertible elements is the real numbers $\mathbb{R}$, and the canonical function used in defining the [[quotient ring]] is the function $\Re:A \to \mathbb{R}$ which takes a number $a \in A$ to its purely real component $\Re(a) \in \mathbb{R}$. Since $A$ is an ordered $\mathbb{R}$-algebra, there is a [[strictly monotone]] [[ring homomorphism]] $h:\mathbb{R} \to A$. A number $a \in A$ is **purely real** if $h(\Re(a)) = a$, and a number $a \in A$ is **purely infinitesimal** if it is in the ideal of zero divisors $D$, the [[fiber]] of $\Re$ at the real number $0$. Zero is the only number in $A$ which is both purely real and purely infinitesimal.

The advantage of working with nilpotent infinitesimals is that the ring homomorphism $h:\mathbb{R} \to A$ preserves [[smooth functions]]: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal number $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$, then for every [[smooth function]] $f \in C^\omega(\mathbb{R})$, there is a function $f_A:A \to A$ such that for all real numbers $x \in \mathbb{R}$, $f_A(h(x)) = h(f(x))$ and
$$f_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i$$

If we restrict to Archimedean ordered local Artinian $\mathbb{R}$-algebras $A$ where every element of the nilradical $D$ is a nilsquare element, where for all $\epsilon \in D$, $\epsilon^2 = 0$, then the ring homomorphism $h:\mathbb{R} \to A$ preserves [[differentiable functions]]; for every [[differentiable function]] $f:\mathbb{R} \to \mathbb{R}$, there is a function $f_A:A \to A$ such that for all real numbers $x \in \mathbb{R}$ and nilpotent elements $\epsilon \in D$, $f_A(h(x)) = h(f(x))$ and
$$f_A(h(x) + \epsilon) = h(f(x)) + h\left(\frac{d f}{d x}(x)\right) \epsilon$$

This allows us to define certain differentiable functions on the [[real numbers]], such as the [[exponential function]], the [[sine function]], and the [[cosine function]], through its property as solutions to systems of first-order [[ordinary differential equations]], without having to define the notion of [[limit of a sequence]], [[series]], [[integral]], and prove the [[fundamental theorem of calculus]] and/or [[convergence]] of [[Taylor series]].

One could also work with partial functions instead of functions. Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A [[partial function]] $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $(D^{(-)} j)(-):\mathbb{N} \times S \to \mathbb{R}$ with $(D^0 j)(a) = a$ for all $a \in S$, such that for all [[Archimedean ordered Artinian local ring|Archimedean ordered Artinian local $\mathbb{R}$-algebras]] $A$ with ring homomorphism $h_A:\mathbb{R} \to A$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$
$$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A((D^i j)(a)) \epsilon^i$$

Special cases include being smooth at an element $a \in I$, which is the same as being smooth at the [[singleton subset]] $\{a\}$, and being **smooth** which is the same as being smooth at the [[improper subset]] of $I$.

This approach is used in [[synthetic differential geometry]], where one adds additional axioms to the foundations stating that the only functions on $A$ are the [[smooth functions]].

## Nonstandard analysis

If the point is to do with infinitesimal numbers everything that we can do with ordinary numbers, then why not use high-powered [[logic]] to do this for us?  That is the approach taken by [[Abraham Robinson]] in [[nonstandard analysis]].

Nonstandard analysis can be applied to [[mathematics]] as a whole, so it treats complex numbers, cardinal numbers, and the rest all together.

...

## Smooth toposes

Another approach is to focus only on what we want to do with infinitesimals in a particular field.  Since infinitesimals were used to do calculus, then let\'s just do calculus.  We can take as axiomatic the familiar properties of [[smooth map|smooth functions]] used in calculus, including the ways these were applied to infinitesimals before analysis was made rigorous (by modern standards) in the 19th century.  This is the approach taken by [[Bill Lawvere]] and others in [[synthetic differential geometry]].

...

## Other approaches

The [[surreal number]]s have infinitesimals, and include both the real numbers and the [[ordinal numbers]].  But they are not much good for ordinary calculus, although there are some indications that they might have applications in asymptotics.

There is also an interpretation of pre-Cauchy calculus in which infinitesimals are interpreted as [[infinite sequences]] that converge to $0$.  I don\'t know much about this.

## Comparisons

### Invertible infinitesimals in NSA and SDG {#NSAvsSDG}

All the infinitesimals appearing in nonstandard analysis are *invertible*, since the [[hyperreal numbers]] form a [[field]].  (This is also true for all the surreal infinitesimals.)  By contrast, most of the infinitesimals appearing in synthetic differential geometry are [[nilpotent]], and hence not invertible.  However, there are some models that do contain invertible infinitesimals, and hence also 'infinite' numbers as their inverses (see [[smooth natural number]]).  Two such models are the [[smooth topos]]es called $\mathcal{Z}$ and $\mathcal{B}$ in

* [[Ieke Moerdijk]] and [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_

These toposes are related to the smooth topos called $\mathcal{G}$ there, which has nilpotent but no invertible infinitesimals, by a transfer theorem (chapter VII, section 4) valid for a certain class of [[coherent formula]]s.  Additionally, the 'object of nonstandard smooth natural numbers' in these toposes is defined by an 'algebra of unbounded sequences', similar in spirit to the unbounded sequences which represent infinitely large numbers in nonstandard analysis.  However, it is not clear whether any more precise comparison can be made.

## Related concepts

* [[infinitesimal object]]

* [[infinitesimal neighbourhood]]

| [[ring]] with [[infinitesimals]] | [[function]] |
---------------------|-----------------|
| [[dual numbers]] | [[differentiable function]] |
| [[Weil ring]] | [[smooth function]] |
| [[power series ring]] | [[analytic function]] |

## References

*  article in the [English Wikipedia](http://en.wikipedia.org/wiki/Infinitesimal) for etymology

* {#Cohen83} [[Hermann Cohen]], _Das Prinzip der Infinitesimal-Methode und seine Geschichte_ , Berlin 1883. ([html](http://www.gleichsatz.de/b-u-t/archiv/Cohen/hc-infinit1.html))

* {#KatzSherry13} [[Mikhail Katz]], David Sherry, _Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond_, D. Erkenn (2013) 78: 571 ([arXiv:1205.0174](https://arxiv.org/abs/1205.0174), [doi:10.1007/s10670-012-9370-y](https://link.springer.com/article/10.1007/s10670-012-9370-y))

  (on [[Gottfried Leibniz]]'s concept of infinitesimals)

[[!redirects notions of infinitesimals]]
[[!redirects infinitesimal]]
[[!redirects infinitesimals]]
[[!redirects infinitesimal number]]
[[!redirects infinitesimal numbers]]

[[!redirects purely infinitesimal]]
[[!redirects purely infinitesimal number]]
[[!redirects purely infinitesimal numbers]]


infinitesimal object

+-- {: .rightHandSide}
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#### Formal geometry
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[[!include formal geometry -- contents]]
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#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Compact objects
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[[!include compact object - contents]]
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

An _infinitesimal quantity_ is supposed to be a quantity that is infinitely small in size, yet not necessarily perfectly small (zero).  An _infinitesimal space_ is supposed to be a space whose extension is infinitely small, yet not necessarily perfectly small (pointlike).

Infinitesimal objects have been conceived and used in one way or other for a long time, notably in [[algebraic geometry]], where [[Grothendieck]] emphasized the now familiar role of [[duality|formal dual]]s ([[affine scheme]]s) of commutative rings $R$ with nilpotent ideals $J\subset R$ as _infinitesimal thickenings_ of the formal dual of the quotient ring $R/J$.

See also [[infinitesimally thickened point]].

### Formalization in synthetic differential geometry

A proposal for formalizing the [[category theory|abstract nonsense]] behind the notion of the _infinitesimal_ such that these algebraic constructions become _models_ for more general axioms was given by [[William Lawvere]] in his 1967 lecture (see the references below).

Lawvere observed that a simple yet powerful characterization of the notion of _infinitesimal space_ $D$ is that $D$ is an [[object]] in a [[topos]] $\mathcal{T}$ of [[space]]s such that the [[inner hom]] [[functor]] $(-)^D : \mathcal{T} \to \mathcal{T}$ has a [[right adjoint]].

If the [[topos]] in question furthermore is equipped with a [[lined topos|line object]] $R$ that plays the role of the _real line_ $\mathbb{R}$ then a sensible notion of _infinitesimal quantities_ in $R$ is obtained when all [[morphism]]s $D \to R$ from infinitesimal spaces $D$ are necessarily _linear_ maps. This is now known as the [[Kock-Lawvere axiom]] on [[lined topos]]es $(\mathcal{T}, R)$. When it is satisfied, $(\mathcal{T}, R)$ is called a [[smooth topos]]. The study of these is known as [[synthetic differential geometry]].

The notion of infinitesimal object and infinitesimal space then makes sense in any [[smooth topos]], and may be reasoned about generally for all smooth toposes. In any concrete [[Models for Smooth Infinitesimal Analysis|model]] for the axioms there will accordingly be concrete realizations of these infinitesimal objects.

### Realizations in algebraic geometry

Notably, for instance the [[Grothendieck topos]] of [[presheaf|presheaves]] on the [[opposite category]] $k CAlg^{op}$ of that of commutative $k$-[[algebra]]s (over some [[field]] $k$) is a simple realization of a [[smooth topos]] (see for instance [Kock-SGM, section 93](http://home.imf.au.dk/kock/SGM-final.pdf)). This topos and its variants and in particular their [[category of sheaves|sheaf]]-[[localization]]s provide the context in which [[algebraic geometry]] takes place.

Therefore the notion of infinitesimals in [[algebraic geometry]] may be understood as being models of the general notion of infinitesimals in [[synthetic differential geometry]] in context such as $\mathcal{T} = Sh(k CAlg^{op})$ or similar.

The vast majority of existing work on infinitesimals and infinitesimal neighbourhoods comes from [[algebraic geometry]]. It is the foundation of Grothendieck's approach to [[regular differential operators]], to costratifications, [[crystalline cohomology]] and [[Grothendieck connection|de Rham descent]].

Similar infinitesimal thickenings also appear in the [[noncommutative geometry]] of Kapranov, and in the language of [[abelian category|abelian categories]] of [[quasicoherent sheaf|quasicoherent sheaves]] in the work of Lunts and Rosenberg on regular [[differential operator]]s in the content of [[noncommutative geometry]], which strongly takes into account [[tensor product]]s.

### Comparison to infinitesimals in nonstandard analysis

Another notion of infinitesimals has arisen in the context of [[nonstandard analysis]]. The infinitesimal quantities considered there differ from the general ones in [[synthetic differential geometry]] in that they are all _invertible_ (their inverses being "infinitely large").   Nevertheless, one can construct models of [[synthetic differential geometry]] which, in addition to nilpotent infinitesimals, contain invertible infinitesimals; see for instance [[Models for Smooth Infinitesimal Analysis|MSIA, chapters VI and VII]].  Such invertible infinitesimals can be applied in some of the same ways as the infinitesimals of nonstandard analysis.

However, as pointed out in MSIA (intro. to Chapter VII), "there are some obvious differences."  The primary tool used in nonstandard analysis is a completely general [[transfer principle]], saying that any statement in the ordinary world is also true in the nonstandard world.  In particular, this implies that the infinitesimal and infinitely large quantities in nonstandard analysis obey all the same rules of arithmetic and analysis as do the standard ones.  By contrast, a limited sort of transfer principle relating a pair of specific models for SDG is proven in MSIA, but it applies only to statements of a certain logical form.  Moreover, the arithmetic of invertible infinitesimals in SDG has some unfamiliar aspects: for instance, mathematical induction is only valid for statements of a certain logical form, and the axiom of finite choice fails.

The construction of models for nonstandard analysis does, however, have a topos-theoretic description, using [[filterpower]]s.

## Definition

### Atomic object

+-- {: .num_defn}
###### Definition (Lawvere)

In a [[cartesian closed category]] $C$ an [[object]] $D$ is called **infinitesimal** **atomic** if the [[hom-functor]]
$(-)^D : C \to C$ for maps out of $D$ (i.e. the functor of
[[exponential object|exponentiation]] by $D$) is a [[left adjoint]], i.e.
if it has a [[right adjoint]].
=--

An object which is infinitesimal atomic is small from the internal perspective of the category. In contrast, an object is a [[tiny object]] when it is small from the external perspective, that is when the functor $C(D,-):C\to Set$ into the category of sets preseves colimits.

When $C$ is a local Grothendieck topos over $Set$, then the two notions agree, but they do not in general. For example the terminal object $\mathbf 1$ in $C$ is always internally small (infinitesimal atomic), since the functor $(-)^{\mathbf 1}$ always has a right adjoint. But it is not necessarily a [[tiny object]] from the external perspective, since $C(\mathbf 1,-)$ need not preserve colimits. This happens in situations where it is not appropriate to view the terminal object $\mathbf 1$ in a categoy as a small object from the external perspective, for example when we are in a slice topos $C/S$. In such cases the correct intuitive external meaning of an object being internally small is that it is small in each fiber.

+-- {: .num_remark}
###### Remark
**(intuitive interpretation)**

Here is how to think of what this definition means intuitively. For that, notice how maps out of an ordinary space fail to preserve [[colimit]]s:

for definiteness, consider the case of a [[cover]] of a [[space]] $X$ by spaces $\{U_i \to X\}_i$ so that $X$ is the [[coequalizer]]

$$
  (\coprod_{i,j} U_i \times_X U_j)
  \stackrel{\to}{\to}
  (\coprod_i U_i)
  \to X
$$

as discussed in detail at [[sieve]] and [[sheaf]]. This says effectively that every point of $X$ is element of at least one of the covering spaces $U_i$ and that one obtains $X$ by identifying the points in the covering spaces that correspond to the same one in $X$.

Now let $\Sigma$ be any other space. We may assume here that the [[internal hom]] $[\Sigma,-] : T \to T$ at least preserves [[coproduct]]s, so that applying this functor to the above diagram yields

$$
  (\coprod_{i,j} [\Sigma,U_i \times_X U_j])
  \stackrel{\to}{\to}
  (\coprod_i [\Sigma,U_i])
  \to [\Sigma,X]
  \,.
$$

Now notice how this will in general fail to still be a [[coequalizer]]: if it were, for one the morphism $  (\coprod_i [\Sigma,U_i])
  \to [\Sigma,X]$ would have to be an [[epimorphism]]. But this can't be in general, because it would mean that every map $\Sigma \to X$ factors through one of the covering spaces. The problem here is that in general the image of $\Sigma \to X$ may be _larger_ than any of the $U_i$.

This is maybe most familiar in the context of [[loop space]]s (for $\Sigma$ the circle): the loop space of a cover of $X$ is not in general a cover of the loop space.

But suppose that $\Sigma$ were infinitesimal. One thing that should mean is that there is no other space that is "effectively smaller" in some useful sense. For $\Sigma$ infinitesimal, we do expect that every map $\Sigma \to X$ can always be factored through at least one of the $U_i$: because $\Sigma$ is so small, the image of a map out of it can never be too large.

So only if $\Sigma$ qualifies as having infinitesimal extension can the functor $[\Sigma,-]$ be expected to preserve colimits.

=--

### Formal infinitesimal space

+-- {: .num_defn}
###### Definition **(formal infinitesimal space)**

An object $\Delta$ in a [[smooth topos]] $(\mathcal{T}, R)$
is called a **formally infinitesimal object** if it is the
algebra-spectrum of (what in the [[synthetic differential geometry|sdg]]-literature is usually called) a _-$R$-[[infinitesimally thickened point|Weil algebra]]_ in $\mathcal{T}$

$$
  \Delta \simeq Spec_R(W)
  \,.
$$

=--

Here

* $W = R \oplus J$ is an [[internalization|internal]] $R$-algebra object in $\mathcal{T}$ with $J$ an $R$-finite dimensional nilpotent ideal

* $Spec_R(W) := R Alg_{\mathcal{T}}(W,R) \subset R^W$ is the [[subobject]] of the [[internal hom]] of morphisms that respect the $R$-algebra structure on $W$ and $R$.

All the spaces that are described as collection of degree $n$ infinitesimal neighbours are of this form. Infinitesimal spaces not of this form are germ-spaces (see the examples below). These violate the finite-dimensionality assumption on $J$.

## Examples

### Infinitesimal intervals

There are several different objects that one may think
of as an infinitesimal interval.

The smallest of them is often denoted $D$ and sometimes called the **disembodied tangent vector** or the **[[walking]] tangent vector** .

This is described in more detail at

* [[infinitesimal interval object]].

It is such that a morphism $D \to X$ into a [[manifold]] $X$ is the same as
a choice of point $x \in X$ and of a [[tangent vector]]
$v \in T_x X$. Equivalently, it is such that restricting a smooth function $f : \mathbb{R} \to \mathbb{R}$ along the inclusion  $D \hookrightarrow \mathbb{R}$ produces the first-order [[jet]] defined
by $f$ at the point $0 \hookrightarrow D \to \mathbb{R}$.

Accordingly, for each $k \in \mathbb{N}$
there is a "slightly bigger" infinitesimal interval often denoted $D_k$, which is such that restricting a smooth function $f : \mathbb{R} \to \mathbb{R}$
along $D_k \to \mathbb{R}$ produces the order-$k$ [[jet]] represented by this
function at the given point.

Still infinitesimal but bigger than all these is the object
$\Lambda_0 := \cap_{0 \in U \subset \mathbb{R}} U$ of intersections of all neighbourhods of the origin of $\mathbb{R}$. This is such that the restriction of
a map $f : \mathbb{R} \to \mathbb{R}$ along $\Lambda_0 \hookrightarrow \mathbb{R}$
produces the [[germ]] of $f$ at $0$.

### The standard infinitesimal interval

#### Models

The classical example of a realization of an infinitesimal object is in terms of what is (traditionally but undescriptively) called the _ring of [[dual number]]s_. For that we place ourselves in some context in which [[space]]s are characterized [[duality|dually]]
in terms of the [[space and quantity|quantities]] on them, i.e. in terms of their would-be function algebras.

For some real number $t \in \mathbb{R}$,
functions on the closed [[interval]] $[-t,t] \subset \mathbb{R}$  of length $2 t$ may be thought of
as represented by functions on the whole real line $\mathbb{R}$, where two representatives represent the same function on the interval if they differ
by a function that vanishes on the interval.

Precisely:

+-- {: .num_lemma}
###### Lemma

The ([[generalized smooth algebra|generalized smooth]]) algebra of smooth
functions $C^\infty([-t,t])$ on $[-t,t]$ is isomorphic to the quotient of the
algebra of smooth functions $C^\infty(\mathbb{R})$ on all of $\mathbb{R}$ by the
functions that vanish on $[-t,t]$

$$
  C^\infty([-t,t]) \simeq C^\infty(\mathbb{R})/\{f \in C^\infty(\mathbb{R})| \forall x \in [-t,t]: f(x) = 0\}
    \,.
$$
=--

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

This is a corollary of the smooth version of the [[Tietze extension theorem]],
which says that for $U \subset \mathbb{R}^n$ a closed subset, every smooth
function on $U$ extends to a smooth function on all of $\mathbb{R}^n$.

See page 20 of [[Models for Smooth Infinitesimal Analysis|MSIA]].
=--

As we think of the length of the interval shrinking to an infinitesimal  value, the notion of derivative of functions is such that we want to say
that the statement "a function vanishes on the infinitesimal interval" is equivalent to "a function vanishes at the origin and its first derivative there vanishes,  too". This in turn is usually equivalent (in a smooth context) to "a function is a square of a function that vanishes at the origin".

Accordingly, in a context where one considers [[polynomial]] functions over the [[ground field]]
$k$, the infinitesimal
interval is given by the [[space]] -- usually called $D$ --
that is [[duality|dual]] to the [[ring]]
$k[\epsilon] := k[Z]/Z^2$ which is the quotient of the polynomial ring in
one variable $Z$ modulo the polynomial $Z^2$. This is often called the
**ring of dual numbers** (where the term 'dual' historically refers to its being $2$-dimensional). In terms of generators and relations this is the ring
generated by a single element $\epsilon$ subject to the relation that $\epsilon^2 = 0$.

Similarly, in the smooth context of, for instance, Moerdijk--Reyes
_[[Models for Smooth Infinitesimal Analysis]]_, $D$ is the [[space]] [[duality|dual]]
to the [[generalized smooth algebra]] $C^\infty(\mathbb{R})/J^2$ obtained as the
smooth functions on the real line modulo squares of functions that vanish at the origin.

+-- {: .num_defn}
###### Definition (the 1-dimensional infinitesimal space)

In the context of [[generalized smooth algebra]], the $1$-dimensional infinitesimal
space is the space $D$ whose function algebra is the quotient

$$
  C^\infty(D) := C^\infty(\mathbb{R})/\{x^2\}
$$

of all functions on the real line, modulo those that are a product with the function $x \mapsto x^2$.
=--

This does reproduce the above ring of dual numbers due to the [[Hadamard lemma]], which says that for $g \in C^\infty(\mathbb{R})$ a smooth function, there exists
a smooth function $h \in C^\infty(\mathbb{R})$ such that for all $x \in \mathbb{R}$
we have $g(x) = g(0) + x g'(x) + x^2 h(x)$. So modulo $x^2$, every smooth function is in fact
a polynomial function.

See pages 19-20 of _[[Models for Smooth Infinitesimal Analysis|MSIA]]_.

In this dual generators-and-relations description, the infinitesimal interval is very familiar in many mathematically less sophisticated contexts. It prevails for instance
in the basic physics textbook treatment since [[Isaac Newton]] up to this day. [[Sophus Lie]] is famously
quoted as having said that he found many of his famous insights by such
"synthetic reasoning" and only a lack of proper formalization prevented him
from writing them up in this way instead of in the more wide-spread way of
differential calculus.

#### Axiomatics

More generally, one may abstract the above properties of concrete realizations
of the infinitesimal interval such as to get such a notion in an arbitrary suitable
context. A suitable context for [[synthetic differential geometry]] is any
[[topos]] $C$ equipped with an [[internalization|internal]] [[commutative ring]] $R$.

Using the [[topos]]-[[internal logic]] we may speak of both $R$ and $D$ as
if they were [[set]]s, where "element" means [[generalized element]]. This
way we have:

+-- {: .num_example #FirstOrderInfinitesimalInterval}
###### Example

Let $(\mathcal{T}, R)$ be a [[smooth topos]]. Then the
first order
**[[infinitesimal interval object]]**
$D$ is the [[subobject]] of $R$ of all those elements whose square is 0.

$$
  D = \{x \in R | x^2 = 0\}
$$
=--

It may be helpful to recall that in terms of [[limit]]s this notation means that
$D$ is the [[equalizer]] of

$$
  R \stackrel{(-)^2}{\to} R
$$

and

$$
  R \stackrel{}{\to} 0 \stackrel{}{\to} R
  \,.
$$

For with $x : U \to D$ any morphism embodying a generalized element
$x \in D$, the universal property of the [[limit]] identifies this
uniquely with a morphism $U \to R$, hence with a generalized element
of $R$, such that $U \to R \stackrel{(-)^2}{\to} R$ is the $0$ element
of $R$ with domain of definition $U$ : $\cdots = U \to 0 \to R$.

### The cartesian product of infinitesimal intervals

This works analogously to how the $k$-[[cube]] is the $k$-fold [[cartesian product]]
$D^k$ of the [[unit interval]] $[-1,1]$ with itself.

+-- {: .num_example}
###### Example

The $k$-fold [[cartesian product]] $D^k$ of the first-order infinitesimal interval $D$, example \ref{FirstOrderInfinitesimalInterval},
with itself might be called the "infinitesimal $k$-cube".

By the discussion at [[smooth algebra]], we have

$$
  C^\infty(D^k) \simeq C^\infty(D)^{\otimes^k} \simeq (\mathbb{R}[\epsilon]/(\epsilon^2))^{\otimes^k}
  \,.
$$

=--

### The $k$-dimensional infinitesimal disk

+-- {: .num_example}
###### Example

For $n\in \mathbb{N}$ the $k$-dimensional infinitesimal disk is

$$
  D(k)
   \coloneqq
  \{ (x_1,\cdots, x_k) \in R^k | \forall j,k \colon x_j \cdot x_k = 0 \}
$$

=--

+-- {: .num_remark}
###### Remark

Since in particular $x_j^2 = 0$ for all elements of the
infinitesimal $n$-disk, we have an inclusion

$$
  D(n) \subset D^n
$$

which is proper if $n \gt 1$. For $n = 1$ we have $D(1) = D$.

While $D(n)$ is closed under multiplication by elements of $R$, it is
not in general closed under addition of its elements. For instance for
$d_1,d_2 \in D(1) = D$ we have that $d_1 + d_2$ (the operation being in $R$) is still in $D$ precisely if $(d_1,d_2)$ is in $D(2)$.

=--

### The infinitesimal neighbourhood

For $x \in X$ a point in a manifold, the infinitesimal [[neighbourhood]] $U_p$ is the intersection of all open neighbourhoods of $x$. This is such that the restriction of a function $f : X \to \mathbb{R}$ along the inclusion $U_p \to X$ is precisely the "infinitesimal [[germ]]" of the function $f$.

All of the infinitesimal spaces above are contained in the corresponding infinitesimal neighbourhood. So this is the "largest" of the infinitesimal spaces discussed here.

### Spaces of infinitesimal $k$-simplices
  {#SpacOfInfSimpl}

> Much of this is being reworked at [[infinity-Lie algebroid]].

#### Idea

An **infinitesimal $k$-[[simplex]]** in $R^n$ based at the origin is a collection $(\vec \epsilon_i \in R^n)_{i = 1}^k$ of points in $R^n$, such that each is an infinitesimal neighbour of the origin

$$
  \forall i : \;\; \vec \epsilon_i \sim 0
$$

and each are infinitesimal neighbours of each other

$$
  \forall i,j: \;\; (\vec \epsilon_i - \vec \epsilon_j) \sim 0
  \,.
$$

Following section 1.2 of

* [[Anders Kock]],  _Synthetic differential geometry of manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

we write $\tilde D(k,n)$ for the space of all infinitesimal $k$-simplices in $R^n$. More precisely, this is defined as the formal dual of the algebra $C^\infty(\tilde D(k,n))$ defined as follows.

Functions on spaces of infinitesimal $k$-simplices turn out to be degree $k$-differential forms. This provides a "synthetic" way of precisely thinking of wedge produts $d x \wedge dy$ etc as products of infinitesimals. As the following computations do show, the skew-commutativity of the wedge product is an inherent consequence of the nature of infinitesimals.

#### Definition

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

The algebra $C^\infty(\tilde D(k,n))$ is the
commutative $\mathbb{R}$-algebra generated from $k \times n$ generators $(\epsilon_i^j)_{1 \leq i \leq n, 1 \leq j \leq n}$ subject to the relations

$$
  \forall i, j,j' : \;\; \epsilon_i^{j} \epsilon_i^{j'} = 0
$$

and

$$
  \forall i,i',j,j'   : \;\;\;
  (\epsilon_i^j - \epsilon_{i'}^j) (\epsilon_i^{j'} - \epsilon_{i'}^{j'}) = 0
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

By multiplying out the latter set of relations and using the former, these relations are seen to be equivalent to the set of relations

\[
  \forall i,i',j,j' : \;\;\;
  \epsilon_i^j \epsilon_{i'}^{j'}
  + \epsilon_{i'}^j \epsilon_{i}^{j'} = 0
  \,.
\]

Notice that this implies also that

$$
  \forall i,i', j: \;\;\;  \epsilon_i^{j} \epsilon_{i'}^j = 0
  \,.
$$

=--

A general element $f$ of this algebra we think of as a function on a certain infinitesimal neightbourhood of the origin of $R^{k \cdot n}$, interpreted as the space of infinitesimal $k$-simplices in $R^n$ based at 0.

Since $C^\infty(\tilde D(k,n))$ is a [[infinitesimally thickened point|Weil algebra]] in the sense of [[synthetic differential geometry]], its structure as an $\mathbb{R}$-algebra extends uniquely to the structure of a [[smooth algebra]] (as discussed there) and we may think of $\tilde D(k,n)$ as an infinitesimal [[smooth locus]].

+-- {: .num_example}
###### Example

For $n = 2$ and $k = 2$ we have that $C^\infty(\tilde D(2,2))$ consists of elements of the form

$$
  \begin{aligned}
    f
    +
    a \cdot \epsilon_1
     + b \cdot \epsilon _2
     + (\omega \cdot \epsilon_1) (\lambda \cdot \epsilon_2)
    &=
    f +
    a_1 \epsilon_1^1
     + a_2 \epsilon_1^2
     + b_1 \epsilon_2^1
     + b_2 \epsilon_1^2
    \\
    & + (\omega_1 \lambda_2 - \omega_2 \lambda_1)
      \frac{1}{2}(\epsilon_1^1 \epsilon_2^2 - \epsilon_1^2 \epsilon_2^1)
  \end{aligned}
$$

for $f \in \mathbb{R}$ and $(a, b, \omega, \lambda \in (\mathbb{R}^n)^*)_{1 \leq i \leq n}$ a collection of ordinary covectors and with "$\cdot$" denoting the evident contraction, and where in the last step we used the above relations.

It is noteworthy here that the coefficient of the term which is multilinear in each of the $\epsilon_i$ is the [[wedge product]] of two [[covector]]s $\omega$ and $\lambda$: we may naturally identify the subspace of $C^\infty(\tilde D(2,2))$ on those elements that vanish if either $\epsilon_1$ or $\epsilon_2$ are set to 0 as the space $\wedge^2 T_0^* \mathbb{R}^2$ of 2-forms at the origin of $\mathbb{R}^2$.

Of course for this identification to be more than a coincidence we need that this is the beginning of a pattern that holds more generally. But this is indeed the case.

=--

#### Properties

Let $E$ be the set of _square_ submatrices of the $k \times n$-matrix $(\epsilon_i^j)$. As a set this is isomorphic to the set of pairs of subsets of the same size of $\{1, \cdots, k\}$ and $\{1, \cdots , n\}$, respectively. For instance the square submatrix labeled by $\{2,3,4\}$ and  $\{1,4,5\}$ is

$$
  e
  =
  \left(
    \array{
       \epsilon_1^2 & \epsilon_4^2 & \epsilon_5^2
       \\
       \epsilon_1^3 & \epsilon_4^3 & \epsilon_5^3
       \\
       \epsilon_1^4 & \epsilon_4^4 & \epsilon_5^4
    }
  \right)
  \,.
$$

For $e \in E$ an $r\times r$ submatrix, we write

$$
  det(e) =
    \sum_{\sigma} sgn(\sigma)
    \epsilon_{1}^{\sigma(1)} \epsilon_2^{\sigma(2)}
    \cdots
    \epsilon_r^{\sigma(r)}
  \in
  C^\infty(\tilde D(k,n))
  \,.
$$

for the corresponding [[determinant]], given as a product of generators in $C^\infty(\tilde D(k,n))$. Here the sum runs over all [[permutation]]s $\sigma$ of $\{1, \cdots, r\}$ and $sgn(\sigma) \in \{+1, -1\} \subset \mathbb{R}$ is the [[signature]] of the permutation $\sigma$.

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

The elements $f \in C^\infty(\tilde D(k,n))$ are precisely of the form

$$
  f = \sum_{e \in E} f_e \; det(e)
$$

for _unique_ $\{f_e \in \mathbb{R} | e \in E\}$. In other words, the map of [[vector space]]s

$$
  \mathbb{R}^{|E|} \to C^\infty(\tilde D(k,n))
$$

given by

$$
  (f_e)_{e \in E} \mapsto \sum_{e \in E} f_e det(e)
$$

is an [[isomorphism]].

=--

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

This is a direct extension of the argument in the above example: a general product of $r$ generators in $C^\infty(\tilde D(k,n))$ is

$$
  \epsilon_{i_1}^{j_1} \epsilon_{i_2}^{j_2}
   \cdots \epsilon_{i_r}^{j_r}
  \,.
$$

By the relations in $C^\infty(\tilde D(k,n))$, this is non-vanishing precisely if none of the $i$-indices repeats and none of the $j$-indices repeats. Furthermore by the relations, for any permutation $\sigma$ of $r$ elements, this is equal to

$$
  \cdots = sgn(\sigma)
   \epsilon_{i_1}^{j_{\sigma(1)}}
   \epsilon_{i_1}^{j_{\sigma(2)}}
   \cdots
   \epsilon_{i_1}^{j_{\sigma(r)}}
  \,.
$$

It follows that each such element may be written as

$$
  \cdots = \frac{1}{r!} det(e)
  \,,
$$

where $e$ is the $r \times r$ subdetermined given by the subset $\{i_1, \cdots, i_r\}$ and $(\{j_1, \cdots, j_r\})$ as discussed above.
=--

+-- {: .num_remark}
###### Remark

In section 1.3 of

* [[Anders Kock]],  _Synthetic differential geometry of manifold_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

effectively this proposition appears as the "[[Kock-Lawvere axiom]] scheme for $\tilde D(k,n)$" when $\tilde D(k,n)$ is regarded as an object of a suitable [[smooth topos]]. It is useful to record this simple but very crucial observation of Anders Kock here in the category $Alg_{\mathbb{R}}^{op}$ or in the category $C^\infty Alg^{op}$ of [[smooth loci]], as we do here, where it is just a simple observation. The point of the Kock-Lawvere axiom scheme is effectively to ensure that the properties of $C^\infty(\tilde D(k,n)) \in C^\infty Alg^{op}$ are preserved under [[Yoneda embedding]] into a corresponding [[sheaf topos]]. But it has been observed that it serves to clarify what is going on in parts of Ander Kock's book by separating the combinatorial and algebraic arguments from their internalization into suitable [[smooth topos]]es.

=--

Let $C^\infty(\tilde D(k,n))_{top}$ be the sub-[[vector space]] of the underlying vector space of $C^\infty(\tilde D(k,n))$ on those elements that vanish if the collection of generators $\epsilon_i = (\epsilon_i^1 , \epsilon_i^2, \cdots, \epsilon_i^n)$ is set to 0, for all $i$. This are those elements that are linear combinations of the form $\sum_{e_{top} \in E_{top}} det(e_{top}) f_{e_{top}}$, for $e_{top}$ ranging over the maximal square submatrices of $(\epsilon_i^j)$.

+-- {: .num_cor}
###### Corollary

The map

$$
  \wedge^k (\mathbb{R}^n)^* \to C^\infty(\tilde D(k,n))_{top}
$$

given by

$$
  \omega^1 \wedge \omega^2 \wedge
  \cdots \wedge \omega^k
  \mapsto
  (\omega^1 \cdot \epsilon_1) (\omega^2 \cdot \epsilon_2) \cdots
  (\omega^k \cdot \epsilon_k)
$$

is well defined and constitutes an [[isomorphism]] of vector spaces.

=--

So inside the space of functions on infinitesimal simplices, we find the [[differential form]]s. The next crucial observation now is that there is a _natural reason_ , from the [[nPOV]],  to restrict to $C^\infty(\tilde D(k,n))_{top} \subset C^\infty(\tilde D(k,n))$.

#### The tangent Lie algebroid and differential forms

The collection of the spaces $R^n \times \tilde D(k,n)$ for all $k \in \mathbb{N}$ naturally forms a [[simplicial object|simplicial]] [[smooth locus]] $(\mathbb{R}^n)^{(\Delta^\bullet_{inf})}$, which represents the [[schreiber:infinitesimal path ∞-groupoid]] of $\mathbb{R}^n$, equivalently the [[tangent Lie algebroid]] of $\mathbb{R}^n$.

Dually this is a [[smooth algebra|smooth]] [[cosimplicial algebra]]. Under the _normalized cochain complex functor_ of the dual [[Dold-Kan correspondence]] this identifies with a [[dg-algebra]]. The fact that this is the _normalized_ cochain complex algebra means that it consists in degree $k$ only of a subspace of the space that the cosimplicial algebra has in degree $k$. This subspace is precisely that of differential $k$-forms.

This we now describe in detail. All the arguments involved are still (with slightly different parameterization, possibly) due to [[Anders Kock]], the only new thing here being the observation that the restriction to the joint kernel of the degeneracy maps exhibts the Dold-Kan map, and that this way using the simplicial picture everything acquires a nice [[nPOV]] interpretation as being about the [[schreiber:infinitesimal path ∞-groupoid]] of $\mathbb{R}^n$, regarded either as an infinitesimal [[Lie ∞-groupoid]] or as a [[∞-Lie algebroid]].

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

Consider the [[simplicial object|simplicial]] [[smooth locus]]

$$
  (R^n)^{(\Delta^\bullet_{inf})}
  :=
  \left(
  \cdots
  R^n \times \tilde D(2,n)
  \stackrel{\to}{\stackrel{\to}{\to}} R^n \times \tilde D(1,n) \stackrel{\to}{\to} R^n
  \right)
  \,,
$$

where

* the face maps $d_i : R^n \times \tilde D(k+1,n) \to R^n \times \tilde D(k,n)$ are

  * for $0 \lt i \lt k+1$ given by

    $$
      d_i : (x, (v_1, v_2, \cdots, v_{k+1}))
      \mapsto
      (x, v_1, \cdots , v_{i} , v_{i+1} + v_{i+2}, v_{i+3}, \cdots, v_{k+1})
    $$

  * for $i = k+1$ given by

    $$
      d_{k+1} : (x, (v_1, v_2, \cdots, v_{k+1}))
      \mapsto
      (x, v_1, \cdots , v_{k})
    $$

  * for $i = 0$ given by

    $$
      d_0 : (x, (v_1, v_2, \cdots, v_k))
      \mapsto
      (x + v_1, v_2, \cdots , v_{k+1})
      \,.
    $$

* the degeneracy maps are

  $$
    s_i : (x, (v_1, \cdots, v_k))
    \mapsto
    (x, (v_1, \cdots, v_{i}, 0, v_{i+1}, \cdots, v_k))
    \,.
  $$

Dually (and this may be taken now as the precise definition of what the above simplicial object is), this is the [[smooth algebra|smooth]] [[cosimplicial algebra]]

$$
  C^\infty((R^n)^{(\Delta^\bullet_{inf})})
  :=
  \left(
  \cdots
  C^\infty(\mathbb{R}^n) \otimes C^\infty(\tilde D(2,n))
  \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
  C^\infty(\mathbb{R}^n) \otimes C^\infty(\tilde D(1,n))
   \stackrel{\leftarrow}{\leftarrow}
   C^\infty(\mathbb{R}^n)
  \right)
  \,,
$$

where

* the co-degeneracy maps $s_i^*$ are given on generators by

  * for $r \lt i+1$ sending
    $\epsilon_r = (\epsilon_r^1, \cdots, \epsilon_r^{k+1})
     \in C^\infty(\tilde D(k+1,n))$
    to itself regarded as an element of $C^\infty(\tilde D(k,n))$;

  * for $r = i +1$ sending $\epsilon_r$ to 0;

  * for $r \gt i+1$ sending $\epsilon_r$ to $\epsilon_{r-1}$.

* the co-face maps $d_i^*$ are given on generators

  * for $0 \lt i \lt k +1$ by sending

    $\epsilon_{1} \mapsto \epsilon_1$, $\cdots$, $\epsilon_{i+1} \mapsto
    \epsilon_{i+1} + \epsilon_{i+2}$, $\epsilon_{i+2} \mapsto \epsilon_{i+3}$,
    $\cdots$;

  * for $i = k+1 $ by sending each $\epsilon_{j}$ to the generator of the same
    name;

  * for $i = 0$ by sending

    $$
      h \otimes (\sum_{e} det(e) f_e)
      \mapsto
      (h \otimes (\sum_{e} det(\tilde e) f_e)
      +
      \sum_{i=1}^n
      (\frac{\partial}{\partial x^i} h)
      \otimes
      \epsilon_1^i
      (\sum_{e} det(\tilde e) f_e)
      \,,
    $$

    where $\tilde e$ is obtained from $e$ by replacing each $\epsilon_{i}^j$
    it contains with $\epsilon_{i+1}^j$.

=--

The way to think of how the face and degeracy maps here work is to imagine that a collection of elements $(v_1, \cdots, v_k) \in \tilde D(k,n)$ spans an infinitesimal $k$-parallelepiped, and that inside that the face and degeneracy maps slice out a $k$-simplex. The proof that this is indeed a (co)simplicial object is entirely analogous to the discussion of the simplicial object of finite simplices at [[interval object]].

For instance for $k = 3$ we have six 3-simplices sitting inside each 3-cube

<img src="http://ncatlab.org/nlab/files/3simplex_in_3cube.jpg" width = "550"/>

and the face maps identify one of these:

<img src="http://ncatlab.org/nlab/files/3simplex_facemaps.jpg" width = "550"/>.

Now observe that under the dual [[Dold-Kan correspondence]] the _normalized cochain complex_ of this cosimplicial algebra is, up to isomorphism, the complex that in degree $k$ has the joint [[kernel]] of the co-degeneracy maps.
But by the above remarks, this joint kernel is precisely

$$
  C^\infty(\mathbb{R}^n) \otimes C^\infty(\tilde D(k,n))_{top} \simeq C^\infty(\mathbb{R}^n) \otimes \wedge^k (\mathbb{R}^n)^*
  \simeq
  \Omega^k(\mathbb{R}^n)
  \,,
$$

the space of differential $k$-forms on $\mathbb{R}^n$.

+-- {: .num_theorem}
###### Theorem

The normalized cochain complex of  the cosimplicial algebra $C^{\infty}((\mathbb{R}^n)^{(\Delta^n_{diff})})$ is [[isomorphism|isomorphic]] as a cochain complex to the [[de Rham complex]] of $\mathbb{R}^n$.

Equipped with the [[cup product]] induced from $C^{\infty}((\mathbb{R}^n)^{(\Delta^n_{diff})})$ is is isomorphic to the de Rham complex even as a [[dg-algebra]].

=--

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

We have already seen that degreewise the vector spaces in question are isomorphic.

It remains to check that the differentials agree. The alternating sum of the face maps acts on an element

$$
  h \otimes (\omega^1 \cdot \epsilon_1)(\omega^2 \cdot \epsilon_2) \cdots
      (\omega^k \cdot \epsilon_k)
$$

as

$$
  \begin{aligned}
     \cdots \mapsto &
     (d_0^* + \sum_{r = 1}^n (-1)^r d_r^*)(
       h \otimes (\omega^1 \cdot \epsilon_{1})
       (\omega^2 \cdot \epsilon_{2}) \cdots
       (\omega^k \cdot \epsilon_{k}))
      \\
      = &
       h \otimes (\omega^1 \cdot \epsilon_{2})
      (\omega^2 \cdot \epsilon_{3}) \cdots
       (\omega^k \cdot \epsilon_{k+1}))
      \\
      & +
       (d h \cdot \epsilon_1)
       (\omega^1 \cdot \epsilon_{2})
       (\omega^2 \cdot \epsilon_{3}) \cdots
       (\omega^k \cdot \epsilon_{k+1}))
      \\
      & -
       h \otimes (\omega^1 \cdot (\epsilon_{1} + \epsilon_{2}))
       (\omega^2 \cdot \epsilon_{3}) \cdots
       (\omega^k \cdot \epsilon_{k+1}))
      \\
      & +
       h \otimes (\omega^1 \cdot \epsilon_{1})
       (\omega^2 \cdot (\epsilon_2 + \epsilon_{3})) \cdots
       (\omega^k \cdot \epsilon_{k+1}))
      \\
      & - \cdots
      \\
      =&
       (d h \cdot \epsilon_1)
       (\omega^1 \cdot \epsilon_{2})
       (\omega^2 \cdot \epsilon_{3}) \cdots
       (\omega^k \cdot \epsilon_{k+1}))
  \end{aligned}
  \,.
$$

Under our identification
$C^\infty(\mathbb{R}^n)\otimes C^\infty(\tilde D(k,n))_{top}
  \simeq \wedge^k \Gamma(T^* \mathbb{R}^n)$ this means that the alternating sum of the face maps sends

$$
  h \wedge \omega^1 \wedge \omega^2 \wedge \cdots \wedge \omega^k
  \mapsto
  d h \wedge \omega^1 \wedge \omega^2 \wedge \cdots \wedge \omega^k
  \,,
$$

where each $\omega^j$ is a _constant_ 1-form on $\mathbb{R}^n$. So this is indeed the action of the [[de Rham complex|de Rham differential]].

=--

The construction of $(\mathbb{R}^n)^{(\Delta^\bullet_{inf})}$ is manifestly natural and extends to a functor

$$
  (-)^{(\Delta^\bullet_{inf})}
  :
  CartSp \mapsto [\Delta^{op}, \mathbb{L}]
$$

from [[CartSp]] to [[simplicial object|simplicial]] [[smooth loci]]. Effectively just by ([[derived functor|derived]]) [[Yoneda extension]] of this functor to a functor on [[simplicial presheaves]] on [[CartSp]] one obtains a definition of the [[schreiber:infinitesimal path ∞-groupoid]] of any [[smooth manifold]], any [[diffeological space]] and even every [[∞-Lie groupoid]].

## Related concepts

[[!include infinitesimal and local - table]]

## References

Discussion of [[infinitesimals]] goes back to [[Leibniz]].

* {#Cohen83} [[Hermann Cohen]], _Das Prinzip der Infinitesimal-Methode und seine Geschichte_ , Berlin 1883. ([html](http://www.gleichsatz.de/b-u-t/archiv/Cohen/hc-infinit1.html))

Nowadays infinitesimal spaces and their properties were familiar in all
those areas of mathematics where spaces are characterized by the algebras of functions on them.

It was in a seminal lecture

* [[William Lawvere]], _Categorical Dynamics_ lectures at the University of Chicago (1967)

reproduced in

* [[Anders Kock]], _Topos theoretic methods in Geometry_, Aarhus Universitet (1979)

that the proposal was made
to axiomatize the properties of infinitesimal objects by making use
of the fact that they are supposed to be objects of a
[[cartesian closed category]].

It was from this insight that [[synthetic differential geometry]] was
eventually developed.

* [[William Lawvere]], _Outline of synthetic differential geometry_,
([web](http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html))

This is a classical case of [[general abstract nonsense]] used to
understand a subtle situation.

A summary and discussion of the axiomatically defined standard infinitesimal
objects $D$, $D_k$, $D_k(n)$ is in section 1.2 of

* Anders Kock, _Synthetic Geometry of Manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

### Atomic spaces

The proposal to call objects $D$ such that $[D,-]$ has an [[amazing right adjoint]] "atomic objects" is due to

* [[William Lawvere]], _Toposes of Laws of Motion_

and repeated in

* [[William Lawvere]], _Open problems in topos theory_ ([pdf](http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf))

Details on the right adjoint to the exponentiation functor $(-)^X$ for
$X$ an infinitesimal object are in appendix 4 of

* Moerdijk, Reyes, _[[Models for Smooth Infinitesimal Analysis]]_

### Formally infinitesimal spaces

For formal infinitesimal objects and [[infinitesimally thickened point|Weil algebras]] see

section I.16 of

* [[Anders Kock]], _Synthetic differential geometry_ ([pdf]())

and chapter I, section 4 and chapter II, theorem 1.13 and onwards in

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], [[Models for Smooth Infinitesimal  Analysis]]

A discussion on terminology and share of the content between infinitesimal object and [[infinitesimal quantity]] is saved at $n$Forum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2696&Focus=22805#Comment_22805).

For an approach to infinitesimal thickenings in the context of abelian categories of quasicoherent sheaves see [[differential monad]] and [[regular differential operator in noncommutative geometry]].

[[!redirects infinitesimal objects]]
[[!redirects infinitesimal quantity]]
[[!redirects infinitesimal quantities]]
[[!redirects infinitesimal space]]
[[!redirects infinitesimal spaces]]


infinitesimal path infinity-groupoid in a smooth topos > history

< [[infinitesimal path infinity-groupoid in a smooth topos]]


infinitesimal path ∞-groupoid

#Contents#
* table of contents
{:toc}

## Idea

In a context of [[infinitesimal cohesion]] the **infinitesimal path $\infty$-groupoid** $\Pi_{inf}X \coloneqq \Im(X)$ of a type $X$ is the result of identifying infinitesimally close points in $X$ by adding in further equivalences between all objects (points of $X$) that are infinitesimal neighbours.

## Definition

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

Given [[differential cohesion]]

$$
  (\mathbf{H} \coloneqq CohesiveType)
    \stackrel{i}{\hookrightarrow}
  (\mathbf{H}_{th} \coloneqq InfThickenedCohesiveType)
$$

define the [[reduction modality]]/[[infinitesimal shape modality]] [[adjunction]]

$$
  (\Re \dashv \Im)
  \colon
  \mathbf{H}_{th}
  \stackrel{\overset{i_!}{\leftarrow}}{\underset{i^\ast}{\to}}
  \mathbf{H}
  \stackrel{\overset{i^\ast}{\leftarrow}}{\underset{i_\ast}{\to}}
  \mathbf{H}_{th}
  \,.
$$

We call $\Pi_{inf}(X) \coloneqq \Im(X)$ the **infinitesimal path ∞-groupoid** of $X$ and  $\Re(X)$ the **[[reduced type]]** of $X$.

For the $(i_* \dashv i^*)$-[[unit of an adjunction|unit]] we write

$$
  InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)
$$

and call it the **constant infinitesimal path inclusion** on $X$.

The $(i_* \dashv i^*)$-[[unit of an adjunction|counit]]

$$
  \Re (X) \to X
$$

we call the **inclusion of the reduced part** of $X$.

=--

## Examples

* [[de Rham space]]

[[!redirects infinitesimal path infinity-groupoid]]
[[!redirects infinitesimal path ∞-groupoids]]
[[!redirects infinitesimal path infinity-groupoids]]


infinitesimal shape modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

In a context of [[differential cohesion]] the _infinitesimal shape modality_, _crystalline modality_ or _étale modality_ $\Im $ characterizes [[coreduced objects]]. It is itself the [[right adjoint]] in an [[adjoint modality]] with the [[reduction modality]] and the [[left adjoint]] in an [[adjoint modality]] with the [[infinitesimal flat modality]].

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im $ is an [[idempotent monad]].

Here $\Im $ is the **infinitesimal shape modality**. The [[reflective subcategory]] that it defines is that of [[coreduced objects]].

## Properties

### Relation for formally étale morphisms

The [[modal types]] of $\Im $ in the context of some $X$, i.e. those $(Y\to X) \in \mathbf{H}_{/X}$ for which the naturality square of the $\Im $-[[unit of a monad|unit]]

$$
  \array{
     Y &\longrightarrow& \Im Y
     \\
     \downarrow && \downarrow
     \\
     X &\longrightarrow& \Im X
  }
$$

is a ([[homotopy pullback|homotopy]]) [[pullback]] square, are the [[formally étale morphisms]] $Y \to X$.

### Relation to de Rham spaces

For $X$ a [[geometric homotopy type]], the result of applying the infinitesimal shape modality yields a type $ \Im X$ which has the interpretation of the [[de Rham space]] of $X$. See there for more.

### Relation to jet bundles

For any object $X$ in [[differential cohesion]], the [[base change]] [[comonad]] $Jet \coloneqq i^\ast i_\ast$ along the [[unit of a monad|unit]] $i \colon X \to \Im X$ has the interpretation of being the [[jet comonad]] which sends [[bundles]] over $X$ to their [[jet bundles]].

### Relation to crystalline cohomology

The [[cohomology]] of $\Im X$ has the interpretation of [[crystalline cohomology]] of $X$. See there for more.

## Related concepts

[[!include cohesion - table]]

## References
 {#References}

The infinitesimal shape modality as part of an extension of [[cohesive (infinity,1)-toposes|cohesive $\infty$-toposes]] to [[differential cohesive (infinity,1)-toposes|differential cohesive $\infty$-toposes]] was introduced in

* [[Urs Schreiber]], Def. 3.5.4 in: _[[schreiber:differential cohomology in a cohesive topos]]_ [[arXiv:1310.7930](https://arxiv.org/abs/1310.7930)]

mimicking an analogous construction (cf. *[[de Rham space]]*) for [[schemes]] in

* {#SimpsonTeleman} [[Carlos Simpson]], [[Constantin Teleman]], Section 3 of: _deRham theorem for $\infty$-stacks_ (1997) [[pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf), [[SimpsonTeleman-deRham.pdf:file]]]

and further developed (in relation to [[jet comonads]] and [[partial differential equations]]) in

* [[Igor Khavkine]], [[Urs Schreiber]], *[[schreiber:Synthetic geometry of differential equations]]* [[arXiv:1701.06238](https://arxiv.org/abs/1701.06238)]

reviewed in

* {#Loregian19} [[Fosco Loregian]], pp. 30 in: *Cohesion in Rome*, talk notes (Dec. 2019)  [[pdf](https://tetrapharmakon.github.io/stuff/cohesive_rome.pdf), [[LoregianCohesion.pdf:file]]]

Its implementation in [[HoTT]] and application to [[Cartan geometry]] is considered in

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

It is called the *crystalline modality* in

* [[David Jaz Myers]], *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* ([arXiv:2205.15887](https://arxiv.org/abs/2205.15887))

[[!redirects infinitesimal shape modalities]]

[[!redirects infinitesimal shape]]

[[!redirects crystalline modality]]
[[!redirects crystalline modalities]]


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###Context###
#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#

* table of contents
{:toc}

## Idea

The _infinitesimal singular simplicial complex_ of a [[space]] $X$ in a [[smooth topos]] $(\mathcal{T},R)$ is the infinitesimal analogue of the singular simplicial complex $X^{\Delta_R^\bullet}$ (see [[interval object]]) that in degree $k$ is the space of $k$-simplices $\Delta^k_R \to X$ in $X$: the infinitesimal singular simplicial complex has in degree $k$ the [[infinitesimal space|infinitesimal]] $k$-simplices in $X$.

There are several ways to make the notion of "infinitesimal $k$-simplex in $X$" precise. Here we describe a notion promoted by [[Anders Kock]], where an "infinitesimal $k$-simplex" in $X$ for $X$ a suitably _locally linear space_ , is a $(k+1)$-tuple $(x_0,\cdots, x_k) \in X^{k+1}$ of points in $X$ that are pairwise _infinitesimal neighbours_ in $X$.

One central application of the singular simplicial complex is in the definition of [[differential forms in synthetic differential geometry]].

## Definition

The basic definition applies to spaces of the form $R^n$ and is generalized from there to spaces that "locally look like" $R^n$ in one way or other.

Let here and in the following $(\mathcal{T},R)$ be a [[smooth topos]].

### In $R^n$

Write, as usual

$$
  D(n) := \{(\epsilon_i) \in n | \epsilon_i \epsilon_j = 0\} \hookrightarrow R^n
$$

for the [[infinitesimal space]] of first order infinitesimal neighbours of the origin of $R^n$, with its canonical inclusion into $R^n$.

Two elements $x , y \in R^n$ are called **first order infinitesimal neighbours**, denoted $x \sim_1 y$, if their difference is in the image of this inclusion.

$$
  (x \sim_1 y) \Leftrightarrow
  (\exists \epsilon \in D(n) :
  (x-y) = \epsilon) \,.
$$

Write

$$
  (R^n)^{\Delta^k_{inf}}
  :=
  \{
    (x_0 \in R^n, \cdots, x_k \in R^n) |
    x_i \sim_1 x_j
  \}
  \,.
$$

This naturally forms a [[simplicial object]] $X^{\Delta_{inf}^{\bullet}} : \Delta^{op} \to \mathcal{T}$.
This is the infinitesimal simplicial singular complex of $R^n$.

A more detailed discussion of this is in the entry [[infinitesimal object]] in the section [Spaces of infinitesimal simplices](http://ncatlab.org/nlab/show/infinitesimal+object#SpacOfInfSimpl).

### In smooth loci

>**warning** this section is as such not drawn from the literature, it seems

Recall that a [[smooth locus]] in $\mathcal{T}$ is an object $\ell A$ that is the joint [[limit]] over some

$$
  R^n \stackrel{\stackrel{f \in J}{\to}}{\stackrel{0}{\to}}
  R
$$

here $J \subset Hom_{\mathcal{T}}(R^n,R)$ is an ideal.

Declare that two [[generalized element]]s $x,y \in \ell A$ are infinitesimal neighbours if their image under the injection

$$
  \ell A \hookrightarrow R^n
$$

is a pair of infinitesimal neighbour in $R^n$. Then let

$$
  (\ell A)^{\Delta^\bullet_{inf}}
  \hookrightarrow
  (R^n)^{\Delta_R^\bullet}
$$

be the sub-simplicial object of infinitesimal neighbours in $R^n$ that are points in $\ell A$.

\begin{prpn}
**(linearity of space of infinitesimal neighbours)** If $p, q \in \ell A$ are infinitesimal neighbours in the [[smooth locus]] $\ell A \subset R^n$, then for all $t \in R$ also the element $p + t(q-p)$ formed by linear combination in $R^n$ is in $\ell A$ and hence is an infinitesimal neighbour of $p$ there.

\end{prpn}

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

Because by the [[Kock-Lawvere axiom]] valid in the [[smooth topos]] $(\mathcal{T},R)$ we have for all $f : R^n \to R$

$$
  0 = f(q) = f(p + (q-p)) =
  f(p) + \sum_i (q-p)_i (\partial_i f)(q)
  = \sum_i (q-p)_i (\partial_i f)(q)
  \,.
$$

Therefore also

$$
  f(p + t(q-q)) = t \sum_i (q-p)_i (\partial_i f)(q) = 0
  \,.
$$

=--

#### Examples

Consider the circle $S^1$ regarded as the [[smooth locus]] $S^1 = \{(x, y) \in R^2 | x^2 + y^2 = 1\}$.

For $a \in S^1 \subset R^2$ an infinitesimal neighbour $(a + \epsilon)$ in $R^2$ is again a point on the circle, and hence an infinitesimal neighbour of $a$ in $S^1$, if

$$
  a_x^2 + 2 a_x \epsilon_x  + a_y^2 + 2 a_y \epsilon_y = 1
$$

which, due to $a_x^2 + a_y^2 = 1$ is equivalent to

$$
  2 a_x \epsilon_x + 2 a_y \epsilon_y = 0
  \,.
$$

This is solved by $\epsilon$ of the form

$$
  \epsilon = \delta \left(
    \array{
       a_y
       \\
       - a_x
    }
  \right)
$$

for some fixed $\delta \in D$.

### In formal manifolds

>use that each manifold is locally isomorphic to an $R^n$ and that the neighbourhood relation only needs an infinitesimal neighbourhood. Proceed locally as above and then patch. See references below.

## Inclusion into the finite singular simplicial complex {#InclusionIntoFinPaths}

The [[lined topos]] $(\mathcal{T}, R)$ also comes canonically for every object $X \in \mathcal{T}$ with the finite singular simplicial complex $\Pi(X) : [n] \mapsto X^{R^n}$ induced from regarding

$$
  (0_*, 1_*) : {*} \coprod {*} \to R
$$

as an [[interval object]] (see there for details).

\begin{defn}
**(inclusion of infinitesimal into finite simplices)** For $\ell A =: X \hookrightarrow R^n$ a [[smooth locus]]
define for all $n \in \mathbb{N}$ a morphism

$$
  \iota_n
  :
  X^{\Delta^k_{inf}}
  \to
  X^{\Delta_R^k} = X^{R^k}
$$

by defining it on [[generalized element]]s as

$$
  (x^0, \cdots, x^{k})
  \mapsto
  (\vec t \mapsto x^0 + \sum_{i=1}^{k} t_i (x^i - x^{i-1}) )
  \,.
$$

\end{defn}

\begin{prpn} The morphisms $\iota_n$ constitute a morphism of [[simplicial object]]s

$$
  \iota : X^{\Delta^\bullet_{inf}}
  \hookrightarrow
  X^{\Delta_R^k}
$$

in that they respects the face and degenracy maps on each side.

\end{prpn}

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

Straightforward checking:

For instance

* The inner face maps $d_i$ on $X^{\Delta_{inf}^{k}}$ omit the $i$th point in the $(k+1)$-tuple of points, while on $X^{\Delta^{k}}$ they act by pullback along $(t_1, \cdots, t_k) \mapsto (t_1, \cdots, t_{i-1}, t_{i}, t_i, t_{i+1}, cdots, t_k)$. That means that in the sum above $t_i$ appears twice to yield

  $$
    \cdots + t_i(x^i - x^{i-1}) + t_i(x^{i+1} - x^i) + \cdots
    =
    \cdots + t_i(x^{i+1} - x^{i-1} ) + \cdots
  $$

  which indeed corresponds to omission of the $i$th point $x^i$.

=--

## Related concepts

* The collection of first order infinitesimal neighbours of a space $X$ arranges itself into the [[schreiber:infinitesimal path ∞-groupoid]] $\Pi^{inf}(X)$. Various concepts derive from this one. [[differential form|Differential forms]] may be understood in terms of functions on $\Pi(x)^{inf}$. This is described at [[differential forms in synthetic differential geometry]]. A [[deRham space]] is the colimit over a $\Pi^{inf}(X)$.

* [[formal neighbourhood of the diagonal]]

## References

In the language of [[synthetic differential geometry]] the infinitesimal singular complex for "formal manifolds" (internally defined manifolds with an infinitesimal thickening to all orderes) is described (with the simplicial structure not made explicit) in

section I.18 of

* [[Anders Kock]], _Synthetic Differential Geometry_ ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

and in [section 2.8](http://home.imf.au.dk/kock/SGM-final.pdf#page=89) of

* Anders Kock, _Synthetic geometry of manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

Discussion of this that does make the simplicial structure explicit and relates it to the [[Dold-Kan correspondence]] is in

* Herman Stel, _[[schreiber:master thesis Stel|∞-Stacks and their Function Algebras ? with applications to ∞-Lie theory]]_.

* Herman Stel, _Cosimplicial C-infinity rings and the de Rham complex of Euclidean space_ ([arXiv:1310.7407](http://arxiv.org/abs/1310.7407))

The details of what $X^{\Delta^k_{inf}}$ is like concretely on representables in the [[smooth topos]] $PSh(k-Alg^{op})$ of [[algebraic geometry]], i.e. on [[affine schemes]] is worked out in detail in

* [[Larry Breen]], [[William Messing]], _Combinatorial differential forms_ ([pdf](http://arxiv.org/abs/math/0005087))

The formulas given there should more or less directly carry over to [[smooth topos]]es with [[smooth locus|smooth loci]] by replacing ordinary rings with [[smooth algebra]]s.

> to be discussed

As the title suggests, the infinitesimal singular simplicial complex is tightly related to [[differential forms in synthetic differential geometry]]: the deRham complex is the normalized [[Moore complex|Moore cochain complex]] of the [[cosimplicial algebra]] $C^\infty(X^{\Delta^\bullet_{inf}})$ of functions on the spaces of infinitesimal simplices.

There is also

* [[Eduardo Dubuc]], [[Anders Kock|Kock]], _On 1-form classifiers_ , Communications in Algebra 12
(1984)

* Dubuc, $C^\infty$-schemes, Amer. J. of Math. 103 (1981)

* Kumpera, Spencer, _Lie Equations_ , Annals of Math. Studies 73 (1973)

There is also a version of the infinitesimal singular simplicial context in the context of [[nonstandard analysis]]. See

* Zivaljevic, _On a cohomology theory based on hyperfinite sums of microcomplexes_ .

[[!redirects infinitesimal path ∞-groupoid in a lined topos]]
[[!redirects infinitesimal path ∞-groupoid in a smooth topos]]
[[!redirects infinitesimal path infinity-groupoid in a smooth topos]]
[[!redirects infinitesimal path infinity-groupoid in a lined topos]]
[[!redirects infinitesimal neighbour]]


infinitesimally thickened point

> This entry is about the notion of _Weil algebra_ as the algebra of functions on an infinitesimally thickened point. For the concept of Weil algebra in Lie theory see [[Weil algebra]].

***

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###Context###
#### Formal geometry
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[[!include formal geometry -- contents]]
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#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

### Abstractly

In [[differential cohesion]] an [[object]]/[[type]] $D$ is an _infinitesimally thickened point_ if its corresponding [[reduced object]] is the [[terminal object]],

$$
  \Re(D) \simeq *
  \,,
$$

hence it is an [[anti-reduced object]].

### In the standard type of model

An _infinitesimally thickened point_ is -- under [[Isbell duality]] -- the formal dual of an $R$-[[algebra]] of the form

$$
  A = R \oplus W
  \,,
$$

where $W$ is a [[module]] of finite rank over $R$ and consisting of nilpotent elements in the algebra $A$.

+-- {: .num_remark #WeilAlgebra}
###### Remark
**on terminology**

In the literature on [[synthetic differential geometry]] an algebra $A$ of this form is also called a **Weil algebra**. Notice that this is unrelated to the notion of Weil algebra in [[Lie theory]]. For more on that, see [[Weil algebra]].

Over more general base fields, this is called a **[[local Artinian algebra]]**.

=--

## Examples

The smallest nontrivial example is the space dual to the [[ring of dual numbers]]. This is the point with "minimal infinitesimal thickening".

This is a special case of the following

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

For $n , k \in \mathbb{N}$, the _jet algebra_ $C^\infty(\mathbb{D}^n(k))$ is the [[quotient]]

$$
  C^\infty(\mathbb{D}^n(k))
  =
  C^\infty(\mathbb{R}^n)/ I^{k+1}
  \,,
$$

where $I = (x_1, \cdots, x_n)$.

The [[formal dual]] [[smooth locus]] $\mathbb{D}^n(k)$ is the "order-$k$ infinitesimal $n$-disk".

=--

For $X$ an $n$-dimensional [[smooth manifold]] and $E \overset{p}{\to} X$ a [[bundle]], and $\mathbb{D}^n(k) \hookrightarrow X$ the order-$k$ infinitesimal $n$-disk (def. \ref{JetAlgebra}) in $X$ around a point $x \in X$, then a lift

$$
  \array{
     \mathbb{D}^n(k) && \longrightarrow && E
     \\
     & \searrow && \swarrow_{\mathrlap{p}}
     \\
     && X
  }
$$

is a $k$-[[jet]] of a [[section]] of $E$ at $x$. The collection of all of these constitutes the order-$k$ [[jet bundle]] of $E$.

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

Every Weil algebra (remark \ref{WeilAlgebra}) is a [[quotient]] of a jet algebra (def. \ref{JetAlgebra}).

=--

(e.g. [Carchedi-Roytenberg 12, prop. 4.43](#CarchediRoytenberg12))

A class of examples are the spaces $\tilde D(n,r)$ of $r$-tuples of infinitesimal neighbours of the origin of $R^n$, that are each also infinitesimal neighbours of each other. Their Weil algebras of functions are a model for the degree $r$-[[differential forms]]. Details on this are at spaces of infinitesimal k-simplices.

## Applications

The [[site]] of definition for the [[Cahiers topos]] is the category of spaces that are products of an $R^n$ with the dual of a  Weil algebra. So these are infinitesimally thickened [[Cartesian space]]s. These are typically sufficient as test spaces for more general spaces.

## Related concepts

* [[superpoint]]

* [[formal disk]]

* [[infinitesimal extension]]

## References

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]], _On Theories of Superalgebras of Differentiable Functions_ ([arXiv:1211.6134](http://arxiv.org/abs/1211.6134))

[[!include cohesion - table]]

[[!redirects infinitesimally thickened points]]

[[!redirects fat point]]
[[!redirects fat points]]


#Contents#
* table of contents
{:toc}

## Definition

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

The **infinitesimally thickened Sierpinski topos** is the topos of presheaves on the ordinal category $\mathbf{3}: \{ 0 \to 1 \to 2 \}$. The name emphasizes the view of this topos as exhibiting [[differential cohesion]] over the cohesive [[Sierpinski topos]] using the coreflective embedding $i : \mathbf{2} \hookrightarrow \mathbf{3}$ sending $0$ to $0$ and $1$ to $2$.

The objects of the category are diagrams $A_2 \to A_1 \to A_0$ in $Set$ and arrows are commuting squares.

=--

## Properties

### Differential cohesion
 {#DifferentialCohesion}

We can view objects of the topos as exhibiting a simple notion of differential cohesion: in $A_r \to A_{inf} \to A_p$ the $A_r$ are the "real" points, the $A_{inf}$ are the "infinitesimal" points and the $A_p$ are the "pieces", i.e., connected components.

The coreflective embedding above results in an adjoint quadruple

$$
 ( i_! \dashv i^* \dashv i_* \dashv i^! )
  :
  PSh(\mathbf{3})
  \stackrel{\overset{i_!}{\hookleftarrow}}{\stackrel{\overset{i^*}{\to}}{\stackrel{\overset{i_*}{\hookleftarrow}}{\stackrel{i^!}{\to}}}}

  PSh(\mathbf{2})
  \,.
$$

which are defined explicitly as follows:

$$ i_!(A_1 \to A_0) = A_1 \to A_1 \to A_0 $$
$$i^*(A_2 \to A_1 \to A_0) = A_2 \to A_0$$
$$i_*(A_1 \to A_0) =  A_1 \to A_0 \to A_0$$
$$ i^!(A_2 \to A_1 \to A_0) = A_2 \to A_1$$

This gives us elementary characterizations of the various subcategories of $Sh(\mathbf{3})$ involved in differential cohesion:

* The [[reduced object|reduced objects]] are the image of $i_!$: so they have the same real and infinitesimal points.
* The [[coreduced object|coreduced objects]] are the image of $i_*$: pieces all consist of exactly one infinitesimal point.

and the differential modalities are defined as:
$$ \Re(A_2 \to A_1 \to A_0) = A_2 \to A_2 \to A_0$$
$$ \Im(A_2 \to A_1 \to A_0) = A_2 \to A_0 \to A_0$$
$$ \&(A_2 \to A_1 \to A_0) = A_2 \to A_1 \to A_1$$

The cohesive structure

$$
  (\Pi \dashv \Disc \dashv \Gamma \dashv coDisc)
  :
  Psh(\mathbf{3})
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  Set
  \,.
$$
factors through the cohesive structure of the Sierpinski topos and is defined as:

$$\Pi(A_2\to A_1 \to A_0) = A_0$$
$$\Disc(A) = A \to A \to A$$
$$\Gamma(A_2 \to A_1 \to A_0) = A_2$$
$$coDisc(A) = A \to 1 \to 1$$

Giving us two more subcategories:

* The [[discrete object| discrete objects]] are the image of $\Disc$: every piece consists of exactly one real point.
* The [[codiscrete object|codiscrete objects]] are the image of $coDisc$: they consist of exactly one piece with exactly one infinitesimal point that all of the real points are over.

This is reflective of the general phenomenon that discrete objects are reduced and coreduced, in fact here the discrete objects are *exactly* the intersection of reduced and coreduced objects.

And the cohesive modalities are defines as:
$$ ʃ(A_2 \to A_1 \to A_0) = A_0 \to A_0 \to A_0$$
$$ \flat(A_2 \to A_1 \to A_0) = A_2 \to A_2 \to A_2$$
$$ \sharp(A_2 \to A_1 \to A_0) = A_2 \to 1 \to 1$$

#### Constructions in Differential Cohesion

A morphism $f : A \to B$ is [[formally etale morphism|formally etale]] when the square

$$
  \array{
    A_1 &\to& A_0
    \\
    \downarrow^{\mathrlap{f_1}} && \downarrow^{\mathrlap{f_0}}
    \\
    B_1 &\to& B_0
  }
$$

is a pullback. This makes it easy to see explicitly that coreduced objects are exactly the objects with $A \to 1$ formally etale.

The [[infinitesimal disk bundle]] of $A_2 \overset{inf}{\to}A_1 \overset{pc}{\to} A_0$ is the pullback of the unit of $\Im$ along itself. Since pullbacks are computed pointwise in a presheaf topos, this is just:

$$TA = A_2 \to \{(x_i,y_i) \in A_1 | pc(x_i) = pc(y_i) \} \to A_0$$

with the fiber of some real point $x_r \in A_2$ given by:

$$\{x_r \} \to \{y_i \in A_1 | pc(y_i) = pc(inf(x_r)) \} \to \{pc(inf(x_r)) \}$$

which is an [[infinitesimally thickened point]].