# nLab infinitesimal interval object

## Theorems

differential geometry

synthetic differential geometry

# Contents

## 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 ∞-groupoids, maps out of infinitesimal interval object model infinitesimal paths that may arrange themselves into infinitesimal path ∞-groupoid.

## Definition

In any lined topos $\left(𝒯,\left(R,+,\cdot \right)\right)$ the line object $R$ is naturally regarded as a cartesian interval object.

$\begin{array}{ccc}& & R\\ & {}^{{0}_{*}}↗& & {↖}^{{1}_{*}}\\ *& & & & *\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && R \\ & {}^{0_{*}}\nearrow && \nwarrow^{1_{*}} \\ {*} &&&& {*} } \,.

If the lined topos $\left(𝒯,R\right)$ 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

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

or equivalently expressed in topos logic as

$D=\left\{x\in R\mid {x}^{2}=0\right\}\phantom{\rule{thinmathspace}{0ex}}.$D = \{x \in R | x^2 = 0 \} \,.

and think of $D$ as the infinitesimal interval object of the smooth topos $\left(𝒯,R\right)$ 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.

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

Urs Schreiber: the following should be checked

### Introduction

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

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

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

$\Pi \left(X\right):={X}^{{\Delta }_{R}^{•}}\phantom{\rule{thinmathspace}{0ex}}.$\Pi(X) := X^{\Delta_R^\bullet} \,.

Here as an object in $𝒯$ 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 ${\Pi }^{\mathrm{inf}}\left(X\right)$ 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 elements.

The natural way to encode an infinitesimal path between two elements in an object $X$ in the smooth topos $𝒯$ is therefore not as an element of ${X}^{D}$ but of ${X}^{D}×D$, 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

$\begin{array}{ccc}& & {X}^{D}×D\\ & {}^{{p}_{TX}\circ {p}_{1}}↙& & {↘}^{\mathrm{ev}}\\ X& & & & X\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 ${X}^{D}×D\to {X}^{D}$ followed by the tangent bundle projection ${p}_{TX}={X}^{\left(*\to D\right)}:{X}^{D}\to X$

• the right leg is the evaluation map, i.e. the inner hom-adjunct of $\mathrm{Id}:{X}^{D}\to {X}^{D}$.

With this setup a pair of (generalized) elements $x,y\in X$ may be thought of as connected by an infinitesimal path if there is an element $\left(v,ϵ\right)\in {X}^{D}×D$ such that

${\delta }_{1}\left(v,ϵ\right)=v\left(0\right)=x$\delta_1(v,\epsilon) = v(0) = x

and

${\delta }_{0}\left(v,ϵ\right)=v\left(ϵ\right)=y\phantom{\rule{thinmathspace}{0ex}}.$\delta_0(v,\epsilon) = v(\epsilon) = y \,.

But not all elements $\left(v,ϵ\right)$ 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 $\left(v,tϵ\right)$ and $\left(tv,ϵ\right)$ define the same pair of infinitesimal neighbours, $x=v\left(0\right)$ and $y=\left(tv\right)\left(ϵ\right)=v\left(tϵ\right)$.

We may identify such elements $\left(tv,ϵ\right)$ and $\left(v,tϵ\right)$ by passing to the tensor product ${X}^{D}{\otimes }_{R}D$, i.e. the coequalizer of

${X}^{D}×R×D\stackrel{\stackrel{\cdot ×\mathrm{Id}}{\to }}{\underset{\mathrm{Id}×\cdot }{\to }}{X}^{D}×D$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 $\left(R,\cdot \right)$ on $D$ (by the embedding $D↪R$) and on ${X}^{D}$ (by the fact that $X$ is assumed to be microlinear).

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

### The definition

###### Definition

(infinitesimal path simplicial object)

Let $X\in 𝒯$ be a microlinear space in the smooth topos $\left(𝒯,R\right)$ with infinitesimal interval object $D$.

Then define the simplicial object

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

as follows:

• in degree $n$ it assigns the object

$\left[n\right]↦{X}^{D\left(n\right)}{\otimes }_{{R}^{n}}D\left(n\right)↪\left({X}^{D}{\otimes }_{R}D{\right)}^{{×}_{X}^{n}}$[n] \mapsto X^{D(n)} \otimes_{R^n} D(n) \hookrightarrow (X^D \otimes_R D)^{\times_X^n}

whose generalized elements we write $\left({ϵ}_{i}{v}_{i}{\right)}_{x}$ with $\stackrel{⇀}{ϵ}\in D\left(n\right)$ or $\left({\nu }_{i}{\right)}_{x}$ for short; where $x\in X$ indicates the fiber of ${X}^{D\left(n\right)}\to X$ that the element lives in

• the face maps ${d}_{i}:\left({X}^{D}{×}_{R}D{\right)}^{{×}_{X}^{n+1}}\to \left({X}^{D}{×}_{R}D{\right)}^{{×}_{X}^{n}}$ are

• for $0

given on generalized elements by

${d}_{i}:\left({\nu }_{i}{\right)}_{x}↦\left({\nu }_{0},\cdots ,{\nu }_{i-2},{\nu }_{i-1}+{\nu }_{i},{\nu }_{i+1},\cdots ,{\nu }_{m+1}{\right)}_{x}$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

${d}_{0}:\left({\nu }_{i}\right)↦\left({v}_{0}\left({ϵ}_{0}\right)+{\nu }_{i}\right)$d_0 : (\nu_i) \mapsto (v_0(\epsilon_0) + \nu_i)

where the element on the right denotes the evaluation of the map $\left({\nu }_{i}\right):D\left(n\right)\to X$ in its first argument on ${ϵ}_{0}$, regarded as an element in the fiber over ${v}_{0}\left({ϵ}_{0}\right)$.

• for $i=n+1$

given by

${d}_{n+1}:\left({\nu }_{0},\cdots ,{\nu }_{n+1}\right)↦\left({\nu }_{0},\cdots ,{\nu }_{n}\right)$

• the degeneracy maps ${\sigma }_{i}$ act by inserting the 0-vector in position i:

${\sigma }_{i}:\left({\ni }_{i}\right)↦\left({\nu }_{0},\cdots ,{\nu }_{i-1},0,{\nu }_{i+1},\cdots ,{\nu }_{n}\right)$.

###### Proposition

These face and degeneracy maps indeed satisfy the simplicial identities.

###### Proof

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

###### Remark

By thinking of the ${v}_{i}:D\to X$ as infinitesimal collared curves in $X$ with source ${v}_{i}\left(0\right)$ and target ${v}_{i}\left({ϵ}_{i}\right)$ the above definition is an immediate analog of the definition of the path $\infty$-groupoid $\Pi \left(X\right)$ of finite paths as discussed at interval object.

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

### The inclusion of infinitesimal paths into finite paths

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

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

as discussed there. On object this assigns

$\Pi \left(X\right):\left[n\right]↦{X}^{{R}^{n}}\phantom{\rule{thinmathspace}{0ex}}.$\Pi(X) : [n] \mapsto X^{R^n} \,.
###### Definition

(inclusion of infinitesimal into finite paths)

For $n\in ℕ$ define a morphism

${X}^{\left(}D\left(n\right)\right){\otimes }_{{R}^{n}}D\left(n\right)\to {X}^{{R}^{n}}$X^(D(n)) \otimes_{R^n} D(n) \to X^{R^n}

on generalized elements by

${\iota }_{n}:\left({ϵ}_{i}{v}_{i}\right)↦\left(\left({t}_{0},\cdots ,{t}_{n-1}\right)↦\sum _{i=0}^{n-1}{v}_{i}\left({t}_{i}{ϵ}_{i}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\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)) \,.
###### Proposition

These morphism $\left({\iota }_{n}\right)$ respect the face and degeneracy maps on both sides and hence induce an inclusion of simplicial objects

${\Pi }^{\mathrm{inf}}\left(X\right)↪\Pi \left(X\right)$\Pi^{inf}(X) \hookrightarrow \Pi(X)
###### Proof

Straightforward checking on generalized elements.

### Properties

under construction

Let $X$ be a micorlinear space.

Sketch of Proposition

We want to show that the morphism of simplicial sets

$\left[{\Delta }^{\mathrm{op}},𝒯\right]\left(U×D,{X}^{\left({\Delta }_{\mathrm{inf}}^{•}\right)}\right)\to \left[{\Delta }^{\mathrm{op}},𝒯\right]\left(U,{X}^{\left({\Delta }_{\mathrm{inf}}^{•}\right)}\right)$[\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 $U\simeq U×*\to U×D$ is a weak homotopy equivalence.

Sketch of proof

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

$\begin{array}{ccc}\partial \Delta \left[n\right]& \to & \left[{\Delta }^{\mathrm{op}},𝒯\right]\left(U×D,{X}^{\left({\Delta }_{\mathrm{inf}}^{•}\right)}\right)\\ ↓& & ↓\\ \Delta \left[n\right]& \to & \left[{\Delta }^{\mathrm{op}},𝒯\right]\left(U,{X}^{\left({\Delta }_{\mathrm{inf}}^{•}\right)}\right)\end{array}$\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 $U×D\to U×*\simeq U\to X$.

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

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

The case for $U×D$ replaced with $U×{D}^{n}$ works analogously.

Revised on March 23, 2010 07:29:11 by Urs Schreiber (87.212.203.135)