nLab infinitesimal interval object

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 $(\mathcal{T},(R,+,\cdot))$ the line object $R$ is naturally regarded as a 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

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 elements.

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) 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 neighbours, $x = v(0)$ 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).

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

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 elements 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 elements 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:

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

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 $\Pi(X)$ 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 $v_i(0)$ 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

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

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 elements by

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

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

Straightforward checking on generalized elements.

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.