# Contents

## 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

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

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 we write

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

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

The $(i_* \dashv i^*)$-counit

$\Re (X) \to X$

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

## Examples

Last revised on May 13, 2015 at 08:38:26. See the history of this page for a list of all contributions to it.