nLab infinitesimal path ∞-groupoid

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

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 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

• de Rham space