nLab
infinitesimal path ∞-groupoid

Contents

Idea

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

Definition

Definition

Given differential cohesion

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

define the reduction modality/infinitesimal shape modality adjunction

():H thi *i !Hi *i *H th. (\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 Π inf(X)(X)\Pi_{inf}(X) \coloneqq \Im(X) the infinitesimal path ∞-groupoid of XX and (X)\Re(X) the reduced type of XX.

For the (i *i *)(i_* \dashv i^*)-unit we write

InfinitesimalPathInclusion X:XΠ inf(X) InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)

and call it the constant infinitesimal path inclusion on XX.

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

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

we call the inclusion of the reduced part of XX.

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.