nLab
reduced space

Contents

Idea

In a context of infinitesimal cohesion the reduced type Red(X)Red(X) of a type XX is the result of removing all infinitesimal thickening in XX.

Definition

Definition

Given differential cohesion

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

define the monad/comonad adjunction

(RedΠ inf):H thi *i *Hi *i !H th. (Red \dashv \Pi_{inf}) \colon \mathbf{H}_{th} \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}} \mathbf{H} \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H}_{th} \,.

We call Red(X)Red(X) the reduced type of XX and Π inf(X)\Pi_{inf}(X) the infinitesimal path ∞-groupoid of XX.

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

RedXX Red X \to X

we call the inclusion of the reduced part 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.

Examples

(…)

Last revised on January 5, 2014 at 12:23:54. See the history of this page for a list of all contributions to it.