nLab
infinitesimal path ∞-groupoid
Contents
Idea
In a context of infinitesimal cohesion the infinitesimal path ∞ \infty -groupoid Π inf X ≔ ℑ ( X ) \Pi_{inf}X \coloneqq \Im(X) of a type X X is the result of identifying infinitesimally close points in X X by adding in further equivalences between all objects (points of X X ) that are infinitesimal neighbours.
Definition
Definition
Given differential cohesion
( H ≔ CohesiveType ) ↪ i ( H th ≔ InfThickenedCohesiveType )
(\mathbf{H} \coloneqq CohesiveType)
\stackrel{i}{\hookrightarrow}
(\mathbf{H}_{th} \coloneqq InfThickenedCohesiveType)
define the reduction modality /infinitesimal shape modality adjunction
( ℜ ⊣ ℑ ) : H th → i * ← i ! H → i * ← 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 ) ≔ ℑ ( X ) \Pi_{inf}(X) \coloneqq \Im(X) the infinitesimal path ∞-groupoid of X X and ℜ ( X ) \Re(X) the reduced type of X X .
For the ( i * ⊣ i * ) (i_* \dashv i^*) -unit we write
InfinitesimalPathInclusion X : X → Π inf ( X )
InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)
and call it the constant infinitesimal path inclusion on X X .
The ( i * ⊣ i * ) (i_* \dashv i^*) -counit
we call the inclusion of the reduced part of X X .
Examples
Last revised on May 13, 2015 at 08:38:26.
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