nLab
simplicial fundamental groupoid

Contents

Contents

Definitions

Definition using the category of simplices

The fundamental groupoid of a simplicial set XX is the localization C[C 1]C[C^{-1}] of its category of simplices CC (a special case of the category of elements of a presheaf).

Definition using the fundamental category

The fundamental groupoid of a simplicial set XX is the localization C[C 1]C[C^{-1}] of its fundamental category CC, defined as the left adjoint functor to the nerve functor

N:CatsSet.N\colon Cat \to sSet.

Definition using generators and relations

The fundamental groupoid of a simplicial set XX is a groupoid defined via the following system of generators and relations for a groupoid:

  • objects are precisely the vertices of XX;

  • generating isomorphisms are precisely the edges of XX;

  • for every 2-simplex of XX we have a relation

    d 1(σ)=d 0(σ)d 2(σ).d_1(\sigma)=d_0(\sigma)\circ d_2(\sigma).

See also

Created on October 25, 2019 at 17:47:17. See the history of this page for a list of all contributions to it.