Contents

Contents

Definitions

Definition using the category of simplices

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

Definition using the fundamental category

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

$N\colon Cat \to sSet.$

Definition using generators and relations

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

• objects are precisely the vertices of $X$;

• generating isomorphisms are precisely the edges of $X$;

• for every 2-simplex of $X$ we have a relation

$d_1(\sigma)=d_0(\sigma)\circ d_2(\sigma).$