Fundamental groupoid

In mathematics , especially in algebraic topology , the fundamental groupoid of a topological space is supposed to summarize the set of path connection components and the fundamental groups (for all ) in a single algebraic object. ${\displaystyle \Pi (X)}$ ${\displaystyle X}$ ${\displaystyle \pi _{0}(X)}$ ${\displaystyle \pi _{1}(X,x)}$ ${\displaystyle x\in X}$

The fundamental groupoid is a groupoid , i.e. a category in which every morphism is an isomorphism . The objects are the points from , the morphisms from to are the homotopy classes (relative ) of continuous paths with . ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \partial \left[0,1\right]}$ ${\displaystyle p\colon \left[0,1\right]\to X}$ ${\displaystyle p(0)=x,p(1)=y}$

In this groupoid, the set corresponds to the isomorphism classes of objects, while the automorphism group corresponds to the object . ${\displaystyle \pi _{0}(X)}$ ${\displaystyle \pi _{1}(X,x)}$ ${\displaystyle x}$

Web links

• Fundamental Groupoid (nLab)