Underlying anima of a topological space

Example 1.1.6. Every topological space \(X\) gives rise to an anima \(\Pi _{\infty }(X)\), called its underlying anima (or fundamental \(\infty \)-groupoid). Every point in \(X\) defines an element of \(\Pi _{\infty }(X)\). Given points \(x,y \in X\), every path \(p\colon [0,1] \to X\) from \(x\) to \(y\) defines an equality in \(\Pi _{\infty }(X)\) between \(x\) and \(y\). Given two such paths \(p\) and \(q\), every homotopy between \(p\) and \(q\) defines an equality between them in \(\Pi _{\infty }(X)(x,y)\). Every homotopy between homotopies defines an equality of equalities. This pattern repeats ad infinitum. We will study the assignment \(X \mapsto \Pi _{\infty }(X)\) in more detail in Chapter 2.