Algebraic topology and homotopical equality

Example 1.1.1 (Algebraic topology). Algebraic topology provides the most natural example of the Fundamental Principle:

• Two continuous maps \(f,g\colon X \to Y\) are considered ‘equal up to homotopy’ if we provide a homotopy \(H\colon X \times [0,1] \to Y\) between \(f\) and \(g\);
• Two spaces \(X\) and \(Y\) are then considered ‘homotopically the same’ when equipped with a homotopy equivalence: continuous maps \(f\colon X \to Y\) and \(g\colon Y \to X\) together with chosen homotopies \(gf \sim \id _X\) and \(fg \sim \id _Y\).
• When studying diagrams of spaces, we similarly replace strict commutativity by homotopy commutativity. For instance, given a square

  

  we do not ask for a strict equality \(h \circ f = k \circ g\), but instead require a specified homotopy \(H\colon h \circ f \sim k \circ g\) witnessing how the two compositions are ‘the same’. This perspective naturally leads to homotopy coherent diagrams, where we must also specify compatibilities between different choices of homotopies.