Homotopy pullback of spaces

Definition 2.1.6. Consider maps \(f\colon X \to Z\) and \(g\colon Y \to Z\) of topological spaces. We define their homotopy pullback \(X \times _Z^h Y\) as the subspace of the product \(X \times Z^{[0,1]} \times Y\) given as \begin {align*} X \times _Z^h Y \quad &:= \quad \{(x,p,y) \in X \times Z^{[0,1]} \times Y \mid p(0) = f(x), p(1) = g(y)\}. \end {align*}

Here \(Z^{[0,1]} = \Map ([0,1], Z)\) is the space of paths in \(Z\) with the compact-open topology, which in this case is completely determined by the requirement that a map \(T \to Z^{[0,1]}\) is continuous if and only if the curried map \(T \times [0,1] \to Z\) is continuous.