Geometric realization of simplicial sets

Definition 1.6.9 (Geometric realization). Let \(K\colon \simp \catop \to \Set \) be a simplicial set. Its geometric realization is the quotient space \[ \abs {K}:=\left (\bigsqcup _{n\geq 0}K_n\times \abs {\Delta ^n}\right )/\sim , \] where each set \(K_n\) carries the discrete topology and the equivalence relation is generated by \[ (\phi ^*(x),y)\sim (x,\abs {\Delta ^\phi }(y)), \] for all maps \(\phi \colon [n]\to [m]\) in \(\simp \), elements \(x\in K_m\), and points \(y\in \abs {\Delta ^n}\). This construction defines a functor \[ \abs {-}\colon \sSet \to \Top \] which is left adjoint to the singular complex functor \(\Sing \colon \Top \to \sSet \).