Singular simplicial complex

Example 1.6.8 (Singular simplicial complex). Let \(X\) be a topological space. We will construct a simplicial set \(\Sing (X) \in \sSet \) called the singular simplicial complex of \(X\).

• For every \(n \geq 0\), let \(\abs {\Delta ^n}\) denote the topological \(n\)-simplex: \[ \abs {\Delta ^n} := \{ (x_0, \dots , x_n)\in \R ^{n+1} \mid x_{0} + \dots + x_n=1, \,\, x_{i}\geq 0{\text { for }}i=0,\dots ,n\}. \]
• For every order-preserving map \(\phi \colon [n] \to [m]\) we get a continuous map \[ \abs {\Delta ^{\phi }}\colon \abs {\Delta ^n} \to \abs {\Delta ^m}, \quad (x_0, \dots , x_n) \mapsto (y_0, \dots , y_m), \qquad y_i := \sum _{j \in \phi ^{-1}(i)} x_j. \] This defines a functor \[ \abs {\Delta ^{\bullet }} \colon \simp \to \Top , \qquad [n] \mapsto \abs {\Delta ^n}, \qquad \phi \mapsto \abs {\Delta ^{\phi }}. \]
• We now define the simplicial set \(\Sing (X)\) as \[ \Sing (X)_n := \Hom _{\Top }(\abs {\Delta ^n}, X), \] i.e. it is the composite \(\simp \catop \xrightarrow {\abs {\Delta ^{\bullet }}} \Top \catop \xrightarrow {\Hom _{\Top }(-,X)} \Set \).

The simplicial set \(\Sing (X)\) is a familiar object from algebraic topology, and is used for example in the definition of the singular (co)homology of \(X\). It is not difficult to see that \(\Sing (X)\) is functorial in \(X\), so that it defines a functor \[ \Sing \colon \Top \to \sSet . \]