Moduli anima of real vector bundles

Definition 10.2.10. We define the moduli anima of real vector bundles over \(X\) by \[ \Vect _{\R }(X)^{\simeq } := \colim _{[q]\in \simp \catop } \Vect _{\R }^{\disc }(X\times \abs {\Delta ^q})^{\simeq } \qin \CMon (\An ). \] This colimit is computed on underlying animae and is functorial in \(X\), giving a functor \(\Vect _{\R }(-)^{\simeq }\colon \Top \catop \to \CMon (\An )\). When \(X=\pt \), we refer to \(\Vect _{\R }(\pt )^{\simeq }\) as the moduli anima of finite-dimensional real vector spaces.