Stable Grassmannian as a stable classifying anima

Proof. The principal bundles \(V_n(\C ^\infty ) \to \Gr _n(\C ^\infty )\) identify \(\Pi _{\infty }\Gr _n(\C ^\infty )\) with \(\bB U(n)\) by Corollary 9.2.15. These identifications are compatible with the stabilization maps \(\Gr _n(\C ^\infty ) \to \Gr _{n+1}(\C ^\infty )\) and the block-sum inclusions \(U(n) \to U(n+1)\). Since the Grassmannian stabilization maps are closed embeddings between Hausdorff spaces, Proposition 9.2.19 gives \[ \Pi _{\infty }(BU) \simeq \colim _n \bB U(n). \] The classifying-anima equivalence \(\bB \colon \Grp (\An )\iso \An _{*,\geq 1}\) preserves colimits. Since connected pointed animae are closed under filtered colimits, the filtered colimit of the \(\bB U(n)\) agrees whether computed in \(\An \), \(\An _*\), or \(\An _{*,\geq 1}\). Moreover, filtered colimits in \(\Grp (\An )\) are computed on underlying animae, since finite limits commute with filtered colimits in \(\An \). Applying Proposition 9.2.19 also to the closed embeddings \(U(n)\hookrightarrow U(n+1)\) therefore gives \[ \colim _n\bB U(n) \simeq \bB \!\left (\colim _n\Pi _{\infty }U(n)\right ) \simeq \bB \Pi _{\infty }(U), \] which is the asserted classifying anima \(\bB U\). □