My problem is from Bruno Harris paper "Bott Periodicity via Simplicial Spaces"
Let $\mathcal{G}$ be the category, $Ob(\mathcal{G})=Gr= \coprod_n Gr_n(\mathbb{C^\infty})$ with only identity morphisms. Then B$\mathcal{G}=Gr$.
Next we define the category $\mathcal{Z}$. Let $ob(\mathcal{Z}) = \{\theta: \mathbb{C}^n \rightarrow \mathbb{C}^\infty| ||\theta(x)||=||x||, \text{ and } \theta \text{ is injective}\}$. This is just the Stiefel manifold $V_n(\mathbb{C}^\infty)$, wich is the collection of $n$-tuples of orthonormal vectors in $\mathbb{C}^\infty$. And morphisms $\alpha: \theta \rightarrow \theta'$ only being defined when $\theta$ and $\theta'$ have the same image in $\mathbb{C}^\infty$ and being the unique unitary transformation such that $\theta' \circ \alpha = \theta$.
Let $H$ be the functor $H:\mathcal{Z} \rightarrow \mathcal{G}$ takes the object $\theta:\mathbb{C}^n \rightarrow \mathbb{C}^\infty$, which is a $n$-frame, to the projection corresponding to the image of $\theta$, which is an $n$-plane. And morphisms to the identity morphism.
My problem is then as follows:Show that $BH: B\mathcal{Z} \rightarrow B\mathcal{G}$ is a serre fibration. My problem is that I don't know what the space $B\mathcal{Z}$ look like other then that it is the geometric realization of the simplicial space with $q$ simplices being $\coprod_n V_n(\mathbb{C}^\infty) \times U_n^q(\text{unitary group})$. So far I tried to solve it by showing that is is a Kan fibration without any luck.