path.

Finally in the last chapter we construct the classifying map for a given set of compatible transition functions. For this we construct a prismatic map from $\bar{P}(S)$ to the prismatic model for the classifying space constructed in chapter 6.

**Acknowledgements:** We would like to thank Marcel Bökstedt for his interests and comments during the preparation of this paper.

**[Prismatic Sets]{}\[two\]**
=============================

Prismatic sets are similar to simplicial sets but they are realized by using prisms instead of only simplices.

Let $\Delta^p=\{(t_0,\dots,t_p)\in{{\Bbb{R}}^{p+1}}\text{ }|\text{
}\sum_i t_i=1,t_i\leq{1}\}$ be a standard $p$-simplex given with barycentric coordinates. A prism is a product of simplices, that is, a set of the form $\Delta^{q_0...q_p}=\Delta^{q_0}\times{\dots}\times{\Delta^{q_p}}$.

The motivating example is triangulated fibre bundles:

\[2.1\]

Given a smooth fibre bundle $\pi:Y\to{Z}$ with dim$Y=m+n$, dim$Z=m$ and compact fibres possibly with boundary. By a theorem of Johnson [@J], there are smooth triangulations $K$ and $L$ of $Y$ and $Z$, respectively and a simplicial map $\pi
^\prime:K\to{L}$ in the following commutative diagram

$$\xymatrix{ |K| \ar[d]^{|\pi^{\prime}|} \ar[r]^{\approx}
& Y  \ar[d]^\pi \\
|L| \ar[r]^{\approx} & Z }$$

and the horizontal maps are homeomorphisms which are smooth on each simplex, here $|K|=\bigcup_{\tau\in{K_k}}\Delta^k\times{\tau}/\sim$, $k=0,...,\text{dim}K$, is the geometric realization.

One can extend a given such triangulation of 