a theorem of Johnson [@J], the base and total space of a locally trivial smooth fibre bundle with projection $\pi:E\to{B}$ can be triangulated in such a way that $\pi$ is a simplicial map. But obviously even in this case a general fibre is not a simplicial complex in any natural way. However such a fibre has a natural decomposition into prisms, i.e., products of simplices, and the whole triangulated bundle gives the basic example of a prismatic set, analogous to the notion of a simplicial set derived from a simplicial complex. Prismatic sets were introduced and used by the second author and R. Ljungmann in [@DLj] (see also Ljungmann’s thesis [@Lj]) in order to construct an explicit fibre integration map in smooth Deligne cohomology, see also [@DK]. But the important special case of the prismatic subdivision of a simplicial set was used in Akyar [@B] in connection with “Lattice Gauge Theory” in the sense of Phillips and Stone [@PS], [@PS1], [@PS5] and similar constructions have been used in other connections, see e.g. [@MS]. One can see [@L] for further information about Lattice Gauge Fields.

In this paper we shall give a more systematic treatment of prismatic sets and their properties 