Let P be an n-dimensional  simple convex polytope. A small cover over P is a smooth closed manifold with a locally standard $\mathbb{Z}_2^n$-action whose orbit space can be identified P. Small covers over P can be classified using characteristic functions from the set of facets of P to  $\mathbb{Z}_2^n$, up to weakly $\mathbb{Z}_2^n$-equivariant homeomorphisms covering the identity on $P$.
In this talk, we consider the case where P is a product of simplices. Using characteristic functions, we give a weakly $\mathbb{Z}_2^n$-equivariant classification of small covers over a product of simplices in terms of associated digraphs.
This is a joint work with S. Kaan Gürbüzer.