Before revision:  Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one simplicial and the other polyhedral , which are both topologically equivalent to the multicover bifiltration and far smaller than a \v{C model considered in prior work of Sheehy.
After revision:  Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other polyhedral , which are both topologically equivalent to the multicover bifiltration and far smaller than a \v{C model considered in prior work of Sheehy.