Combinatorial abstractions of the graphs of polyhedra were
considered by Adler, Dantzig, and Murty (abstract polytopes) and Kalai
(ultraconnected set systems). More recently Eisenbrand, H\"ahnle,
Razborov, and Rothvo\ss{} introduced a slightly more general class of
objects (called connected layer families) and proved a superlinear lower
bound on their diameters. In this talk, we give a survey of these
abstractions, giving complete definitions and showing how they fit into
a more general framework, which naturally leads to some variants of
these earlier abstractions. We then focus on special subclasses in the
new framework. In particular, we define a specific kind of combinatorial
prismatoid, inspired by the use of prismatoid widths in Santos'
counterexample to the Hirsch Conjecture. We define a notion of width for
these combinatorial prismatoids and prove that it is superlinear.
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