  In recent years, several authors have been investigating simplicial models, a
model of epistemic logic based on higher-dimensional structures called
simplicial complexes. In the original formulation, simplicial models were
always assumed to be pure, meaning that all worlds have the same dimension.
This is equivalent to the standard S5n semantics of epistemic logic, based on
Kripke models. By removing the assumption that models must be pure, we can go
beyond the usual Kripke semantics and study epistemic logics where the number
of agents participating in a world can vary. This approach has been developed
in a number of papers, with applications in fault-tolerant distributed
computing where processes may crash during the execution of a system. A
difficulty that arises is that subtle design choices in the definition of
impure simplicial models can result in different axioms of the resulting logic.
In this paper, we classify those design choices systematically, and axiomatize
the corresponding logics. We illustrate them via distributed computing examples
of synchronous systems where processes may crash.
