Semisimplicial and augmented semisimplicial modules are equivalent to chain-complex homotopy theories at localization and Quillen levels, while semicubical modules induce a Quillen adjunction but not equivalence due to an obstruction in augmented homology at degree -1.
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Combinatorial Models for Linear Homotopy Theories
Semisimplicial and augmented semisimplicial modules are equivalent to chain-complex homotopy theories at localization and Quillen levels, while semicubical modules induce a Quillen adjunction but not equivalence due to an obstruction in augmented homology at degree -1.