Combinatorial Models for Linear Homotopy Theories

For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical algebras. We prove that semisimplicial modules and augmented semisimplicial modules are equivalent to appropriate chain-complex homotopy theories, both at the Gabriel–Zisman localization and the Quillen model-categorical level. The semicubical sign embedding gives a natural comparison from semicubical modules to augmented semisimplicial modules and induces a Quillen adjunction, but not a Quillen equivalence on the full semicubical category since there is an obstruction in augmented homology at degree $-1$.