nLab
model structure on semi-simplicial sets

Idea

There exists the model category structure on the category of semi-simplicial sets which is transferred along the right adjoint to the forgetful functor from the classical model structure on simplicial sets (van den Berg 13). See this discussion, which seems to conclude that this is Quillen equivalent to the classical model structure on simplicial sets.

There is also a weak model category structure (Henry 18), for which the Quillen equivalence to simplicial sets is proven as Henry 18, Thm 5.5.6 (iv).

Also there is the structure of a semimodel category (Rooduijn 2018) and of a fibration category and cofibration category on semisimplicial sets (Sattler 18, Th, 3.18 & 3.43).

References

As a model category-structure:

• Benno van den Berg, A note on semisimplicial sets, 2013 (pdf)

As a weak model category:

• Simon Henry, Theorem 5.5.6 of: Weak model categories in classical and constructive mathematics, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958. (arXiv:1807.02650, tac:35-24)

As a right semimodel category:

• Jan Rooduijn, A right semimodel structure on semisimplicial sets, Amsterdam 2018 (pdf, mol:4787)

As a fibration category and cofibration category:

• Christian Sattler, Constructive homotopy theory of marked semisimplicial sets (arXiv:1809.11168)