nLab
model structure on semi-simplicial sets

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

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:

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:

Last revised on June 26, 2021 at 06:19:39. See the history of this page for a list of all contributions to it.