# nLab model structure on semi-simplicial sets

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-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 on semisimplicial sets (Sattler 18, Th, 3.18) (and cofibration category on fibrant-cofibrant objects).

## 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:

The above note contains a mistake in Theorem 3.43: semisimplicial sets do not form a cofibration category as considered there: (cofibration, weak equivalence)-factorizations do not generally exist (cf. the paragraph at the end of Subsection 3.2). Instead, the notion of cofibration category has to be weakened using a notion of pseudofactorizations (similar to the notion of weak model category).

Last revised on June 12, 2022 at 15:43:32. See the history of this page for a list of all contributions to it.