model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
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).
As a model category-structure:
As a weak model category:
As a right semimodel category:
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 19:43:32. See the history of this page for a list of all contributions to it.