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
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Every strict 2-category $K$ with finite strict 2-limits and finite strict 2-colimits becomes a model category (or, rather, its underlying 1-category does) in a canonical way, where:
The weak equivalences are the equivalences.
The fibrations are the morphisms that are representably isofibrations, i.e. the morphisms $e\to b$ such that $K(x,e)\to K(x,b)$ is an isofibration for all $x\in K$.
The cofibrations are determined.
We call it the 2-trivial model structure, as it is a 2-categorical analogue of the trivial model structure on any 1-category. It can be said to regard $C$ as an (∞,1)-category with only trivial k-morphisms for $k \geq 3$.
Every object is fibrant and cofibrant.
By duality, any such category has another model structure, with the same weak equivalences but where the cofibrations are the iso-cofibrations and the fibrations are determined. In $Cat$, the two model structures are the same.
In Cat, this produces the canonical model structure.
If $T$ is an accessible strict 2-monad on a locally finitely presentable strict 2-category $K$. Then the category $T Alg_s$ of strict $T$-algebras admits a transferred model structure from the 2-trivial model structure on $K$. The cofibrant objects therein are the flexible algebras.
Last revised on December 20, 2021 at 04:14:02. See the history of this page for a list of all contributions to it.