nLab 2-trivial model structure

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \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

2-Category theory

Contents

Definition

Every strict 2-category KK 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 ebe\to b such that K(x,e)K(x,b)K(x,e)\to K(x,b) is an isofibration for all xKx\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 CC as an (∞,1)-category with only trivial k-morphisms for k3k \geq 3.

Properties

  • 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 CatCat, the two model structures are the same.

Examples

  • In model categories built from various kinds of topological spaces, there are often analogous Hurewicz model structures. These are not actually examples of a 2-trivial model structure (for instance, the 2-category of spaces, continuous functions and homotopy classes of homotopies does not have finite limits as a 2-category, or even as a 1-category), but they share a common intuition and can sometimes be obtained as two instances of a more general construction.

References

Last revised on December 19, 2021 at 23:14:02. See the history of this page for a list of all contributions to it.