nLab
2-trivial model structure

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 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

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

References

Created on January 28, 2012 09:44:36 by Mike Shulman (71.136.231.206)