# nLab 2-trivial model structure

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### 2-Category theory

2-category theory

## Definition

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 $k \geq 3$.

## Properties

• Every object is fibrant and cofibrant.

• In Cat, this produces the canonical model structure.

• 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.

• 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 algebra?s.

## References

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