nLab 2-trivial model structure

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.

• 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.

Examples

• 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.

Related pages

• 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

• Steve Lack, Homotopy-theoretic aspects of 2-monads, arXiv