Proposition Trivial Model Structure [efr-MRQ3]

Let \mathcal {C} be a 2-category. Then the underlying category \mathcal {C}_0 carries a model structure defined as follows:

1. A morphism A \to B is a weak equivalence, respectively a fibration if, for each X \in \mathcal {C}, the induced functor \mathcal {C}(X,A) \to \mathcal {C}(X,B) is a weak equivalence, respectively an isofibration
2. A morphism is a cofibration if and only if it has the left lifting property against every morphism which is both a weak equivalence and a fibration

Moreover, when \mathsf {Cat} is equipped with its folk model structure, and \mathcal {C} is considered as enriched over \mathsf {Cat}, this is an enriched model structure in the usual sense.