nLab local fibration

Idea

In a category of simplicial presheaves it makes sense to ask if a morphism is locally a Kan fibration, in that the required lifting property holds after refinement along some cover.

These local fibrations are typically not the fibrations in any of the model structures on simplicial presheaves. (They may instead form the fibrations in the structure of a category of fibrant objects on simplicial sheaves.) Nevertheless, it is useful to consider them, and they satisfy various properties otherwise known from genuine fibrations.

For instance a pullback diagram of simplicial presheaves is a homotopy pullback already if one of the two morphisms in the cospan is a local fibration (e.g. Jardine, lemma 5.16)

References

• Rick Jardine, Local homotopy theory (2011) (pdf)