local fibration

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

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)

- Rick Jardine,
*Local homotopy theory*(2011) (pdf)