A Grothendieck fibration over a base previewcategory $\mathrm{ℬ︀}$ is a generalization of the structure of pullback over $\mathrm{ℬ︀}$. Rather than pulling back a morphism along a morphism, one can pull back any object in the total category $\mathrm{ℰ︀}$ along a morphism in $\mathrm{ℬ︀}$, assuming the “codomain” of that object is the same as that of the morphism.

The notion of a “pullback square” is replaced with that of cartesian morphism, which is the “universal” pull back of an object along a morphism.

Fibrations are widely used in categorical semantics of dependent type theory.