In topology, any continuous function can be considered as a fibration over space , with total space , where the fibres are glued together according to the topology of .
Likewise, any functor can be regarded as a kind of fibration over . Each object determines a fibre category with arrows which are mapped to .
In this generality, the fibres connect up by means of profunctors between them and then and the original composition of are encapsulated in the arising mapping .
More specifically, given any arrow , its preimage determines [is] a profunctor between categories and , all within .
For a composable pair of arrows , we obtain a comparison map , which is just the composition defined in of the occuring pairs of arrows.
The comparison maps make this mapping a ‘normal lax functor’.
Under certain hypothesis on , all the arising profunctors become functorial (when each object from the source category has a reflection on the target category), yielding a normal lax functor . Such an is called an opposite Grothendieck prefibration. The reflection arrows are also called (weak) cartesian morphisms of .
If they are closed under composition in , the comparison maps become invertible, so that composition of the profunctors is and we talk about Grothendieck opfibration.
Its dual notion, when all profunctors are opfunctorial, is the Grothendieck (pre-)fibration.