Grothendieck Construction

The Grothendieck construction glues the fibres of an indexed category into a single category. An indexed category is a (pseudo)functor assigning to each object a fibre category and to each morphism a restriction functor. An object of is a pair with and , and a morphism is a pair , composing by composing the base maps and transporting along .

The projection is a fibration

The projection , , is a fibration. Every base morphism has cartesian lifts, and the fibre over is . The construction is the categorical form of a family of spaces indexed by a base assembling into one total space. The total category records both the position in the base and the fibre object over that position.

Iterating gives nested indexing

When the fibres are themselves indexed, so that lands in indexed categories, the construction can be applied twice, , producing a total category fibred over whose fibres are each fibred in turn. Two independent directions of variation, an outer index and an inner one that varies with it, are then held in one object. The two layers are separated homologically by a spectral sequence.

References

• Alexander Grothendieck, “Categories fibrees et descente”, SGA 1, Expose VI (1971)