Grothendieck construction


The Grothendieck construction is a construction used in the mathematical field of category theory.

Definition

Let be a functor from any small category to the category of small categories. The Grothendieck construction for is the category , with
Composition of morphisms is defined by.

Slogan

"The Grothendieck construction takes structured, tabulated data and flattens it by throwing it all into one big space. The projection functor is then tasked with remembering which box each datum originally came from."

Example

If is a group, then it can be viewed as a category, with one object and all morphisms invertible. Let be a functor whose value at the sole object of is the category a category representing the group in the same way. The requirement that be a functor is then equivalent to specifying a group homomorphism where denotes the group of automorphisms of Finally, the Grothendieck construction, results in a category with one object, which can again be viewed as a group, and in this case, the resulting group is the semidirect product