The Grothendieck construction is a construction used in the mathematical field of category theory.
Let F : C → C a t be a functor from any small category to the category of small categories. The Grothendieck construction for F is the category Γ ( F ) (also written C ∫ F or F ⋊ C ), with
objects being pairs ( c , x ) , where c ∈ obj ( C ) and x ∈ obj ( F ( c ) ) ; andmorphisms in hom Γ ( F ) ( ( c 1 , x 1 ) , ( c 2 , x 2 ) ) being pairs ( f , g ) such that f : c 1 → c 2 in C , and g : F ( f ) ( x 1 ) → x 2 in F ( c 2 ) .Composition of morphisms is defined by ( f , g ) ∘ ( f ′ , g ′ ) = ( f ∘ f ′ , g ∘ F ( f ) ( g ′ ) ) .
