Dear All,
Where does the Grothendieck construction come from? What is the original reference? Here is the construction.
A standard reference is (after Wikipedia, http://en.wikipedia.org/wiki/Grothendieck's_S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique): Grothendieck, Alexandre, Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Lecture notes in mathematics 224) (in French). Berlin; New York: Springer-Verlag, xxii+447. ISBN 3540056149. An updated version has been put in the arxiv: http://www.arxiv.org/abs/math.AG/0206203 The construction itself is defined in Section 8, as far as I remember. Artur
Take a functor H:I-->Cat (the category of small categories)
The objects are the pairs (i,a) where a is an object of H(i). A morphism (i,a)-->(j,b) consists of a morphism f:i-->j of I and a morphism H(f)(a)-->b of H(j).
pg.