Gluing Axiom - Removing Restrictions On C

Removing Restrictions On C

To rephrase this definition in a way that will work in any category C that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing":

Here the first map is the product of the restriction maps

res_U,Ui,:F(U)→F(U_i)

and each pair of arrows represents the two restrictions

res_Ui,Ui∩Uj:F(U_i)→F(U_i∩U_j)

and

res_Uj,Ui∩Uj:F(U_j)→F(U_i∩U_j).

It is worthwhile to note that these maps exhaust all of the possible restriction maps among U, the U_i, and the U_i∩U_j.

The condition for F to be a sheaf is exactly that F is the limit of the diagram. This suggests the correct form of the gluing axiom:

A presheaf F is a sheaf if for any open set U and any collection of open sets {U_i}_i∈I whose union is U, F(U) is the limit of the diagram (G) above.

One way of understanding the gluing axiom is to notice that "un-applying" F to (G) yields the following diagram:

Here U is the colimit of this diagram. The gluing axiom says that F turns colimits of such diagrams into limits.