Editing Gluing axiom (section)

==Removing restrictions on ''C''==
To rephrase this definition in a way that will work in any category <math>C</math> 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":

:<math>{\mathcal F}(U)\rightarrow\prod_i{\mathcal F}(U_i){{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}}\prod_{i,j}{\mathcal F}(U_i\cap U_j)</math>

Here the first map is the product of the restriction maps

:<math>{res}_{U,U_{i}}:{\mathcal F}(U)\rightarrow{\mathcal F}(U_{i})</math>

and each pair of arrows represents the two restrictions

:<math>res_{U_i,U_i\cap U_j}:{\mathcal F}(U_i)\rightarrow{\mathcal F}(U_i\cap U_j)</math>

and

:<math>res_{U_j,U_i\cap U_j}:{\mathcal F}(U_j)\rightarrow{\mathcal F}(U_i\cap U_j)</math>.

It is worthwhile to note that these maps exhaust all of the possible restriction maps among <math>U</math>, the <math>U_i</math>, and the <math>U_i\cap U_j</math>.

The condition for <math>\mathcal F</math> to be a sheaf is that for any open set <math>U</math> and any collection of open sets <math>\{U_i\}_{i\in I}</math>  whose union is <math>U</math>, the diagram (G) above is an [[Equaliser (mathematics)|equalizer]].

One way of understanding the gluing axiom is to notice that <math>U</math> is the [[colimit]] of the following diagram:

:<math>\coprod_{i,j}U_i\cap U_j{{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}}\coprod_iU_i</math>

The gluing axiom says that <math>\mathcal F</math> turns colimits of such diagrams into limits.