Gluing axiom

In mathematics, the gluing axiom is introduced to define what a sheaf ${\displaystyle {\mathcal {F}}}$ on a topological space ${\displaystyle X}$ must satisfy, given that it is a presheaf, which is by definition a contravariant functor

${\displaystyle {\mathcal {F}}:{\mathcal {O}}(X)\rightarrow C}$

to a category ${\displaystyle C}$ which initially one takes to be the category of sets. Here ${\displaystyle {\mathcal {O}}(X)}$ is the partial order of open sets of ${\displaystyle X}$ ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism

${\displaystyle U\rightarrow V}$

if ${\displaystyle U}$ is a subset of ${\displaystyle V}$, and none otherwise.

As phrased in the sheaf article, there is a certain axiom that ${\displaystyle F}$ must satisfy, for any open cover of an open set of ${\displaystyle X}$. For example, given open sets ${\displaystyle U}$ and ${\displaystyle V}$ with union ${\displaystyle X}$ and intersection ${\displaystyle W}$, the required condition is that

${\displaystyle {\mathcal {F}}(X)}$ is the subset of ${\displaystyle {\mathcal {F}}(U)\times {\mathcal {F}}(V)}$ With equal image in ${\displaystyle {\mathcal {F}}(W)}$

In less formal language, a section ${\displaystyle s}$ of ${\displaystyle F}$ over ${\displaystyle X}$ is equally well given by a pair of sections :${\displaystyle (s',s'')}$ on ${\displaystyle U}$ and ${\displaystyle V}$ respectively, which 'agree' in the sense that ${\displaystyle s'}$ and ${\displaystyle s''}$ have a common image in ${\displaystyle {\mathcal {F}}(W)}$ under the respective restriction maps

${\displaystyle {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(W)}$

and

${\displaystyle {\mathcal {F}}(V)\rightarrow {\mathcal {F}}(W)}$.

The first major hurdle in sheaf theory is to see that this gluing or patching axiom is a correct abstraction from the usual idea in geometric situations. For example, a vector field is a section of a tangent bundle on a smooth manifold; this says that a vector field on the union of two open sets is (no more and no less than) vector fields on the two sets that agree where they overlap.

Given this basic understanding, there are further issues in the theory, and some will be addressed here. A different direction is that of the Grothendieck topology, and yet another is the logical status of 'local existence' (see Kripke–Joyal semantics).