In mathematics, the gluing axiom is introduced to define what a sheaf
must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category
ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism if
, the required condition is that In less formal language, a section
under the respective restriction maps and 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).
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 and each pair of arrows represents the two restrictions and It is worthwhile to note that these maps exhaust all of the possible restriction maps among
In some categories, it is possible to construct a sheaf by specifying only some of its sections.
is a contravariant functor which satisfies the gluing axiom for sets in
specifies all of the sections of a sheaf, and on the other open sets, it is undetermined.
is a sheaf, essentially because every element of every open cover of
is a union of basis elements (by the definition of a basis), and every pairwise intersection of elements in an open cover of
The first needs of sheaf theory were for sheaves of abelian groups; so taking the category
In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central.
This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of local rings.
as a parametrised family of local rings, depending on
One can speak freely of a sheaf of abelian groups, or rings, because those are algebraic structures (defined, if one insists, by an explicit signature).
In the case of this kind of purely algebraic structure, we can talk either of a sheaf having values in the category of abelian groups, or an abelian group in the category of sheaves of sets; it really doesn't matter.
At a foundational level we must use the second style of definition, to describe what a local ring means in a category.
This is a logical matter: axioms for a local ring require use of existential quantification, in the form that for any
, there is a standard device called sheafification or sheaving.
The rough intuition of what one should do, at least for a presheaf of sets, is to introduce an equivalence relation, which makes equivalent data given by different covers on the overlaps by refining the covers.
This use of language strongly suggests that we are dealing here with adjoint functors.
Therefore, we get an abstract characterisation of sheafification as left adjoint to the inclusion.
form a reflective subcategory of the presheaves (Mac Lane–Moerdijk Sheaves in Geometry and Logic p. 86).
In topos theory, for a Lawvere–Tierney topology and its sheaves, there is an analogous result (ibid.
The gluing axiom of sheaf theory is rather general.
One can note that the Mayer–Vietoris axiom of homotopy theory, for example, is a special case.