Gluing axiom

In mathematics, the gluing axiom is introduced to define what a sheaf formula_1 on a topological space formula_2 must satisfy, given that it is a presheaf, which is by definition a contravariant functor
to a category formula_4 which initially one takes to be the category of sets. Here formula_5 is the partial order of open sets of formula_2 ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism
if formula_8 is a subset of formula_9, and none otherwise.
As phrased in the sheaf article, there is a certain axiom that formula_10 must satisfy, for any open cover of an open set of formula_2. For example, given open sets formula_8 and formula_9 with union formula_2 and intersection formula_15, the required condition is that
In less formal language, a section formula_19 of formula_10 over formula_2 is equally well given by a pair of sections :formula_22 on formula_8 and formula_9 respectively, which 'agree' in the sense that formula_25 and formula_26 have a common image in formula_18 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).
Removing restrictions on "C".
To rephrase this definition in a way that will work in any category formula_4 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 formula_8, the formula_36, and the formula_37.
The condition for formula_1 to be a sheaf is exactly that formula_1 is the limit of the diagram. This suggests the correct form of the gluing axiom:
One way of understanding the gluing axiom is to notice that "un-applying" formula_1 to (G) yields the following diagram:
Here formula_8 is the colimit of this diagram. The gluing axiom says that formula_1 turns colimits of such diagrams into limits.
Sheaves on a basis of open sets.
In some categories, it is possible to construct a sheaf by specifying only some of its sections. Specifically, let formula_2 be a topological space with basis formula_50. We can define a category to be the full subcategory of formula_5 whose objects are the formula_52. A B-sheaf on formula_2 with values in formula_4 is a contravariant functor
which satisfies the gluing axiom for sets in formula_56. That is, on a selection of open sets of formula_2, formula_1 specifies all of the sections of a sheaf, and on the other open sets, it is undetermined.
B-sheaves are equivalent to sheaves (that is, the category of sheaves is equivalent to the category of B-sheaves). Clearly a sheaf on formula_2 can be restricted to a B-sheaf. In the other direction, given a B-sheaf formula_1 we must determine the sections of formula_1 on the other objects of formula_5. To do this, note that for each open set formula_8, we can find a collection formula_64 whose union is formula_8. Categorically speaking, this choice makes formula_8 the colimit of the full subcategory of formula_56 whose objects are formula_64. Since formula_1 is contravariant, we define formula_70 to be the limit of the formula_71 with respect to the restriction maps. (Here we must assume that this limit exists in formula_4.) If formula_8 is a basic open set, then formula_8 is a terminal object of the above subcategory of formula_56, and hence formula_76. Therefore, formula_77 extends formula_1 to a presheaf on formula_2. It can be verified that formula_77 is a sheaf, essentially because every element of every open cover of formula_2 is a union of basis elements (by the definition of a basis), and every pairwise intersection of elements in an open cover of formula_2 is a union of basis elements (again by the definition of a basis).
The logic of "C".
The first needs of sheaf theory were for sheaves of abelian groups; so taking the category formula_4 as the category of abelian groups was only natural. 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. It is the "stalks" of the sheaf that are local rings, not the collections of "sections" (which are rings, but in general are not close to being "local"). We can think of a locally ringed space formula_2 as a parametrised family of local rings, depending on formula_85 in formula_2.
A more careful discussion dispels any mystery here. 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). Any category formula_4 having finite products supports the idea of a group object, which some prefer just to call a group "in" formula_4. 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.
In the local ring case, it does 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 formula_89 in the ring, one of formula_89 and formula_91 is invertible. This allows one to specify what a 'local ring in a category' should be, in the case that the category supports enough structure.
Sheafification.
To turn a given presheaf formula_92 into a sheaf formula_1, 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. One approach is therefore to go to the stalks and recover the sheaf space of the "best possible" sheaf formula_1 produced from formula_92.
This use of language strongly suggests that we are dealing here with adjoint functors. Therefore, it makes sense to observe that the sheaves on formula_2 form a full subcategory of the presheaves on formula_2. Implicit in that is the statement that a morphism of sheaves is nothing more than a natural transformation of the sheaves, considered as functors. Therefore, we get an abstract characterisation of sheafification as left adjoint to the inclusion. In some applications, naturally, one does need a description.
In more abstract language, the sheaves on formula_2 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. p. 227).
Other gluing axioms.
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.