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

F: O(X) → C

to a category C which initially one takes to be the category of sets. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are Here O(X) is the partial order of open sets of X ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism

U → V

if U is a subset of V, and none otherwise. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

As phrased in the sheaf article, there is a certain axiom that F must satisfy, for any open cover of an open set of X. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection For example given open sets U and V with union X and intersection W, the required condition is that

F(X) is the subset of F(U)×F(V) with equal image in F(W). In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently

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

F(U) → F(W)

and

F(V) → 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. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus.

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). In Category theory, a branch of Mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics

Contents 1 Removing restrictions on C 2 Sheaves on a basis of open sets 3 The logic of C 4 Sheafification 5 Other gluing axioms

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. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts 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. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The gluing axiom says that F 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 X be a topological space with basis {B_i}_i∈I. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets We can define a category O ′(X) to be the full subcategory of O(X) whose objects are the {B_i}. A B-sheaf on X with values in C is a contravariant functor

F: O ′(X) → C

which satisfies the gluing axiom for sets in O ′(X). We would like to recover the values of F on the other objects of O(X).

To do this, note that for each open set U, we can find a collection {B_j}_j∈J whose union is U. Categorically speaking, U is the colimit of the {B_j}_j∈J. Since F is contravariant, we define F(U) to be the limit of the {F(B)}_j∈J. (Here we must assume that this limit exists in C. ) It can be shown that this new object agrees with the old F on each basic open set, and that it is a sheaf.

The logic of C

The first needs of sheaf theory were for sheaves of abelian groups; so taking the category C as the category of abelian groups was only natural. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called 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. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on 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). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real We can think of a locally-ringed space X as a parametrised family of local rings, depending on x in X.

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). In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Logic, especially Mathematical logic, a signature lists and describes the Non-logical symbols of a Formal language. Any category C having finite products supports the idea of a group object, which some prefer just to call a group in C. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets. 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 r in the ring, one of r and 1 − r is invertible. In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to 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 P into a sheaf F, 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 F produced from P. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

This use of language strongly suggests that we are dealing here with adjoint functors. Therefore it makes sense to observe that the sheaves on X form a full subcategory of the presheaves on X. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in Implicit in that is the statement that a morphism of sheaves is nothing more than a natural transformation of the sheaves, considered as functors. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal 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 X form a reflective subcategory of the presheaves (Mac Lane-Moerdijk Sheaves in Geometry and Logic p. In Mathematics, a Subcategory A of a category B is said to be reflective in B when the Inclusion functor from 86). In topos theory, for a Lawvere-Tierney topology and its sheaves, there is an analogous result (ibid. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets 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. In Mathematics, Brown's representability theorem in Homotopy theory gives Necessary and sufficient conditions on a Contravariant functor In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

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