# Maths - Sheaf

The sheaf structure is a powerful idea that goes across many mathematical subjects and therefore it can be approached from different directions. We can go bottom-up from sets or top-down from category theory. We will start with the origins in topology.

## Presheaf on a Topological Space

We can have a presheaf of any structure 'A' on a topological space 'X'.

A presheaf of sets 'A' on 'X' is valid but it does not give much structure to relate the open sets. Something with an abelian group structure for example (such as vectors) gives us more structure to work with. More about open sets on page here.

 A presheaf of abelian group 'A' on 'X' assigns to each open set U X an abelian group A(U). and that assigns to each pair U V of open sets a homomorphism called the restriction rU,V : A(V) -> A(U) in such a way that rU,U = 1 (identity map) rU,VrV,W = rU,W (composition) ## Presheaf in Category Theory

In category theory terms:

• X is a topological space
• A presheaf of abelian groups on X is a contravariant functor from the category of open subsets of X and inclusions to the category of abelian groups.
 Let C=Top(X) the category whose objects are the open subsets of X. C can be represented as a poset of open sets in a topological space with the morphisms being inclusion maps. There is a single arrow from V to U if V is included in U or V=U (V is a subset of U). Hom(V,U) has one element. If V is not a subset of U there is no arrow. Hom(V,U) is empty. There is never more than one arrow between any two open sets, if it exists the arrorw is unique. In the same way as for fibres we usually reverse the arrow so that it is a contravarient arrow from Cop to Set: A presheaf on a category C is a functor F : Cop-> Set

For instance a presheaf can be a contravariant functor from the category Top(X) to the category Ab of Abelian groups (which may also have more structure).

Sheaves are discussed from a category theory point of view on the page here.

## Developing Fibre Bundle Approach

The page here introduced fibre bundles.

For example, where there is a family of types indexed by elements of another type. The type families in fibre bundles are disjoint. One way to extend that concept is to allow an overlapping type family. open sets simplectical complex In some cases we can convert between open sets and simplices. see Wikipedia articles:  ### Sheaf and Co-Sheaf

Allow us to translate between physical sources of data and open sets or simpicies.

sheaf Co-sheaf
vertex weighted hyperedge weighted
vertex has non-trvial stalk. toplex has non-trvial stalk.
All restrictions are zero maps All extensions are zero maps
The resulting sheaf is flabby The resulting cosheaf is coflabby

### Sheaf

A sheaf is a presheaf that satisfies the following two additional axioms:

• (Locality) If U is an open covering of an open set U , and if s , t F(U) have the property s | U i = t | U i for each set Ui of the covering, then s = t and
• (Gluing) If U is an open covering of an open set U, and if for each i I a section si is given such that for each pair Ui ,Uj of the covering sets the restrictions of si and sj agree on the overlaps, so s i | U i ∩ U j = s j | U i ∩ U j, then there is a section s F ( U ) such that s | U i = sifor each i I.

Where the notation used is:

• s | U i means s restricted to U i
• s is a section of the space over U

more notation see box on right.

### Diagrams

 We can think of this in a combinatorics way. The relations are shown more clearly in an attachment diagram here. This shows subset relations with the arrows going from lower dimension to higher dimension. A sheaf assigns some data spaces to the attachment diagram above. In this case reals (ℜ) - A sheaf of vector spaces. Each such set is called a stalk over the simplex. ℜ in all the points gives a vector ℜ² a matrix links the vectors, each such function is known as a restriction. Section - an assignment of values from each of the stalks that is consistent with the restrictions.

Sections can be:

• Global - defined everywhere.
• Local - defined for some part.

If all local sections extend to global sections the sheaf is called flabby/flasque (don't have interesting invariants).

 metadata block see also: Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory: Correspondence about this page Book Shop - Further reading . Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.       Introduction to Topological Manifolds (Graduate Texts in Mathematics S.) Other Books about Curves and Surfaces

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