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 UcontainsX an abelian group A(U).

and that assigns to each pair UcontainsV of open sets a homomorphism called the restriction

rU,V : A(V) -> A(U)

sheaf on topological space

in such a way that

rU,U = 1 (identity map)

rU,VrV,W = rU,W (composition)

Presheaf in Category Theory

In category theory terms:

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.

fibre bundle
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:

nerve1 nerve2

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:

Where the notation used is:

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.

attachment diagram

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.
sheaf

Section - an assignment of values from each of the stalks that is consistent with the restrictions.

Sections can be:

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

 


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see also:

Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory:

  1. Lecture 1
  2. Lecture 2
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
Correspondence about this page

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