Maths - Sheaf

• As a way to relate global structure to local structure.

• As a generalisation of a fibre bundle.

• As a pre-sheaf with some additional axioms.

• As a way to attach data to a space.

• Yoneda Embedding

A topological space is quite restrictive and there may be ways to loosen the requirements to work with other spaces.

Topological spaces are discussed on the pages here.

Pre-sheaf

A topology consists of open sets with some structure and we need to relate this to sets and functions between them. This is done using a contravarient functor.

The arrows in the topological space are injective and they map to sets and functions between them in the opposite direction.

To have a diagram that scales up to many sets it seems to be necessary to separate the sets from the logic.

This diagram is less intuitive and compact than Venn diagrams but it is still very powerful. It must obey certain axioms so that the sets are consistent with the logic part.

For more information see the page about sheaves.

Fibre With Structure

The page about fibre bundles here shows how reversing a map between sets gives some interesting structures. Here we extend that idea.

A restriction of 'u' to 'v' is notated res_v,u : F(u)->F(v)

s|_v restricts s to v

see Wiki

Example - Subsets

In order to try to find a simple example lets say that A and B are sets and B is a subset of A. If we look at it in terms of logic we can say that A implies B so the arrow goes from A to B.

When we look at the elements on the right it is simplest to show it as an injective function.

Example - Graph and Simplical Complexes

As a start with this approach we want to probe the red object on the right:

To do this start by 'probing' it with arrows from a graph with only one vertex and no edges. All the possible arrows from the single vertex object form a homset.

This homset is a set (where the elements are arrows) so we can use our set theoretic intuition on it. This set has one object for every vertex in the object.

Now we need to 'probe' the edges. We can do this with an arrow from a graph with a single edge between two vertices. The bottom part of the graph on the left shows arrows (green) to each edge. So this time we have a homset which gives the number of edges.

However, the mapping of the edges needs to play well with the mapping of the vertices (top part of diagram).

The set of arrows from vertex 1 is covariant x -> Hom(1,x) : Grf -> Set this is usually written Hom(1,_)

Similarly with vertex 2
x -> Hom(2,x) : Grf -> Set

However the mapping between these two sets Hom(2,_) -> Hom(1,_) is contravarient (it goes down on the diagram where the other mappings go up).

It is easier to see this contravarience from a simpler diagram.

If 'F' is given we can derive 'h' from 'g' by composing with 'F' but we can't derive 'g' from 'h'.

So if we want to get the usual functions in set we need to start with Grf^op

A contravarient functor of the form C^op->set occurs in other structures and is called a presheaf as described on page here.

This graph object that we are using to 'probe' another graph object is known as a representable object in Grf (see Yoneda lemma page).

This representable object is a graph with two vertices and an arrow between them.

Example - Relational Database

see this page.

Example - Equalities

see this page.

Example - Cubical Type Theory

see this page.

A cover on an object U in a category C is a collection of morphisms
{U_i-> U} _iI.

A specification of a collection of covers for each object of the category, subject to some compatibility condition, makes a coverage on C

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 invariant's).

global section 's'.

Locality

If U is an open covering of an open set U , and if s , tF(U) have the property s | _{U i} = t | _{U i} for each set U_i of the covering, then s = t

Gluing

If U is an open covering of an open set U, and if for each iI a section s_i is given such that for each pair U_i ,U_j of the covering sets the restrictions of s_i and s_j agree on the overlaps, so s _i | _{U i ∩ U j} = s _j | _{U i ∩ U j}, then there is a section sF ( U ) such that s | _{U i} = s_ifor each iI.

What if we start with sets which already have some structure?

The diagram on the right is intended to show a morphism between partial orders. So the blue arrows represent ≥ and the arrow 'ab' means b ≥ a.

The red arrows show the morphism between the partial orders and these all need to play nicely together.

So how can this morphism be reversed?

The diagram on the right is the best I can do although somehow it doesn't seem completely correct, for instance it seems to imply that c ≥ a whereas the original diagram above doesn't imply that.

In order to be consistent this morphism needs to be a local homeomorphism which is a function between topological spaces that preserves local but not necessarily global structure.

Local and Global Structures

The interesting thing is that this is dividing the structure into 'local' structure and 'global' structure.

It seems natural to have morphisms that preserve global structure but may compress local structure. This does give a morphism that is reversible and we get a fibre bundle with some extra structure.

However, that's not what we want for a sheaf, what we want is a morphism that preserves local structure but may compress global structure.

Local Homeomorphism

In order to be reversible this morphism needs to be a local homeomorphism, as in the example on the right, here the local structure is preserved.

Reversing this morphism gives a fibre structure, as described on this page, but with additional structure. Now the local structure inside the xy type (on the right of the diagram) maps to the same structure between the family of types.

This is also a comma category as described on this page.

Here I have just rotated the above diagram by 90 degrees just to show in the same way we did with fibres.

So each element x,y of x->y indexes a type.

Section

If we have a subset (subtype?) of the indexing type then this translates to a subset of the type family.

A presheaf of abelian group 'A' on 'X' assigns to each open set UX an abelian group A(U).

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

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

in such a way that

r_U,U = 1 (identity map)

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 arrow is unique.

In the same way as for fibres we usually reverse the arrow so that it is a contravarient arrow from C^op to Set:

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.

In some cases we can convert between open sets and simplices.

see Wikipedia articles:

• Leyray Theorem
• Nerve of a covering
• Alexandrov topologies

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.