Maths - Sheaf Example - Simplicial Complex

This is an example for the page about sheaves which is here.

Example - Simplicial Complex

More about simplicial complexes on this page.

A simplicial complex can define a space, however this space is completely different from the topological space of the sheaf.


Example - Graph

I think we can see the sheaf idea more clearly if we first restrict ourselves to the simpler case of a graph.

More about graph theory on this page.

If we were coding a graph structure in a computer language we would probably have a set of edges (E) and a set of vertices (V). Each edge has two pointers into the vertices set. One for the source(s) and one for the target(t). diagram graph

So this is what it looks like in set, but for a sheaf we need a map out of a topology-like structure. So what does the topology look like?

For a topology (see page about topological space) we tend to think of the arrows as inclusions, so the arrows would go from V to E, perhaps we can think of them as a set (homset) of functions.

Although the topology encodes a form of nearness, it is the proximity of V to E, not anything to do with the nearness of the vertices in the graph. So we need to keep this distinction in mind.

diagram graph

So now we can show the sheaf as a contravarient functor and hopefully get a better feel for its inner workings.

Here Cop is a category with two objects E (for edge) and V (for vertex) also two arrows s (blue for source) and t (red for target).

This allows us to build a structure on top of set where the diagram on the right commutes.

We can therefore build up complex graphs from individual vertices and edges.

diagram of graph

Example - Simplical Complexes

In the graph example above I think we can start to see the structure but, in that case, the topology is so simple that we might not see all the interactions. So it might be worth looking at a more complicated case.

Here we extend from verticies and edges to include triangles and tetrahedrons up to any dimension.

There are various ways to encode these structures. Here each dimension is coded in terms of the dimension below it, this is known as a delta complex.

In terms of topology this gives a sense of nearness since a vertex is nearer to the edge than to the triangle.







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see also:

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

  1. Lecture 1
  2. Lecture 2 - about simplicies & topology (sound on video is not good - quiet and delayed)
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
Correspondence about this page

Book Shop - Further reading .

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cover Introduction to Topological Manifolds (Graduate Texts in Mathematics S.)

Other Books about Curves and Surfaces

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