Maths - Sheaf

Sheaf is an important subject in mathematics and there are many ways to approach the subject, for example:

Presheaf

A 'presheaf' category is a special case of a functor category (see page here). It is a contravarient functor from a category 'C' to Set.

Since it is contravarient it is usually written:

CopSet

or

SetCop

There is more about presheaves on the page here.

Example - Fibre Bundle as Endo Function

C diagram endo function

Every element has 1 outgoing arrow.

May have zero or multiple incoming arrows.

Cop diagram endo function

Every element has 1 incoming arrow.

May have zero or multiple outgoing arrows.

Cop->set diagram endo function Every element maps to a set.

This page explains Fibre bundles.

Presheaf Example - Single Element Set

A very simple example would be where Cop is a single element set (terminal object in set).

Hom( Cop, Set) therefore contains set of single arrows, one for every element of the set.

presheaf set

Presheaf Example - Graph

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

presheaf graph

Presheaf Example - Relational Database

Here Cop is a database schema.

This imposes a structure on the sets which are the database tables.

This implements a category of simplical databases.

presheaf database

Sheaf

We can think of a sheaf as building a more complicated structure from simpler components. These simpler components come from the presheaf.

To investigate this take the graph example from above. We can take individual edges and glue them together to form a more complidated shape.

An individual edge is defined by its source and target verticies. So this diagram must commute.
   
Multiple edges can be glued together. Here the source of one edge is connected to the target of another edge. Note, although the definitions in Cop are the same the squares don't all commute unless we keep them separate.

Generalisation of a Fibre Bundle

This is discussed from a topological point of view:

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. sheaf type

Sheaf - Tracking Locally Defined Data

A sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

Sheaves, with mappings which preserve this open set structure, form a category. This open set structure can be extended to groups, abelian groups, or commutative rings. To make it more general and apply it to more general categories we use the concept of presheaf which uses 'restriction morphisms'.

More about presheaf on page here.

sheaf category

Algebraic Topology


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see also:

 

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.

flag flag flag flag flag flag The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.

 

Terminology and Notation

Specific to this page here:

 

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