# Maths - Presheaves

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:

Cop->Set

or

SetCop

So we have a functor from a category to the category of sets. This sends objects to sets and arrows to functions in a way that plays well with the objects. In category theory we don't tend to look inside objects but in set theory we look inside sets. S how can we define these morphisms? In order to work with sets it needs to preserve something to do with subsets.

## Presheaf Category

 In a presheaf category the object is a functor. Morphisms are structure preserving maps between these functors.

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

## Presheaf Examples

• A simplicial set is a presheaf on the simplex category
• A globular set is a presheaf on the globe category.
• A cubical set is a presheaf on the cube category.

See page here.

See page here.

### Further Information

For a more general introduction to sheaves see the page here.

The following two videos discuss some of these concepts in a slightly topological way. For instance maps into sets related to fibres.

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.

 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.