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:
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.
- 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.
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.
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.
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.