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
Presheaf Category
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In a presheaf category the object is a functor. |
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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.
Example - Single Element SetA 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. |
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Example - GraphHere 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. |
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Example - Relational DatabaseHere Cop is a database schema. This imposes a structure on the sets which are the database tables. This implements a category of simplical databases. |
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