Category theory MOC

Functor

A (covariant) functor is a structure-preserving map between categories. cat It associates:

• An object for every
• A morphism for every

with the following compatibility conditions

• for any and
• for any

A functor behaves like a functor but flips arrows, and is called a Contravariant functor. Sometimes these are also just referred to as functors,¹ however in these notes all functors will be assumed to be covariant, and contravariant functors will be made explicit by invoking the opposite category.

Types of functors

As defined above, a functor associates a mapping to every hom-set in its codomain:

Functors are categorised based on the behaviour of this mapping (for all possible hom-sets)

• A Faithful functor is injective on hom-sets.
• A Full functor is surjective on hom-sets.
• A Fully faithful functor is bijective on hom-sets (an embedding of a category into another, however it need not be injective on objects.

Further classification

• Continuity and cocontinuity
• Exact functor
• Adjoint functor

Properties

• Functors encode invariants of isomorphism classes, i.e. functors are invariants.

Typical functors

• Hom-functor

See also

• Multifunctor

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Footnotes

1. 2020, Topology: A categorical approach, p. 10 ↩