Category theory MOC

Functor

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

• An object 𝐹𝑋 ∈𝖣 for every 𝑋 ∈𝖢
• A morphism 𝐹𝑓 ∈𝖣(𝐹𝑋,𝐹𝑌) for every 𝑓 ∈𝖢(𝑋,𝑌)

with the following compatibility conditions

• (𝐹𝑔)(𝐹𝑓) =𝐹(𝑔𝑓) for any 𝑓 ∈𝖢(𝑋,𝑌) and 𝑔 ∈𝖢(𝑌,𝑍)
• 𝐹id𝑋 =id𝐹𝑋 for any 𝑋 ∈𝖢

A functor 𝐹 :𝖢𝐨𝐩 →𝖣 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 𝖢(𝑋,𝑌) in its codomain:

𝐹:𝖢(𝑋,𝑌)→𝖣(𝐹𝑋,𝐹𝑌)𝑓↦𝐹𝑓

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 ↩