Category theory MOC

Functor

A (covariant) functor 𝐹 :𝖢 →𝖣 is a structure-preserving map between categories. This is given by #m/def/cat

1. A map 𝐹0 :𝖢0 →𝖣0 :𝑋 ↦𝐹𝑋;
2. For every 𝑋,𝑌 ∈𝖢0, a function 𝐹1 :𝖢(𝑋,𝑌) →𝖣(𝐹𝑋,𝐹𝑌);

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 ∈𝖢0

A functor 𝐹 :𝖢𝐨𝐩 →𝖣 behaves like a functor but flips arrows, and is called a contravariant functor from 𝖢 to 𝖣. 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

Functors are categorised based on the behaviour of 𝐹1 (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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