本科低年级Category Theory计划中

Functors

Functors as structure-preserving maps between categories: covariant and contravariant functors, forgetful and free functors, power-set and hom functors, and functoriality as a tool for proving that things are impossible.

math.cat.functors

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前置知识

你将学会

Verify that the power-set, forgetful and hom functors are functors, and that a contravariant functor from C to D is the same as a functor Cᵒᵖ → D.
Explain why functors preserve isomorphisms and use this to show that objects are not isomorphic.
Describe the free monoid on a set (finite lists) and the free vector space on a set as functors.

定义

A functor F : C → D assigns to each object A of C an object F(A) of D and to each morphism f : A → B a morphism F(f) : F(A) → F(B), such that F(g ∘ f) = F(g) ∘ F(f) and F(id_A) = id_{F(A)}.
A contravariant functor from C to D is a functor Cᵒᵖ → D.
For an object A of a locally small category C, the hom functor Hom(A, −) : C → Set sends each object B to the set of morphisms A → B.

定理

Functors preserve isomorphisms: if f is an isomorphism, so is F(f).
Functors compose, and small categories with functors form a category Cat.
The fundamental group is a functor from pointed spaces to groups; functoriality shows there is no retraction of the disc onto its boundary circle.

常见误解

Thinking a functor must be injective on objects or on morphisms.
Believing every functor reflects isomorphisms, so that F(f) invertible would force f invertible; this fails in general.