The Concept of Functor in Category Theory

San José State University

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The Concept of Functor in Category Theory

A category is a collection of objects and a collection of arrows between those object. The collection of arrows is closed under composition. A functor is mapping between two categories. Because any cateory involves two collections, objects and arrows, a functor has to involve two mappings; one mapping between the object collections of the two categories and another between the arrow collections.
The functor mappings have satisfy further conditions. Let the two categories for a functor be denoted as C and D and a functor between them as F:C → D. Just for this presentation think of C and D as the order pairs (C_O, C_A) and (D_O, D_A). Likewise F is an ordered pair (F_O, F_A) where
if f ∈ F_O then f:C_O → D_O
and
if φ ∈ F_A then φ:C_A → D_A

The conditions which F must satisfy are:
(i) If φ ∈ C_A and φ:O→O
then the ψ ∈ D_A such that
ψ:F_O(O)→F_O(O) is F_A(φ)
(ii) F_A(f°g) = F_A(f) ° F_A(g)
(iii) For any object O in C:
F_O(1_O₁) = 1_{F_O(O₁)}

The remarable and beautiful thing is that the collection of categories with functors as the collection of arrows is a CATEGORY!

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if f ∈ FO then f:CO → DO and if φ ∈ FA then φ:CA → DA

(i) If φ ∈ CA and φ:O→O then the ψ ∈ DA such that ψ:FO(O)→FO(O) is FA(φ) (ii) FA(f°g) = FA(f) ° FA(g) (iii) For any object O in C: FO(1O1) = 1FO(O1)

if f ∈ F_O then f:C_O → D_O
and
if φ ∈ F_A then φ:C_A → D_A

(i) If φ ∈ C_A and φ:O→O
then the ψ ∈ D_A such that
ψ:F_O(O)→F_O(O) is F_A(φ)
(ii) F_A(f°g) = F_A(f) ° F_A(g)
(iii) For any object O in C:
F_O(1_O₁) = 1_{F_O(O₁)}