San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

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 (CO, CA) and (DO, DA). Likewise F is an ordered pair (FO, FA) where

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

The conditions which F must satisfy are:

(i) If φ ∈ CA and φ:O1>→O2
then the ψ ∈ DA such that
ψ:FO(O1)→FO(O2) is FA(φ)
(ii) FA(f o g) = FA(f) o FA(g)
(iii) For any object O in C:
FO(1O1) = 1FO(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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