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

#### if f ∈ FO then f:CO → DOand 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!