Hom-Sets in Category Theory

A hom-set, denoted by hom(A, B), is a collection of all morphisms that map an object A as the domain to an object B as the codomain. $$ \text{hom}(A,B) $$

The hom-set hom(A, B) includes only those morphisms that operate from A to B.

Morphisms mapping from B to A are part of a separate morphism set, denoted by hom(B, A).

Furthermore, morphism sets are distinct unless they involve the same domain and codomain.

Put simply, each morphism is unique to the morphism set defined by its domain and codomain, leading to disjoint sets unless the domains and codomains are identical. For instance, hom(A, B) and hom(C, D) are separate unless A equals C and B equals D.

Morphisms within the same hom-set for objects $ A $ and $ B $ are known as "parallel morphisms."

This classification means that both morphisms start from $ A $ and terminate at $ B $, sharing their initial and final points.

In essence, these morphisms are elements of the set that maps $ A $ to $ B $.

The term "parallel" simply refers to their common start and end points in the category structure, without implying other specific relationships such as equivalence or distinction in properties. Moreover, it doesn’t suggest that they are part of the same specific "morphism" $ f_{AB} $ between $ A $ and $ B $, since $ f_{AB} $ represents just one potential morphism within the hom-set $ \text{Hom}(A, B) $.

Example

Consider a practical example of Hom-Sets within the category of sets.

Take two finite sets $ A $ and $ B $:

$$ A = \{1, 2\} $$

$$ B = \{a, b, c\} $$

The Hom-Set $ \text{Hom}(A, B) $ encompasses all possible functions mapping $ A $ to $ B $.

Each function assigns an element of $ B $ to every element of $ A $.

example of morphisms

Here are some potential functions that might fall within the $ \text{Hom}(A, B) $ class:

A possible function $ f $ could map both 1→a and 2→a:
$ f(1) = a $
$ f(2) = a $
Another function $ g $ might assign 1→b and 2→c:
$ g(1) = b $
$ g(2) = c $
A third option $ h $ could map 1→a and 2→b:
$ h(1) = a $
$ h(2) = b $

With three choices each for $ f(1) $ and $ f(2) $, up to $ 3 \times 3 = 9 $ different functions are possible from $ A $ to $ B $.

These functions collectively form the hom-set $ \text{Hom}(A, B) $, encapsulating all potential mappings from $ A $ to $ B $.

This example between two finite sets illustrates that a hom-set is not a solitary morphism but rather an aggregate of all possible mappings from elements of $ A $ to those of $ B $, adhering to function rules.