Category. Definition.
Category
A category
- a class of objects, denoted as
- a set, denoted as
, for any objects and of . The elements are called morphisms.
The set of morphisms must have the following properties:
- Identity
- For each object
, there exists (at least) one morphism , called the identity on . - Composition
- Morphisms can be composed: any two morphisms
and determine the existance of another morphism . - Associativity of composition
- For any
, and , we have: - Identity law
- The identity morphisms are identities with respect to composition. For
any
, we have: - Morphism sets are disjoint
- For any
, then and are disjoint unless and .
Composition as the existance of a function
The statement of morphism composition above can be stated alternatively as the existance of a function.
For any
The existance of a function imposes the requirement that for any element of
the domain
Using functions to imply non-empty sets
I find the following bi-implication interesting:
Let
This bi-implication allows function existance to be used instead of the LHS. How does the nature of morphism composition change when the RHS is replaced with the LHS of the bi-implication above? Maybe this has an effect on what we view as the datum of a category.
Example
Category of sets
There is a category whose objects are sets and whose morphisms are all the
functions between sets. This set often is given special syntax, such as
Category from relation
Let
- The elements of
are the objects of the category. - For some
, let iff , else let .
It's best to keep in mind the relation
This example demonstrates the flexibility of the definition of morphisms: the morphisms do not need to be functions, they can be elements of a set (or any other type of object) as long as the requirements of the morphisms in the above definition are satisfied.