# Category. Definition.

### Category

A *category* \( \cat{C} \) consists of:

- a class of objects, denoted as \( \catobj{C} \)
- a set, denoted as \( \cathom{C}(A, B) \), for any objects \( A \) and \( B \) of \( \cat{C} \). The elements are called
*morphisms.*

The set of morphisms must have the following properties:

- Identity
- For each object \( A \in \catobj{C} \), there exists (at least) one
morphism \( 1_A \in \cathom{C}(A, A) \), called the
*identity*on \( A\). - Composition
- Morphisms can be composed: any two morphisms \( f \in \cathom{C}(A, B) \) and \( g \in \cathom{C}(B, C) \) determine the existance of another morphism \( gf \in \cathom{C}(A, C) \).
- Associativity of composition
- For any \( f \in \cathom{C}(A, B) \), \( g \in \cathom{C}(B, C) \) and
\( h \in \cathom{C}(C, D) \), we have:
\[ (hg)f = h(gf). \]
- Identity law
- The identity morphisms are identities with respect to composition. For
any \( f \in \cathom{C}(A, B) \), we have:
\[ f1_A = f, \quad 1_Bf = f\]
- Morphism sets are disjoint
- For any \( A, B, C, D \in \catobj{C} \), then \( \cathom{C}(A, B) \) and \( \cathom{C}(C, D) \) are disjoint unless \( A = C \) and \( B = D \).

### 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 \(A, B, C \in \catobj{C} \), there exists a function \( \cathom{C}(A, B) \times \cathom{C}(B, C) \to \cathom{C}(A, C) \) which is denoted as \( fg \).

The existance of a function imposes the requirement that for any element of the domain \( \cathom{C}(A, B) \times \cathom{C}(B, C) \) there must be an element in the co-domain \( \cathom{C}(A, C) \). The function existance requirement allows for an empty domain if one or both of \( \cathom{C}(A, B) \) or \( \cathom{C}(B, C) \) are empty.

#### Using functions to imply non-empty sets

I find the following bi-implication interesting:

Let \( S_a \) and \( S_b \) be sets. Then

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 \( \mathrm{Set} \) or \( \mathcal{Set} \).

### Category from relation

Let \( S \) be a set and \( \sim \) be a relation on \( S \) which is both reflexive and transitive. Then we can form a category like so:

- The elements of \( S \) are the objects of the category.
- For some \( a, b \in S \), let \( \cathom{}(a, b) = (a, b) \) iff \( a \sim b \), else let \( \cathom{}(a, b) = \emptyset \).

It's best to keep in mind the relation \( \sim \) as being represented by
the graph \( \Gamma \subseteq S \times S \), where \( (a,b) \in \Gamma \) iff
\( a \sim b \). The case where \( \sim \) is the equivalence relation '=' and thus all morphisms are identity morphisms, produces a category which is said to be *discrete*.

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.