\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)

Math and science::Algebra::Aluffi

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.

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Source

Aluffi, p19