Basic constructions in the categories of sets, sets with a binary relation on them, preorders, and posets

TL;DR

This paper details how fundamental categorical constructions such as products, coproducts, and exponentials are concretely realized in categories like sets, sets with relations, preorders, and posets, clarifying their structure.

Contribution

It provides explicit descriptions of key categorical constructions in basic categories, enhancing understanding of their concrete implementations.

Findings

Explicit formulations of categorical constructions in each category.

Clarification of how these constructions differ across categories.

Foundational insights for categorical theory and applications.

Abstract

The purpose of this note is to work out the details of the concrete incarnation of a few categorical constructions (products, coproducts, pullbacks, pushouts, equalizers, coequalizers, and exponentials) in some useful and basic categories: the categories of sets, sets endowed with a binary relation, preorders, and posets.