8.4.1 Foundations

Let $A$ be a set.

Definition 8.4.1.1.1 ▶ Equivalence Relations

A relation $R$ is an equivalence relation if it is reflexive, symmetric, and transitive.1

1Further Terminology: If instead $R$ is just symmetric and transitive, then it is called a partial equivalence relation.

Example 8.4.1.1.2 ▶ The Kernel of a Function

The kernel of a function $f\colon A\to B$ is the equivalence relation $\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )}$ on $A$ obtained by declaring $a\sim _{\mathrm{Ker}\webleft (f\webright )}b$ iff $f\webleft (a\webright )=f\webleft (b\webright )$.1

1The kernel $\mathrm{Ker}\webleft (f\webright )\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ of $f$ is the underlying functor of the monad induced by the adjunction $\text{Gr}\webleft (f\webright )\dashv f^{-1}\colon A\mathbin {\rightleftarrows }B$ in $\textbf{Rel}$ of Chapter 7: Constructions With Relations, Item 2 of Proposition 7.3.1.1.2.

Definition 8.4.1.1.3 ▶ The Po/Set of Equivalence Relations on a Set

Let $A$ and $B$ be sets.

  1. The set of equivalence relations from $A$ to $B$ is the subset $\smash {\mathrm{Rel}^{\mathrm{eq}}\webleft (A,B\webright )}$ of $\mathrm{Rel}\webleft (A,B\webright )$ spanned by the equivalence relations.
  2. The poset of relations from $A$ to $B$ is is the subposet $\smash {\mathbf{Rel}^{\text{eq}}\webleft (A,B\webright )}$ of $\mathbf{Rel}\webleft (A,B\webright )$ spanned by the equivalence relations.

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