A classification of congruences in the category of sets and relations

We will give a classification of congruences in $\Rel$, the category of sets and relations. This classification will prove in particular that $\Rel$ has quotients of congruences and that congruences are effective.

Let $i : E \hookrightarrow X \times X$ be a congruence in $\Rel$. Recall that products in $\Rel$ are actually given by disjoint unions at the level of underlying sets. Also, recall that a morphism $R : X \to Y$ in $\Rel$ is a monomorphism if and only if $$R_* : P(X) \to P(Y),\, S \mapsto \bigl\{ y\in Y \mid \exists x\in S, (x, y) \in R \bigr\}$$ is injective, and it is an isomorphism if and only if $R$ is the graph of a bijection $X \to Y$ in $\Set$. In particular, we get $$i_* : P(E) \to P(X + X) \cong P(X) \times P(X)$$ which must be injective. It must also preserve arbitrary unions and in particular be inclusion-preserving. From the assumption that $i$ is a congruence, since the functor $(P, {-}_*) : \Rel \to \Set$ is representable by the object 1, we see that $i_*$ must have image given by an equivalence relation $\sim$ on $P(X)$. Note also that since $i_*$ preserves arbitrary unions, we must have that $\sim$ respects arbitrary unions.

Since the symmetry morphism $s : E \to E$ satisfies $s^2 = \id$, it must be the graph of an involution $s_0$ on $E$, where $s_0$ is a morphism in $\Set$.

To see this, $r_*(\{ x \})$ certainly cannot be empty, or we would have $$i_*(r_*(\{ x \})) = (\varnothing, \varnothing) \ne (\{ x \}, \{ x \}).$$ On the other hand, if we had at least three distinct elements $a,b,c \in r_*(\{ x \})$, then $$i_*(\varnothing), i_*(\{ a \}), i_*(\{ a, b \}), i_*(\{ a, b, c \})$$ would have to be a chain of four distinct subsets of $i_*(r_*(\{ x \})) = (\{ x \}, \{ x \})$, which is impossible.

To see this, if $r_*(\{ x \}) = \{ e, f \}$ with $e\ne f$, then we must have $$\varnothing = i_*(\varnothing) \subsetneqq i_*(\{ e \}) \subsetneqq i_*(\{ e, f \}) = (\{ x \}, \{ x \}),$$ which means that $i_*(\{ e \})$ is equal to either $(\{ x \}, \varnothing)$ or $(\varnothing, \{ x \})$. By the same token, $i_*(\{ f \})$ is also one of those choices, and $i_*(\{ e \}) \ne i_*(\{ f \})$. Since $$i_*(\{ s_0(e) \}) = i_*(s_*(\{ e \})) = i_*(\{ f \})$$ in either choice of order, by injectivity of $i_*$ we must have that $s_0(e) = f$. Similarly, $s_0(f) = e$. Otherwise, if $r_*(\{ x \}) = \{ e \}$ is a singleton, $$i_*(\{ s_0(e) \}) = i_*(s_*(\{ e \})) = i_*(\{ e \}),$$ so injectivity of $i_*$ gives $s_0(e) = e$.

If we did have $\{ x \} \sim \varnothing$, then there would have to be some subset of $E$ which maps to $(\varnothing, \{ x \})$. Similarly to previous steps, injectivity of $i_*$ implies the subset is non-empty, and if the subset had two distinct elements $a,b$, then we would have $$i_*(\varnothing) \subsetneqq i_*(\{a\}) \subsetneqq i_*(\{a,b\}) \subseteq (\varnothing, \{ x \}),$$ giving a contradiction. Therefore, this subset of $E$ would have to be a singleton $\{ f_1 \}$. Similarly, $(\{ x \}, \varnothing)$ would have to be the image of a singleton $\{ f_2 \}$ with $f_1 \ne f_2$. But then $i_*(\{ f_1, f_2 \}) = (\{ x \}, \{ x \}) = i_*(\{ e \})$, contradicting the injectivity of $i_*$.

To show this, let $e\in E$, and suppose that $i_*(\{ e \}) = (S, T)$. Then $i_*(\{ s_0(e) \}) = (T, S)$, so $$\textstyle i_*(\{ e, s_0(e) \}) = (S \cup T, S\cup T) = i_*\bigl(\bigcup_{x\in S\cup T} r_*(\{ x \})\bigr).$$ It follows that $$\textstyle\bigcup_{x\in S\cup T} r_*(\{ x \}) = \{ e, s_0(e) \}.$$ Therefore, $e \in r_*(\{ x \})$ for some $x \in X$, and by Claims 2 and 3, we get the desired conclusion.

For the forward direction, suppose $(S, T) = i_*(U)$ for $U \subseteq E$. Then $$\textstyle (S, T) = \bigcup_{e\in U} i_*(\{ e \}).$$ The set $i_*(\{ e \})$ satisfies the relation of having equal intersections with $A$ for each $e\in E$ in any case from Claim 4; and this relation respects unions. For the reverse implication, whenever $x\notin A$, we have $\{ x \} \sim \varnothing$. Therefore, since $$\textstyle S = (S \cap A) \cup \bigcup_{x\in S \setminus A} \{ x \},$$ we must have $$\textstyle S \sim (S \cap A) \cup \bigcup_{x\in S \setminus A} \varnothing = S \cap A.$$ Similarly, $T \sim T \cap A$, so if $S\cap A = T\cap A$, then $S \sim T$.

We can define a bijection $E \to E'$ as follows: let $e \in E$. If $i_*(\{ e \}) = (\{ x \}, \varnothing)$ with $x\notin A$, then send $e \mapsto i_1(x)$; if $i_*(\{ e \}) = (\varnothing, \{ x \})$ with $x\notin A$, then send $e \mapsto i_2(x)$; and if $i_*(\{ e \}) = (\{ x \}, \{ x \})$ with $x \in A$, then send $e \mapsto i_1(x) = i_2(x)$. Transferring the congruence to $E'$, we see that it is exactly of the given form.

Now, without loss of generality we will assume from now on that the congruence is of exactly this form.

With this classification, we can show that $A$ is a quotient for the congruence, with quotient morphism $\Delta_A \subseteq A \times A \subseteq X \times A$. In fact, we can define a morphism $\Delta_A^\circ : A \to X$ as the inverse relation to $\Delta_A$, and a morphism $X \to E$ as the graph of $i_1$. It is then straightforward to verify that these maps together make $A$ a split coequalizer of $p_1 \circ i$ and $p_2 \circ i$.

We can also conclude that $E$ is the kernel pair of this quotient, by the general result below.