I came across this definition while reading Jaap van Oosten's Topos Theory lecture notes (pg 29, def 1.3).
In a category with finite limits, an equivalence relation on an object $X$ is a subobject $R$ of $X \times X$ for which the following statements hold:
- The diagonal embedding $X \to X \times X$ factors through $R$.
- The composition $R \hookrightarrow X \times X \xrightarrow{tw} X \times X$ factors through $R$, where $tw$ denotes the twist map $\langle p_1, p_0 \rangle : X \times X \to X \times X$. Here $p_0, p_1 : X \times X \to X$ are the projections.
- The map $\langle p_0s, p_1t \rangle : R' \to X \times X$ factors through $R$, where we assume that the subobject $R$ is represented by the arrow $\langle r_0, r_1 \rangle : R \to X \times X$, and the arrows $s$ and $t$ are defined by the pullback diagram,
$\require{AMScd} \begin{CD} R' @>{t}>> R\\ @V{s}VV @VV{r_0}V\\ R @>{r_1}>> X \times X \end{CD}$
The subobject $R'$ is the “object of $R$-related triples”.
I was able to understand the first two conditions. However, I'm stuck on the third one. If my understanding is correct, $r_0, r_1$ are arrows from $R$ to $X$. But in the diagram we have $r_0,r_1 : R \to X \times X$. Also the projections $p_0, p_1$ are from $X \times X$ to $X$ and from the pullback diagram $s,t :R' \to R$. So I am not able to understand that how $\langle p_0s, p_1t \rangle : R' \to X \times X$ even makes sense.
Is it a typo or am I missing something?
There is one typo and one slight abuse of notation. The correct diagram is a pullback diagram $$ \require{AMScd} \begin{CD} R' @>{t}>> R\\ @V{s}VV @VV{r_0}V\\ R @>{r_1}>> X \end{CD} $$ Then $R'$ can be thought of as triples $(a,b,c)$ for which $(a,b)\in R$ and $(b,c)\in R$. Moreover, it is more precise to say that it is the map $\langle r_0s, r_1t\rangle\colon R'\to X\times X$ that factors through $R\hookrightarrow X\times X$. Informally, the former map sends $(a,b,c)$ to $(a,c)$, so the requirement that this map factors through $R$ is the usual transitivity condition on equivalence relations.