I'm looking into the various types of epimorphism given here https://ncatlab.org/nlab/show/epimorphism. The nlab entry on descent morphism https://ncatlab.org/nlab/show/descent+morphism contains the following: Let $C$ be a category with pullbacks. For any morphism $p: A \to B$ there is an associated internal category $\ker(p)$ defined by $\pi_{1},\pi_{2}: A \times_{B} A \to A$, the kernel pair of $p$. The category of descent data for $p$ is the category $C^{ker(p)}$ (the "descent object") of internal diagrams on this internal category. Explicitly, an object of $C^{\ker(p)}$ is a morphism $C \to A$ together with an action $A \times_{B} C \to C$ satisfying suitable axioms.
I know what kernel pairs and internal categories are but don't see how the kernel pair defines an internal category. What are the "suitable axioms" that define the morphisms in $C^{\ker(p)}$?
Best Answer
Maybe it would be easier to see that $\langle \pi_1,\pi_2\rangle:A\times_BA\rightarrow A\times A$ is an internal equivalence relation in $\mathcal{C}$.
One way of explaining this is the following. First note that this is obvious if $\mathcal{C} = \textbf{Set}$. Indeed, $A\times_BA$ consists of pairs $(a_1,a_2)\in A\times A$ such that $p(a_1) = p(a_2)$ and this clearly gives rise to an equivalence relation on $A$. Now for general category $\mathcal{C}$ you can use essentially the same argument as for $\textbf{Set}$ if you apply Yoneda embedding.
Hence $\langle \pi_1,\pi_2\rangle:A\times_BA\rightarrow A\times A$ is an internal equivalence relation in $\mathcal{C}$. Recall that we may view every equivalence relation as a rather simple groupoid and thus as a category. Therefore, $\langle \pi_1,\pi_2\rangle:A\times_BA\rightarrow A\times A$ admits a structure of an internal category.