Category theoretic analog of choosing a representative of an equivalence class In a category $\mathcal{C}$ with binary products, a congruence on an object $X$ is a monomorphism $i : R \to X \times X$ satisfying reflexivity, symmetry and transitivity, which take the form of morphisms and commuting squares. In the category of sets, the coequaliser for the projection morphisms 
$$
\pi_1\circ i, \pi_2\circ i : R \to X 
$$
exists and is the set of equivalence classes. Then I can choose a representative for each equivalence class to get a section of the quotient map.
Is there a name for a category where something similar can always be done, i.e. the coequaliser for a congruence exists and there always exists a section?
 A: For starters, let's distinguish three types of epimorphisms. 


*

*$f : X \to Y$ is a split epimorphism if it has a section. 

*$f : X \to Y$ is a regular epimorphism if it is the coequalizer of some pair of maps to $X$ (not necessarily a congruence).

*$f : X \to Y$ is an effective epimorphism if it has a kernel pair $X \times_Y X$ (which I believe is always a congruence on $X$) and is the coequalizer of its kernel pair.


Split epimorphisms are always regular, and in any category with pullbacks, regular and effective epimorphisms coincide (see e.g. this blog post). I understand these conditions somewhat better than your condition about congruences (which I expect ought to be equivalent to both regular and effective under mild hypotheses) so I'll work with them instead. So one version of the question is: 

When are effective epimorphisms always split?

The answer is almost never, and I don't know a name for categories with this property. For example, 


*

*effective epimorphisms in $\text{Mod}(R)$ are just quotient maps $M \to N$, but split epimorphisms are projections onto direct summands $N \oplus M/N \to N$. Every effective epimorphism is split iff $R$ is semisimple. More generally, in any abelian category, every effective epimorphism is split iff $\text{Ext}^1(-, -)$ vanishes identically iff every object is projective iff every object is injective. 

*effective epimorphisms in $\text{Grp}$ are just quotient maps $G \to H$, which correspond to extensions $N \to G \to H$, but split epimorphisms correspond to semidirect products, and most extensions are not semidirect products. The simplest example is $\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2$.

*effective epimorphisms in $\text{Top}$ are quotient maps $f : X \to Y$ (that is, $Y$ must have the quotient topology), but split epimorphisms are, well, quotient maps with sections. Most quotient maps don't have sections. For example, a fiber bundle is an effective epimorphism, but most fiber bundles don't have sections; think of the bundle of nonzero tangent vectors on a smooth manifold. 


As Daniel Schepler says, even the claim that this is possible in $\text{Set}$ is more or less equivalent to the axiom of choice.
