quotient of an object

Question

In category theory What from these 2 things is called quotient:

1. epi

or rather

2. split epi

Whats the difference of a usage of these 2.

Answer 1

Usually epis (or, as Ittay Weiss comments, equivalence classes of epis) are called quotient objects.

However, in my opinion this is inadequate as a general approach. For example, in the category $\mathbf{Top}$ of topological spaces and continuous maps the epis are nothing else than the continuous surjections. See https://en.wikipedia.org/wiki/Epimorphism. But a quotient object in $\mathbf{Top}$ should be a quotient map which is much more restrictive than being an epi. In fact, the quotient topology on $Y$ is uniquely determined by the space $X$ and the (surjective) function $p$. If $p$ is only required to be an epi, then there are many topologies on the set $Y$ making $p$ continuous.

The dual concept is that of a subobject. Usually these are understood to be monos, but again this is problematic. See my answer to Attempt to define the notion of subobjects.