-1

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

  1. epi

or rather

  1. split epi

Whats the difference of a usage of these 2.

user122424
  • 3,926
  • 1
    Well, perhaps regular epi would do it best in general: it's by definition the coequalizer of a pair of arrows, where pairs of arrows models binary relations, and coequalizer would model the quotient of the generated equivalence relation. – Berci Sep 11 '21 at 06:57
  • 1
    Technically, neither. A quotient is an equivalence class of epis. A single epi may represent a quotient. Once can place various further conditions on the epis. – Ittay Weiss Sep 11 '21 at 08:21

1 Answers1

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.

Paul Frost
  • 76,394
  • 12
  • 43
  • 125