The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), the two characterizations of universal quotient maps below are mentioned.

Proposition. For a continuous map $p$, TFAE.

  • $p$ is a universal quotient map (i.e., a regular epi that is stable under pullback).
  • Fibers of adherence points of filters $\mathscr F$ contain adherence points of $p^\ast \mathscr F$ ($x$ is an adherence point of $\mathscr{F}$ if every neighborhood of $x$ has nonempty intersection with every element of $\mathscr{F}$).
  • For every open cover $ \left\{ U_i \right\}_{i\in I}$ of a fiber $p^\ast \left\{ b \right\}$, there's a finite $I_0\subset I$ satisfying $b\in\operatorname{int} \left( p_\ast \bigcup_{i\in I_0}U_i\right)$.

I can't seem to find a geometrical way to think of either of these conditions, so I would like some help in finding intuition..

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    $\begingroup$ Some more references should be given; I added one. $\endgroup$ – Todd Trimble Apr 30 '16 at 12:25
  • $\begingroup$ Indeed, complete categorical definitions are often easy to understand but--for non-categorists--a big nuance to search for them. I feel that the materials on MO should be meant also for non-specialists, especially that they have a good chance to contribute answers now and then. It's more generally, a question of communication and mathematical customs and culture. $\endgroup$ – Włodzimierz Holsztyński Apr 30 '16 at 13:59
  • $\begingroup$ In general, preferably one person provides a service, and many enjoy it; or everybody is supposed to provide a similar service each for themselves, and not many people do it. Instead of general mathematical discussions we have often but minor cliques. $\endgroup$ – Włodzimierz Holsztyński Apr 30 '16 at 14:02

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