# Intuition for universal quotient maps

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..

• Some more references should be given; I added one. – Todd Trimble Apr 30 '16 at 12:25
• 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. – Włodzimierz Holsztyński Apr 30 '16 at 13:59
• 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. – Włodzimierz Holsztyński Apr 30 '16 at 14:02