Continuation of namings (1) - Is there a standard name for the squares where the canonical map is monic , i.e. the pair of maps A --->B and A --->C is jointly monic. I propose semi-pullback (2)- In most cases the canonical map being epic is not what one really wants. Of course Joyal assumes the category where the maps live to be a pre-topos, then it's enough, otherwise one cannot "compose" such squares. Do we have to rename the squares where the canonical map is a universal epi, or those where its a universal regular epi? In view of (1), one would like to say that a square is a pullback iff it is both a quasi and semi pullback Début du message réexpédié :
De: Eduardo Dubuc <edubuc@dm.uba.ar> Date: Lun 5 déc 2005 17:16:13 Europe/Paris À: categories@mta.ca (Categories list) Objet: categories: Re: Name for a concept
Is there a standard name for a square A ----> B | | | | | | v v C ----> D in which the canonical map A ---> B x_D C is epic?
These are called "quasi-pullbacks" by Joyal, and they form a class of "open maps" in the category of squares. The pullbacks form the corresponding class of etal maps. These two classes are essential for the development of the theory (etal class and open class in the sense of Joyal). There are published articles by Joyal and Moerdijk on the subject.