RE: A brief survey of cartesian functors

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Jean,

I apologise for my ignorance of your work.

I guess that an equivalence of categories P:X-->S is always a foliation, but not
a fibration, unless it is surjective on objects.

-André
  
__________________________________
From: Jean Bénabou [jean.benabou@wanadoo.fr]
Sent: Monday, July 28, 2014 5:54 AM
To: Categories
Subject: categories: A brief survey of cartesian functors

Dear Ross, Dear all,

In a recent mail I asked Ross if pseudo cartesian functors between pseudo fibrations had been studied.
There are many generalizations of fibrations. Pseudo fibrations are only one of them. But there are also prefibrations, defined by Grothendieck, but almost never considered, and pre foliations, which I  define here, which generalise greatly pre fibrations. For such pre foliatons, I define cartesian functors  and show that they have striking properties, most of which are not known, even in the very special case of fibrations.
I thought this brief survey might interest you, in case you decide to study  seriously the properties of pseudo cartesian functors.

Best regards to all,
Jean


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