Dear Jon, for constructing the opposite of a fibration you do not need at all that the chosen cleavage is split. It is easy to see that fixing the cartesian arrow also fixes the vertical arrow. If everybody is happy with this solution I am fine. It is in the same spirit as chosen pullbacks or chosen finite limits. The reason for my reluctance is that nobody would consider as a natural notion a vector space with a chosen basis. The virtue of strong choice is that then such additional structure can be pulled out of the hat at demand. Thomas ---------- You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. Leave group: https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27