Hi Dusko, if you say that both local smallness and definability are formulated as requirements that certain elementary fibrations are representable fibrations are reprsentable then I agree. Elementary fibrations are fibrations of posetal groupoids and representable means that the total category of fibrations has a terminal object. Where I disagree is that local smallness can be formulated as a certain subfibration being definable. But, as I said already previously, a posetal fibration P : X --> B of cats with 1 has comprehension if the subfibration 1 : Id --> P is definable. Moreover, Lawvere comprehension means "small global elements" which in case of a fibration of cc's is equivalent to local smallness in the sense of Benabou. Best wishes, 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