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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 mai= ling list group from Macquarie University. Leave group: https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.as= hx?source=3DEscalatedMessage&action=3Dleave&GuestId=3D4eb9b40c-9b3a-48a5-97= 81-836e5a171e8b