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Univalence asserts (among other things) that all operations on the u= niverse (even the universe of sets) are "natural" under equivalences, but a= global choice operator cannot be natural in this sense. Thanks for this clarification! One needs global choice in CT for establishing that full and faithful functors which are essentially surjective are actually categorical equivalences. In practice you just have the stronger notion. In my eyes HoTT is an attempt to live with the weaker notion. In basic topos theory global choice is assumed when identifying inverse images of geometric morphisms with cocontinuous functors preserving finite limits. Therefore, I never believed that you can do topos theory in a weak meta theory without strong choice principles. At least not the way it is done in the traditional texts. Maybe HoTT is an attempt to avoid these hidden uses of strong choice... But it is a mystery to me why people have overlooked that they were using very strong choice principle. Well, it is the usual difference betwee= n ideology and practice as it seems to me... 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