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Things like connecting f.f.+e.s.o functors to eq= uivalences in classical math do not require global choice when stated in th= e way that a careful classical mathematician understands (or is supposed to= understand) them. The statement is "There exists a mapping that takes any f.f.+e.s.o F : C ->= D functor to a functor G : D -> F together with natural isomorphisms GF=3D= 1 and FG=3D1". This statement follows from "local" choice easily without ma= king any global choices. I understand that you need global choice for this = claimed function to form a 'definitional extension' of ZFC, but that is sim= ply not what is meant by the original claim; it is for this reason that the= practical formalisation of mathematics in ZFC emphasises consecutive conse= rvative extensions and not only definitional extensions. Avoiding this dist= inction is one of the reasons that some people (like Hilbert and Bourbaki) = have introduced global choice, but I must emphasise that essentially all th= e theorems of classical mathematics also work fine with only local choice. = This is because the ground statements of classical mathematics are always p= ropositional: when we want to introduce a construction, we describe the spe= cification of that construction by means of a predicate, and then say that = there exists something satisfying that predicate. After this, we work with = an arbitrary thing satisfying that specification. Likewise I do not agree with the discussion of geometric morphisms, for rea= sons that are identical to what I have mentioned above regarding weak and s= trong equivalences of categories. For these reasons, I personally would not think of HoTT as an attempt to av= oid hidden uses of strong/global choice, because the rest of classical math= ematics (with the exception of Bourbaki) also does a perfectly fine job of = avoiding global choice. (I'm not arguing against the convenience and utilit= y of global choice though.) Best, Jon On Tue, Feb 13, 2024, at 11:38 AM, Thomas Streicher wrote: >> In fact, the reference shows that global choice really is inconsistent i= n HoTT. Univalence asserts (among other things) that all operations on the = universe (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 betw= een > 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