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Univalence asserts (among other things) that all operations on the uni= verse (even the universe of sets) are "natural" under equivalences, but a g= lobal choice operator cannot be natural in this sense. Best, Jon On Tue, Feb 13, 2024, at 10:04 AM, Thomas Streicher wrote: > Dear Nath and Jon, > > I am naturally sympathizing with Nath's remark. > Thanks, Jon, for giving precise reference for why global choice > implies classical logic in HoTT (it is not inconsistent as far as I know)= . > I am grateful for the quite non-ideological discussion of the issue BTW ! > > My scepticism w.r.t. HoTT is not ideological (or "self destructive" as > some people may have thought) but was rather motivated by the fact that i= t > caused problems with one of my favourite notions of category theory, name= ly > Grothendieck fibrations. There is the notion of Street fibration which > you obtain when closing Groth. fibrations by closing under categorical > equivalences. But there are not too many examples as far as I can see. > > There is a way out of it suggested by Ahrens and Lumsdaine, namely > "displayed categories", which are systematically employed in Jon's very > nice notes on relative category theory with C. Angiuli. This is certainly= a way > how to proceed in case one wants to keep the univalence principle without > enforcing classical logic. > > But it makes things technically more complicated and I personally prefer = a > non-univalent setting. Lurie also can live without univalence both in > his book and the Kerodon. > This confirms my impression that univalence is a radical thought experime= nt > whose consequences still have to be evaluated. For me (and many other peo= ple) > foundations is not a religious business but rather a question which setti= ng is > the most convenient one for doing the things one wants to do in maths. Th= is > can be evaluated only a posteriori and a certain pluralism might be a hea= lthy > attitude. > > In any case I think w.r.t. fibered categories the most reasonable thing > seems to be cleaved fibrations and arbitrary cartesian functors. It is > foundationally tolerant and close to the practice of fibered category the= ory. > > 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