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charset="utf-8" Content-Transfer-Encoding: quoted-printable Hi Thomas et al, I believe the proposed non-quotiented construction works even with a cleavi= ng that is not split. Does anyone know if this is true? (I thought I had ch= ecked this a while ago, but now I cannot find my notes and am less confiden= t; however, it is claimed by Von Glehn in Example 3.9 here: http://www.tac.= mta.ca/tac/volumes/33/36/33-36.pdf.) Glancing at it briefly, it looks ra= ther like the identity and associativity laws follow from the higher identi= ties governing associators and unitors in a pseudofunctor. I agree it is inelegant to assume a splitting, but I think it is rather ele= gant to assume a (non-split) cleaving as property-like structure (so we do = not ask that it be preserved) =E2=80=94 Indeed, this is precisely what you = get from the Chevalley criterion for fibrations in a 2-category. I think th= is is analogous to the way that in an internal setting we must assume chose= n structures but not typically ask to preserve the choices; when we do not = wish to assume chosen structures at all, we can either pass to stacks or wo= rk in a univalent / fully saturated setting. Best, Jon On Wed, Jan 31, 2024, at 11:41 PM, streicher@mathematik.tu-darmstadt.de wro= te: > If we work with split fibrations and arbitrary cartesian functors between > them we can construct the opposite of a fibration without quotienting. > That is possible but in my eyes less elegant than the usual approach wher= e > one assumes that one can factorize modulo equivalence relations even if > they are big. > > Thomas You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --MCBoundary=_12402012043560051 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=utf-8 Hi Thomas et al,

I believe the proposed non-quotiented construction works even with a cleavi= ng that is not split. Does anyone know if this is true? (I thought I had ch= ecked this a while ago, but now I cannot find my notes and am less confiden= t; however, it is claimed by Von Glehn in Example 3.9 here: http://www.tac.mta.ca/tac/volumes/33/36/33-36.pdf.) Glancing at it brie= fly, it looks rather like the identity and associativity laws follow from t= he higher identities governing associators and unitors in a pseudofunctor.<= br>
I agree it is inelegant to assume a splitting, but I think it is rather ele= gant to assume a (non-split) cleaving as property-like structure (so we do = not ask that it be preserved) =E2=80=94 Indeed, this is precisely what= you get from the Chevalley criterion for fibrations in a 2-category. I think this is analogous to the way that in an internal = setting we must assume chosen structures but not typically ask to preserve = the choices; when we do not wish to assume chosen structures at all, we can= either pass to stacks or work in a univalent / fully saturated setting.

Best,
Jon


On Wed, Jan 31, 2024, at 11:41 PM, streicher@mathematik.tu-darmstadt.de wro= te:
> If we work with split fibrations and arbitrary cartesian functors betw= een
> them we can construct the opposite of a fibration without quotienting.=
> That is possible but in my eyes less elegant than the usual approach w= here
> one assumes that one can factorize modulo equivalence relations even i= f
> they are big.
>
> Thomas
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
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