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charset=UTF-8 Content-Transfer-Encoding: quoted-printable i did, of course, miss something (as i suspected before hitting return, but didn't resist :) but it seems that fixing the error makes the benabou/lawvere link even nicer. (even if i am still missing something i am sure that the link is there.) the one-to-one correspondence between definable subfibrations of E---->B and the "comprehensions" E---->Sub(B) stands. the claim that the subfibration C---->B is obtained by pullback is wrong, because we don't want to pull back along the "truth" into Sub(B) but along the "comprehension". remember how lawvere defined comprehension of the bifibration E---->B to be (in equivalent to hyperdoctrines) the right adjoint of the cartesian functor Arr(B)---->E mapping (i--f-->j) to f_!(top_i). [[i even introduced Arr(B) in the previous message but oversimplified and didn't use it.]] now i can of course restrict that functor to Sub(B)---->E. a definable subfibration C of E, determines a right adjoint E---->Sub(B) and the adjunction localizes on C. how can i say this for fibrations, without using the direct images? well (now i write Arr(E) as E/E and B/P is the comma into P) E--P-->B is a fibration if and only if the induced functor E/E---->B/P is a reflection (the cartesian liftings induce a right adjoint right splitting). a subcategory C of E is a definable subfibration of P if and only E/E---->B/P restricts to C/E---->Sub/P. i think. apologies that i am thinking so much in public, but there is no shortage of private thinking and there is a shortage of public thinking and we should all think together how to bring lawvere and benabou together :) -- dusko On Mon, Jan 22, 2024 at 3:05=E2=80=AFPM Dusko Pavlovic = wrote: > as we are talking about lawvere's and benabou's nachlasse, it occurred to > me to ask what would benabou's definability be in lawvere's terms. maybe > they both knew this and maybe other people do, but i hope it doesn't hurt > to spell it out. > > let B be a category with pullbacks so we have the fibrations > Arr(B)--Cod-->B and > Sub(B)--Cod-->B, > where Arr(B) is the category of arrows and Sub(B) is the category of > subobjects. the cartesian functors B>--Ids-->Sub(B) and B>--Ids-->Arr(B) > are the right adjoints. > > Prop. B>--Ids-->Sub(B) is the classifier of definable subfibrations. more > precisely, for any fibration E--P-->B and a subfibration C>---->E here is= a > unique cartesian functor E--c-->Sub(B) such that the fibration C---->B is > the pullback of c along B>--Ids-->Sub(B). > > i did check this, but i didn't check it twice, so i may still be missing > something. > > -- dusko > > On Mon, Jan 22, 2024 at 10:00=E2=80=AFAM Francis Borceux < > francis.borceux@uclouvain.be> wrote: > >> >> Sorry, and thanks to Jon for noticing the slip of terminology in my mail= . >> >> Indeed, B=C3=A9nabou was insisting on the importance of his notion of >> definability. >> >> Francis >> >> *********************************** >> Francis Borceux >> 6 rue Fran=C3=A7ois >> 1490 Court-Saint-=C3=89tienne >> Belgique >> Fixe: +32(0)10614205 >> Mobile: +32(0)478390328 >> >> >> ---------- >> >> You're receiving this message because you're a member of the Categories >> mailing list group from Macquarie University. >> >> Leave group: >> >> https://protect-au.mimecast.com/s/JhEqC5QP8ySGq6xkfzFqDC?domain=3Doutloo= k.office365.com >> > --0000000000000901ff060f944ef3 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
i did, of course, miss something (as i suspected befo= re hitting return, but didn't resist :)

but it= seems that fixing the error makes the benabou/lawvere link even nicer. (ev= en if i am still missing something i am sure that the link is there.)
=

the one-to-one correspondence between definable subfibr= ations of E---->B=C2=A0 and the "comprehensions" E---->Sub(= B) stands. the claim that the subfibration C---->B is obtained by pullba= ck is wrong, because we don't want to pull back along the "truth&q= uot; into Sub(B) but along the "comprehension".

remember how lawvere defined comprehension of the bifibration E----= >B to be (in equivalent to hyperdoctrines) the right adjoint of the cart= esian functor Arr(B)---->E=C2=A0 mapping (i--f-->j) to f_!(top_i). [[= i even introduced Arr(B) in the previous message but oversimplified and did= n't use it.]]

now i can of course restrict= that functor to Sub(B)---->E. a definable subfibration C of E, determin= es a right adjoint E---->Sub(B) and the adjunction localizes on C.
=

how can i say this for fibrations, without using the di= rect images? well (now i write Arr(E) as E/E and B/P is the comma into P) E= --P-->B is a fibration if and only if the induced functor E/E---->B/P= is a reflection (the cartesian liftings induce a right adjoint right split= ting). a subcategory C of E is a definable subfibration of P if and only E/= E---->B/P restricts to C/E---->Sub/P. i think.

apologies that i am thinking so much in public, but there is no shortage= of private thinking and there is a shortage of public thinking and we shou= ld all think together how to bring lawvere and benabou together :)

-- dusko

On Mon, Jan 22, 2024 at 3:05=E2=80=AFPM = Dusko Pavlovic <duskgoo@gmail.com> wrote:
as we are talking about lawvere's and b= enabou's nachlasse, it occurred to me to ask what would benabou's d= efinability be in lawvere's terms. maybe they both knew this and maybe = other people do, but i hope it doesn't hurt to spell it out.

let= B be a category with pullbacks so we have the fibrations
Arr(B)--Cod--= >B and
Sub(B)--Cod-->B,
where Arr(B) is the category of arrow= s and Sub(B) is the category of subobjects. the cartesian functors B>--I= ds-->Sub(B) and B>--Ids-->Arr(B) are the right adjoints.

P= rop. B>--Ids-->Sub(B) is the classifier of definable subfibrations. m= ore precisely, for any fibration E--P-->B and a subfibration C>----&g= t;E here is a unique cartesian functor E--c-->Sub(B) such that the fibra= tion C---->B is the pullback of c along B>--Ids-->Sub(B).

i= did check this, but i didn't check it twice, so i may still be missing= something.

-- dusko

On Mon, Jan 22, 2024 at 10:00=E2= =80=AFAM Francis Borceux <francis.borceux@uclouvain.be> wrote:

Sorry, and thanks to Jon for noticing the slip of terminology in my mail.
Indeed, B=C3=A9nabou was insisting on the importance of his notion of defin= ability.

Francis

***********************************
Francis Borceux
6 rue Fran=C3=A7ois
1490 Court-Saint-=C3=89tienne
Belgique
Fixe: +32(0)10614205
Mobile: +32(0)478390328


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