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charset="utf-8" Content-Transfer-Encoding: quoted-printable hi thomas, i am not managing to find time to think of this during the day and wise peo= ple make sure to be asleep at 2am but i am way too young for that ;) i do think that there is more to the link between definability and comprehe= nsion than just my confusions. the reason is that the two of them are tryin= g to say closely related things and there must be a way to spell it out, in= variant under any mistakes that i could ever make. lawvere said early on that categories could be used as the foundation of ma= thematics. eilenberg and maclane were shaking their heads about the crazy i= dea of sets without elements (as he recounted) but then, as we know, they i= nvited him to write the introductory paper for la jolla proceedings. after = that he went on: "the main foundational operation is comprehension. **how a= re categorical predicates (in hyperdoctrines) comprehended as categorical c= onstructs?**" --- and he spelled out comprehension as the right adjoint. (y= es, he used the fibrewise terminals and direct images to display the logica= l intuitions behind this particular adjunction in the foundations. but the = construction is more general than that.) benabou, on the other hand, says (i just looked up the paper): "In 1970 I r= ealised the possibility of doing all of naive category theory, without sets= , in the context of fibrations, and started work on proving that claim." an= d his naive category theory is the usual category theory, for which he want= s to provide a non-set-theoretic categorical axiomatics. his main question = was: "*how are categorical predicates (in fibrations) definable as categori= cal constructs?**. "morally" (as they say in cambridge) the two of them are trying to do the s= ame thing! there must be some truth in morality (contrary to all evidence := ))) the way definability is defined in your benabou notes, as far as i can tell= in the middle of the night, seems to be saying that * a subfibration CC>--->XX--P-->BB is definable * when there is a cartesian functor XX--->Sub(BB) ** whose image is a subfibration SC>--->Sub(BB) such that ** the reflection XX/XX--->>BB/P (whose right adjoint makes P into fibratio= n) ** restricts along the inclusion SC/P>--->BB/P ** to the reflection CC/XX--->>SC/P. there is probably a better way to say all this. but the main point is that = there is, i think, a bijective correspondence between * definable subfibrations of XX--->BB and * subfibrations of Sub(B)--->BB. g'night and sorry about the typos ;) -- dusko On Tue, Jan 23, 2024 at 12:12=E2=80=AFAM Thomas Streicher > wrote: Dear Dusko, I think in your mail you confuse small global sections and definability of 1. What I mean is the following. Let P : XX --> BB be a fibration of cats with 1, i.e. P has a right adjoint right inverse 1. Lawvere"s notion of comprehension means that 1 has a right adjoint right inverse G. The counit eps_X : 1_{GX} --> X of 1 --| G at X has the following universal property: for every f : 1_I --> X (over u : I --> P= X) there exists a unique \check{f} : I --> GX with eps_X \circ 1_{\check{f}} = =3D f. This is an instance local smallness for for P in the sense that GX =3D hom(= 1,X). This is more general than Lawvere's notion of comprehension which assumes that P has also internal sums in which case maps f : 1_I --> X over u : I --> PX correspond uniquely to maps \coprod_u 1_I --> X. But f also corresponds uniquely to \check{f} : I --> GX with P(eps_X) \circ \check{f} =3D u . But all this has nothing to do with definablity in the sense of Benabou. But one may consider Id_BB as a full subfibration of P via 1. This being definable in the sense of Benabou would mean that for every X in P(I) there exists a greatest subobject m of I such that m^*X is terminal in its fiber. But notice that P(eps_X) is not monic for Lawvere comprehension as considered above. But for posetal fibrations P(eps_X) is always monic, of course. Indeed for posetal fibrations having small global sections may be thought of as a kind of comprehension. But for non-posetal fibrations P the map P(eps_X) is better thiought of as the P(X)-indexed family whose fiber at i \in PX is thought of as the "set of global elements of X_i". 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 --000000000000d862bd0610de4612 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
hi thomas,

i am not managing to find time to think of this during the day and wise peo= ple make sure to be asleep at 2am but i am way too young for that ;)

i do think that there is more to the link between definability and comprehe= nsion than just my confusions. the reason is that the two of them are tryin= g to say closely related things and there must be a way to spell it out, in= variant under any mistakes that i could ever make.

lawvere said early on that categories could be used as the foundation of ma= thematics. eilenberg and maclane were shaking their heads about the crazy i= dea of sets without elements (as he recounted) but then, as we know, they i= nvited him to write the introductory paper for la jolla proceedings. after that he went on: "the main foun= dational operation is comprehension. **how are categorical predicates (in h= yperdoctrines) comprehended as categorical constructs?**" --- and he s= pelled out comprehension as the right adjoint. (yes, he used the fibrewise terminals and direct images to display the log= ical intuitions behind this particular adjunction in the foundations. but t= he construction is more general than that.)

benabou, on the other hand, says (i just looked up the paper): "In 197= 0 I realised the possibility of doing all of naive category theory, without= sets, in the context of fibrations, and started work on proving that claim= ." and his naive category theory is the usual category theory, for which he wants to provide a non-set-theoretic c= ategorical axiomatics. his main question was: "*how are categorical pr= edicates (in fibrations) definable as categorical constructs?**.

"morally" (as they say in cambridge) the two of them are trying t= o do the same thing! there must be some truth in morality (contrary to all = evidence :)))

the way definability is defined in your benabou notes, as far as i can tell= in the middle of the night, seems to be saying that
* a subfibration CC>--->XX--P-->BB is definable
* when there is a cartesian functor XX--->Sub(BB)
** whose image is a subfibration SC>--->Sub(BB) such that
** the reflection XX/XX--->>BB/P (whose right adjoint makes P into fi= bration)
** restricts along the inclusion SC/P>--->BB/P
** to the reflection CC/XX--->>SC/P.

there is probably a better way to say all this. but the main point is that = there is, i think, a bijective correspondence between
* definable subfibrations of XX--->BB and
* subfibrations of Sub(B)--->BB.

g'night and sorry about the typos ;)
-- dusko


On Tue, Jan 23, 2024 at 12:12=E2=80= =AFAM Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
Dear Dusko,

I think in your mail you confuse small global sections and
definability of 1.

What I mean is the following. Let P : XX --> BB be a fibration of cats with 1, i.e. P has a right adjoint right inverse 1.
Lawvere"s notion of comprehension means that 1 has a right adjoint
right inverse G. The counit eps_X : 1_{GX} --> X of 1 --| G at X has
the following universal property: for every f : 1_I --> X (over u : I --= > PX)
there exists a unique \check{f} : I --> GX with  eps_X \circ 1_{\ch= eck{f}} =3D f.
This is an instance local smallness for for P in the sense that GX =3D hom(= 1,X).

This is more general than Lawvere's notion of comprehension which
assumes that P has also internal sums in which case maps f : 1_I --> X over u : I --> PX correspond uniquely to maps  \coprod_u 1_I -->= X.
But f also corresponds uniquely to  \check{f} : I --> GX  with=
P(eps_X) \circ \check{f} =3D u .

But all this has nothing to do with definablity in the sense of Benabou. But one may consider Id_BB as a full subfibration of P via 1. This being definable in the sense of Benabou would mean that for every X in P(I)
there exists a greatest subobject m of I such that m^*X is terminal in
its fiber.

But notice that P(eps_X) is not monic for Lawvere comprehension as
considered above.
But for posetal fibrations P(eps_X) is always monic, of course.
Indeed for posetal fibrations having small global sections may be
thought of as a kind of comprehension. But for non-posetal fibrations P
the map P(eps_X) is better thiought of as the P(X)-indexed family whose
fiber at i \in PX is thought of as the "set of global elements of X_i&= quot;.

Thomas

 
 
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