Dear Andr'e, of course, logos would be a natural name when understood as elementary toposes and logical morphisms. In the Elephant one finds Top, BTop and LogTop as names for elementary toposes and various different notions of morphisms. There is a problem with just geometric morphisms since they dont correspond to Grothendieck toposes - only bounded ones do. I personally find logical morphisms the basic notion and geometric morphisms as a derived notion. As I learnt from Benabou quite some time ago one should consider geometric morphisms as fibred toposes. If SS is a (base) topos then P : EE -> SS is a fibration of toposes iff P is a Grothendieck fibration whose fibers are toposes and whose reindexing functors are logical. One may characterize them as fibrations P such that P is a logical morphism of toposes. Fibrations of cocomplete toposes over SS correspond to finite limit preserving functors from SS to some topos (namely Delta(I) = \coprod_I 1_I). Fibrations of locally small cocomplete toposes correspond to finite limit preserving functors to some topos which, moreover, have a right adjoint. The bounded gm's to SS correspond to fibrations of locally small cocomplete toposes which moreover admot a small generating family. This, for me justifies the notion of bounded gm's. The fibrational point of view doesn't suggest how to compose them (see discussion pp.66-67 of my notes on fibrations). But the corresponding Delta's are obviously closed under composition. I then would add the "observation" that these functors maybe understood as "glorified" frame morphisms. This, of course, doesn't reflect history but that's not the issue here. It's about "rational reconstruction" of a notion. Thomas PS That's what I think Benabou had in mind with his remark after one of your lectures at IHES last November. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear friends and colleagues, In Spring 1981, near a lavender field in Southern France, Alexander Grothendieck greeted me at the door of his home. He wasted no time and immediately put the question: 'What is the relationship between the two uses of the term 'topos'?' This led to a very interesting discussion.The first thing that was established as a basis was that SGA4 never defined 'topos', but rather spoke always of 'U-topos', where U was a certain kind of model of set theory. All the categories so arising have common features, such as cartesian closure, and the U itself can be construed as such a category. (TAC Reprints no. 11). Thus we arrived at the notion of 'U-topos' as a special geometric morphism E →U of 'elementary' toposes. Grothendieck's general method of relativization suggests the usefulness of a general topos as a codomain or base U. (see Giraud, SLN 274). But to focus more specifically on the original case, various special properties of the base U could also be considered: Booleanness (note for example, that Booleanness distinguishes algebraic points among algebraic figures) Axiom of choice; Lack of measurable cardinals; et cetera. One of the many topics we discussed was the 'Medaille de Chocolat' exercise in SGA4, and its basic importance for understanding applications of topos theory: the gros and petit sheaves of an object point out that there should be a qualitative distinction between a topos of SPACES and a topos of set-valued sheaves on a generalized space. I believe that considerable progress is now being made on the characterization of 'gros' toposes under the name of Cohesion. Grothendieck made a big step towards the characterization of 'petit' under the name of 'etendu' (sometimes known as 'locally localic'). Concerning Grothendieck's most famous contribution, the 'petit etale' topos, what are it's distinguishing properties as a topos? We also discussed the Grauert direct image theorem as a relativization of the Cartan-Serre theorem. It is important to note that Grothendieck's work was not limited to the Weil conjectures but, for example, involved around 1960 several categories related to complex analysis which were perhaps part of his inspiration for the notion of topos. Separation? Actually, separation has been one of the main sources of confusion. I wish that someone with internet confidence would correct the Wikipedia article that claims that pre-1970 toposes were about geometry, but that post-1970 toposes were about logic. Certainly, that discourages students from studying either. Omitted was the fact that logic has always been used to sharpen the study of geometry; in the last 50 years we have been able to make this relation more explicit, with the help of categories. Of course, separating a certain kind of object from a certain kind of map would be basic 'grammar'. But we cannot separate the legacy of Grothendieck from the inspiration it gives to the continuing development of topos theory. Best wishes Bill Lawvere [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Thomas, -
On 6 Nov 2016, at 10:26, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
If SS is a (base) topos then P : EE -> SS is a fibration of toposes iff P is a Grothendieck fibration whose fibers are toposes and whose reindexing functors are logical. One may characterize them as fibrations P such that P is a logical morphism of toposes. Fibrations of cocomplete toposes over SS correspond to finite limit preserving functors from SS to some topos (namely Delta(I) = \coprod_I 1_I). Fibrations of locally small cocomplete toposes correspond to finite limit preserving functors to some topos which, moreover, have a right adjoint. The bounded gm's to SS correspond to fibrations of locally small cocomplete toposes which moreover admot a small generating family.
This is to check that I understand what you said correctly. First, technically, the topos EE that is the domain of P is not itself the domain of the geometric morphism, but an arrow topos. For example, for the identity geometric morphism, EE is [->,SS] and P is the codomain functor. Then the fibres are the slices of SS, still toposes, and the pullback functors are logical. (I think you said that before, but I didn't understand it then.) Then what is gained from using fibrations is a more secure understanding of "SS-cocomplete", having all SS-small colimits, in other words how SS supplies the infinities used in the geometric logic. Elephant B1.4.2 says for indexed categories that having all SS-indexed coproducts is to say that the reindexing functors have left adjoints, with Beck-Chevalley, and I imagine there's an analogous and possibly improved way to do it with fibrations. Is that right? All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[Note from moderator: Late last week the listserv software used by the categories list experienced an outage just now resolved, whence the flurry of messages from the list, for which I apologize. This should be the last. I also suggest that this thread should end soon. ] On 05/11/16 16:04, Joyal, Andr? wrote:
I am considering using the word "logos" instead of "elementary topos". The word "logos" has seldom been used in mathematics. It is a noble word in philosophy where it means: reason, discourse, logic, knowledge, principle of order.
The idea is actually very good, but I'm afraid it comes 40 years too late. It seems to me that at this point it would only dramatically increase terminological chaos... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The term 'logos' already has a well-established meaning. See Tholen's review of the 1990 book by Freyd and Scedrov: Categories, allegories ...(a logos is a regular category in which the subobjects of an object form a lattice, and in which each inverse-image map has a right adjoint) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
What Eduardo means by "Giraud topos" is a category validating all conditions of the Giraud theorem with the exception of having a small generating family. These guys can be elementary toposes or not. There is Freyd's example where objects are set X with an ordinal indexed family of bijections from X to X and morphisms are maps commuting with all these ordinal many maps. This is a locally small cocomplete elementary topos lacking a s mall generating family. In 2011 on this list Johnstone gave the example of a category which superficially is like Freyd's example but the class many endomaps are not required to be bijections. This category is also bicomplete and locally small but it hasn't got a subobject classifier. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 09/11/16 10:48, Thomas Streicher wrote:
What Eduardo means by "Giraud topos" is a category validating all conditions of the Giraud theorem with the exception of having a small generating family. These guys can be elementary toposes or not. There is Freyd's example where objects are set X with an ordinal indexed family of bijections from X to X
So they don't form a class. Is there an example where the objects form a class? (I like categories to have this property!) Paul
and morphisms are maps commuting with elementary topos lacking a s mall generating family. In 2011 on this list Johnstone gave the example of a category which superficially is like Freyd's example but the class many endomaps are not required to be bijections. This category is also bicomplete and locally small but it hasn't got a subobject classifier.
Thomas
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
-- Paul Blain Levy School of Computer Science, University of Birmingham http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Bill, I'd really like to understand these issues about gros and petit toposes better. My own current direction with arithmetic universes is - I believe - a reasonable one to try in respect to geometric theories and classifying toposes, but it loses sight of gros toposes and synthetic approaches, which, after all, were part of the founding ideas of toposes. I need to know better what it is that I risk losing. However, my attempts to understand gros toposes run into difficulties with geometricity. This is illustrated by the Zariski topos for algebras over a field k. If k is R or C then the geometric methodology expects it and its algebras to be locales (and non-discrete), not sets. But does the polynomial ring R[x] exist localically? The degree of a polynomial looks like being part of the geometric structure, and removing leading zeros (e.g. after a subtraction) is not continuous. Do you know if anyone has investigated a localic form of these methods in algebraic geometry? All the best, Steve. p.s. I like to think I'm following Grothendieck's insights, but the truth is I understand only a tiny fraction of them.
On 6 Nov 2016, at 15:41, wlawvere =0A= <wlawvere@buffalo.edu> wrote:
Dear friends and colleagues,
In Spring 1981, near a lavender field in Southern France, Alexander Grothendieck greeted me at the door of his home. He wasted no time and immediately put the question:
'What is the relationship between the two uses of the term 'topos'?'
This led to a very interesting discussion.The first thing that was established as a basis was that SGA4 never defined 'topos', but rather spoke always of 'U-topos', where U was a certain kind of model of set theory. All the categories so arising have common features, such as cartesian closure, and the U itself can be construed as such a category. (TAC Reprints no. 11).
Thus we arrived at the notion of 'U-topos' as a special geometric morphism E →U of 'elementary' toposes. Grothendieck's general method of relativization suggests the usefulness of a general topos as a codomain or base U. (see Giraud, SLN 274). But to focus more specifically on the original case, various special properties of the base U could also be considered: Booleanness (note for example, that Booleanness distinguishes algebraic points among algebraic figures) Axiom of choice; Lack of measurable cardinals; et cetera.
One of the many topics we discussed was the 'Medaille de Chocolat' exercise in SGA4, and its basic importance for understanding applications of topos theory: the gros and petit sheaves of an object point out that there should be a qualitative distinction between a topos of SPACES and a topos of set-valued sheaves on a generalized space. I believe that considerable progress is now being made on the characterization of 'gros' toposes under the name of Cohesion. Grothendieck made a big step towards the characterization of 'petit' under the name of 'etendu' (sometimes known as 'locally localic'). Concerning Grothendieck's most famous contribution, the 'petit etale' topos, what are it's distinguishing properties as a topos?
We also discussed the Grauert direct image theorem as a relativization of the Cartan-Serre theorem. It is important to note that Grothendieck's work was not limited to the Weil conjectures but, for example, involved around 1960 several categories related to complex analysis which were perhaps part of his inspiration for the notion of topos.
Separation? Actually, separation has been one of the main sources of confusion. I wish that someone with internet confidence would correct the Wikipedia article that claims that pre-1970 toposes were about geometry, but that post-1970 toposes were about logic. Certainly, that discourages students from studying either. Omitted was the fact that logic has always been used to sharpen the study of geometry; in the last 50 years we have been able to make this relation more explicit, with the help of categories.
Of course, separating a certain kind of object from a certain kind of map would be basic 'grammar'. But we cannot separate the legacy of Grothendieck from the inspiration it gives to the continuing development of topos theory.
Best wishes
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Paul B Levy -
Steve Vickers -
Tadeusz Litak -
Thomas Streicher -
wlawvere