Timeline for Notation classifying topos
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 2, 2022 at 15:08 | comment | added | user483320 | @MatthiasHutzler Thank you very much for your great help! I learned a lot. :-) | |
| Jun 2, 2022 at 11:54 | comment | added | Matthias Hutzler | Also, it makes sense to say that the equations in a "system of equations" correspond to the axioms of a theory. But then, what do the variables in the "system of equations" correspond to? Geometric theories consist more naturally of three layers instead of two: (1) sorts, (2) function- and relation symbols and (3) axioms. So are systems of equations more like propositional geometric theories (those with zero sorts), and rings more like localic toposes (or just locales)? These are the kind of questions I wanted to avoid. ;-) | |
| Jun 2, 2022 at 11:47 | comment | added | Matthias Hutzler | On the other hand, I don't think the notation $R[\ldots]$ is used for systems of equations. And if for example we just divide out an ideal in $R$, I think you would usually not call that "adjoining" something. | |
| Jun 2, 2022 at 11:44 | comment | added | Matthias Hutzler | Re "Why does the analogy start breaking down here?": Well, maybe it doesn't. :-) I agree that geometric theories behave like "systems of equations" (a set of variables and a set of equations in these variables) under this analogy. And while a "system of equations" is just another name for a (not necessarily finite) presentation of some $R$-algebra, geometric theories can be seen as syntactic presentations for toposes. | |
| Jun 1, 2022 at 17:21 | comment | added | user483320 | @MatthiasHutzler I asked a few questions but then could answer them myself, so I deleted them ... there's one remaining though (see above). Thank you very much for your answer! | |
| Jun 1, 2022 at 17:20 | vote | accept | user483320 | ||
| Jun 1, 2022 at 11:49 | comment | added | user483320 | @MaximeRamzi "I think some people give a different name to "the left adjoint in the geometric morphism" like "logical morphism" precisely for this reason" I don't think the left adjoint of a geometric morphism is a logical morphism of toposes. | |
| Jun 1, 2022 at 11:47 | comment | added | user483320 | "But the analogy arguably starts breaking down here." why? can't you adjoin to a ring any set of variables together with a set of equations these variables should satisfy. I mean if you can adjoin X and Y with the property that X2=5Y to the ring R and get R[X,Y]/(X2−5Y), can't you do that with every equation? Why does the analogy breaks down here? | |
| Jun 1, 2022 at 7:18 | comment | added | Maxime Ramzi | I think some people give a different name to "the left adjoint in the geometric morphism" like "logical morphism" precisely for this reason | |
| May 31, 2022 at 19:45 | history | answered | Matthias Hutzler | CC BY-SA 4.0 |