Timeline for Can triangulated categories be "approximated by countable subcategories" (that are triangulated but not full!)?
Current License: CC BY-SA 3.0
10 events
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| Aug 27, 2015 at 12:00 | comment | added | David White | Yes, it's of interest. Thanks for editing | |
| Aug 27, 2015 at 11:12 | history | edited | Mikhail Bondarko | CC BY-SA 3.0 |
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| Jun 29, 2015 at 20:08 | history | edited | David White | CC BY-SA 3.0 |
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| Jun 29, 2015 at 15:30 | comment | added | Eric Wofsey | @ZhenLin: Yes, of course. Indeed, I suspect that not very many interesting properties of triangulated categories are first-order. Still, I wanted to give some indication of how far the idea in Jeremy Rickard's comment could be pushed. | |
| Jun 29, 2015 at 15:23 | comment | added | Zhen Lin | @EricWofsey Of course, the usefulness of this observation is directly proportional to the expressiveness of the first-order language in question. But I am sure you know that. | |
| Jun 29, 2015 at 14:54 | history | edited | Mikhail Bondarko |
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| Jun 28, 2015 at 16:56 | comment | added | Mikhail Bondarko | Yes, such a logical argument should work. Yet I would prefer to avoid it since it is not quite "inner mathematical". | |
| Jun 28, 2015 at 12:45 | comment | added | Eric Wofsey | More strongly, all of the structure and axioms of a triangulated category can be expressed in first-order logic with a finite language, so by Lowenheim-Skolem given any set $S$ of morphisms, there is a subcategory of size at most $\aleph_0+|S|$ containing $S$ which is not just triangulated but an elementary submodel of $C$. | |
| Jun 28, 2015 at 10:42 | comment | added | Jeremy Rickard | Each instance of every axiom says that, given some finite set of objects and morphisms, there is a finite set of objects and morphisms satisfying some property. So starting with a countable set of objects and morphisms, can you not just add countably many objects and morphisms so that all instances of axioms whose hypotheses involve only the original set are satisfied? And then you can iterate and take the union, to get a countable triangulated subcategory. | |
| Jun 28, 2015 at 8:35 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |