Timeline for How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2022 at 10:09 | comment | added | godelian | @TimCampion It is encoded, but for the duality I had in mind the morphisms should not necessarily preserve all of them but only some, so in that sense it's a bit different. | |
| Mar 30, 2022 at 1:52 | comment | added | Tim Campion | @AlecRhea Thanks, fixed! I agree, before today, really, I never really thought of sketches as anything more than an occasionally-convenient language. But the internal hom of sketches presents a kind of intriguing way to enhance the "semantic" category of models in a way which remembers something extra about the syntax of the "theory" presenting it. On the flip side, it might allow for extracting various "canonical" theories from a category of models which might be "smaller" than the sort of thing one tends to get from, say, ordinary Gabriel-Ulmer duality. | |
| Mar 30, 2022 at 1:46 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 220 characters in body; edited title
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| Mar 30, 2022 at 1:32 | comment | added | Alec Rhea | I think you meant for the domain of $(-)\times T$ to be $A_S$. Nice question, this is the first time I’ve clicked the word ‘sketch’ and found something interesting! | |
| Mar 30, 2022 at 1:01 | comment | added | Tim Campion | @varkor I see what you mean in the context of limit sketches. But I really do want to allow colimits in my sketches, which would seem to make things more complicated. | |
| Mar 30, 2022 at 0:56 | comment | added | Tim Campion | @godelian I think I’m talking about something a bit different because in my setting, mod(s) is not just a category but rather still has some cone/cocone information. But maybe that information is encoded somehow in the approach you describe? | |
| Mar 29, 2022 at 19:51 | comment | added | varkor | A duality relative to a limit doctrine gives a general answer to "To what extent can Gabriel–Ulmer duality be generalised to different classes of limits and colimits?". It's not in the terminology of sketches, but that should be a straightforward conversion. The enriched version will follow from Accessibility and presentability in 2-categories. | |
| Mar 29, 2022 at 19:16 | comment | added | godelian | (if yes, I can post it as an answer detailing what are the morphisms on each side) | |
| Mar 29, 2022 at 19:08 | comment | added | godelian | Since the models of a sketch are the same as the models of an infinitary coherent theory coding the (co)limiting (co)cones, the duality theory can be understood by introducing the notion of k-pretopos representing the sketch, which would correspond to the syntactic side. Then one recovers a nice duality theory which makes the maps in questions 1 and 2 always equivalences. Is this what you were looking for? | |
| Mar 29, 2022 at 19:08 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator` etc.
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| Mar 29, 2022 at 18:51 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 162 characters in body
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| Mar 29, 2022 at 18:42 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 179 characters in body
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| Mar 29, 2022 at 18:36 | history | asked | Tim Campion | CC BY-SA 4.0 |