Timeline for Which categories are the categories of models of a Lawvere theory?
Current License: CC BY-SA 3.0
8 events
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| May 9, 2014 at 2:45 | comment | added | Tim Campion | A duality that actually involves classical 1-sorted Lawvere theories rather than their idempotent completions is given in Adamek, Rosicky, and Vitale's book Algebraic Theories, Thm 11.39, but they consider concrete categories, i.e. the forgetful functor to $\mathbf{Set}$ is part of the data. Barring such a move, I suspect it might be impossible to get a functorial duality because there autoequivalences of the category of algebras which don't fix the free algebras. | |
| May 9, 2014 at 2:13 | comment | added | Tim Campion | A small subtlety: if small categories with finite products are your "theories" and algebraic categories are your "models", then you can only recover the theory from the models up to Morita equivalence. If your Lawvere theory was the finitely-generated free algebras, then the strongly finitely presentable algebras are the retracts of these. So if we want a precise duality, we need our theories to be idempotent-complete as well as having finite produts. This complication doesn't arise for Gabriel-Ulmer duality because finitely-complete categories are automatically idempotent-complete. | |
| Jun 11, 2013 at 8:02 | comment | added | Qiaochu Yuan | @Zhen: ah, I see. I would also guess that $G$ generated by a single object under finite coproducts is the correct condition. | |
| Jun 11, 2013 at 7:39 | comment | added | Zhen Lin | @Qiaochu: Yes, this is multi-sorted. It will not do to take $\mathcal{G}$ to be a single object: this already fails for concrete examples like $\mathbf{Ab}$. I think instead you need to ask that $\mathcal{G}$ be generated by a single object under finite coproducts – but I have not checked. Certainly it is enough to ask that the full subcategory of strongly finitely presentable objects have this property, because this subcategory becomes the Lawvere theory. | |
| Jun 11, 2013 at 7:05 | comment | added | Justin Noel | Yes I believe Zhen is talking about the multisorted case and you will get the single-sorted case by taking a single object (a small projective generator). Note the algebras over a multi/singly-sorted theory correspond to cocomplete algebraic categories with a set of projective generators/a single projective generator in the sense of Quillen. | |
| Jun 11, 2013 at 7:00 | vote | accept | Qiaochu Yuan | ||
| Jun 10, 2013 at 23:20 | comment | added | Qiaochu Yuan | Great! Am I correct in guessing that by "Lawvere theory" here you mean a multisorted Lawvere theory, and that if I was only interested in Lawvere theories with one sort the second condition should be "there exists an object in $C$ such that..."? | |
| Jun 10, 2013 at 22:16 | history | answered | Zhen Lin | CC BY-SA 3.0 |