Timeline for Does every Lawvere theory arise in this way?
Current License: CC BY-SA 3.0
17 events
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| Dec 31, 2022 at 9:38 | answer | added | Maxime Ramzi | timeline score: 1 | |
| Jul 30, 2015 at 13:14 | comment | added | Giorgio Mossa | @goblin is perfect. | |
| Jul 28, 2015 at 12:38 | comment | added | goblin GONE | @GiorgioMossa, is that clearer? We require it to be a full subcategory, if that helps. | |
| Jul 28, 2015 at 12:37 | history | edited | goblin GONE | CC BY-SA 3.0 |
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| Jul 28, 2015 at 8:42 | comment | added | Giorgio Mossa | Can you give a more detail on what $Lawv(X)$ should be? It should be the smallest finite product sub-category of $\mathbf C$ containing $X$? It should be the smallest finite product category generated by $X$ and its algebraic morphism? | |
| Jul 28, 2015 at 7:27 | comment | added | Qiaochu Yuan | @David: no, that already fails in the case of groups. The way you recover a Lawvere theory from the free algebra over it on one generator $X$ is by taking the opposite of the full subcategory on the finite coproducts of $X$, not by taking the full subcategory on the finite products of $X$. | |
| Jul 28, 2015 at 5:36 | answer | added | Georg Lehner | timeline score: 3 | |
| Jul 28, 2015 at 2:42 | comment | added | goblin GONE | @ZhenLin, that's okay; sets are the models of the initial Lawvere theory. | |
| Jul 28, 2015 at 2:41 | comment | added | Zhen Lin | @goblin There's a problem with your question, though – the Lawvere theory of boolean algebras is not generated by $2$ as a boolean algebra but rather $2$ as a set (as you say). This is in contrast to vector spaces; after all, the opposite of the category of finite boolean algebras is the category of finite sets, whereas the opposite of the category of f.d. vector spaces is itself. | |
| Jul 28, 2015 at 2:40 | history | edited | goblin GONE | CC BY-SA 3.0 |
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| Jul 28, 2015 at 2:35 | comment | added | David Roberts♦ | Since the Lawvere theory for a variety of algebras is given by the opposite of the category of finitely presented free algebras, shouldn't $X$ be the "free algebra on one generator" (whatever that is)? | |
| Jul 28, 2015 at 2:33 | comment | added | goblin GONE | @ZhenLin, interesting! | |
| Jul 28, 2015 at 2:33 | comment | added | goblin GONE | @DavidRoberts, thank you, yes. I bounce between $\mathbb{F}$ and $\mathbb{K}$ for my fields haha... | |
| Jul 28, 2015 at 2:32 | history | edited | goblin GONE | CC BY-SA 3.0 |
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| Jul 28, 2015 at 2:32 | comment | added | David Roberts♦ | In that second example, did you mean "Lawvere theory of $\mathbb{K}$-modules"? | |
| Jul 28, 2015 at 2:32 | comment | added | Zhen Lin | What the two examples have in common is a good theory of dualisation. After all, the opposite of any Lawvere theory embeds in the category of algebras in a canonical way. | |
| Jul 28, 2015 at 2:28 | history | asked | goblin GONE | CC BY-SA 3.0 |