Timeline for Lawvere theories versus classical universal algebra
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Jun 30, 2011 at 21:17 | vote | accept | Benjamin Steinberg | ||
| Jun 30, 2011 at 21:13 | vote | accept | Benjamin Steinberg | ||
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| Jun 30, 2011 at 19:40 | vote | accept | Benjamin Steinberg | ||
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| Jun 30, 2011 at 15:50 | history | edited | Todd Trimble | CC BY-SA 3.0 |
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| Jun 30, 2011 at 10:40 | comment | added | Todd Trimble | of Toposes, Triples, and Theories by Barr and Wells (p. 263 of 302 in this pdf file: tac.mta.ca/tac/reprints/articles/12/tr12.pdf). | |
| Jun 30, 2011 at 10:38 | comment | added | Todd Trimble | (cont.) of all locally small Lawvere theories of possibly unbounded rank (or the category of monads on $Set$ has coequalizers; I'm pretty sure it's not finitely cocomplete however. If you restrict to theories of rank bounded by some cardinal, that category is complete and cocomplete properties. As for your other question, it sounds to me you're asking if there is a Birkhoff theorem for general (possibly infinitary) Lawvere theories. The modern categorical interpretation of Birkhoff's theorem comes under the rubric "monadicity" (or "tripleability") theorems. See for example p. 250 ff. (cont.) | |
| Jun 30, 2011 at 10:23 | comment | added | Todd Trimble | Benjamin, sorry not to have responded sooner. As you probably know, the Lawvere theory is the category opposite to the Kleisli category of free algebras the associated monad. (There is no harm in restricting to the finitely generated free algebras if the theory is finitary, or to free algebras generated by less than $\alpha$ elements if $\alpha$ is the rank of the theory.) You can always consider an identity of the theory to be given by two parallel arrows from $F[1]$ (free on one generator) to some other free algebra, as you were suggesting. I don't know if the category of all locally (cont.) | |
| Jun 29, 2011 at 20:38 | comment | added | Todd Trimble | Gerhard, instead of trying to figure out in comments what might work, it might help to talk a little behind the scenes, and if something comes of it, I'd be willing to add to my answer. I can be reached at my gmail account: place a dot between topological and musings, followed by the logical thing. (I forget how to reach you, but my memory is that it's through a proxy.) Todd "I'm Not Cut Out to Be a Salesman" Trimble. | |
| Jun 29, 2011 at 19:17 | comment | added | Gerhard Paseman | I am in the somewhat unusual position (thanks more to me than to my advisors) of knowing a fair amount of "classical universal algebra" while knowing almost no category theory. Of course, use your expository strengths in forming your answer, but if you tailor your pitch to me, you may end up capturing a much larger audience. (I'm willing to fill in the UA details to the best of my ability .) I look forward to your additions. Gerhard "Ask Me About System Design" Paseman, 2011.06.29 | |
| Jun 29, 2011 at 13:07 | comment | added | Todd Trimble | Hi, Gerhard. Thanks for mentioning n-valued Post algebras; this helps me track down some learned allusions that I hadn't picked up on before. My general reaction here is to emphasize that despite the shortcomings of some expositors, perhaps the main practical point of category theory is to make mathematics easier to understand, not harder. Despite some jargon, I actually meant for my answer to be understandable to a wide range of mathematicians, not just category theorists, and ideally to think, "hey, that's pretty cool", and not, "ugh, not more categorical abstract nonsense". More to come... | |
| Jun 29, 2011 at 12:04 | comment | added | Benjamin Steinberg | Continuing... Is there some notion for a theory that an identity is a formal equality between two coterminous arrows of the theory category? Is the collection of all models, say in Set, some natural generalzation of a variety? What closure properties does it have? | |
| Jun 29, 2011 at 12:02 | comment | added | Benjamin Steinberg | @Todd, I mostly like this answer. But I would like to know if the theory viewpoint allows one to apply the equational theory concept to these more general situations. Like if I look at compact Hausdorff spaces, then I can think of an identity over the alphabet X as a formal equality between elements of the Stone-Cech compactification $\beta X$ and say that a compact Hausdorff space to $Y$ satisfies this identity if any map from $X$ to $Y$ has the property that the extension to $\beta X$ identifies these two elements. Is something like this always true for theories? | |
| Jun 29, 2011 at 6:47 | comment | added | Gerhard Paseman | So this pair of comments should be viewed as a request. Can you expand your answer so that we can see more applications of Lawvere theories (or even monads) to other realms which are not (in appearance at least) so category-theoretical in nature? If you can produce one or two examples that are true, I am willing to take your word on something (which I am making up) like "because of the monad perspective, we can lift these properties about Boolean algebras over to a decomposition of iterated forcing over models of ZF". Gerhard "Doesn't Believe Anything Before Coffee" Paseman, 2011.06.28 | |
| Jun 29, 2011 at 6:37 | comment | added | Gerhard Paseman | I felt that the original question was akin to asking for advantages and disadvantages of apples versus oranges, and your answer strengthens the feeling. Lawvere theories might provide a generalization, but until there is a lot more work shown outside of Set, it is hard to see such a generalization as applicable to anything far removed from category theory. Now if your example could be turned pack to Set and say something about n-valued Post algebras or arithmetical (in the UA sense) varieties, then I might be sold. Gerhard "Ignorance Can Help Form Opinion" Paseman, 2011.06.28 | |
| Jun 29, 2011 at 6:24 | history | answered | Todd Trimble | CC BY-SA 3.0 |