Timeline for Topological universal algebra: what is a variety?
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16 events
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| May 29, 2017 at 15:24 | comment | added | Noah Schweber | @UlrikBuchholtz I'm interested, though, in a single underlying space - e.g. a good description of the set of all continuous binary operations on $\mathbb{R}$ with the usual topology. Lawvere theories, or rather functors from them, don't obviously help me with this, so far as I see (but I could be missing something). | |
| May 29, 2017 at 12:22 | answer | added | Keith Kearnes | timeline score: 9 | |
| May 29, 2017 at 8:44 | comment | added | Ulrik Buchholtz | @NoahSchweber maybe I'm not understanding the first question correctly, but don't Lawvere theories give you exactly what you want? First, we can take any algebraic theory and form its Lawvere theory, then its topological algebras are the finite product preserving functors into Top. You also mention Adams' theorem, the setting of which would be homotopy-coherent algebras (i.e., laws are only satisfied up to coherent higher homotopies) and these are modeled by $(\infty,1)$-finite product preserving functors into $\infty$Gpd. (But Lawvere theories are then generalized to the $\infty$-setting.) | |
| May 29, 2017 at 2:28 | comment | added | Noah Schweber | @GerhardPaseman I don't believe hyperassociativity or subclones are what I'm interested in, though I could be wrong. $X$ is a topological space here, and, for $\mathcal{E}$ a set of equations in some fixed functional language, I'm asking about the set of algebras with domain $X$, each of whose operations is continuous in the sense of $X$, which satisfy $\mathcal{E}$. E.g. questions 2.1-2.2 take $\mathcal{E}=\{x*(y*z)=(x*y)*z\}$, and are asking for data about the set of $\mathbb{R}$-algebras satisfying $\mathcal{E}$ (that is, the set of continuous associative binary operations on $\mathbb{R}$). | |
| May 29, 2017 at 2:23 | comment | added | Gerhard Paseman | I think you are really interested in sub clones on X, but I am unsure. Hyperassociativity on MathOverflow refers to all term operations (not just basic operations) of an algebra being associative. If you were asking when all derived (binary) operations of an algebra were associative, I could point you to some literature. Unfortunately, expanding on this must wait til later. Gerhard "May Have A Future Comment" Paseman, 2017.05.28. | |
| May 29, 2017 at 1:07 | comment | added | Nik Weaver | Fair enough. It just confused me at first. | |
| May 29, 2017 at 1:07 | comment | added | Noah Schweber | To clarify on my last sentence, many other questions don't make sense (what does it mean for an $(X, \tau)$-variety to be locally finite, given that the underlying set is always the same?), but the questions that are purely equational-theoretic still make perfect sense. So maybe the right things to talk about are $(X, \tau)$-satisfiable equational theories? But even classically, a variety is just a semantic proxy for a satisfiable equational theory. (Of course, "satisfiable" in the classical case is trivial.) | |
| May 29, 2017 at 1:04 | comment | added | Noah Schweber | @NikWeaver Fair point, but: an $(X, \tau)$-variety is the set of $(X, \tau)$-algebras satisfying some fixed set of equations, so there is a terminological parallel. In each case we have a notion of what a relevant structure is (general algebra or $(X, \tau)$-algebra respectively) and are interested in classes of such algebras picked out by equational theories. It is definitely a very different notion, though, but I think the term is suggestive in an overall good way, especially since the analogues of many of the usual problems make sense (e.g. is an $(X, \tau)$-variety finitely based, etc.). | |
| May 29, 2017 at 1:01 | comment | added | Nik Weaver | I'm not sure the term "variety" is appropriate here --- in universal algebra this refers to the class of all algebras of a given type which satisfy some set of equations. Whereas apparently you are interested in algebraic operations defined on a single space, which seems very different. | |
| May 28, 2017 at 22:46 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| May 28, 2017 at 22:43 | comment | added | Noah Schweber | @GerhardPaseman I'm not familiar with hyperassociativity (and googling is giving me lots of stuff about psychology), what is that? Also, what do you mean by "you might have X as part of the similarity type"? (The analogy with $R$-modules seems incorrect to me, since an $R$-module won't have domain $R$ in general, but I could be missing something.) As for the topology, I am definitely happy to stick to "very nice" spaces like $\mathbb{R}$, but I would also be interested in weirder spaces, or even natural non-$T_2$ spaces like Zariski topologies. I'm eager to hear your thoughts! | |
| May 28, 2017 at 22:30 | comment | added | Gerhard Paseman | I have opinions (not answers) on all of these that I will share later. In brief, you might have X as part of the similarity type, like R-modules over a ring R. To combine two associative operations and expect a third leads to hyperassociativity. Finally, if you expect any homogeneity from the topology (and a reasonable collection of operations), this lands you into normal spaces if you have T_1 (or something like that, maybe T_0 gives T_7/2). Gerhard "More On These Subjects Later" Paseman, 2017.05.28. | |
| May 28, 2017 at 22:22 | history | edited | YCor |
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| May 28, 2017 at 20:50 | comment | added | Noah Schweber | I'm curious, why the downvote? | |
| May 28, 2017 at 20:49 | comment | added | Noah Schweber | The "algebraic-topology" tag here is a guess on my part; feel free to remove it if it seems inappropriate. | |
| May 28, 2017 at 20:48 | history | asked | Noah Schweber | CC BY-SA 3.0 |