Timeline for Monomorphisms in operad algebras
Current License: CC BY-SA 3.0
13 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Mar 6, 2016 at 19:49 | comment | added | Todd Trimble | (For Lawvere theories, see this paper by Martin Hyland for a mention of the coslice result, arxiv.org/pdf/1211.5762v2.pdf, around propositions 2.6 and 2.7.) | |
| Mar 6, 2016 at 14:45 | comment | added | Todd Trimble | For Lawvere algebraic theories, as opposed to operads, I think there's no problem with the construction. But for permutative operads, I seemed to be getting stuck, so I cut my losses and went with a case I could prove to my satisfaction. | |
| Mar 6, 2016 at 13:52 | comment | added | David Spivak | @Todd, I thought the same thing about there being an $\mathcal{O}_A$. In fact, I have the thought that the bifibration $p\colon\mathbf{Algebra}\to\mathbf{Theory}$ (given by $p(T,A) = T)$ has the property that there exists a functor $q\colon\mathbf{Algebra}\to\mathbf{Theory}$ (given by $q(T,A) = T_A$), such that the pullback of $p$ along $q$ is the domain bifibration $\text{dom}\colon\mathbf{Algebra}^2\to\mathbf{Algebra}$. | |
| Mar 6, 2016 at 2:19 | comment | added | Todd Trimble | Regarding the question about characterizing the algebraic theories: I don't know, but a keyword I used to compose my answer was 'amalgamation' which carries model-theoretic resonances (a more specific phrase is 'weak amalgamation property'). See for example here en.wikipedia.org/wiki/Amalgamation_property, or perhaps better yet A Shorter Model Theory by Hodges, for some possible inspiration or food for thought. I thought at first, by the way, that for any operad $O$ and any algebra $A$ there would be an operad $O_A$ governing the co-slice, but couldn't quite prove it. | |
| Mar 6, 2016 at 0:10 | vote | accept | David Spivak | ||
| Mar 5, 2016 at 17:20 | answer | added | Todd Trimble | timeline score: 4 | |
| Mar 4, 2016 at 14:13 | comment | added | David Spivak | I was being a bit ambiguous about whether I intend colored operads or plain operads (similarly multi-sorted vs. single-sorted algebraic theories). But yes, if we're talking single-sorted, then $\kappa=\mathcal{O}(0)$. | |
| Mar 4, 2016 at 6:21 | comment | added | Gabriel C. Drummond-Cole | Can the initial object be something other than $\mathcal{O}(0)?$ | |
| Mar 4, 2016 at 2:30 | history | edited | David Spivak | CC BY-SA 3.0 |
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| Mar 4, 2016 at 2:06 | comment | added | Omar Antolín-Camarena | Oh, wait, your talk of algebraic theories make me think that possibly you meant operads in the category of sets specifically, in which case you can ignore my previous remark, but should add that requirement to the question. | |
| Mar 4, 2016 at 2:05 | comment | added | Omar Antolín-Camarena | I'm confused: aren't $R$-algebras also algebras for an operad (namely, the commutative operad in $R$-modules), so that Zhen Lin's counterexample also applies here? | |
| Mar 4, 2016 at 1:43 | history | asked | David Spivak | CC BY-SA 3.0 |