Timeline for Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are...?
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16 events
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| Jun 2, 2020 at 14:49 | comment | added | Simon Henry | It seems to me if you want this to work, as you use the monoidal structure on pointed sets, you everything to be enriched in pointed sets. It means that you should restrict to $Set_*$-enriched functor, which are exactly the functor preserving the $0$-objects. Once you impose this restriction everything should work. | |
| Jun 2, 2020 at 11:39 | answer | added | Ben MacAdam | timeline score: 3 | |
| Jun 2, 2020 at 10:28 | history | edited | fosco | CC BY-SA 4.0 |
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| May 31, 2020 at 14:11 | comment | added | varkor | It's unclear to me whether or not this is true: the definitions are very abstract, so it's difficult to get an intuition without explicitly working through the example. I notice that Bourke & Garner do not seem to give any examples in this vein, but it's not clear whether this is a deliberate omission. I personally would be interested to know the outcome if you do work through it. | |
| May 31, 2020 at 10:55 | comment | added | fosco | A distinctive feature of algebraic theories is that they are monoidal functors (in the case of classical Lawvere theories, cartesian). These correspond in turn to promonoidal promonads on the associated arity. Is it the case that if an arity ${\cal A} \hookrightarrow {\cal E}$ is chosen to be monoidal (i.e., $\cal E$ is monoidal, e.g. the base of enrichment, and $\cal A$ is $\otimes$-closed), the $\cal A$-nervous monads must preserve this structure (thus, bestowing theories or the associated promonads with monoidality)? | |
| May 30, 2020 at 12:52 | comment | added | varkor | Interesting. Your theories are a little unusual, as they no longer look like a class of categories equipped with specified structure and generators, as in the traditional setting. If you haven't checked already, it would be worth comparing your setting to that of Bourke & Garner's, which establishes theory–monad correspondences in a great level of generality (potentially at the cost of some massaging to check your definitions align with theirs). | |
| May 30, 2020 at 7:26 | comment | added | fosco | @varkor I edited the question! | |
| May 30, 2020 at 7:26 | history | edited | fosco | CC BY-SA 4.0 |
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| May 29, 2020 at 22:15 | comment | added | fosco | @varkor I will address your question tomorrow. Thanks for the attention :) Power's enriched Lawvere theories framework isn't exactly what it's needed here. Because you would have to take the monoidal structure given by smash on Fin_* too | |
| May 29, 2020 at 21:47 | comment | added | Ivan Di Liberti | @მამუკაჯიბლაძე Yes, it is. | |
| May 29, 2020 at 21:39 | comment | added | მამუკა ჯიბლაძე | Set is Ind(Fin). Is Set$_*$ Ind(Fin$_*$)? | |
| S May 29, 2020 at 21:17 | history | suggested | RobPratt |
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| May 29, 2020 at 20:35 | comment | added | varkor | Could it be the case that the equivalence works out in the $\mathrm{Set}_*$-enriched setting, giving a correspondence between $\mathrm{Set}_*$-enriched cartesian operads (or substitution monoids) and finitary $\mathrm{Set}_*$-monads on $\mathrm{Set}_*$? It looks like your setting is one in which sets are being consistently replaced with pointed sets, which seems suggestive of the setting for enriched Lawvere theories. But perhaps there's an obvious reason this doesn't work out. | |
| May 29, 2020 at 20:27 | review | Suggested edits | |||
| S May 29, 2020 at 21:17 | |||||
| May 29, 2020 at 20:26 | history | edited | LSpice | CC BY-SA 4.0 |
MathOperators, and proofreading
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| May 29, 2020 at 20:04 | history | asked | fosco | CC BY-SA 4.0 |