Timeline for Why are operads sometimes better than algebraic theories?
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13 events
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| Nov 21, 2022 at 20:39 | comment | added | Arshak Aivazian | Indeed, but nonetheless. | |
| Nov 21, 2022 at 18:15 | comment | added | Simon Henry | To be clear, even in the $\infty$-case, operads still aren't a full subcategories: morphisms of operads are cartesian natural transformation (meaning natural transformation whose naturality square are pullback square) but when talking about polynomial functor and polynomial monads this is a very natural notion of morphisms. | |
| Nov 21, 2022 at 16:21 | comment | added | Arshak Aivazian | Great, thanks! In my opinion, this means that (1) the classical concept of operads does not make much sense in itself (being some non-intuitive additional structure over the concept of algebraic theory) (2) the really important concept is $\infty$-algebraic theories and, in particular, $\infty$-operad (constituting their full subcategory). | |
| Nov 21, 2022 at 16:07 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 21, 2022 at 15:57 | comment | added | Simon Henry | @AivazianArshak : it is true that the $\infty$-categorical situation is a little different - I wouldn't call it "open" as it is very well understood as well, but it does deserves a different explanation. This was too long for a comment so I added it to the answer at the end. | |
| Nov 21, 2022 at 15:55 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 21, 2022 at 3:22 | comment | added | Arshak Aivazian | Thank you very much for the detailed answer, I learned a lot by meditating on it! I can't find: is it known that $\infty$-operads also induce $\infty$-algebraic theories and $\infty$-categories of algebras over them are equivalent? If so, then the second advantage of operads you mentioned can also be achieved by going general to $\infty$-algebraic theories, and my question is "when is an operad language useful?" still partially open, right? | |
| Nov 21, 2022 at 2:40 | vote | accept | Arshak Aivazian | ||
| Nov 16, 2022 at 20:06 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 16, 2022 at 19:59 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 16, 2022 at 19:15 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 16, 2022 at 19:09 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 16, 2022 at 18:57 | history | answered | Simon Henry | CC BY-SA 4.0 |