Timeline for Why are operads sometimes better than algebraic theories?
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12 events
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| Nov 16, 2022 at 17:04 | comment | added | Arshak Aivazian | I removed the word "algebraic" in the first question (together with its definition in terms of Lover's theory) since indeed monads defined by operads in arbitrary monoidal categories may not be expressed in terms of (cartesian) Lawvere theory (although if the monoidal structure is rich enough, as I write in question 2, then Lawvere theories can be interpreted in it) | |
| Nov 16, 2022 at 12:35 | comment | added | David Roberts♦ | I don't presume to read Ivan's mind, but: algebras for a Lawvere theory are defined in any finite-product category, not just Set, as I'm sure you know. If you mean to say that working in a non-cartesian symmetric monoidal category renders Lawvere theories not applicable, then that is a clear and true statement. But without specifying what monoidal structure you mean to consider algebras relative to, simply saying "category of blahs can't do Lawvere theories" is ambiguous at best. Monoidal structures are genuine structure and don't come for free! | |
| Nov 16, 2022 at 11:54 | comment | added | David White | A comment says this answer is false. I have no idea what the commenter means. In the answer linked below, Qiaochu Yuan draws the analogy Lawvere theories : cartesian monoidal categories as operads : symmetric monoidal categories. The examples I gave are symmetric monoidal but not cartesian, and that's why Lawvere theories don't help you there. mathoverflow.net/questions/303847/… | |
| Nov 16, 2022 at 8:33 | comment | added | Ivan Di Liberti | The second and third sentences of this answer are simply false. | |
| Nov 16, 2022 at 0:21 | comment | added | Arshak Aivazian | Thank you, sorry! Probably, I still do not have enough knowledge to understand how monads corresponding to operads would behave worse than operads in the contexts you named. I will come back to this when I study them. | |
| Nov 16, 2022 at 0:11 | comment | added | David White | In homotopy theory we work with closed symmetric monoidal categories. Every one of these has a functor from operads to monads. Lawvere theories can encode things more general than operads can, like groups (inverses are ok). I think my answer covers the question in the title and Question 1. I don't think much about topoi so I'll leave it to others to think about question 2. This week is very busy for me so I can't promise to write again anytime soon. | |
| Nov 16, 2022 at 0:04 | comment | added | Arshak Aivazian | If Lawvere theories are ill-suited to expressing algebra in some contexts, then a wide class of monads (larger than the monads defined by operads) could be directly distinguished by some property. In my opinion, the issue is not settled yet. | |
| Nov 15, 2022 at 23:54 | comment | added | Arshak Aivazian | Operads define monads in every distributive monoidal category | |
| Nov 15, 2022 at 23:49 | comment | added | Arshak Aivazian | Hmm, or do you mean that not every monoidal category has a functor from operads to monads? I know that it exists, for example, for good topological spaces that are not a topos. But with chain complexes I'm not sure and I can't find a link, I'll think about it. | |
| Nov 15, 2022 at 23:42 | comment | added | Arshak Aivazian | The question is about the case when it is enough. In other words, is it useful every time to consider the maximum amount of algebra that can be expressed. | |
| Nov 15, 2022 at 23:42 | comment | added | Arshak Aivazian | Thank you very much for some new to me important examples of the usefulness of an operad when a monoidal structure is sparse! But your answer doesn't seem to answer the questions asked. I may have made the last edit to the question after you started writing the answer, I apologize in that case. I also already realized that the expressiveness of the language may not be enough for Lawvere theory (in fact, initially when I wrote the question I was thinking about the category of sets). | |
| Nov 15, 2022 at 23:29 | history | answered | David White | CC BY-SA 4.0 |