Timeline for Characterisation of essentially algebraic theories as monads
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2021 at 15:57 | comment | added | varkor | @MartinBrandenburg: a monad correspondence as I originally envisaged it is still open. For a while, partial algebraic theories seemed a possible route to a correspondence, but there are some difficulties with this approach. However, there is a "monad-like" correspondence described in Essentially equational categories. I believe this result can be phrased along the lines of the usual monad–theory correspondence, but have not yet gotten around to working out all the details. | |
| Jul 1, 2021 at 15:41 | comment | added | Martin Brandenburg | @varkor Is it still open? | |
| Mar 5, 2021 at 17:27 | comment | added | varkor | Looking back at this question, I think the point I ought to have emphasised more was that I was interested in essentially algebraic theories for a fixed set of sorts. One can recover finite limit theories using the technique of Simon Henry, but it is not possible to fix a set of sorts this way. This is then somehow less insightful than the traditional correspondence between algebraic theories and strongly finitary monads on $\mathbf{Set}$. | |
| Feb 29, 2020 at 13:53 | vote | accept | varkor | ||
| Feb 26, 2020 at 11:54 | answer | added | varkor | timeline score: 3 | |
| Feb 22, 2020 at 22:02 | comment | added | Tim Campion | To expand on Simon Henry's comment, the category of models $C$ of an essentially algebraic theory form a locally finitely-presentable theory, which is a finitarily reflective subcategory of a presheaf category $Set^{C_0^{op}}$ (in particular, the inclusion is monadic). In turn, the presheaf category $Set^{C_0^{op}}$ is monadic, via the obvious forgetful functor, over $Set^{Ob C_0}$. Thus $C$ admits a functor to a slice of $Set$ which is a composite of monadic functors (but typically not itself monadic). I'm not sure how to get a composite of monadic functors to $Set$ itself though... | |
| Feb 22, 2020 at 20:50 | comment | added | Robert Furber | Of course, I meant to say "torsion-free abelian groups" in my previous comment. | |
| Feb 22, 2020 at 1:46 | comment | added | Robert Furber | You can have two distinct essentially algebraic theories such that the monad defined by the free-forgetful adjunction to $\mathbf{Set}$ is the same. For example, both abelian groups and torsion-free groups are essentially algebraic theories, but they define the same monad on $\mathbf{Set}$, because every free abelian group is torsion-free. | |
| Feb 21, 2020 at 18:38 | comment | added | Simon Henry | Is "finitary monad on a presheaf category" a satisfying answer ? An alternative is iterated monads (considering finitary monads acting on category of another finitary monads on sets) | |
| Feb 21, 2020 at 18:05 | review | First posts | |||
| Feb 21, 2020 at 19:16 | |||||
| Feb 21, 2020 at 18:03 | history | asked | varkor | CC BY-SA 4.0 |