Timeline for Equivalent conditions to be a lax idempotent contravariant monads
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2018 at 8:11 | comment | added | fosco | For example, presheaf constructions of Yoneda structures $P$ satisfy this property when $A\mapsto PA$ really acts as a cocompletion. Some don't (and even though they are uninteresting, they exist anyway). Now, if you ask me "are you sure these uninteresting YS don't fit in your picture for maybe another reason?" my answer is: I'm not :-) but I add assumptions one at a time. | |
| Oct 26, 2018 at 0:44 | comment | added | Todd Trimble | Not sure. But I thought you said in your first of three posts that you had examples where this adjointness didn't hold (which was why you were slightly shying away from the suggestions in my answer there)? If so, then it seems it would be worthwhile to see where this bit of centipede mathematics takes you. I'm slightly curious myself... | |
| Oct 25, 2018 at 23:34 | comment | added | fosco | For the moment, I find a bit counterintuitive that even if the adjunction $T_!f \dashv Tf$ exists, the left adjoint is the image of $f$ under a monad, and $T$ does not have any interesting property. I am really tempted to ignore the problem and define a "contramonad" as a functor $T : C^{coop}\to C$ such that each $\forall f : \exists T_!f \dashv Tf$ and $T_!$ is a monad. Do you know if these objects (functors -not monads anymore- with the property that each $Tf$ has an adjoint) have been studied under a precise name? | |
| Oct 25, 2018 at 23:29 | comment | added | fosco | While I was waiting for an answer to this question I persuaded myself that it's really a good idea to follow your final advice; it's all about using the monad structure of $T_!$ instead of searching for a really convoluted alternative structure for $T$. I will toroughly read your comment tomorrow to test it against my current intuition, but I must say thank you for taking the time to write in such detail! | |
| Oct 25, 2018 at 22:29 | history | answered | Todd Trimble | CC BY-SA 4.0 |