Timeline for What is the universal property of algebras for the codensity monad?
Current License: CC BY-SA 4.0
9 events
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| Feb 10, 2022 at 18:56 | comment | added | Simon Henry | Ah you're right ! sorry I had misread you. Monadic right adjoint are not stable under composition and that why we don't get a factorization system from the existence of the factorization... Thanks, I new there was something fishy, but I couldn't figure out what ^^ | |
| Feb 10, 2022 at 18:51 | comment | added | Tim Campion | Probably you're right about stability under pullbacks -- what I'm saying is that in order to get a factorization system, that's not all you need to check. One of the other things you need to check is stability under composition, which I think fails. For instance, if $C$ is a small category and $K$ is a reflective subcategory of presheaves on $C$, then $K \to Psh(C) \to Psh(Ob C)$ is a composite of monadic functors which is typically not monadic. | |
| Feb 10, 2022 at 18:50 | comment | added | Simon Henry | We wrote a proof of this for $\infty$-categories in arxiv.org/pdf/2106.02706.pdf (see Prop 3.23), so I really hope I'm correct on this ^^ | |
| Feb 10, 2022 at 18:48 | comment | added | Simon Henry | No I think I'm right on that part. Using Beck's criterion you can see that a pullback of a monadic right adjoint functor is monadic when it is a right adjoint. So in situation where the special adjoint functor theorem applies, pullback of monadic functor are monadic ? | |
| Feb 10, 2022 at 18:36 | comment | added | Tim Campion | I think I got confused -- even with presentability assumptions, monadic functors are not closed under composition, right? This leads me to think that the best one can hope for is an "over $B$" statement. Another point leading me to think this is that the functors you describe would be exactly the codense functors, but I'm pretty sure that $\tilde F$ need not be codense -- the factorization functor is not idempotent if we allow $B$ to vary. | |
| Feb 10, 2022 at 18:32 | comment | added | Simon Henry | @TimCampion I'm pretty sure there is a mistake in what I'm going to say but... If I assume all categories involved are locally presentable and all functors accessible then monadic right adjoint functor are stable under pullback, so the argument above is enough to show that one has a unique factorization system. The left class is then exactly the functors whose factorization gives $A \to B = B$, that is the functor whose codensity monad is the identity, that is the codense functor ? That's seem strange though... | |
| Feb 10, 2022 at 17:59 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Feb 10, 2022 at 17:55 | comment | added | Tim Campion | Great, thanks! So I guess that just leaves the question of which functors are left orthogonal to monadic functors... | |
| Feb 10, 2022 at 17:52 | history | answered | Simon Henry | CC BY-SA 4.0 |