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Stream: theory: category theory

Topic: Extending a monad from a co-Kleisli category

Patrick Nicodemus (Feb 24 2023 at 04:30):

@dusko pointed out to me the other day that I have a situation where there is a monad $T$ and a comonad $G$ together with a distributive law $\alpha :GT\to TG$, and I am trying to construct a monad on the category of $G$-coalgebras.

Now unfortunately this distributive law goes in the "wrong direction" for the obvious way we would try to handle this problem. But in this case, $T$ does act on the co-Kleisli category for $G$, or $G$ acts on the Kleisli category for $T$, and so the problem is partially solved, i.e. we know how to define the monad in the case of the free co-algebras. Then a reasonable approach is to try and extend $T$ from the co-Kleisli category to the whole category of coalgebras by (right?) Kan extension, and prove this forms a monad. Alternatively could construct a codensity monad using $T$ which is already a monad automatically.

I am curious, has anything like this been studied in the literature? Do we have conditions under which we can get a monad on a category of coalgebras given a distributive law going the "wrong way"?

Beck's theorem gives conditions to get a right adjoint going from the category of G-coalgebras to the co-Kleisli category, using this it is clear ?) how to get the monad to work. I don't think Beck's conditions are satisfied in my case though.

dusko (Feb 24 2023 at 06:05):

lifting monads to coalgebras often arises in descent theory. a distributivity law exists only when descent data are already representable, i.e. when descent can be avoided. but the no-distributivity situations are what prompted grothendieck to embark on galois descent.

erm, sorry i forget what was your monad (on simplicial sets?) but depending on what the monad is, sometimes having algebras over kleisli is enough, because any algebra over any coalgebra is isomorphic to an algebra over a cofree one.

here is a crazy fact in this that may help. suppose you have a comonad $G:{\cal C}\to {\cal C}$ with two different resolutions $L\dashv R:{\cal C}\to {\cal D}$ and $M\dashv S:{\cal C}\to {\cal E}$ (i.e. $LR = G = MS$). now we have monads $H=RL:{\cal D}\to {\cal D}$ and $K = SM:{\cal E}\to {\cal E}$. then the categories of algebras ${\cal D}^H$ and ${\cal E}^K$ are equivalent. for instance when ${\cal D} = {\cal C}_G$ is the kleisli and ${\cal E} = {\cal C}^G$ is the eilenberg-moore.

i said "crazy fact" because my collaborators didn't believe it, and deriving it from the general descent was tweaky. (i noticed it when i wrote the descent paper in the Como 1990 proceedings.) but if you think about it, it just says that categorical quotients are representation-invariant. as they should be if galois descent makes sense. in the meantime, it quite simple to derive from the fact that the adjunction between algebras and coalgebras is nuclear in my nucleus paper...

(bah i hate it when people push their stuff and here i go with the nucleus for the second time in one day :rolling_eyes: which also says that i am getting addicted to zulip.)

Patrick Nicodemus (Feb 24 2023 at 22:41):

Thank you, Dusko. I think this perspective is helpful.
The monad I am studying is the cone monad, aka Day convolution with the terminal presheaf. I have a distributive law $\alpha :DT\to TD$ of the comonad over the monad and i am wondering how this might come into the situation, as i said it is going the "wrong way" for the obvious approach to apply