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Stream: learning: questions

Topic: Adjoint triples and what is between algebras and coalgebras

Moana Jubert (Jan 10 2025 at 12:16):

Hello everyone,

Suppose we are given an adjoint triple $L \dashv R \dashv K$. It is "well-known" that the category of algebras $RLa \to a$ over the monad $RL$ is isomorphic to the category of coalgebras $a \to RKa$ via successive transpositions

$$\cfrac{RLa \to a}{\cfrac{La \to Ka}{a \to RKa}}$$

Most importantly, we go through an intermediate arrow $\chi : La \to Ka$ which is neither an algebra nor a coalgebra.
It satisfies however the following properties:

1. (Unitality) $\epsilon_a \circ R\chi \circ \eta_a = \mathrm{id}_a$
2. (Multiplication) $R\chi \circ \eta_a \circ \epsilon_a \circ R\chi = R\chi$

where $R\chi : RLa \to RKa$ and $\eta_a$, $\epsilon_a$ are the unit (resp. counit) of $RL$ (resp. $RK$).

There appears to be a category of such arrows with the above properties which is, in turn, isomorphic to the category of algebras and coalgebras from the beginning. Is this known? Has it been studied anywhere?

My end goal is to study when $R$ is either monadic or comonadic, which is equivalent. To avoid having to make a choice, it appears that I can directly study the case of arrows $La \to Ka$ with structure instead of settling with either algebras or coalgebras.

David Corfield (Jan 11 2025 at 19:54):

I have an example of this situation in my Modal HoTT book:

image.png

(For some context of what's going on here, see nLab: necessity and possibility.)

Moana Jubert (Jan 12 2025 at 18:25):

Thank you, I will check it out!