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In the case of monads, where $L$ is the free monad functor to the Eilenberg-Moore category, the Beck-Chevalley condition seems to rarely hold. … "Zerosumfree monads"? …

For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor …

Monads on the category Set of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science. … Can we limit the three possible answers to (1),(2),(3) above for monads? Are any other interesting facts known about $Mnd$? …

I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive. … Is there a good reference for these monads, if they really are monads? Thanks. …

(Note: some people restrict polynomial monads to be the Cartesian ones, i.e. those for which $\eta_t$ and $\mu_t$ are Cartesian natural transformations; e.g. this is the case for any free monad $\mathfrak … For this question, however, I would be happy with either Cartesian or non-Cartesian polynomial monads, as long as they are not free and have a nontrivial quine.) …

For any monoid $(M,e,*)$ in $\mathsf{Set}$ there is a corresponding comonad $y^M$ on $\mathsf{Set}$. It sends a set $A$ to the set of morphisms into $A$ from $M$, $$ A\mapsto A^M. $$ Note that $y=y^1$ …