Monadic Approach to Actions of Internal Categories

Group actions are one of the most fundamental basic tools in algebra and geometry. An important categorical result is that sets equipped with an action of a fixed group can be captured as algebras for a monad on $\mathrm{Set}$. This monad, called the action monad or also the writer monad, has had abundant applications in both mathematics and computer science. Notably, one of the outcomes of the monadicity result for group actions is the consequent notion of internal action of a group object in a category $𝒞$. Indeed, this notion is given by generalizing the action monad to a monad on $𝒞$.

More recently, George Janelidze and Walter Tholen introduced a notion of action of an internal category in a category $𝒞$ on an object of $𝒞$. This was motivated by important applications to descent theory.

In this talk, we prove a monadicity result for these actions of an internal category. We reach this by generalizing the action monad for internal group actions to a monad on the slice category over the object of objects of the internal category. The classical action monad for a group object is recovered when viewing the group object as an internal category with object of objects given by the terminal object.

As an outcome of this monadicity result, we show that the process of associating to an internal category its category of actions is functorial, as it corresponds to a morphism of monads. This is an important ingredient that we needed for applications to categorical Galois theory.

Moreover, we completely characterize which monads arise as monads of actions of internal categories. Such monads have as base category a slice category $𝒞/X$, and they are equipped with a comorphism of monads towards the identity monad. This in particular implies that the category $𝒞$ is embedded in the category of algebras. In the case of actions of a group object $G$, the presence of such embedding corresponds to the fact that every object of the category can be equipped with a trivial action of $G$ given by the projection. Surprisingly, this property turns out to be fundamental in identifying the monads of internal actions.

Finally, we show some applications of this work to categorical Galois theory as well as algebraic geometry, which motivated us to develop these results.