An algebra for a monad $(T, \mu, \eta)$ on a category $\mathbb{C}$ is defined as a morphism $T X \to X$ for some object $X$ such that the obvious diagrams commute.

If I look at the monad as a monoid in the category of endofunctors, actions for a monad seem to be a generalisation of monad algebras: an action for $T$ is a natural transformation $T \circ F \to F$ for some functor $F$. So a monad algebra is an action for which the underlying functor is constant.

Is there a reason why we can concentrate on constant functors when studying monad algebras instead of studying actions in general?

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    What you call an "action" is usually called a module over the monad $T$, and you're right the usual notion of algebra is a special case when you use the terminal category on one side. I guess they probably come up more often in practice? Basically the same reason we often concentrate on elements of an object even though we also know about generalized elements. – Najib Idrissi Sep 10 '15 at 14:48
up vote 4 down vote accepted

So, you know that if you have a monoid $m$ in a monoidal category $(M, \otimes)$ then it's a natural thing to do to look at actions of $m$, namely morphisms $m \otimes c \to c$ satisfying etc. where $c$ is another object in $M$. One way to motivate the definition of monads is that you can actually do something more general than this: $c$ need not be an object of $M$! Instead, if you specify an action of $M$ on a category $C$, or equivalently a monoidal functor

$$M \to \text{End}(C)$$

then you can make sense of what it means to specify an action of $m$ on an object $c \in C$, where instead of $m \otimes c$ you use the action of $M$ on $C$. This is an instance of the microcosm principle. It is possible to describe, for example, algebras over an operad in this language, where $M$ is the monoidal category of species under composition.

The question then arises: what is the most general kind of monoid that can act in the above sense on an object in $C$? The answer is a monoid in $\text{End}(C)$, or equivalently a monad on $C$. So the reason that we think of monads as acting on objects in $C$ and not on objects in $\text{End}(C)$ is that $C$ is where our focus is. We want to understand $C$ and maybe categories monadic over $C$; we aren't really trying to understand monads for their own sake.

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    It may be illuminating to decategorify this discussion down one step. You can think of monads on a category $C$ as decategorifyig to idempotent endomorphisms of an object $c$, and actions / modules as decategorifying to fixed points of those idempotents. Now, an idempotent endomorphism of $c$ acts not only on $c$ but on $\text{End}(c)$, but the reason we restrict our attention to fixed points of the action on $c$ is that $c$ is where our focus is. Idempotent endomorphisms are a tool for understanding objects, rather than something we study for their own sake. – Qiaochu Yuan Sep 10 '15 at 16:27

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