symmetric monoidal (∞,1)-category of spectra
symmetric monoidal (∞,1)-category of spectra
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The action monad or writer monad is a construction generalizing many seemingly different concepts across mathematics and computer science. It could be intuitively understood in the following ways. First of all, fix a group or monoid $M$.
If one, more generally, replaces sets and a monoid with a monoidal category and an internal monoid, a similar construction can be given. This generalizes, for example, the action that rings have on their modules, and that Lie groups have on manifolds.
Let $M$ be a monoid with unit $e:1\to M$ and multiplication $\cdot:M\times M \to M$. The (left) $M$-action monad is a monad on Set where
The endofunctor maps a set $X$ to the set $M\times X$;
The unit is given by
for each object $X$;
for each object $X$.
The monad axioms follow from the monoid axioms for $M$.
There exists an analogous monad for right actions, whose endofunctor maps $X$ to $X\times M$.
More generally, let $(C,\otimes,1)$ be a monoidal category. Let $M$ be a monoid object in $C$ with unit $e:1\to M$ and multiplication $\cdot:M\otimes M \to M$. The (left) $M$-action monad is a monad on $C$ where
The endofunctor maps an object $X$ of $C$ to the object $M\otimes X$;
The unit is given by
for each object $X$;
for each object $X$.
Again, the monad axioms follow from the monoid axioms for $M$. (And again there is an analogous notion for right actions).
The algebras over the action monad for a monoid (or group) $M$ can be seen as the $M$-spaces, i.e. sets or spaces equipped with an action of $M$ (hence the name).
Plugging in the definition, an algebra over this monad is then a set $A$ together with a map $e:M\times A\to A$, such that the following diagrams commute. The algebra diagrams
say equivalently that for all $a\in A$ and $m,n\in M$,
In other words, a $T_M$-algebra is exactly a set equipped with an $M$-action in the traditional group-theoretical sense.
This gives a way to talk about monoid and group actions internally to any monoidal category, giving the notion of a module object.
In the category of manifolds with their cartesian product, an internal group is a Lie group. The algebras of the corresponding action monad are the manifolds equipped with the action of the Lie group, such as homogeneous spaces.
The same can be said about topological spaces and continuous actions of topological groups.
In the category Ab of abelian groups with their tensor product, an internal monoid is a ring. The algebras of the resulting action monad are the modules over that ring.
In the category of endofunctors of a category $C$, an internal monoid is the same as a monad $T$. An algebra of the resulting action monad is an instance of a module over a monad $T$.
The action monad in computer science is known under the name of writer monad. See also writer comonad.
A way to motivate this name is to look at its Kleisli morphisms. The following explanation is taken from Perrone, Example 5.1.14.
Let $M$ be a monoid, and let’s write it additively. Denote by $T_M$ its right writer monad. A Kleisli morphism of $T_M$ is a map $k:X\to Y\times M$. We can interpret it as a process which, when given an input $x\in X$, does not just produce an output $y\in Y$, but also an element of $M$. For example, it could be energy released by a chemical reaction, or waste, or a cost of the transaction. In computer science, this is the behaviour of a function that computes a certain value, but that also writes into a log file (or to the standard output) that something has happened (the monoid operation being the concatenation of strings). For example, when you compile a tex document, a log file is produced alongside your output file. Hence the name “writer monad”.
Let’s now look at the Kleisli composition. If we have processes $k:X\to Y\times M$ and $h:Y\to Z\times M$, then $k\circ_{kl} Y:X\to Z\times M$ is given by
What it does is as follows:
So, for example, the cost of executing two processes one after another is the sum of the costs. The same is true about the release of energy in a chemical reaction, and about waste. Just as well, executing two programs one after another will produce a concatenation of text in a log file (or two log files).
Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)
Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)
Paolo Perrone, Notes on Category Theory with examples from basic mathematics, Chapter 5. (arXiv)
Last revised on March 12, 2021 at 23:35:47. See the history of this page for a list of all contributions to it.